U

'
I

II
I
I
|
‘ll
_

5%
_“_H_'HW_

J)‘

VWQHM’

M
WP
’_'

w,
M1

I N
My WW
_'>_“
_

___I1
~.&M%M‘W_%.
§;_m)¥M

‘
ll“
in
“_'M_
W _
Il
H M
V_

I
‘W
W
"M
“_Pm
_
>_(_
WM“

~v
‘§V_____’
w____
H _

3M
I_
7Y

‘ V

if
g‘
% _ A

WW
v_
g
“_ A

“__

x‘_ ‘“
an
7;
~‘Q
‘Wm
_v
_
M,

Kh

_
xW
'
_INI_\v’)AMW
§‘X
)_z¥§§§_
_
‘ xq
>V
§§A§g§
aY
_
Y
H
WW"

_
M”

W
*’A“
__

> _
WWW
I
v>‘\
\
MW
WW‘ N

__
A_

“W
iU
k’ w
}_ W

%M‘
NW v
__ __

v‘F
Hp’
‘in _(
)\_%$
' >

H\V
H"

IIL
‘\1AM\
1H“
h

NM
V
AWx
MM
A

I *‘ 6
um
U
__
2

_2%
< k

‘W'
VV
‘ y _

M
M“
“V
W
U“
A"
L_
“HA
WA

I‘ H
_

ax~
N
~W_

w
__>___
WW"

_ _
_W_
NW
_ _ >

d
M
lI
}<
~
WWW Y

*My
W _

WY
W
mu
W_
H
(v
x~‘_
0WW
V

;i‘\
_;
Ni‘
_v
_>
>_\
_\
l“_'_l__ V
%mm0H“§§%%§§_‘§\s
\3\!_\_

M
MW
~_‘
v_
‘_
_v§VvvYALV§§kg!"x§€v}_h\_‘_z_‘$S;;_:§23$
__

V
__
Harshad K. D. H. Bhadeshia

Geometry of crystals,
polycrystals, and phase
transformations
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742

© 2018 by Taylor & Francis Group, an Informa business

CRC Press is an imprint of Taylor & Francis Group, an Informa business

No claim to original U.S. Government works

Printed on acid-free paper

Version date 20170802

International Standard Book Number -13: 978-1-138-07078-3 (Hardback)

This book contains information obtained from authentic and highly regarded sources. Reasonable efforts
have been made to publish reliable data and information, but the author and publisher cannot assume
responsibility for the validity of all materials or the consequences of their use. The authors and
publishers have attempted to trace the copyright holders of all material reproduced in this publication
and apologize to copyright holders if permission to publish in this form has not been obtained. If any
copyright material has not been acknowledged please write and let us know so we may rectify in any
future reprint.

Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,
transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or
hereafter invented, including photocopying, microfilming, and recording, or in any information storage
or retrieval system, without written permission from the publishers.

For permission to photocopy or use material electronically from this work, please access
www.copyright.com​ (​http://www.copyright.com/​) or contact the Copyright Clearance Center, Inc. (CCC),
222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that
provides licenses and registration for a variety of users. For organizations that have been granted a
photocopy license by the CCC, a separate system of payment has been arranged.

Trademark Notice:​ Product or corporate names may be trademarks or registered trademarks, and are
used only for identification and explanation without intent to infringe.

Visit the Taylor & Francis Web site at

http://www.taylorandfrancis.com

and the CRC Press Web site at

http://www.crcpress.com
To my late father and mother.
Contents

Preface and Acknowledgments xiii

Author’ xv

Acronyms xvii

I Basic Crystallography 1
1 Introduction and Point Groups 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 The lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Primitive representation of Cubic-F . . . . . . . . . . . . . . . . 7
1.3 Bravais lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Number of equivalent indices . . . . . . . . . . . . . . . . . . . . 15
1.5 Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Weiss zone law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.8 Symmetry operations . . . . . . . . . . . . . . . . . . . . . . . . . 17
Five-fold rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.9 Crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Structure of graphene . . . . . . . . . . . . . . . . . . . . . . . . 22
1.10 Point group symmetry . . . . . . . . . . . . . . . . . . . . . . . . 23
Point symmetry of chess pieces . . . . . . . . . . . . . . . . . . . 24
Octahedral interstices in iron . . . . . . . . . . . . . . . . . . . 26
1.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

References 29

2 Stereographic Projections 31
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Projection of small circle . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Utility of stereographic projections . . . . . . . . . . . . . . . . . 34
2.3 Stereographic projection: construction and characteristics . . . . . 35
Radius of trace of great circle on Wulff net . . . . . . . . . . . . . 39
Traces of plates . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4 Stereographic representation of point groups . . . . . . . . . . . . 42

vii
viii Contents

Mirror plane equivalent to 2 . . . . . . . . . . . . . . . . . . . . 42

The point groups 3m and m3 . . . . . . . . . . . . . . . . . . . 43
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

References 49

3 Stereograms for Low Symmetry Systems 51

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Hexagonal system . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Angles in the hexagonal system . . . . . . . . . . . . . . . . . . 54
Growth direction of cementite laths . . . . . . . . . . . . . . . . 56
3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

References 59

4 Space Groups 61
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Screw axes and glide planes . . . . . . . . . . . . . . . . . . . . . 62
4.3 Cuprite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4 Location of atoms in cuprite cell . . . . . . . . . . . . . . . . . . . 64
Space group of Fe-Si-U compound . . . . . . . . . . . . . . . . . 66
Cementite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Diamond and zinc sulphide . . . . . . . . . . . . . . . . . . . . . 69
4.5 Shape of precipitates . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

References 73

5 The Reciprocal Lattice and Diffraction 75

5.1 The reciprocal basis . . . . . . . . . . . . . . . . . . . . . . . . . 75
Weiss zone law . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2 Crystallography of diffraction . . . . . . . . . . . . . . . . . . . . 78
5.3 Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Solution of electron diffraction pattern . . . . . . . . . . . . . . . 80
Another diffraction pattern solution . . . . . . . . . . . . . . . . 81
Disordered and ordered crystals . . . . . . . . . . . . . . . . . . 82
5.4 Diffraction from thin crystals . . . . . . . . . . . . . . . . . . . . 85
5.5 Neutron diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

References 89

6 Deformation and Texture 91

6.1 Slip in a single-crystal . . . . . . . . . . . . . . . . . . . . . . . . 91
Elongation during single-crystal deformation . . . . . . . . . . . 94
ε-martensite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Contents ix

6.2 Texture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.3 Orientation distribution functions . . . . . . . . . . . . . . . . . . 99
Euler angles relating two frames . . . . . . . . . . . . . . . . . . 100
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

References 103

7 Interfaces, Orientation Relationships 105

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.2 Symmetrical tilt boundary . . . . . . . . . . . . . . . . . . . . . . 106
7.3 Coincidence site lattices . . . . . . . . . . . . . . . . . . . . . . . 108
7.4 Representation of orientation relationships . . . . . . . . . . . . . 109
Coordinate transformation . . . . . . . . . . . . . . . . . . . . . 110
7.5 Mathematical method for determining Σ . . . . . . . . . . . . . . 113
7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

References 115

8 Crystallography of Martensitic Transformations 117

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.2 Shape deformation . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8.3 Bain strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

References 125

II A Few Advanced Methods 127

9 Orientation relations 129
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
9.2 Cementite in steels . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Bagaryatski orientation relationship . . . . . . . . . . . . . . . . 130
9.3 Relations between fcc and bcc crystals . . . . . . . . . . . . . . . 135
Kurdjumov-Sachs orientation relationship . . . . . . . . . . . . . 135
9.4 Relationships between grains of identical structure . . . . . . . . . 139
Axis-angle pairs and rotation matrices . . . . . . . . . . . . . . . 139
Double twinning . . . . . . . . . . . . . . . . . . . . . . . . . . 141
9.5 The metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Plane normals and directions in an orthorhombic structure . . . . 143
9.6 More about the vector cross product . . . . . . . . . . . . . . . . . 146
9.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

References 149
x Contents

10 Homogeneous deformations 151

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
10.2 Homogeneous deformations . . . . . . . . . . . . . . . . . . . . . 152
Bain strain: undistorted vectors . . . . . . . . . . . . . . . . . . 154
10.3 Eigenvectors and eigenvalues . . . . . . . . . . . . . . . . . . . . 156
Eigenvectors and eigenvalues . . . . . . . . . . . . . . . . . . . 157
10.4 Stretch and rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 158
10.5 Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Deformations and interfaces . . . . . . . . . . . . . . . . . . . . 161
10.6 Topology of grain deformation . . . . . . . . . . . . . . . . . . . . 162
10.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

References 169

11 Invariant-plane strains 171

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Tensile tests on single-crystals . . . . . . . . . . . . . . . . . . . 175
Transition from easy glide to duplex slip . . . . . . . . . . . . . . 177
11.2 Deformation twins . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Twins in fcc crystals . . . . . . . . . . . . . . . . . . . . . . . . 179
11.3 Correspondence matrix . . . . . . . . . . . . . . . . . . . . . . . . 182
11.4 An alternative to the Bain strain . . . . . . . . . . . . . . . . . . . 184
11.5 Stepped interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 186
Interaction of dislocations with interfaces . . . . . . . . . . . . . 186
fcc to hcp transformation revisited . . . . . . . . . . . . . . . . . 191
11.6 Conjugate of an invariant-plane strain . . . . . . . . . . . . . . . . 197
Combined effect of two invariant-plane strains . . . . . . . . . . 198
11.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

References 201

12 Martensite 203
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
12.2 Shape deformation . . . . . . . . . . . . . . . . . . . . . . . . . . 203
12.3 Interfacial structure of martensite . . . . . . . . . . . . . . . . . . 206
12.4 Phenomenological theory of martensite crystallography . . . . . . 208
12.5 Stage 1: Calculation of lattice transformation strain . . . . . . . . . 211
Determination of lattice transformation strain . . . . . . . . . . . 212
12.6 Stage 2: Determination of the orientation relationship . . . . . . . . 215
Martensite-austenite orientation relationship . . . . . . . . . . . 215
12.7 Stage 3: Nature of the shape deformation . . . . . . . . . . . . . . 217
Habit plane and the shape deformation . . . . . . . . . . . . . . 218
12.8 Stage 4: Nature of the lattice-invariant shear . . . . . . . . . . . . . 220
Lattice–invariant shear . . . . . . . . . . . . . . . . . . . . . . . 220
12.9 Texture due to displacive transformations . . . . . . . . . . . . . . 222
12.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Contents xi

References 227

13 Interfaces 231
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
13.2 Misfit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
Symmetrical tilt boundary . . . . . . . . . . . . . . . . . . . . . 234
Interface between alpha and beta brass . . . . . . . . . . . . . . 237
13.3 Coincidence site lattices . . . . . . . . . . . . . . . . . . . . . . . 239
Coincidence site lattices . . . . . . . . . . . . . . . . . . . . . . 239
Symmetry and the axis-angle representations of CSL’s . . . . . . 242
13.4 The O-lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
alpha/beta brass interface using O-lattice theory . . . . . . . . . 248
13.5 Secondary dislocations . . . . . . . . . . . . . . . . . . . . . . . . 249
Intrinsic secondary dislocations . . . . . . . . . . . . . . . . . . 249
13.6 The DSC lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
13.7 Some difficulties associated with interface theory . . . . . . . . . . 253
13.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

References 257

Appendices 259

A Matrix methods 261

A.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
A.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

B General rotation matrix 267

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
Preface and Acknowledgments

To state the obvious, crystals contain order. Even when that is disturbed locally, the
perturbations themselves may sometimes form regular patterns. This is evident in
the structure of interfaces where two or more crystals meet in an apparently hap-
hazard manner. Disorder can sometimes be ignored without compromising some of
the consequences of long-range order. Crystalline solid solutions in which atoms
are dispersed at random would fail the strict definition of long-range order, but they
nevertheless show the characteristics of crystals when probed by X-rays. We shall at-
tempt in this book to understand not only the elegance of individual crystals, but also
of clusters of space-filling crystals and transformations between crystalline phases.
Crystallography is hardly a new subject so there are numerous books available,
many of which are beacons of scholarship. So why another text on this much mooted
topic? First, as Buckminster Fuller pointed out, there has been a massive expansion in
human knowledge, with the process continuing at an unabating pace. New subjects
spring up with notorious regularity and some of these have come to be regarded
as essential first steps in the higher education curriculum. Modern undergraduate
students are therefore faced with a much greater palette of distinct courses than has
been the case in the past.
This book is partitioned into two, the first part of which is meant to be self-
contained and deals with what I feel is essential learning for any student in the mate-
rial sciences, physics, chemistry, earth sciences, and the natural sciences in general.
This part, intentionally concise, is based on a set of nine lectures that I give annually
to undergraduate students of the Natural Sciences Tripos in Cambridge University. It
is of generic value and has just sufficient material to deliver concepts. It covers crys-
tals, polycrystals, interfaces and transformations that occur by the disciplined motion
of atoms. I feel that most books on crystallography are too detailed to accommodate
within the schedule of a contemporary undergraduate.
The second part has depth which would be appreciated most in the context of
research. It is a development from a book that I taught and published in 1987 on the
Geometry of Crystals. The treatment is limited to phenomena dominated by crystal-
lography.
The book contains worked examples throughout because crystallography is a sub-
ject that thrives on practice. Video lectures and other electronic materials that can
enhance the content of this book are available freely on
http://www.phase-trans.msm.cam.ac.uk/teaching.html
Crystallography has rules, established by convention, but many of these have ex-
ceptions and there sometimes are multiple conventions. Donnay in his 1943 paper

xiii
xiv Preface and Acknowledgments

proposed rules for defining the crystallographic orientation of a crystal [American

Mineral 28 (1943) 313-328], punctuated by exceptions, special cases, and difficul-
ties. In the Bunge convention for Euler angles, the sample frame is rotated into the
crystal frame, but there is another convention where the reverse is the case. My view
is that in the application of crystallography, only a limited number of conventions are
important. For example, the handedness of cell axes should be consistent through-
out. Similarly, although only three indices are necessary to define vectors in three
dimensions, the four index system for the hexagonal class permeates the literature
and hence needs to be taught. The emphasis of the book is to teach concepts rather
than rules, which necessarily require rote learning – I am uncertain as to whether I
have done enough to minimise the use of conventions.
My interest in crystallography stems primarily from research on solid-state phase
transformations. I have benefitted enormously from the writings of the late J. W.
Christian and C. M. Wayman. And from incisive questions posed by undergraduates.
I have been able to tap routinely into the knowledge and wisdom of John Leake and
Kevin Knowles whenever I felt confused about the fine detail of crystallography.
To all of these characters, I shall remain perpetually grateful for the education and
camaraderie.

H. K. D. H. Bhadeshia
Cambridge
Author’

Harshad K. D. H. Bhadeshia is the Tata Steel Professor of Metallurgy at the Uni-

versity of Cambridge. His main interest has been on the theory of solid-state phase
transformations with emphasis on the prediction and verification of microstructural
development in complex metallic alloys, particularly multicomponent steels. He has
authored or co-authored some 700 publications including several books. He is a Fel-
low of the Royal Society, the Royal Academy of Engineering, and a Foreign Fellow
of the National Academy of Engineering (India). Professor Bhadeshia was in 2015
appointed a Knights Bachelor in the Queen’s Birthday Honours for services to Sci-
ence and Technology.

xv
Acronyms

bcc Body-centred cubic ILS Invariant-line strain

bct Body-centred tetragonal P Primitive
fcc Face-centred cubic I Body-centred
hcp Hexagonal close-packed F Face-centreed
IPS Invariant-plane strain R Trigonal

xvii
 Part I

Basic Crystallography
1
Introduction and Point Groups

Abstract

The term ‘order’ is a fundamental feature of the definition of a crystal, a feature

that distinguishes it from random aggregates of atoms where the description of the
structure requires a specification of the coordinates of every single atom that exists in
the material. In contrast, the crystal can be described completely in terms of a repeat
unit that usually consists of a ‘handful’ of atoms. Another consequence of the order is
that crystals have properties that vary with direction, unlike amorphous materials that
tend to be isotropic. Certain tools are required to deal with the regularity of crystals,
some of which are introduced in this chapter.

1.1 Introduction
Amorphous solids are homogeneous and isotropic because there is no long range
order or periodicity in their internal atomic arrangement. In contrast, the crystalline
state is characterised by a regular arrangement of atoms over large distances. Crystals
are therefore anisotropic - their properties vary with direction. For example, the inter-
atomic spacing varies with orientation within the crystal, as does the elastic response
to an applied stress.
Crystals can be two or three dimensional; graphene is a two-dimensional crystal if
we neglect the fact that it is one atom thick (Fig. 1.1a), whereas polonium (Fig. 1.1b)
is obviously three-dimensional. There is talk about one-dimensional crystals where
a string of equally spaced atoms is in a straight line, made by confining the atoms
into a carbon nanotube; however, there are interactions between these atoms and the
nanotube which violate the definition.1 Four dimensional crystals which are periodic
in both time and space have been postulated [2]. For example, a cylindrical crystal
with a periodic arrangement of atoms that repeats as time is stepped. Such a structure
would require perpetual motion, i.e. must not radiate rotational energy so there are
difficulties in explaining how such a crystal could exist in a non-stationary ground
state, a subject of discussion in the associated literature. Some experimental evidence
1 To quote Ziman [1], “the hypothetical ordered linear chain is physically unrealisable as a genuine

one-dimensional system”.

3
4 1 Introduction and Point Groups

now exists for discrete time crystals [3, 4]; the question remains as to whether the
‘time crystals’ are everlasting [5].

(a) (b)

FIGURE 1.1
(a) The arrangement of carbon atoms in the two-dimensional structure of graphene.
(b) The arrangement of atoms in the three-dimensional structure of polonium.

The crystalline form usually conjures visions of beautiful facetted forms, but it
is worth noting at the outset that they can be solid or liquid,2 and can have arbitrary
shapes. The familiar elegant shapes are a consequence of the minimisation of surface
energy but in engineering applications the crystals are produced in functional shapes
(Fig. 1.2).
The beautiful shapes of many crystals that are found in nature represent an at-
tempt at the minimisation of surface energy per unit volume of material when at
equilibrium, or because the rate of growth varies with direction. For isotropic mate-
rials, a sphere would represent the equilibrium shape but crystals are not isotropic;
some planes of atoms have a lower surface energy than others so there will be a ten-
dency to maximise the area of those planes, in which case the equilibrium shape is
no longer spherical [7].
Engineering materials usually are space-filling aggregates of many crystals of
varying sizes and shapes; these polycrystalline materials have properties which de-
pend on the nature of the individual crystals, but also on aggregate properties such
as the size and shape distributions of the crystals, and the orientation relationships
between the individual crystals. The degree of randomness in the orientations of the
crystals relative to a fixed frame is a measure of texture, which has to be controlled
in the manufacture of transformer steels, uranium fuel rods and beverage cans.
Connecting a pair of crystals together to form a bi-crystal requires a process in-
volving at least five degrees of freedom; two for the choice of the plane along which
the pair are connected and three for the relative orientations of the two crystals. The
connecting plane is referred to as a ‘boundary’ or an ‘interface’ in general. The crys-
tallography of interfaces connecting adjacent crystals can determine the deformation
2 Liquid crystals contain order that is associated with anisotropic molecules that can flow past each

other and yet maintain orientational order [6]. Normal liquids are isotropic.
1.1 Introduction 5

(a) (b)

FIGURE 1.2
(a) Clusters of naturally occurring pyrite crystals (FeS2 ) from Peru. The structure
belongs to the cubic crystal system. We shall see later that the octahedral shapes with
triangular facets are consistent with the defining symmetry of a cube. (b) A nickel
alloy in the shape of a turbine blade; the part above the spiral is the single crystal. Af-
ter removal of the appendages associated with the manufacturing process, the shape
is consistent with the aerodynamic form required for a turbine engine. Blades of this
kind serve in harsh environments and are subjected to large stresses. This can lead to
slow, stress-driven lengthening of the blade driven by the movement of atoms, even-
tually making it unfit for service. In polycrystalline blades, the boundaries between
the crystals are easy paths for atom movement and hence have shorter service lives.
6 1 Introduction and Point Groups

behaviour of the polycrystalline aggregate; it can influence mechanical properties

such as toughness through its effect on the degree of segregation of impurities to
such interfaces [8].

1.2 The lattice

Imagine an infinite, two-dimensional pattern of points such that an observer located
at any single point would see the same pattern all around when translating to any
other point in the pattern. Such a pattern would be periodic, i.e. a basic unit of the
pattern repeats indefinitely to generate the whole pattern. Each of the points in the
pattern therefore has the same environment and could be selected as the origin of
the pattern. Points in this pattern that have identical environments are known as the
lattice points.
The set of these lattice points constitutes a three dimensional lattice. A unit cell
may be defined within this lattice as a space-filling parallelepiped with origin at a
lattice point, and with its edges given by three non-coplanar basis vectors a1 , a2
and a3 , each of which represents translations between two adjacent lattice points.
The entire lattice can then be generated by stacking unit cells in three dimensions.
Any vector representing a translation between lattice points is called a lattice vector,
Fig. 1.3.
The unit cell defined above has lattice points located at its corners. Since these
are shared with seven other such cells, and since each cell has eight corners, there is
only one lattice point per unit cell. Such a unit cell is primitive and has the lattice
symbol ‘P’.
Non-primitive unit cells can have two or more lattice points, in which case, the
additional ones will be located at positions other than the corners of the cell. A cell
with additional lattice points located at the centres of all its faces has the lattice
symbol ‘F’; such a cell would contain four lattice points. Not all the faces of the cell
need to have face-centering lattice points; when a cell containing two lattice points
has the additional point located at the centre of the face defined by a2 and a3 , the
lattice symbol is ‘A’ and the cell is said to be A-centred. B-centred and C-centred
cells have the additional lattice point located on the face defined by a3 & a1 or
a1 & a2 respectively. A unit cell with two lattice points can alternatively have the
additional lattice point at the body-centre of the cell, in which case the lattice symbol
is I. The lattice symbol ‘R’ is for a trigonal cell; the cell usually is defined such that
it contains three lattice points. The detailed shapes of unit cells are considered in
section 1.3.
The basis vectors a1 , a2 and a3 define the unit cell; their magnitudes a1 , a2
and a3 respectively, are the lattice parameters of the unit cell. The angles a !1 a2 ,
!
a 2 a3 and a! 3 a1 are conventionally labelled γ, α and β respectively. Vectors within
this framework can be represented by their components with respect to the basis
1.2 The lattice 7

(a) (b)

FIGURE 1.3
(a) Definition of a unit cell; the lattice points are drawn as blue spheres for clarity
and the red vector is a lattice vector joining two lattice points. (b) The stacking of
unit cells to produce a periodic pattern; the stacking can be continued indefinitely to
fill the space available without leaving unfilled regions.

vectors. For a particular vector, this set of components defines the Miller indices of
that vector.3
Our initial choice of the basis vectors was arbitrary since there is an infinite num-
ber of lattice vectors which could have been used in defining the unit cell. The pre-
ferred choice includes small basis vectors which are as equal as possible, provided
the shape of the cell reflects the essential symmetry of the lattice.

Example 1.1: Primitive representation of Cubic-F

The face-centred cubic (Cubic-F) cell has four lattice points per cell and hence is
not primitive. Construct its primitive version bearing in mind that the definition of
the unit cell requires it to be space filling. Identify its lattice type and calculate its
relative volume.
In what follows, directions are represented by their components (Miller in-
dices) along the basis vectors, enclosed within square brackets. The primitive cell
3 See [9] for a discussion of the history of indexing.
8 1 Introduction and Point Groups
is illustrated in Fig. 1.4 with:
1
[100]P ≡ [110]F
2
1
[010]P ≡ [101]F
2
1
[001]P ≡ [011]F .
2
The primitive cell is trigonal, a1 = a2 = a3 , the angle α = β = γ = 60◦ ,
containing a single triad along [111]P . This is in contrast to the four triads along
the body diagonals of the cubic cell. As discussed later (Table 1.1), the defining
symmetry of a cube is four triads whereas that of a trigonal cell is just one triad.
Since the primitive cell has a quarter of the lattice points of the face-centred
cubic cell, its volume is also smaller by a factor of four.

001F

010 F
100 F

FIGURE 1.4
The conventional face-centred cubic unit cell is in black and its primitive version
in blue.

1.3 Bravais lattices

The number of ways in which points can be arranged regularly in three dimensions,
such that the stacking of unit cells fills space, is not limitless; Bravais showed in 1848
that all possible arrangements can be represented by just fourteen lattices.
The fourteen lattices can be categorised into seven crystal systems (cubic, tetrag-
onal, orthorhombic, trigonal, hexagonal, monoclinic and triclinic, Table 1.1); the cu-
bic system contains for example, the cubic-P, cubic-F and cubic-I lattices. Each crys-
tal system can be characterised uniquely by a set of defining symmetry elements,
1.3 Bravais lattices 9

which any crystal within that system must possess as a minimum requirement (Ta-
ble 1.1). The Bravais lattices are illustrated in Fig. 1.5.

TABLE 1.1
The crystal systems. The symmetry axes monad, diad, triad, tetrad and hexad involve
rotations of 360◦ , 180◦ , 120◦ , 90◦ and 60◦ respectively. These particular rotations
leave the lattice in a state that cannot be distinguished from its starting configuration.
ai are the magnitudes of the basis vectors ai (which form a right-handed set). The
defining symmetry cannot be achieved unless the conditions highlighted in red are
satisfied.

System Conventional unit cell Defining symmetry

Triclinic a1 ̸
= a2 ̸
= a3 α̸=β̸ =γ monad
Monoclinic a1 ̸
= a2 ̸
= a3 α = γ = 90◦ , β ≥ 90◦ 1 diad
Orthorhombic a1 ̸
= a2 ̸
= a3 α = β = γ = 90◦ 3 diads
Tetragonal a1 = a2 ̸
= a3 α = β = γ = 90◦ 1 tetrad
Trigonal a1 = a2 = a3 α=β=γ̸ = 90◦ 1 triad
Hexagonal a1 = a2 ̸
= a3 α = β = 90◦ , γ = 120◦ 1 hexad
Cubic a1 = a2 = a3 α = β = γ = 90◦ 4 triads

1.3.1 Two-dimensional crystals and surfaces

The loss of the third dimension reduces the number of periodic patterns of points that
can be used to define unit cells to just the five possibilities illustrated in Fig. 1.7. All
of these can be derived from the projections shown in Fig. 1.6 in the x-y plane – all
the cubic lattices, for example, reduce to square lattices.
Two-dimensional atomic crystals are rare and frequently do not survive for long
in uncontrolled environments. Graphene is a single layer of carbon atoms with a
hexagonal two-dimensional lattice. It can be extracted from graphite, which con-
sists of parallel layers of graphene with weak bonding between the layers. There
are other examples of layered structures which are amenable to the preparation of
two-dimensional crystals that can reach lateral dimensions that are in the microme-
tre range: BN, MoS2 (often used as a lubricant because of its layered structure),
NbSe2 , Bi2 Sr2 CaCu2 Ox [10]. Complex molecules can precipitate on substrates,
from fluid, as two-dimensional arrays that are arranged periodically, for example
the two-dimensional chiral crystals of phospholipid [11]. There are numerous ex-
amples of periodic patterns developing when molecules (e.g. water) or particles
(polystyrene spheres) are deposited on substrates – whether these can be regarded
as two-dimensional crystals, as they often are, remains doubtful since such patterned
objects cannot be isolated in their own right. Nevertheless, from the point of view
10 1 Introduction and Point Groups

of effects such as diffraction and electrical properties, the periodicity of the particles
deposited on substrates is influential.
The structure of a three-dimensional crystal is in general assumed to persist to
the very surfaces of the crystals, but the forces there are different from those in the
bulk of the material. As a consequence, surfaces are known to reconstruct into one of
the five lattice types illustrated in Fig. 1.7. This of course is important in processes
that rely on surface structure, such as catalysis [12].
1.3 Bravais lattices 11

FIGURE 1.5
The fourteen three-dimensional Bravais lattices.
12 1 Introduction and Point Groups

FIGURE 1.6
Projection of the three-dimensional shapes of the fourteen Bravais lattices on to the
a1 -a2 plane. The numbers indicate the coordinates of lattice points relative to the a3
axis; the unlabelled lattice points are by implication located at coordinates 0 and 1
with respect to the a3 axis.
1.3 Bravais lattices 13

FIGURE 1.7
The five two-dimensional lattices.
14 1 Introduction and Point Groups

1.4 Directions
A vector u can be represented as a linear combination of the basis vectors ai of the
unit cell (i = 1, 2, 3):
u = u1 a1 + u2 a2 + u3 a3 (1.1)
and the scalar quantities u1 , u2 and u3 are the components of the vector u with
respect to the basis vectors a1 , a2 and a3 . Once the unit cell is defined, any direction
u within the lattice can be identified uniquely by its components [u1 u2 u3 ], and the
components are called the Miller indices of that direction. They are by convention
enclosed in square brackets (Fig. 1.8).

(a) (b)

(c)

FIGURE 1.8
(a,b) Miller indices for directions and planes. (c) Notice that in non-cubic systems, a
direction with indices identical to those of a plane is not necessarily normal to that
plane.

It is sometimes the case that the properties along two or more different directions
are identical. These directions are said to be equivalent and the crystal is said to
possess symmetry. For example, the [1 0 0] direction for a cubic lattice is equivalent
to the [0 1 0], [0 0 1], [0 1 0], [0 0 1] and [1 0 0] directions; the bar on top of the
number implies that the index is negative.
The indices of directions of the same form are conventionally enclosed in special
1.6 Planes 15

brackets, e.g. ⟨1 0 0⟩. The number of equivalent directions within the form is called
the multiplicity of that direction, which in this case is 6.
Example 1.2: Number of equivalent indices
Given a direction with indices [u v w] such that u ̸ =v ̸=w ̸ = 0, calculate the
number of equivalent directions possible, for the following lattice types: (a) cubic,
(b) tetragonal.
(a) In the cubic system where a1 = a2 = a3 , there are six choices for the first
index, i.e. u, −u, v, −v, w, −w. Suppose that v is selected as the first in-
dex, then the choice for the second index is limited to one of four options
u, −u, w, −w. If the second index is −u, then the choice for the final in-
dex becomes w or −w. The number of equivalent directions of the form
⟨u v w⟩, 6 × 4 × 2 = 48. However, if antiparallel options such as [u v w]
and [−u − v − w] are not counted then the number of equivalent directions
reduces to 24.
Note that if u = v then the number of equivalent directions reduces to 6 ×
1 × 2 = 12 because the choice of the first index leaves only one choice for the
second. An index that is zero also reduces the options by a factor of two since
there is then no negative or positive aspect.Therefore, in the cubic system,
there are 6 variations of ⟨1 1 0⟩: [1 1 0], [1 1 0], [0 1 1], [0 1 1], [1 0 1] and
[1 0 1] if antiparallel directions are not counted.
(b) For the tetragonal system, a1 = a2 ̸ = a3 . Therefore, the choice of the first
index is limited to four options, u, −u, v, −v,; if u is selected as the first
index then the choice of the second is just two (v, −v) and of the third w
or −w. Therefore, the number of equivalent directions of the form ⟨u v w⟩
becomes 6 × 2 × 2 = 24, reducing to 12 if antiparallel options are removed.

1.5 Planes
If a plane intersects the a1 , a2 and a3 axes at distances x1 , x2 and x3 respectively,
relative to the origin, then the Miller indices of that plane are given by (h1 h2 h3 )
where:
h1 = φa1 /x1 , h2 = φa2 /x2 , h3 = φa3 /x3 (1.2)
φ is a scalar which clears the numbers hi off fractions or common factors. Note that
xi are negative when measured in the −ai directions. The intercept of the plane with
an axis may occur at infinity (∞), in which case the plane is parallel to that axis and
the corresponding Miller index will be zero (Fig. 1.8).
Miller indices for planes are by convention written using round brackets:
(h1 h2 h3 ) with braces being used to indicate planes of the same form: {h1 h2 h3 }.
16 1 Introduction and Point Groups

1.6 Weiss zone law

This law states that if a direction [u1 u2 u3 ] lies in a plane (h1 h2 h3 ) then

u 1 h1 + u 2 h2 + u 3 h3 = 0 (1.3)

and the law applies to any crystal system. This will be proven when we deal with the
reciprocal lattice (Chapter 5).

FIGURE 1.9
A zone refers to a set of planes which share a common
direction, which in turn is known as a zone axis. The
direction illustrated would satisfy the Weiss law for
all the planes shown. For example, the planes (1 1 1),
(1 1 2) and (1 1 0) all belong to the zone [1 1 0].

1.7 Symmetry
Although the properties of a crystal can be anisotropic, there may be different direc-
tions along which they are identical. These directions are said to be equivalent and
the crystal is said to possess symmetry.
That a particular edge of a cube cannot be distinguished from any other is a
measure of its symmetry; an orthorhombic parallelepiped has lower symmetry, since
its edges can be distinguished by length.
Some symmetry operations are illustrated in Fig. 1.10; in essence, they transform
a spatial arrangement into another that is indistinguishable from the original. The
rotation of a cubic lattice through 90◦ about an axis along the edge of the unit cell is
an example of a symmetry operation, since the lattice points of the final and original
lattice coincide in space and cannot consequently be distinguished.
We have encountered translational symmetry when defining the lattice; since the
environment of each lattice point is identical, translation between lattice points has
the effect of shifting the origin. Dislocations whose Burgers vectors are lattice vectors
therefore accomplish slip without changing the crystal structure. Deformations like
these are known as lattice-invariant deformations because they accomplish a shape
1.8 Symmetry operations 17

change without altering the nature of the lattice. In contrast, partial dislocations have
Burgers vectors that are not lattice vectors – they therefore change the local stacking
fault sequence of the planes on which they glide.

FIGURE 1.10
Some symmetry operations. The mirror
6 6 6 6 6 plane involves reflection symmetry,
translation whereas the glide plane is a combination
of reflection and translation parallel to the
6 mirror. Translational symmetry is intrinsic
6 to all lattices whereas the other elements
may or may not exist for all lattices. For
mirror example, the triclinic lattice has no mirror
plane.
rotation
6 Note that the translations involved in
the screw axis and glide plane are rational
6 fractions of the repeat distance along
glide
the translation direction. The screw axis
6 illustrated involves a rotation of 180◦ (≡
screw diad diad) combined with a translation that is
half of the repeat distance parallel to the
axis.

1.8 Symmetry operations

An object possesses an n-fold axis of rotational symmetry if it coincides with itself
upon rotation about the axis through an angle 360◦ /n. The possible angles of rotation,
which are consistent with the translational symmetry of the lattice, are 360◦ , 180◦ ,
120◦ , 90◦ and 60◦ for values of n equal to 1, 2, 3, 4 and 6 respectively. A five-fold
axis of rotation does not preserve the translational symmetry of the lattice and hence
is forbidden. A one-fold axis of rotation is called a monad and the terms diad, triad,
tetrad and hexad correspond to n = 2, 3, 4 and 6 respectively.
All of the Bravais lattices have a centre of symmetry, Fig. 1.11. An observer at
the centre of symmetry sees no difference in arrangement between the directions
[u1 u2 u3 ] and [u1 u2 u3 ]. The centre of symmetry is such that inversion through that
point produces an identical arrangement but in the opposite sense. A rotoinversion
axis of symmetry rotates a point through a specified angle and then inverts it through
the centre of symmetry such that the arrangements before and after this combined
operation are in coincidence. For example, a three-fold inversion axis involves a ro-
18 1 Introduction and Point Groups

tation through 120◦ combined with an inversion, the axis being labelled 3. Some of
the symbols representing rotation and rotoinversion axis are as follows, where 2 is
omitted because it is, as will be shown in Chapter 2, equivalent to a mirror operation:

Cl

centre of
CH3 symmetry

H H

CH 3

(a) Cl (b)

FIGURE 1.11
Illustration of centre of symmetry. (a) A molecule with the centre of symmetry iden-
tified by the dot. (b) The common point where the arrows intersect is the centre of
symmetry.

A rotation operation can be combined with a translation parallel to that axis to

generate a screw axis of symmetry. The magnitude of the translation is a fraction of
the lattice repeat distance along the axis concerned. A 31 screw axis would rotate a
point through 120◦ and translate it through a distance t/3, where t is the magnitude
of the shortest lattice vector along the axis. A 32 operation involves a rotation through
120◦ followed by a translation through 2t/3 along the axis. For a right-handed screw
axis, the sense of rotation is anticlockwise when the translation is along the positive
direction of the axis.
A plane of mirror symmetry implies arrangements which are mirror images. Our
left and right hands are (approximately) mirror images. The operation of a 2 axis
produces a result which is equivalent to a reflection through a mirror plane normal to
that axis.
The operation of a glide plane combines a reflection with a translation parallel to
the plane, through a distance which is a fraction of the lattice repeat in the direction
concerned. The translation may be parallel to a unit cell edge, in which case the glide
is axial; the term diagonal glide refers to translation along a face or body diagonal
of the unit cell. In the latter case, the translation is through a distance which is half
1.9 Crystal structure 19

the length of the diagonal concerned, except for diamond glide, where it is a quarter
of the diagonal length.
Example 1.3: Five-fold rotation
Show that a five-fold rotation axis is inconsistent with the translational sym-
metry of a lattice.
In Fig. 1.12, the a is the repeat distance between adjacent lattice points. The
operation of a five-fold rotation axis by θ = 72◦ would lead to lattice points sep-
arated by a distance x. In order for this rotation to be a symmetry operation, x
must equal na, where n is an integer. Since x = a − 2a cos θ, the governing equa-
tion that defines possible values of θ that are consistent with symmetry operations
become:

x = na = a − 2a cos θ
1−n
cos θ = with − 1 ≤ cos θ ≤ 1 (1.4)
2
It follows that for cos θ = −1, − 12 , 0, 12 and 1, θ takes the values 180◦,
120◦ , 90◦ , 60◦ corresponding to two-fold, three-fold, four-fold and six-fold rota-
tion axes. Equation 1.4 shows, therefore, that a five-, or indeed seven-fold rotations
are not symmetry axes. Another way of approaching this problem is on page 112.

FIGURE 1.12
For θ to represent a rotation consistent with the translational symmetry of the lat-
tice, x must equal an integer multiplied by the translation a. The proposed rotation
axis is normal to the plane of the figure.
We note in passing that ordered structures exist that are not periodic but can
yield diffraction patterns that have five-fold or ten-fold symmetries. These struc-
tures are known as quasicrystals; they lack translational symmetry [13].

1.9 Crystal structure

Lattices are regular arrays of imaginary points in space. A real crystal has atoms
associated with these points. Consider the projection of the primitive cubic cell illus-
trated in Fig. 1.13a. In the sequence Fig. 1.13b-e, a pair of atoms (Cu at 0,0,0 and
20 1 Introduction and Point Groups

Zn at 12 , 12 , 12 ), which also is known as the motif, is placed at each lattice point of

Fig. 1.13a in order to build up the crystal structure of β-brass (Fig. 1.13f).

lattice + motif = crystal structure (1.5)

(a) (b) (c)

(d) (e) (f)

FIGURE 1.13
Building up the crystal structure of β-brass by placing a motif consisting of an ap-
propriate pair of Cu and Zn atoms at each lattice point of a primitive cubic lattice. (a)
represents the primitive lattice with the motif consisting of a pair of distinct atoms
that have yet to be placed. (b-e) The motif is placed at each lattice point. (f) The
three-dimensional representation of the final structure.

The location of an atom of copper at each lattice point of a cubic-F lattice, gen-
erates the crystal structure of copper, with four copper atoms per unit cell (one per
lattice point), representing the actual arrangement of copper atoms in space.
Consider now a motif consisting of a pair of carbon atoms, with coordinates
[0 0 0] and [ 14 14 41 ] relative to a lattice point. Placing this motif at each lattice point
of the cubic-F lattice generates the diamond crystal structure (Fig. 1.14), with each
unit cell containing 8 carbon atoms (2 carbon atoms per lattice point). The figure also
shows the tetrahedral bonding which makes diamond a giant molecule with strong
covalent bonds in three dimensions. Diamonds do contain dislocations [14] but they
have complex cores, are dissociated and incredibly difficult to move, making it very
hard. The structure of silicon is obtained simply by replacing the carbon atoms by sil-
icon (cubic-F, motif of a pair of Si atoms at 0,0,0 and 41 , 14 , 14 ); the strong directional
bonding also results in a large Peierls barrier to dislocation mobility [15], making
silicon single-crystals rather brittle.
Fig. 1.15 shows how a Cubic-F cell changes into a Cubic-P cell when the nickel
and aluminium atoms order in the classical γ/γ ′ superalloy system.
1.9 Crystal structure 21

(a) (b)

(c)

FIGURE 1.14
(a) Projection of the cubic-F lattice. (b) Projection of cubic-F lattice with a motif of a
pair of carbon atoms at 0, 0, 0 and 14 , 14 , 14 placed at each lattice point. (c) Perspective
of the same structure illustrating the tetrahedral bonding of the carbon atoms.

(a) (b)

FIGURE 1.15
Nickel based superalloys. (a) The face-centred cubic crystal structure of disordered
γ. (b) The primitive cubic crystal structure of γ ′ .
22 1 Introduction and Point Groups

Example 1.4: Structure of graphene

Fig. 1.16 shows the atomic arrangement of carbon atoms in a two-dimensional
layer of graphene. Identify the unit cell and lattice type, and calculate the lattice
parameter given that the C-C distance is 0.142 nm (at room temperature).
A single layer of copper atoms is created by depositing copper atoms on
an appropriate substrate. The layer corresponds to the {111} plane of the three-
dimensional structure of copper. Identify the unit cell and lattice type describing
the two-dimensional arrangement of copper atoms and calculate its lattice param-
eter given that the the three-dimensional form of copper has the lattice parameter
0.361 nm at room temperature.
Explain why the {111}Cu plane would make a good substrate for the chemical
vapour deposition of single-layer graphene. Giving reasons, explain the orienta-
tion relationship you might expect between the copper and graphene layers.

0.142 nm

FIGURE 1.16
Atomic structure of graphene sheet without
defects. Entropy effects mean that optically
visible samples cannot ever be made pristine
[16].

The unit cell, identified blue in Fig. 1.17 is primitive hexagonal with a motif
of two carbon atoms per lattice point. The environment about the black atom is
not identical to that around the red atom. The lattice parameter is 0.142 × 2 ×
cos 30◦ = 0.245 nm.
Copper is cubic close-packed. The arrangement of atoms on the {111} close-
packed plane is hexagonal, primitive. The two dimensional cell has the edges
equal to a2 ⟨110⟩, so the lattice parameter is 0.255 nm. This closely matches that of
graphene.
Given that the unit cell edges of the two-dimensional structures of graphene
and copper match to 4% (100[0.255 − 0.245]/0.255), it is logical that their unit
cell edges are exactly aligned, [10]graphene ∥ [10]Cu and [01]graphene ∥ [01]Cu .
1.10 Point group symmetry 23

FIGURE 1.17
The blue lines represent the unit cell, with
a motif of a pair of carbon atoms coloured
black and red, per lattice point.

1.10 Point group symmetry

Consider a molecule such as that illustrated in Fig. 1.18. The symmetry operations on
the molecule constitute a collection known as a point group because there always is
one point in space that is left unchanged by every symmetry operation in that group.
The point group symmetry of the molecule is illustrated in Fig. 1.18a. When exposed
to infrared light the molecules absorb energy and vibrate as illustrated in Fig. 1.18b.
These vibrations result in a characteristic spectrum showing peaks at certain energies
which depend on the modes illustrated. Other molecules with the same point group
symmetry, such as sulphur tetrafluoride, will exhibit similar spectroscopic properties.
The point group of a crystal is also the symmetry common to all of its macroscopic
properties.

(a) (b)

FIGURE 1.18
(a) A molecule with point group symmetry 2m (diad and a mirror plane parallel to the
diad). (b) Vibration modes (symmetrical stretch, asymmetrical stretch and bending)
of the molecule when appropriately stimulated. There is an additional mirror plane
passing through the centres of all three atoms if account is taken of the electron
clouds associated with the molecule.

Translations therefore are necessarily excluded in a point group. Rotations, mir-

24 1 Introduction and Point Groups

ror planes, centre of symmetry and inversion axes are permitted. There are 32 point
groups in three dimensions, classified within the seven crystal classes (Table 1.2).
The triclinic, monoclinic and orthorhombic groups do not contain triads, tetrads or
hexads. For those systems, when the point group symbol contains three elements
(e.g. 2mm), then the symbols are presented in the order of the symmetry elements
parallel to the x, y and z axes respectively, as illustrated in Fig. 1.19a for an object
in the orthorhombic class.

TABLE 1.2
Point group symmetries associated with the seven crystal classes.

Class Non-centrosymmetric Centrosymmetric

⎫
Cubic 23, 432, 43m m3, m3m ⎪
⎪
⎬
Hexagonal 6, 6, 622, 6mm, 6m2 6/m, 6/mmm
With high order axes
Trigonal 3, 32, 3m 3, 3m ⎪
⎪
⎭
Tetragonal 4, 4, 422, 4mm, 42m 4/m, 4/mmm
⎫
Orthorhombic 222, 2mm mmm ⎬
Monoclinic 2, m 2/m Without high order axes
⎭
Triclinic 1 1

For crystal systems with higher order axes, the z direction is assigned to that
higher order axis, with the second symbol corresponding to equivalent secondary
axes that are normal to z, and the third also normal to z to equivalent tertiary direc-
tions passing between the secondary ones. Fig. 1.19b shows an example for an object
in the tetragonal class, with the tetrad placed along the z axis, the two mirror planes
with normals along the x and y axes and the additional mirror planes generated by
this symmetry also illustrated.
In the case of the cubic system the triad is always the second symbol given that
the defining symmetry is four triads.
Some further details on the notation are as follows:
• m ≡ 2 represents a mirror plane.
• 2, 3, 4, 6 are 2-fold, 3-fold, 4-fold and 6-fold rotation axes, respectively.
• 1, 2, 3 etc. are inversion axes; the one fold inversion axis is equivalent to a centre
of symmetry; 2 signifies a rotation of 360◦/2 combined with an inversion through
the centre point.
4
• m mm is a 4-fold axis with a mirror normal to it and four mirror planes containing
it. The symbol is usually written without the space as 4/mmm.
• 432 refers to a point group with 3-fold axes which are not parallel to the z axis,
where a unit cell has x, y and z axes. This is limited to the cubic system.
1.10 Point group symmetry 25
zxy
xyz additional
mirrors 4mm
2mm
z
x
y
x
y
z
mirrors
(a) mirrors (b)

FIGURE 1.19
Convention for point group notation. (a) Groups without high order axes. (b) Groups
with high order axes.

Example 1.5: Point symmetry of chess pieces

Determine the point group symmetries of the chess pieces illustrated in Fig. 1.20
and identify the crystal class to which each belongs. Both shapes have been ma-
chined from solid cubes.

The bishop has a three-fold axis passing through the cut corner of the cube
and a mirror parallel to that axis, so the point group symmetry is 3m which fits
the trigonal crystal class. The castle clearly has a single tetrad with two mirrors
parallel and one normal to the tetrad so in this case, the point group is 4/mmm
which is tetragonal class (Table 1.2).

(a) (b)

FIGURE 1.20
Avant garde chess pieces. (a) Bishop. (b) Castle.

Some of the point group notation comes from the classification of the macro-
scopic shapes of crystals. If a crystal exhibits well-developed faces, its point group
symmetry can be derived from its external form. Fig. 1.21 illustrates this for two
cases. In the case of the gypsum there is a diad normal to (010) and a mirror plane
normal to that diad so that the shape has the point group 2/m belonging to mono-
26 1 Introduction and Point Groups

clinic class. With the epsomite there is a diad along the vertical axis of the diagram
and two diads perpendicular to the vertical axis passing through the vertical edges
between the planes (110) and (110), i.e. it belongs to the orthorhombic crystal class.
With appropriate analysis of facetted crystals it therefore becomes possible to
deduce the crystal class simply from symmetry. Fig. 1.22 shows two such examples.
The quartz crystal clearly has a hexad which is the defining symmetry of the hexag-
onal crystal class. The celestite has three mutually perpendicular diads, placing it in
the orthorhombic system.

FIGURE 1.21
(a) Shape of gypsum
(CaSO4 .2H2 O) with point
group symmetry 2/m. (b) Shape
of epsomite (MgSO4 .7H2 O)
with point group symmetry 222.
Note that the angle between the
(110) and (110) faces is 89.37◦C,
because the lattice parameters a,
(a) (b) b, and c are 1.1866, 1.1998 and
0.6855 nm respectively.

(a) (b)

FIGURE 1.22
Each figure shows an idealised model on the left and the corresponding crystal on the
right. (a) Quartz. (b) Celestite. Photographs taken at the Colorado School of Mines.

Example 1.6: Octahedral interstices in iron

Draw structure projections of the unit cells of austenite (cubic-F) and ferrite
(cubic-I) in iron and mark on these the locations of the octahedral interstices.
Identify the point group symmetry of an octahedral interstice in austenitic iron.
Repeat this for the octahedral interstice in ferritic iron. Identify using Table 1.2,
the symmetry class to which each interstice belongs.
Why does carbon strengthen ferrite much more than it does austenite?
1.11 Point group symmetry 27
The octahedral interstices in the two structures are illustrated in Fig. 1.23 and
the following points are revealing:
• As can be seen from the structure projection in Fig. 1.23a, there is one octahe-
dral interstice per iron atom in austenite. In contrast, there are three octahedral
interstices per iron atom in the ferrite. This has consequences when the lattice
of austenite is homogeneously deformed into that of ferrite because carbon
atoms distributed at random in the austenite end up in just one subset of octa-
hedral sites in the ferrite, in effect leading to an ordering of the carbon along
one of the cube axes, causing the ferrite to become tetragonal rather than cubic
[17]. This reduction in symmetry has consequences on the thermodynamics of
the lattice, leading to an increase in the solubility of carbon in tetragonal fer-
rite that is in equilibrium with the austenite [18, 19].
• The octahedral interstice in austenite is regular (all sides of equal length) with
a point group symmetry m3m. In contrast, the three axes of the irregular√oc-
√ interstice in ferrite are a (magnitude of the lattice parameter), 2a
tahedral
and 2a with the tetragonal point group 4/mmm.
• For both austenite and ferrite, the octahedral interstice is smaller than a carbon
atom, resulting in a straining of the lattice. In the austenite, the cubic symme-
try of the interstice means that the strain is isotropic whereas in the ferrite it is
anisotropic (tetragonal). An isotropic strain interacts weakly with dislocation
strain fields so the hardening caused by carbon in austenite is also weak. This
is because the hydrostatic components of the strain fields of dislocations are
much smaller than the deviatoric terms. Given this, there is a powerful inter-
action of the tetragonal strain with dislocation strain fields, leading to intense
hardening.

(a) (b) (c) (d)

FIGURE 1.23
Octahedral interstices in austenite (cubic-F) and ferrite (cubic-I). The red objects
are located at a height half, with filled circles representing iron atoms and open
circles the positions of the octahedral interstices. (a) A regular-octahedral inter-
stice in austenite; (b) projection of the austenite unit cell to show the positions of
all such interstices. (a) An irregular-octahedral interstice in ferrite; (b) projection
of the ferrite unit cell to show the positions of all such interstices.
28 1 Introduction and Point Groups

1.11 Summary
It is extraordinary that all crystals without exception can be described in terms of
just fourteen imaginary lattices, the points of which can then be loaded identically
with motifs that may contain one or more atoms that are not necessarily identical.
On the same logic, the apparently great variety of wallpapers available for decora-
tive purposes have just five basic lattices, with the variations generated by changing
the motif. A sixth is of course possible if random two-dimensional patterns can be
printed. One advantage of a random-printed wallpaper is that there would be no effort
required in matching designs at the edges.
The idea that there is an identical configuration of atoms around each lattice point
in a crystal does not in fact sit well with the vast majority of real crystals. Materials
are not pure; random mixtures of atoms in solid solutions cannot result in an identical
environment at each lattice point. Translational periodicity does not then exist, rather,
there is a probability of finding a particular species at a particular location. The con-
sequence is that the crystal must be considered in terms of its averaged properties
and this suffices for most purposes, for example in X-ray diffraction, where an aver-
aged atomic scattering factor is used to calculate intensities. The strain fields around
solute atoms in a random solid solution may then lead to X-ray peak broadening.
The anisotropy of crystals, moderated by symmetry, has major consequences on
how they can be exploited for engineering purposes. The crystallographic direction
along which single-crystal turbine blades are grown is selected in order to minimise
the possibility of vibrations during the operation of gas turbines.
References

1. J. M. Ziman: Models of disorder: Cambridge, U. K.: Cambridge University

Press, 1979.
2. F. Wilczek: ‘Quantum time crystals’, Physical Review Letters, 2012, 109,
160401.
3. J. Zhang, P. W. Hess, A. Kyprianidis, P. Becker, A. Lee, J. Smith, G. Pagano,
I.-D. Potirniche, A. C. Potter, A. Vishwanath, N. Y. Yao, and C. Monroe: ‘Ob-
servation of a discrete time crystal’, Nature, 2017, 543, 217–220.
4. S. Choi, J. Choi, R. Landig, G. Kucsko, H. Zhou, J. Isoya, F. Jelezko, S. Onoda,
H. Sumiya, V. Khemani, C. von Keyserlingk, N. Y. Yao, E. Demler, and M. D.
Lukin: ‘Observation of discrete time-crystalline order in a disordered dipolar
many-body system’, Nature, 2017, 543, 221–225.
5. C. Nayak: ‘Time crystals: new form of matter through to break laws of physics
created by scientists’: Quoted in ‘The Independent’ newspaper, 2017.
6. P. G. de Gennes, and J. Prost: The physics of liquid crystals: 2nd ed., Oxford,
U.K.: Oxford University Press, 2003.
7. C. Herring: ‘The use of classical macroscopic concepts in surface energy prob-
lems’, In: R. Gomer, and C. S. Smith, eds. Structure and Properties of Solid
Interfaces. Chicago, U.S.A.: University of Chicago Press, 1953:5–81.
8. T. Watanabe: ‘Grain boundary engineering: historical perspective and future
prospects’, Journal of Materials Science, 2011, 46, 4095–4115.
9. R. J. Howarth: ‘History of the stereographic projection and its early use in
geology’, Tera Nova, 1996, 8, 499–513.
10. K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V.
Morozov, and A. K. Geim: ‘Two-dimensional atomic crystals’, Proceedings of
the National Academy of Sciences, 2005, 120, 10451–10453.
11. R. M. Weis, and H. . McConnell: ‘Two-dimensional chiral crystals of phospho-
lipid’, Nature, 1983, 310, 47–49.
12. G. A. Somorjai: ‘The surface structure of and catalysis by platinum single
crystal surfaces’, Catalysis Reviews, 1972, 7, 87–120.

29
30 1 References

13. D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn: ‘Metallic phase with

long-range orientational order and no translational symmetry’, Physical Review
Letters, 1984, 53, 1951–1953.
14. F. C. Frank: ‘Observation by X-ray diffraction of dislocations in a diamond’,
Philosophical Magazine, 1959, 39, 383–384.
15. V. V. Bulatov, S. Yip, and A. S. Argon: ‘Atomic modes of dislocation mobility
in silicon’, Philosophical Magazine, 1995, 72, 453–496.
16. H. K. D. H. Bhadeshia: ‘Large chunks of very strong steel’, Materials Science
and Technology, 2005, 21, 1293–1302.
17. E. Honda, and Z. Nishiyama: ‘On the nature of the tetragonal and cubic marten-
sites’, Science Reports of Tohoku Imperial University, 1932, 21, 299–331.
18. J. H. Jang, H. K. D. H. Bhadeshia, and D. W. Suh: ‘Solubility of carbon in
tetragonal ferrite in equilibrium with austenite’, Scripta Materialia, 2012, 68,
195–198.
19. H. K. D. H. Bhadeshia: ‘Carbon in cubic and tetragonal ferrite’, Philosophical
Magazine, 2013, 93, 3714–3715.
2
Stereographic Projections

Abstract

A stereographic projection maps angular relationships on to a two-dimensional sur-

face; it can illustrate the beauty of symmetry and its consequences, or when combined
with an appropriate net, allow the quantitative determination of angles between plane
normals or directions. The focus of this chapter is on the particular kind of projection
that retains angular truth. Many more applications of stereographic projections will
become apparent in the chapters that follow.

2.1 Introduction
We have seen already that the projection of a three-dimensional crystal structure into
two dimensions, along the z-coordinate, can simplify the perception and representa-
tion of atomic arrangements. There is no loss of information given that the fractional
z-coordinate is identified clearly on the projection. This can be seen in Fig. 1.14
where the three dimensional representation of the structure of diamond lacks clarity
whereas the projected cell is readily visualised. The ease of interpretation using the
projected structure is illustrated for the more complex structure of ε-carbide, which
has a chemical formula between Fe2 C and Fe3 C, in Fig. 2.1. There are six iron atoms
in the cell (3 at z = 21 and the six others that are shared at faces with other cells at
z = 0, 1 and therefore contributing a further 3 to the cell). The illustrated cell con-
tains a full complement of carbon atoms but to achieve the composition Fe2.4 , the
sites coloured blue would only be occupied partially [1].
The projection described in Fig. 2.1 is a straightforward linear operation and there
are no distortions of that linearity. In contrast, Fig. 2.2 shows non-linear projections
of circular arcs, in one case a simple extrapolation onto a horizontal line, and in
the other case via a pole located below the horizontal line where the projections are
recorded. The purpose here is to record angles rather than spatial coordinates.
Imagine now that instead of quadrants of a circle, the diagrams in Fig. 2.2 rep-
resent the surfaces of spheres. A circle drawn on the surface of the sphere would in
general project as an ellipse on the horizontal plane of Fig. 2.2a, whereas it would
project as a true circle in the case of the projection method used in Fig. 2.2b.

31
32 2 Stereographic Projections

(a) (b)

FIGURE 2.1
The structure of hexagonal ε-carbide with lattice parameters a = 0.4767 nm and
c = 0.4353 nm. The large atoms are iron. The small atoms are carbon but not all
the sites designated blue are occupied. (a) The three-dimensional cell with the z-axis
vertical. (b) Projection of the cell along the z-axis, with the fractional z-coordinates
listed.
2.1 Introduction 33
90 80
70
60
50
90 80
70 40
60
50 30
40 20
10
20 0
10
0

90 70 50 30 0
(a) 10 (b)

FIGURE 2.2
Projections of angles made by lines intersecting a circular quadrant onto a horizontal
line.

Example 2.1: Projection of small circle

A sphere has north and south poles. Show that if a circle is constructed on the
northern hemisphere with each point on its locus projected on to the equatorial
plane by lines originating from the south pole, then the projection itself will re-
main a circle. Fig 2.3a illustrates the circle ab on the surface of the sphere, to-
gether with its projection a′ b′ on the equatorial plane. A cross-section of the same
diagram through the centre of the circle ab is shown in Fig 2.3b. The conjugate
circular section cd is symmetrically inclined to the axis ST .
The line eb is constructed to be parallel to the equatorial plane. The arcs eS
and Sb are equal in length so it follows that φ1 = φ2 . Since cd is symmetrically
inclined to ST , it follows that φ3 = φ2 and therefore φ3 = φ1 . In other words,
the circle cd is parallel to the equatorial plane. If the circle cd is now translated
towards the equatorial plane, it remains a circle and ends up as a circle a′ b′ .
34 2 Stereographic Projections

(a) (b)

FIGURE 2.3
(a) Circle on the surface of a sphere projected on to the equatorial plane via the
south pole. (b) Construction showing that the projection is also a circle.

This example shows that the stereographic projection method illustrated in

Fig. 2.3 preserves angular truth; if two circles (e.g. a longitude and latitude) inscribed
on the sphere cross at 90◦ then the projections of those two circles will also cross at
that angle. We shall now consider these stereographic projections in more detail,
following a brief description of why they are of importance.

2.2 Utility of stereographic projections

There is an excellent review of the history of the stereographic projection by Howarth
[2], tracing the origin to the second century BCE in the context of astronomy.
Howarth describes how projections were then used in mineralogy and structural ge-
ology, where ‘the spatial orientations of a crystal face, betting plane, fault surface
etc. can be considered by imagining a plane passing through the centre of a sphere’.
Stereographic projections are two-dimensional representations which now have
found numerous applications in materials science, with the method often embedded
in the software that controls experimental crystallography, for example,
• the deformation of single crystals whence crystal planes rotate in order to com-
ply with the external stress; such rotations in polycrystalline materials lead to
non-random aggregates of crystals, the properties of which are in between those
of single and random polycrystals. In both of these cases, it becomes necessary
to define the orientations of individual crystals relative to the deformation axes,
which is where stereographic projections become useful [3].
• When phase transformations occur in the solid state, the probability is that the
2.3 Stereographic projection: construction and characteristics 35

product phase will form such that the atomic arrangements match as much as
possible, along the parent/product interface. This often leads to a reproducible ori-
entation relationship between the two crystals, one which can be represented on a
stereographic projection in order to understand the mechanism of transformation
or the deformation behaviour of the two-phase mixture. Stereographic projections
can be used to decide whether the orientation relationship is reproducible or oc-
curs by chance [4].
• Diffraction data can be presented on stereographic projections. It is now routine to
to examine both structure and crystallographic information from polycrystalline
samples on a single image [5], with the crystallographic data presented in the
form of corresponding colours on the microstructural image and stereographic
projection.

2.3 Stereographic projection: construction and characteristics

Imagine a sphere. Any plane that passes through its centre will intersect the surface
of the sphere at a circle whose diameter is that of the sphere; this is known as a great
circle, Fig. 2.4a. The primitive (great) circle is where the equatorial plane intersects
the sphere. The diameter of a great circle is the same as that of the sphere. Lines of
longitude are all great circles in the spherical earth approximation.

small
great circle
circle

(a) (b) (c)

FIGURE 2.4
Intersections of planes with sphere. (a) Great circle inclined with respect to the north
and south poles of the sphere, (b) small circle; (c) latitude and longitude lines (image
courtesy of vectortemplates.com).

When an intersecting plane does not pass through the centre of the sphere it
results in a small circle at the intersection, as illustrated in Fig. 2.4b. Lines of latitude
are in general small circles, the exception being the equator.
The shortest distance between two points on a sphere is along the great circle
which passes through both of the points.
Fig. 2.5a shows a crystal with a cubic lattice, placed at the centre of a sphere
with the [100] ∥ x-axis, [010] ∥ y-axis and [001] ∥ z-axis of the sphere; this is the
36 2 Stereographic Projections

conventional orientation of the crystal in the sphere. Similarly, the orientation of an

arbitrary plane can be defined by its normal, which in general is known as the pole
of the plane. Suppose now that the normal to the (011) plane is projected so that it
intersects the sphere at the point a (Fig. 2.5b), which then is projected to the south
pole. The intersection on the equatorial plane being the point b which represents the
pole of the plane on the stereographic projection.
The plane (011) when extended so that it intersects the sphere will result in a
great circle on the surface of the sphere, the trace of which can be projected onto the
equatorial plane as illustrated in Fig. 2.5d. The part of the plane that is in the northern
hemisphere projects as a line, and that from the southern hemisphere as a dashed-line.
All points on the trace are at 90◦ to the pole of (011) since the pole represents the
normal to the plane, another indication of how the stereographic projection maintains
angular truth.
A projection of the longitudes and latitudes illustrated in Fig. 2.4c on to the equa-
torial plane would lead to a net, known as the ‘Wulff net’ in which the resulting great
circles are spaced at equal angles as illustrated in Fig. 2.6 where the angular intervals
are 2◦ . The net can be used to measure angles between two poles on a stereographic
projection by rotating the net such that both poles lie on the same great circle, fol-
lowed by counting the degrees separating them. Note that the angle between two
great circles will be the same as the angle between their poles.
2.3 Stereographic projection: construction and characteristics 37

010 001

100

010

100

(a) (b)

(north) (north)
010 001 010 001

100 100
trace

010 x 010
x
100 100

(c) (south) (d) (south)

FIGURE 2.5
Plotting plane normals on a stereographic projection. The insets, top-left in each
case, show a plan view of the equatorial plane. (a) Crystal placed at the centre of the
sphere. (b) Projection of the pole of (011). (c) Projection of (011). (d) Projection of
the trace of the great circle which is (011) on to the equatorial plane. The trace of the
plane in the southern hemisphere is marked as a dashed curve. The crosses identify
the intersections of the projection lines with the equatorial plane.
38 2 Stereographic Projections

trace

90°
pole

(a) (b)

(c) (d)

FIGURE 2.6
The Wulff net. (a) Relation between pole and trace of a plane. (b) The net is rotated
until the poles a and b lie on the same great circle to measure the angle in between.
(c) The geometrical centre c of the small circle is different from its angular centre b.
The angles ab and bd as measured on the great circle are identical. The distances ac
and cd are identical when measured on a ruler. (d) The poles identified with arrows
are both located at angles φ1 to p1 and φ2 to p2 .
2.3 Stereographic projection: construction and characteristics 39

Example 2.2: Radius of trace of great circle on Wulff net

The curve representing a great circle on a Wulff net is itself an arc of a circle, a
result which follows from the fact that the projection method preserves angular
truth. This arc is from a circle that in general has a radius r greater than that of the
net (ro ). Derive the relationship between r, ro and the offset x of the trace of the
great circle from the origin.
Referring to the figure below, the distance ab is 2ro ; the Pythagoras theorem
then gives

r2 − ro2 = (r − x)2
so it follows that r = (x2 + ro2 )/2x with ro ≤ r ≤ ∞.

Fig. 2.7a,b show the stereographic projection for the cubic system, with the ro-
tational symmetry elements related to the crystallographic directions about which
the rotations occur. Fig. 2.7c shows the stereographic projection being used to illus-
trate how the elastic modulus of a single crystal of ferritic iron varies between 131
and 284 GPa as a function of the crystallographic orientation, in a manner consistent
with its crystalline symmetry. Even aggregates of crystals often do not show collec-
tive macroscopic-properties that are isotropic because the crystals are not distributed
at random. Polycrystalline ferritic steel sheet in which the crystals are aligned with
the ⟨001⟩ direction and with the {110} planes aligned to the rolling plane, exhibit
modulus variation in the range 140 to 210 GPa [6]. The level of anisotropy in aggre-
gates can be controlled, as we shall see in later chapters, to suit specific purposes such
as the formability of metal sheets or the magnetic properties of transformer steels.
40 2 Stereographic Projections

(a) (b)

(c)

FIGURE 2.7
Stereographic projections for the cubic system. (a) Showing the standard way of
drawing the projection with the locations of [100] and [010] on the perimeter and
[001] at the centre. (b) The symmetry elements. The filled squares are tetrads, the
ellipses are diads and triangles are triads. The tetrads are at ⟨001⟩, triads at ⟨111⟩ and
diads at ⟨011⟩. Anything with cubic symmetry must have four triads. (c) Variation
in the elastic modulus of a single crystal of bcc iron as a function of orientation.
Diagram courtesy of Shaumik Lenka, constructed using data from Dieter [7].
2.3 Stereographic projection: construction and characteristics 41

Example 2.3: Traces of plates

Meteorites are essentially iron-nickel alloys that cool ridiculously slowly from
red heat as they traverse through space. Their structure at high temperatures is
austenitic (cubic-F) but on cooling, plates of ferrite (cubic-I) precipitate to form
a Widmanstätten pattern as illustrated in Fig. 2.8. Because of the slow cooling
rate, the coarse structure that forms is visible to the naked eye when the mete-
orite is sectioned and etched. Assuming that theses plates precipitate exactly on
{111}F, and that all the plates are of identical dimensions and shapes, sketch the
macrostructures that would be observed when the meteorite is sectioned on the
(001)F , (111)F and (112)F planes.

(a) (b)

FIGURE 2.8
(a) A meteorite at the Smithsonian Museum in Washington. (b) The Wid-
manstätten pattern observed in the meteorite, of ferrite plates in an austenitic ma-
trix.

Fig. 2.9 shows the stereographic projection referred to the crystallographic

axes of austenite, containing traces drawn in blue, of the {111}F planes on which
the Widmanstätten ferrite precipitates. The intersections of these {111}F planes
with the plane of observation defines the orientations of the plates that would be
observed. Such intersections are identified by letters a,b,c,. . . on the stereographic
projection.
The (001)F plane will reveal a square pattern of plates, because pairs of
{111}F planes have common intersections at the points marked ‘a’ and ‘d’.
Plates precipitating on the (111)F appear as discs when observed on that plane.
The (111)F , (111)F and (111)F intersect (111)F at ‘b’, ‘a’ and ‘c’ respectively,
with these three intersections making angles of 60◦ to each other, generating the
equilateral arrangement of plates in addition to the disc.
On the (112)F plane of observation, there will be an arrangement of three
plates corresponding to the intersections at ‘a’, ‘e’ and ‘f’, making angles 67◦
(between ‘a’ and ‘e’), 46◦ (between ‘e’ and ‘f’) and 67◦ (between ‘e’ and ‘a’).
42 2 Stereographic Projections

d (112) a
f
-111
- -111
c
e
001
010
b

-
111 111

a d
b f
a 100 a
e
c a
(111) (001)
d (112)

FIGURE 2.9
Stereographic projection referred to the crystallographic axes of the austenite. The
blue lines are traces of {111}F planes, and the dashed lines the planes on which the
macrostructure is observed. The insets at the bottom show what the macrostruc-
tures should look like if all the plates are of identical dimension and shape, as a
function of the plane of observation.

2.4 Stereographic representation of point groups

Example 2.4: Mirror plane equivalent to 2
Show using sketch stereograms, that the operation of a diad followed by an in-
version (i.e. 2) is equivalent to a mirror plane. Do the operations 3 and 6 lead to
centres of symmetry?
A general pole on a stereographic projection is one that is not located at par-
ticular symmetry elements such as the rotation axis (other than a monad) or on a
mirror plane. The black dot on Fig. 2.10a is a general pole which on the operation
of a diad results in a second pole as shown in Fig. 2.10b. When inverted through
the centre, this second pole intersects the stereographic sphere in the southern
2.4 Stereographic representation of point groups 43
hemisphere and hence is shown as an open circle in Fig. 2.10c, which shows that
the net operation of a diad followed by an inversion is a mirror plane parallel to
the plane of the diagram. It is seen that 2 ≡ m.
The operation of a triad on a general pole leads to three such poles, which
after inversion result in the situation illustrated in Fig. 2.10c, which shows that the
point group 3 belongs to a class that has centres of symmetry whereas 2 and 6 do
not.

(a)

(b) (c) (d)

FIGURE 2.10
(a-c) An arbitrary pole is rotated by 180◦ about the diad normal to the plane of the
diagram, to generate the second pole. This second pole is then inverted through
the centre into the southern hemisphere to to result in inversion diad 2 that is
equivalent to a mirror normal to that axis. (b) Triad, (c) inversion triad and (d)
inversion hexad. Mirror planes in (a) and (d) are identified using bold circles.

Example 2.5: The point groups 3m and m3

Referring to Table 1.2, explain using sketch stereograms, the difference be-
tween the point groups 3m and m3.
According to the conventions described on page 24, the point group 3m im-
plies that there is a single triad with the secondary axis being the normal of the
mirror plane. The triad ensures that there would be three such mirror planes. There
is no mirror plane normal to the triad (Fig. 2.11a), which means that the atomic
arrangement along one end of the triad is different from the opposite end. Such
a crystal may display spontaneous polarisation and the point group 3m is said to
be a polar point group, which includes 1 (triclinic); 2, m (monoclinic); mm2 (or-
thorhombic); 4, 4mm (tetragonal); 3, 3m (trigonal); 6, 6mm (hexagonal). There
are no polar groups in cubic crystals. Gallium nitride as a polar material (6mm) is
of huge interest in the manufacture of light emitting diodes – Fig. 2.11b shows that
the top surface normal to the hexad is different from the bottom surface. During
deposition of GaN films on substrates, the layers deposit predominantly with uni-
44 2 Stereographic Projections
form polarity except some regions which have opposite polarity and are regarded
as defects that lead to poor control of electrical properties [8].

(a) (b)

(c) (d)

FIGURE 2.11
Sketch stereograms showing what happens to a general pole on the operation of
the following point groups (bold lines represent mirror planes): (a) trigonal 3m.
(b) The structure of GaN with point group symmetry 6mm, viewed with the 6-fold
axis along z. The gallium atoms are labelled green. (c) Cubic m3. (d) Projection
of the unit cell of pyrite (FeS2 ) with the fractional z coordinates indicated, glide
planes marked with dashed lines and screw diads. Not all the symmetry elements
are illustrated for simplicity. For example, there will be glide planes parallel to the
plane of the diagram at heights 14 z and 34 z.

The point group m3 belongs to the cubic crystal class with the first symbol m
parallel to the z axis (page 24) and the mandatory symbol ‘3’ to follow since the
defining symmetry of a cubic system is four triads, in this case inversion triads.
It is evident from the plotting of a general pole and its symmetry related poles
shown in Fig. 2.11c that there is a centre of symmetry. Pyrite belongs to the cubic
class with point group m3; the actual crystals are illustrated in Fig. 1.2 where
the four triads associated with the regular octahedral shape of the crystals are
obvious. Fig. 2.11d shows a projection of the crystal structure of pyrite, which
2.4 Stereographic representation of point groups 45
is primitive cubic. The structure can be compared with the stereogram showing
the point group symmetry in Fig. 2.11c. Bearing in mind that point groups do
not include translations, the glide planes and screw axes indicated in Fig. 2.11d
become mirrors and diads respectively in Fig. 2.11c.

Crystals which lack a centre of symmetry can be piezoelectric, i.e. they develop
a dipole on being deformed and will change shape on the application of an elec-
trical field. Under normal conditions the crystal has positive and negative electrical
charges which are symmetrically distributed, leaving it in a neutral state when homo-
geneously deformed. The application of a stress causes charge asymmetry, and the
development of a voltage across the crystal. PbTiO3 is such a substance (Fig. 2.12)
where for example, the titanium atom is not located at exactly 21 , 12 , 12 . The structure
is primitive tetragonal, with point group 4mm.1
The stereogram illustrated in Fig. 2.12b shows that the structure does not have a
centre of symmetry. Secondly, if an atom is placed at a general position (marked a)
then there will be seven other identical atoms in the unit cell, inconsistent with the
structure illustrated in Fig. 2.12a. An atom placed on a tetrad (marked c) will generate
just one atom per unit cell; thus, there is just one Ti at 21 , 12 , 0.572. The oxygens at
1 1 1 1
2 , 2 , 0 and 2 , 2 , 1 also lie on a tetrad, and since they are shared between two cells,
there is only one such oxygen in the unit cell. An atom, such as the oxygen at 0, 21 , 12 ,
placed on the mirror plane as illustrated by the poles marked b, would generate two
such atoms in the cell (the four on the vertical faces are each shared by two cells).
This example shows how the placing of an atom within a unit cell must be con-
sistent with the symmetry elements of the cell because there must be identical atoms
at all symmetry related positions.
There are many other consequences of whether or not a crystal has a centre of
symmetry. The free energy of a small volume element containing a one-dimensional
composition gradient varies as follows [9]:
% &2
dc d2 c dc
g = g{c} + κ1 + κ2 2 + κ3 (2.1)
dz dz dz

where c is the concentration, g{c} is the free energy of a homogeneous solution with
composition c, z is the distance and κi are specific functions of the dependence of
free energy on the gradients of concentration as described elsewhere [9]. In this, κ1
is zero for a centrosymmetric crystal since the free energy must then be invariant to
a change in the sign of the coordinate z.

1 That is, by the convention quoted on page 24 the z direction is assigned to the four-fold rotation axis

followed by m for the mirrors with normals parallel to the x and y axes, and the final symbol for the
tertiary mirrors with normals between the x and y axes.
46 2 Stereographic Projections

es indicate
mirror planes

(a) (b)

FIGURE 2.12
(a) Lead titanate (PbTiO3 ) is tetragonal with a = 0.3904 nm and c = 0.4152 nm.
The Pb is located at coordinates (0,0,0.116), the Ti at ( 12 , 21 , 0.572), an oxygen atom
at (0, 12 , 12 ) and another oxygen at ( 12 , 21 , 0). The structure is primitive tetragonal
with point group 4mm. Note that each of the lead atoms is on a cell edge (and hence
shared with three other adjacent cells), not a cell corner so there is only one lead
atom per cell. (b) Stereogram showing the symmetry operations associated with the
point group 4mm.

2.5 Summary
A circle on a sphere projects as a circle on the stereographic projection; angular
relationships are therefore preserved. The method is therefore well suited to look
at angles between planes or directions in a crystal. The symmetry of the crystal in
its three dimensions becomes easy to visualise by plotting the symmetry elements
in their correct angular orientations on a stereographic projection. The placing of
atoms inside the unit cell during attempts to solve structure must clearly be consistent
with the symmetry determined, for example, from the external form of well-formed
crystals.
The stereographic projection can also be used quantitatively, by superimposing
a Wulff net to directly measure angles between a pair of poles. The net consists of
projections of great and small circles, rather like the longitudes and latitudes on a
globe representing the earth, The shortest distance between two points on the surface
of a sphere is when the two points are located on an arc of a circle which has the same
diameter as the sphere. When using a Wulff net to measure the angle, it therefore is
necessary to locate the two poles on a great circle.
We have seen that some quite elementary considerations of symmetry can indi-
2.5 Summary 47

cate the occurrence of phenomena such as piezoelectricity, or polar materials. There

is much more to be explored in the chapters that follow.
References

1. S. Nagakura: ‘Study of metallic carbides by electron diffraction part III. iron

carbides’, Journal of the Physical Society of Japan, 1959, 14, 186–195.
2. R. J. Howarth: ‘History of the stereographic projection and its early use in geol-
ogy’, Tera Nova, 1996, 8, 499–513.
3. S. Suwas, and R. K. Ray: Crystallographic texture of materials: New York, USA:
Springer, 2014.
4. H. S. Fong: ‘Chance occurrence of orientation relationships between phases
in solid transformations and distribution of grain misorientations in a polycrys-
talline single phase material’, Metallurgical Transactions A, 1981, 12, 2057–
2062.
5. A. F. Gourgues-Lorenzon: ‘Application of electron backscatter diffraction to
the study of phase transformations’, International Materials Reviews, 2007, 52,
65–128.
6. S. Pramanik, S. Suwas, and R. K. Ray: ‘Influence of crystallographic texture and
microstructure on elastic modulus of steels’, Canadian Metallurgical Quarterly,
2014, 53, 274–281.
7. G. E. Dieter: Mechanical Metallurgy: McGraw-Hill, 1988.
8. K. Bertness: ‘GaN nanowires: knowing which end is up’: 2015: URL
http://www.nist.gov/pml/div686/grp10/polarity-of-gan-nanowires.cfm.
9. J. E. Hilliard: ‘Spinodal decomposition’, In: V. F. Zackay, and H. I. Aaron-
son, eds. Phase Transformations. Metals Park, Ohio, USA: ASM International,
1970:497–560.

49
3
Stereograms for Low Symmetry Systems

Abstract

Most of the metals in the periodic table have a cubic structure under ambient condi-
tions, but there are many that are in the hexagonal class. The stereographic projection
for non-cubic systems depends on the lattice parameters and the angles between the
basis vectors. And it can no longer be assumed that directions and planes with the
same indices are parallel. We shall see that in the case of the hexagonal system, it
can be useful to use four basis vectors to define directions and planes.

3.1 Introduction
In the cubic system, a plane normal (pole) with indices hkl is parallel to a direction
uvw when h = u, k = v, l = w; thus the normal to (123) is parallel to [123]. This
is generally not the case for less symmetrical systems. Fig. 3.1 illustrates this for a
two–dimensional unit cell in which b > a.
normal to (11)
x
FIGURE 3.1
The direction [11] is not parallel to the normal to the
tra plane (11). On the other hand, the directions parallel
ce
] of ( to the cell edges, i.e. [01] and [10] are parallel to the
[11 11)
y
normals of (01) and (10) respectively.

The principles for the plotting of poles on stereographic projections are never-
theless identical for any lattice. However, the cubic system exhibits extraordinary
symmetry so angular relationships on the stereographic projection are not dependent
on the lattice parameter. This is not the case for other lattices with lesser symmetry,
the stereograms of which will not only appear different from a distance, but for the
same lattice type, the angular positions become dependent on the lattice parameters.
Fig. 3.2a,b show how the projections differ in the case of the cubic and orthorhombic
lattices where the absence of tetrads in the latter is obvious. Fig. 3.2c shows how the
projection changes when one of the lattice parameters of the orthorhombic lattice is
changed relative to that illustrated in Fig. 3.2b.

51
52 3 Stereograms for Low Symmetry Systems

The fact that directions are not in general parallel to plane normals with the same
indices is apparent from a comparison of Fig. 3.2c,d.

(a) (b)

(c) (d)

FIGURE 3.2
Stereographic projections centred on the 001 pole. (a) Cubic, lattice parameter 5
units. (b) Orthorhombic, lattice parameters 2, 3, 8 units. (c) Orthorhombic, lattice
parameters 2, 3, 5 units. (d) The orthorhombic cell as in (c) but with directions plotted
instead of poles.
3.2 Hexagonal system 53

3.2 Hexagonal system

The unit cell has a = b ̸ = c and α = β = 90◦ with γ = 120◦ . The normals to
the (100) and (010) planes therefore make an angle of 60◦ (Fig. 3.3a). This is shown
on the projection in Fig. 3.3, which highlights a special difficulty with the hexagonal
system.

(a) (b)

(c) (d)

FIGURE 3.3
(a) Showing the hexagonal cell with the normals to the (100) and (010) planes iden-
tified with blue-coloured arrows. (b) Stereographic projection of hexagonal structure
(lattice parameters a = 2, b = 2 and c = 3 units with γ = 120◦ ) centred on the 001
pole. (c) Showing the hexagonal cell, now in four index notation. (d) Corresponding
stereogram in four index notation.

It is evident from Fig. 3.3a that the (100), (010) and (110) are planes belong-
ing to the same form {100}, i.e., they are crystallographically equivalent. And yet,
54 3 Stereograms for Low Symmetry Systems

they have different indices. To eliminate this discrepancy, a four index system is
frequently used for hexagonal lattices by introducing a fourth axis (labelled t in
Fig. 3.3c), which gives plane indices as hkil, where from geometry, i = −(h + k).
This has the effect of giving equivalent planes the same combination of digits in
their indices. Thus, (100), (010) and (110) become (1010), (0110) and (1100) re-
spectively. The four components are referred to as the Miller-Bravais indices and the
system is reserved for hexagonal lattices [1].
A similar conversion for directions from uvw to U V JW is a little more difficult
because it is designed in such a way that J = −(U + V ) and at the same time, the
Weiss zone law in which a plane hkl containing a direction uvw is satisfied

hu + kv + lw = 0 (3.1)

so that
hU + kV + iJ + lW should equal zero.
On substituting for i = −(h + k) we get,

hU + kV + (−h − k)J + lW = 0 (3.2)

h(U − J) + k(V − J) + lW = 0

On comparing the coefficients of h, k and l, and noting that J = −(U + V ),

U −J =u u = 2U + V U = 13 (2u − v)
V − J = v → v = U + 2V and V = 13 (2v − u) (3.3)
W =w w=W J = − 13 (u + v)

TABLE 3.1
Directions of the same form now have similar combinations of digits in the four
figure notation.

Three-index notation Four-index notation

1
[100] 3 [2110]
1
[010] 3 [1210]
1
[110] 3 [1120]
1
[uvw] 3 [2u − v, 2v − u, − (u + v), 3w]
3.2 Hexagonal system 55

Example 3.1: Angles in the hexagonal system

Iron-manganese alloys can be hexagonal, class 6/mmm with lattice parameters
a = 0.2951 nm and c = 0.4679 nm. The material is ductile because slip can oc-
cur not only on the basal plane (0001) but also the prism {1010} and pyramidal
{1011} planes. Calculate the angles that these planes make with the [0001] direc-
tion.
Draw a stereogram representing the upper hemisphere and mark and index the
expected slip directions. Show and index the poles of the slip planes.

The normal method of taking scalar products between vectors in order to de-
termine angles is difficult in non-cubic systems without a deeper understanding of
real and reciprocal vectors, as explained in Chapter 9. Noting that angles between
planes are also the angles between their normals, the simple geometry illustrated
in Fig. 3.4a,b shows that the required angle is arctan(c/a cos 30) = 61.36◦. Note
that the [0001] direction is normal to (0001) and parallel to the (1010) plane, so
the normals to these planes make angles of 0◦ and 90◦ respectively to [0001].
The slip directions and slip planes are plotted on Fig. 3.4c. The slip direction
lying in a particular slip plane can be identified by applying the Weiss zone rule.
Thus, [2110] lies in each of the planes (0001), (0111) and (0110).

(1011)
[0001]

(1011)

c normal to
(1011)
61.36°

p o p
(a) (b) a cos 30° (c)

FIGURE 3.4
(a) Hexagonal unit cell showing the (1011) plane. (b) The section poq which
contains the normals to the (0001) and (1011) planes. (c) Stereographic projec-
tion showing the poles of the slip planes (filled circles), and the slip directions
(filled squares). The plot is based on the lattice parameters a = 0.2951 nm and
c = 0.4679 nm.

Many polycrystalline metals with hexagonal close-packed crystal structures

are brittle. This is because only three slip systems are active, the (0001) close-
packed plane with the three close-packed ⟨1120⟩ directions on that plane. To pro-
duce an arbitrary deformation requires five independent slip systems. Imagine a
56 3 Stereograms for Low Symmetry Systems
strain tensor representing strains with respect to a coordinate system:
⎛ ⎞
ϵ11 ϵ12 ϵ13
⎜ ⎟
⎜ ⎟
⎝ϵ21 ϵ22 ϵ33 ⎠
ϵ31 ϵ32 ϵ33

where ϵij with i = j are normal strain and the rest shear strains such that ϵij = ϵji .
Assuming that there is no volume change, ϵ11 + ϵ22 + ϵ33 = 0 so only two of the
normal strains are independent and there are three shear strains. Therefore there
are five independent strains needed in order to produce an arbitrary deformation.
Consider now a polycrystalline material. The application of stress will cause the
individual crystals to respond differently because they have differing orientations
relative to the stress. Their changes in shape are therefore arbitrary as far as neigh-
bouring crystals are concerned, so these differences must be accommodated, with-
out which fracture occurs. Hence the need for five independent slip systems for a
polycrystalline material to maintain structural integrity.

Iron, by far the most used metallic material in the world, under ambient condi-
tions is cubic-I with one iron atom located at each lattice point. The common slip
system is {011}⟨111⟩, each containing two independent slip directions of the form
⟨111⟩. There are therefore, twelve independent slip systems, making the iron ductile
in polycrystalline form. The same can be demonstrated for the cubic-F form of iron.
This makes iron very ductile.
The cubic-I form of iron is stabilised under ambient conditions by its ferromag-
netic properties; ruthenium and osmium, which in terms of their outer-electron struc-
ture are iron analogues in the periodic table, do not exhibit ferromagnetism and
hence have hexagonal crystal structures. In the absence of its ferromagnetic prop-
erties, the stable form of iron would also be hexagonal with a motif of a pair of
identical Fe-atoms, one at 0,0,0 and another at 31 , 23 , 12 . The easy-slip system is then
{0001}⟨1120⟩. Since there is only one (0001) plane in which there are three direc-
tions of the form ⟨1120⟩, polycrystalline hexagonal iron is relatively brittle since the
number of slip systems in each crystal is less than the required five. So in the absence
of ferromagnetism, we would not have civilisation in the form that we know today.
There are caveats to this story. Deformation can occur by mechanisms other than
slip, for example, mechanical twinning, so that the five independent slip system re-
quirement is not sufficient to determine the ductility of a polycrystalline material.
And iron below its Curie temperature is not strictly cubic, but tetragonal because
the magnetic spins in any given crystal tend to be aligned along a ⟨100⟩ axis [2].
However, the tetragonality is quite small and can be neglected in most experiments,
although it does manifest on a macroscopic scale via the magnetostrictive effect,
which is the variation in the length of a sample subjected to a magnetic field [3].
3.2 Hexagonal system 57

Example 3.2: Growth direction of cementite laths

Cementite (Fe3 C, orthorhombic) can precipitate in certain iron alloys in the
form of coarse laths at temperatures where the iron is in its austenitic (cubic-F)
state. A lath is a shape where two of its dimensions are much smaller than its
length, and therefore can be associated with a direction along which it exhibits
rapid growth. An idealised shape of a lath of cementite is shown in Fig. 3.5. The
precipitates when observed on two-dimensional sections can give a misleading in-
dication of the major growth direction. For example, when observed on the section
‘abcd’, the longest axis of the section is not parallel to the actual growth direction.
To resolve this, a number of experiments revealed traces of the longest di-
mensions of cementite laths to be parallel to [011], [012], [013]. Plot these traces
on a stereogram in order to see whether they intersect at a common direction,
which corresponds to the true growth direction. Bearing in mind that cementite
is orthorhombic with lattice parameters a = 0.45165 nm, b = 0.50837 nm and
c = 0.67475 nm. It is first necessary to calculate the angles that these traces make
with respect to the basis vectors of the unit cell. This can be done by converting
the components of each vector into an orthonormal basis:

[100]Fe3C ≡ [a 0 0]orthonormal (3.4)

[010]Fe3C ≡ [0 b 0]orthonormal
[001]Fe3C ≡ [0 0 c]orthonormal
[011]Fe3C ≡ [0 b c]orthonormal
[012]Fe3C ≡ [0 b 2c]orthonormal
[013]Fe3C ≡ [0 b 3c]orthonormal

Vector scalar products can then be implemented using elementary methods as fol-
lows to calculate the required angles (φ in the case illustrated):

[0 0 c]Fe3C ! [0 b c]orthonormal c2
cos φ = - =- = 37◦
c2 × (b2 + c2 ) c2 × (b2 + c2 )

After calculating all three angles, a stereographic projection is constructed with

the traces marked as in Fig. 3.5; the traces intersect at [100]Fe3C . The growth oc-
curs most rapidly therefore, along the [100]Fe3C direction (to avoid confusion in
the order in which the axis are labelled, we note that this direction corresponds to
the lattice parameter a = 0.45165 nm). This example is illustrative, but the growth
direction deduced is as is observed experimentally for cementite laths precipitat-
ing in alloyed austenite at high temperature (650 ◦ C) [4].
Finally, it is worth noting that the method used to calculate angles is cumber-
some and limited to orthogonal lattices when compared with the use of metric
tensors, as described in chapter 9.
58 3 Stereograms for Low Symmetry Systems

a c
n

[001] [013]

[010]

[001]

FIGURE 3.5
Discovery of the growth direction of cementite laths in austenitic iron using trace
analysis. The stereogram shows directions in cementite that has the lattice param-
eters a = 0.45165 nm, b = 0.50837 nm and c = 0.67475 nm.

3.3 Summary
Complications arise once the defining symmetry of the cube, i.e., four triads, is lost.
Directions and plane normals with the same indices are then not necessarily parallel
and hence have to be distinguished on a stereographic projection. Furthermore, the
extent of the discrepancy relative to the cubic system depends on the anisotropy of the
lattice parameters and the angles between the basis vectors of the system concerned.
The hexagonal system poses a particular problem of communication because of
the nature of the special symmetry of the hexad. For example, the crystallographi-
cally equivalent planes (100), (010) and (110) have indices that are not permutations
of the same set of digits. The Miller-Bravais four-index system eliminates this am-
biguity but requires additional work to ensure that the Weiss zone rule is satisfied.
Whether the intellectual cost of the four-index system over the usual three-index no-
tation is justified by its elegance is a matter of taste.
References

1. H. M. Otte, and A. G. Crocker: ‘Crystallographic formulae for hexagonal lat-

tices’, Physica Status Solidi B, 1965, 9, 441–450.
2. R. M. White: Quantum Theory of Magnetism: New York, USA: Springer, 2007.

3. J. P. Joule: ‘On the effects of magnetism, &c. upon the dimensions of iron and
steel bars.’, The London, Edinburgh, and Dublin Philosophical Magazine and
Journal of Science, 1847, 30, 225–241.
4. M. V. Kral, and R. W. Fonda: ‘The primary growth direction of Widmanstätten
cementite laths’, Scripta Materialia, 2000, 43, 193–198.

59
4
Space Groups

Abstract

Some of the symmetry elements of a crystal involve translations that are simple frac-
tions of a repeat distance. These translations have consequences, for example on the
number of symmetry related positions in the unit cell, and hence on the number of
atoms in the unit cell. This chapter deals with space groups, which, unlike point
groups, include screw axes and glide planes.

4.1 Introduction
A point group is a collection of symmetry elements passing through a point, and
therefore, necessarily does not include translations. Space groups, in contrast, in-
clude translations that are fractions of a repeat unit, for example, a 21 axis which
involves a rotation of 180◦ followed by a translation of 21 of the repeat distance along
the axis. Glide planes involve reflections followed by fractional translations. These
translations are small, and hence do not manifest when the point group is determined
from the external shape of well-formed crystals. However, they have consequences
in structure determination and on other properties of crystals. There are 230 space
groups which are made from combinations of the 32 crystallographic point groups
with the 14 Bravais lattices.
The relationships between the placing of atoms in the unit cell and the symmetry
of the structure of lead titanate was described on page 45. The structure of PbTiO3
is primitive tetragonal with point group 4mm. Since there are no screw axes or glide
planes, the space group symbol for the titanate is P 4mm, where the first symbol is
to identify the lattice type as primitive. In this chapter, we will consider a case where
translational symmetry elements1 are present.
1 other than the normal periodicity of a lattice

61
62 4 Space Groups

4.2 Screw axes and glide planes

A proper rotation axis designated N brings a motif into coincidence after a rotation of
2π p
N . A screw axis, designated Np , does the same but only when a translation of N × t
2π
follows the rotation by N . Here t is the distance between lattice points parallel to
the axis. A screw diad (21 ) is illustrated in Fig. 4.1, involving a rotation of 180◦
followed by a translation of 21 t parallel to the axis. Other rotation axes following the
same logic in terms of notation are: 31 , 32 , 41 , 42 , 43 , 61 , 61 , 62 , 63 , 64 and 65 .

FIGURE 4.1
An illustration of the operation of a diad
where the structure is restored by a rota-
tion of 180◦ , and a screw diad which in-
volves the same rotation but then a trans-
lation along the axis by 0.5 (hence the no-
tation 21 ) of the repeat distance t.

Glide planes are like mirror planes but a reflection is followed by a translation
in order to recover the symmetry.The terminology for glide planes is explained in
Table 4.1; the translations following reflection are parallel to the glide plane, the
exception begin the diamond glide in a body-centred cubic structure.

TABLE 4.1
Symbols for glide planes. ‘fcc’ and ‘bcc’ are abbreviations for face–centred cubic
and body–centred cubic respectively. The structure of pyrite, which is primitive cubic
and has a point group symmetry m3 is illustrated on page 44, containing axial glide
planes; its space group therefore is P a3 where the symbol a replaces the m in the
point group.

Glide character Direction Magnitude Symbol

1
Axial glide parallel to a, b or c axis 2 (a, b, or c) a, b, or c
1
Diagonal glide parallel to face diagonal 2 (a + c), 12 (b + c) or 21 (a + b) n
1 1 1
Diamond glide (fcc) parallel to face diagonal 4 (a + c), 4 (b + c) or 4 (a + b) d
1
Diamond glide (bcc) parallel to body diagonal 4 (a + b + c) d
4.3 Cuprite 63

4.3 Cuprite
Consider now the structure of cuprite (Cu2 O) which is primitive cubic, with a motif
of four copper atoms at 000, 21 0 12 , 0 12 21 and 12 21 0, and two oxygen atoms at 14 14 43
and 34 34 41 located at each lattice point.

FIGURE 4.2
Projection along the [001] direction of four unit cells of cuprite (Cu2 O). The frac-
tional height along [001] are indicated, with unlabelled atoms at 0 and 1.

Fig. 4.2 shows a projection of four such unit cells along the [001] direction, with
fractional coordinates along that direction indicated. The environment of each of the
six atoms within the unit cell is clearly different, consistent with the cubic-P lattice
type. There are mirror planes parallel to {110} – the trace of only one of these is
illustrated in Fig. 4.2 for the sake of clarity. Note also that there are 42 axes located
at 14 41 z and 34 43 z and their equivalent positions. A 42 axis involves a rotation of 90◦
followed by a translation by 24 of the repeat distance along that axis. The diagram
also shows that the structure has a centre of symmetry at 12 12 12 .
64 4 Space Groups

The glide plane parallel to {100} (Fig. 4.2) involves a reflection followed by two
orthogonal translations parallel to the plane by 12 the lattice parameter. It is, therefore,
an n-glide plane according to the notation in Table 4.1.
Since cuprite is cubic, the defining symmetry is four triads. However, Fig. 4.3
shows that the triads are in fact inversion triads (3), meaning that a rotation of 120◦
is followed by an inversion through the centre to recover the structure. This is evident
from the projection along ⟨111⟩ that there is no triad since the atoms marked with
z coordinates are in two sets of three, with each set at a different height along z. A
simple three-fold operation would not therefore work. On the other hand, the rotation
combined with the inversion reproduces the structure.

(a) (b)

FIGURE 4.3
(a) The unit cell of cuprite. (b) A projection along ⟨111⟩, with fractional coordinates
along [001] indicated for some of the atoms.

4.4 Location of atoms in cuprite cell

The number of atoms in the unit cell depends on where an atom is placed first be-
cause all other symmetry related equivalent positions must also contain that atom.
The space group P n3m reduces to the point group m3m when translations are omit-
ted. Fig. 4.4a shows that the placing of an atom at a general location (i.e. not on a
symmetry element) must lead to a total of 48 such atoms in the cell since there are
that many crystallographically equivalent positions in the cell. On the other hand, by
placing the atom on a mirror plane, the number of positions halves to 24 since an
atom that is on a mirror plane does not have a reflection from that plane, Fig. 4.4b.
Referring to Fig. 4.4c,d, suppose that a copper atom is located at 0,0,0 which has
4.4 Location of atoms in cuprite cell 65

(a) (b)

(c) (d)

FIGURE 4.4
Stereograms consistent with the point group m3m, showing the number of equiva-
lent positions when (a) an atom is placed at general location x, y, z and all equivalent
positions; (b) an atom is placed at general location x, x, z and all equivalent posi-
tions. (c) Projection of the cuprite unit cell with the traces of glide planes marked
as dashed lines and those of mirror planes as heavy lines. Initial placing of atoms
copper in black and oxygen in red. (d) New set of atoms generated by the operation
of the glide planes. The relation between the original and new atoms is marked by
blue lines.
66 4 Space Groups

the symmetry elements 3m passing through it. The operation of the n-glide plane
marked by the horizontal line generates the atom at the opposite face-centre at a
height 12 and the other symmetry elements then generate atoms at all face-centres
and corners of the cube. There are therefore, four equivalent copper atoms in the unit
cell.
Suppose that an oxygen atom is placed at the location 41 , 41 , 34 which has the sym-
metry elements 43m passing through it. The operation of the same n-glide plane on
that oxygen atom leads to an oxygen atom at 34 , 34 , 14 . All other symmetry operations
have the same effect so there are only two oxygen atoms per cell. The locations of the
copper and oxygen atoms are therefore consistent with the chemical formula Cu2 O
and the density of the compound can be measured to verify this. Placing a copper
atom at the general position x, y, z or x, x, z would not lead to the correct density
or stoichiometry. The results are summarise in Table 4.2. The International Union
of Crystallography publishes much more comprehensive space group tables which
list the multiplicity of coordinates as a function of symmetry at that coordinate, for
all possible scenarios and for all space groups. Table 4.2 can be regarded as a much
simplified version of this kind of information.

TABLE 4.2
Space group for cuprite, P n3m. The origin of the cell is set on a copper atom.

Point symmetry Number Coordinates

1 48 x, y, z x, y, z x, y, z ...

m 24 x, x, z x, x, z x, x, z ...
..
.
..
.
1 1 1 1
3m 4 0,0,0 2, 2, 0 2 , 0, 2 0, 12 , 12
1 1 3 3 3 1
43m 2 4, 4, 4 4, 4, 4

Space groups!P m

Example 4.1: Space group of Fe-Si-U compound

• An intermetallic compound containing Fe, Si and U has the lattice parameters
a = 0.7 nm, b = 0.6 nm, c = 0.5 nm parallel to the x, y and z axes respec-
tively, with the angle β = 105◦ between the x and z axes. What crystal class
does this belong to?
• The space group symmetry of the crystal is P m with the mirror planes normal
4.4 Location of atoms in cuprite cell 67

to b at y = 0 and y = 21 . Find the number of equivalent sites (multiplicity) for

the following coordinates: (x, y, z), (x, 21 , z) and (x, 0, z).
• A silicon atom is placed at 0, 0, 0, Fe at (0.5, 0.1, 0.3), U at (0.75, 0.5, 0.75).
Accounting for multiplicity, draw a projection of the unit cell normal to the z
axis. Hence write the chemical formula for the compound. What are the point
group symmetries of the iron and uranium atoms?
• Describe a shear that could convert the monoclinic cell into one that is or-
thorhombic, stating the plane and direction of the shear. Calculate the magni-
tude of the shear.
The crystal class is monoclinic since none of the lattice parameters are equal,
and one of the angles does not equal 90◦ . The cell is primitive with a mirror
planes at y = 0 or 21 . Anything located at a general coordinate x, y, z therefore
has a multiplicity of of 2, i.e., x, y, z and x, −y, z, due to reflection in mirror. Any-
thing located on a mirror plane clearly does not have a reflection, and therefore, a
multiplicity of just 1, which applies to x, 21 , z and x, 0, z.
The resulting structure projection is illustrated below, giving the chemical for-
mula Fe2 USi. Note that there are two iron atoms due to the mirror plane, but only
one uranium atom that is located on the mirror plane. The point group symmetry
of the uranium atom is therefore m. That of iron is just 1 (monad).
In order to convert the monoclinic cell into one that is orthorhombic, it is
necessary to convert the angle β to 90◦ . The required shear is therefore on the
(001) plane, in the [100] direction, with a magnitude tan(105 − 90) where the
angles are expressed in degrees.

FIGURE 4.5
Projection of the unit cell of Fe2 USi normal to the z-axis.
68 4 Space Groups

Space groups!P 21/c

Example 4.2: Cementite

Cementite is an iron carbide (Fe3 C) and the commonly used convention
in metallurgical studies is to set the order of the lattice parameters as a =
0.50837 nm, b = 0.67475 nm and c = 0.45165 nm. There are twelve atoms of
iron in the unit cell and four of carbon, as illustrated in Fig. 4.6. Deduce the space
group of the structure.
n n

m
b

a
x

FIGURE 4.6
The crystal structure of cementite, consisting of twelve iron atoms (large) and four
carbon atoms (small). The fractional z coordinates of the atoms are marked.

The lattice type is primitive (P). There are n-glide planes normal to the x-axis,
at 41 x and 34 x involving translations of 2b + 2c . There are mirror planes normal to the
y-axis and a-glide planes normal to the z-axis, at heights 14 z and 34 z with fractional
translations of a2 parallel to the x-axis. The space group symbol is therefore P nma
[1].
The structure of cementite clearly is very anisotropic so its properties, such as
the elastic moduli, also vary strongly with the direction within the crystal [2]. The
shear modulus c44 is exceptionally small, some two times smaller than the cor-
responding term for aluminium. Nevertheless, the cementite has an exceptionally
large ideal shear strength because elastic deformation reduces its symmetry from
orthorhombic to monoclinic (space group P 21 /c), with an accompanying increase
in three dimensional covalent bonding that stiffens the material [3].
4.4 Location of atoms in cuprite cell 69
a
In the context of plastic deformation, the slip systems on which dislocation
glide has been observed include (001)[100], (100)[010], (100)[001], (010)[001]
and (010)[100] [4] but the complex arrangement of atoms means that slip is ac-
tually quite difficult to achieve, making the cementite very hard. There may be
other slip systems that operate when the cementite is forced to deform in a phase
mixture such as pearlite, but the Burgers vectors of slip dislocations would never-
theless be large when compared with dislocations in the allotropic forms of iron
[5].
a It is important to note the labelling of the orthogonal axes of the unit cell, i.e., a = 0.50837 nm,

b = 0.67475 nm and c = 0.45165 nm. since different conventions are used in the published literature.
The indices referred to below have been adjusted from the original sources, for the unit cell as defined
here.

Example 4.3: Diamond and zinc sulphide

Both diamond and zinc sulphide have a cubic-F lattice; in diamond a motif of
a pair of carbon atoms at 0,0,0 and 14 , 14 , 14 is placed at each lattice point whereas
in ZnS, the motif consists of a Zn atom and a S atom. The space group of diamond
is F d3m whereas that of ZnS is F 43m. Draw structure projections and explain
why the diamond glide is missing in the structure of ZnS.

Diamond consists only of carbon atoms, each of which is symmetrically re-

lated to all the others in the cell. For example, the operation of the diamond glide
plane illustrated in Fig. 4.7 on the carbon atom at 0,0,0 leads to the carbon atom at
1 1 1
4 , 4 , 4 . In ZnS, this operation would not lead to a symmetrically related Zn atom
because a sulphur atom is located at 41 , 14 , 14 .
The space group of ZnS is in fact a sub-set of the space group of diamond, with
all the symmetry elements of the diamond structure that lead a zinc atom into the
position of a sulphur atom, missing. Therefore, the inversion triad in diamond is
reduced to a triad and there is no centre of symmetry. In diamond the centre of
symmetry is at 81 , 18 , 81 and equivalent positions.
Referring to Fig. 1.15, when the randomly arranged nickel and aluminium
atoms in γ, which has the space group F m3m on ordering to the γ ′ phase becomes
the sub-group P m3m. In γ the point symmetry elements passing through each
atom are m3m, whereas in γ ′ the point group symmetries of the nickel atoms
are 4/mmm and the aluminium atoms, m3m. There is, therefore, no three-fold
axis through the nickel atoms. The reduction in symmetry on ordering has huge
consequences for the deformation behaviour of γ ′ because larger displacements
are necessary before the structure is reproduced. A typical Burgers vector of a
dislocation doubles in length in the γ ′ relative to the γ.
70 4 Space Groups

(a) (b)

FIGURE 4.7
Structure projections. (a) Diamond (carbon). The dashed line shows a d-glide
plane involving two orthogonal translations by a quarter of the repeat distance,
parallel to the glide plane. (c) Cubic form of zinc sulphide.

4.5 Shape of precipitates

Crystallography is one of the characteristics that can influence the shape of a precip-
itate, but its influence is usually overwhelming when the precipitation occurs in the
solid state [6–9]. In some circumstances, it is symmetry which determines the shape
of precipitates which form inside a solid.
Suppose that Gm and Gp represent the symmetry groups of the matrix and pre-
cipitate respectively. The common symmetry (intersection group) between these two
phases in the observed orientation relationship is defined as H. Given a planar in-
terface between them, symmetry operations consistent with H leads to other ori-
entations of the interface until the particle is fully enclosed and its shape defined.
Note that any operation of H is a symmetry operation of the parent and product and
hence each crystal orientation is brought into congruence so the orientation of the
two crystals is not affected by H.
Aluminium has the space group F m3m and the silver–rich precipitate Ω has the
space group F mmm [10]. The orientation relation has been measured to be:

[001]Ω ∥ [111]Al

[100]Ω ∥ [101]Al
Fig. 4.8 shows the stereograms of the symmetry elements for the two crystals in the
4.6 Summary 71

correct relative orientation. It is evident the common point group symmetry is 2/m,
which is consistent with the observed plate shape of the Ω both with respect to the
diad and the mirror plane.

(a) (b) (c)

FIGURE 4.8
(a) Stereographic representation of the symmetry elements in the point group m3m
for aluminium. (b) Stereographic representation of the symmetry elements in the
point group mmm for Ω. The symmetry elements shared by Al and Ω are highlighted
in red. (c) The precipitate shape observed, corresponding to the shared symmetry
group 2/m.

4.6 Summary
We have touched on the notion of space groups which represent all the symmetry
elements of the crystal. The appreciation of space groups is essential in the solution
of structures, transformations which lead to changes in symmetry, and the properties
of crystals. A great deal of the work in identifying the number of symmetry related
positions as a function of the space group has already been documented, with the
information routinely available either from the tables published by the International
Union of Crystallography or via the world wide web.
References

1. E. J. Fasiska, and G. A. Jeffrey: ‘On the cementite structure’, Acta Crystallo-

graphica, 1965, 19, 463–471.
2. C. Jiang, S. G. Srinivasan, A. Caro, and S. A. Maoly: ‘Structural, elastic, and
electronic properties of Fe3 C from first principles’, Journal of Applied Physics,
2008, 103, 043502.
3. C. Jiang, and S. G. Shrinivasan: ‘Unexpected strain-stiffening in crystalline
solids’, Nature, 2013, 496, 339–342.
4. A. Inoue, T. Ogura, and T. Muramatsu: ‘Burgers vectors of dislocations in
cementite crystal’, Scripta Metallurgica, 1977, 11, 1–5.
5. J. Gil-Sevillano: ‘Room temperature plastic deformation of pearlitic cementite’,
Materials Science & Engineering, 1975, 21, 221–225.
6. G. R. Hugo, and B. C. Muddle: ‘Role of symmetry in determining precipitate
morphology’, Materials Forum, 1989, 13, 147–152.
7. G. R. Hugo, and B. C. Muddle: ‘The morphology of precipitates in an Al-Ge
alloy-ii. analysis using symmetry’, Acta Materialia, 1990, 38, 365–374.
8. B. C. Muddle, and I. J. Polmear: ‘The precipitate omega phase in Al Cu Mg
Ag alloys’, Acta Metallurgica, 1989, 37, 777–789.
9. L. S. Chumbley, B. C. Muddle, and H. L. Fraser: ‘Crystallography of precipi-
tation of Ti5 Si3 in Ti-Si alloys’, Acta Metallurgica, 1988, 36, 299–310.
10. K. M. Knowles, and W. M. Stobbs: ‘The structure of {111} age-hardening
precipitates in Al-Cu-Mg-Ag alloys’, Acta Crystallographica B, 1988, B44,
207–227.

73
5
The Reciprocal Lattice and Diffraction

Abstract

The Miller indices for directions and planes are defined differently. For directions
the indices are simply the components of a vector corresponding to that direction
with respect to the basis vectors, but those of a plane are defined rather strangely at
first sight. The intercepts of the plane with the basis vectors are determined, and their
reciprocals form the indices of the plane. The mystery behind this scheme is revealed
in this chapter, whereby a plane is also defined by a vector that is normal to it and
has a magnitude that is the reciprocal of the plane spacing. Diffraction too becomes
easier to visualise when expressed in terms of reciprocal space.

5.1 The reciprocal basis

The basis vectors ai were defined in Fig. 1.3; a vector u in real space can then be
written as a linear combination of the basis vectors:

u = u1 a1 + u2 a2 + u3 a3 ,

where u1 , u2 and u3 are its components, when u is referred to the basis A.

The reciprocal lattice constitutes a special coordinate system, designed originally
to simplify the study of diffraction phenomena. If we consider a lattice, represented
by a basis symbol ‘A’ and an arbitrary set of non-coplanar basis vectors a1 , a2 and a3 ,
then the corresponding reciprocal basis A∗ has basis vectors a∗1 , a∗2 and a∗3 , defined
by the following equations:

a∗1 = (a2 ∧ a3 )/(a1 .a2 ∧ a3 )

a∗2 = (a3 ∧ a1 )/(a1 .a2 ∧ a3 )
a∗3 = (a1 ∧ a2 )/(a1 .a2 ∧ a3 ) (5.1)

The term (a1 .a2 ∧ a3 ) represents the volume of the unit cell formed by ai , while the
magnitude of the vector (a2 ∧ a3 ) represents the area of the (1 0 0)A plane (Fig. 5.1).
Since (a2 ∧ a3 ) points along the normal to the (1 0 0)A plane, it follows that a∗1 also
points along the normal to (1 0 0)A and that its magnitude |a∗1 | is the reciprocal of
the spacing of the (1 0 0)A planes (Fig. 5.1).

75
76 5 The Reciprocal Lattice and Diffraction

FIGURE 5.1
The relationship between a∗1 and ai . The vec-
tor a∗1 lies normal to the basal plane wxyz and
the volume of the parallelepiped formed by the
basis vectors ai is given by a1 .a2 ∧ a3 . The
area of the basal plane is |a2 ∧ a3 |. The height
is the spacing between planes parallel to the
basal plane, given by 1/|a∗1 |.

The components of any vector referred to the reciprocal basis represent the
Miller indices of a plane whose normal is along that vector, with the spacing of the
plane given by the inverse of the magnitude of that vector. For example, the vector
(u; A∗ ) = (1 2 3) is normal to planes with Miller indices (1 2 3) and interplanar
spacing 1/|u|. We see that

ai .a∗j = 1 when i = j, and ai .a∗j = 0 when i ̸

=j (5.2)

(a) (b) (c)

FIGURE 5.2
Sections of the reciprocal lattice of a cubic-P lattice. (a) Section normal to [001].
(b) Section normal to [011]. (c) Section normal to [111]. Notice how the distance
varies inversely with the spacing of the planes. The origin of the reciprocal lattice
is not labelled, but if necessary, can be designated 000. Each of the spots in these
patterns should strictly be points, but they are weighted here with intensity to show
the relationship with electron diffraction patterns discussed later.

Example 5.1: Weiss zone law

Using real and reciprocal lattice vectors, prove the Weiss zone rule that when a
direction with real space indices [u v w] lies in a plane with reciprocal space
indices (h k l),
uh + vk + wl = 0 (5.3)
irrespective of the nature of the unit cell.
5.2 The reciprocal basis 77
The area of the polygon defined by a pair of vectors is given by the magnitude
of the cross-product, i.e., |a ∧ b|, |b ∧ c| and |c ∧ a|. The cross-product leads to a
vector which is normal to the plane enclosed by the pair of vectors.
The volume is given by the area of the base multiplied by the height. When
the base is considered to be defined by a and b, the area is |a ∧ b| and the height
of the cell normal to the a-b plane is

a∧b
c. (5.4)
|a ∧ b|

If follows that the volume of the cell is

a∧b
c. × |a ∧ b| = c.a ∧ b (5.5)
|a ∧ b|

Every vector in a reciprocal lattice defines a normal to a plane of the same indices
as the components of the reciprocal lattice vector, and the vector has a magnitude
which is the reciprocal of the spacing of those planes. Therefore, for each of the
basis vectors,
a∧b
c∗ = etc. (5.6)
c.a ∧ b
With this equation it is evident that ai .a∗j = 0 when i ̸
= j and ai .a∗ j = 1 when
∗ ◦
i = j. The angle between a and b is clearly 90 . Therefore, if a direction [u v w]
lies in a plane (h k l),

[ua + vb + wc].(ha∗ + kb∗ + lc∗ ) = 0 = uh + vk + wl (5.7)

Note that this is a generic proof for the Weiss zone rule since there are no restric-
tions placed on the basis vectors.

A corollary to this example is a general method for taking a scalar product be-
tween two vectors. The magnitude of a vector u is given by

|u|2 = (u1 a1 + u2 a2 + u3 a3 ).(u∗1 a∗1 + u∗2 a∗2 + u∗3 a∗3 )

= u1 u1 ∗ +u2 u∗2 + u3 u∗3 ≡ (u; A)[A∗ ; u] (5.8)

where (u; A) is a row matrix of the components ui of u referred to the basis vectors
ai represented by the basis symbol ‘A’, and [A∗ ; u] is a column matrix containing
the components u∗i of u referred to the reciprocal basis vectors a∗i represented by the
basis symbol ‘A∗ ’. This is an important results that gives a different interpretation
to the scalar (or ‘dot’) product between any two vectors referred to any basis. To
evaluate the dot product, one of the vectors must be referred to the real basis and the
other to the corresponding reciprocal basis.
78 5 The Reciprocal Lattice and Diffraction

5.2 Crystallography of diffraction

Consider waves of length λ incident on planes of atoms. The beams reflected from
different planes in the parallel set illustrated in Fig. 5.3 must be in phase to avoid
destructive interference. The path difference between beams a and b, i.e. the distance
xyz, must then be an integral number of wavelengths. Since xyz = 2d sin θ, the
diffraction condition is
nλ = 2d sin θ (5.9)
where n is an integer; this equation is the Bragg law [1], with θ designated the Bragg
angle.

FIGURE 5.3
Electromagnetic waves inci-
dent on a set of parallel crys-
tal planes with an interplanar
spacing d. The angle of emer-
gence of the scattered waves
is the same as that of inci-
dence.

The Bragg law can be expressed conveniently in terms of vectors in reciprocal

space. In Fig. 5.4, the Bragg law is satisfied if

k′ − k = g (5.10)

Here g is a reciprocal lattice vector beginning at the origin (0,0,0) and representing
the crystal which is being illuminated by the radiation k. From the geometry of the
triangle, it is evident that sin θ = 12 g/k so that sin θ = (0.5/d)/(1/λ) which on
rearrangement gives the Bragg law.
5.3 Intensities 79

(a) (b)

FIGURE 5.4
(a) The Ewald sphere construction in reciprocal space. k represents the direction of
the incident beam, k′ that of the diffracted beam and g the normal to the diffracting
planes. Both k and k′ have a magnitude λ−1 and g has a magnitude d−1 . The beam is
incident on a crystal which is represented by its reciprocal lattice with origin at 0,0,0.
(b) Another illustration of the Ewald sphere superimposed on the reciprocal lattice.
In this case only the 101 and 101 planes are in Bragg condition. The reciprocal lattice
vectors which do not touch the sphere are not in Bragg orientation.

5.3 Intensities
The electrons from an individual atom can coherently scatter an X-ray beam; the
atomic scattering factor f is the ratio of the amplitude scattered by an atom to that
scattered by one electron. We are interested in coherent scattering by all the atoms
in a unit cell, in which case the resultant amplitude for reflections from hkl planes is
given by a summation known as the structure factor:
n atoms
.
Fhkl = fn exp{2πi(hun + kvn + lwn )} (5.11)
1
n atoms
.
≡ fn [cos 2π(hun + kvn + lwn ) + i sin 2π(hun + kvn + lwn )]
1

where fn is the scattering factor of atom n and the sum is over all atoms in the
unit cell.† The magnitude |F | is now the ratio of the amplitude scattered by a unit
cell to that scattered by one electron. |F |2 is proportional to the scattered intensity.
Notice that for a centrosymmetric system, the sine term can be neglected since that
function is not symmetric about zero. As an example, for the structure of Cu (cubic
† eπi = e3πi = −1, e2πi = e4πi = +1, eπi/2 = i, e3πi/2 = −i, (1 + i)(1 − i) = 2
80 5 The Reciprocal Lattice and Diffraction
11 1 1
close-packed, with four atoms in the cell at 000, 2 2 0, 2 0 2 , and 0 21 12 ),

F = f [1 + eπi(h+k) + eπi(h+l) + eπi(k+l) ] (5.12)

so that the {100} and {110} reflections would have zero structure factor.
Fig. 5.5 shows how the electron diffraction patterns of austenite, which has a
cubic-F lattice, have zero intensities for {100} and {110} planes, as expected by
substitution of hkl into equation 5.12.

(a) (b) (c)

FIGURE 5.5
Electron diffraction patterns from austenite (cubic-F). (a) Zone axis parallel to [001].
(b) Zone axis parallel to [011]. (c) Zone axis parallel to [111]. Notice that in any
pattern, it is necessary to index just two reciprocal lattice vectors in order for all
the others to be determined. For the cases illustrated, the symmetry of the patterns
is consistent with that of the zone axes. Contrast these patterns with the reciprocal
lattice sections presented in Fig. 5.2.

Example 5.2: Solution of electron diffraction pattern

Fig. 5.6a shows an electron diffraction pattern from a body-centred cubic lattice
with a motif of just one atom per lattice point. Index the labelled g-vectors.

The pattern is rectangular so one possibility is that the axis normal to the
pattern corresponds to a ⟨110⟩, a diad. The g-vectors normal to this direction could
include the normals to {001}, {110}, {112} and {11n} where n is an integer. The
d-spacing of planes in a cubic system is given by

1 h2 + k 2 + l 2
=
d2 a2
where a is the lattice parameter. So the d-spacing decreases in the order {001},
{110}, {112} and {11n}. Fig. 5.6b shows that the intensity diffracted from the
{001} planes would be zero given that there is an atom at the body-centre which
would scatter out of phase with the atoms on the cube faces. However, {002}
planes would scatter in phase. If it is assumed that the g-vector labelled ‘a’ is
a {110} reflection (largest d-spacing and hence closest to the incident beam in
reciprocal space) and that at the reflection at ‘b’ is due to the {002} planes then
5.3 Intensities 81
ratios of the lengths of the ‘a’ and ‘b’ and the 90◦ angle between gives consistent
indexing. Any other choice would not.
Once two of the reflections are labelled, all others are linear combinations of
those two. For example ‘c’ is given by 002 + 110 = 112.
In this particular example, it was easy by examining the structure projection
to show that {001} reflections would be systematically absent. However, if the
indices are of higher order, e.g. {532}, then it is easier to use the structure factor
equation to decide whether these planes would result in diffracted intensity.

(a) (b)

FIGURE 5.6
(a) An electron diffraction pattern from a body-centred cubic crystal. (b) Structure
projection of bcc, with traces of {110}and {200} planes indicated.

Example 5.3: Another diffraction pattern solution

The electron diffraction pattern in Fig. 5.7 is from a cubic crystal. Index the pattern
for all three types of cubic lattices.

The pattern has three-fold symmetry because there are three different g-
vectors if their opposites are not counted. This indicates that there is a triad ⟨111⟩
normal to the pattern. All the g-vectors will be normal to the ⟨111⟩ axis and of
equal length, so the following solutions can be deduced:
• For cubic-P, ‘a’= {011} and ‘b’= {110}. The choice of the first solution as
{110} is arbitrarily made from the three possible, but the second will then be
determined as being at 60◦ from the first. Their right-handed cross product
gives [111] so the beam direction is [111].
• For cubic-I, the solution would be the same as for cubic-P since {011} type
reflections are permitted.
• In the case of cubic-F, {011} is a systematic absence but {022} is not so all
the indices should be doubled.
82 5 The Reciprocal Lattice and Diffraction

(a) (b)

FIGURE 5.7
(a) An electron diffraction pattern from a crystal with a cubic structure. (b) Struc-
ture projection for cubic-F showing that {011} would result in zero diffracted
intensity because of the atoms in-between.

Notice that it is not possible, from the shape of the diffraction pattern alone,
to determine the lattice type. However, if the lattice parameters of the different
forms are known, then the measured d-spacings on the pattern can be compared
with interplanar spacings calculated from the parameters to fix the lattice type.
The relationship between the distance measured on the diffraction pattern and the
camera length of the electron microscope is illustrated in Fig. 5.8. It is evident that
sin θ = R/2L so with the Bragg equation 5.9, it is easy to show that Rd = Lλ,
with the term Lλ referred to as the ‘camera constant’.

FIGURE 5.8
Geometry describing the relationship between the camera length on the transmis-
sion electron microscope and distance measured on an electron diffraction pattern.

Example 5.4: Disordered and ordered crystals

Austenitic Fe-Al-C alloys containing about 25 at% Al are heat-treated to precip-
itate the δ-phase within the aluminium-rich solid solution (γ). The δ-phase is an
ordered compound with the chemical composition Al3 Fe. Both phases contain a
total of four atoms per unit cell.
The γ has a cubic-F lattice; index its electron diffraction pattern. What is the
5.3 Intensities 83
shortest lattice vector? Judging from the symmetry of its diffraction pattern, indi-
cate the possible lattice types for the δ phase.
Now assume that the correct lattice type for δ is that which contains four triads
and its lattice parameter is almost identical to that of the solid solution γ. Bearing
in mind its chemical composition, draw a projection of the crystal structure of
δ along a convenient axis, identify the lattice type and the shortest lattice vector
along the ⟨110⟩ direction. Index the diffraction pattern.
Their electron diffraction patterns are presented in Fig. 5.9 in the correct rela-
tive orientation. Comment on the orientation relationship between the two phases
and why dislocations in γ would need to move in pairs in order to propagate
through the δ-phase. Obtain expressions for the structure factors for the 100 and
200 reflections for both phases.
Similar ordering of atoms is observed in a wide range of alloys, often involv-
ing atoms with similar atomic numbers. Discuss the relative merits of different
diffraction techniques that could be used to characterise order in such systems.

(a) (b)

FIGURE 5.9
(a) Al-Fe solid solution (γ). (b) δ-phase. The relative orientations of the two pat-
terns are preserved.

The γ is said to be cubic-F; since the pattern has four-fold symmetry, the direc-
tion normal to the pattern must be parallel to ⟨001⟩ so the spots marked ‘a’ and ‘b’
can be labelled {020} and {200} respectively, with all other reflections obtained
by linear combinations of these two. Therefore, ‘c’ becomes {220}. The shortest
lattice vector in cubic-F is a2 ⟨110⟩. Note that in the cubic-F lattice, reflections of
the type {100} are systematic absences (equation 5.12).
The pattern for the δ-phase also has four-fold symmetry and hence could come
from a cubic or tetragonal lattice. However, it is stated that the lattice has four
triads so it must be cubic-P, cubic-F or cubic-I. The composition and number of
atoms in the unit cell is consistent with cubic-P with a motif of four atoms per
lattice point, iron at 000 and three aluminium atoms at 12 21 0, 0 12 12 and 21 0 12 . As a
result, the spots labelled ‘a’, ‘b’, ‘c’ and ‘d’ in Fig. 5.9b are identified as {010},
{100},{110},{020} respectively.
The structure projections of the two phases are shown in Fig. 5.10, together
84 5 The Reciprocal Lattice and Diffraction
with the shortest lattice vectors along the ⟨110⟩ direction. The significance of this
is that a dislocation with Burgers vector a2 ⟨110⟩ in γ would create faults in the δ
because it would not be a lattice vector in the ordered phase. However, a pair of
γ dislocations moving together as a superdislocation in the δ would achieve slip
without leaving a fault in their wake. The γ and δ are in a ‘cube-cube’ orientation,
i.e.
[100]γ ∥ [100]δ and [010]γ ∥ [010]δ
and have similar lattice parameters, which would facilitate slip that initiates in the
γ to traverse into δ with the caveat that lattice dislocations of γ would need to pair
up when moving into δ.

(a) (b)

FIGURE 5.10
(a) Structure projection of γ. (b) Structure projection of δ.

Using equation 5.12 the required structure factors are as follows:

γ
F100 = f [e0 + e0 + eπi + eπi ] = 0
γ
F200 = f [e0 + e0 + e2πi + e2πi ] = 4f
δ
F100 = fFe e0 + fAl eπi + fAl e0 + fAl eπi = fFe − fAl
δ
F200 = fFe e0 + fAl e2πi + fAl e0 + fAl e2πi = fFe + 3fAl
(5.13)

where f is an average atomic scattering factor for the random solid solution γ.

Since intensity is proportional to F 2 , the {100}δ superlattice reflection should be

much weaker than {200}δ as is evident in Fig. 5.9b.
Techniques in which the atomic scattering factor scales with the atomic number
include electron and X-ray diffraction. Reflections such as {100}δ that are dependent
on the differences in the atomic scattering factors of unlike species, will therefore be
weak. In neutron diffraction, the scattering factors can be quite different for atoms
with similar atomic numbers so superlattice reflections can be detected more readily.
5.4 Diffraction from thin crystals 85

5.4 Diffraction from thin crystals

When an incident beam deviates by ∆θ from the ideal Bragg angle θ consistent with
equation 5.9, it is possible to find another beam from within the depth of the crystal
which has a path difference of 21 λ and which will destructively interfere with it.
This depth, needed to find the wave which is out of phase, increases as ∆θ becomes
smaller; hence for thin crystals, it may not be possible to destroy deviant rays. For
thin crystals it becomes possible to obtain diffracted intensity even when the incident
beam is not at the exact Bragg orientation.
The samples in transmission electron microscopy are constrained to be thin so
that electrons can penetrate the typically < 100 nm thick foils. It follows that diffrac-
tion becomes possible even when the incident electron beam is not at the exact Bragg
orientation. This can be expressed on the Ewald sphere construction by extending
the reciprocal lattice points in the direction normal to the thin foil (Fig. 5.11, 5.12,
5.13). The fact that electrons have a very small wavelength compared with X-rays
also helps achieve a greater chance of diffraction because the Ewald sphere is then
relatively flat.

(a) (b)

FIGURE 5.11
(a) Ewald sphere construction for diffraction from a thin foil. The origin of the re-
ciprocal lattice is at ‘O’ and the layer designated ‘P’ is the next layer normal to the
incident beam, a distance 1/t∗ from the first layer. The arrows mark reciprocal lattice
points which contribute to diffraction even though they are not at the exact Bragg ori-
entation. (b) ⟨111⟩ axis ReO3 showing the higher order Laue zone (the second layer
of the type sketched in (a).) - pattern courtesy of A. Eggeman, T. Chang and P. Midg-
ley.
86 5 The Reciprocal Lattice and Diffraction

(a) (b) (c)

FIGURE 5.12
Fe with cubic-F structure. (a) Electron diffraction from a foil which is 200 nm thick,
beam direction [111]. (b) Diffraction from the same foil after thinning it to just 50 nm
thickness, beam direction [111]. Because of this minute thickness, it becomes pos-
sible to pick up reflections from planes of the form {224}. (c) as (a) but with the
beam tilted towards [1 1 0.9]. The Ewald sphere is now tilted away from some of the
reciprocal lattice points on the left hand side.

FIGURE 5.13
The effect of crystal size (in all directions) on an X-ray diffraction pattern. The ma-
terial concerned is halite. The peaks broaden as the crystal gets smaller.

5.5 Neutron diffraction

Neutrons are scattered by the nuclei of atoms so the scattering factor is not sensitive
to scattering angle; unlike X-rays, the scattering power does not diminish with the
Bragg angle θ, so good results are obtained even at high θ. Also unlike X-rays, there
is little correlation between the scattering factor and atomic number. Because the
interactions are with the nuclei, it becomes possible to distinguish between isotopes
5.6 Summary 87

because they now have different neutron scattering factors (for example, hydrogen
and deuterium).
X-rays typically penetrate a few micrometers of the surface of a sample, whereas
neutrons can penetrate several centimetres. Neutrons can therefore be used to deter-
mine the state of residual stress that exists in a macroscopic sample that is otherwise
at equilibrium [2, 3]. Fig 5.14 shows residual stress contours that are determined by
measuring lattice spacings, comparing them with a stress-free sample to calculate
strains, and then converting strains into stresses using the elastic properties of the
material.

FIGURE 5.14
Stress distribution in a steel
weld determined using neu-
tron diffraction [4]. This
would not be possible using
X-rays. The numbers on the
contours represent MPa.

5.6 Summary
It takes a leap of imagination to understand why it is necessary to deal with a space
in which dimensions are inverted. But the fact is that in crystallography, it can be
regarded simply as a lattice in which all reciprocal lattice vectors represent normals
to planes in real space, with the magnitudes of these vectors representing the inverse
of the spacing of those planes. Whereas a plane in real space is defined by a pair of
vectors that lie in that plane, in reciprocal space the plane is represented by a vector.
A plane in reciprocal space represents all real space planes that share a common
direction (a zone axis).
Imagine now that we wish to find a direction that is parallel to a plane normal.
This is trivial for a cubic system where plane normals and directions with the same
indices are parallel. So a [123] direction is parallel to the normal of the (123) plane.
This is not generally true for non-cubic systems, but there exists a metric tensor
which makes it easy to find vectors in reciprocal space that are parallel to those in
real space (and vice versa), a subject reserved for Chapter 9.
References

1. W. H. Bragg, and W. L. Bragg: ‘The reflection of X-rays by crystals’, Proceed-

ings fo the Royal Society A, 1913, 88, 428–438.
2. P. J. Withers, and H. K. D. H. Bhadeshia: ‘Residual stress part 1 – measurement
techniques’, Materials Science and Technology, 2001, 17, 355–365.
3. P. J. Withers, and H. K. D. H. Bhadeshia: ‘Residual stress part 2 – nature and
origins’, Materials Science and Technology, 2001, 17, 366–375.
4. J. A. Francis, H. J. Stone, S. Kundu, R. B. Rogge, H. K. D. H. Bhadeshia, P. J.
Withers, and L. Karlsson: ‘Transformation temperatures and welding residual
stresses in ferritic steels’, In: Proceedings of PVP2007, ASME Pressure Vessels
and Piping Division Conference. American Society of Mechanical Engineers,
San Antonio, Texas: ASME, 2007:1–8.

89
6
Deformation and Texture

Abstract
Single crystals are now used routinely in engineering turbine blades for service in
hottest part of an aeroengine; it is astonishing that the design of the blade permits it
to operate in an environment where the temperature is greater than its melting point.
The deformation of a single crystal on the application of a stress or a system of
stresses is interesting in its own right, but the understanding that emerges from single
crystal studies can useful in explaining the rotation of crystals in a polycrystalline
material during plastic deformation. Real polycrystalline materials rarely consist of
crystals that are randomly oriented relative to the sample frame of reference. Instead,
the long-range order in a polycrystalline material is somewhere between that of a
hypothetical aggregate consisting of randomly oriented crystals, and a single crystal.
This is because the crystallographic axes of the different grains in the polycrystal tend
to align in specific ways. We shall learn in this chapter the methods for representing
such texture.

6.1 Slip in a single-crystal

In a tensile test on the sample of cross-sectional area A as illustrated in Fig. 6.1a, the
normal to the slip plane lies at an angle φ to the force F , and the slip direction is at an
angle λ to the tensile axis. The area of the slip plane is A/ cos φ and the force in the
slip direction is F cos λ so that the shear stress on the slip plane in the slip direction
is
F
τ= cos φ cos λ . (6.1)
A / 01 2
Schmid factor
The Schmid factor [1–3] has a maximum value of 0.5 when φ = λ = 45◦ . Sup-
pose that a tensile axis is along [213] in a cubic system, as plotted on the shaded
stereographic triangle in Fig. 6.1b. There is a simple way of determining which slip
system will have the largest resolved shear stress. The slip system will be defined by
a slip plane and a slip direction with the latter lying in the former. This means that
the two components of the slip system must come from different stereographic trian-
gles. The two stereographic triangles that are adjacent to the shaded triangle satisfy
this condition. Any other combination gives angles that are too large, giving Schmid

91
92 6 Deformation and Texture

factors that are smaller. The slip system with the highest resolved shear stress when
the tensile lies in the shaded triangle will therefore be [011](111). The construction
illustrated is known as Diehl’s rule.

(a) (b)

FIGURE 6.1
(a) Deformation of a single crystal. The force is resolved onto the slip plane and
towards the slip direction. Dividing by the area of the slip plane gives the shear stress
along the slip direction. (b) There are 24 stereographic triangles corresponding to the
24 slip systems of the type {111}⟨110⟩. The orientation of a tensile axis is plotted in
the shaded triangle.

The unconstrained shear deformation of a single crystal as illustrated in Fig. 6.2

leads to the rotation of the axis along which the force is applied. The normal to the
slip plane, the tensile axis and the slip direction are all in the same plane and remain
so during rotation. The tensile axis rotates towards the slip direction, on this common
plane, during the course of deformation. If the axis along which the force is applied
is not allowed to change orientation, then it is the slip system which rotates during
deformation, Fig. 6.2.
6.1 Slip in a single-crystal 93

FIGURE 6.2
Slip of an unconstrained single-
crystal. Notice how the axis along
which the force is applied rotates as a
consequence of the slip. The amount
of shear implemented is written αs. If
the force axis is constrained to be ver-
tical then the sample will rotate rela-
tive to the testing machine by the ex-
tent illustrated.
94 6 Deformation and Texture

Example 6.1: Elongation during single-crystal deformation

Referring to Fig. 6.3, derive an equation for the elongation of a single crystal due
to shear on a plane with unit normal n.
The definition of a shear strain γ is:

displacement parallel to shear plane |d|

γ= = so the magnitude of d = γx.n.
height over which displacement acts x.n
It follows therefore that
|y|
y = x + d = x + γ(x.n)s so the elongation is = .
|x|

s
d

FIGURE 6.3
Schematic diagram showing the shear of a single crystal. n is a unit normal vector
to the plane on which the shear occurs, d is vector describing the displacement
of the far end of the crystal due to the shear, x is its initial length above the slip
plane and y the corresponding length following the deformation. s is a unit vector
parallel to the shear direction.

A more general method for calculating the elongation during single-crystal deforma-
tion is described on page 175.
6.2 Slip in a single-crystal 95

Example 6.2: ε-martensite

A simple shear can transform the cubic-F structure of austenite into that of the
hexagonal close-packed ε martensite. This consists of a displacement of a6 ⟨112⟩
on every second close-packed plane. Calculate the shear strain associated with the
transformation. (a is the lattice parameter of austenite.)
How many different crystallographic variants of ε-martensite can in principle
form within a single austenite grain?
Fig. 6.4 shows a stereographic projection of austenite showing the ε habit
planes of the form {111} and shear directions of the form ⟨112⟩. Which is the
most favoured crystallographic variant that is stimulated to grow by applying a
tensile stress along the direction [123]?

FIGURE 6.4
Stereographic projection of austenite showing the planes and shear directions cor-
responding to the deformation that leads to the formation of ε martensite.

The shear strain is given by the displacement divided by the height. The height
2a
is the d-spacing of two close packed planes of austenite, i.e., √ 3
. The displacement
1
is the magnitude of a6 ⟨112⟩, i.e. √a6 . Therefore, the shear strain is 2√ 2
.
The habit plane is {111} (four variants) and there are three directions of the
form ⟨112⟩ in each of these planes. Therefore, twelve variants in all.
The strain is a shear deformation on a close-packed plane, but with the shear
direction along ⟨112⟩. Therefore, either using Diehl’s rule, or by calculation, the
plane stimulated would be (111). In that plane, there are three shear directions,
[211], [112] and [211]. Of these, the [211] direction gives the largest Schmid factor.
96 6 Deformation and Texture

6.2 Texture
Imagine that there is a polycrystalline substance in which the individual crystals are
randomly oriented relative to an external frame of reference‡ . Suppose that this poly-
crystalline substance is subjected to rolling deformation. The individual crystals will
then tend to rotate such that the slip planes comply with the external forces in a
manner akin to that illustrated in Fig. 6.2b. This means that the original random dis-
tribution of crystals in the undeformed sample becomes non-random. The deformed
material is said to become crystallographically textured [4].
Most polycrystalline materials show texture due to processing. Texture can arise
during solidification when those crystals which have a fast growth direction parallel
to that of the heat flow will dominate the final structure. It can arise during recrys-
tallisation and phase transformation when selective nucleation leads to the formation
of a biased distribution of crystals.
A convenient method for communicating texture caused by deformation, for ex-
ample, in rolled sheet, is by stating the set {h k l}⟨u v w⟩, the planes which lie
roughly parallel to the rolling plane, and direction in the rolling plane that tends to
be parallel to the rolling direction. The overall texture can be represented as the sum
of components: .
texture = λi {h k l}i ⟨u v w⟩i (6.2)
i

where λ represents the weighting given to a particular type of texture. The ma-
jor components of the deformation texture of austenite are {1 1 0}⟨1 1 2⟩ and
{1 1 2}⟨1 1 1⟩, the so-called brass and copper textures respectively. Recrystalli-
sation in a ferritic steel often leads to a cube texture {1 0 0}⟨0 0 1⟩ cube component
and special thermomechanical processing of the type used to produce magnetically
soft metal is associated with the Goss texture {1 1 0}⟨0 0 1⟩.
Texture can be plotted on a stereogram. A pole figure in this context consists of a
stereogram with its axes defined relative to the external frame of reference, and with
particular hkl poles from each of the crystals in the polycrystalline aggregate plotted
on to it. If the distribution of crystals is random, the pole figure would appear as in
Fig. 6.5a, and if the distribution is non-random then this would be apparent in the
pole figure, as shown in Fig. 6.5b.
An inverse pole figure is one in which the sample frame is plotted relative to the
crystal axes of a reference crystal. Fig. 6.6 illustrates an example where the rolling,
normal and transverse directions (RD, ND and TD respectively) are plotted relative
to the crystal axes as defined by the stereographic triangle. In a cubic system, the 24
stereographic triangles with ⟨100⟩, ⟨101⟩, and ⟨111⟩ at the corner of each triangle, are
crystallographically equivalent. Therefore, the orientation of the rolling direction, for
example, from each of the 24 triangles can be plotted on just one of the stereographic
‡ This frame may, for example, consist of a set of vectors parallel to the rolling direction, transverse

direction and thickness of a rolled plate; alternatively, the principal axes of an applied system of stresses.
6.2 Texture 97

(a) (b)

FIGURE 6.5
(a) Figure showing a random distribution of 100 poles from a polycrystalline sample
containing 500 crystals, plotted relative to the rolling direction (RD) and transverse
direction (T D). Notice that the distribution of poles does not appear random but this
is an artifact due to the angular distortion on a stereographic projection. (b) Corre-
sponding pole figure after texture is introduced by phase transformation. Here there
is a strong tendency for the 100 poles to cluster around the sample axes. The distri-
bution of poles can be plotted as contours rather than the dots illustrated. Diagrams
courtesy of Saurabh Kundu.

triangles in the form of poles, or contour plots representing the design of poles in
angular space.
Crystallographic texture can be measured using a variety of techniques such as
X-ray diffraction or electron back-scattered diffraction (EBSD) [5]. In the former
case, the polycrystalline sample is exposed in a system set to detect X-rays from a
particular reflection (Bragg angle) and the sample is systematically tilted and rotated
in order to capture intensity at a variety of orientations. In EBSD, an electron beam is
rocked about a fixed point on the surface of a grain using a scanning electron micro-
scope. At particular angles, the beam is Bragg diffracted, resulting in a reduction in
intensity picked up by the detector, leading to the formation of channelling patterns
of the type illustrated in Fig. 6.7. These patterns can be interpreted to determine the
orientation of the crystal relative to the sample axes.
98 6 Deformation and Texture

1 1
1 1

FIGURE 6.6
Inverse pole figures for a cubic crystal. The diagrams at the top have poles corre-
sponding to the rolling, normal and transverse directions plotted relative to the crys-
tal axes, whereas those at the bottom have the same data represented as contours
representing the density of poles.

FIGURE 6.7
A simulated electron channelling pattern from silicon carbide. Reproduced from
Winkelmann et al. [6], with permission of Elsevier.
6.3 Orientation distribution functions 99

6.3 Orientation distribution functions

A pole figure does not give information about the orientation of a particular crystal
relative to another, since the poles are not, in the final plot, identified with particular
crystals. An orientation distribution function is a more rigorous method of expressing
the orientations of crystals relative to the frame of reference. It uses three Euler angles
(ϕ1 , Φ and ϕ2 ) which define the relative orientation of the crystal and sample frames
of reference, as illustrated in Fig. 6.8.

ND

001
010
normal to ND

(a) 100 (b)

(c) (d)

FIGURE 6.8
Operations that define the three (Bunge) Euler angles. (a) Initial relationship between
crystal and sample coordinates. (b) Rotation by ϕ1 about ND to generate RD’ in a
position normal to the plane containing ND and 001. (c) Rotate by Φ about RD’ to
generate ND’ parallel to 001. (d) Rotate by ϕ2 about 001 to bring bring crystal and
sample axes into coincidence. The insets show the rotations relative to an aircraft.

The set of measured Euler angles made by the RD, T D and N D directions for a
100 6 Deformation and Texture

(a) (b)

FIGURE 6.9
(a) An orientation distribution plotted on a cube with axes defined by the Euler an-
gles. Each dot represents a crystal orientation relative to sample axes. (b) Sections of
the cube to show how the density of orientations varies.

particular crystal is plotted as a point in a cube whose axes scale with ϕ1 , Φ and ϕ2 .
Each point inside this cube is a single crystal.
There are very many industrial processes which control and exploit texture on
a grand scale. Beverage cans are incredibly thin even though their bodies are made
without fracture using extreme deformation including deep drawing. The texture of
the material prior to drawing is controlled such that plastic instability is avoided in
the thickness direction. For similar reasons, the steels used in the manufacture of
formed automobile bodies are texture-controlled. Iron alloys used to make electric-
ity transformers have to be magnetically soft. This is achieved by controlling the
crystallographic texture to minimise magnetic losses.

Example 6.3: Euler angles relating two frames

By comparison with the operations illustrated in Fig. 6.8a, deduce the set ϕ1 ,
Φ and ϕ2 relating the bases A and B in Fig. 6.10. Represent the relative orientation
of the two frames on a section of an orientation distribution function.
6.4 Summary 101

010

010

100

(a) 100 (b)

FIGURE 6.10
(a) Note that [001]A ∥ [001]B . (b) Construction showing that the projection is also
a circle.
The Euler angles define the operations needed to bring the two frames of refer-
ence into coincidence. Referring to Fig. 6.8b, since [001]A ∥ [001]B , ϕ1 = 0◦ . For
the same reason, Φ = 0◦ ( Fig. 6.8c) and the remaining angle ϕ2 is clearly 45◦ .
Fig. 6.8b shows the ODF section φ1 = 0, referred to the basis-B as the reference
frame.

6.4 Summary
Crystallographic texture can be understood by a thought experiment in which a large
single crystal is broken up into many smaller ones which are then joined together in
somewhat different orientations that are not random. The material then behaves as if
it has some of the characteristics of a single crystal (such as anisotropy) while other
properties are more a feature of random polycrystals (such as the multiple slip sys-
tems needed per crystal in order to achieve plasticity without the creation of voids).
We have discussed how texture can arise from plastic deformation, but many
kinds of processing can lead to its development. The heat treatment of deformed
materials to induce recrystallisation can lead to recrystallisation textures because not
all ‘nuclei’ grow at the same rate [7]. The nuclei have to be sufficiently different in
orientation from their surroundings to accomplish rapid growth. Similar selection can
occur in phase transformations where the product phase forms at a grain boundary
because certain boundaries make it possible for the product to adopt a favourable
orientation relationship with the parent phase on either side of the interface [8, 9].
102 6 Deformation and Texture

Both magnetic [10] and electrical fields [11] can also lead to orientational order in
appropriate materials.
So how does one produce a polycrystalline material with a random arrangement
of grains? The short answer is, with enormous difficulty. Powder techniques can
partially reach this goal; metallic or ceramic powders that are sintered together tend
to undergo very limited deformation or rotation during fabrication, and hence may
approach a random orientational distribution.
References

1. E. Schmid, and W. Boas: Plasticity of Crystals (translated from the 1935 edition
of Kristalplastizitaet): London, U.K.: F. A. Hughes and Co., 1950.
2. J. W. Christian: ‘Deformation by moving interfaces’, Metallurgical Transac-
tions A, 1982, 13, 509–538.
3. A. Kelly, and K. M. Knowles: Crystallography and Crystal Defects: 2nd ed.,
New York, USA: John Wiley & Sons, Inc., 2012.
4. D. N. Lee: Texture and related phenomena: Seoul, Republic of Korea: Korean
Institute of Metals and Materials, 2006.

5. A. F. Gourgues-Lorenzon: ‘Application of electron backscatter diffraction to

the study of phase transformations’, International Materials Reviews, 2007, 52,
65–128.
6. A. Winkelmann, B. Schröter, and W. Richter: ‘Dynamical simulations of zone
axis electron channelling patterns of cubic silicon carbide’, Ultramicroscopy,
2003, 98, 1–7.
7. W. B. Hutchinson: ‘Recrystallisation textures in iron resulting from nucleation
at grain boundaries’, Acta Metallurgica, 1989, 37, 1047–1056.
8. O. Hashimoto, S. Satoh, and T. Tanaka: ‘α → γ → α transformation behaviour
during heating from the α + γ region in 0.18%C steel’, Tetsu-to-Hagané, 1980,
66, 102–111.
9. H. K. D. H. Bhadeshia: ‘Diffusional formation of ferrite in iron and its alloys’,
Progress in Materials Science, 1985, 29, 321–386.
10. A. D. Sheikh-Ali, D. A. Molodov, and H. Garemstani: ‘Magnetically induced
texture development in zinc alloy sheet’, Scripta Materialia, 2002, 46, 857–
862.
11. D. Coates, W. A. Crossland, J. H. Morrissy, and B. Needham: ‘Electrically
induced scattering textures in smectic A phases and their electrical reversal’,
Journal of Applied Physics D, 1978, 11, 2025–2035.

103
7
Interfaces, Orientation Relationships

Abstract

An interface is a region where the long-range translational order of a crystal is dis-

rupted because there is an abrupt change in orientation. It is in general a very narrow
region within which the atoms may not have a strong sense of belonging to the crys-
talline material on either side. The density of the interfacial region is likely to be
smaller than that of the neighbouring crystals with consequences on diffusion, impu-
rity concentrations and cohesion. Nevertheless, we shall see that the structure of an
interface can have elements of periodicity, including regularity in the arrangement of
the defect structure within.

7.1 Introduction
Much of the science and technology of polycrystalline materials depends on the na-
ture of interfaces between crystals and the relative orientations of adjacent crystals.
The corrosion resistance of a boundary depends on how coherent it is [1], the conti-
nuity of deformation varies with texture [2] . . . the list is seemingly endless!
Atoms in the boundary between crystals must in general be displaced from po-
sitions they would occupy in the undisturbed crystal, but it is now well established
that many interfaces have a periodic structure. In such cases, the misfit between the
crystals connected by the boundary is not distributed uniformly over every element
of the interface; it is localised periodically into discontinuities that separate patches
of the boundary where the fit between the two crystals is good or perfect. When these
discontinuities are well separated, they may individually be recognised as interface
dislocations which separate coherent patches in the boundary, which is macroscop-
ically said to be semi-coherent. The simple example of a symmetrical tilt boundary
illustrates this.

105
106 7 Interfaces, Orientation Relationships

7.2 Symmetrical tilt boundary

The structure of an interface can be understood by creating a boundary beginning
with a single crystal that is sliced and the two halves then rotated relative to generate
two crystals in different orientations. The rotation is illustrated in Fig. 7.1 about
an axis normal to the diagram through an angle θ. This leaves a gap between the
bicrystal which can be filled with edge dislocations that after all, have extra half-
planes. The misorientation θ between the grains can therefore be described in terms
of dislocations. Inserting an edge dislocation of Burgers vector b is like forcing a
wedge into the lattice, so that each dislocation is associated with a small change in
the orientation of the lattice on either side of the extra half plane. If the spacing of
dislocations is d, then it follows from Fig. 7.1b that
θ b
tan = (7.1)
2 2d

(a) (b)

FIGURE 7.1
The creation of an interface. (a) Single crystal cut in two halves which are then tilted
through an angle θ about an axis normal to the diagram. (b) The gap within the tilt is
filled with edge dislocations, each with a Burgers vector b and spacing d to create an
interface.
The structure of the interface in terms of dislocations provides a mechanism for
calculating the energy per unit area of the interface as a function of the misorientation
θ. The energy per unit length Ws of a dislocation is given by [3]
3
1 r∞ Gb2 dr
Ws = Wc +
2 r0 2π(1 − ν) r
4 5
Gb2 r∞
= Wc + ln (7.2)
4π(1 − ν) r0
where Wc is the core energy per unit length, covering the region of the dislocation
where elasticity theory fails; G and ν are the shear modulus and Poisson’s ratio,
respectively; r0 is the radius of the dislocation core and r∞ is a cut-off radius beyond
which the dislocation strain field can be neglected or is limited by the size of the
7.3 Symmetrical tilt boundary 107

sample. In arrays of dislocations of the type found in interfaces, the strain fields of
individual dislocations are partly compensated by their neighbours so it is a good
approximation that r∞ ≈ d in which case r∞ ∝ 1/θ. It follows that equation 7.2 can
be written Ws = A(B − ln θ). Therefore, the interfacial energy per unit area σi is
related to the number of dislocations per unit area (1/d):
1
σi = × Ws
d
2 θ
= tan × A(B − ln θ). (7.3)
b 2
This gives a model for the energy of grain boundaries [4], based on the density of
dislocations in the boundary (∝ d−1 ). The variation in energy as a function of mis-
orientation in the symmetrical tilt boundary described above is illustrated in Fig. 7.2.
The model is not valid for large misorientations where the dislocation spacing be-
comes comparable with the magnitude of the Burgers vector, since the dislocation
cores then begin to overlap.

FIGURE 7.2
The interfacial energy as a function of the misorientation for a tilt boundary. The
dashed curve represents cusps when the orientation forms a coincidence site lattice.

The symmetrical tilt boundary is an illustration of how interfaces can be de-

scribed in terms of defect structures with coherent patches in-between the defects.
More complex misorientations would require more than one array of dislocations
to represent the fit between the crystals, and there will come a point where the de-
fect density becomes so large that it is no longer reasonable to talk of a dislocation
structure.
108 7 Interfaces, Orientation Relationships

7.3 Coincidence site lattices

For high-misorientation boundaries, the predicted spacings of dislocations may turn
out to be so small that the misfit is highly localised with respect to the boundary,
and the dislocation model of the interface has only formal significance (it often is
said that the dislocations get so close to each other that their cores overlap). The
arrangement of atoms in such incoherent boundaries may be haphazard, with little
correlation of atomic positions across the boundary.
On the other hand, it is unreasonable to assume that all high-angle boundaries
have the disordered structure. There is clear experimental evidence which shows that
certain high-angle boundaries exhibit the characteristics of low-energy coherent or
semi-coherent interfaces; for example, they exhibit strong faceting, have low mobil-
ity in pure materials and the boundary diffusion coefficient may be abnormally small.
These observations imply that there are special orientations, which would usually be
classified as large mis-orientations, where it is possible to obtain boundaries that have
a distinct structure. They contain regions of good fit, which occur at regular intervals
in the boundary plane, giving a pattern of good fit points in the interface. It is along
these points that the two crystals match exactly at their mutual interface.
The lattice points that are common to the two crystals can be identified by al-
lowing the two lattices, with a common origin, notionally to interpenetrate and fill
all space. The set of these coincidence points forms a coincidence site lattice (CSL),
and the fraction of lattice points which are also coincidence sites is a rational fraction
1/Σ. Σ is thus the reciprocal density of coincidence sites relative to ordinary lattice
sites [5, 6]. The value of Σ is a function of the relative orientation of the two grains
and not of the orientation of the boundary plane. The boundary simply intersects the
CSL and will contain regions of good fit that have a periodicity of a planar net of the
CSL along which the intersection occurs. Boundaries parallel to low-index planes
of the CSL are two-dimensionally periodic with relatively small repeat cells. Those
boundaries with a high planar coincidence site density may have a relatively low
energy per unit area.
Fig. 7.3 shows the projection on (100) of two primitive cubic lattices with a com-
mon origin, each with a unit lattice parameter, and an orientation relationship de-
scribed by a rotation of 36.9◦ about ⟨100⟩ which results in a√Σ5 √ coincidence site
lattice. It can be seen that the lattice parameters of the CSL are 5, 5, and 1 (along
the rotation axis) so its volume is exactly five times that of the cubic lattice.
7.4 Representation of orientation relationships 109

(a) (b)

FIGURE 7.3
(a) A rotation of 36.9◦ about ⟨100⟩ defines the orientation relationship between two
primitive cubic crystals, each of lattice parameter 1, leading to a Σ5 coincidence site
lattice. The origin is labelled ‘o’. (b) Simplified illustration of the fact that there are
only one in five of the lattice points of each of the cubic crystals in coincidence – the
four in the middle of the cell are not.

7.4 Representation of orientation relationships

In Fig. 7.4a, the choice of basis vectors ai is arbitrary though convenient; Fig. 7.4b
illustrates an alternative basis, a body-centred tetragonal (bct) unit cell describing
the same lattice. We label this as basis ‘B’, consisting of the vectors b1 , b2 and b3
which define the bct unit cell. It is obvious that [B; u] = [0 2 1], compared with
[A; u] = [1 1 1]. The following vector equations illustrate the relationships between
the basis vectors of A and those of B (Fig. 7.4):

a1 = 1b1 + 1b2 + 0b3 (7.4)

a2 = 1b1 + 1b2 + 0b3
a3 = 0b1 + 0b2 + 1b3 .

These equations can also be presented in matrix form as follows:

⎛ ⎞
1 1 0
⎜ ⎟
(a1 a2 a3 ) = (b1 b2 b3 ) × ⎜⎝ 1 1 0 ⎠.
⎟ (7.5)
0 0 1

This 3 × 3 matrix representing the coordinate transformation is denoted (B J A)

110 7 Interfaces, Orientation Relationships

(a) (b)

FIGURE 7.4
(a) Cubic coordinate system with orthogonal basis vectors of equal magnitude. (b)
Tetragonal coordinate system (red) in which the basis vectors on the basal plane end
at the face-centres of the cubic system. The relationship to the cubic cells (black) is
also illustrated.

and transforms the components of vectors referred to the A basis to those referred to
the B basis. The first column of (B J A) represents the components of the basis vector
a1 , with respect to the basis B, and so on.
The components of a vector u can now be transformed between bases using the
matrix (B J A) as follows:

[B; u] = (B J A)[A; u] (7.6)

Notice the juxtapositioning of like basis symbols. Writing (A J B) as the inverse of

(B J A):
[A; u] = (A J B)[B; u]

Example 7.1: Coordinate transformation

Two adjacent crystals that have an identical cubic lattice, are represented by
bases ‘A’ and ‘B’ respectively. The basis vectors ai of A and bi of B respec-
tively define the cubic unit cells of the crystals concerned. The lattice parameter
is identical for the two bases, |ai | = |bi | = a. The grains are orientated such that
[0 0 1]A ∥ [0 0 1]B , and [1 0 0]B makes an angle of 45◦ with both [1 0 0]A and
[0 1 0]A , with [001]A ∥ [001]B , as illustrated in Fig. 7.5. Prove√that if
√u is a vector
such that its components in crystal A are given by [A; u] = [ 2 2 2 0], then in
the basis B, [B; u] = [3 1 0]. Show that the magnitude of u (i.e. |u|) does not
depend on the choice of the basis.
7.4 Representation of orientation relationships 111

010

010

100

100

FIGURE 7.5
Diagram illustrating the relation between the bases A and B. Note that [001]A ∥
[001]B .

Referring to Fig. 7.5, and recalling that the matrix (B J A) consists of three
columns, each column being the components of one of the basis vectors of A,
with respect to B, we have

[B; a1 ] = [cos 45 − sin 45 0]

[B; a2 ] = [sin 45 cos 45 0]
[B; a3 ] = [0 0 1]
so that ⎛ ⎞
cos 45 0 sin 45
⎜ ⎟
(B J A) = ⎜ ⎟
⎝− sin 45 cos 45 0⎠
0 0 1
√ √
Now, [B; u] = (B J A)[A; u]. On substituting for [A; u] = [ 2 2 2 0], we get
[B; u] = [3 1 0]. Both the bases A and B are orthogonal so that the magnitude
of u can be obtained using the Pythagoras theorem. Hence, choosing components
referred to the basis B, we get:

|u|2 = (3|b1 |)2 + (|b2 |)2 = 10a2 (7.7)

With respect to basis A,

√ √
|u|2 = ( 2|a1 |)2 + (2 2|a2 |)2 = 10a2 (7.8)

Hence, |u| is invariant to the coordinate transformation. This is a general result,

since a vector is a physical entity, whose magnitude and direction cannot depend
on the choice of a reference frame.
112 7 Interfaces, Orientation Relationships

7.4.1 General Rotation Matrix

If the axis of rotation is represented as a unit vector, then only three independent
quantities are needed to define a misorientation between grains: two components of
the axis of rotation, and an angle of rotation. It follows that a rotation matrix must
also have only three independent terms. The components of any rotation matrix can
be written in terms of a vector u = [u1 u2 u3 ] which lies along the axis of rotation
(such that u1 u1 + u2 u2 + u3 u3 = 1), and in terms of the right-handed angle of
rotation θ as follows (see Appendix B for derivation):
⎛ ⎞
u1 u1 (1 − m) + m u1 u2 (1 − m) + u3 n u1 u3 (1 − m) − u2 n
⎜ ⎟
(Y J X) = ⎜ ⎝ u1 u2 (1 − m) − u3 n u2 u2 (1 − m) + m u2 u3 (1 − m) + u1 n ⎠
⎟

u1 u3 (1 − m) + u2 n u2 u3 (1 − m) − u1 n
u3 u3 (1 − m) + m
(7.9)
where m = cos θ and n = sin θ. The right-handed angle of rotation can be obtained
from the fact that
J11 + J22 + J33 = 1 + 2 cos θ (7.10)
and the components of the vector u along the axis of rotation are given by

u1 = (J23 − J32 )/2 sin θ

u2 = (J31 − J13 )/2 sin θ (7.11)
u3 = (J12 − J21 )/2 sin θ.

If the rotation axis in equation 7.9 is set to be [001]X then

⎛ ⎞
cos θ sin θ 0
⎜ ⎟
(Y J X) = ⎜ ⎝ − sin θ cos θ 0 ⎠
⎟ (7.12)
0 0 1

Referring again to equation 7.9, if the rotation is a symmetry operation then it

should map lattice points onto lattice points, in which case all components of the
rotation matrix should be integers. It follows that the trace of the matrix should also
be an integer, and since −1 ≤ cos θ ≤ 1, the set of possible integers is limited:

J11 + J22 + J33 = 1 + 2 cos θ = −1, 0, 1, 2, 3 (7.13)

or
2π 2π 2π 2π
θ = 2π, , , , , (7.14)
2 3 4 6
another way of showing that a five-fold rotation axis is not permitted when transla-
tional symmetry is to be maintained.
7.5 Mathematical method for determining Σ 113

7.5 Mathematical method for determining Σ

We now consider a mathematical method of determining the CSL formed by allowing
the lattices of crystals A and B notionally to interpenetrate; A and B are assumed to
be related by a transformation (A S A) which deforms the A lattice into that of B; A
and B need not have the same crystal structure or lattice parameters, so that (A S A)
need not be a rigid body rotation. Consider a vector u which is a lattice vector whose
integral components do not have a common factor. As a result of the transformation
(A S A), u becomes a new vector x such that

[A; x] = (A S A)[A; u] (7.15)

x does not necessarily have integral components in the A basis (i.e. it need not be
a lattice vector of A). CSL vectors, on the other hand, identify lattice points which
are common to both A and B, and therefore are lattice vectors of both crystals. It
follows that CSL vectors have integral indices when referred to either crystal. Hence,
x is only a CSL vector if it has integral components in the basis A. We note that x
always has integral components in B, because a lattice vector of A (such as u) always
deforms into a lattice vector of B.
The meaning of Σ is that 1/Σ of the lattice sites of A or B are common to both
A and B. It follows that any primitive lattice vector of A or B, when multiplied
by Σ, must give a CSL vector. Σx must therefore always be a CSL vector and if
equation 7.15 is multiplied by Σ, then we obtain an equation in which the vector u
always transforms into a CSL vector:

Σ[A; x] = Σ(A S A)[A; u] (7.16)

i.e.. given that u is a lattice vector of A, whose components have no common factor,
Σx is a CSL vector with integral components in either basis. This can only be true
if the matrix Σ(A S A) has elements which are all integral since it is only then that
Σ[A; x] has elements which are all integral.
It follows that if an integer H can be found such that all the elements of the matrix
H(A S A) are integers (without a common factor), then H is the Σ value relating A
and B.
Applying this to the problem considered in section 7.3, the rotation matrix corre-
sponding to the rotation 180◦ about [1 1 2]A is given by (equation 7.9)
⎛ ⎞
2 1 2
1⎜ ⎟
(A J A) = ⎜ 1 2 2
⎟ (7.17)
3 ⎝ ⎠
2 2 1

and since 3 is the integer which when multiplied with (A J A) gives a matrix of
integral elements (without a common factor), the Σ value for this orientation is given
by Σ = 3.
114 7 Interfaces, Orientation Relationships

Coincidence site lattices have in recent years become popular because of the
advent of orientation imaging in the scanning electron microscope. The microscope
is usually associated with software that enables the estimation of the Σ value at every
junction between two grains. The accuracy of the technique is limited so such Σ maps
should be taken with a pinch of salt, and large Σ values are not very meaningful as
representations of low-energy boundaries.

7.6 Summary
Interfaces clearly have a structure that can be periodic. Simple boundaries between
misoriented crystals that have identical structures can be described in terms of arrays
of dislocations which become visible during transmission electron microscopy. The
spacing of these boundary dislocations can be related to the degree of misorientation
whereas their line vectors, in the case of a tilt boundary, lie along the tilt axis. The
dislocation structure can be used also to estimate the interfacial energy per unit area
as long as the dislocation description remains valid.
It is possible that at certain values of the misorientations, a significant fraction
of the lattice points between the two crystals that form a boundary, are coincident.
Furthermore, the coincident sites form a lattice that is periodic, known as a coinci-
dence site lattice. It is important to realise that this is a reflection of the orientation
relationships between the crystals rather than the boundary plane. The properties of a
boundary change significantly when the fraction of coincident points, 1/Σ, becomes
large. For example, the interfacial energy per unit area declines dramatically as might
the boundary diffusion coefficient.
References

1. T. Watanabe: ‘Grain boundary engineering: historical perspective and future

prospects’, Journal of Materials Science, 2011, 46, 4095–4115.
2. A. F. Gourgues, H. M. Flower, and T. C. Lindley: ‘Electron backscattering
diffraction study of acicular ferrite, bainite, and martensite steel microstructures’,
Materials Science and Technology, 2000, 16, 26–40.
3. J. S. Koehler: ‘On the dislocation theory of plastic deformation’, Physical Re-
view, 1941, 60, 397–410.
4. W. T. Read, and W. Schockley: ‘Dislocation models of crystal grain boundaries’,
Physical Review, 1950, 78, 275–289.
5. M. L. Kronberg, and F. H. Wilson: ‘Secondary recrystallization in copper’,
Trans. A.I.M.E., 1949, 185, 501–514.
6. S. Ranganathan: ‘On the geometry of coincidence-site lattices’, Acta Crystallo-
graphica, 1966, 21, 197–199.

115
8
Crystallography of Martensitic
Transformations

Abstract

Martensitic transformations are exceptionally simple and precise. This is because

they do not involve diffusion, which has a tendency to create confusion in the ar-
rangement of atoms. With martensite, the atoms move in a disciplined manner that
leads to strong crystallographic predictability, including that of irrational quantities
and exact changes in shape. This chapter contains a basic explanation of the theory
needed to explain the fine details of the crystallography of martensite and associated
consequences.

8.1 Introduction
Martensitic transformations, and displacive transformations in general, remain the
only mechanism for the manufacture of bulk nanostructured materials for engineer-
ing applications [1, 2]. The transformation is diffusionless and hence can occur at
incredibly low temperatures [3], or can propagate at the speed of sound in the ma-
terial (a few thousand m s−1 ) [4]. There is no composition change when it grows to
consume the parent phase.
There are, however, a number of peculiarities associated with martensitic trans-
formations, which are important in the design of alloys. The interface plane between
austenite and martensite is known as the habit plane and measurements show that the
crystallographic indices of that plane are irrational and strange (Table 8.1). Another
difficulty is that the martensite-austenite (α/γ) interface has to be glissile, that is, it
must be able to move without diffusion. This requires that the interface must contain
no more than one set of dislocations. Any more than one array of line vectors can lead
to interference between the dislocations which renders the interface sessile. In crys-
tallographic terms, this means that there must be at least one line fully coherent in
the α/γ interface. The transformation strain relating the two lattices must therefore,
as a minimum, be an invariant-line strain.

117
118 8 Crystallography of Martensitic Transformations

TABLE 8.1
Habit plane indices for martensite. The quoted indices are approximate because the
habit planes are in general irrational. There is an interesting study on the regularity
of the habit plane indices as a function of, for example, strain in the austenite [5].

Composition /wt.% Approximate habit plane indices

Low-alloy steels, Fe-28Ni {1 1 1}γ

Plate martensite in Fe-1.8C {2 9 5}γ
Fe-30Ni-0.3C {3 15 10}γ
Fe-8Cr-1C {2 5 2}γ

8.2 Shape deformation

During martensitic transformation, the pattern in which the atoms in the parent crys-
tal are arranged is deformed into that appropriate for martensite, there must be a
corresponding change in the macroscopic shape of the crystal undergoing transfor-
mation. The dislocations responsible for the deformation are in the α/γ interface,
with Burgers vectors such that in addition to deformation they cause the change in
crystal structure. The deformation is such that an initially flat surface becomes uni-
formly tilted about the line formed by the intersection of the interface plane with
the free surface (Fig. 8.1). Any scratch traversing the transformed region is similarly
deflected though the scratch remains connected at the α/γ interface.
8.3 Bain strain 119

FIGURE 8.1
The measured shape deformation due to martensitic transformation is an invariant-
plane strain with a large shear component (≃ 0.22) and a small dilatational strain
(≃ 0.03) directed normal to the habit plane in steels.

8.3 Bain strain

We now consider the nature of the strain necessary to transform the fcc lattice of γ
into the bcc lattice of α. Such a strain was proposed by Bain in 1924 [6] and hence
is known as the ‘Bain Strain’ (Fig. 8.2). There is a compression along the z axis and
a uniform expansion along the x and y axes.
The Bain strain implies the following rational orientation relationship between
the parent and product lattices:

[0 0 1]fcc | |[0 0 1]bcc [1 1 0]fcc | |[1 0 0]bcc [1 1 0]fcc | |[0 1 0]bcc (8.1)

and although the orientations actually observed are consistent with good matching
between the two lattices (Fig. 8.3) but they are inconsistent with the Bain relation-
ship. Typical observed orientation relationships are irrational but are represented ap-
proximately by:

Kurdjumov-Sachs orientation relationship:

{1 1 1}γ ∥ {0 1 1}α ,

⟨1 0 1⟩γ ∥ ⟨1 1 1⟩α ,
120 8 Crystallography of Martensitic Transformations

(a)

(b) (c)

FIGURE 8.2
The Bain strain. The iron atoms at the centres of the faces of the austenite cell paral-
lel to the plane of the diagram, are omitted for clarity. (a) The lattice correspondence
for formation of martensite from austenite, showing a single carbon atom in an octa-
hedral interstice on the [001]γ axis. (b) Tetragonal unit cell outlined in austenite. (c)
Lattice deformation (compression along c-axis) to form martensite with an appropri-
ate c/a ratio. Note that the martensite in iron alloys can be body-centred tetragonal
(bct) or body-centred cubic (bcc); none of the descriptions in this chapter are sensi-
tive to this.

Nishiyama-Wasserman orientation relationship:

{1 1 1}γ ∥ {0 1 1}α ,

⟨1 0 1⟩γ about 5.3◦ from⟨1 1 1⟩α towards ⟨1 1 1⟩α ,

Greninger-Troiano orientation relationship:

{1 1 1}γ about 0.2◦ from {0 1 1}α ,

⟨1 0 1⟩γ about 2.7◦ from ⟨1 1 1⟩α towards ⟨1 1 1⟩α .

A conceptual difficulty therefore is to explain why the observed orientation rela-

tions differ from that implied by the Bain strain. The second inconsistency with the
Bain strain does not leave any line invariant, thus violating the essential condition
8.3 Bain strain 121

FIGURE 8.3
Electron diffraction pattern from bcc martensite and fcc austenite lattices in steels.

for the existence of a glissile interface that facilitated martensitic transformation. .

In Fig. 8.4a,b, the austenite is represented as a sphere which, as a result of the Bain
strain B, is deformed into an ellipsoid of revolution which represents the martensite.
There are no lines which are left undistorted or unrotated by B. There are no lines in
the (0 0 1)fcc plane which are undistorted. The lines wx and yz are undistorted but
are rotated to the new positions w′ x′ and y ′ z ′ . Such rotated lines are not invariant.
However, the combined effect of the Bain strain B and the rigid body rotation R is
indeed an invariant-line strain (ILS) because it brings yz and y ′ z ′ into coincidence
(Fig. 8.4c). This is the reason why the observed irrational orientation relationship
differs from that implied by the Bain strain. The rotation required to convert B into
an ILS precisely corrects the Bain orientation into that which is observed experimen-
tally.

FIGURE 8.4
(a) and (b) show the effect of the Bain strain on austenite, which when undeformed
is represented as a sphere of diameter wx = yz in three-dimensions. The strain trans-
forms it into an ellipsoid of revolution. (c) Shows the ILS obtained by combining
the Bain strain with a rigid body rotation through an angle θ. a1 , a2 and a3 refer to
[100]γ , [010]γ and [001]γ axes respectively.
122 8 Crystallography of Martensitic Transformations

As can be seen from Fig. 8.4c, there is no rotation which can make B into an
invariant-plane strain since this would require two non-parallel invariant-lines. Thus,
for the fcc→bcc transformation, austenite cannot be transformed into martensite by
a homogeneous strain which is an invariant-plane strain. And yet, the observed shape
deformation leaves the habit plane undistorted and unrotated, i.e., it is an invariant-
plane strain.
The phenomenological theory of martensite crystallography solves this remain-
ing problem (Fig. 8.5) [7–10]. The Bain strain converts the structure of the parent
phase into that of the product phase. When combined with an appropriate rigid body
rotation, the net homogeneous lattice deformation RB is an invariant-line strain (step
a to c in Fig. 8.5). However, the observed shape deformation is an invariant-plane
strain P1 (step a to b in Fig. 8.5), but this gives the wrong crystal structure. If a
second homogeneous shear P2 is combined with P1 (step b to c), then the correct
structure is obtained but the wrong shape since

P1 P2 = RB.

These discrepancies are all resolved if the shape changing effect of P2 is cancelled
macroscopically by an inhomogeneous lattice-invariant deformation, which may be
slip or twinning as illustrated in Fig. 8.5.
The theory explains all the observed features of the martensite crystallography.
The orientation relationship is predicted by deducing the rotation needed to change
the Bain strain into an invariant-line strain. The habit plane does not have rational
indices because the amount of lattice-invariant deformation needed to recover the
correct macroscopic shape is not usually rational. The theory predicts a substruc-
ture in plates of martensite (either twins or slip steps) as is observed experimentally.
The transformation goes to all the trouble of ensuring that the shape deformation
is macroscopically an invariant-plane strain because this reduces the strain energy
when compared with the case where the shape deformation might be an invariant-
line strain.
When austenite transforms martensitically into ε (hcp), the entire change is
achieved by a deformation that is an invariant-plane strain as illustrated in Fig. 8.6.
As a consequence, the habit plane of the martensite is rational, i.e., precisely the
close-packed plane, the orientation relationship is rational:

(111)γ ∥ (0001)H [101]γ ∥ [1010]H .

The strain not only transforms the lattice but the resulting shape of the final transfor-
mation product is as observed, so there is no need for any lattice-invariant deforma-
tion (slip or twinning).
8.4 Summary 123

FIGURE 8.5
The phenomenological theory of martensite crystallography.

FIGURE 8.6
aγ
The fcc→hcp transformation involving a shear of 6 [112] on alternate (111)γ planes
to generate the hcp structure.

8.4 Summary
It is hard to exaggerate the success of the crystallographic theory of martensitic trans-
formations, both in terms of the conceptual leap in explaining away the apparent dif-
124 8 Crystallography of Martensitic Transformations

ficulties accumulated via careful experiments, and in the quantitative closure between
experiment and theory. This has not, in many respects, been achieved for any other
solid-state transformations, particularly those that involve diffusion and composition
change. The theory has generic value because it is not restricted to changes between
particular crystal systems.
References

1. R. Ueji, N. Tsuji, Y. Minamino, and Y. Koizumi: ‘Ultragrain refinement of plain

low carbon steel by cold-rolling and annealing of martensite’, Acta Materialia,
2002, 50, 4177–4139.
2. H. K. D. H. Bhadeshia: ‘The first bulk nanostructured metal’, Science and
Technology of Advanced Materials, 2013, 14, 014202.
3. L. Kaufman, and M. Cohen: ‘The martensitic transformation in iron-nickel
system’, Trans. AIME Journal of Metals, 1956, 206, 1393–1401.
4. R. F. Bunshah, and R. F. Mehl: ‘Rate of propagation of martensite’, Trans.
A.I.M.E., 1953, 193, 1251–1258.
5. D. P. Dunne, and N. F. Kennon: ‘The martensite interface - how regular is the
habit?’, Materials Science Forum, 1994, 189-190, 273–278.
6. E. C. Bain: ‘The nature of martensite’, Trans. AIME, 1924, 70, 25–46.
7. J. S. Bowles, and J. K. Mackenzie: ‘The crystallography of martensite transfor-
mations, part I’, Acta Metallurgica, 1954, 2, 129–137.

8. J. K. Mackenzie, and J. S. Bowles: ‘The crystallography of martensite transfor-

mations II’, Acta Metallurgica, 1954, 2, 138–147.
9. J. K. Mackenzie, and J. S. Bowles: ‘The crystallography of martensite trans-
formations III FCC to BCT transformations’, Acta Metallurgica, 1954, 2, 224–
234.
10. M. S. Wechsler, D. S. Lieberman, and T. A. Read: ‘On the theory of the forma-
tion of martensite’, Trans. AIME Journal of Metals, 1953, 197, 1503–1515.

125
 Part II

A Few Advanced Methods

9
Orientation relations

Abstract
Macroscopic materials consist of aggregates of crystals. The order that typifies sin-
gle crystals is disrupted at boundaries so the properties of polycrystalline samples
can be sensitive to the relative orientations of neighbouring crystals. Or even to the
averaged disposition of the myriads of crystals relative to some macroscopic frame
of reference. Orientation relationships between crystals are useful in understanding
the structure of the interfaces between crystals and the evolution of the shapes of pre-
cipitates as they grow in a solid matrix. An entire subject now exists on grain bound-
ary engineering, where, for example, the corrosion resistance of a stainless steel can
be improved dramatically by ensuring a preponderance of good-fit boundaries. And
there are remarkable experimental techniques that facilitate the crystallographic char-
acterisation of millions of crystals in a long blink of an eye.

9.1 Introduction
A substantial part of research on polycrystalline materials is concerned with the accu-
rate determination, assessment and theoretical prediction of orientation relationships
between adjacent crystals. There are obvious practical applications, such as in the
study of anisotropy and texture, or in defining the role of interfaces in controlling
the mechanical properties. The existence of a reproducible orientation relation be-
tween the parent and product phases might determine the ultimate morphology of
any precipitates, by allowing the development of low interfacial-energy facets. It is
possible that vital clues about nucleation in the solid state might emerge from a de-
tailed examination of orientation relationships, even though these can usually only
be measured when the crystals concerned are well into the growth stage. Needless to
say, the properties of interfaces depend critically on the relative dispositions of the
crystals that they connect.
Perhaps the most interesting experimental observation is that the orientation rela-
tionships that are found to develop during phase transformations (or during deforma-
tion or recrystallisation experiments) usually are not random [1–3]. The frequency of
occurrence of any particular orientation relation usually far exceeds the probability
of obtaining it simply by taking the two separate crystals and joining them up in an
arbitrary way.

129
130 9 Orientation relations

This indicates that there are favoured orientation relations, perhaps because it is
these which allow the best fit at the interface between the two crystals. This would in
turn reduce the interfacial energy, and hence the activation energy for nucleation. Nu-
clei which, by chance, happen to be orientated in this manner would find it relatively
easy to grow, giving rise to the non-random distribution mentioned earlier.
On the other hand, it could be the case that nuclei actually form by the homo-
geneous deformation of a small region of the parent lattice. The deformation, which
transforms the parent structure to that of the product (e.g. Bain strain), would have
to be of the kind which minimises strain energy. Of all the possible ways of accom-
plishing the lattice change by homogeneous deformation, only a few might satisfy the
minimum strain energy criterion - this again would lead to the observed non-random
distribution of orientation relationships. It is a problem to understand which of these
factors really determine the existence of rational orientation relations. In this chap-
ter we deal with the methods for adequately describing the relationships between
crystals.

9.2 Cementite in steels

Cementite, Fe3 C, referred to as θ, is the most common precipitate to be found in
steels. It has an orthorhombic crystal structure and can precipitate from supersatu-
rated ferrite or austenite.1 It sometimes can precipitate at temperatures below 400 K
in times too short to allow any substantial diffusion of iron atoms [4–6] although
the long range diffusion of carbon atoms is clearly necessary and indeed possible. It
has therefore been suggested that the cementite lattice may sometimes be generated
by the deformation of the ferrite lattice, combined with the necessary diffusion of
carbon into the appropriate sites [4, 7].
Shackleton and Kelly [8] showed that the plane of precipitation of cementite from
ferrite is {1 0 1}θ ∥ {1 1 2}α . This is consistent with the habit plane containing the
direction of reasonable fit between the θ and α lattices [4], i.e. ⟨0 1 0⟩θ ∥ ⟨1 1 1⟩α .
Cementite formed during the tempering of martensite adopts many crystallographic
variants of this habit plane in any given plate of martensite; in lower bainite it is
usual to find just one such variant, with the habit plane inclined at some 60◦ to the
plate axis. The problem is discussed in detail in [9]. Cementite which forms from
austenite usually exhibits the Pitsch [10] orientation relation with [0 0 1]θ ∥ [2 2 5]γ
and [1 0 0]θ ∥ [5 5 4]γ and a mechanism which involves the intermediate formation
of ferrite has been postulated [4] to explain this orientation relationship.

1 The structure of cementite is described on page 68 but we will assume in what follows that the lattice

parameters increase in the order a > b > c.

9.2 Cementite in steels 131

Example 9.1: Bagaryatski orientation relationship

Cementite (θ) has an orthorhombic crystal structure, with lattice parameters
a = 4.5241, b = 5.0883 and c = 6.7416 Å along the [1 0 0], [0 1 0] and [0 0 1]
directions respectively. This differs from the convention used on page 68, but is
equally valid although we note that the space group symbol for the structure now
becomes P bnm instead of P nma (the original solution of the crystal structure
of cementite by Lipson and Petch used the convention described here [11]). The
reason some prefer the convention that leads to the space group P nma may have
something to do with the fact that it is the most abundant space group of known
inorganic crystals and minerals [12].
When cementite precipitates from ferrite (α, bcc structure, lattice parameter
aα = 2.8662 Å), the lattices are related by the Bagaryatski orientation relation-
ship, given by:

[1 0 0]θ ∥ [0 1 1]α , [0 1 0]θ ∥ [1 1 1]α , [0 0 1]θ ∥ [2 1 1]α (9.1)

(i) Derive the coordinate transformation matrix (α J θ) representing this orienta-

tion relationship, given that the basis vectors of θ and α define the orthorhom-
bic and bcc unit cells of the cementite and ferrite, respectively.
(ii) Published stereograms of this orientation relation have the (0 2 3)θ plane
exactly parallel
√ to the (1 3 3)α plane. Show that this can only be correct if the
ratio b/c = 8 2/15.

The information given indicates parallelisms between vectors in the two lat-
tices. In order to find (α J θ), it is necessary to ensure that the magnitudes of
the parallel vectors are also equal, since the magnitude of a vector must remain
invariant to a coordinate transformation. If the constants k, g and m are defined as
|[1 0 0]θ | a√
k= |[0 1 1]α |
= aα 2
= 1.116120
|[0 1 0]θ | b√
g= |[1 1 1]α |
= aα 3
= 1.024957 (9.2)
|[0 0 1]θ | c√
m= |[2 1 1]α | = aα 6
= 0.960242

then multiplying [0 1 1]α by k makes its magnitude equal to that of [1 0 0]θ ; the
constants g and m can similarly be used for the other two α vectors.
Recalling now the definition of a coordinate transformation matrix, each col-
umn of (α J θ) represents the components of a basis vector of θ in the α basis.
For example, the first column of (α J θ) consists of the components of [1 0 0]θ in
the α basis, [0 k k]α . It follows that (α J θ) can be derived simply by inspection
of the relations 9.1, 9.2, so that
⎛ ⎞
0.000000 1.024957 1.920485
⎜ ⎟
(α J θ) = ⎜ ⎝−1.116120 −1.024957 0.960242⎠
⎟ (9.3)
1.116120 −1.024957 0.960242
132 9 Orientation relations
The transformation matrix can therefore be deduced by inspection when the
orientation relationship 9.1 is stated in terms of the basis vectors of one of the
crystals concerned (in this case, the basis vectors of θ are specified in relation-
ship 9.1). On the other hand, orientation relationships can, and often are, specified
in terms of vectors other than the basis vectors. Furthermore, electron diffraction
patterns may not include direct information about basis vectors. A more general
method of solving for (α J θ) is now presented, a method independent of the
vectors used in specifying the orientation relationship:
From the relations 9.1, 9.2 we see that
6 7
0 k k α = (α J θ)[1 0 0]θ
6 7
g g g α = (α J θ)[0 1 0]θ
6 7
2m m m α = (α J θ)[0 0 1]θ . (9.4)

These equations can written as:

⎛ ⎞ ⎛ ⎞⎛ ⎞
0 g 2m J J12 J13 1 0 0
⎜ ⎟ ⎜ 11 ⎟⎜ ⎟
⎜ ⎟=⎜ ⎟⎜ ⎟ (9.5)
⎝k g m ⎠ ⎝J21 J22 J23 ⎠ ⎝0 1 0⎠
k g m J31 J32 J33 0 0 1

where the Jij (i = 1, 2, 3 & j = 1, 2, 3) are the elements of the matrix (α J θ).
From equation 9.5, it follows that

⎛ ⎞ ⎛ ⎞
0 g 2m 0 1.024957 1.920485
⎜ ⎟ ⎜ ⎟
(α J θ) = ⎜
⎝k g m ⎠ = ⎝−1.116120 −1.024957 0.960242⎠ . (9.6)
⎟ ⎜ ⎟

k g m 1.116120 −1.024957 0.960242

It is possible to accumulate rounding off errors in such calculations so the use

at least six figures after the decimal point is advisable.
To consider the relationships between plane normals (rather than directions)
in the two lattices, we have to discover how vectors representing plane normals,
(always referred to a reciprocal basis) transform. From equation 5.8, if u is any
vector,

|u|2 = (u; α∗ )[α; u] = (u; θ∗ )[θ; u]

or (u; α∗ )(α J θ)[θ; u] = (u; θ∗ )[θ; u]
giving (u; α∗ )(α J θ) = (u; θ∗ ) (9.7)
∗ ∗
(u; α ) = (u; θ )(θ J α)
where (θ J α) = (α J θ)−1 .
9.2 Cementite in steels 133

⎛ ⎞
0 −3gm 3gm
1 ⎜ ⎟
(θ J α) = ⎜
2mk −2mk −2mk
⎟
6gmk ⎝ ⎠
2gk gk gk
⎛ ⎞
0 −0.447981 0.447981
⎜ ⎟
= ⎜
⎝0.325217 −0.325217 −0.325217⎠
⎟

0.347135 0.173567 0.173567

If (u; θ∗ ) = (0 2 3) is now substituted into equation 9.8, the corresponding vector

1
(u; α∗ ) = (6gk − 4mk 3gk + 4mk 3gk + 4mk).
6gmk
For this to be parallel to a (1 3 3) plane normal in the ferrite, the second and third
components must equal three times√ the first; i.e. 3(6gk − 4mk) = (3gk + 4mk),
which is equivalent to b/c = 8 2/15, as required.

It should be noted that the determinant of (α J θ) gives the ratio of the volume
of the θ unit cell to that of the ferrite α unit cell. If the coordinate transformation
simply involves a rotation of bases, e.g. when it describes the relation between two
grains of identical structure, then the matrix is orthogonal and its determinant has a
value of unity for all proper rotations (i.e. not involving inversion operations). Such
coordinate transformation matrices are called rotation matrices.
A stereographic representation of the Bagaryatski orientation is presented in
Fig. 9.1, to visualise the angular relationships between poles (plane normals) of crys-
tal planes and give some indication of symmetry; the picture is of course distorted
because distance on the stereogram does not scale directly with angle. Angular mea-
surements on stereograms are in general made using Wulff nets and may typically
be in error by a few degrees, depending on the size of the stereogram. Space and
aesthetic considerations usually limit the number of poles plotted on stereograms,
and those plotted usually have rational indices. Separate plots are needed for cases
where directions and plane normals of the same indices have a different orientation
in space. A coordinate transformation matrix is, by comparison, a precise way of rep-
resenting orientation relationships; angles between any plane normals or directions
can be calculated to any desired degree of accuracy and information on both plane
normals and directions can be derived from just one matrix. With a little nurturing,
it is possible to picture the meaning of the elements of a coordinate transformation
matrix: each column of (α J θ) represents the components of a basis vector of θ in
the basis α, and the determinant of this matrix gives the ratio of volumes of the two
unit cells.
134 9 Orientation relations

(a)

(b) (c)

FIGURE 9.1
(a) Stereographic representation of the Bagaryatski orientation relationship between
ferrite and cementite in steels, where [1 0 0]θ ∥ [0 1 1]α , [0 1 0]θ ∥ [1 1 1]α , and
[0 0 1]θ ∥ [2 1 1]α . All the poles plotted are in the upper hemisphere. (b) Typical
electron diffraction pattern consistent with the Bagaryatski orientation relation with
[1 0 0]θ ∥ [0 1 1]α normal to the plane of the pattern. (c) An actual electron diffraction
pattern corresponding to (b).
9.3 Relations between fcc and bcc crystals 135

9.3 Relations between fcc and bcc crystals

The ratio of the lattice parameters of austenite and ferrite in steels is about 1.24,
and there are several other alloys (e.g. Cu-Zn brasses, Cu-Cr alloys) where fcc and
bcc precipitates of similar lattice parameter ratios coexist. The orientation relations
between these phases vary within a well defined range, but it is usually the case
that a close-packed {1 1 1}fcc plane is approximately parallel to a {0 1 1}bcc plane
(Fig. 9.2). Within these planes, there can be a significant variation in orientation,
with ⟨1 0 1⟩fcc ∥ ⟨1 1 1⟩bcc for the Kurdjumov-Sachs orientation relation [13],
and ⟨1 0 1⟩fcc about 5.3◦ towards ⟨1 1 1⟩bcc for the Nishiyama-Wasserman relation
[14, 15]. It is experimentally difficult to distinguish between these relations using
ordinary electron diffraction or X-ray techniques, but accurate work, such as that
of Crosky et al. [16], shows a spread of fcc-bcc orientation relationships (roughly
near the Kurdjumov-Sachs and Nishiyama-Wasserman orientations) within the same
alloy. When discussing these small variations in the orientation relationship, it is im-
portant to realise that the samples studied are at an advanced stage of transformation.
The state of the austenite actually changes as transformation evolves, particularly
with respect to local plastic strains. As a result, the reference frame from which the
martensite forms is no longer identical to that of the undeformed austenite, which
may be the real cause of the spread in orientations. Ideally, a comparison needs to be
made of the observed orientation relationship as a function of the number of marten-
site plates per unit volume. Example 9.3 deals with the exact Kurdjumov-Sachs ori-
entation relationship.
Example 9.2: Kurdjumov-Sachs orientation relationship
Two plates of Widmanstätten ferrite (α ferrite, basis symbols X and Y respec-
tively) growing in the same grain of austenite (basis symbol γ) are found to ex-
hibit two different variants of the Kurdjumov-Sachs orientation relationship with
the austenite; the data below show the sets of parallel vectors of the three lattices:

[1 1 1]γ ∥ [0 1 1]X [1 1 1]γ ∥ [0 1 1]Y

[1 0 1]γ ∥ [1 1 1]X [1 0 1]γ ∥ [1 1 1]Y .
[1 2 1]γ ∥ [2 1 1]X [1 2 1]γ ∥ [2 1 1]Y
Derive the matrices (X J γ) and (Y J γ). Hence obtain the rotation matrix
(X J Y) describing the orientation relationship between the two Widmanstätten
ferrite plates.
The information given relates to parallelisms between vectors in different lat-
tices. It is necessary to equalise the magnitudes of parallel vectors in order to solve
for the various coordinate transformation matrices. Defining the constants k, g and
m as √ √ √
aγ 3 aγ 2 6aγ aγ
k= √ g= √ m= √ = ,
aα 2 aα 3 6aα a α
136 9 Orientation relations
we obtain:
[0 k k]X = (X J γ)[1 1 1]γ
[g g g]X = (X J γ)[1 0 1]γ
[2m m m]X = (X J γ)[1 2 1]γ
or in other words,
⎛ ⎞ ⎛ ⎞⎛ ⎞
0 g 2m J11 J12 J13 1 1 1
⎜ ⎟ ⎜ ⎟⎜ ⎟
m ⎠ = ⎝J21
⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝k g J22 J23 ⎠ ⎝1 0 2⎠
k g m J31 J32 J33 1 1 1

⎛ ⎞⎛ ⎞
0 g 2m 2/6 2/6 2/6
⎜ ⎟⎜ ⎟
(X J γ) = ⎜
⎝k g
⎟⎜
m ⎠ ⎝3/6 0 3/6⎠
⎟

k g m 1/6 2/6 1/6

⎛ ⎞
3g + 2m 4m 3g + 2m
1⎜ ⎟
= ⎜
2k + 3g − m 2k + 2m 2k − 3g − m
⎟
6 ⎝ ⎠
2k − 3g + m 2k − 2m 2k + 3g + m
⎛ ⎞
0.741582 −0.666667 −0.074915
aγ ⎜ ⎟
= ⎜
0.649830 0.741582 −0.166667
⎟.
aα ⎝ ⎠
0.166667 0.074915 0.983163

In an similar manner,
⎛ ⎞
0.741582 −0.666667 0.074915
aγ ⎜ ⎟
(Y J γ) = ⎜
−0.983163⎠ .
⎟ (9.8)
aα ⎝0.166667 0.074915
0.649830 0.741582 0.166667

To find the rotation matrix relating X and Y, we proceed as follows:

[X; u] = (X J γ)[γ; u] and [Y ; u] = (Y J γ)[γ; u] and [X; u] = (X J Y)[Y; u]

it follows that
(X J γ)[γ; u] = (X J Y)[Y; u]
substituting for [Y;u], we get

(X J γ)[γ; u] = (X J Y)(Y J γ)[γ; u]

so that
(X J Y) = (X J γ)(γ J Y)
9.3 Relations between fcc and bcc crystals 137
carrying out this operation, the required X-Y orientation relation is obtained as
follows:
⎛ ⎞
0.988776 0.147308 −0.024972
⎜ ⎟
(X J Y) = ⎜ ⎝−0.024972 0.327722 0.944445 ⎠ .
⎟

0.147308 −0.933219 0.327722

We see that the matrix (X J Y) is orthogonal because it represents an axis trans-

formation between like orthogonal bases. In fact, (X J γ) and (Y J γ) each equal an
orthogonal matrix times a scalar factor aγ /aα ; this is because the bases X, Y and γ
are themselves orthogonal.
In this example, we chose to represent the Kurdjumov-Sachs orientation relation-
ship by a coordinate transformation matrix (X J γ). Named orientation relationships
like this usually assume the parallelism of particular low index planes and directions.
In the example, these parallelisms are independent of the lattice parameters of the
fcc and bcc structures concerned. In such cases, the orientation relationship may be
represented by a pure rotation matrix, relating the orthogonal, but not necessarily
orthonormal, bases of the two structures. Orientation relationships are indeed often
specified in this way, or in some equivalent manner such as an axis-angle pair or a
set of three Euler angles. This provides an economic way of representing orientation
relations, but it should be emphasised that there is a loss of information in doing this.
For example, a coordinate transformation matrix like (X J γ) not only gives informa-
tion about vectors which are parallel, but also gives a ratio of the volumes of the two
unit cells. Axis-angle pairs and Euler angles assume orthonormal axes.
138 9 Orientation relations

(a)

(b)

FIGURE 9.2
(a) Stereographic representation of the Kurdjumov-Sachs orientation relationship.
Note that the positions of the base vectors of the γ lattice are consistent with the ma-
trix (X J γ) derived in example 9.3, equation 9.8, where each column of (X J γ) rep-
resents the components of a basis vector of γ in the basis X, so that [1 0 0]γ , [0 1 0]γ
and [0 0 1]γ are approximately parallel to [1 1 0]α , [1 1 0]α and [0 0 1]α respec-
tively, as seen in the stereographic representation. (b) Stereographic representation of
the Nishiyama-Wasserman orientation relationship which can be generated from the
Kurdjumov-Sachs orientation by a rotation of 5.26◦ about [0 1 1]α . The necessary ro-
tation makes [1 1 2]γ exactly parallel to [0 1 1]α . The Nishiyama-Wasserman relation
is midway between the two variants of the Kurdjumov-Sachs relation which share the
same parallel close-packed plane. The stereograms also show that the Kurdjumov-
Sachs and Nishiyama-Wasserman orientation relationships do not differ much from
the γ/α orientation relationship implied by the Bain strain illustrated in Fig. 8.2.
9.4 Relationships between grains of identical structure 139

9.4 Relationships between grains of identical structure

The relationship between two crystals which are of identical structure but which are
misoriented with respect to each other is described in terms of a rotation matrix rep-
resenting the rigid body rotation which can be imagined to generate one crystal from
the other. As discussed below, any rotation of this kind, which leaves the common
origin of the two crystals fixed, can also be described in terms of a rotation of 180◦
or less about an axis passing through that origin.
Example 9.3: Axis-angle pairs and rotation matrices
Two ferrite grains X and Y can be related by a rotation matrix
⎛ ⎞
2 2 1
1⎜ ⎟
(Y J X) = ⎜ 1 2 2
⎟
3 ⎝ ⎠
2 1 2

where the basis vectors of X and Y define the respective bcc unit cells. Show that
the crystal Y can be generated from X by a right-handed rotation of 60◦ about an
axis parallel to the [1 1 1]X direction.
A rigid body rotation leaves the magnitudes and relative positions of all vec-
tors in that body unchanged. For example, an initial vector u with components
[u1 u2 u3 ]X relative to the X basis, due to rotation becomes a new vector x, with
the same components [u1 u2 u3 ]Y , but with respect to the rotated basis Y. The fact
that x represents a different direction than u (due to the rotation operation) means
that its components in the X basis, [w1 w2 w3 ]X must differ from [u1 u2 u3 ]X .
The components of x in either basis are obviously related by

[Y; x] = (Y J X)[X; x]
in other words,
[u1 u2 u3 ] = (Y J X)[w1 w2 w3 ]. (9.9)
However, if u happens to lie along the axis of rotation relating X and Y, then
not only will [X;u] = [Y;x] as before, but its direction also remains invariant to
the rotation operation, so that [X;x] = [Y;x] and

(Y J X)[X; x] = [Y; x]

so that
(Y J X)[X; u] = [X; u]
and hence
{(Y J X) − I}[X; u] = 0 (9.10)
140 9 Orientation relations
where I is a 3 × 3 identity matrix. Any rotation axis must satisfy an equation of
this form; expanding equation 9.10, we get
1 2 1
− u1 + u2 − u3 =0
3 3 3
1 1 2
− u1 − u2 + u3 =0
3 3 3
2 1 1
u1 − u2 − u3 = 0.
3 3 3
Solving these simultaneously gives u1 = u2 = u3 , proving that the required
rotation axis lies along the [1 1 1]X direction, which also is the [1 1 1]Y direction.
The angle, and sense of rotation can be determined by examining a vector
v which lies at 90◦ to u. If, say, v = [1 0 1]X , then as a result of the rotation
operation it becomes z = [1 0 1]Y = [0 1 1]X , making an angle of 60◦ with v,
giving the required angle of rotation. Since v ∧z gives [1 1 1]X , it is also a rotation
in the right-handed sense.

9.4.1 Comments
(i) The problem illustrates the fact that the orientation relation between two grains
can be represented by a matrix such as (Y J X), or by an axis-angle pair such as
[1 1 1]X and 60◦ . Obviously, the often used practice of stating a misorientation
between grains in terms of just an angle of rotation is inadequate and incorrect.
(ii) The Euler theorem for rigid rotations [17] states that “regardless of the way a
coordinate system is rotated from its original orientation, it always is possible to
find a fixed axis in space about which a single rotation of the initial coordinates
ends at the final orientation” (the original theorem is in Latin - the English quote
is from [18]). If the axis of rotation is constrained to be a unit vector, then only
three independent quantities are needed to define a misorientation between grains:
two components of the axis of rotation, and an angle of rotation. It follows that
a rotation matrix must also have only three independent terms. The relationship
between the axis-angle pair and rotation matrix is given in equation 7.9.
From the definition of a coordinate transformation matrix, each column of (Y J X)
gives the components of a basis vector of X in the basis Y. It follows that
[1 0 0]X ∥ [2 1 2]Y [0 1 0]X ∥ [2 2 1]Y [0 0 1]X ∥ [1 2 2 ]Y .
Suppose now that there exists another ferrite crystal (basis symbol Z), such that
[0 1 0]Z ∥ [2 1 2]Y [1 0 0]Z ∥ [2 2 1]Y [0 0 1]Z ∥ [1 2 2]Y
⎛ ⎞
2 2 1
1⎜ ⎟
(Y J Z) = ⎝2 1
⎜ ⎟
2⎠
3
1 2 2
9.4 Relationships between grains of identical structure 141

with the crystal Y being generated from Z by a right-handed rotation of 70.52◦ about
[1 0 1]Z direction. It can easily be demonstrated that
⎛ ⎞
0 1 0
⎜ ⎟
(Z J X) = ⎜ ⎝1 0 0⎠
⎟ from (Z J X) = (Z J Y)(Y J X)
0 0 1
so that crystal X can be generated from Z by a rotation of 90◦ about [0 0 1]X axis.
However, this clearly is a symmetry operation of a cubic crystal, and it follows that
crystal X can never be distinguished experimentally from crystal Z, so that the matri-
ces (Y J X) and (Y J Z) are crystallographically equivalent, as are the corresponding
axis-angle pair descriptions. In other words, Y can be generated from X either by a
rotation of 60◦ about [1 1 1]X , or by a rotation of 70.52◦ about [1 0 1]X . The two axis-
angle pair representations are equivalent. There are actually 24 matrices like (Z J X)
which represent symmetry rotations in cubic systems. It follows that a cubic bicrys-
tal can be represented in 24 equivalent ways, with 24 axis-angle pairs. Any rotation
matrix like (Y J X) when multiplied by rotation matrices representing symmetry op-
erations (e.g. (Z J X)) will lead to the 24 axis-angle pair representations. The degen-
eracy of other structures is as follows [19, 20]: cubic (24), hexagonal (12), hexagonal
close-packed (6), tetragonal (8), trigonal (6), orthorhombic (4), monoclinic (2) and
triclinic (1). In general, the number N of axis-angle pairs is given by
N = 1 + N2 + 2N3 + 3N4 + 5N6
where N2 , N3 , N4 and N6 refer to the number of diads, triads, tetrads and hexads in
the symmetry elements of the structure concerned.
Fig. 9.3 is an electron diffraction pattern taken from an internally twinned marten-
site plate in a Fe-4Ni-0.4C wt% steel. It contains two ⟨0 1 1⟩ bcc zones, one from
the parent plate (m) and the other from the twin (t). The pattern clearly illustrates
how symmetry makes it possible to represent the same bi-crystal in terms of more
than one axis-angle pair. This particular pattern shows that the twin crystal can be
generated from the parent in at least three different ways: (i) rotation of 70.52◦ about
the ⟨0 1 1⟩ zone axes of the patterns, (ii) rotation of 180◦ about the {1 1 1} plane
normal, and (iii) rotation of 180◦ about the {2 1 1} plane normal. It is apparent that
these three operations would lead to the same orientation relation between the twin
and the parent lattices.
Example 9.4: ‘Double’ twinning
Plates of bcc martensite in Fe-30.4Ni wt% contain {1 1 2} transformation
twins, the two twin orientations X and Y being related by a rotation of 60◦ about
a ⟨1 1 1⟩ axis. Deformation of this martensite at low temperatures leads to the
formation of twins on {5 8 11} planes, the direction of the twinning shear being
⟨5 1 3⟩. This is a very rare mode of twinning deformation in bcc systems; show
that its occurrence may be related to the fact that such twins can propagate without
any deviation, across the already existing transformation twins in the martensite
[21].
142 9 Orientation relations

(a) (b)

FIGURE 9.3
(a) Electron diffraction pattern from a martensite plate (m) and its twin (t). Spots not
connected by lines (e.g. “dd”) arise from double diffraction. (b) Interpretation of the
diffraction pattern.

The orientation relationship between the transformation twins is clearly the

same as the matrix (Y J X) of Example 9.4. It is easy to show therefore, that:

[5 1 3]X ∥ ⟨5 3 1⟩Y

(5 8 11)X ∥ {5 11 8}Y .
It follows that {5 8 11} deformation twins can propagate without deviation across
the transformation twins, since the above planes and directions, respectively, are
crystallographically equivalent and indeed parallel. This may explain the occur-
rence of such an unusual deformation twinning mode.

9.5 The metric

For cubic crystals, it is a familiar result that if the indices [u1 u2 u3 ] of a direction
u in the lattice are numerically identical to the Miller indices (h1 h2 h3 ) of a plane
in the lattice, then the normal to this plane (h) is parallel to the direction mentioned.
In other words, u ∥ h, and since [A; u] = [u1 u2 u3 ] and (h; A∗ ) = (h1 h2 h3 ), we
have [u1 u2 u3 ] = [h1 h2 h3 ]. (‘A’ represents the basis of the cubic system concerned,
and ‘A∗ ’ the corresponding reciprocal basis, in the usual way).
However, this is a special case reflecting the high symmetry of cubic systems,
and it is generally not true that if ui = hi , then u ∥ h. For example, the [1 2 3]
direction in cementite is not parallel to the (1 2 3) plane normal.
Consider an arbitrary crystal system, defined by a basis A (basis vectors ai ),
and by the corresponding reciprocal basis A∗ consisting of the basis vectors a∗i . To
find the angle between the direction u and the plane normal h, it would be useful
to have a matrix (A∗ G A), which allows the transformation of the components of
9.5 The metric 143

a vector referred to the basis A, to those referred to its reciprocal basis A∗ . (The
symbol G is used, rather than J, to follow convention). This matrix, called the metric,
with components Gij can be determined in exactly the same manner as any other
coordinate transformation matrix. Each column of (A∗ G A) thus consists of the
components of one of the basis vectors of A, when referred to the basis A∗ . For
example,
a1 = G11 a∗1 + G21 a∗2 + G31 a∗3
Taking successive scalar dot products with a1 , a2 and a3 respectively on both sides
yields:
G11 = a1 .a1 , G21 = a1 .a2 G31 = a1 .a3
since ai .a∗j = 0 when i ̸= j. The rest of the elements of (A∗ G A) can be determined
in a similar way, so that
⎛ ⎞
a1 .a1 a2 .a1 a3 .a1
⎜ ⎟
(A∗ G A) = ⎜ ⎝a1 .a2 a2 .a2 a3 .a2 ⎠ .
⎟ (9.11)
a1 .a3 a2 .a3 a3 .a3
It is easily demonstrated that the determinant of (A∗ G A) equals the square of the
volume of the cell formed by the basis vectors of A. We note in passing that for
orthonormal coordinates, (Z∗ G Z) =I. A list of metric tensors is given in Table 9.1.
Example 9.5: Plane normals and directions in an orthorhombic structure
A crystal with an orthorhombic structure has lattice parameters a, b and c. If
the edges of the orthorhombic unit cell are taken to define the basis θ, determine
the metric (θ∗ G θ). Hence derive the equation giving the angle φ between a plane
normal (h; θ∗ ) = (h1 h2 h3 ) and any direction [θ; u] = [u1 u2 u3 ].
From the definition of a scalar dot product, h.u/|h||u| = cos φ. Now,
⎛ ⎞ ⎛ ⎞
a2 0 0 a−2 0 0
⎜ ⎟ ⎜ ⎟
(θ∗ G θ) = ⎜⎝ 0 b2 0 ⎠
⎟ (θ G θ∗ ) = ⎜
⎝ 0 b−2 0 ⎠.
⎟

0 0 c2 0 0 c−2

From equation 5.8,

|h|2 = h.h = (h; θ∗ )[θ; h]

= (h; θ∗ )(θ G θ∗ )[θ∗ ; h]
= h21 /a2 + h22 /b2 + h23 /c2 .

Similarly,

|u|2 = u.u = (u; θ)[θ∗ ; u]

= (u; θ)(θ∗ G θ)[θ; u]
= u1 a2 + u2 b2 + u23 c2
144 9 Orientation relations
Since
h.u = (h; θ∗ )[θ; u] = h1 u1 + h2 u2 + h3 u3
it follows that
(h1 u1 + h2 u2 + h3 u3 )
cos φ = - 2 .
(h1 / a2 + h22 /b2 + h23 /c2 )(u1 a2 + u2 b2 + u23 c2 )
 9.5 The metric

TABLE 9.1
Elements of metric tensor (A∗ G A) and its inverse. The term φ = (−2 cos{β} cos{α} cos{γ} + cos2 {β} + cos2 {α} + cos2 {γ} − 1)−1

(A∗ G A)
Class G11 G12 G13 G21 G22 G23 G31 G32 G33

Cubic a2 0 0 0 a2 0 0 0 a2
Tetragonal a2 0 0 0 a2 0 0 0 c2
2 2
Orthorhombic a 0 0 0 b 0 0 0 c2
Hexagonal a2 −0.5a2 0 −0.5a2 a2 0 0 0 c2
Monoclinic a2 0 ac cos{β} 0 b2 0 ac cos{β} 0 c2
Triclinic a2 ab cos{γ} ac cos{β} ab cos{γ} b2 bc cos{α} ac cos{β} bc cos{α} c2

(A G A ) ∗

Class G11 G12 G13 G21 G22 G23 G31 G32 G33

Cubic a −2
0 0 0 a −2
0 0 0 a−2
Tetragonal a−2 0 0 0 a−2 0 0 0 c−2
Orthorhombic a−2 0 0 0 b−2 0 0 0 c−2
b2 2 2 4
Hexagonal a2 b2 −0.25a4 − a2 −4b2 0 − a2 −4b2 − a2 −4b2 0 0 0 c−2
csc2 {β} csc2 {β}
Monoclinic a2 0 0 b−2 0 0 c2
2 2 2
− cot{β}accsc{β} − cot{β}accsc{β}
cos{γ} cos{γ} cos{γ} cos{γ}
Triclinic 2
−φ sina{α} ab
−φ cos{β} cos{α}−cos{γ} φ cos{β}−cos{α}
ac ab
−φ cos{β} cos{α}−cos{γ} −φ sinb2{β} φ cos{α}−cos{β}
bc φ cos{β}−cos{α}
ac φ cos{α}−cos{β}
bc −φ sinc2{γ}
145
146 9 Orientation relations

9.6 More about the vector cross product

Suppose that the basis vectors a, b and c of the basis θ define an orthorhombic unit
cell, then the cross product between two arbitrary vectors u and v referred to this
basis may be written:

u ∧ v = (u1 a + u2 b + u3 c) ∧ (v1 a + v2 b + v3 c)

where [θ; u] = [u1 u2 u3 ] and [θ; v] = [v1 v2 v3 ]. This equation can be expanded to
give:

u ∧ v = (u2 v3 − u3 v2 )(b ∧ c) + (u3 v1 − u1 v3 )(c ∧ a) + (u1 v2 − u2 v1 )(a ∧ b)

Since a.b∧c = V , the volume of the orthorhombic unit cell, and since b∧c = V a∗ ,
it follows that
8 9
∗ ∗ ∗
u ∧ v = V (u2 v3 − u3 v2 )a + (u3 v1 − u1 v3 )b + (u1 v2 − u2 v1 )c .

Hence, u ∧ v gives a vector whose components are expressed in the reciprocal basis.
Writing x = u ∧ v, with (x; θ∗ ) = (w1 w2 w3 ), it follows that w1 = V (u2 v3 −
u3 v2 ), w2 = V (u3 v1 − u1 v3 ) and w3 = V (u1 v2 − u2 v1 ). The cross product of two
directions thus gives a normal to the plane containing the two directions. If necessary,
the components of x in the basis θ can easily be obtained using the metric, since
[θ; x] = (θ G θ∗ )[θ∗ ; x]. Similarly, the cross product of two vectors h and k which
are referred to the reciprocal basis θ∗ , such that (h; θ∗ ) = (h1 h2 h3 ) and (k; θ∗ ) =
(k1 k2 k3 ), can be shown to be:
8 9
1
h∧k = (h2 k3 − h3 k2 )a + (h3 k1 − h1 k3 )b − (h2 k1 − h1 k2 )c .
V

Hence, h ∧ k gives a vector whose components are expressed in the real basis. The
vector cross product of two plane normals gives a direction (zone axis) which is
common to the two planes represented by the plane normals.

9.7 Summary
The term ‘orientation relationship’ has more than one meaning. When the relation-
ship is reproducible, for example when precipitates form in a solid matrix away from
heterogeneous nucleation sites, the strong implication is that there is something that
favours that behaviour. The reasons could include:
1. The reduction in interfacial energy between the precipitate and matrix
9.7 Summary 147

when a specific relationship is adopted. This facilitates nucleation because

the activation energy for nucleation scales with the cube of the interfacial
energy per unit area.
2. That the nucleus is generated by a homogeneous deformation of the ma-
trix, in which case the deformation that minimises the strain energy will
determine the observed orientation relationship.
The second kind of relationship is not reproducible but arises because deformation or
other external influences induce the alignment of crystals in a polycrystalline sam-
ple. In this case, the average orientation and the (different) relative orientations of
adjacent crystals matter in determining the anisotropy and macroscopic properties of
the material as a whole. This subject falls in the domain of crystallographic texture
as introduced in Chapter 6, but in geology a different descriptor is used: fabric. Ice
at ambient pressure has a hexagonal structure with the resistance to slip being some
60 times larger on non-basal planes [22]. Initially isotropic polycrystalline ice then
becomes anisotropic when fabric develops because the basal slip-planes of differ-
ent crystals tend to align during flow [23]. A similar effect happens when hexagonal
titanium alloys are friction stir welded [24].
References

1. P. L. Ryder, and W. Pitsch: ‘The crystallographic analysis of grain boundary

precipitation’, Acta Metallurgica, 1966, 14, 1437–1448.
2. P. L. Ryder, W. Pitsch, and R. F. Mehl: ‘Crystallography of the precipitation of
ferrite on austenite grain boundaries in a Co-20%Fe alloy’, Acta Metallurgica,
1967, 15, 1431–1440.
3. J. W. Christian: ‘Martensitic transformations: a current assessment’, In: The
Mechanism of Phase Transformations in Crystalline Solids, Monography 33.
London, U.K.: Institute of Metals, 1969:129–142.
4. K. W. Andrews: ‘The structure of cementite and its relation to ferrite’, Acta
Metallurgica, 1963, 11, 939–946.
5. H. K. D. H. Bhadeshia: Bainite in steels: theory and practice: 3rd ed., Leeds,
U.K.: Maney Publishing, 2015.
6. A. I. Tyshchenko, W. Theisen, A. Oppenkowski, S. Siebert, O. N. Razumov,
A. P. Skoblik, V. A. Sirosh, Y. N. Petrov, and V. G. Gavriljuk: ‘Low-temperature
martensitic transformation and deep cryogenic treatment of a tool steel’, Mate-
rials Science & Engineering A, 2010, 527, 7027–7039.
7. W. Hume-Rothery, G. V. Raynor, and A. T. Little: ‘On the carbide and nitride
particles in titanium steels’, Journal of the Iron and Steel Institute, 1942, 145,
129–141.
8. D. N. Shackleton, and P. M. Kelly: ‘The crystallography of cementite precipi-
tation in the bainite tranformation’, Acta Metallurgica, 1967, 15, 979–992.
9. H. K. D. H. Bhadeshia: ‘The lower bainite transformation and the significance
of carbide precipitation’, Acta Metallurgica, 1980, 28, 1103–1114.
10. W. Pitsch: ‘Der orientierungszusammenhang zwischen zementit und austenit’,
Acta Metallurgica, 1962, 10, 897–900.
11. H. Lipson, and N. J. Petch: ‘The crystal structure of cementite Fe3 C’, Journal
of the Iron and Steel Institute, 1940, 142, 95P–103P.
12. V. S. Urusov, and T. N. Nadezhina: ‘Frequency distribution and selection of
space groups in inorganic crystal chemistry’, Journal of Structural Chemistry,
2009, 50, S22–S37.

149
150 9 References

13. G. V. Kurdjumov, and G. Sachs: ‘über den mechanismus der stahlhärtung’,

Zietschrift für Physik A Hadrons and Nuclei, 1930, 64, 325–343.
14. Z. Nishiyama: ‘X-ray investigation of the mechanism of the transformation
from face centered cubic lattice to body centered cubic’, Science Reports of
Tohoku Imperial University, 1934, 23, 637–634.
15. G. Wassermann: ‘Einflusse der α-γ-umwandlung eines irreversiblen nickel-
stahls auf kristallorientierung und zugfestigkeit’, Arch. Eisenhüttenwes., 1933,
6, 347–351.
16. A. Crosky, P. G. McDougall, and J. S. Bowles: ‘The crystallography of the
precipitation of alpha rods from beta Cu Zn alloys’, Acta Metllurgica, 1980,
28, 1495–1504.
17. L. Euler: ‘Formulae generales pro trandlatione quacunque corporum rigi-
dorum’, Novi commentarii Academiae Scientiarum Imperialis Petropolitanae,
1975, 20, 189–207.
18. B. Palais, and R. Palais: ‘Euler’s fixed point theorem: The axis of a rotation’,
Journal of Fixed Point Theory and Applications, 2007, 2, 215–220.
19. F. F. Lange: ‘Mathematical characterization of a general bicrystal’, Acta Met-
allurgica, 1967, 15, 311–318.
20. C. Goux: ‘Structure of joined grains-crystallographic consideration and meth-
ods of structure estimation’, Canadian Metallurgical Quarterly, 1974, 13, 9–
31.
21. P. C. Rowlands, E. O. Fearon, and M. Bevis: ‘Deformation twinning in Fe-Ni
and Fe-Ni-C martensites’, Trans. A.I.M.E., 1968, 242, 1559–1562.
22. P. Duval, M. F. Ashby, and I. Andereman: ‘Rate-controlling processes in the
creep of polycrystalline ice’, The Journal of Physical Chemistry, 1983, 87,
4066–4074.

23. O. Castelnau, and P. Duval: ‘Viscoplastic modelling of texture development

in polycrystalline ice with a self-consistent approach: comparison with bound
estimates’, Journal of Geophysical Research, 1996, 101, 13851–13868.
24. R. Fonda: ‘Texture development in friction stir welds’, Science and Technology
of Welding and Joining, 2011, 16, 288–294.
10
Homogeneous deformations

Abstract

Many operations in crystallography involve physical deformations that lead to an

observable change in shape of the affected region of the crystal. Such deformations
can in general be factorised mathematically into components that involve only a rigid
body rotation, and another that is a pure strain which involves the stretching of the
body along three orthogonal directions. The sign and magnitude of the stretch can
vary according to the nature of the deformation. Homogeneous deformations help
represent changes in crystal structure or the reorientation of the crystal. They can
also be used to define the nature of interfaces between phases.

10.1 Introduction
A homogeneous deformation is one in which points that initially are collinear remain
so after the deformation. It follows that a plane will similarly remain a plane in
such circumstances. Each volume element in the solid therefore experiences the same
change in shape. Fig. 10.1 illustrates the difference between a homogeneous and
heterogeneous deformation. In this chapter, we describe deformations that alter the
length and/or direction of vectors, i.e., physical deformations, with the initial and
resultant vectors referred to a fixed reference frame.
Elastic strain energy calculations often are based on the assumption that the de-
formations are homogeneous [1] and this usually is justified given that the elastic
strain fields may extend over large distances whereas atomic perturbations are just
that. Similarly, the stability of crystal lattices has been studied by subjecting them
to homogeneous deformation along the direction of one of its axes, and to another
small deformation along arbitrary direction [2].

151
152 10 Homogeneous deformations

n
atio

initial

ation

FIGURE 10.1
An illustration of the difference between homogeneous deformations and those that
are heterogeneous. Examples of the former include shear, hydrostatic expansion and
uniaxial expansion, whereas slip caused by the motion of a dislocation, or the for-
mation of a crack, or the motion of rigid continental plates, involve heterogeneous
deformation.

10.2 Homogeneous deformations

We can now return to the question of martensite, and how the homogeneous defor-
mation known as the Bain strain [3] might transform the austenite lattice (parameter
aγ ) to a bcc or bct martensite (parameters aα′ , cα′ where aα′ ≤ cα′ ). Referring to
Fig. 10.2, the basis ‘A’ is defined by the basis vectors ai , each of magnitude aγ , and
basis ‘B’ is defined by basis vectors bi as illustrated in Fig. 10.2b. Focussing atten-
tion on the bct representation of the austenite unit cell, it is evident that a compression
along the [0 0 1]B axis, coupled with expansions along [1 0 0]B and [0 1 0]B would
accomplish the transformation of the bct austenite unit cell into a bcc or bct α′ cell.
This deformation, referred to the basis B, can be written as:
√
η1 = η2 = 2(aα′ /aγ )

along [1 0 0]B and [0 1 0]B respectively and

η3 = cα′ /aγ

along the [0 0 1]B axis.

The deformation just described can be written as a 3 × 3 matrix, referred to
the austenite lattice. In other words, we imagine that a part of a single crystal of
austenite undergoes the prescribed deformation, allowing us to describe the strain
in terms of the remaining (and undeformed) region, which forms a fixed reference
basis. Hence, the deformation matrix does not involve any change of basis, and this
point is emphasised by writing it as (A S A), with the same basis symbol on both
sides of S:
[A; v] = (A S A)[A; u].
10.2 Homogeneous deformations 153

(a) (b)

FIGURE 10.2
(a) Two austenite unit cells, one the conventional cubic-F (basis A) and the other a
body-centred tetragonal cell (basis B). Some of the atoms in face-centering positions
on the vertical faces of the the cubic cell of austenite are not illustrated for the sake of
clarity. (b) How the body centred tetragonal cell can be deformed by the Bain strain
into that of body-centred cubic or body-centred tetragonal cell of martensite.

(A S A) converts the vector [A;u] into a new vector [A;v], with v still referred to the
basis A.
The difference between a co-ordinate transformation (B J A) and a deformation
matrix (A S A) is illustrated in Fig. 10.3, where ai and bi are the basis vectors of the
bases A and B respectively.

FIGURE 10.3
Difference between coordinate transformation (B J γ) and deformation matrix
(γ S γ). The latter is defined with respect to a single reference frame whereas the
former deals with a change in the reference frame.

We see that an advantage of the Mackenzie-Bowles notation is that it enables a

clear distinction to be made between 3 × 3 matrices which represent changes of axes
and those which represent physical deformations referred to just one axis system.
154 10 Homogeneous deformations

The following additional information can be deduced from Fig. 10.2:

Vector components before Bain strain Vector components after Bain strain
[1 0 0]A [η1 0 0]A
[0 1 0]A [0 η2 0]A
[0 0 1]A [0 0 η3 ]A

so the matrix (A S A) can be written as follows, with each column consisting of the
components of a vector equal to the basis vector of A, following deformation:
⎛ ⎞
η1 0 0
⎜ ⎟
(A S A) = ⎜ ⎝ 0 η2 0 ⎠ .
⎟ (10.1)
0 0 η3

Each column of the deformation matrix represents the components of the new
vector formed as a result of the deformation of a vector equal to one of the basis
vectors of A.
The strain represented by (A S A) is called a pure strain because it involves
stretching along three orthogonal axes; for this reason, the matrix representation
(A S A) is symmetrical. Stretching means extensions or contractions but no rota-
tions, and the three initially orthogonal directions are referred to as the principal axes
which obviously remain orthogonal and unrotated during the deformation.
The ratio of its final length to initial length is the principal deformation associated
with that principal axis. The directions [1 0 0]B , [0 1 0]B and [0 0 1]B are therefore
the principal axes of the Bain strain, and ηi the respective principal deformations. In
the particular example under consideration, η1 = η2 so that any two perpendicular
lines in the (0 0 1)B plane could form two of the principal axes. Since a3 ∥ b3 and
since a1 and a2 lie in (0 0 1)B , it is clear that the vectors ai must also be the principal
axes of the Bain deformation.
Since the deformation matrix (A S A) is referred to a basis system which co-
incides with the principal axes, the off-diagonal components of this matrix must be
zero. Column 1 of the matrix (A S A) represents the new co-ordinates of the vector
[1 0 0], after the latter has undergone the deformation described by (A S A), and a
similar rationale applies to the other columns. (A S A) deforms the fcc γ lattice into a
bcc or bct α′ lattice, and the ratio of the final to initial volume of the material is sim-
ply η1 η2 η3 ; the ratio is more generally given by the determinant of the deformation
matrix. Finally, it should be noted that any tetragonality in the martensite can readily
be taken into account by writing η3 = cα′ /aγ , where cα′ /aα is the aspect ratio of
the bct martensite unit cell.
Example 10.1: Bain strain: undistorted vectors
Given that the principal distortions of the Bain strain (A S A), referred to the
crystallographic axes of the fcc γ lattice (lattice parameter aγ ), are η1 = η2 =
10.2 Homogeneous deformations 155
1.123883, and η3 = 0.794705, show that the vector

[A; x] = [−0.645452 0.408391 0.645452]

remains undistorted, though not unrotated as a result of the operation of the Bain
strain. Furthermore, show that for x to remain unextended as a result of the Bain
strain, its components xi must satisfy the equation

(η12 − 1)x21 + (η22 − 1)x22 + (η32 − 1)x23 = 0. (10.2)

As a result of the deformation (A S A), the vector x becomes a new vector y,

according to the equation

(A S A)[A; x] = [A; y] = [η1 x1 η2 x2 η3 x3 ] = [−0.723412 0.458983 0.512944]

Now,
|x|2 = (x; A∗ )[A; x] = a2γ (x21 + x22 + x23 ) (10.3)
and,
|y|2 = (y; A∗ )[A; y] = a2γ (y12 + y22 + y32 ). (10.4)
Using these equations, and the numerical values of xi and yi obtained above,
it is easy to show that |x| = |y|. It should be noted that although x remains
unextended, it is rotated by the strain (A S A), since xi ̸
= yi . Using equations 10.3
and 10.4 with yi = ηi xi , yields the required equation 10.2. Since η1 and η2 are
equal and greater than 1, and since η3 is less than unity, equation 10.2 amounts
to the equation of a right-circular cone, the axis of which coincides with [0 0 1]A .
Any vector initially lying on this cone will remain unextended as a result of the
Bain strain.
This process was illustrated in Fig. 8.4 where a a spherical volume of austenite
was deformed into an ellipsoid of revolution. Notice that the principal axes (ai )
remain unrotated by the deformation, and that lines such as xw and yz which
become x′ w′ and y ′ z ′ respectively, remain unextended by the deformation (since
they are all diameters of the original sphere), although rotated through the angle θ.
The lines xw and yz of course lie on the initial cone described by equation 10.2.
Suppose now, that the ellipsoid resulting from the Bain strain is rotated through
a right-handed angle of θ, about the axis a2 , then Fig. 8.4c illustrates that this
rotation will cause the initial and final cones of unextended lines to touch along
yz, bringing yz and y ′ z ′ into coincidence. If the total deformation can therefore
be described as (A S A) combined with the above rigid body rotation, then such
a deformation would leave the line cd both unrotated and unextended; such a
deformation is called an invariant-line strain. Notice that the total deformation,
consisting of (A S A) and a rigid body rotation is no longer a pure strain, since the
vectors parallel to the principal axes of (A S A) are rotated into the new positions
a′i Fig. 8.4c.

It will later be shown that the lattice deformation in a martensitic transformation

156 10 Homogeneous deformations

must contain an invariant line. Therefore, the Bain strain must be combined with a
suitable rigid body rotation in order to define the total lattice deformation. This ex-
plains why the experimentally observed orientation relationship between martensite
and austenite does not correspond to that implied by Fig. 10.2. The need to have an
invariant line in the martensite-austenite interface means that the Bain Strain does
not in itself constitute the total transformation strain, which can be factorised into
the Bain Strain and a rigid body rotation. It is this total strain which determines the
final orientation relationship although the Bain Strain accomplishes the total fcc to
bcc lattice change.

10.3 Eigenvectors and eigenvalues

It is useful to establish a method of determining the directions which remain un-
rotated, though not necessarily undistorted, as a consequence of the deformation
concerned. Vectors lying along such unrotated directions are called eigenvectors of
the deformation matrix, and the ratios of their final to initial lengths are the corre-
sponding eigenvalues of the matrix. Considering the deformation matrix (A S A),
the unrotated directions may be determined by solving the equations

(A S A)[A; u] = λ[A; u] (10.5)

where u is a unit vector lying along an eigenvector, A is a convenient orthonormal

basis and λ is a positive scalar quantity. This equation shows that the vector u does
not change in direction as a result of (A S A), although its length changes by the ratio
λ. If I is a 3 × 3 identity matrix, then on rearranging equation 10.5, we obtain

{(A S A) − λI}[A; u] = 0 (10.6)

which can be written more fully as:

⎛ ⎞⎛ ⎞
S −λ S12 S13 u1
⎜ 11 ⎟⎜ ⎟
S23 ⎠ ⎝u2 ⎠ = 0 (10.7)
⎜ ⎟⎜ ⎟
⎝ S21 S22 − λ
S31 S32 S33 − λ u3

where [A; u] = [u1 u2 u3 ]. This system of homogeneous equations has non-trivial

solutions if the determinant
: :
: :
:S11 − λ S12 S13 :
: :
S23 : = 0. (10.8)
: :
: S21 S22 − λ
: :
: S S32 S33 − λ:
31

The expansion of this determinant yields an equation which is in general cubic

10.3 Eigenvectors and eigenvalues 157

in λ; the roots of this equation are the three eigenvalues λi . Associated with each of
the eigenvalues is a corresponding eigenvector whose components may be obtained
by substituting each eigenvalue, in turn, into equation 10.7. Of course, since every
vector which lies along the unrotated direction is an eigenvector, if u is a solution of
equation 10.7 then ku must also satisfy equation 10.7, k being a scalar constant. If the
matrix (A S A) is real then there must exist three eigenvalues, at least one of which
is necessarily real. If (A S A) is symmetrical then all three of its eigenvalues are real;
the existence of three real eigenvalues does not however imply that the deformation
matrix is symmetrical. Every real eigenvalue implies the existence of a corresponding
vector which remains unchanged in direction as a result of the operation of (A S A).
Example 10.2: Eigenvectors and eigenvalues
Find the eigenvalues and eigenvectors of
⎛ ⎞
18 −6 −6
⎜ ⎟
(A S A) = ⎜⎝−6 21
⎟
3⎠
−6 3 21
To solve for the eigenvalues, we use equation 10.8 to form the determinant
: :
: :
:18 − λ −6 −6 :
: :
3 :=0
: :
: 6 21 − λ
: :
: −6 3 21 − λ:
which on expansion gives the cubic equation
(12 − λ)(λ − 30)(λ − 18) = 0
with the roots
λ1 = 12, λ2 = 30 and λ3 = 18.
To find the eigenvector u = [A; u] = [u1 u2 u3 ] corresponding to λ1 , we
substitute λ1 into equation 10.7 to obtain
6u1 − 6u2 − 6u3 =0
−6u1 + 9u2 + 3u3 =0
−6u1 + 3u2 + 9u3 = 0.
These equations can be solved simultaneously to show that u1 = 2u2 = 2u3 . The
other two eigenvectors, v and x, corresponding to λ2 and λ3 respectively, can be
determined in a similar way. Hence, it is found that:
1
[A; u] = (6− 2 )[2 1 1]
1
[A; v] = (3− 2 )[1 1 1]
1
[A; x] = (2− 2 )[0 1 1].
All vectors parallel to u, v or x remain unchanged in direction, though not in
magnitude, due to the deformation (A S A).
158 10 Homogeneous deformations

10.3.1 Comments
(i) Since the matrix (A S A) is symmetrical, we find three real eigenvectors, which
form an orthogonal set.
(ii) A negative eigenvalue implies that a vector initially parallel to the corresponding
eigenvector becomes antiparallel (changes sign) on deformation. A deformation
like this is physically impossible.
(iii) If a new orthonormal basis B is defined, consisting of unit basis vectors parallel
to u, v and x respectively, then the deformation (A S A) can be expressed in the
new basis with the help of a similarity transformation. Keeping like basis symbols
adjacent, it follows that

(B S B) = (B J A)(A S A)(A J B) (10.9)

where the columns of (A J B) consist of the components (referred to the basis A)

of the eigenvectors u, v and x respectively, so that
⎛ ⎞⎛ ⎞⎛ ⎞
u1 u2 u3 18 −6 −6 u1 v1 w1
⎜ ⎟⎜ ⎟⎜ ⎟
(B S B) = ⎜ ⎝ v1 v2 v3 ⎠ ⎝ 6
⎟⎜
21
⎟⎜
3 ⎠ ⎝u2 v2 w2 ⎠
⎟

w1 w2 w3 −6 3 21 u3 v3 w3
⎛ ⎞
18 0 0
⎜ ⎟
= ⎝ 0 30 0 ⎠ .
⎜ ⎟

0 0 12

Notice that the off diagonal terms of (B S B) are zero because it is referred to a
basis formed by the principal axes of the deformation - i.e. the three orthogonal
eigenvectors. The matrix representing the Bain strain (page 154) is also diagonal
because it too is referred to the principal axes of the strain. Any real symmetrical
matrix can be diagonalised using the procedure illustrated above. (B S B) is called
the diagonal representation of the deformation and this special representation will
henceforth be identified by placing a bar over the matrix symbol: (B S B).

10.4 Stretch and rotation

A pure strain consists of simple extensions or contractions along the principal axes.
The ratios of the final to initial lengths of vectors parallel to the principal axes are
called the principal deformations, and the change in length per unit length the prin-
cipal strains. It has a symmetrical matrix representation irrespective of the choice of
basis. It is possible to find three real and orthogonal eigenvectors.
10.4 Stretch and rotation 159

On the other hand, the shear (Z P2 Z) illustrated in Fig. 10.4a is not a pure de-
formation because it is possible to identify just two mutually perpendicular eigen-
vectors, both of which must lie in the invariant-plane. All other vectors are rotated
by the shearing action. The original object, represented as a sphere, is sheared into
an ellipsoid. The invariant-plane of the deformation contains the z1 and z2 axes. The
deformation can be imagined to occur in two stages, the first one involving simple
extensions and contractions along the y1 and y3 directions respectively (Fig. 10.4b)
and the second involving a rigid body rotation of the ellipsoid, about the axis z2 ∥ y2 ,
through a right-handed angle φ.

(a) (b)

FIGURE 10.4
Factorisation of a simple shear (Z P2 Z) into a pure deformation (Z Q Z) and a right
handed rigid body rotation of φ about z2 . In (a), ac is the trace of the invariant plane.
(Z Q Z) leaves ac undistorted but rotated to a′ c′ and rigid body rotation brings a′ c′
into coincidence with ac. The axes y1 , y2 and y3 are the principal axes of the pure
deformation. The undeformed shape is represented as a sphere in three dimensions.

In essence, we have just carried out an imaginary factorisation of the impure

strain (Z P2 Z) into a pure strain (Fig. 10.4b) and a rigid body rotation. If the pure
strain part is referred to as (Z Q Z) and the rotation part as (Z J Z), then

(Z P2 Z) = (Z J Z)(Z Q Z). (10.10)

It was arbitrarily chosen that the pure strain would occur first and be followed by the
rigid body rotation, but the reverse order is equally acceptable,

(Z P2 Z) = (Z Q2 Z)(Z J2 Z)

where in general,

(Z Q2 Z) ̸
= (Z Q Z) and (Z J2 Z) ̸
= (Z J Z).

Any real deformation can in general be factorised into a pure strain and a rigid body
160 10 Homogeneous deformations

rotation, but it is important to realise that the factorisation is simply a mathematical

convenience and the deformation does not actually occur in the two stages. The fac-
torisation in no way indicates the path by which the initial state reaches the final state
and is merely phenomenological.
The actual factorisation can be considered in terms of the arbitrary deformation
(Z S Z), referred to an orthonormal basis Z. Bearing in mind that (Z S′ Z) is the
transpose of (Z S Z),

(Z S′ Z)(Z S Z) = (Z Q′ Z)(Z J′ Z)(Z J Z)(Z Q Z)

2
or (Z S′ Z)(Z S Z) = (Z Q Z) (10.11)

since (Z J′ Z)(Z J Z) = I and (Z Q′ Z) = (Z Q Z), since (Z Q Z) is a pure defor-

mation with a symmetrical matrix representation. If the product (Z S′ Z)(Z S Z) is
written as the symmetrical matrix (Z T Z), then the eigenvalues λi of (Z√ T Z) are
2
also the eigenvalues of (Z Q Z) , so that the eigenvalues of (Z Q Z) are λi . If the
eigenvectors of (Z T Z) are u, v and w (with corresponding eigenvalues λ1 , λ2 and
λ3 respectively), then (Z T Z) can be diagonalised by similarity transforming it to
another orthonormal basis Y formed by the vectors u,v and w. From equation 10.9,
⎛ ⎞
λ1 0 0
⎜ ⎟
(Y T Y) = ⎜ ⎝ 0 λ2 0 ⎠ = (Y J Z)(Z T Z)(Z J Y)
⎟

0 0 λ3

where the columns of (Z J Y) consist of the components of u,v and w, respectively,

2
when the latter are referred to the Z basis. It follows that since (Y Q Y) = (Y T Y),
1
(Z Q Z) = (Z J Y)(Y T Y) 2 (Y J Z)
1 1
where the square root of a diagonal matrix (Y T Y) is such that (Y T Y) 2 (Y T Y) 2 =
(Y T Y). It follows that:
⎛ ⎞ ⎛√ ⎞⎛ ⎞
u1 v1 w1 λ1 0 0 u1 u2 u3
⎜ ⎟⎜ √ ⎟⎜ ⎟
(Z Q Z) = ⎜ ⎝u2 v2 w2 ⎠ ⎝ 0
⎟⎜
λ2 0 ⎠ ⎝ v1 v2 v3 ⎠ . (10.12)
⎟⎜ ⎟
√
u3 v3 w3 0 0 λ3 w1 w2 w3

It is worth repeating that in equation 10.12, λi are the eigenvalues of the matrix
(Z S′ Z)(Z S Z) and ui , vi and wi are the components, in the basis Z of the eigen-
vectors of (Z S′ Z)(Z S Z). The rotation part of the strain (Z S Z) is simply

(Z J Z) = (Z S Z)(Z Q Z)−1 . (10.13)

10.5 Interfaces 161

10.5 Interfaces
A vector parallel to a principal axis of a pure deformation may become extended
but is not changed in direction by the deformation. The ratio η of its final to initial
length is called a principal deformation associated with that principal axis and the
corresponding quantity (η − 1) is called a principal strain. We have seen that when
two of the principal strains of the pure deformation differ in sign from the third,
all three being non-zero, it is possible to obtain a total strain which leaves one line
invariant (page 121). It intuitively seems advantageous to have the invariant-line in
the interface connecting the two crystals, since their lattices would then match exactly
along that line.
A completely undistorted interface would have to contain two non-parallel direc-
tions which are invariant to the total transformation strain. The following example il-
lustrates the characteristics of such a transformation strain, called an invariant-plane
strain, which allows the existence of a plane which remains unrotated and undistorted
during the deformation.
Example 10.3: Deformations and interfaces
A pure strain (A Q A), referred to an orthonormal basis ‘A’ whose basis vectors
are parallel to the principal axes of the deformation, has the principal deformations
η1 = 1.192281, η2 = 1 and η3 = 0.838728. Show that (A Q A) combined
with a rigid body rotation gives a total strain which leaves a plane unrotated and
undistorted.
Because (A Q A) is a pure strain referred to a basis composed of unit vectors
parallel to its principal axes, it consists of simple extensions or contractions along
the basis vectors a1 , a2 and a3 . Hence, Fig. 10.5 can be constructed by analogy for
the Bain strain illustrated on page 121. Since η2 = 1, ef ∥ a2 remains unextended
and unrotated by (A Q A), and if a rigid body rotation (about fe as the axis of
rotation) is added to bring yz into coincidence with y ′ z ′ , then the two vectors
ef and ab remain invariant to the total deformation. Any combination of ef and
ab will also remain invariant, and hence all lines in the plane containing ef and
y ′ z ′ are invariant, giving an invariant plane. Thus, a pure strain when combined
with a rigid body rotation can only generate an invariant-plane strain if two of its
principal strains have opposite signs, the third being zero. Since it is the pure strain
which actually accomplishes the lattice change (the rigid body rotation causes
no further lattice change), any two lattices related by a pure strain with these
characteristics may be joined by a fully coherent interface.
(A Q A) actually represents the pure strain part of the total transformation
strain required to change an fcc lattice to an hcp (hexagonal close-packed) lattice,
assuming that there is no volume change, by shearing on the {1 1 1}γ plane, in
the ⟨1 1 2⟩γ direction, the magnitude of the shear being equal to half the twinning
shear (see Chapter 11). Consistent with the proof given above, a fully coherent in-
terface is observed experimentally when hcp martensite is formed in this manner.
162 10 Homogeneous deformations
It is worth noting that the total deformation, i.e. the combination of the pure
and rotational strains, completely defines the set of crystallographic features of the
hcp martensite [4–6]. The set consists of the habit plane, orientation relationship
and shape deformation associated with the transformation.

(a) (b)

(c)

FIGURE 10.5
(a,b) Illustration of the strain (A Q A), the undeformed crystal represented initially
as a sphere of diameter ef which also is an invariant line since η2 = 1. The strain
(c) illustrates that a combination of (A Q A) with a rigid body rotation about the
axis ef generates another invariant line z ′ y ′ , so that the net deformation becomes an
invariant-plane strain.

10.6 Topology of grain deformation

Metallic alloys are produced in very large quantities using plastic-deformation in or-
der to achieve particular shapes and properties. The microstructure changes during
deformation, with an increase in the defect density and in the amount of grain bound-
ary area per unit volume (SV ) and grain edge length per unit volume (LV ). All of
these changes are important in determining the course of phase transformations and
recrystallisation processes in general.
10.6 Topology of grain deformation 163

The equiaxed grain structure typical of most undeformed materials can be repre-
sented using Kelvin tetrakaidecahedra which accomplish the ‘division of space with a
minimum of partitional area’ [7]. A tetrakaidecahedron has 8 hexagonal and 6 square
faces, Fig. 10.6, with 36 edges, each of length a. All of the edges can be described in
terms of just six vectors, as listed in Table 10.1. It then becomes possible to operate
on these vectors by a deformation matrix in order to calculate consequential changes
in grain parameters.

2 1
FIGURE 10.6
The Kelvin tetrakaidecahedron.

TABLE 10.1
Vectors defining the edges of a tetrakaidecahedron.

Vector Components

1 [a 0 0]

2 [0 a 0]

3 [− a2 − a2 √a ]
2

4 [ a2 − a2 √a ]
2

5 [ a2 a
2
√a ]
2

6 [− a2 a
2
√a ]
2
164 10 Homogeneous deformations

10.6.1 Plane strain deformation

A general deformation matrix S acts on a vector u to give a new vector v as follows
[8–10]: ⎛ ⎞⎛ ⎞ ⎛ ⎞
S S12 S13 u v
⎜ 11 ⎟ ⎜ 1⎟ ⎜ 1⎟
⎜ ⎟⎜ ⎟ = ⎜ ⎟ (10.14)
⎝S21 S22 S23 ⎠ ⎝u2 ⎠ ⎝v2 ⎠
S31 S32 S33 u3 v3
Consider first the orientation of the tetrakaidecahedron as illustrated in Fig. 10.6.
The tetrakaidecahedron is completely specified by the six initial vectors listed in
Table 10.1. For plane strain deformation, all Sij in equation 10.14 are zero except
that S11 × S22 × S33 = 1 to conserve volume, and S11 × S33 = 1 since S22 = 1.
For a diagonal matrix, the terms S11 , S22 and S33 represent the principal distortions,
i.e., the ratios of the final to initial lengths of unit vectors along the principal axes.
It follows that for a diagonal S, the true strains are given by ϵ11 = ln{S11 }, ϵ22 =
ln{S22 } and ϵ33 = ln{S33 }.
The application of the deformation to the initial set of vectors results in the new
set of vectors listed in Table 10.2. The latter are used to calculate the area and edge-
lengths of the deformed object. Using equation 10.14 and the conditions for plane
strain deformation, it can be shown that the final to initial area (A/A0 ) and edge-
length (L/L0 ) ratios for the deformed tetrakaidecahedron are give by:
- - -
A SV S11 + 3(S11 1 + 2S33 2 + 2 + 2S 2 ) + S
S11 2 )
2(1 + S11
33 33
≡ = √
A0 SV0 3(2 3 + 1)
(10.15)
-
L LV 1 + S11 + 2 1 + S11 2 + 2S 2
33
≡ = (10.16)
L0 LV0 6
Here SV0 and LV0 are the values at zero strain, of grain surface area and edge-length
per unit volume. These equations apply strictly to the grain orientation illustrated in
Fig. 10.6, relative to S. From stereology [11], SV0 = 2/L so that SSVV ≡ 2SV L, and
0
2
LV0 = 9.088/L , where L is the mean linear intercept commonly used to define the
grain size [10, 12]. It follows that equations 10.15 and 10.16 implicitly contain the
grain size as an input variable.
The grain-orientation illustrated in Fig. 10.6 may not be representative. Suppose
that we wish to orient the tetrakaidecahedron randomly with respect to the deforma-
tion. A rotation matrix R can be generated using random numbers to rotate the object
relative to the axes defining S. Equation 10.14 then becomes
⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞
S11 S12 S13 R11 R12 R13 u v
⎜ ⎟⎜ ⎟ ⎜ 1⎟ ⎜ 1⎟
⎜ ⎟⎜ ⎟⎜ ⎟ = ⎜ ⎟. (10.17)
⎝S21 S22 S23 ⎠ ⎝R21 R22 R23 ⎠ ⎝u2 ⎠ ⎝v2 ⎠
S31 S32 S33 R31 R32 R33 u3 v3

The results are illustrated in Fig. 10.7. For comparison purposes, the results are
10.6 Topology of grain deformation 165

TABLE 10.2
Components of the six vectors listed in Table 10.1, following plane strain or axisym-
metric deformation.

Deformed Vector Components

1 [aS11 0 0]

2 [0 aS22 0]

3 [− aS211 − aS222 aS
√33 ]
2

4 [ aS211 − aS222 aS
√33 ]
2

5 [ aS211 aS22
2
aS
√33 ]
2

6 [− aS211 aS22
2
aS
√33 ]
2

plotted against the equivalent strain:

% & 12 % &1
2 1 2 1 2 1 2 2
ϵ= ϵ211 + ϵ222 + ϵ233 + γ13 + γ12 + γ23 (10.18)
3 2 2 2
where ϵ11 , ϵ22 , and ϵ33 are the normal components and γ13 , γ12 and γ23 are the
shear components of strain (the
√ tangents of the shear angles). For homogeneous plane
strain compression, ϵ = (2/ 3)ϵ11 . In Fig. 10.7, the dashed line represents the out-
come for the orientation illustrated in Fig. 10.6 and the points are for the 99 other
results of randomly oriented tetrakaidecahedra. It is clear that the orientation of the
tetrakaidecahedron does not make much of a difference to the outcome as far as the
surface and edge-lengths per unit volume are concerned. This is probably because
the tetrakaidecahedron is almost isotropic in shape.

10.6.2 Sequential deformations

The method here is general - all that is needed is to define the matrix S for the
appropriate circumstances. There are cases where two or more different kinds of
deformation are used in sequence, for example, cross-rolling in which the plate is
rotated through 90◦ after a degree of reduction. This is readily tackled by generalising
equation 10.14. Rotation through 90◦ about the compression axis [0 0 1] is given by
[10]: ⎛ ⎞
0 1 0
⎜ ⎟
R = ⎝−1 0
⎜ ⎟
0⎠ (10.19)
0 0 1
166 10 Homogeneous deformations

(a) (b)

FIGURE 10.7
Calculations for plane strain deformation. The curve represents the data for the
tetrakaidecahedron oriented as illustrated in Fig. 10.6. The small-points are 99 other
cases where the tetrakaidecahedron is randomly oriented relative to S.

Writing the first rolling pass as S and the cross-rolling pass as T, the net deformation
U is given by T × R × S:
⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞
T11 0 0 0 1 0 S11 0 0 0 T11 0
⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟
U=⎜ ⎝ 0 1
⎟⎜
0 ⎠ ⎝−1 0 0⎠ ⎝ 0 1
⎟⎜
0 ⎠ = ⎝−S11
⎟ ⎜ 0 0 ⎟ ⎠.
1
0 0 1/T11 0 0 1 0 0 1/S11 0 0 T11 S11
(10.20)

10.6.3 Complex deformations

Flat product rolling is approximated as plane strain compression, but friction with
the rolls leads to shears. The plane strain condition is strictly satisfied only at the
centre of the rolled material. The matrix S can be used to deal with the simultaneous
actions of plane strain compression and simple shear (on the rolling plane and in the
rolling direction) by generalising the deformation matrix equation 10.14 as follows:
⎛ ⎞
′
S11 0 S13
⎜ ⎟
0 ⎠. (10.21)
⎜ ⎟
⎝ 0 1
0 0 1/S11
′
The final shear strain S13 arises from the imposed shear strain, S13 , modified by the
compression S11 and is represented by S13 × S11 .

10.6.4 Multi-directional deformations

The effect of sequential deformations in different directions, as for example in the
cross rolling of plate, is illustrated in Fig. 10.8 which represents calculations done
10.7 Summary 167

(a) (b)

FIGURE 10.8
Comparison of cross-rolling (identical strains in the two rolling directions), axisym-
metric compression and single-direction rolling.

using equation 10.20. In this graph, the rolling strains are the same in both directions
(T11 = S11 ) and data for axisymmetric compression are also included. To allow a
comparison between these deformation modes, the data are plotted as a function of
the compressive strain. Notice that the axisymmetric compression and cross-rolling
with equal strains in both directions give exactly identical results, illustrating that
the results depend on the final strain components and not on the strain path to reach
them.
The area ratio for single-direction rolling is only slightly larger than for the cross-
rolling when plotted against the compressive strain. This is not surprising given
that for the same rolling reduction, the length along the rolling direction for single-
direction rolling will be much larger than obtained by cross-rolling. In contrast, the
edge ratio becomes much larger for the single-direction scenario, reflecting its greater
microstructural anisotropy.

10.7 Summary
Solid state phase transformations often are limited by the fact that the diffusion of
atoms becomes difficult at low homologous temperatures. Nevertheless, there may
be a tendency for transformation to occur to a lower free energy state, but by a mech-
anism in which the structure of the parent phase is changed into that of the product
by a deformation which is driven not by external forces, but by the favourable change
in free energy. The deformation is real in the sense that the shape of the crystal will
change in order to reflect the difference between the atomic patterns of the parent
and product crystals.
Transformations of this kind are described generically as ‘displacive’ because the
structural change is accomplished by a coordinated shift of atoms [13]. The deforma-
168 10 Homogeneous deformations

tions described in this chapter are homogeneous; in the case illustrated in Fig. 10.9a,
the atoms themselves are placed in the correct final positions by the homogeneous
deformation of the lattice. However, the shuffle transformation in Fig. 10.9 cannot
be described using a homogeneous deformation matrix. The term shuffle refers here
to small displacements within the unit cell. There are cases, as we shall see in Chap-
ter 11 where the transformation is a mixture of lattice distortion and shuffles, so the
homogeneous deformation places some of the atoms in the correct position whereas
others have to move small distances so that overall structure of the product phase is
correct.

(a)

(b)

FIGURE 10.9
An illustration of a lattice distortive and shuffle transformation [13].
References

1. M. Mooney: ‘A theory of large elastic deformation’, Journal of Applied

Physics, 1940, 11, 582–592.
2. M. Born, and R. Fürth: ‘The stability of crystal lattices III’, Mathematical
Proceedings of the Cambridge Philosophical Society, 1940, 36, 454–465.
3. E. C. Bain: ‘The nature of martensite’, Trans. AIME, 1924, 70, 25–46.
4. J. G. Parr: ‘The crystallographic relationship between the phases γ and ε in the
system iron-manganese.’, Acta Crystallographica, 1952, 5, 842–843.
5. T. Anantharaman, and J. Christian: ‘The existence of a macroscopic shear in the
transformation in cobalt’, The London, Edinburgh, and Dublin Philosophical
Magazine and Journal of Science, 1952, 43, 1338–1342.
6. J. S. Bowles, and C. S. Barrett: ‘Crystallography of transformations’, Progress
in Metal Physics, 1952, 3, 1–41.
7. W. Thomson: ‘On the division of space with minimum partitional area’, Acta
mathematica, 1887, 11, 121–134.
8. I. Czinege, and T. Reti: ‘Determination of local deformation in cold formed
products by a measurement of the geometric characteristics of the crystallites’,
In: Eighteenth International Machine Tool Design and Research Conference,
Forming, vol. 1. 1977:159–163.
9. S. B. Singh, and H. K. D. H. Bhadeshia: ‘Topology of grain deformation’,
Materials Science and Technology, 1998, 15, 832–834.
10. H. K. D. H. Bhadeshia: Geometry of Crystals: 2nd edition, Institute of Materi-
als, 2001.
11. E. E. Underwood: Quantitative Stereology: Addison-Wesley Publication Com-
pany, 1970.
12. C. Mack: ‘On clumps formed when convex laminae or bodies are placed at ran-
dom in two or three dimensions’, Proceedings of the Cambridge Philosophical
Society, 1956, 52, 246–250.
13. M. Cohen, G. B. Olson, and P. C. Clapp: ‘On the classification of displacive
transformations (what is martensite?)’, In: G. B. Olson, and M. Cohen, eds.
International Conference on Martensitic Transformations ICOMAT ’79. Mass-
chusetts, USA: Alpine Press, 1979:1–11.

169
11
Invariant-plane strains

Abstract

An invariant-plane strain is a deformation that leaves a plane unchanged and in its

original orientation. A simple shear deformation applied to a solid is an example,
with all the displacements parallel to the shear plane which itself is unaffected. The
same deformation can be imagined as a combination of a pure strain that involves just
expansions and contractions along orthogonal axes, and a rigid body rotation. There
is much to be learned from such factorisations, which although phenomenological,
reveal the degeneracies in the combinations of pure strain and rotation that can lead
to similar outcomes. If a strain leaves a plane invariant, then two such strains on dif-
ferent planes leave just a line undistorted and unrotated. Both kinds of deformations
produced shape changes but are also important in defining the structure of interfaces,
as we shall see in the chapters that follow.

11.1 Introduction
The deformation of crystals by the conservative glide of dislocations on a single set
of crystallographic planes causes shear in the direction of the resultant Burgers vector
of the dislocations concerned, a direction which lies in the slip plane. The slip plane
and slip direction constitute a slip system. The material in the slip plane remains
crystalline during slip and since there is no reconstruction of this material during slip
(e.g. by localised melting followed by resolidification), there can be no change in
the relative positions of atoms in the slip plane. The atomic arrangement on the slip
plane is thus completely unaffected by the deformation.
Another mode of conservative plastic deformation is mechanical twinning, in
which the parent lattice is homogeneously sheared into the twin orientation. The
plane on which the twinning shear occurs is again unaffected by the deformation and
forms a coherent boundary between the parent and its twin.
If a material which has a Poisson’s ratio equal to zero is uniaxially stressed below
its elastic limit, then the plane that is normal to the stress axis is unaffected by the
deformation since the only non-zero strain is that parallel to the stress axis. Polycrys-
talline beryllium has a Poisson’s ratio close to zero [1] so that stretching would lead
to little transverse strain.

171
172 11 Invariant-plane strains

All these strains belong to a class of deformations called invariant-plane strains.

The operation of an invariant-plane strain (IPS) always leaves one plane of the parent
crystal completely undistorted and unrotated; this plane is the invariant plane. The
condition for a strain to leave a plane undistorted is, as described on page 161, is
that the principal deformations of its pure strain component, η1 , η2 and η3 are greater
than, equal to and less than unity, respectively. However, as seen in Figs. 10.5, this
does not ensure that the undistorted plane is also unrotated; it is necessary to combine
the pure strain with an appropriate rotation to generate the invariant plane.
A homogeneous deformation (A S A) strains a vector u into another vector v
which in general may have a different direction and magnitude:
[A; v] = (A S A)[A; u] (11.1)
However, the deformation could also have been defined with respect to another basis,
say ‘B’ (basis vectors bi ), to give the deformation matrix (B S B), using a similarity
transformation:
(B S B) = (B J A)(A S A)(A J B) (11.2)
The nature of the physical deformation does not, of course, change, the strain is
simply referred to another coordinate system [2]. We can now proceed to examine
the nature of invariant-plane strains. Fig. 11.1 illustrates three such strains, defined
with respect to a right-handed orthonormal basis Z, such that z3 is parallel to the
unit normal p of the invariant plane; z1 and z2 lie within the invariant plane, z1
being parallel to the shear component of the strain concerned. Fig. 11.1a illustrates
an invariant-plane strain which is purely dilatational, and is of the type to be expected
when a plate-shaped precipitate grows diffusionally [3, 4]. The change of shape, as
illustrated in Fig. 11.1a, due to the growth of this precipitate then reflects the volume
change accompanying transformation.
In Fig. 11.1b, the invariant-plane strain corresponds to a simple shear, involving
no change of volume, as in the macroscopically homogeneous deformation of crys-
tals by slip. The shape of the parent crystal alters in a way which reflects the shear
character of the deformation.
The most general invariant-plane strain (Fig. 11.1c) involves both a volume
change and a shear; if d is a unit vector in the direction of the displacements in-
volved, then md represents the displacement vector, where m is a scalar giving the
magnitude of the displacements. md may be factorised as md = sz1 + δz3 , where s
and δ are the shear and dilatational components, respectively, of the invariant-plane
strain. The strain illustrated in Fig. 11.1c is of the type associated with the marten-
sitic transformation of γ iron into hcp iron. This involves a shear √ on the {1 1 1}γ
planes in ⟨1 1 2⟩γ direction, the magnitude of the shear being 1/ 8. However, there
also is a dilatational component to the strain, since hcp iron is more dense than fcc
iron, consistent with the fact that the former is the stable form at high pressures [5].
There is therefore a volume contraction on martensitic transformation, an additional
displacement δ normal to the {1 1 1} austenite planes [6, 26].
It has often been suggested that the passage of a single Shockley partial disloca-
tion on a close-packed plane of austenite leads to the formation of a 3-layer thick re-
gion of hcp, since this region contains the correct stacking sequence of close-packed
11.1 Introduction 173

FIGURE 11.1
Three kinds of invariant-plane strains. The cubes indicate the shape before deforma-
tion. δ, s and m represent the magnitudes of the dilatational strain, shear strain and
general displacement respectively. p is a unit vector, the shear strain s is parallel to
z1 , whereas δ is parallel to z3 .

planes for the hcp lattice. It has not been possible until recently to prove this, because
such a small region of hcp material gives very diffuse and unconvincing hcp reflec-
tions in electron diffraction experiments. However, the δ component of the fcc-hcp
martensite transformation strain has now been detected to be present for single stack-
ing faults, proving the hcp model of such faults [6]. In Fig. 11.2, the residual contrast
that is visible in part (b) in spite of the fact that the diffraction vector is normal to the
displacement in the plane of the fault, is indicative of a change in interplanar spacing,
i.e. a volume change normal to the fault plane, consistent with the presence of the
hcp phase. The contrast from a fault should otherwise be invisible in the absence of
a change in interplanar spacing.
Turning now to the description of the strains illustrated in Fig. 11.1, we follow
the procedure of Chapter 8, to find the matrices (Z P Z); the symbol ‘P’ in the matrix
representation is used to identify specifically, an invariant-plane strain, the symbol
‘S’ being the representation of any general deformation. Each column of such a ma-
trix represents the components of a new vector generated by deformation of a vector
equal to one of the basis vectors of Z. It follows that the three matrices representing
the deformations of Fig. 11.1a-c are, respectively,
⎛ ⎞ ⎛ ⎞
1 0 0 1 0 s
⎜ ⎟ ⎜ ⎟
(Z P1 Z) = ⎜ ⎝0 1 0 ⎠,
⎟ (Z P2 Z) = ⎜ ⎝0 1 0⎠ ,
⎟

0 0 1+δ 0 0 1
⎛ ⎞
1 0 s
⎜ ⎟
(Z P3 Z) = ⎝0
⎜
1 0 ⎠.
⎟

0 0 1+δ
174 11 Invariant-plane strains

(a) (b)

FIGURE 11.2
Stacking faults in austenitic stainless steel. (a) Bright field image of faults. (b) Image
with g.R equal to an integer for both of the faults illustrated, where g is the diffraction
vector and R is the displacement vector. After Brooks et al. [6], reproduced with
permission of Elsevier.

These have a particularly simple form, because the basis Z has been chosen care-
fully, such that p ∥ z3 and the direction of the shear is parallel to z1 . However, it
is often necessary to represent invariant-plane strains in a crystallographic basis, or
in some other basis X. This can be achieved with the similarity transformation law,
equation 11.2. If (X J Z) represents the coordinate transformation from the basis Z to
X, we have
(X P X) = (X J Z)(Z P Z)(Z J X).
Expansion of this equation gives [7]
⎛ ⎞
1 + md1 p1 md1 p2 md1 p3
⎜ ⎟
(X P X) = ⎜ ⎝ md2 p1 1 + md2 p2
⎟
md2 p3 ⎠ (11.3)
md3 p1 md3 p2 1 + md3 p3

where di are the components of d in the X basis, such that (d; X∗ )[X; d] = 1. The
vector d points in the direction of the displacements involved; a vector which is
parallel to d remains parallel following deformation, although the ratio of its final to
initial length may be changed. The quantities pi are the components of the invariant-
plane normal p, referred to the X∗ basis, normalised to satisfy (p; X∗ )[X; p] = 1.
Equation 11.3 may be simplified as follows:

(X P X) = I + m[X; d](p; X∗ ). (11.4)

The multiplication of a single-column matrix with a single-row matrix gives a 3 ×

3 matrix, whereas the reverse order of multiplication gives a scalar quantity. The
matrix (X P X) can be used to study the way in which vectors representing directions
11.1 Introduction 175

(referred to the X basis) deform. In order to examine the way in which vectors which
are plane normals (i.e. referred to the reciprocal basis X∗ ) deform, we proceed in the
following manner.
The property of a homogeneous deformation is that points which originally are
collinear remain collinear after the deformation [8]. Lines that initially are coplanar
remain so following deformation. It follows that an initial direction u which lies in
a plane whose normal is initially h, becomes a new vector v within a plane whose
normal is k, where v and k result from the deformation of u and h, respectively. Now,
h.u = k.v = 0, so that:

(h; X∗ )[X; u] = (k; X∗ )[X; v] = (k; X∗ )(X P X)[X; u]

i.e.
−1
(k; X∗ ) = (h; X∗ )(X P X) (11.5)
Equation 11.5 describes the way in which plane normals are affected by the defor-
mation (X P X). From equation 11.4, it can be shown that
−1
(X P X) = I − mq[X; d](p; X∗ ) (11.6)

where 1/q = det(X P X) = 1 + m(p; X∗ )[d; X]. The inverse of (X P X) is thus

another invariant-plane strain in the opposite direction.

Example 11.1: Tensile tests on single-crystals

A thin cylindrical single-crystal specimen of α iron is tensile tested at
−140◦C, the tensile axis being along the [4 4 1] direction (the cylinder axis).
On application of a tensile stress, the entire specimen deforms by twinning on the
(1 1 2) plane and in the [1 1 1] direction, the magnitude of the twinning shear being
1
2− 2 . Calculate the plastic strain recorded along the tensile axis, assuming that the
ends of the specimen are always maintained in perfect alignment. (Refs. [9–11],
contain details on single crystal deformation).

011

112 p
011
100
u
441

d
(a) (b) (c) 111
176 11 Invariant-plane strains
FIGURE 11.3
Longitudinal section of the tensile specimen illustrating the (1 1 0) plane. All
directions refer to the crystal basis. The tensile axis rotates towards d, in the plane
containing the original direction of the tensile axis (u) and d.

Fig. 11.3a illustrates the deformation involved; the parent crystal basis α con-
sists of basis vectors which form the conventional bcc unit cell of α-iron. The
effect of the mechanical twinning is to alter the original shape abcd to a′ b′ c′ d′ .
ef is a trace of (1 1 2)α on which the twinning shear occurs in the [1 1 1]α di-
rection. However, as in most ordinary tensile tests, the ends of the specimen are
constrained to be vertically aligned at all times; a′ d′ must therefore be vertical and
the deforming crystal must rotate to comply with this requirement. The configura-
tion illustrated in Fig. 11.3c is thus obtained, with ad and a′ d′ parallel, the tensile
strain being (a′ d′ − ad)/(ad).
As discussed earlier, mechanical twinning is an invariant-plane strain; it in-
volves a homogeneous simple shear on the twinning plane, a plane which is
not affected by the deformation and which is common to both the parent and
twin regions. Equation 11.3 can be used to find the matrix (α P α) describ-
ing the mechanical twinning, given that the normal to the invariant-plane is
1
− 12
(p; α∗ ) = aα 6− 2 (1 1 2), the displacement direction is [α; d] = a−1 α 3 [1 1 1]
1
and m = 2− 2 . It should be noted that p and d respectively satisfy the conditions
(p; α∗ )[α; p] = 1 and (d; α∗ )[α; d] = 1, as required for equation 11.3. Hence
⎛ ⎞
7 1 2
1⎜ ⎟
(α P α) = ⎜ 1 7 2
⎟.
6 ⎝ ⎠
1 1 4

Using this, we see that an initial vector [α; u] = [4 4 1] becomes a new vector
[α; v] = (α P α)[α; u] = 61 [34 34 4] due to the deformation. The need to maintain
the specimen ends in alignment means that v is rotated to be parallel to the tensile
axis. Now, |u| = 5.745aα where aα is the lattice parameter of the ferrite, and
|v| = 8.042aα, giving the required tensile strain as (8.042−5.745)/5.745 = 0.40.

11.1.1 Comments
(i) From Fig. 11.3 it is evident that the end faces of the specimen will also undergo
deformation (ab to a′ b′ ) and if the specimen gripping mechanism imposes con-
straints on these ends, then the rod will tend to bend into the form of an ‘S’. For
thin specimens this effect may be small.
(ii) The tensile axis at the beginning of the experiment was [4 4 1], but at the end is
1
6 [34 34 4]. The tensile direction clearly has rotated during the course of the exper-
iment. The direction in which it has moved is 16 [34 34 4] − [4 4 1] = 16 [10 10 10],
parallel to [1 1 1], the shear direction d. In fact, any initial vector u will be dis-
11.1 Introduction 177

placed towards d to give a new vector v as a consequence of the IPS. Using equa-
tion 11.4, we see that

[α; v] = (α P α)[α; u] = [α; u] + m[α; d](p; α∗ )[α; u] = [α; u] + β[α; d]

where β is a scalar quantity β = m(p; α∗ )[α; u].

Evidently, v = u+βd, with β = 0 if u lies in the invariant-plane. All points in the
lattice are thus displaced in the direction d, although the extent of displacement
depends on β.
(iii) Suppose now that only a volume fraction VV of the specimen underwent the
twinning deformation, the remainder being unaffected by the applied stress. The
tensile strain recorded over the whole specimen as the gauge length would simply
be 0.40 VV , which is obtained by replacing m in equation 11.3 by VV m [10, 12].
(iv) If the shear strain is allowed to vary, as is usually the case during slip deforma-
tion, then the position of the tensile axis is still given by v = u + βd, with β and
v both varying as the test progresses. Since v is a linear combination of u and βd,
it must always lie in the plane containing both u and d. Hence, the tensile axis
rotates in the direction d within the plane defined by the original tensile axis and
the shear direction, as illustrated in Fig. 11.3.

Considering further the deformation of single-crystals, an applied stress σ can be

resolved into a shear stress τ acting on a slip system. The relationship between σ
and τ can be shown (page 91) to be τ = σ cos φ cosλ, where φ is the angle between
the slip plane normal and the tensile axis, and λ is the angle between the slip direc-
tion and the tensile axis. Glide will first occur in the particular slip system for which
τ exceeds the critical resolved shear stress necessary to initiate dislocation motion
on that system. In austenite, glide is easiest on {1 1 1}⟨0 1 1⟩ and the γ standard
projection (Fig. 11.4a) can be used [9] to determine the particular slip system which
has the maximum resolved shear stress due to a tensile stress applied along u. For
example, if u falls within the stereographic triangle labelled a2, then (1 1 1)[0 1 1]
can be shown to be the most highly stressed system. Hence, when τ reaches a criti-
cal value (the critical resolved shear stress), this system alone operates, giving “easy
glide” since there is very little work hardening at this stage of deformation; the dis-
locations which accomplish the shear can simply glide out of the crystal and there
is no interference with this glide since none of the other slip systems are active. Of
course, the tensile axis is continually rotating towards d and may eventually fall on
the boundary between two adjacent triangles in Fig. 11.4a. If u falls on the boundary
between triangles a2 and d4, then the slip systems (1 1 1)[0 1 1] and (1 1 1)[1 0 1]
are both equally stressed. This means that both systems can simultaneously operate
and duplex slip is said to occur; the work hardening rate drastically increases as
dislocations moving on different planes interfere with each other in a way which hin-
ders glide and increases the defect density. It follows that a crystal which is initially
orientated for single slip eventually deforms by multiple slip.
178 11 Invariant-plane strains

--
6 5 --- 101 --
b4 d4 111 111
a6 c5
4 ---
b2 c a d1 --- 213
a2 c1 112
d6 2 d4 b4 1 b5 a2 c1
010 -- d4 b4
c5 c3 a3 a6 011 011
b2 d1 c3 a3
a2 d
3
b
b2 d1
b5 d6 c3
5 6
--
111 111
(a) 100 (b) 101

FIGURE 11.4
Stereographic analysis of slip in fcc single-crystals. The dashed curve in (b) is the
trace of (111).

Example 11.2: Transition from easy glide to duplex slip

A single-crystal of austenite is tensile tested at 25◦ C, the stress being applied
along [2 1 3] direction; the specimen deforms by easy glide on the (1 1 1)[0 1 1]
system. If slip can only occur on systems of this form, calculate the tensile strain
necessary for the onset of duplex slip. Assume that the ends of the specimen are
maintained in alignment throughout the test.
The tensile axis (v) is expected to rotate towards the slip direction, its motion
being confined to the plane containing the initial tensile axis (u) and the slip di-
rection (d). In Fig. 11.4b, v will therefore move on the trace of the (1 1 1) plane.
Duplex slip is expected to begin when v reaches the great circle which separates
the stereographic triangles a2 and d4 of Fig. 11.4b, since the (1 1 1)[1 0 1] slip
system will have a resolved shear stress equal to that on the initial slip system. The
tensile axis can be expressed as a function of the shear strain m as in the example
on the tensile testing of single crystals (page 175):
[γ; v] = [γ; u] + m[γ; d](p; γ ∗ )[γ; u]
a
where (p; γ ∗ ) = √γ (1 1 1) and [γ; d] = √ 1 [0 1 1], so that
3 2aγ

4m 4m
[γ; v] = [γ; u] + √ [0 1 1] = [2 1 3] + √ [0 1 1]
6 6
When duplex slip occurs, v must lie along the intersection of the (1 1 1) and (1 1 0)
planes, the former being the plane on which v is confined to move and the latter
being the boundary between triangles a2 and d4. It follows that v ∥ [1 1 2] and
must be of the form v = [v v 2v]. Substituting this into the earlier equation gives
4m
[v v 2v] = [2 1 3] + √ [0 1 1]
6
11.2 Deformation twins 179
and on comparing coefficients from both sides of this equation, we obtain

[γ; v] = [2 2 4]

so that the tensile strain required is (|v| − |u|)/|u| = 0.31.

11.2 Deformation twins

We can now proceed to study twinning deformations [13, 14], in greater depth, noting
that a twin is said to be any region of a parent which has undergone a homogeneous
shear to give a re-orientated region with the same crystal structure. The example
below illustrates some of the important concepts of twinning deformation.
Example 11.3: Twins in fcc crystals
Show that the austenite lattice can be twinned by a shear deformation on the
{1 1 1} plane and in the ⟨1 1 2⟩ direction. Deduce the magnitude of the twinning
shear, and explain why this is the most common twinning mode in fcc crystals.
Derive the matrix representing the orientation relationship between the twin and
parent lattices.

(a) (b)

(c) (d)

FIGURE 11.5
Twinning in the fcc austenite crystal structure. The diagrams represent sections in
a
the (1 1 0) plane. The vector v is 6γ [112]. In (c), K2′ and η2′ are the final positions
of the undistorted plane K2 and the undistorted direction η2 , respectively. (d) A
180 11 Invariant-plane strains
close-packed {111} plane showing that a shear along v is not equivalent to one
parallel to −v, since the magnitude of the reverse displacement has to be twice as
large.

The {1 1 1} planes in fcc crystals have a stacking sequence . . . ABCABC . . .

The region of the parent which becomes re-orientated due to the twinning
shear can be generated by reflection across the twinning plane; the stacking
sequence across the plane B which is the coherent twin interface is therefore
. . . ABCABACBA . . . Fig. 11.5a illustrates how a stack of close-packed planes
(stacking sequence . . . ABC . . .) may be labelled . . . − 1, 0, +1 . . . Reflection
across 0 can be achieved by shearing atoms in the +1 plane into positions which
are directly above (i.e. along ⟨1 1 1⟩) the atoms in the −1 plane.
Fig. 11.5a is a section of the lattice on the (1 1 0) plane; it is evident that a
displacement of all the atoms on +1 through a distance v = |v| along ⟨1 1 2⟩
gives the required reflection across the twinning plane 0. The twinning shear is
given by the equation s2 = (v/d)2 , where d is the spacing of the (1 1 1) planes.
Since v 2 = u2 − 4d2 , we may write

s2 = (u/d)2 − 4 (11.7)

where u = |u| and u connects a site on the +1 plane to an equivalent site on

the −1 plane (Fig.√11.5a). Hence, the fcc lattice can be twinned by a shear of
magnitude s = 1/ 2 on {1 1 1}.

To answer why a crystal twins in a particular way, it is necessary to make the

physically reasonable assumption that the twinning mode should lead to the small-
est possible shear. When the twin is forced to form in a constrained environment,
such as within a polycrystalline material, the shape change resulting from the shear
deformation causes elastic distortions in both the twin and the matrix. The conse-
quent strain energy increase (per unit volume of material) is given approximately
by E = (c/r)µs2 [15–17], where c and r represent the thickness and length of the
twin respectively, and µ is the shear modulus. This also is the reason why mechan-
ical twins have a lenticular morphology, since the small thickness to length ratio of
thin-plates minimises E. Annealing twins grow diffusionally so there is no physical
deformation involved in their growth. Hence, their shape is not restricted to that of a
thin plate, the morphology being governed by the need to minimise interface energy.
It is interesting that annealing and mechanical twins are crystallographically equiva-
lent (if we ignore the absence of a shape change in the former) but their mechanisms
of growth are very different.
Equation 11.7 indicates that s can be minimised by choosing twinning planes
with large d-spacings and by choosing the smallest vector u connecting a site on the
+1 plane to an equivalent site on the −1 plane; for the (1 1 1) plane the smallest u
is 21 [1 1 2], as illustrated in Fig. 11.5a. Equation 11.7 can also be used to show that
none of the planes of slightly smaller spacing than {1 1 1} can lead to twins with
1
s < 2− 2 ; two of these planes are also mirror planes and thus cannot serve as the
invariant-plane (K1 , Fig. 11.5b) of the reflection twin.
11.2 Deformation twins 181

From Fig. 11.5a,d we see that the twin lattice could also have been obtained by
displacing the atoms
√ √ in the +1 plane through
√ a distance 2v along [1 1 2] had u been
chosen to equal [ 2 2 0], giving s = 2. This larger shear is of course inconsis-
tent with the hypothesis that the favoured twinning mode involves the smallest shear,
and indeed, this mode of twinning is not observed. To obtain the smallest shear, the
magnitude of the vector v must also be minimised; in the example under considera-
tion, the correct choice of v will connect a lattice site of plane +1 with the projection
of its nearest neighbour lattice site on plane −1. The twinning direction is therefore
expected to be along [1 1 2]. It follows that the operative twin mode for the fcc lattice
1
should involve a shear of magnitude s = 2− 2 on {1 1 1}⟨1 1 2⟩.
The matrix-twin orientation relationship (M J T) can be deduced from the fact
that the twin was generated by a shear which brought atoms in the twin into positions
which are related to the parent lattice points by reflection across the twinning plane
(the basis vectors of M and T define the fcc unit cells of the matrix and twin crystals
respectively). From Fig. 11.5 we note that:

[1 1 2]M ∥ [1 1 2]T [1 1 0]M ∥ [1 1 0]T [1 1 1]M ∥ [1 1 1]T

It follows that
⎛ ⎞ ⎛ ⎞⎛ ⎞
1 1 1 J11 J12 J13 1 1 1
⎜ ⎟ ⎜ ⎟⎜ ⎟
1 1⎠ = ⎝J21
⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝1 J22 J23 ⎠ ⎝1 1 1⎠
2 0 1 J31 J32 J33 2 0 1

Solving for (M J T), we get

⎛ ⎞⎛ ⎞ ⎛ ⎞
1 1 1 1 1 2 1 2 2
1⎜ ⎟⎜ ⎟ 1⎜ ⎟
(M J T) = ⎜ ⎟⎜ ⎟= ⎜ ⎟
6 ⎝1 1 1⎠ ⎝3 3 0⎠ 3 ⎝2 1 2⎠
2 0 1 2 2 2 2 2 1

11.2.1 Comments
(i) Equations like equation 11.7 can be used to predict the likely ways in which dif-
ferent lattices might twin, especially when the determining factor is the magnitude
of the twinning shear [13].
(ii) There actually are four different ways of generating the twin lattice from the
parent crystal: (a) by reflection about the K1 plane on which the twinning shear
occurs, (b) by a rotation of π about η1 , the direction of the twinning shear, (c) by
reflection about the plane normal to η1 and (d) by a rotation of π about the normal
to the K1 plane.

Since most metals are centrosymmetric, operations (a) and (d) produce crystallo-
graphically equivalent results, as do (b) and (c). In the case of the fcc twin discussed
182 11 Invariant-plane strains

above, the high symmetry of the cubic lattice means that all four operations are crys-
tallographically equivalent. Twins which can be produced by the operations (a) and
(d) are called type I twins; type II twins result form the other two twinning opera-
tions. The twin discussed in the above example is called a compound twin, since type
I and type II twins cannot be crystallographically distinguished.
Fig. 11.5b illustrates some additional features of twinning. The K2 plane is the
plane which (like K1 ) is undistorted by the twinning shear, but unlike K1 , is rotated
by the shear. The ‘plane of shear’ is the plane containing η1 and the perpendicular
to K1 ; its intersection with K2 defines the undistorted but rotated direction η2 . In
general, η2 and K1 are rational for type I twins, and η1 and K2 are rational for
type II twins. The set of four twinning elements K1 , K2 , η1 and η2 are all rational
for compound twins. From Fig. 11.5b, η2 makes an angle of arctan(s/2) with the
normal to K1 and simple geometry shows that η2 = [1 1 2] for the fcc twin of
Example 11.2. The corresponding K2 plane which contains η2 and η1 ∧η2 is therefore
(1 1 1), giving the rational set of twinning elements
1
K1 = (1 1 1) η2 = [1 1 2] s = 2− 2 η1 = [1 1 2] K2 = (1 1 1)
In fact, it only is necessary to specify either K1 and η2 or K2 and η1 to completely
describe the twin mode concerned.
The deformation matrix (M P M) describing the twinning shear can be deduced
using equation 11.3 and the information that [M;d] ∥ [1 1 2], (p; M∗ ) ∥ (1 1 1) and
1
s = 2− 2 to give
⎛ ⎞ ⎛ ⎞
7 1 1 5 1 1
1⎜ ⎟ 1⎜ ⎟
(M P M) = ⎜ 1 7 1
⎟ and (M P M)−1 = ⎜ 1 5 1
⎟ (11.8)
6 ⎝ ⎠ 6 ⎝ ⎠
2 2 4 2 2 8
and if a vector u is deformed into a new vector v by the twinning shear, then
(M P M)[M;u] = [M;v] (11.9)
and if h is a plane normal which after deformation becomes k, then
−1
(h; M∗ )(M P M) = (k; M∗ ) (11.10)
These laws can be used to verify that p and d are unaffected by the twinning shear,
and that the magnitude of a vector originally along η2 is not changed by the defor-
mation; similarly, the spacing of the planes initially parallel to K2 remains the same
after deformation, although the planes are rotated.

11.3 Correspondence matrix

The property of the homogeneous deformations we have been considering is that
points which are initially collinear remain so in spite of the deformation, and lines
11.4 Correspondence matrix 183

which are initially coplanar remain coplanar after the strain. Using the data from
equation 11.8, we see that the deformation (M P M) alters the vector [M;u] = [0 0 1]
to a new vector [M;v] = 16 [1 1 4] i.e.

1
(M P M)[0 0 1]M = [1 1 4]M (11.11a)
6
The indices of this new vector v relative to the twin basis T can be obtained using the
coordinate transformation matrix (T J M), so that
1 1
(T J M) [1 1 4]M = [T; v] = [1 1 0]T (11.11b)
6 2
Hence, the effect of the shear stress is to deform a vector [0 0 1]M of the parent lattice
into a vector 12 [1 1 0]T of the twin. Equations 11.11a,b could have been combined to
obtain this result, as follows:

(T J M)(M P M)[M;u] = [T; v]

or
(T C M)[M;u] = [T; v]
where
(T J M)(M P M) = (T C M) (11.12)
The matrix (T C M) is called the correspondence matrix; the initial vector u
with indices [M;u] in the parent basis, due to deformation becomes a corresponding
vector v with indices [T;v] in the twin basis. The correspondence matrix tells us that a
certain vector in the twin is formed by deforming a particular corresponding vector of
the parent. In the example considered above, the vector u has rational components in
M (i.e. the components are small integers or fractions) and v has rational components
in T. It follows that the elements of the correspondence matrix (T C M) must also be
rational numbers or fractions. The correspondence matrix can usually be written from
inspection since its columns are rational lattice vectors referred to the second basis
produced by the deformation from the basis vectors of the first basis.
We can similarly relate planes in the twin to planes in the parent, the correspon-
dence matrix being given by
−1
(M C T) = (M P M) (M J T)

where
(h; M∗ )(M C T) = (k; T∗ )
so that the plane (k; T∗ ) of the twin was formed by the deformation of the plane
(h; M∗ ) of the parent.
184 11 Invariant-plane strains

11.4 An alternative to the Bain strain

The pure strain proposed originally by Bain [18] (page 120) is one way of homo-
geneously deforming the austenite lattice into that of a body-centred cubic or body-
centred tetragonal unit cell of martensite. The deformation involves√a uniform ex-
pansion along two of the cubic-austenite cell axes by the ratio η1 = 2aα /aγ , and
a compression along the third axis by η3 = aα /aγ , where aα and aγ are the lat-
tice parameters of the cubic unit cells of martensite and austenite respectively. The
Bain strain is not, however, a unique way of deforming the cubic-austenite unit cell
into that of martensite. Fig. 11.6 shows an alternative correspondence in terms of
a triclinic body-centred cell of austenite [19, 20] that can be deformed into a body-
centred cubic cell. In what follows, all vectors and operations are referred to the cubic
austenite lattice.
[001]
[013]
[111]

60° [110]
[101]

[010]

[100]

FIGURE 11.6
An alternative correspondence between the austenite, conventionally represented by
the cubic unit cells illustrated, can also be represented by the body-centred triclinic
cell. The main diagram is adapted from [19]. Not all the iron atoms in the cubic cells
are illustrated for clarity. The directions oa, ob and oc are, respectively, parallel to
[101]γ , [110]γ and [013]γ .
11.4 An alternative to the Bain strain 185
a
The basal plane of the triclinic cell consists of vectors oa = 2γ [101] and ob =
aγ
2 [110], making an angle of 60◦ with each other. The normal to the basal plane is
a
therefore parallel to the [111] direction but the vector oc = 2γ [013]. A shear (γ S1 γ)
of s1 = tan 43◦ parallel to [415] on (111) would make oc normal to the basal plane
of the triclinic cell without affecting any vectors in the basal plane defined by the
triclinic cell edges oa and ob. Defining d = √142 [415] as the unit vector parallel
to the shear direction, and p = √13 [111] as the unit normal to the shear plane, and
defining φ = s1 √13 √142 , it follows that

(γ S1 γ) = I + s1 [γ; d](p; γ ∗ )
⎛ ⎞
1 + 4φ −4φ 4φ
⎜ ⎟
= ⎜
⎝ −φ 1+φ
⎟
−φ ⎠ (11.13)
−5φ 5φ 1 − 5φ

This deformation makes the original vector [013] rotated and distorted to
[1.33 1.33 1.33] ∥ [111]. It would need to be compressed by a ratio η3 =
aα /2.3076aγ so that its magnitude becomes the lattice parameter aα of martensite.
To change the angle 60◦ between oa and ob to 90◦ without affecting any vectors
normal to the basal plane requires a second shear (γ S2 γ) of magnitude s2 = tan 30◦
with d = √12 [101] and p = √16 [121]. Taking ψ = s2 √12 √16 :

(γ S2 γ) = I + s2 [γ; d](p; γ ∗ )
⎛ ⎞
1 − ψ −2ψ −ψ
⎜ ⎟
= ⎜
⎝ 0 1 0
⎟.
⎠ (11.14)
ψ 2ψ 1+ψ

This shear leaves [101] unaffected, whereas a vector originally parallel to [110] be-
comes [ 21 1 12 ] which is at 90◦ to [101]. To match the lattice parameter aα , the vector
1
√
2 [101] needs to expanded by η1 = 2aα /aγ whereas 12 [ 12 1 12 ] would need to be ex-
panded by the larger quantity η2 = 1.633aα/aγ because the shear (γ S2 γ) reduces
the length of 21 [110].
Therefore, the total deformation required to convert the triclinic austenite cell
into the body-centred cubic cell of martensite is
⎛ ⎞⎛ ⎞⎛ ⎞
1 + 4φ −4φ 4φ 1 − ψ −2ψ −ψ η1 0 0
⎜ ⎟⎜ ⎟⎜ ⎟
(γ S γ) = ⎜ ⎝ −φ 1 + φ −φ
⎟⎜
⎠ ⎝ 0 1 0
⎟⎜
⎠ ⎝ 0 η2 0
⎟.
⎠
−5φ 5φ 1 − 5φ ψ 2ψ 1+ψ 0 0 η3
Since shears do not lead to any change in volume, the ratio of the volumes of the
martensite and ferrite unit cells is given by the determinant of the distortion matrix,
i.e. η1 η2 η3 = a3α /a3γ as expected. These distortions, even though they are not mea-
sured along the principal axes, are large when compared with the ones associated
186 11 Invariant-plane strains

with the Bain strain. As pointed out by Wayman, it might be concluded on this basis
and and with the backing of other experimental evidence, that the Bain strain is the
homogeneous deformation that leads to a change from fcc→bcc structure.

11.5 Stepped interfaces

A planar coherent twin boundary (unit normal p) can be generated from a single crys-
tal by shearing on the twinning plane p, the unit shear direction and shear magnitude
being d and m respectively.
On the other hand, to generate a similar boundary but containing a step of height
h requires additional virtual operations (Fig. 11.7) [10, 21, 22]. The single crystal
is first slit along a plane which is not parallel to p (Fig. 11.7b), before applying the
twinning shear. The shear which generates the twinned orientation also opens up the
slit (Fig. 11.7c), which then has to be rewelded (Fig. 11.7d) along the original cut;
this produces the required stepped interface. A Burgers circuit constructed around
the stepped interface will, when compared with an equivalent circuit constructed
around the unstepped interface exhibit a closure failure. This closure failure gives
the Burgers vector bI associated with the step:

bI = hmd

The operations outlined above indicate one way of generating the required stepped
interface. They are simply the virtual operations which allow us to produce the re-
quired defect - similar operations were first used by Volterra [23] in describing the
elastic properties of cut and deformed cylinders, operations which were later recog-
nised to generate the ordinary dislocations that metallurgists are so familiar with.
Having defined bI , we note that an initially planar coherent twin boundary can
acquire a step if a dislocation of Burgers vector bm crosses the interface. The height
of the step is given by [10]:
h = bm .p
so that
bI = m(bm .p)d. (11.15)
From equation 11.3, the invariant plane strain necessary to generate the twin from
the parent lattice is given by (M P M) = I + m[M;d](p; M∗ ) so that equation 11.15
becomes
[M;bI ] = (M P M)[M;bm ] − [M;bm ] (11.16)
11.5 Stepped interfaces 187

(a) (c)

FIGURE 11.7
The virtual operations [24] used in determining bI . (a) Single crystal. (b) Crystal cut
along ‘xy’. (c) Gap opens on shearing. The closure failure bI is the Burgers vector
of the final step in the coherent twin boundary, that results on rewelding the gap as
shown in (d).
188 11 Invariant-plane strains

Example 11.4: Interaction of dislocations with interfaces

Deduce the correspondence matrix for a fcc deformation twin and hence show
that there are no geometrical restrictions to the passage of slip dislocations across
coherent twin boundaries in fcc materials.
It was shown on page 179 that
⎛ ⎞ ⎛ ⎞
1 2 2 7 1 1
1⎜ ⎟ 1⎜ ⎟
(T J M) = ⎜ ⎟ and (M P M) = ⎜ ⎟.
3 ⎝2 1 2 ⎠ 6 ⎝1 7 1⎠
2 2 1 2 2 4

The correspondence matrix (T C M) which associates each vector of the parent

with a corresponding vector in the twin is, from equation 11.12, given by

(T C M) = (T J M)(M P M)

(M C T) = (M P M)−1 (M J T)
so that ⎛ ⎞
1 1 1
1⎜ ⎟
(T C M) = (M C T) = ⎜ 1 1⎠ .
⎟
2 ⎝1
2 2 0
The character of a dislocation will in general be altered on crossing an in-
terface. This is because the crossing process introduces a step in the interface,
rather like the slip steps which arise at the free surfaces of deformed crystals. We
consider the case where a dislocation crosses a coherent twin boundary. The in-
terfacial step has dislocation character so that the original dislocation (Burgers
vector bm ) from the parent crystal is in effect converted into two dislocations,
one being the step (Burgers vector bI ) and the other the dislocation (Burgers vec-
tor bt ) which has penetrated the interface and entered into the twin lattice. If the
total Burgers vector content of the system is to be preserved then it follows that
in general, bt ̸= bm , since bm = b+ bt . Using this equation and equation 11.16,
we see that
[M;bt ] = (M P M)[M;bm ]
or
[T; bt ] = (T J M)(M P M)[M;bm ]
so that
[T; bt ] = (T C M)[M;bm ].
Clearly, dislocation glide across the coherent interface will not be hindered if
bt is a perfect lattice vector of the twin. If this is not the case and bt is a partial
dislocation in the twin, then glide across the interface will be hindered because
the motion of bt in the twin would leave a stacking fault trailing all the way from
the interface to the position of the partial dislocation in the twin.
11.5 Stepped interfaces 189
There is an additional condition to be fulfilled for easy glide across the inter-
face; the corresponding glide planes pm and pt of dislocations bm and bt in
the parent and twin lattices respectively, must meet edge to edge in the interface.
Now,
(pt ; T∗ ) = (pm ; M∗ )(M C T).
If the interface plane normal is pi , then the edge to edge condition is satisfied
if pm ∧ pi ∥ pt ∧ pi .
Dislocations in fcc materials usually glide on close-packed {1 1 1} planes
and have Burgers vectors of type a2 ⟨1 1 0⟩. Using the data of Table 11.1 it can
easily be verified that all the close-packed planes of the parent lattice meet the
corresponding glide planes in the twin edge to edge in the interface, which is
taken to be the coherent (1 1 1)M twinning plane. Furthermore, all the a2 ⟨1 1 0⟩
Burgers vectors of glide dislocations in the parent correspond to perfect lattice
dislocations in the twin. It must be concluded that the coherent twin boundary
for {1 1 1} twins in fcc metals does not offer any geometrical restrictions to the
transfer of slip between the parent and product lattices.

TABLE 11.1
Corresponding glide planes and Burgers vectors

Parent Twin
a
2 [1 1 0] a[0 0 1]
a a
2 [1 0 1] 2 [0 1 1]
a a
2 [0 1 1] 2 [1 0 1]
a a
2 [1 1 0] 2 [1 1 0]
a a
2 [1 0 1] 2 [1 0 1]
a a
2 [0 1 1] 2 [0 1 1]
(1 1 1) (1 1 1)
(1 1 1) (1 1 1)
(1 1 1) (0 2 0)
(1 1 1) (2 0 0)

These data (Table 11.1) show also that all dislocations with Burgers vectors in
the (1 1 1)M plane are unaffected, both in magnitude and direction, as a result of
crossing into the twin. For example, a2 [1 1 0]M becomes a2 [1 1 0]T so that |bm | =
|bt |, and using (T J M) it can be demonstrated that [1 1 0]M ∥ [1 1 0]T . This result
is expected because these particular dislocations cannot generate a step in the
(1 1 1)M interface when they cross into the twin lattice (see equation 11.15). Only
dislocations with Burgers vectors not parallel to the interface cause the formation
of steps.
190 11 Invariant-plane strains
The data further illustrate the fact that when bm lies in the (1 1 1)M plane,
there is no increase in energy due to the reaction bm → bI + bt , which occurs
when a dislocation crosses the interface. This is because bI = 0 and bt = bm . For
all other cases bI is not zero and since |bt | is never less than |bm | , bm → bI + bt
is always energetically unfavourable. In fact, in the example being discussed, there
can never be an energy reduction when an a2 ⟨1 1 0⟩ dislocation penetrates the co-
herent twin boundary. The dislocations cannot therefore spontaneously cross the
boundary. A trivial case where dislocations might spontaneously cross a boundary
is when the latter is a free surface, assuming that the increase in surface area (and
hence surface energy) due to the formation of a step is not prohibitive. Sponta-
neous penetration of the interface might also become favourable if the interface
separates crystals with very different elastic properties.

FIGURE 11.8
The passage of a slip dislocation across a coherent twin boundary in a bcc crystal.
1
The twinning system is {1 1 2}⟨1 1 1⟩, s = 2− 2 . The subscripts m and t refer
11.5 Stepped interfaces 191
to the twin and matrix respectively; the arrows at the dislocation symbols indicate
the sense of the Burgers vectors and the dislocation line vectors are all parallel to
[1 1 0]m,t where m refers to the matrix and t to the twin.

The results obtained show that single dislocations can glide into twins in fcc
crystals without leaving a fault; there are no geometrical restrictions to the passage
of slip dislocations across the coherent twin boundaries concerned. It can similarly
be demonstrated that slip dislocations can comfortably traverse the coherent twin
boundaries of {1 1 2} twins in bcc or bct lattices and this has implications on the
interpretation of the strength of martensite [22]. As will be discussed later, the sub-
structure of martensite plates in steels (and in many non-ferrous alloys) often consists
of very finely spaced {1 1 2} transformation twins. It was at one time believed that
the twins were mainly responsible for the high strength of ferrous martensites, be-
cause the numerous twin boundaries should hinder slip – the analysis above clearly
suggests otherwise. Indeed, twinned martensites which do not contain carbon also
do not exhibit exceptionally high strengths and it is now generally accepted that
the strength of virgin ferrous martensites is largely due to interstitial solid solution
hardening by carbon atoms, or in the case of lightly autotempered martensites due
to carbon atom clustering or fine precipitation. Consistent with this, it is found that
Fe-30Ni wt% twinned martensites are not particularly hard.
Finally, it should be mentioned that even when glide across coherent twin bound-
aries in martensites should be unhindered, the boundaries will cause a small amount
of hardening, partly because the corresponding slip systems in the matrix and twin
will in general be differently stressed [22, 25] (simply because they are not necessar-
ily parallel) and partly due to the work necessary to create the steps in the interfaces.
It is emphasised, however, that these should be relatively small contributions to the
strength of martensite. Fig. 11.8 illustrates the passage of a slip dislocation across a
coherent {1 1 2} twin interface in a bcc material.

Example 11.5: fcc to hcp transformation revisited

A Co-6.5Fe wt% alloy transforms from an fcc (γ) structure to a hcp martensite
structure with zero change in density [26]. The invariant plane of the transforma-
tion is the close-packed {1 1 1}γ plane, the shear direction being ⟨1 1 2⟩γ . The
1
magnitude of the shear is 8− 2 , which is half the normal twinning shear for fcc
crystals. By factorising the total transformation strain into a pure strain and a rigid
body rotation, show that the maximum extension or contraction suffered by any
vector of the parent lattice, as a result of the transformation, is less than 20%.
Representing the fcc parent lattice in an orthonormal basis Z, consisting of
unit basis vectors parallel to [1 0 0], [0 1 0] and [0 0 1] fcc directions respectively,
1 1 1
and substituting (p; Z∗ ) = (3− 2 )(1 1 1), [Z; d] = (6− 2 )[1 1 2] and m = 8− 2
192 11 Invariant-plane strains
into equation 11.3, the total transformation strain (Z P Z) is found to be:
⎛ ⎞
13 1 1
1 ⎜ ⎟
(Z P Z) = ⎜ ⎟
12 ⎝ 1 13 1 ⎠
−2 −2 10

This can be factorised into a pure strain (Z Q Z) and a rigid body rotation (Z J Z)
(Fig. 10.5). Writing (Z T Z) = (Z P′ Z)(Z P Z), we obtain:
⎛ ⎞
174 30 −6
1 ⎜ ⎟
(Z T Z) = ⎜
30 174 −6
⎟.
144 ⎝ ⎠
−6 −6 102

The eigenvalues and eigenvectors of (Z T Z) are:

λ1 = 1.421535 [Z; u] = [0.704706 0.704706

−0.082341]
λ2 = 1.000000 [Z; v] = [0.707107 −0.707107 0.000000] .
λ3 = 0.703465 [Z; x] = [0.058224 0.058224 0.996604]

Notice that the eigenvectors form an orthogonal set and that consistent with the
fact that v lies in the invariant plane, λ2 has a value of unity. u, v and x are also the
eigenvectors of (Z Q Z). The eigenvalues of (Z Q Z) are given by the square roots
of the eigenvalues of (Z T Z); they are 1.192282, 1.0 and 0.838728. Hence, the
maximum extensions and contractions are less than 20% since each eigenvalue is
the ratio of the final to initial length of a vector parallel to an eigenvector. The
maximum extension occurs along u and the maximum contraction along x.

The matrix (Z Q Z) is given by equation 10.12 as:

⎛ ⎞⎛ ⎞
0.70471 0.70711 0.05822 1.19228 0.0 0.0
⎜ ⎟⎜ ⎟
(Z Q Z) = ⎜ ⎝ 0.70471 −0.70711 0.05822⎠ ⎝ 0.0
⎟⎜
1.00000
⎟
0.0 ⎠
−0.08234 0.00000 0.99660 0.0 0.0 0.83873
⎛ ⎞
0.70471 0.70471 −0.08234
⎜ ⎟
×⎜ ⎟
⎝0.70711 −0.70711 0.00000 ⎠
0.05822 0.05822 0.99660
⎛ ⎞
1.094944 0.094943 −0.020515
⎜ ⎟
= ⎜
⎝ 0.094943 1.094944
⎟
−0.020515⎠ (11.17)
−0.020515 −0.020515 0.841125
11.5 Stepped interfaces 193

and ⎛ ⎞
0.920562
−0.079438 0.020515
−1 ⎜ ⎟
(Z Q Z) =⎜
⎝−0.079438 0.920562 0.020515⎠
⎟

0.020515 0.020515 1.189885

−1
From equation 10.13, (Z J Z) = (Z P Z)(Z Q Z)
⎛ ⎞
0.992365 −0.007635 0.123091
⎜ ⎟
(Z J Z) = ⎜⎝−0.007635 0.992365 0.123091⎠ .
⎟ (11.18)
−0.123092 −0.123092 0.984732

The matrix (Z J Z) represents a right-handed rotation of 10.03◦ about [1 1 0]Z axis.

It is interesting to examine what happens to the vector [1 1 2]Z due to the opera-
tions (Z Q Z) and (Z J Z):

(Z Q Z)[1 1 2]Z = [1.230916 1.230916 − 1.723280]Z

where the new vector can be shown to have the same magnitude as [1 1 2] but points
in a different direction. The effect of the pure rotation is

(Z J Z)[1.230916 1.230916 − 1.723280]Z = [1 1 2]Z .

Thus, the pure strain deforms [1 1 2]Z into another vector of identical magnitude
and the pure rotation brings this new vector back into the [1 1 2]Z direction, the
net operation leaving it invariant, as expected, since [1 1 2]Z is the shear direction
which lies in the invariant plane. Referring to Fig. 10.5, the direction f e = [1 1 0]Z ,
yz = [1 1 2]Z and y ′ z ′ = [1.230916 1.230916 − 1.723280]Z. y ′ z ′ is brought into
coincidence with yz by the rigid body rotation (Z J Z) to generate the invariant plane
containing f e and yz.
Physically, the fcc to hcp transformation occurs by the movement of a single set
of Shockley partial dislocations, Burgers vector b = a6 ⟨1 1 2⟩γ on alternate close-
packed {1 1 1}γ planes. To produce a fair thickness of hcp martensite, a mechanism
has to be sought which allows Shockley partials to be generated on every other slip
plane. Some kind of a pole mechanism (see for example, p. 310 of [27]) would allow
this to happen, but there is as yet no experimental evidence confirming this. Motion
of the partials would cause a shearing of the γ lattice, on the system {1 1 1}γ ⟨1 1 2⟩γ ,
the average magnitude s of the shear being s = |b|/2d, where d is the spacing of the
1 1 1
close-packed planes. Hence, s = 6− 2 a/2(3− 2 a) = 8− 2 . This is exactly the shear
system we used in generating the matrix (Z P Z) and the physical effect of the shear
on the shape of an originally flat surface is, in general, to tilt the surface (about a
line given by its intersection with the hcp habit plane) through some angle dependant
on the indices of the free surface. By measuring such tilts it is possible to deduce s,
which has been experimentally confirmed to equal half the twinning shear.
In fcc crystals, the close-packed planes stack in the sequence . . . ABCABC . . .;
the passage of a single Shockley partial causes the sequence to change to . . . ABA . . .
194 11 Invariant-plane strains

creating a three layer thick region of hcp phase since the stacking sequence of close-
packed planes in the hcp lattice has a periodicity of 2. This then is the physical
manner in which the transformation occurs, the martensite having a {1 1 1}γ habit
plane – if the parent product interface deviates slightly from {1 1 1}γ , then it will
consist of stepped sections of close-packed plane, the steps representing the Shockley
partial transformation dislocations. The spacing of the partials along ⟨1 1 1⟩γ would
be 2d. In other words, in the stacking sequence ABC, the motion of a partial on B
1
would leave A and B unaffected though C would be displaced by 6− 2 a⟨1 1 2⟩γ
to a new position A, giving ABA stacking. Partials could thus be located on every
alternate plane of the fcc crystal.
Hence, we see that the matrix (Z P Z) is quite compatible with the microscopic
dislocation based mechanism of transformation. (Z P Z) predicts the correct macro-
scopic surface relief effect and its invariant plane is the habit plane of the martensite.
However, if (Z P Z) is considered to act homogeneously over the entire crystal, then
it would carry half the atoms into the wrong positions. For instance, if the habit
plane is designated A in the sequence ABC of close packed planes, then the effect
of (Z P Z) is to leave A unchanged, shift the atoms on plane C by 2sd and those on
plane B by sd along ⟨1 1 2⟩γ . Of course, this puts the atoms originally in C sites into
A sites, as required for hcp stacking. However, the B atoms are located at positions
a
half way between B and C sites, through a distance 12 ⟨1 1 2⟩γ . Shuffles are thus
necessary to bring these atoms back into the original B positions and to restore the
. . . ABA . . . hcp sequence. These atomic movements in the middle layer are called
shuffles because they occur through very small distances (always less than the inter-
atomic spacing) and do not affect the macroscopic shape change [21]. The shuffle
here is a purely formal concept; consistent with the fact that the Shockley partials
glide over alternate close-packed planes, the deformation (Z P Z) must in fact be
considered homogeneous only on a scale of every two planes. By locking the close-
packed planes together in pairs, we avoid displacing the B-site atoms to the wrong
positions and thus automatically avoid the reverse shuffle displacement.
In the particular example discussed above, the dislocation mechanism is estab-
lished experimentally and physically reasonable shear systems were used in deter-
mining (Z P Z). However, in general it is possible to find an infinite number of defor-
mations [8, 21] which may accomplish the same lattice change and slightly empirical
criteria have to be used in selecting the correct deformation. One such criterion could
involve the selection of deformations which involve the minimum principal strains
and the minimum degree of shuffling, but intuition and experimental evidence is al-
most always necessary to reach a decision. The Bain strain which transforms the fcc
lattice to the bcc lattice is believed to be the correct choice because it seems to in-
volve the least atomic displacements and zero shuffling of atoms [20]. The absence
of shuffles can be deduced from the Bain correspondence matrix (α C γ) which can
be deduced from inspection since its columns are rational lattice vectors referred to
the α basis, produced by the deformation of the basis vectors of the γ basis; since
[1 0 0]γ is deformed to [1 1 0]α , [0 1 0]γ to [1 1 0]α and [0 0 1]γ to [0 0 1]α , by the
11.5 Stepped interfaces 195

Bain strain (Fig. 8.2), the correspondence matrix is simply:

⎛ ⎞
1 1 0
⎜ ⎟
(α C γ) = ⎜
⎝1 1 0⎠
⎟ (11.19)
0 0 1

If u is a vector defining the position of an atom in the γ unit cell, then it can be
verified that (α C γ)[γ; u] always gives a corresponding vector in the α lattice which
terminates at a lattice point. For example, 12 [1 0 1]γ corresponds to 21 [1 1 1]α ; both
these vectors connect the origins of their respective unit cells to an atomic position.
The Bain correspondence thus defines the position of each and every atom in the α
lattice relative to the γ lattice. It only is possible to obtain a correspondence matrix
like this when the primitive cells of each of the lattices concerned contain just one
atom [8].
The primitive cell of the hcp lattice contains two atoms and any lattice corre-
spondence will only define the final positions of an integral fraction of the atoms, the
remainder having to shuffle into their correct positions in the product lattice. This can
be demonstrated with the correspondence matrix for the example presented above.
It is convenient to represent the conventional hcp lattice (basis H) in an alternative
orthorhombic basis (symbol O), with basis vectors:
1 1
[1 0 0]O = [0 1 1]γ = [1 1 2 0]H
2 3
1
[0 1 0]O = [2 1 1]γ = [1 1 0 0]H
2
2
[0 0 1]O = [1 1 1]γ = [0 0 0 1]H
3
The orthorhombic unit cell thus contains three close-packed layers of atoms par-
allel to its (0 0 1) faces. The middle layer has atoms located at [0 31 21 ]O , [1 13 21 ]O
and [ 12 65 12 ]O . The other two layers have atoms located at each corner of the unit cell
and in the middle of each (0 0 1) face, as illustrated in Fig. 11.9.
From our earlier definition of a correspondence matrix, (O C γ) can be written
directly from the relations (between basis vectors) stated earlier:
⎛ ⎞
0 2 1
1⎜ ⎟
(γ C O) = ⎜ −1 −1 1
⎟
2 ⎝ ⎠
1 −1 2

Alternatively, the correspondence matrix (O C γ) may be derived (page 182) as fol-

lows:
(O C γ) = (O J γ)(γ P γ)
The matrix (γ P γ) is the total strain, which transforms the fcc lattice into the hcp
lattice; it is equal to the matrix (Z P Z) derived previously (page 191), since the basis
196 11 Invariant-plane strains

FIGURE 11.9
Representation of bases O, H, and γ. The directions in the hexagonal cell are ex-
pressed in the Weber notation.

vectors of the orthonormal basis Z are parallel to the corresponding basis vectors of
the orthogonal basis γ. It follows that:
⎛ ⎞⎛ ⎞
0 −1 1 13/12 1/12 1/12
⎜ ⎟⎜ ⎟
(O C γ) = ⎜ ⎝2/3 −1/3 −1/3⎠ ⎝ 1/12
⎟⎜
13/12 1/12 ⎠
⎟

1/2 1/2 1/2 −2/12 −2/12 10/12

⎛ ⎞
−1/4 −5/4 3/4
⎜ ⎟
= ⎜
⎝ 3/4 −1/4 −1/4⎠
⎟

1/2 1/2 1/2

and ⎛ ⎞
1 0 2
1⎜ ⎟
(γ C O) = ⎜ ⎟.
2 ⎝−1 −1 1⎠
1 −1 2
Using this correspondence matrix, we can show that all the atoms, except those in the
middle close-packed layer in the unit cell, have their positions relative to the parent
lattice defined by the correspondence matrix. For example, the atom at the position
[1 0 0]O corresponds directly to that at [0 1 1]γ in the fcc lattice. However, [0 31 12 ]O
corresponds to 12 [7 1 4]γ and there is no atom located at these coordinates in the γ
11.6 Conjugate of an invariant-plane strain 197
1
lattice. The generation of the middle layer thus involves shuffles of 12 [1 1 2]γ , as
1 1 1
discussed earlier; we note that 12 [7 1 4]γ − 12 [1 1 2]γ = 2 [1 0 1]γ . Thus, the atom
at [0 13 12 ]O is derived from that at 21 [1 0 1]γ in addition to a shuffle displacement
a
through 12 [1 1 2]γ .

11.6 Conjugate of an invariant-plane strain

We have seen already that an fcc lattice can be transformed to an hcp lattice by
1
shearing the former on the system {1 1 1}⟨1 1 2⟩, s = 8− 2 . This shear represents
an invariant-plane strain (Z P Z) which can be factorised into a pure strain (Z Q Z)
and a rigid body rotation (Z J Z), as in section 10.4. The pure deformation (Z Q Z)
accomplishes the required lattice change from fcc to hcp, but is not an invariant-plane
strain. As illustrated in Fig. 10.5 and from equation 11.18, it is the rigid body rotation
of 10.03◦ about ⟨1 1 0⟩ that makes the {1 1 1} plane invariant and in combination
with (Z Q Z) produces the final orientation relation implied by (Z P Z).
Referring to Fig. 10.5a,b, we see that there are in fact two ways [21] in which
(Z Q Z) can be converted into an invariant-plane strain which transforms the fcc
lattice to the hcp lattice. The first involves the rigid body rotation (Z J Z) in which
y ′ z ′ is brought into coincidence with yz, as shown in Fig. 10.12c. The alternative
would be to employ a rigid body rotation (Z J2 Z), involving a rotation of 10.03◦
about ⟨1 1 0⟩, which would bring x′ w′ into coincidence with xw, making xw the
trace of the invariant-plane. Hence, (Z Q Z) when combined with (Z J2 Z) would
result in a different invariant-plane strain (Z P2 Z) which also shears the fcc lattice to
the hcp lattice. From equation 7.9, (Z J2 Z) is given by:
⎛ ⎞
0.992365 −0.007635 −0.123091
⎜ ⎟
(Z J2 Z) = ⎜⎝−0.007635 0.992365 −0.123091⎠
⎟

0.123092 0.123092 0.984732

From example 16, (Z Q Z) is given by:

⎛ ⎞
1.094944 0.094943 −0.020515
⎜ ⎟
(Z Q Z) = ⎜⎝ 0.094943 1.094944
⎟
−0.020515⎠
−0.020515 −0.020515 0.841125

From equation 10.10, (Z P2 Z) = (Z J2 Z)(Z Q Z)

⎛ ⎞
1.0883834 0.088384 −0.123737
⎜ ⎟
(Z P2 Z) = ⎜⎝ 0.088384 1.088384 −0.123737⎠ .
⎟

0.126263 0.126263 0.823232

198 11 Invariant-plane strains

On comparing this with equation 11.3, we see that (Z P2 Z) involves a shear of

1
magnitude s = 8− 2 on {5 5 7}Z ⟨7 7 10⟩Z . It follows that there are two ways of
accomplishing the fcc to hcp change:
1
Mode 1: Shear on {5 5 7}Z ⟨7 7 10⟩Z s = 8− 2
1
Mode 2: Shear on {1 1 1}Z ⟨1 1 2⟩Z s = 8− 2
Both shears can generate a fully coherent interface between the fcc and hcp lattices
(the coherent interface plane being coincident with the invariant-plane). Of course,
while the {1 1 1} interface of mode 2 would be atomically flat, the {5 5 7} interface
of mode 1 must probably be stepped on an atomic scale. The orientation relations
between the fcc and hcp lattices would be different for the two mechanisms. In fact,
(Z J2 Z) is
⎛ ⎞
−0.2121216 −1.2121216 0.6969703
⎜ ⎟
(Z J2 Z) = ⎜⎝ 0.7373739 −0.2626261 −0.2323234⎠ .
⎟

0.3484848 0.3484848 0.7121212

It is intriguing that only the second mode has been observed experimentally, even
though both involve identical shear magnitudes.
A general conclusion to be drawn from the above analysis is that whenever two
lattices can be related by an IPS (i.e. whenever they can be joined by a fully coherent
interface), it is always possible to find a conjugate IPS which in general allows the
two lattices to be differently orientated but still connected by a fully coherent inter-
face. This is clear from Fig. 10.5 where we see that there are two ways of carrying
out the rigid body rotation in order to obtain an IPS which transforms the fcc lattice
to the hcp lattice. The deformation involved in twinning is also an IPS so that for a
given twin mode it ought to be possible to find a conjugate twin mode. In Fig. 11.5c,
a rigid body rotation about [1 1 0], which brings K2 into coincidence with K2′ would
give the conjugate twin mode on (1 1 1)[1 1 2].
We have used the pure strain (Z Q Z) to transform the fcc crystal into a hcp crys-
tal. However, before this transformation, we could use any of an infinite number of
operations (e.g. a symmetry operation) to bring the fcc lattice into self-coincidence.
Combining any one of these operations with (Z Q Z) then gives us an alternative
deformation which can accomplish the fcc→hcp lattice change without altering the
orientation relationship. It follows that two lattices can be deformed into one another
in an infinite number of ways. Hence, prediction of the transformation strain is not
possible in the sense that intuition or experimental evidence has to be used to choose
the ‘best’ or ’physically most meaningful’ transformation strain.
Example 11.6: Combined effect of two invariant-plane strains
Show that the combined effect of the operation of two arbitrary invariant-plane
strains is equivalent to an invariant-line strain (ILS). Hence prove that if the two
invariant-plane strains have the same invariant-plane, or the same displacement
direction, then their combined effect is simply another IPS [2].
11.6 Conjugate of an invariant-plane strain 199
The two invariant-plane strains are referred to an orthonormal basis X and are
designated (X P X) and (X Q X), such that m and n are their respective mag-
nitudes, d and e their respective unit displacement directions and p and q their
respective unit invariant-plane normals. If (X Q X) operates first, then the com-
bined effect of the two strains is

(X P X)(X Q X) = {I + m[X; d](p; X∗ )}{I + n[X; e](q; X∗ )}

= I + m[X; d](p; X∗ ) + n[X; e](q; X∗ )
+mn[X; d](p; X∗ )[X; e](q; X∗ )
= I + m[X; d](p; X∗ ) + n[X; e](q; X∗ )
+g[X; d](q; X∗ ) (11.20)

where g is the scalar quantity g = mn(p; X∗ )[X; e].

If u is a vector which lies in both the planes represented by p and q, i.e. it is
parallel to p∧q, then it is obvious (equation 11.20) that (X P X)(X Q X)[X; u] =
[X; u], since (p; X∗ )[X; u] = 0 and (q; X∗ )[X; u] = 0. It follows that u is parallel
to the invariant line of the total deformation (X S X) = (X P X)(X Q X). This is
logical since (X P X) should leave every line on p invariant and (X Q X) should
leave all lines on q invariant. The line that is common to both p and q should there-
fore be unaffected by (X P X)(X Q X). Hence, the combination of two arbitrary
invariant-plane strains (X P X)(X Q X) gives and invariant-line strain (X S X).
If d = e, then from equation 11.20

(X P X)(X Q X) = I + [X; d](r; X∗ )

where (r; X∗ ) = m(p; X∗ ) + n(q; X∗ ) + g(q; X∗ )

which is simply another IPS on a plane whose normal is parallel to r. If p = q,

then it follows that

(X P X)(X Q X) = I + [X; f ](p; X∗ )

where [X; f ] = m[X; d] + n[X; e] + g[X; d]

which is an IPS with a displacement direction parallel to [X;f].

Hence, in the special case where the two IPSs have their displacement direc-
tions parallel, or have their invariant-plane normals parallel, their combined effect
is simply another IPS.
It is interesting to examine how plane normals are affected by invariant-line
strains. Taking the inverse of (X S X), we see that

(X S X)−1 = (X Q X)−1 (X P X)−1

or from equation 11.6,

−1
(X S X) = I − an[X; e](q; X∗ )I − bm[X; d](p; X∗ )
= I − an[X; e](q; X∗ ) − bm[X; d](p; X∗ )
+cnm[X; e](p; X∗ ) (11.21)
200 11 Invariant-plane strains
where a, b and c are scalar constants given by 1/a = det(X Q X), 1/b =
det(X P X) and c = ab(q; X∗ )[X; d].
If h = e ∧ d, then h is a reciprocal lattice vector representing the plane which
−1
contains both e and d. It is evident from equation 11.21 that (h; X∗ )(X S X) =
∗ ∗ ∗
(h; X ), since (h; X )[X; e] = 0 and (h; X )[X; d] = 0. In other words, the plane
−1
normal h is an invariant normal of the invariant-line strain (X S X) .

We have found that an ILS has two important characteristics: it leaves a line u
invariant and also leaves a plane normal h invariant. If the ILS is factorised into
two IPS’s, then u lies at the intersection of the invariant-planes of these component
IPS’s, and h defines the plane containing the two displacement vectors of these IPS’s.
These results will be useful in gaining a deeper understanding of martensitic trans-
formations.

11.7 Summary
In mathematics, the term invariance generally refers to a property that is unchanged
by a transformation, which in the present context can be a deformation. The deforma-
tions considered here leave planes or lines invariant and have physical consequences
that can be detected experimentally. Ordinary slip in solids leaves traces at free sur-
faces, that for crystalline materials reflect the anisotropy of atomic arrangements but
there is no change in volume. In contrast, solid-state phase transformations are usu-
ally associated with changes in density, but the displacements associated with the
volume change can be directed normal to the broad interface between the parent and
product crystal, so that an invariant plane is preserved. An important result from
the methods described in this chapter is that there is unique connection between the
orientation relationship, invariant plane (habit plane) and the shape deformation.
References

1. L. Gold: ‘Evaluation of the stiffness coeffcients for beryllium from ultrasonic

measurements in polycrystalline and single crystal specimens’, Physical Re-
view, 1949, 77, 390–395.
2. J. S. Bowles, and J. K. Mackenzie: ‘The crystallography of martensite transfor-
mations, part I’, Acta Metallurgica, 1954, 2, 129–137.
3. J. W. Christian: ‘The origin of surface relief effects in phase transformations’,
In: V. F. Zackay, and H. I. Aaronson, eds. Decomposition of austenite by diffu-
sional processes. New York, USA: Interscience, 1962:371–386.
4. J. W. Christian, and D. V. Edmonds: ‘The bainite transformation’, In: A. R.
Marder, and J. I. Goldstein, eds. Phase Transformations in Ferrous Alloys. War-
rendale, Pennsylvania, USA: TMS-AIME, 1984:293–327.
5. F. P. Bundy: ‘Pressure-temperature phase diagram of iron to 200 kbar, 900◦ C’,
Journal of Applied Physics, 1965, 36, 616–620.
6. J. W. Brooks, M. H. Loretto, and R. E. Smallman: ‘Direct observations of
martensite nuclei in stainless steel’, Acta Metallurgica, 1979, 27, 1839–1847.
7. C. M. Wayman: Introduction to the crystallography of martensitic transforma-
tions: New York: Macmillan, 1964.
8. J. W. Christian: Theory of Transformations in Metals and Alloys, Part I: 2 ed.,
Oxford, U. K.: Pergamon Press, 1975.
9. E. Schmid, and W. Boas: Plasticity of Crystals (translated from the 1935 edition
of Kristalplastizitaet): London, U.K.: F. A. Hughes and Co., 1950.
10. J. W. Christian: ‘Deformation by moving interfaces’, Metallurgical Transac-
tions A, 1982, 13, 509–538.
11. A. Kelly, and K. M. Knowles: Crystallography and Crystal Defects: 2nd ed.,
New York, USA: John Wiley & Sons, Inc., 2012.
12. H. K. D. H. Bhadeshia: ‘TRIP-assisted steels?’, ISIJ International, 2002, 42,
1059–1060.
13. J. W. Christian: Theory of Transformations in Metals and Alloys: Oxford, U.
K.: Pergamon Press, 1965.

201
202 11 References

14. J. W. Christian, and S. Mahajan: ‘Deformation twinning’, Progress in Materials

Science, 1995, 39, 1–157.
15. J. D. Eshelby: ‘The determination of the elastic field of an ellipsoidal inclusion
and related problem’, Proceedings fo the Royal Society A, 1957, 241, 376–396.
16. J. W. Christian: ‘Accommodation strains in martensite formation, the use of the
dilatation parameter’, Acta Metallurgica, 1958, 6, 377–379.
17. J. W. Christian: ‘Thermodynamics and kinetics of martensite’, In: G. B. Olson,
and M. Cohen, eds. International Conference on Martensitic Transformations
ICOMAT ’79. Massachusetts, USA: Alpine Press, 1979:220–234.
18. E. C. Bain: ‘The nature of martensite’, Trans. AIME, 1924, 70, 25–46.
19. C. M. Wayman: ‘The crystallography of martensitic transformations in alloys
of iron’: In: H. Hermans, ed. Advances in Materials Research, vol. 3. John
Wiley & Sons, Inc., 1968:147–304.
20. J. S. Bowles, and C. M. Wayman: ‘The Bain strain, lattice correspondences,
and deformations related to martensitic transformations’, Metallurgical Trans-
actions, 1972, 3, 1113–1121.
21. J. W. Christian, and A. G. Crocker: Dislocations in Solids, ed. F. R. N. Nabarro:
Amsterdam, Holland: North Holland, 1980.
22. J. W. Christian: ‘The strength of martensite’, In: A. Kelly, and R. Nicholson,
eds. Strengthening Methods in Crystals. Netherlands: Elsevier, 1971:261–329.
23. V. Volterra: ‘Sur l’équilibre des corps élastiques multiplement connexes’, An-
nales scientifiques de l’École normale supérieure, 1907, 24, 401–517.
24. J. W. Christian, and A. G. Crocker: Dislocations in Solids, ed. F. R. N. Nabarro,
vol. 3, chap. 11: Amsterdam, Holland: North Holland, 1980:165–252.
25. P. M. Kelly, and G. Pollard: ‘The movement of slip dislocations in internally
twinned martensite’, Acta Metallurgica, 1969, 17, 1005–1008.
26. J. W. Brooks, M. H. Loretto, and R. E. Smallman: ‘In situ observations of
martensite formation in stainless steel’, Acta Metallurgica, 1979, 27, 1829–
1838.
27. A. Kelly, and G. W. Groves: Crystallography and Crystal Defects: London,
U.K.: Longmans, 1970.
12
Martensite

Abstract
Martensitic transformations can be quite simple to understand in systems where the
transformation strain is equal to the observed shape deformation. This is the case
when hcp martensite grows in fcc austenite. The odd observation that requires ex-
planation is that during the fcc→bcc or bct transformation, the shape deformation
is inconsistent with the strain needed to change the change the lattice. The crystallo-
graphic theory described in this Chapter deals with this anomaly using a most elegant
formalism that is quantitative.

12.1 Introduction
The basic principles of the characteristics of martensitic transformation have already
been covered in Chapter 8, albeit qualitatively. The purpose here is to explain the
mathematical framework developed by Bowles, MacKenzie, Wechsler, Lieberman
and Read [1–4], a framework that represents one of the most elegant and complete
theories that helped explain an accumulation of findings that could not be reconciled.
And there were predictions made which were only to be verified when advanced in-
strumentation such as transmission electron microscopy became available. The his-
tory of this wonderful theory is described by Wayman [5]; a classic example of a
subject receptive to a new way of thinking when a discipline is in a state of crisis
[6, 7]. The theory has had many consequences, implicitly or directly, in unrelated
subjects such as in the estimation of interfacial structure [8–11], the calculation of
stress-induced crystallographic texture [12–14] and the general understanding of the
relative orientations of crystals that precipitate in the solid state.

12.2 Shape deformation

All martensitic transformations involve co-ordinated movements of atoms and are
diffusionless. Since the shape of the pattern in which the atoms in the parent crystal
are arranged nevertheless changes to generate the product phase, it follows that there

203
204 12 Martensite

must be a visible change in the macroscopic shape of the parent crystal during trans-
formation [15]. The shape deformation and its significance can best be illustrated
by reference to Fig. 12.1, where a comparison is made between reconstructive and
diffusionless transformations. For simplicity, the diagram refers to a case where the
transformation strain is an invariant-plane strain and a fully coherent interface exists
between the parent and product lattices, irrespective of the mechanism of transfor-
mation.
Considering the shear transformation first, we note that since the pattern of
atomic arrangement is changed on transformation, and since the transformation is
diffusionless, the macroscopic shape of the crystal changes. The shape deformation
has the exact characteristics of an IPS. The initially flat surface normal to da becomes
tilted about the line formed by the intersection of the interface plane with the surface
normal to da. The straight line ab is bent into two connected and straight segments
ae and eb. Hence, an observer looking at a scratch that is initially along ab and in
the surface abcd would note that on martensite formation, the scratch becomes ho-
mogeneously deflected about the point e where it intersects the trace of the interface
plane. Furthermore, the scratches ae and eb would be seen to remain connected at
the point e. This amounts to proof that the shape deformation has, on a macroscopic
scale, the characteristics of an IPS and that the interface between the parent and prod-
uct lattices does not contain any distortions (i.e. it is an invariant-plane). Observing
the deflection of scratches is one way of deducing the nature of shape deformations
accompanying transformations.
In Fig. 12.1 it also is implied that martensitic transformation is diffusionless;
labelled rows of atoms in the parent crystal are expected to remain in the correct
sequence in the martensite lattice. It is possible therefore to suggest that a particular
atom in the martensite must have originated from a corresponding particular atom in
the parent crystal. A formal way of expressing this property is to say that there exists
an atomic correspondence between the parent and product lattices.
In the case of the reconstructive transformation illustrated in Fig. 12.1, it is ev-
ident that the product phase can be of a different composition from the parent. In
addition, there has been much mixing up of atoms during transformation and the or-
der of arrangement of atoms in the product lattice is different from that in the parent
lattice - the atomic correspondence has been destroyed. Because the transformation
involves a reconstruction of the parent lattice, atoms are able to diffuse around in
such a way that the IPS shape deformation and its accompanying strain energy, do
not arise. The scratch ab remains straight across the interface and is unaffected by
the transformation.
In summary, martensitic transformations always are accompanied by a change
in the shape of the parent crystal, characterised as an invariant-plane strain when
examined on a macroscopic scale. The occurrence of such a shape deformation is
taken to imply the existence of an atomic correspondence between the parent and
product lattices. It is possible to state that a particular atom in the product occupied
a particular corresponding site in the parent lattice.
These results have some interesting consequences. The formation of martensite
in a constrained environment causes distortions in its surroundings. The strain energy
12.2 Shape deformation 205

FIGURE 12.1
Schematic illustration of the mechanisms of diffusional and shear transformations.

due to this distortion, per unit volume of martensite, is given by [16–18]

c
E ≈ µ(s2 + δ 2 ) (12.1)
r
where µ is the shear modulus of the parent lattice, c/r is the thickness to length ratio
of the martensite plate and s and δ are the shear and dilatational components of the
shape deformation strain. It follows that martensite must always have a thin plate
morphology consistent with the minimising of strain energy but achieving transfor-
mation at the same time. E usually amounts to about 600 J mol−1 for martensite in
steels [18], when the shape deformation is entirely elastically accommodated. If the
austenite is soft, then some plastic accommodation driven by the shape deformation
may relieve some of the strains, but the E value calculated on the basis of purely
206 12 Martensite

elastic accommodation should be taken to be the upper limit of the stored energy
due to the shape change accompanying martensitic transformation. This is because
the plastic accommodation is driven by the shape deformation [18]. In the event that
plastic accommodation occurs, dislocations and other defects may be generated both
in the parent and product lattices, which would make it impossible to reverse the
motion of the interface, thus eliminating the possibility of a shape memory effect.
Martensitic transformation does not require diffusion, can occur at very low tem-
peratures and the interface can move rapidly. The interface between martensite and
the parent phase must therefore be glissile, so we begin with a discussion of what the
appropriate structure of such an interface might be.

12.3 Interfacial structure of martensite

All the mathematical operations that we describe in the context of martensite occur
at the transformation interface, which separates the perfect austenite from the perfect
martensite. The transformation is thermodynamically of first order [19] which im-
plies a sharp interface between the parent and product crystals which can therefore
co-exist. This is obvious when we observe routinely a mixture of martensite plates
embedded in austenite.
That martensite can form at cryogenic temperatures [20] indicates that the pro-
cess cannot rely on thermal activation. In particular, the (sharp) interface connecting
the martensite with the parent phase must be able to move in the manner of dislo-
cation glide rather than dislocation climb. It therefore cannot be incoherent but it is
possible for certain semi-coherent or fully coherent boundaries to be glissile [21].
However, stress-free coherence is only possible when the parent and product lattices
can be related by a strain which is an invariant-plane strain [22] so such interfaces
are rare for particles of appreciable size. The fcc→hcp transformation is one exam-
ple where a fully coherent interface is possible. Martensitic transformation in ordered
Fe3 Be occurs by a simple shearing of the lattice (an IPS) [23], so a fully coherent
interface is again possible. More generally, the interfaces tend to be semi-coherent.
It was demonstrated in Chapter 8 that a fcc austenite lattice cannot be transformed
into a bcc martensite lattice by a strain which is an IPS, so that the as semi-coherent
interface is expected.
The semi-coherent interface should consist of coherent regions separated period-
ically by discontinuities which prevent the misfit in the interface plane from accu-
mulating over large distances, in order to minimise the elastic strains associated with
the interface. There are two kinds of semi-coherency [22, 24]; if these discontinuities
are intrinsic dislocations with Burgers vectors in the interface plane, not parallel to
the dislocation line, then the interface is said to be epitaxially semi-coherent. The
term ‘intrinsic’ means that the dislocations are a necessary part of the interface struc-
ture and have not simply strayed into the boundary - they do not have a long-range
strain field. The normal displacement of such an interface requires the thermally
12.3 Interfacial structure of martensite 207

activated climb of intrinsic dislocations, so that the interface can only move in a non-
conservative manner, with relatively restricted or zero mobility at low temperatures.
A martensite interface cannot therefore be epitaxially semi-coherent.
In the second type of semi-coherency, the discontinuities discussed above are
screw dislocations, or dislocations whose Burgers vectors do not lie in the interface
plane. This kind of semi-coherency is of the type associated with glissile martensite
interfaces, whose motion is conservative (i.e. the motion does not lead to the cre-
ation or destruction of lattice sites). Such an interface should have a high mobility
since the migration of atoms is not necessary for its movement. Actually, two further
conditions must be satisfied before even this interface can be said to be glissile:

(i) A glissile interface requires that the glide planes of the intrinsic dislocations
associated with the product lattice must meet the corresponding glide planes of
the parent lattice edge to edge in the interface [24], along the dislocation lines.
(ii) If more than one set of intrinsic dislocations exist, then these should either have
the same line vector in the interface, or their respective Burgers vectors must be
parallel [24]. This condition ensures that the interface can move as an integral unit.
It implies in addition, that the deformation caused by the intrinsic dislocations,
when the interface moves, can always be described as a simple shear (caused by
a resultant intrinsic dislocation which is a combination of all the intrinsic dislo-
cations) on some plane which makes a finite angle with the interface plane, and
intersects the latter along the line vector of the resultant intrinsic dislocation.

Obviously, if the intrinsic dislocation structure consists of just a single set of

parallel dislocations, or of a set of different dislocations which can be summed to
give a single glissile intrinsic dislocation, then it follows that there must exist in the
interface, a line which is parallel to the resultant intrinsic dislocation line vector,
along which there is zero distortion. Because this line exists in the interface, it is
also unrotated. It is an invariant-line in the interface between the parent and product
lattices. When full coherency between the parent and and martensite lattices is not
possible, then for the interface to be glissile, the transformation strain relating the two
lattices must be an invariant-line strain, with the invariant-line being in the interface
plane.
The interface between the martensite and the parent phase usually is called the
‘habit plane’; when the transformation occurs without any constraint, the habit plane
is macroscopically flat, as illustrated in Fig. 12.2. When the martensite forms in
a constrained environment, it grows in the shape of a thin lenticular plate or lath
and the habit plane is a little less clear in the sense that the interface is curved on
a macroscopic scale. However, it is found that the average plane of the plate (the
plane containing the major circumference of the lens) corresponds closely to that
expected from crystallographic theory, and to that determined under conditions of
unconstrained transformation. The aspect ratio (maximum thickness to length ratio)
of lenticular plates is usually less than 0.05, so that the interface plane does not de-
part very much from the average plane of the plate. Some examples of habit plane
indices (relative to the austenite lattice) are given in Table 8.1.
208 12 Martensite

martensite

ined

martensite

constrained

FIGURE 12.2
The habit plane of martensite (α′ ) under conditions of unconstrained and constrained
transformation, respectively. In the latter case, the dashed line indicates the trace of
the habit plane.

12.4 Phenomenological theory of martensite crystallography

We have emphasised that an essential feature of martensitic transformation is its
shape deformation, which on a macroscopic scale has the characteristics of an
invariant-plane strain. The magnitude m of the shape deformation can be determined
experimentally as can its unit displacement vector d. The habit plane of the marten-
site (unit normal p) is the invariant-plane of the shape deformation. The shape de-
formation can be represented by means of a shape deformation matrix (F P F) such
that:
(F P F) = I + m[F;d](p; F∗ )
where the basis F is for convenience chosen to be orthonormal, although the equation
is valid for any basis.
For the shear transformation illustrated in Fig. 12.1 and for the fcc→hcp marten-
site reaction, the lattice transformation strain is itself an IPS and there is no difficulty
in reconciling the transformation strain and the observed shape deformation. In other
words, if the parent lattice is operated on by the shape deformation matrix, then the
correct product lattice is generated if shuffles are allowed; the transformation strain
is the same as the shape deformation.
This is not the case [25] for the fcc→ bcc martensite reaction and for many other
martensite transformations where the lattice transformation strain (F S F) does not
equal the observed shape deformation (F P F). Fig. 8.4 illustrated the fact that the
Bain strain (F B F) when combined with an appropriate rigid body rotation (F J F)
gives an invariant-line strain which when applied to the fcc lattice generates the bcc
martensite lattice. However, the shape deformation that accompanies the formation of
12.4 Phenomenological theory of martensite crystallography 209

bcc martensite from austenite is nevertheless found experimentally to be an invariant-

plane strain. This is the major anomaly that the theory of martensite crystallography
attempts to resolve: the experimentally observed shape deformation is inconsistent
with the lattice transformation strain. If the observed shape deformation is applied
to the parent lattice then the austenite lattice is deformed into an intermediate lattice
(not experimentally observed) but not into the required bcc lattice.

FIGURE 12.3
Schematic illustration of the phenomenological theory of martensite. (a) represents
the austenite crystal and (c) has a bcc structure. (b) has a structure between fcc and
bcc, p is the habit plane unit normal and q is the unit normal to the plane on which
the lattice-invariant shear occurs. The heavy horizontal lines in (e) are coherent twin
boundaries. Note that the vector e is normal to q but does not lie in the plane of the
diagram.

This problem is illustrated in Figs. 12.3. The shape of the starting austenite crystal
with the fcc structure is shown in Fig. 12.3a. On martensitic transformation its shape
alters to that illustrated in Fig. 12.3b and the shape deformation on going from (a)
to (b) is clearly an IPS on the plane with unit normal p and in the unit displacement
direction d. However, the structure of the crystal in Fig. 12.3b is some intermediate
lattice which is not bcc, since an IPS cannot on its own change the fcc structure to
the bcc structure. An invariant-line strain can however transform fcc to bcc, and since
an ILS can be factorised into two invariant-plane strains, it follows that the further
deformation (F Q F) needed to change the intermediate structure of Fig. 12.3b to the
bcc structure (Fig. 12.3c) is another IPS. If the deformation (F Q F) is of magnitude
210 12 Martensite

n on a plane with unit normal q and in a unit direction e, then:

(F Q F) = I + n[F;e](q; F∗ )

(F Q F) has to be chosen in such a way that (F P F)(F Q F) = (F S F), where

(F S F) is an invariant-line strain which transforms the fcc lattice to the bcc lattice.
Hence, a combination of two invariant-plane strains can accomplish the necessary
lattice change but this then gives the wrong shape change as the extra shape change
due to (F Q F), in changing (b) to (c), is not observed.
Numerous experiments have demonstrated that the shape deformation due to the
fcc→bcc martensite transformation is an IPS [11], and it seems that the effect of
(F Q F) on the macroscopic shape is invisible. If a way can be discovered of making
the shape altering effect of (F Q F) invisible, then the problem is essentially deter-
mined.
(F Q F) can be made invisible by applying another deformation to (c) such that
the shape of (c) is brought back to that of (b), without altering the bcc structure of (c).
Such a deformation must therefore be lattice-invariant because it must not alter the
symmetry or unit cell dimensions of the parent crystal structure. Ordinary slip does
not change the nature of the lattice and is one form of a lattice-invariant deformation.
Hence, slip deformation on the planes q and in the direction -e would make the shape
change due to (F Q F) invisible on a macroscopic scale, as illustrated in Fig. 12.3d.
The magnitude of this lattice-invariant slip shear is of course determined by that
of (F Q F) and we know that it is not possible to continuously vary the magnitude
of slip shear, since the Burgers vectors of slip dislocations are discrete. (F Q F) on
the other hand can have any arbitrary magnitude. This difficulty can be overcome by
applying the slip shear inhomogeneously, by the passage of a discrete slip dislocation
on say every nth plane, which has the effect of allowing the magnitude of the lattice-
invariant shear to vary as a function of n. In applying the lattice-invariant shear to
(c) in order to obtain (d), the bcc structure of (c) is completely unaffected, while
is shape is deformed inhomogeneously to correspond to that of (b), as illustrated in
Fig. 12.3d.
This then is the essence of the theory of martensite crystallography [1–4], which
explains the contradiction that the lattice transformation strain is an ILS but the
macroscopic shape deformation is an IPS. The lattice transformation strain when
combined with an inhomogeneous lattice-invariant shear produces a macroscopic
shape change which is an IPS.
Twinning is another deformation which does not change the crystal structure,
simply reorientates it; the shape (c) of Fig. 12.3 could be deformed to correspond
macroscopically to that of (b), without changing its bcc nature, by twinning, as il-
lustrated in Fig. 12.3e. The magnitude of the lattice-invariant deformation can be
adjusted by varying the volume fraction of the twin. This explains the twin substruc-
ture found in many ferrous martensites, and such twins are called transformation
twins. The irrationality of the habit planes arises because the indices of the habit
plane depend on the amount of lattice-invariant deformation, a quantity which does
not necessarily correlate with displacements equal to discrete lattice vectors.
Since the effect of the homogeneous strain (F Q F) on the shape of the parent
12.5 Stage 1: Calculation of lattice transformation strain 211

crystal has to be cancelled by another opposite but inhomogeneously applied lattice-

invariant deformation, it follows that (F Q F) is restricted to being a simple shear
with the displacement vector e being confined to the invariant-plane of (F Q F). In
other words, (F Q F) must have a zero dilatational component, since it cannot oth-
erwise be cancelled by another IPS which preserves the lattice2 . As noted earlier,
lattice-invariant deformations cannot alter the volume or symmetry of the lattice.
The determinant of a deformation matrix gives the ratio of the volume after deforma-
tion to that prior to deformation, so that det(F Q F) = 1. This means that the total
volume change of transformation is given by det(F P F) = det(F S F).
In summary, the martensite transformation in iron requires an invariant-line
strain (F S F) to change the fcc lattice to the bcc martensite lattice and to ob-
tain the experimentally observed orientation relation. This can be imagined to con-
sist of two homogeneous invariant-plane strains (F P F) and (F Q F), such that
(F S F) = (F P F)(F Q F). However, the shape change due to the simple shear
(F Q F) is rendered invisible on a macroscopic scale since there is also an inhomo-
geneous lattice-invariant deformation (which can be slip or twinning) which cancels
out the shape change due to (F Q F), without altering the lattice structure. It follows
that the macroscopic shape change observed is solely due to (F P F) and therefore
has the characteristics of an invariant-plane strain, as experimentally observed. We
have seen already that the transformation strain (F S F) can be factorised into a Bain
strain (F B F) combined with an appropriate rigid body rotation (F J F), such that
(F S F)=(F J F)(F B F) and is an invariant-line strain, with the invariant-line lying in
the planes p and q, and the invariant-normal of (F S F) defining a plane containing d
and e. Hence, the theory of martensite can be summarised in terms of the equation
(F S F) = (F J F)(F B F) = (F P F)(F Q F). (12.2)

12.5 Stage 1: Calculation of lattice transformation strain

Two arbitrary lattices can be transformed into one another by an infinite number of
different transformation strains, but only some of these may have reasonably small
principal deformations. The choice available can be further reduced by considering
only those strains which involve the minimum degree of shuffling of atoms and by
considering the physical implications of such strains. In the case of martensitic trans-
formations, a further condition has to be satisfied; the lattice transformation strain
must also be an invariant-line strain if the interface is to be glissile [26].
For the fcc→bcc martensitic transformation, the Bain strain, which is a pure de-
formation, involves the smallest atomic displacements during transformation. When
it is combined with an appropriate rigid body rotation, the total strain amounts to an
invariant-line strain. For martensitic transformations, the rigid body rotation has to be
chosen in such a way that the invariant-line lies in the plane of the lattice-invariant
shear and also in the habit plane of the martensite; these planes are the invariant
planes of (F Q F) and (F P F) respectively, so that the line common to these planes
212 12 Martensite

is not affected by these deformations. Furthermore, the invariant normal of the ILS
must define a plane which contains the displacement directions of the lattice invari-
ant shear and of the shape deformation. This ensures that the spacing of this plane is
not affected by (F Q F) or (F P F).
The next example illustrates how the transformation strain can be determined
once the pure deformation which accomplishes the lattice change is deduced. To
ensure that the invariant-line and invariant-normal of the transformation strain are
compatible with the mode of lattice-invariant shear, we first need to specify the lat-
ter. It is assumed here that the plane and direction of the lattice-invariant shear are
(1 0 1)F and [1 0 1]F respectively. One variant of the Bain strain is illustrated in
Fig. 8.2, where we see that [1 0 0]γ is deformed into [1 1 0]α , [0 1 0]γ to [1 1 0]α and
[0 0 1]γ to [0 0 1]α , so that the variant of the Bain correspondence matrix is given by
equation 11.19. We will use this variant of the Bain correspondence matrix through-
out the text, but we note that there are two other possibilities, where [0 0 1]α can be
derived from either [1 0 0]γ or [0 1 0]γ respectively.

Example 12.1: Determination of lattice transformation strain

The deformation matrix representing the Bain strain, which carries the fcc
austenite lattice (Fig. 8.2) to the bcc martensite lattice is given by
⎛ ⎞
η1 0 0
⎜ ⎟
(F B F) = ⎜⎝ 0 η2 0 ⎠
⎟

0 0 η3

where F is an orthonormal basis consisting of unit basis vectors fi parallel to the

crystallographic axes of the conventional fcc austenite unit cell (Fig. 8.2), f1 ∥ a1 ,
f2 ∥ a2 & f3 ∥ a3 ). ηi are the principal deformations of the Bain strain, given by
η1 = η2 = 1.136071 and η3 = 0.803324.
Find the rigid body rotation (F J F) which when combined with the Bain strain
gives an invariant-line strain (F S F) = (F J F)(F B F), subject to the condition
that the invariant-line of (F S F) must lie in (1 0 1)F and that the plane defined by
the invariant-normal of (F S F) contains [1 0 1]F .
Writing the invariant-line as [F;u] = [u1 u2 u3 ], we note that for u to lie in
(1 0 1)F , its components must satisfy the equation

u1 = −u3 (12.3)

Prior to deformation,
|u|2 = (u; F)[F;u] = 1 (23b)
u, as a result of deformation becomes a new vector x with

|x|2 = (x; F)[F;x]

= (u; F)(F B′ F)(F B F)[F;u]
2
= (u; F)(F B F) [F;u].
12.5 Stage 1: Calculation of lattice transformation strain 213
If the magnitude of u is not to change on deformation then |u| = |x| or

u21 + u22 + u23 = η12 u21 + η22 u22 + η32 u23 . (12.4)
Equations 12.3 and 12.4 can be solved simultaneously to give two solutions for
undistorted lines:

[F;u] = [−0.671120 − 0.314952 0.671120],

[F;v] = [−0.671120 0.314952 0.671120].

To solve for the invariant normal of the ILS, we proceed as follows. Writing
(h; F∗ ) = (h1 h2 h3 ), we note that for h to contain [1 0 1]F , its components must
satisfy the equation
h1 = h3 (12.5)
and furthermore,
(h; F∗ )[F∗ ; h] = 1.
h, on deformation becomes a new plane normal l and if |h| = |l| then

|l|2 = (l; F∗ )[F ∗ ; l]

−1 −1
= (h; F∗ )(F B F) (F B′ F) [F ∗ ; h]

so that
h21 + h22 + h23 = (l1 /η1 )2 + (l2 /η2 )2 + (l3 /η3 )2 (12.6)
Solving equations 12.5 and 12.6 simultaneously, we obtain the two possible solu-
tions for the undistorted-normals as

(h; F∗ ) = (0.539127 0.647058 0.539127),

(k; F∗ ) = (0.539127 − 0.647058 0.539127).

To convert (F B F) into an invariant-line strain (F S F) we have to employ a rigid

body rotation (F J F) which simultaneously brings an undistorted line (such as x)
and an undistorted normal (such as l) back into their original directions along u
and h respectively. This is possible because the angle between x and l is the same
as that between u and h, as shown below:

l.x = (l; F∗ )[F;x]

= (h; F∗ )(F B F)−1 (F B F)[F;u]
= (h; F∗ )[F;u]
= h.u

Hence, one way of converting (F B F) into an ILS is to employ a rigid body ro-
tation which simultaneously rotates l into h and x into u. We have found that there
are two undistorted lines and two undistorted normals which satisfy the conditions
of the original question, so there are four ways of choosing pairs of undistorted
214 12 Martensite
lines and undistorted normals, but in this case, the four solutions are crystallo-
graphically equivalent. There are therefore four solutions (different in general) to
the problem of converting (F B F) to (F S F), subject to the condition that the
invariant-line should be in (1 0 1) and that the invariant normal defines a plane
containing [1 0 1]. We will concentrate on the solution obtained using the pair u
and h:
l = (h; F∗ )(F B F)−1 = (0.474554 0.569558 0.671120)
x = (F B F)[F;u] = [−0.762440 − 0.357809 0.539127]
a = u ∧ h = (−0.604053 0.723638 − 0.264454)
b = x ∧ l = (−0.547197 0.767534 − 0.264454)
The required rigid body rotation should rotate x back to u, l back to h and b to a,
giving the three equations:

[F;u] = (F J F)[F;x]

[F;h] = (F J F)[F;l]
[F;a] = (F J F)[F;b]
which can be expressed as a 3 × 3 matrix equation
⎛ ⎞ ⎛ ⎞⎛ ⎞
u 1 h1 a1 J11 J12 J13 w l1 b1
⎜ ⎟ ⎜ ⎟⎜ 1 ⎟
⎜ ⎟=⎜ ⎟⎜ ⎟
⎝u2 h2 a2 ⎠ ⎝J21 J22 J23 ⎠ ⎝w2 l2 b2 ⎠
u3 h3 a3 J31 J32 J33 w3 l3 b3
it follows that
⎛ ⎞ ⎛ ⎞
−0.671120 0.539127 −0.604053 −0.762440 0.474554 −0.547197
⎜ ⎟ ⎜ ⎟
⎝−0.314952 0.647058 0.723638 ⎠ = (F J F) ⎝−0.357808 0.569558 0.767534 ⎠
⎜ ⎟ ⎜ ⎟

0.671120 0.539127 −0.264454 0.539127 0.671120 −0.264454

which on solving gives
⎛ ⎞
0.990534 −0.035103 0.132700
⎜ ⎟
(F J F) = ⎜
⎝ 0.021102 0.994197 0.105482⎠
⎟

−0.135633 −0.101683 0.985527

which is a rotation of 9.89◦ about [0.602879 − 0.780887 0.163563]F. The

invariant-line strain (F S F) = (F J F)(F B F) is thus
⎛ ⎞
1.125317 −0.039880 0.106601
⎜ ⎟
(F S F) = ⎜⎝ 0.023973
⎟
1.129478 0.084736⎠ (12.7a)
−0.154089 −0.115519 0.791698
12.6 Stage 2: Determination of the orientation relationship 215
−1 −1 −1
and we note that (F S F) = (F B F) (F J F)
is given by
⎛ ⎞
0.871896 0.018574 −0.119388
−1 ⎜ ⎟
(F S F) =⎜
⎝−0.030899 0.875120 −0.089504⎠ .
⎟ (12.7b)
0.165189 0.131307 1.226811

12.6 Stage 2: Determination of the orientation relationship

The orientation relationship between the austenite and martensite is best expressed
in terms of a co-ordinate transformation matrix (α J γ). Any vector u or any plane
normal h can then be expressed in either crystal basis by using the equations

[α; u] = (α J γ)[γ; u]
[γ; u] = (γ J α)[α; u]
(h; α∗ ) = (h; γ ∗ )(γ J α)
(h; γ ∗ ) = (h; α∗ )(α J γ).

Example 12.2: Martensite-austenite orientation relationship

For the martensite reaction considered in example on page 212, determine the
orientation relationship between the parent and product lattices.
The orientation relationship can be expressed in terms of a co-ordinate trans-
formation matrix (α J γ), which is related to the transformation strain (γ S γ) via
equation 11.12, so that

(α J γ)(γ S γ) = (α C γ)

where (α C γ) is the Bain correspondence matrix (equation 11.19), and it follows

that
(α J γ) = (α C γ)(γ S γ)−1
In equation 12.7 the matrices representing the transformation strain and its inverse
were determined in the basis F, and can be converted into the basis γ by means
of a similarity transformation. However, because fi are parallel to ai , it can easily
−1
be demonstrated that (F S F) = (γ S γ) and (F S F) = (γ S γ)−1 . Hence,
using the data from equation 12.7 and the Bain correspondence matrix from equa-
216 12 Martensite
tion 12.1a, we see that
⎛ ⎞⎛ ⎞
1 1 0 0.871896
0.018574 −0.119388
⎜ ⎟⎜ ⎟
(α J γ) = ⎜ ⎝1 1 0⎠ ⎝−0.030899 0.875120 −0.089504⎠(12.8)
⎟⎜ ⎟

0 0 1 0.165189 0.131307 1.226811

⎛ ⎞
0.902795 −0.856546 −0.029884
⎜ ⎟
(α J γ) = ⎜
⎝0.840997 0.893694 −0.208892⎠
⎟

0.165189 0.131307 1.226811

⎛ ⎞
0.582598 0.542718 0.106602
⎜ ⎟
(γ J α) = ⎜
⎝−0.552752 0.576725 0.084736⎠ .
⎟

−0.019285 −0.134804 0.791698

These co-ordinate transformation matrices can be used to show that

(1 1 1)γ = (0.016365 1.525799 1.523307)α

and
[1 0 1]γ = [−0.932679 − 1.049889 1.061622]α
which means that (1 1 1)γ is very nearly parallel to (0 1 1)α and [1 0 1]γ is about
3◦ from [1 1 1]α . The orientation relationship is illustrated in Fig. 12.4.
12.7 Stage 3: Nature of the shape deformation 217

FIGURE 12.4
Stereographic representation of the orientation relationship between martensite and
austenite, as deduced in equation 12.8. The lattice-invariant shear plane (q) and direc-
tion (-e), and the habit plane (p) and unit displacement vector (d) also are illustrated.

12.7 Stage 3: Nature of the shape deformation

The shape deformation (F P F) can be obtained by factorising the transformation
strain (F S F). The deformation (F P F) is physically significant because it can be
experimentally determined and describes the macroscopic change in shape of the
parent crystal. In practise, all the changes necessary for transformation occur simul-
taneously at the moving interface [27]. Transformation dislocations (atomic height
steps) in the interface cause the fcc lattice to change to the bcc lattice as the interface
moves and the deformation that this produces is described by (F S F). The intrinsic
dislocations which lie along the invariant-line in the interface have Burgers vectors
which are perfect lattice vectors of the parent lattice. They cannot therefore take part
in the actual transformation of the lattice, but as the interface moves, they inhomoge-
neously shear the volume of the material swept by the interface [24]. This of course is
the lattice-invariant shear discussed above, which in combination with the shape de-
formation due to (F S F) gives the experimentally observed IPS surface relief (F P F).
This then is the physical interpretation of the transformation process.
The crystallographic theory of martensite is on the other hand called phenomeno-
logical; the steps into which the transformation is factorised (e.g. the two “shears”
(F P F) & (F Q F)) are not unique and do not necessarily describe the actual path by
which the atoms move from one lattice to the other. The theory simply provides a
218 12 Martensite

definite link between the initial and final states without being certain of the path in
between.
Example 12.3: Habit plane and the shape deformation
For the martensite reaction considered on page 212, determine the habit plane
of the martensite plate, assuming that the lattice-invariant shear occurs on the sys-
tem (1 0 1)γ [1 0 1]γ . Comment on the choice of this shear system and determine
the nature of the shape deformation.
The lattice invariant shear is on (1 0 1)[1 0 1] and since its effect is to cancel
the shape change due to (F Q F), the latter must be a shear on (1 0 1)[1 0 1].
To solve for the habit plane (unit normal p) it is necessary to factorise (F S F)
into the two invariant-plane strains (F P F) = I + m[F;d](p; F∗ ) and (F Q F) =
I + n[F;e](q; F∗ ).
The transformation strain (F S F) of equation 12.7 was calculated by phe-
nomenologically combining the Bain strain with a rigid body rotation, with the
latter chosen to make (F S F) an invariant line strain, subject to the condition that
the invariant line u of (F S F) must lie in (1 0 1)γ and that the invariant-normal
h of (F S F) must define a plane containing [1 0 1]γ . This is of course compat-
ible with the lattice-invariant shear system chosen in the present example since
u.q = e.h = 0. From equation 12.2,

(F S F) = (F P F)(F Q F) = {I + m[F;d](p; F∗ )}{I + n[F;e](q; F∗ )} (12.9)

and using equation 11.6, we see that

(F S F)−1 = (F Q F)−1 (F P F)−1

= {I − n[F;e](q; F∗ )}{I − am[F;d](p; F∗ )} (12.10)

where 1/a = det(F P F) and det(F Q F) = 1. Using equation 12.10, we obtain

−1
(q; F∗ )(F S F) = (q; F∗ ){I − n[F;e](q; F∗ )}{I − am[F;d](p; F∗ )}
= {(q; F∗ ) − n (q; F∗ )[F;e](q; F∗ )} {I − am[F;d](p; F∗ )}
/ 01 2
=0
= (q; F ){I − am[F;d](p; F∗ )}
∗

= (q; F∗ ) − b(p; F∗ )

where b is a scalar constant given by b = am(q; F∗ )[F;d].

12.7 Stage 3: Nature of the shape deformation 219
Hence,
−1
b(p; F∗ ) = (q; F∗ ) − (q; F∗ )(F S F)
; <
= √1 0.000 √1
2 2
⎛ ⎞
; < ⎜0.871896 0.018574 −0.119388⎟
− √1 0.000 √1 × ⎜ ⎟
⎝0.030894 0.875120 −0.089504⎠
2 2
0.165189 0.131307 1.226811
; <
= −0.026223 −0.105982 −0.075960 . (12.11)

This can be normalised to give p as a unit vector:

(p; F∗ ) ∥ (0.197162 0.796841 0.571115)

(p; F∗ ) of course represents the indices of the habit plane of the martensite plate.
As expected, the habit plane is irrational. To completely determine the shape de-
formation matrix (F P F) we also need to know m and d. Using equation 12.9, we
see that
(F S F)[F;e] = [F;e] + m[F;d](p; F∗ )[F;e]
Writing c as the scalar constant c = (p; F∗ )[F;e], we get

cm[F;d] = (F S F)[F;e] − [F;e]

⎛ ⎞⎛ ⎞ ⎛ ⎞
√1 √1
1.125317 −0.039880 0.106601 ⎜ 2 ⎟ ⎜ 2 ⎟
⎜ ⎟⎜ ⎟ ⎜ ⎟
= ⎜
⎝ 0.023973
⎟
1.129478 0.084736⎠ ⎜ ⎟−⎜
⎝ 0 ⎠ ⎝ 0 ⎠
⎟
−0.154089 −0.115519 0.791698 − √12 − √12

= [0.013234 − 0.042966 0.038334]. (12.12)

Now, c = (0.197162 0.796841 0.571115)[ √12 0 − √12 ] = −0.26442478 so

that m[F;d] = [−0.050041 0.162489 − 0.144971]. Since d is a unit vector, it can
be obtained by normalising md to give

[F;d] = [−0.223961 0.727229 − 0.648829]

and
m = |md| = 0.223435
The magnitude m of the displacements involved can be factorised into a shear
component s parallel to the habit plane and a dilatational component δ normal to
1
the habit plane. Hence, δ = md.p = 0.0368161 and s = (m2 −δ 2 ) 2 = 0.220381.
These are typical values of the dilatational and shear components of the shape
strain found in ferrous martensites.
220 12 Martensite
Using these data, the shape deformation matrix is given by
⎛ ⎞
0.990134 −0.039875 −0.028579
⎜ ⎟
(F P F) = ⎜ ⎝ 0.032037 1.129478 0.092800 ⎠ .
⎟ (12.13)
−0.028583 −0.115519 0.917205

12.8 Stage 4: Nature of the lattice-invariant shear

We have already seen that the shape deformation (F P F) cannot account for the
overall conversion of the fcc lattice to that of bcc martensite. An additional homoge-
neous lattice varying shear (F Q F) is necessary, which in combination with (F P F)
completes the required change in structure. However, the macroscopic shape change
observed experimentally is only due to (F P F); the effect of (F Q F) on the macro-
scopic shape change must thus be offset by a system of inhomogeneously applied
lattice-invariant shears. Clearly, to macroscopically cancel the shape change due to
(F Q F), the lattice-invariant shear must be the inverse of (F Q F). Hence, the lattice
invariant shear operates on the plane with unit normal q but in the direction −e, its
magnitude on average being the same as that of (F Q F).
Example 12.4: Lattice–invariant shear
For the martensite reaction discussed in the preceding examples, determine
the nature of the homogeneous shear (F Q F) and hence deduce the magnitude
of the lattice-invariant shear. Assuming that the lattice-invariant shear is a slip
deformation, determine the spacing of the intrinsic dislocations in the habit plane,
which are responsible for this inhomogeneous deformation.
Since (F S F) = (F P F)(F Q F), it follows that (F Q F) = (F P F)−1 (F S F),
so that
⎛ ⎞
1.009516 0.038459 0.027564
⎜ ⎟
(F Q F) = ⎜ ⎝−0.030899 0.875120 −0.089505⎠
⎟

0.027568 0.111417 1.079855

⎛ ⎞
1.125317 −0.039880 0.106601
⎜ ⎟
×⎜
⎝ 0.023973
⎟
1.129478 0.084736⎠
−0.154089 −0.115519 0.791698
⎛ ⎞
1.132700 0.000000 0.132700
⎜ ⎟
= ⎝ 0.000000 1.000000 0.000000⎠ .
⎜ ⎟

−0.132700 0.000000 0.867299

12.8 Stage 4: Nature of the lattice-invariant shear 221
Comparison with equation 11.3 shows that this is a homogeneous shear on the
system (1 0 1)[1 0 1]F with a magnitude n = 0.2654.
The lattice-invariant shear is thus determined since it is the inverse of (F Q F),
having the same average magnitude but occurring inhomogeneously on the system
(1 0 1)[1 0 1]F . If the intrinsic interface dislocations which cause this shear have
a Burgers vector b = (aγ /2)[1 0 1]γ , and if they occur on every K’th slip plane,
then if the spacing of the (1 0 1)γ planes is given by d, it follows that

n = |b|/Kd = 1/K so that K = 1/0.2654 = 3.7679

Of course, K must be an integral number, and the non-integral result must be taken
to mean that there will on average be a dislocation located on every 3.7679th slip
plane; in reality, the dislocations will be non-uniformly placed, either 3 or 4 (1 0 1)
planes apart.
The line vector of the dislocations is the invariant-line u and the spacing of the
intrinsic dislocations, as measured on the habit plane is Kd/(u ∧ p.q) where all
the vectors are unit vectors. Hence, the average spacing would be
1
3.7679(aγ 2− 2 )/0.8395675 = 3.1734aγ

and if aγ = 3.56Å, then the spacing is 11.3Å on average.

If on the other hand, the lattice-invariant shear is a twinning deformation
(rather than slip), then the martensite plate will contain very finely spaced trans-
formation twins, the structure of the interface being radically different from that
deduced above, since it will no longer contain any intrinsic dislocations. The mis-
match between the parent and product lattices was in the slip case accommodated
with the help of intrinsic dislocations, whereas for the internally twinned marten-
site there are no such dislocations. Each twin terminates in the interface to give
a facet between the parent and product lattices, a facet which is forced into co-
herency. The width of the twin and the size of the facet is sufficiently small to
enable this forced coherency to exist. The alternating twin related regions thus
prevent misfit from accumulating over large distances along the habit plane.
If the (fixed) magnitude of the twinning shear is denoted S, then the volume
fraction V of the twin orientation, necessary to cancel the effect of (F Q F), is
given by V = n/S, assuming that n < S. In the above example, the lattice-
invariant shear occurs on (1 0 1)[1 0 1]γ which corresponds to (1 1 2)α [1 1 1]α ,
and twinning on this latter system involves a shear S = √12 , giving V =
0.2654/0.707107 = 0.375.

It is important to note that the twin plane in the martensite corresponds to a mirror
plane in the austenite; this is a necessary condition when the lattice-invariant shear
involves twinning. The condition arises because the twinned and untwinned regions
of the martensite must undergo Bain strain along different though crystallographi-
cally equivalent principal axes [1, 28].
The above theory clearly predicts a certain volume fraction of twins in each
martensite plate, when the lattice-invariant shear is twinning as opposed to slip. How-
222 12 Martensite

ever, the factors governing the spacing of the twins are less quantitatively established;
the finer the spacing of the twins, the lower will be the strain energy associated with
the matching of each twin variant with the parent lattice at the interface. On the other
hand, the amount of coherent twin boundary within the martensite increases as the
spacing of the twins decreases.
A factor to bear in mind is that the lattice-invariant shear is an integral part of
the transformation; it does not happen as a separate event after the lattice change has
occurred. The transformation and the lattice-invariant shear all occur simultaneously
at the interface, as the latter migrates. It is well known that in ordinary plastic defor-
mation, twinning rather than slip tends to be the favoured deformation mode at low
temperatures or when high strain rates are involved. It is therefore often suggested
that martensite with low MS temperatures will tend to be twinned rather than slipped,
but this cannot be formally justified because the lattice-invariant shear is an integral
part of the transformation and not a physical deformation mode on its own. Indeed, it
is possible to find lattice-invariant deformation modes in martensite which do not oc-
cur in ordinary plastic deformation experiments. The reasons why some martensites
are internally twinned and others slipped are not clearly understood [29]. When the
spacing of the transformation twins is roughly comparable to that of the dislocations
in slipped martensite, the interface energies are roughly equal. The interface energy
increases with twin thickness and at the observed thicknesses is very large compared
with the corresponding interface in slipped martensite. The combination of the rel-
atively large interface energy and the twin boundaries left in the martensite plate
means that internally twinned martensite is never thermodynamically favoured rela-
tive to slipped martensite. It is possible that kinetic factors such as interface mobility
actually determine the type of martensite that occurs.

12.9 Texture due to displacive transformations

There is enormous utility in treating displacive transformations such as martensite,
bainite, Widmanstätten ferrite [30] and in some cases, cementite [31] as physical
deformations in which the crystallography is known or can be calculated using the
theory described in this chapter, with great rigour. Displacive transformations are
confined to individual grains of the parent phase and have a reproducible crystal-
lography with respect to that phase. If there are circumstances in which particular
crystallographic variants are favoured, the resulting texture can be estimated from a
knowledge of the orientations of the parent crystals and their orientation relationship
with the product phase. We shall consider this in the context of the transformation of
austenite into martensite in steels.
Given the irrationality of the macroscopic crystallographic features of martensite,
cubic symmetry dictates that there always will be 24 variants of the transformation
product per austenite grain. If all 24 variants are allowed to form in each austenite
grain, then the transformation texture becomes essentially random for any sizeable
12.9 Texture due to displacive transformations 223

sample of material. However, circumstances may force variant selection, i.e. where
certain crystals which are more compliant to an external influence, such as an exter-
nally applied stress.
It is the interaction of the applied stress with the shape deformation of individual
martensite plates P (page 217 which determines variant selection. The interaction
energy which provides the mechanical driving force for transformation is given by
[32]:
U = σN ζ + τ s (12.14)
where σN is the stress component normal to the habit plane, τ is the shear stress
resolved on the habit plane in the direction of shear and ζ and s are the respective
normal and shear strains associated with transformation. The energy U can be used as
a rigorous variant selection criterion when the role of any plastic strain is unimportant
[12].
With displacive transformations it is possible to calculate the macroscopic
plastic-strains as a function of texture; they can also be used to characterise the
texture because they are in general anisotropic [12, 33, 34]. An arbitrary vector u
traversing a grain of austenite prior to transformation (Fig. 12.5a) makes an intercept
∆u with a domain of austenite that eventually transforms, after which it becomes a
new vector v given by:
v = P∆u + (u − ∆u) (12.15)
When many plates form in many austenite grains, u traverses a polycrystalline sam-
ple of austenite so this equation is generalised as follows [12]:

. 24
n . % . 24
n . &
v= Pkj ∆ukj + u − ∆ukj (12.16)
k=1 j=1 k=1 j=1

where j = 1 . . . 24 represents the 24 crystallographic variants possible in each

austenite grain, and k = 1 . . . n represents the n austenite grains traversed by the
vector u. In this scenario of a large number of bainite plates, the intercepts ∆ukj can
be approximated by fjk u where fjk is the fraction of sample transformed by variant j
in austenite grain k.
The deformation due to a particular plate j in austenite grain k, i.e. (γk Pj γk ) ≡
Pkj . The remaining 23 such matrices for grain 1 of austenite are deduced from this
using symmetry operations. They can then be expressed in the reference frame of the
sample using a similarity transformation (page 172) as follows:

(S Pkj S) = (S R γk )(γk Pj γk )(γk R S) (12.17)

where (S R γk ) is the rotation matrix relating the basis vectors of the kth austenite
grain to the sample axes, and (γk R S) is the inverse of that rotation matrix. In this
way, the calculation described in equation 12.15 can be conducted in the sample
frame of reference.
Calculations illustrating the anisotropy of strains as a function of the number of
crystallographic variants of martensite allowed are illustrated in Fig 12.6a for uni-
axial tension when transformation occurs from a randomly oriented set of austenite
224 12 Martensite

(a) (b)

FIGURE 12.5
The deformation of an initial vector u by the formation of bainite. (a) An austenite
grain prior to transformation, with the ultimate location of a plate of bainite marked.
(b) Following displacive transformation.

grains. That displacive transformations produce highly anisotropic strains when vari-
ant selection is significant has been demonstrated experimentally [33–35].
Fig 12.6b shows that transformation texture is absent when 24 variants form in
each austenite grain, for typical intensities of texture in the austenite; the only strain
visible in these circumstances is the an averaged isotropic volume expansion. The
strength of the transformation texture increases as the number of variants per austen-
ite grain decreases.
It is emphasised here that the calculation of texture as presented here neglects the
fact that favoured variants will in fact have greater volume fractions in the microstruc-
ture. The calculations refer simply to the orientations of the martensite plates, not to
the intensities as might be observed in X-ray experiments where dominant variants
will contribute most. To deal with this problem, it is necessary to couple thermody-
namics and crystallography as described elsewhere [36, 37].
12.10 Summary 225

(a) (b)

FIGURE 12.6
These diagrams show the plastic strain that develops in a sample of 500 austenite
grains placed under tension along the longitudinal axis. The differences in the orthog-
onal strains correlate with the intensity of transformation texture, assuming that each
variant that forms contributes equally to the fraction of transformation. (a) Strains
developed due to transformation along the [1 0 0]S direction (labelled longitudinal,
along the stress axis), and the transverse directions [0 1 0]S and [0 0 1]S . (b) Tensile
stress, but transformation beginning from a variety of starting austenite textures and
illustrating only the longitudinal stress.

12.10 Summary
The crystallographic theory described is one of the most complete methods of pre-
dicting observable features of martensitic transformations, including aspects of the
lattice-invariant deformation (slip or twinning), the orientation relationship, the shape
deformation and even the lenticular shape of the martensite plates. The irrational na-
ture of some of the characteristics of martensite are also predicted to a high precision.
The inputs required are the lattice parameters of the parent and produce unit cells and
possible modes of lattice-invariant deformation.
It is worth emphasising that the theory is based on an elastically accommodated
plate that is not interacting with any other martensite plates. In particular, plastic
deformation driven by the shape deformation, and the impingement of the strain
fields of other plates can alter the morphology and crystallography [38, 39].
References

1. J. S. Bowles, and J. K. Mackenzie: ‘The crystallography of martensite transfor-

mations, part I’, Acta Metallurgica, 1954, 2, 129–137.
2. J. K. Mackenzie, and J. S. Bowles: ‘The crystallography of martensite transfor-
mations II’, Acta Metallurgica, 1954, 2, 138–147.
3. J. K. Mackenzie, and J. S. Bowles: ‘The crystallography of martensite trans-
formations III FCC to BCT transformations’, Acta Metallurgica, 1954, 2, 224–
234.
4. M. S. Wechsler, D. S. Lieberman, and T. A. Read: ‘On the theory of the forma-
tion of martensite’, Trans. AIME Journal of Metals, 1953, 197, 1503–1515.
5. C. M. Wayman: ‘The growth of martensite since E C Bain (1924) - some
milestones’, Materials Forum, 1990, 56-58, 1–32.
6. T. S. Kuhn: The structure of scientific revolutions: Chicago, U.S.A.: University
of Chicago Press, 1996.
7. D. K. Simonton: ‘Scientific genius is extinct’, Nature, 2013, 493, 602.

8. R. Bullough, and B. A. Bilby: ‘Continuous distributions of dislocations: Sur-

face dislocations and the crystallography of martensitic transformations’, Proc.
Physics Society B, 1956, 69, 1276–1286.
9. W. Bollmann: Crystal defects and crystalline interfaces: Berlin, Germany:
Springer Verlag, 1970.
10. J. W. Christian: Theory of Transformations in Metals and Alloys, Part I: 3 ed.,
Oxford, U. K.: Pergamon Press, 2003.
11. J. W. Christian: Theory of Transformations in Metals and Alloys, Part II: 3 ed.,
Oxford, U. K.: Pergamon Press, 2003.
12. S. Kundu, K. Hase, and H. K. D. H. Bhadeshia: ‘Crystallographic texture of
stress-affected bainite’, Proceedings of the Royal Society A, 2007, 463, 2309–
2328.
13. H. K. D. H. Bhadeshia: ‘Problems in the calculation of transformation texture
in steels’, ISIJ International, 2010, 50, 1517–1522.

227
228 12 References

14. H. K. D. H. Bhadeshia: ‘Multiple, simultaneous, martensitic transformations:

implications on transformation texture intensities’, Materials Science Forum,
2013, 762, 9–13.
15. J. W. Christian: ‘The origin of surface relief effects in phase transformations’,
In: V. F. Zackay, and H. I. Aaronson, eds. Decomposition of austenite by diffu-
sional processes. New York, USA: Interscience, 1962:371–386.
16. J. D. Eshelby: ‘The determination of the elastic field of an ellipsoidal inclusion
and related problem’, Proceedings fo the Royal Society A, 1957, 241, 376–396.
17. J. W. Christian: ‘Accommodation strains in martensite formation, the use of the
dilatation parameter’, Acta Metallurgica, 1958, 6, 377–379.
18. J. W. Christian: ‘Thermodynamics and kinetics of martensite’, In: G. B. Olson,
and M. Cohen, eds. International Conference on Martensitic Transformations
ICOMAT ’79. Massachusetts, USA: Alpine Press, 1979:220–234.
19. P. Ehrenfest: ‘Phasenumwandlungen im ueblichen und erweiterten sinn, classi-
fiziert nach dem entsprechenden sigularitaeten des thermodynamischen poten-
tiales’, Verhandelingen der Koninklijke Akademie van Wetenschappen (Amster-
dam), 1933, 36, 153–157.
20. L. Kaufman, and M. Cohen: ‘Thermodynamics and kinetics of martensitic
transformation’, Progress in Metal Physics, 1958, 7, 165–246.
21. J. W. Christian, and K. M. Knowles: ‘Interface structures and growth mech-
anisms’, In: H. I. Aaronson, D. E. Laughlin, M. Sekerka, and C. M. Way-
man, eds. Solid-Solid Phase Transformations. Warrendale, Pennsylvania, USA:
TMS-AIME, 1982:1185–1208.
22. J. W. Christian: Theory of Transformations in Metals and Alloys, Part I: 2 ed.,
Oxford, U. K.: Pergamon Press, 1975.
23. G. F. Bolling, and R. H. Richman: ‘Continual mechanical twinning Parts III,
IV’, Acta Metallurgica, 1965, 13, 745–757.
24. J. W. Christian, and A. G. Crocker: Dislocations in Solids, ed. F. R. N. Nabarro:
Amsterdam, Holland: North Holland, 1980.
25. A. B. Greninger, and A. R. Troiano: ‘The mechanism of martensite formation’,
Trans. A.I.M.E., 1949, 185, 590–598.
26. J. W. Christian, and A. G. Crocker: Dislocations in Solids, ed. F. R. N. Nabarro,
vol. 3, chap. 11: Amsterdam, Holland: North Holland, 1980:165–252.
27. G. B. Olson, and M. Cohen: ‘Interphase boundary dislocations and the concept
of coherency’, Acta Metallurgica, 1979, 27, 1907–1918.
28. J. W. Christian: Theory of Transformations in Metals and Alloys: Oxford, U.
K.: Pergamon Press, 1965.
12.10 References 229

29. G. B. Olson, and M. Cohen: ‘Theory of martensitic nucleation: a current

assessment’, In: M. S. H. I. Aaronson, D. E. Laughlin:, ed. Proc. Int. Conf.
Solid→Solid Phase Transformations. Warrendale, Pennsylvania, USA: TMS-
AIME, 1981:1209–1213.
30. H. K. D. H. Bhadeshia, and R. W. K. Honeycombe: Steels: Microstructure and
Properties: 4th ed., Elsevier, 2017.
31. J. W. Stewart, R. C. Thomson, and H. K. D. H. Bhadeshia: ‘Cementite pre-
cipitation during tempering of martensite under the influence of an externally
applied stress’, Journal of Materials Science, 1994, 29, 6079–6084.
32. J. R. Patel, and M. Cohen: ‘Criterion for the action of applied stress in the
martensitic transformation’, Acta Metallurgica, 1953, 1, 531–538.
33. H. K. D. H. Bhadeshia, S. A. David, J. M. Vitek, and R. W. Reed: ‘Stress
induced transformation to bainite in a Fe-Cr-Mo-C pressure vessel steel’, Ma-
terials Science and Technology, 1991, 7, 686–698.
34. A. Matsuzaki, H. K. D. H. Bhadeshia, and H. Harada: ‘Stress-affected bainitic
transformation in a Fe-C-Si-Mn alloy’, Acta Metallurgica and Materialia,
1994, 42, 1081–1090.
35. H. K. D. H. Bhadeshia: ‘Possible effects of stress on steel weld microstruc-
tures’, In: H. Cerjak, and H. K. D. H. Bhadeshia, eds. Mathematical Modelling
of Weld Phenomena – II. London, U.K.: Institute of Materials, 1995:71–118.

36. H. N. Han, C. G. Lee, C.-S. Oh, T.-O. Lee, and S.-J. Kim: ‘A model for de-
formation behaviour and mechanically induced martensitic transformation of
metastable austenitic steel’, Acta Materialia, 2004, 52, 5203–5214.
37. H. K. D. H. Bhadeshia, A. Chintha, and S. Kundu: ‘Model for multiple stress
affected martensitic transformations, microstructural entropy and consequences
on scatter in properties’, Materials Science and Technology, 2014, 30, 160–165.
38. G. Miyamoto, A. Shibata, T. Maki, and T. Furuhara: ‘Precise measurement
of strain accommodation in austenite matrix surrounding martensite in ferrous
alloys by electron backscatter diffraction analysis’, Acta Materialia, 2009, 57,
1120–1131.
39. J. H. Yang, and C. M. Wayman: ‘Self-accommodation and shape memory
mechanism of epsilon martensite - I. experimental observations’, Materials
Characterization, 1992, 28, 23–35.
13
Interfaces

Abstract

That interfaces have structure, some of which may be periodic, is now established
and observable using techniques that have sufficient spatial resolution. Small misori-
entations between like-crystals that are connected at a boundary, can be related to
arrays of dislocations, which in turn lead to an estimation of the boundary energy
per unit area. In this chapter, we describe a method for estimating the total Burgers
vector content of an arbitrary interface that in general connects crystals with differ-
ent structures, as a function of the degrees of freedom that specify the interface. The
concept of the coincidence site lattice is then generalised as Bollmann’s O-lattice.
The ‘secondary’ dislocations that describe how a boundary may move and yet pre-
serve the level of fit add to the impressive array of tools available in understanding
interfaces. There remain some difficulties, which are highlighted towards the end of
the discussion.

13.1 Introduction
Atoms located at the boundary between crystals must in general be displaced from
positions they would occupy in the undisturbed crystal. Nevertheless, it is now well
established that many interfaces have a periodic structure. In such cases, the misfit be-
tween the crystals connected by the boundary is not distributed uniformly over every
element of the interface; it is localised periodically into discontinuities which sepa-
rate patches of the boundary where the fit between the two crystals is good or perfect.
When these discontinuities are well separated, they may individually be recognised
as interface dislocations which separate coherent patches in the boundary, which is
macroscopically said to be semi-coherent.
Stress-free coherent interfaces can exist only between crystals which can be re-
lated by a transformation strain which is an invariant-plane strain. This transforma-
tion strain may be real or notional as far as the calculation of the interface structure
is concerned, but a real strain implies the existence of an atomic correspondence and
an associated macroscopic shape change of the transformed region between the two
crystals, which a notional strain does not.
Incoherency presumably sets in when the misfit between adjacent crystals is so

231
232 13 Interfaces

high that it cannot satisfactorily be localised into identifiable interface dislocations,

giving a boundary structure which is difficult to physically interpret, other than to
say that the motion of such an interface must always occur by the uncoordinated
and haphazard transfer of atoms across the interface. This could be regarded as a
definition of incoherency. As will become clear later, the intuitive feeling that all
‘high-angle’ boundaries are incoherent is not correct.

13.2 Misfit
The misfit across an interface can formally be described in terms of the net Burgers
vector bt crossing a vector p in the interface [1–3]. If this misfit is sufficiently small
then the boundary structure may relax into a set of discrete interfacial dislocations
where the misfit is concentrated)( which are separated by patches of good fit.
In any case, bt may be deduced by constructing a Burgers circuit across the inter-
face, and examining the closure failure when a corresponding circuit is constructed
in a perfect reference lattice. The procedure is illustrated in Fig. 13.1, where crystal
A is taken to be the reference lattice. An initial right-handed Burgers circuit OAPBO
is constructed such that it straddles the interface across any vector p=OP in the in-
terface; the corresponding circuit in the perfect reference lattice is constructed by
deforming crystal B (of the bi-crystal A-B) in such a way that it is converted into
the lattice of A, thereby eliminating the interface. If the deformation (A S A) con-
−1
verts the reference lattice into the B lattice, then the inverse deformation (A S A)
converts the bi-crystal into a single A crystal, and the Burgers circuit in the perfect
reference lattice becomes OAPP′ B, with a closure failure PP′ , which is identified as
bt . Inspection of the vectors forming the triangle OPP′ of Fig. 13.1b shows that:

[A; bt ] = {I − (A S A)−1 }[A; p]. (13.1)

Hence, the net Burgers vector content bt crossing an arbitrary vector p in the
interface is given formally by equation 13.1. The misfit in any interface can in general
be accommodated with three arrays of interfacial dislocations, whose Burgers vectors
bi (i = 1, 2, 3) form a non-coplanar set. Hence, bt can in general be factorised into
three arrays of interfacial dislocations, each array with Burgers vector bi , unit line
vector li and array spacing di , the latter being measured in the interface plane. If the
unit interface normal is n, then a vector mi may be defined as (the treatment that
follows is due to Knowles [4] and Read [5]):

mi = n ∧ li /di (13.2)

We note the |mi | = 1/di , and that any vector p in the interface crosses (mi .p)
dislocations of type I (see Fig. 13.1c). Hence, for the three kinds of dislocations we
have
bt = (m1 .p)b1 + (m2 .p)b2 + (m3 .p)b3
13.2 Misfit 233

(a) (b)

(c) (d)

FIGURE 13.1
(a,b) Burgers circuit used to define the formal dislocation content of an interface [3],
(c) the vector p in the interface, (d) relationship between li , mi , ci and n.

We note that the Burgers vectors bi of interface dislocations are generally lattice
translation vectors of the reference lattice in which they are defined. This makes them
perfect in the sense that the displacement of one of the crystals through bi relative
to the other does not change the structure of the boundary. On the basis of elastic
strain energy arguments, bi should be as small as possible. On substituting this into
equation 13.1, we get:

(m1 ; A∗ )[A; p][A; b1 ] + (m2 ; A∗ )[A; p][A; b2 ]

+ (m3 ; A∗ )[A; p][A; b3 ] = (A T A)[A; p] (13.3)
−1
where (A T A) = I − (A S A) .
If a scalar dot product is taken on both sides of equation 13.3 with the vector b∗1 ,
where b∗1 is
b2 ∧ b3
b∗1 = (13.4)
b1 .b2 ∧ b3
then we obtain
(b∗1 ; A∗ )(A T A)[A; p] = (m1 ; A∗ )[A; p] (13.5)
If p is now taken to be equal to l1 in equation 13.5, we find that

(b∗1 ; A∗ )(A T A)[A; l1 ] = 0

234 13 Interfaces

and
(l1 ; A) (A T′ A)[A∗ ; b∗1 ] = 0 (13.6)
/ 01 2
[A∗ ;c1 ]

where the vector defined as c1 is normal to l1 . If m1 is now substituted for p in

equation 13.5, then we find:

(b∗1 ; A∗ )(A T A)[A; m1 ] = (m1 ; A∗ )[A; m1 ] = |m1 |2

′ ∗
∴ (m1 ; A)(A T A)[A ; b∗1 ] = |m1 |2

and
(m1 ; A)[A∗ ; c1 ] = |m1 |2 i.e. m1 .c1 = |m1 |2
These equations indicate that the projection of c1 along m1 is equal to the magnitude
of m1 . Armed with this and the earlier result that c1 is normal to l1 , the vector
diagram illustrated in Fig. 13.1d can be constructed to illustrate the relations between
m1 , l1 , n and c1 . From this diagram, it is evident that

m1 = c1 − (c1 .n)n
so that |m1 ∧ n| = |c1 ∧ n| = 1/d1 (13.7)

and in addition, Fig. 13.1d shows that

l1 ∥ c1 ∧ n

alternatively,
(1/d1 )l1 = c1 ∧ n (13.8)
Relations of this type are, after appropriately changing indices, applicable also to
the other two arrays of interfacial dislocations, so that a method has been achieved
of deducing the line vectors and array spacings to be found in an interface (of unit
normal n) connecting two arbitrary crystals A and B, related by the deformation
(A S A) which transforms the crystal A to B.
Example 13.1: Symmetrical tilt boundary
Given that the Burgers vectors of interface dislocations in low-angle bound-
aries are of the form ⟨1 0 0⟩, calculate the dislocation structure of a symmetrical
tilt boundary formed between two grains, A and B, related by a rotation of 2θ
about the [1 0 0] axis. The crystal structure of the grains is simple cubic with a
unit lattice parameter.
The definition of a tilt boundary is that the unit boundary normal n is per-
pendicular to the axis of rotation which generates one crystal from the other; a
symmetrical tilt boundary has the additional property that the lattice of one crys-
tal can be generated from the other by reflection across the boundary plane.
If we choose the orthonormal bases A and B (with basis vectors parallel to the
cubic unit cell edges) to represent crystals A and B respectively, and also arbitrar-
ily choose B to be the reference crystal, then the deformation which generates the
13.2 Misfit 235
A crystal from B is a rigid body rotation (B J B) consisting of a rotation of 2θ
about [1 0 0]B . Hence, (B J B) is given by (see equation 7.9):
⎛ ⎞
1 0 0
⎜ ⎟
(B J B) = ⎜ ⎝0 cos 2θ − sin 2θ⎠
⎟

0 sin 2θ cos 2θ
−1
and (B T B) = I − (B J B) is given by
⎛ ⎞
0 0 0
⎜ ⎟
(B T B) = ⎜
⎝0 1 − cos 2θ
⎟
sin 2θ ⎠
0 − sin 2θ 1 − cos 2θ

so that ⎛ ⎞
0 0 0
⎜ ⎟
(B T′ B) = ⎜
⎝0 1 − cos 2θ − sin 2θ ⎠ .
⎟

0 sin 2θ 1 − cos 2θ
Taking [B; b1 ] = [1 0 0], [B; b2 ] = [0 1 0] and [B; b3 ] = [0 0 1], from
equation 13.4 we note that [B∗ ; b∗1 ] = [1 0 0], [B∗ ; b∗2 ] = [0 1 0] and [B∗ ; b∗3 ] =
[0 0 1]. Equation 13.6 can now be used to obtain the vectors ci :
6 7
[B∗ ; c1 ] = (B T′ B)[B∗ ; b∗1 ] = 0 0 0
6 7
[B∗ ; c2 ] = (B T′ B)[B∗ ; b∗2 ] = 0 1 − cos 2θ sin 2θ
6 7
[B∗ ; c3 ] = (B T′ B)[B∗ ; b∗3 ] = 0 − sin 2θ 1 − cos 2θ

Since c1 is a null vector, dislocations with Burgers vector b1 do not exist in the
interface; furthermore, since c1 is always a null vector, irrespective of the bound-
ary orientation n, this conclusion remains valid for any n. This situation arises
because b1 happens to be parallel to the rotation axis, and because (B J B) is an
invariant-line strain, the invariant line being the rotation axis; since any two crys-
tals of identical structure can always be related by a transformation which is a rigid
body rotation, it follows that all grain boundaries (as opposed to phase boundaries)
need only contain two sets of interface dislocations. If b1 had not been parallel to
the rotation axis, then c1 would be finite and three sets of dislocations would be
necessary to accommodate the misfit in the boundary.
To calculate the array spacings di it is necessary to express n in the B∗ basis.
A symmetrical tilt boundary always contains the axis of rotation and has the same
indices in both bases. It follows that

(n; B∗ ) = (0 cos θ − sin θ)

236 13 Interfaces
and from equation 13.8,

(1/d2 )l2 = c2 ∧ n = [−2 sin θ 0 0]B

so that
[B; l2 ] = [1 0 0]
and
d2 = 1/2 sin θ
similarly,
(1/d3 )l3 = c3 ∧ n = [0 0 0]
so that dislocations with Burgers vector b3 have an infinite spacing in the inter-
face, which therefore consists of just one set of interface dislocations with Burgers
vector b2 .

These results are of course identical to those obtained from a simple geometri-
cal construction of the symmetrical tilt boundary [3]. We note that the boundary is
glissile (i.e. its motion does not require the creation or destruction of lattice sites)
because b2 lies outside the boundary plane, so that the dislocations can glide con-
servatively as the interface moves. In the absence of diffusion, the movement of the
boundary leads to a change in shape of the ‘transformed’ region, a shape change
described by (B J B) when the boundary motion is towards the crystal A.
If on the other hand, the interface departs from its symmetrical orientation (with-
out changing the orientation relationship between the two grains), then the boundary
ceases to be glissile, since the dislocations with Burgers vector b3 acquire finite spac-
ings in the interface. Such a boundary is called an asymmetrical tilt boundary. For
example, if n is taken to be (n; B∗ ) = (0 1 0), then

[B; l2 ] = [1 0 0] and d2 = 1/ sin 2θ

[B; l3 ] = [1 0 0] and d3 = 1/(1 − cos 2θ)

The edge dislocations with Burgers vector b3 lie in the interface plane and there-
fore have to climb as the interface moves. This renders the interface sessile.
Finally, we consider the structure of a twist boundary, a boundary where the axis
of rotation is parallel to n. Taking (n; B∗ ) = (1 0 0), we find:

[B; l2 ] ∥ [0 sin 2θ cos 2θ − 1],

[B; l3 ] ∥ [0 1 − cos 2θ sin 2θ],

1
and d2 = d3 = [2 − 2 cos 2θ] 2 .
It is noteworthy that dislocations with Burgers vector b1 do not exist in the in-
terface since b1 is parallel to the rotation axis. The interface thus consists of a cross
grid of two arrays of dislocations with Burgers vectors b2 and b3 respectively, the
array spacings being identical. Although it usually is stated that pure twist bound-
aries contain grids of pure screw dislocations, we see that both sets of dislocations
13.2 Misfit 237

actually have a small edge component. This is because the dislocations, where they
mutually intersect, introduce jogs into each other so that the line vector in the region
between the points of intersection does correspond to a pure screw orientation, but
the jogs make the macroscopic line vector deviate from this screw orientation.
Example 13.2: Interface between alpha and beta brass
The lattice parameter of the fcc alpha phase of a 60 wt% Cu-Zn alloy is 3.6925
Å, and that of the bcc beta phase is 2.944 Å [6]. Two adjacent grains A and B
(alpha phase and beta phase respectively) are orientated in such a way that

[1 1 1]A ∥ [1 1 0]B

[1 1 2]A ∥ [1 1 0]B
[1 1 0]A ∥ [0 0 1]B
and the grains are joined by a boundary which is parallel to (1 1 0)A . Assuming
that the misfit in this interface can be fully accommodated by interface disloca-
tions which have Burgers vectors [A; b1 ] = 21 [1 0 1], [A; b2 ] = 12 [0 1 1] and
[A; b3 ] = 21 [1 1 0], calculate the misfit dislocation structure of the interface. Also
assume that the smallest pure deformation which relates fcc and bcc crystals is the
Bain strain.
The orientation relations provided are first used to calculate the coordinate
transformation matrix (B J A), using the procedure given in Example 9.3. (B J A)
is thus found to be:
⎛ ⎞
0.149974 0.149974 1.236185
⎜ ⎟
(B J A) = ⎜ ⎝0.874109 0.874109 −0.212095⎠ .
⎟

0.874109 0.874109 −0.212095

The smallest pure deformation relating the two lattices is stated to be the Bain
strain, but the total transformation (A S A), which carries the A lattice to that of
B, may include an additional rigid body rotation. Determination of the interface
structure requires a knowledge of (A S A), which may be calculated from the
equation (A S A)−1 = (A C B)(B J A), where (A C B) is the correspondence
matrix. Since (B J A) is known, the problem reduces to the determination of the
correspondence matrix; if a vector equal to a basis vector of B, due to transforma-
tion becomes a new vector u, then the components of u in the basis A form one
column of the correspondence matrix.
For the present example, the correspondence matrix must be a variant of the
Bain correspondence. The vector that [1 0 0]B must become, as a result of trans-
formation, either a vector of the form ⟨1 0 0⟩A , or a vector of the form 12 ⟨1 1 0⟩A ,
although we do not know its final specific indices in the austenite lattice. How-
ever, from the matrix (B J A) we note that [1 0 0]B is close to [0 0 1]A , making it is
reasonable to assume that [1 0 0]B corresponds to [0 0 1]A , so that the first column
of (A C B) is [0 0 1]. Similarly, using (B J A) we find that [0 1 0]B is close to
238 13 Interfaces

[1 1 0]A and [0 0 1]B is close to [1 1 0]A , so that the other two columns of (A C B)
are [ 12 21 0] and [− 21 21 0]. Hence, (A C B) is found to be:
⎛ ⎞
0 1 1
1⎜ ⎟
(A C B) = ⎜ 0 1 1
⎟
2 ⎝ ⎠
2 0 0
−1
so that (A S A) = (A C B)(B J A) is given by:
⎛ ⎞
0.880500 −0.006386 −0.106048
−1 ⎜ ⎟
(A S A) = ⎜ ⎝ −0.006386
⎟
0.880500 −0.106048⎠
0.149974 0.149974 1.236183
and since
−1
(A T A) = I − (A S A) ,
′
so that (A T A) becomes:
⎛ ⎞
0.119500 0.006386 −0.149974
⎜ ⎟
(A T′ A) = ⎜
⎝0.006386 0.119500 −0.149974⎠ .
⎟

0.016048 0.016048 −0.236183

The vectors b∗i , as defined by equation 13.4, are given by:
6 7
[A∗ ; b∗1 ] = 1 1 1
6 7
[A∗ ; b∗2 ] = 1 1 1
6 7
[A∗ ; b∗3 ] = 1 1 1

and applying equation 13.6 results in

[A∗ ; c1 ] = [0.263088 0.036860 0.236183]

∗
[A ; c2 ] = [0.036860 0.263088 0.236083]
∗
[A ; c3 ] = [−0.024088 −0.024088 −0.024087]

and from equation 13.8 and the fact that (n; A∗ ) = 3.6925(−0.707107 0.707107 0),

(1/d1 )[A; l1 ] = [−0.012249 −0.012249 0.015556]

(1/d2 )[A; l2 ] = [−0.012249 −0.012249 0.015556]
(1/d3 )[A; l3 ] = [ 0.001249 0.001249 −0.002498]

so that d1 = d2 = 11.63 Å, and d3 = 88.52 Å.

13.3 Coincidence site lattices 239

13.3 Coincidence site lattices

It has been emphasised that equation 13.1 defines the formal Burgers vector con-
tent of an interface; it sometimes is possible to interpret this in terms of physically
meaningful interfacial dislocations, if the misfit across the interface is sufficiently
small.
For large misorientation boundaries, the predicted spacings of dislocations may
turn out to be so small that the misfit is highly localised with respect to the boundary,
and the dislocation model of the interface has only formal significance – it often
is said that the dislocations get so close to each other that their cores overlap. The
arrangement of atoms in such incoherent boundaries may be haphazard, with little
correlation of atomic positions across the boundary.
On the other hand, it is unreasonable to assume that all high-angle boundaries
have the disordered structure suggested above. There is clear experimental evi-
dence which shows that certain high-angle boundaries exhibit the characteristics of
low-energy coherent or semi-coherent interfaces; for example, they exhibit strong
faceting, have very low mobility in pure materials and the boundary diffusion coeffi-
cient may be abnormally low. These observations imply that at certain relative crystal
orientations, which would usually be classified as large misorientations, it is possible
to obtain boundaries which have a distinct structure – they contain regions of good
fit, which occur at regular intervals in the boundary plane, giving a pattern of good
fit points in the interface. It is along these points that the two crystals connected by
the boundary match exactly.
If we consider the good fit points to correspond to lattice points in the interface
that are common to both crystals, then the following procedure allows us to deduce
the pattern and frequency of these points for any given orientation relationship.
If the two lattices with a common origin are notionally allowed to interpenetrate
and fill all space, then there may exist lattice points (other than the origin) which are
common to both the crystals. The set of these coincidence points forms a coincidence
site lattice (CSL) [7, 8]; the fraction of lattice points which are also coincidence
sites is a rational fraction 1/Σ, where the denominator is the reciprocal density of
coincidence sites relative to ordinary lattice sites. The value of Σ is a function of
the relative orientation of the two grains and not of the orientation of the boundary
plane. The boundary simply intersects the CSL and will contain regions of good fit
which have a periodicity of a planar net of the CSL along which the intersection
occurs. Boundaries parallel to low-index planes of the CSL are two dimensionally
periodic with relatively small repeat cells, and those boundaries with a high planar
coincidence site density should have a relatively low energy per unit area.
Example 13.3: Coincidence site lattices
The axis-angle pair describing the orientation relationship between the two
grains (A and B) of austenite is given by:

axis of rotation parallel to [1 1 2]A

240 13 Interfaces
right-handed angle of rotation 180◦
Show that a CSL with Σ = 3 can be formed by allowing the two lattices
sharing an origin to notionally interpenetrate. Show also that 1 in 3 of the sites in
an interface parallel to (0 2 1)A are coincidence sites.
Because of the centrosymmetric nature of the austenite lattice, a rotation of
180◦ about [1 1 2]A , which generates the B grain from A, is equivalent to a reflec-
tion of the A lattice about (1 1 2)A . We may therefore imagine that B is generated
by reflection of the A lattice about (1 1 2)A as the mirror plane. The stacking
sequence of (1 1 2)A is . . . ABCDEF ABCDEF ABCDEF . . . and grain A is
represented below as a stack of (1 1 2) planes, B being generated by reflecting the
A lattice about one of the (1 1 2)A planes.

Grain A
! "# $
ABCDEF ABC DEF ABCDEF ABC DEF ABC DEF
| | | | | | [1 1 2]A →

# B C D E$!F A B C D" C
A # B A F E D C B A F$!E D C B A F E D C B"
Grain A Grain B

The first sequence represents a stack of (1 1 2)A planes, as do the first 9 layers of the
second sequence. The remainder of the second sequence represents a stack of (1 1 2)B
planes (note that B lattice is obtained by reflection of A about the 9th layer of the second
sequence). Since the two sequences have the same origin, comparison of the first sequence
with the B part of the second sequence amounts to allowing the two crystals to interpene-
trate in order to identify coincidences. Clearly, every 3rd layer of grain B coincides exactly
with a layer from the A lattice (dashed vertical lines), giving Σ = 3.
A boundary parallel to (1 1 2)A will be fully coherent; at least 1 in 3 of the sites in any
other boundary, such as (0 2 1)A will be coincidence sites.

We now consider a mathematical method [3, 9] of determining the CSL formed

by allowing the lattices of crystals A and B to notionally interpenetrate. A and B are
assumed to be related by a transformation (A S A) which deforms the A lattice into
that of B; A and B need not have the same crystal structure or lattice parameters, so
that (A S A) need not be a rigid body rotation. Consider a vector u which is a lattice
vector whose integral components do not have a common factor. As a result of the
transformation (A S A), u becomes a new vector x such that

[A; x] = (A S A)[A; u] (13.9)

x does not necessarily have integral components in the A basis (i.e. it need not be
a lattice vector of A). CSL vectors, on the other hand, identify lattice points which
are common to both A and B, and therefore are lattice vectors of both crystals. It
follows that CSL vectors have integral indices when referred to either crystal. Hence,
x is only a CSL vector if it has integral components in the basis A. We note that x
always has integral components in B, because a lattice vector of A (such as u) always
deforms into a lattice vector of B.
13.3 Coincidence site lattices 241

The meaning of Σ is that 1/Σ of the lattice sites of A or B are common to both
A and B. It follows that any primitive lattice vector of A or B, when multiplied
by Σ, must give a CSL vector. Σx must therefore always be a CSL vector and if
equation 13.9 is multiplied by Σ, then we obtain an equation in which the vector u
always transforms into a CSL vector:

Σ[A; x] = Σ(A S A)[A; u] (13.10)

i.e. given that u is a lattice vector of A, whose components have no common factor,
Σx is a CSL vector with integral components in either basis. This can only be true
if the matrix Σ(A S A) has elements which are all integral since it is only then that
Σ[A; x] has elements which are all integral.
It follows that if an integer H can be found such that all the elements of the matrix
H(A S A) are integers (without a common factor), then H is the Σ value relating A
and B.
The rotation matrix corresponding to the rotation 180◦ about [1 1 2]A is given by
(equation 7.9) ⎛ ⎞
2 1 2
1⎜ ⎟
(A J A) = ⎜ ⎟
3 ⎝1 2 2⎠
2 2 1
and since 3 is the integer which when multiplied with (A J A) gives a matrix of in-
tegral elements (without a common factor), the Σ value for this orientation is given
by Σ = 3 . For reasons of symmetry (see Chapter 9), the above rotation is crystal-
lographically equivalent to a rotation of 60◦ about [1 1 1]A with the rotation matrix
given by ⎛ ⎞
2 2 1
1⎜ ⎟
(A J A) = ⎜ ⎟
3 ⎝1 2 2⎠
2 1 2
◦
so it is not surprising that a rotation of 60 about [1 1 1]A also corresponds to a Σ = 3
value.
Finally, we see from equation 13.10 that if the integer H (defined such that
H(A S A) has integral elements with no common factor) turns out to be even, then
the Σ value is obtained by successively dividing H by 2 until the result H’ is an
odd integer. H’ then represents the true Σ value. This is because (equation 13.10) if
H[A; x] is a CSL vector and if H is even, then Hx has integral even components in
A, but H ′ x would also have integral components in A and would therefore represent
a smaller CSL vector. From page 191, the transformation strain relating fcc-austenite
and hcp-martensite is given by:
⎛ ⎞ ⎛ ⎞
1.083333 0.083333 0.083333 13 1 1
⎜ ⎟ 1 ⎜ ⎟
(Z P Z) = ⎜ ⎝ 0.083333 1.083333 0.083333⎠ = 12 ⎝ 1
⎟ ⎜
13 1⎠
⎟

−0.166667 −0.166667 0.833333 −2 −2 9

242 13 Interfaces

so that H = 12, but H ′ = Σ = 3. This can be illustrated by considering the stacking

sequence of the close-packed planes. The first sequence below represents a stack of
the (1 1 1) planes of the fcc lattice, as do the first 9 planes of the second stacking
sequence. The other planes of the second sequence represent the basal planes of the
hcp lattice. Since two out of every 6 layers are in exact coincidence, Σ = 3 , as
shown earlier.

fcc
1 2/ 0
ABCABCABCABCABCABCABCAB
| | | | | | [1 1 1]γ ∥ [0 0 0 1]hcp →
A
/ BCAB
01 C A B C2 A
/ CACAC A01C A C A C A C2
fcc hcp

Example 13.4: Symmetry and the axis-angle representations of CSL’s

Show that the coincidence site lattice associated with two cubic crystals related
by a rotation of 50.5◦ about ⟨1 1 0⟩ has Σ = 11. Using the symmetry operations
of the cubic lattice, generate all possible axis-angle pair representations which
correspond to this Σ value.
The rotation matrix corresponding to the orientation relation 50.5◦ about
⟨1 1 0⟩ is given by (equation 7.9):
⎛ ⎞ ⎛ ⎞
0.545621 −0.545621 0.636079 6 −6 7
⎜ ⎟ 1 ⎜ ⎟
(A J A) = ⎜ ⎝0.181961 0.818039 0.545621 ⎠ = 11 ⎝2 9
⎟ ⎜
6⎠
⎟

0.818039 0.181961 −0.545621 9 2 −6

so that Σ = 11 (A is the basis symbol representing one of the cubic crystals). The
24 symmetry operations of the cubic lattice are given by:
13.3 Coincidence site lattices 243

Angle (degrees) Axis

0,90,180,270 ⟨1 0 0⟩
90,180,270 ⟨0 1 0⟩
90,180,270 ⟨0 0 1⟩
180 ⟨1 1 0⟩
180 ⟨1 0 1⟩
180 ⟨0 1 1⟩
180 ⟨1 1 0⟩
180 ⟨1 0 1⟩
180 ⟨0 1 1⟩
120,240 ⟨1 1 1⟩
120,240 ⟨1 1 1⟩
120,240 ⟨1 1 1⟩
120,240 ⟨1 1 1⟩

If any symmetry operation is represented as a rotation matrix which then pre-

multiplies (A J A), then the resulting new rotation matrix gives another axis-
angle pair representation. Hence, the alternative axis-angle pair representations
of Σ = 11 are found to be:

Angle (degrees) Axis

82.15 ⟨1 3 3⟩
162.68 ⟨3 3 5⟩
155.37 ⟨1 2 4⟩
180.00 ⟨2 3 3⟩
62.96 ⟨1 1 2⟩
129.54 ⟨1 1 4⟩
129.52 ⟨0 1 1⟩
126.53 ⟨1 3 5⟩
180.00 ⟨1 1 3⟩
100.48 ⟨0 2 3⟩
144.89 ⟨0 1 3⟩

This procedure can be used to derive all the axis-angle pair representations of
any Σ value, and the table below gives some of the CSL relations for cubic crys-
244 13 Interfaces

tals, quoting the axis-angle pair representations which have the minimum angle of
rotation, and also those corresponding to twin axes.

TABLE 13.1
Some CSL relations for Cubic crystals [3]

Σ Angle Axis Twin axes

3 60.0 ⟨1 1 1⟩ ⟨1 1 1⟩, ⟨1 1 2⟩
5 36.9 ⟨1 0 0⟩ ⟨0 1 2⟩, ⟨0 1 3⟩
7 38.2 ⟨1 1 1⟩ ⟨1 2 3⟩
9 38.9 ⟨1 1 0⟩ ⟨1 2 2⟩, ⟨1 1 4⟩
11 50.5 ⟨1 1 0⟩ ⟨1 1 3⟩, ⟨2 3 3⟩
13a 22.6 ⟨1 0 0⟩ ⟨0 2 3⟩, ⟨0 1 5⟩
13b 27.8 ⟨1 1 1⟩ ⟨1 3 4⟩
15 48.2 ⟨2 1 0⟩ ⟨1 2 5⟩
17a 28.1 ⟨1 0 0⟩ ⟨0 1 4⟩, ⟨0 3 5⟩
17b 61.9 ⟨2 2 1⟩ ⟨2 2 3⟩, ⟨3 3 4⟩
19a 26.5 ⟨1 1 0⟩ ⟨1 3 3⟩, ⟨1 1 6⟩
19b 46.8 ⟨1 1 1⟩ ⟨2 3 5⟩
21a 21.8 ⟨1 1 1⟩ ⟨2 3 5⟩, ⟨1 4 5⟩
21b 44.4 ⟨2 1 1⟩ ⟨1 2 4⟩

We note the following further points about CSL relations:

(i) All of the above CSL relations can be represented by a rotation of 180◦ about
some rational axis which is not an even axis of symmetry. Any such operation
corresponds to a twinning orientation (for centrosymmetric crystals), the lattices
being reflected about the plane normal to the 180◦ rotation axis. It follows that
a twin orientation always implies the existence of a CSL, but the reverse is not
always true [3, 10]; for example, in the case of Σ = 39, there is no axis-angle pair
representation with an angle of rotation of 180◦ , so that it is not possible to find a
coherent interface between crystals related by a Σ = 39 orientation relation.
(ii) Boundaries containing a high absolute density of coincidence sites (i.e. a large
value of number of coincidence sites per unit boundary area) can in general be
expected to have the lowest energy.
(iii) Calculations of atomic positions [11–13] in the boundary region, using inter-
atomic force laws, suggest that in materials where the atoms are hard (strong
13.4 Coincidence site lattices 245

repulsive interaction at short range), coincidence site lattices may not exist. For
example, in the case of a Σ3 twin in a ‘hard’ bcc material, with a {1 1 2} coherent
a
twin plane, it is found that a small rigid translation (by a vector 12 ⟨1 1 1⟩) of the
twin lattice lowers the energy of the interface (Fig. 13.2) [3, 14]. Because of this
relaxation, the lattices no longer have a common origin and so the coincidences
vanish. Nevertheless, boundaries which contain high densities of coincidence sites
before relaxation may be expected to represent better fit between the lattices, and
thus have low energies relative to other boundaries. This is because the periodic
nature and the actual repeat period of the structure of the interface implied by the
CSL concept is not destroyed by the small translation. The rigid body translations
mentioned above have been experimentally established in the case of aluminium;
although not conclusively established, the experiments suggest that the translation
may have a component outside the interface plane, but the atomistic calculations
cannot predict this since they always seem to be carried out at constant volume
[14].
(iv) The physical significance of CSL’s must diminish as the Σ value increases,
because only a very small fraction of atoms in an interface can then be common
to both the adjacent crystals.
(v) The Σ = 3 value is independent [3] of symmetry considerations. This is conve-
nient since such matrices do not uniquely relate the grains; it is usually necessary
to impose criteria to allow physically reasonable choices to be made.

(a) (b)

FIGURE 13.2
{112} coherent twin boundary in a bcc material, initially with an exact Σ3 CSL. The
figure on the right illustrates the structure after a rigid body translation is included
[3, 14]
246 13 Interfaces

13.4 The O-lattice

We have seen that coincidence site lattices can be generated by identifying lattice
points which are common to both of the adjacent crystals, when the lattices of these
crystals are notionally allowed to interpenetrate and fill all space. The totality of the
common lattice points then forms the coincidence site lattice. This procedure is only
a formal way of identifying coincidence points; for a real interface connecting the
two crystals, only atoms from crystal 1 will exist on one side of the interface and
those from crystal 2 on the other. There will, however, be good fit regions in the
interface, corresponding to the lattice points that are common to both crystals; the
pattern formed by these points consists of a two dimensional section of the CSL.
The coincidence site lattice thus concentrates on just lattice points, and this is not the
most general case to consider. When the transformation carrying one grain into the
other is a rigid body rotation, all points along the rotation axis will represent points
of perfect fit between the two crystals, not just the lattice points which may lie on
the rotation axis. When the transformation relating two grains is an invariant-plane
strain, all points in the invariant-plane represent points of perfect fit between the two
lattices and not just the lattice points which lie in the invariant-plane.
Any point x within a crystal may be represented as the sum of a lattice vector u
(which has integral components) and a small vector β whose components are frac-
tional and less than unity. The internal coordinates of the point x are then defined
to be the components of the vector β. The O-lattice method takes account of all co-
incidences, between non-lattice sites of identical internal coordinates as well as the
coincidence lattice sites.
All lattice points in a crystal are crystallographically equivalent and any lattice
point may be used as an origin to generate the three dimensional crystal lattice. To
identify points of the CSL we specify that any lattice vector u of crystal A must, as
a result of the transformation to crystal B, become another lattice vector x of A - i.e.
x = u + v where v is a lattice vector of A. The CSL point x can then be considered
to be a perfect fit point in an interface between A and B, because it corresponds to a
lattice point in both crystals.
Non-lattice points in a crystal are crystallographically equivalent when they have
the same internal coordinates. To identify O-points [15, 16], we specify that any non-
lattice vector x of crystal A must, as a result of transformation to crystal B, become
another non-lattice vector y of A such that y = x + v where v is a lattice vector
of A; the points x and y thus have the same internal coordinates in A. The O-point
y can then be considered to be a perfect fit point in the interface between A and B.
Note that when x becomes a lattice vector, y becomes a CSL point. The totality of
O-points obtained by allowing crystals A and B to notionally interpenetrate forms
the O-lattice [15, 16], which may contain the CSL as a sub-lattice if A and B are
suitably orientated at an exact CSL orientation. Any boundary between A and B
cuts the O-lattice and will contain regions of good fit which have the periodicity
corresponding to the periodicity of a planar net of the O-lattice along which the
13.4 The O-lattice 247

intersection occurs. Boundaries parallel to low-index planes of the O-lattice are in

general two dimensionally periodic with relatively small repeat cells, and those with
a high planar O-point density should have a relatively low energy per unit area.
Consider two crystals A and B which are related by the deformation (A S A)
which converts the reference lattice A to that of B; an arbitrary non-lattice point x in
crystal A thus becomes a point y in crystal B, where

[A; y] = (A S A)[A; x]

If the point y is crystallographically equivalent to the point x, in the sense that it has
the same internal coordinates as x, then y is also a point of the O-lattice, designated
O. This means that y = x + u, where u is a lattice vector of A. Since y is only an
O-point when y = x + u, we may write that y is an O-lattice vector O if [15, 16]

[A; y] = [A; O] = [A; x] + [A; u] = (A S A)[A; x]

or in other words,
[A; u] = (A T A)[A; O] (13.11)
−1
where (A T A) = I − (A S A) . It follows that
−1
[A; O] = (A T A) [A; u]. (13.12)

By substituting for u the three basis vectors of A in turn [3], we see that the columns
of (A T A)−1 define the corresponding base vectors of the O-lattice.
Since O-points are points of perfect fit, mismatch must be at a maximum in be-
tween neighbouring O-points. When O is a primitive O-lattice vector, equation 13.12
states that the amount of misfit in between the two O-points connected by O is given
by u. A dislocation with Burgers vector u would thus accommodate this misfit and
localise it at a position between the O-points, and these ideas allow us to consider a
dislocation model of the interface in terms of the O-lattice theory.
Three sets of dislocations with Burgers vectors bi (which form a non-coplanar
set) are in general required to accommodate the misfit in any interface. If the Burgers
vectors b1 , b2 and b3 are chosen to serve this purpose, and each in turn substituted
into equation 13.12, then the corresponding O-lattice vectors O1 , O2 and O3 are
obtained. These vectors Oi thus define the basis vectors of an O-lattice unit cell ap-
propriate to the choice of bi . If the O-points of this O-lattice are separated by ‘cell
walls’ which bisect the lines connecting neighbouring O-points, then the accumulat-
ing misfit in any direction can be considered to be concentrated at these cell walls
[15, 16]. When a real interface (unit normal n) is introduced into the O-lattice, its
line intersections with the cell walls become the interface dislocations with Burgers
vectors bi and unit line vectors li parallel to the line of intersection of the interface
with the cell walls.
The three O-lattice cell walls (with normals O∗i ) have normals parallel to O∗1 =
O2 ∧ O3 , O∗2 = O3 ∧ O1 and O∗3 = O1 ∧ O2 , so that the line vectors of the
dislocations are given by l1 ∥ O∗1 ∧ n, l2 ∥ O∗2 ∧ n and l3 ∥ O∗3 ∧ n. Similarly,
1/di = |O∗i ∧ n| . These results are exactly equivalent to the theory developed at
248 13 Interfaces

the beginning of this chapter (equation 13.1-13.8) but the O-lattice theory perhaps
gives a better physical picture of the interface, and follows naturally from the CSL
approach [3]. The equivalence of the two approaches arises because equation 13.11
is identical to equation 13.1 since u and O are equivalent to bt and p respectively.
Example 13.5: alpha/beta brass interface using O-lattice theory
Derive the structure of the alpha/beta brass interface described on page 237
using O-lattice theory.
From page 237, the matrix (A T A) is given by:
⎛ ⎞
0.119500 0.006386 0.016048
⎜ ⎟
(A T A) = ⎜ ⎝ 0.006386 0.119500 0.016048 ⎠
⎟

−0.149974 −0.149974 −0.236183

so that ⎛ ⎞
−52.448 −61.289 −51.069
−1 ⎜ ⎟
(A T A) =⎜
⎝−61.289 −52.448 −51.064⎠
⎟

72.222 72.222 60.622

If the vectors b1 , b2 and b3 are each in turn substituted for u in equation 13.11,
then the corresponding O lattice vectors are found to be:

[A; O1 ] = [−0.689770 − 5.110092 5.799861]

[A; O2 ] = [−5.110092 − 0.689770 5.799861]
[A; O3 ] = [−56.86861 − 56.86861 72.22212]

[A∗ ; O∗1 ] = [ 0.263088 0.036860 0.236183]

[A∗ ; O∗2 ] = [ 0.036860 0.263088 0.236183]
[A∗ ; O∗3 ] = [−0.024088 − 0.024088 − 0.024088]

O∗1 ∧ n ∥ l1 = [−0.012249 − 0.012249 0.015556]A

O∗2 ∧ n ∥ l2 = [−0.012249 − 0.012249 0.015556]A
O∗3 ∧ n ∥ l3 = [ 0.001249 0.001249 − 0.002498]A

d1 = d2 = 11.63 Å and d3 = 88.52 Å

The results obtained therefore are identical to those described beginning page 237.
13.5 Secondary dislocations 249

13.5 Secondary dislocations

Low-angle boundaries contain interface dislocations whose role is to localise and ac-
commodate the misfit between adjacent grains, such that large areas of the boundary
consist of low-energy coherent patches without mismatch. These intrinsic interface
dislocations are called primary dislocations because they accommodate the misfit
relative to an ideal single crystal as the reference lattice. Boundaries between grains
which are at an exact CSL orientation have relatively low-energy. It then seems rea-
sonable to assume that any small deviation from the CSL orientation should be ac-
commodated by a set of interface dislocations which localise the misfit due to this
deviation, and hence allow the perfect CSL to exist over most of the boundary area.
These intrinsic interface dislocations [17–20] are called secondary dislocations be-
cause they accommodate the misfit relative to a CSL as the reference lattice. High-
angle boundaries between crystals which are not at an exact CSL orientation may
therefore consist of dense arrays of primary dislocations and also relatively widely
spaced arrays of secondary dislocations. The primary dislocations may be so closely
spaced, that their strain fields virtually cancel each other and in these circumstances
only secondary dislocations would be visible using conventional transmission elec-
tron microscopy.
Example 13.6: Intrinsic secondary dislocations
The axis-angle pair describing the orientation relationship between the two
grains (A and B) of austenite is given by:

axis of rotation parallel to [1 1 2]A

right-handed angle of rotation equal to 175◦

Calculate the secondary dislocation structure of an interface lying normal to the
axis of rotation, given that the Burgers vectors of the interface dislocations are
[A; b1 ] = 12 [1 0 1], [A; b2 ] = 12 [0 1 1] and [A; b3 ] = 12 [1 1 0], where the basis A
corresponds to the conventional fcc unit cell of austenite, with a lattice parameter
a = 3.56 Å.
The secondary dislocation structure can be calculated with respect to the near-
est CSL, which is a Σ3 CSL obtained by a 180◦ rotation about [1 1 2]A . The rigid
body rotation matrix corresponding to this exact CSL orientation is thus:
⎛ ⎞
2 1 2
1⎜ ⎟
(A J A) = ⎜ 1 2 2
⎟
3 ⎝ ⎠
2 2 1

The rotation matrix (A J2 A) describing the actual transformation of A to B, cor-

250 13 Interfaces
responding to a rotation of 175◦ about [1 1 2]A is given by:
⎛ ⎞
−0.663496 0.403861 0.629817
⎜ ⎟
(A J2 A) = ⎜ ⎝ 0.261537 −0.663496 0.700979⎠ .
⎟

0.700979 0.629817 0.334602

The matrix (A J3 A) describing the deviation from the exact CSL is given by [16]
−1
(A J3 A) = (A J2 A) (A J A), so that
⎛ ⎞
0.996829 0.071797 −0.034313
⎜ ⎟
(A J3 A) = ⎜ ⎝−0.070528 0.996829 0.036850 ⎠ .
⎟

0.036850 −0.034313 0.998732

−1
If (A T A) = I − (A J3 A) , then (A T’ A) becomes:
⎛ ⎞
0.003171 −0.071797 0.034313
⎜ ⎟
(A T′ A) = ⎜⎝ 0.070528 0.003171 −0.036850⎠ .
⎟

−0.036850 0.034313 0.001268

The secondary dislocation structure can now be calculated using the procedures
described on page 233:

(n; A∗ ) = a(0.408248 0.408248 0.816497)

[A∗ ; b∗1 ] = [1 1 1]
[A∗ ; b∗2 ] = [1 1 1]
[A∗ ; b∗3 ] = [1 1 1]

From equation 13.6,

[A∗ ; c1 ] = [0.040655 0.104207 − 0.072341]

[A∗ ; c2 ] = [−0.0109280 − 0.030507 0.069894]
[A∗ ; c3 ] = [−0.034313 0.036850 − 0.001268]

so that

d1 = 26.71Å, d2 = 26.71Å, and d3 = 70.68 Å

l1 ∥[ 0.860386 − 0.470989 − 0.194695]A

l2 ∥[−0.401046 0.883699 − 0.241327]A
l3 ∥[ 0.607647 0.545961 − 0.576794]A.
13.6 The DSC lattice 251

13.6 The DSC lattice

In a boundary between two crystals, the Burgers vector b of an interface dislocation
must be such that the displacement of one of the crystals through b relative to the
other does not change the structure of the interface. Lattice translation vectors of the
reference lattice always satisfy this condition, so that b can always equal a lattice
translation vector. However, additional possibilities arise in the case of secondary
dislocations, whose Burgers vectors are generally DSC lattice translation vectors
[16, 21]. Secondary dislocations represent the deviation from a particular coinci-
dence site lattice, and a corresponding DSC lattice may be generated such that the
translation vectors of the DSC lattice are possible Burgers vectors of secondary inter-
face dislocations. The interesting point about the DSC lattice is that its lattice vectors
need not be crystal lattice translation vectors. Fig. 13.3 illustrates a Σ5 CSL between
two fcc grains (A & B), related by a rotation of 36.87◦ about [1 0 0]A .

001 001
001 001
(a) 010 010 (b) 010 010

FIGURE 13.3
Σ5 coincidence system for fcc crystals [22]. Filled symbols are lattice A, unfilled
ones lattice B and coincidence sites are a mixture of the two. lattice sites in the
plane of the diagram are represented as circles whereas those displaced by 12 [100]
are represented as triangles. The [100] axis is normal to the plane of the diagram.
1
Fig. 13.3b is obtained by displacing lattice B by [A; b] = 10 [0 3 1] relative to
lattice A and it is obvious that the basic pattern of lattice sites and CSL sites remains
unaffected by this translation, despite the fact that b is not a lattice vector of A or B.
It is thus possible for secondary dislocations to have Burgers vectors which are not
lattice translation vectors, but are vectors of the DSC lattice. The DSC lattice, or the
Displacement Shift Complete lattice, is the coarsest lattice which contains the lattice
points of both A and B, and any DSC lattice vector is a possible Burgers vector for
252 13 Interfaces

a perfect secondary dislocation. We note that the displacement b causes the original
coincidences (Fig. 13.3a) to disappear and be replaced by an equivalent set of new
coincidences (Fig. 13.3b), and this always happens when b is not a lattice translation
vector. This shift of the origin of the CSL has an important consequence on the
topography [22] of any boundary containing secondary dislocations with non-lattice
Burgers vectors.
Considering again Fig. 13.3, suppose that we introduce a boundary into the CSL,
with unit normal [A; n] ∥ [0 1 2], so that its trace is given by XX on Fig. 13.3a. The
effect of the displacement b of crystal B relative to A, due to the presence of a sec-
ondary dislocation, is to shift the origin of the CSL; if the boundary originally at XX
is to have the same structure after the displacement then it has to shift to the position
YY in Fig. 13.3b. Because a dislocation separates slipped from unslipped regions,
the shift of the boundary occurs at the position of the secondary dislocation so that
the boundary is stepped at the core of this dislocation. One such step is illustrated in
Fig. 13.4.

b
001

010

001

010
FIGURE 13.4
The presence of a step [23] in a Σ5, (3 1 0)A boundary of an fcc crystal containing
a secondary interface dislocation with [A; b] = (a/10)[1 3 0]. The symbolism is
identical to that of Fig. 13.3.

The following further points about the DSC lattice and its consequences should
be noted:
(i) The DSC lattice can be constructed graphically simply by inspection, bearing in
mind that it is the coarsest lattice containing lattice sites from both the crystals
orientated at an exact CSL orientation. Rather detailed analytical methods for
computing the basis vectors of the DSC lattice have been presented elsewhere
13.7 Some difficulties associated with interface theory 253

[24], and tabulations of these DSC lattice calculations as a function of Σ3 can

also be found for cubic systems [25].
(ii) For the primitive cubic system, the components of the three basis vectors of the
DSC lattice are the columns of a 3 × 3 matrix (DSC) obtained by taking the trans-
pose of the inverse of another 3 × 3 matrix (CSL). The columns of CSL represent
the components of the basis vectors of the coincidence site lattice concerned.
(iii) Crystal lattice vectors (of both the adjacent crystals) form a sub-set of DSC
lattice vectors.
(iv) The volume of a CSL unit cell is Σ3 times larger than that of the crystal lattice
unit cell, whereas the DSC lattice unit cell has a volume 1/Σ times that of the
crystal lattice unit cell [24].
(v) The homogeneous deformation (A S A) of equation 13.1, describing the trans-
formation from the reference lattice to another crystal is not unique [3], but the
CSL and DSC concepts are independent of the choice of (A S A). The O-lattice
on the other hand, depends critically on the form of (A S A).
(vi) Primitive DSC lattice vectors can be much smaller than primitive crystal lattice
vectors [26]. On the basis of elastic strain energy arguments, smaller interface
dislocation Burgers vectors should be favoured. This is not confirmed experimen-
tally since secondary dislocations often have Burgers vectors which are crystal
lattice vectors.
(vii) Although DSC lattices are defined relative to the unrelaxed CSL, small rigid
body translations which destroy exact coincidence (but preserve the CSL period-
icity) do not affect the essentials of the DSC concept [14].

13.7 Some difficulties associated with interface theory

Consider two crystals A and B, both of cubic structure, related by a rigid body rota-
tion. Any boundary containing the axis of rotation is a tilt boundary; for the special
case of the symmetrical tilt boundary, lattice B can be generated from A by reflec-
tion across the boundary plane. By substituting the rigid body rotation for (A S A) in
equation 13.1, the dislocation structure of the symmetrical tilt boundary may be de-
duced (page 234) to consist of a single array of dislocations with line vectors parallel
to the tilt axis.
Symmetry considerations imply that the rigid body rotation has up to 23 fur-
ther axis-angle representations. If we impose the condition that the physically most
significant representation is that which minimises the Burgers vector content of the
interface, then the choice reduces to the axis-angle pair involving the smallest angle
of rotation.
254 13 Interfaces

On the other hand, (A S A) can also be a lattice-invariant twinning shear on the

symmetrical tilt boundary plane [3]; crystal B would then be related to A by reflection
across the twin plane so that the resulting bicrystal would be equivalent to the case
considered above. The dislocation content of the interface then reduces to zero since
the invariant-plane of the twinning shear is fully coherent.
This ambiguity in the choice of (A S A) is a major difficulty in interface theory
[3]. The problem is compounded by the fact that interface theory is phenomenologi-
cal - i.e. the transformation strain (A S A) may be real or notional as far as interface
theory is concerned. If it is real then we expect to observe a change in the shape of
the transformed region, and this may help in choosing the most reasonable deforma-
tion (A S A). For example, in the case of mechanical twinning in fcc crystals, the
surface relief observed can be used to deduce that (A S A) is a twinning shear rather
than a rigid body rotation. In the case of an fcc annealing twin, which grows from
the matrix by a diffusional mechanism (during grain boundary migration), the same
twinning shear (A S A) may be used to deduce the interface structure, but the defor-
mation is now notional, since the formation of annealing twins is not accompanied
by any surface relief effects. In these circumstances, we cannot be certain that the
deduced interface structure for the annealing twin is correct.
The second major problem follows from the fact that the mathematical Burgers
vector content bt given by equation 13.1 has to be factorised into arrays of physi-
cally realistic dislocations with Burgers vectors which are vectors of the DSC lattice.
There is an infinite number of ways in which this can be done, particularly since the
interface dislocations do not necessarily have Burgers vectors which minimise their
elastic strain energy.
Secondary dislocations are referred to an exact CSL as the reference lattice.
Atomistic calculations suggest that boundaries in crystals orientated at exact coinci-
dence contain arrays of primary dislocations (whose cores are called structural units)
[27]. The nature of the structural units varies with the Σ3 value, but some favoured
CSL’s have boundaries with just one type of structural unit, so that the stress field of
the boundary is very uniform. Other CSL’s have boundaries consisting of a mixture
of structural units from various favoured CSL’s. It has been suggested that it is the
favoured CSL’s which should be used as the reference lattices in the calculation of
secondary dislocation structure, but the situation is unsatisfactory because the same
calculations suggest that the favoured/unfavoured status of a boundary also depends
of the boundary orientation itself. (We note that the term ‘favoured’ does not imply
a low interface energy).
Referring now to the rigid body translations which exist in materials with ‘hard’
atoms, it is not clear whether the calculated translations might be different if the
relaxation were to be carried out for a three dimensionally enclosed particle.
13.8 Summary 255

13.8 Summary
The Burgers vector content of an interface between two crystals that do not neces-
sarily have the same structure, can be obtained by the closure failure of a Burgers
circuit constructed across the interface, with the comparison made against the same
circuit constructed after transforming the bi-crystal into a single reference-crystal.
The Burgers vector content can then be de-convoluted into arrays of interfacial dis-
locations, with each array defined by a Burgers and line vector. This process requires
a prior choice of individual dislocation Burgers vectors but once this is done, the
calculations permit a comparison against experimental observations.
There exist special orientations, the coincidence site lattices, where there is a
fraction of lattice points that is common to both crystals when their lattices are al-
lowed to interpenetrate and fill all space, assuming they have a common origin. The
single crystal may be regarded at Σ = 1 CSL, and when divided to create a bi-crystal
with a small misorientation, leads to the classical dislocation description of the tilt
or twist boundary. Other CSL orientations such as Σ = 3 etc. can also be used as
reference lattices, with deviations from the exact CSL described in terms of dislo-
cation arrays. Furthermore, a boundary located on a CSL plane can be displaced so
that it maintains the coincidences using secondary dislocations as described in the
‘complete pattern shift’ or DSC lattice.
The idea of coincidences of lattice points can be generalised to coincidences of
points that have the same internal coordinates but are not necessarily lattice sites,
as in Bollmann O-lattice theory. This again is a useful concept that permits the in-
terface structure to be calculated after making assumptions about the nature of the
dislocations in the interface.
All the theory described here refers to pure interfaces that have no segregants and
which are not relaxed in the sense of rigid body translations parallel to the interface
plane. Furthermore, when the crystals are solid solutions, there will be a chemical
component of interfacial energy that would need to be taken into account.
References

1. B. A. Bilby: ‘Types of dislocation source’, In: Bristol Conference Report on

Defects in Crystalline Solids. London, U.K.: The Physical Society, 1955:124–
133.
2. F. C. Frank: ‘Report of the pittsburgh symposium’, In: Symposium on the Plas-
tic Deformation of Crystalline Solids. Pittsburgh, USA: Office of Naval Re-
search, 1950:150.
3. J. W. Christian: Theory of Transformations in Metals and Alloys, Part I: 2 ed.,
Oxford, U. K.: Pergamon Press, 1975.
4. K. M. Knowles, D. A. Smith, and W. A. T. Clark: ‘On the use of geometric
parameters in the theory of interphase boundaries’, Scripta Metallurgica, 1982,
16, 413–416.
5. W. T. Read: Dislocations in Crystals: New York, USA: McGraw-Hill, 1953.
6. G. Bäro, and H. Gleiter: ‘On the structure of incoherent interphase boundaries
between FCC/BCC crystals’, Acta Metallurgica, 1973, 21, 1405–1408.
7. M. L. Kronberg, and F. H. Wilson: ‘Secondary recrystallization in copper’,
Trans. A.I.M.E., 1949, 185, 501–514.
8. K. T. Aust, and J. W. Rutter: ‘Temperature dependence of grain migration in
high-purity lead containing small additions of tin’, Trans. A.I.M.M.E., 1959,
215, 820–831.
9. D. H. Warrington, and P. Bufalini: ‘The coincidence site lattice and grain
boundaries’, Scripta Metallurgica, 1971, 5, 771–776.
10. M. A. Fortes: Revista de fisica Quimica e Engenharia, 1972, 4a, 7.
11. V. Vitek: ‘Multilayer stacking faults and twins on {211} planes in bcc metals’,
Scripta Metallurgica, 1970, 4, 725–732.
12. D. A. Smith, V. Vitek, and R. C. Pond: ‘Computer simulation of symmetrical
high angle boundaries in aluminium’, Acta Metallurgica, 1977, 25, 475–483.
13. R. C. Pond: ‘Periodic grain boundary structures in aluminium. II. a geometrical
method for analysing periodic grain boundary structure and some related trans-
mission electron microscope observations’, Proceedings fo the Royal Society
A, 1977, 357, 471–483.

257
258 13 References

14. J. W. Christian, and A. G. Crocker: Dislocations in Solids, ed. F. R. N. Nabarro:

Amsterdam, Holland: North Holland, 1980.
15. W. Bollmann: ‘On the geometry of grain and phase boundaries: I. general
theory’, Philosophical Magazine, 1967, 16, 363–381.
16. W. Bollmann: Crystal defects and crystalline interfaces: Berlin, Germany:
Springer Verlag, 1970.
17. W. T. Read, and W. Schockley: ‘Dislocation models of crystal grain bound-
aries’, Physical Review, 1950, 78, 275–289.
18. D. G. Brandon, B. Ralph, S. T. Ranganathan, and M. S. Wald: ‘A field ion mi-
croscope study of atomic configuration at grain boundaries’, Acta Metallurgica,
1964, 12, 813–821.
19. T. Schober, and R. W. Balluffi: ‘Extraneous grain boundary dislocations in low
and high angle (001) twist boundaries in gold’, Philosophical Magazine, 1971,
24, 165–180.
20. J. P. Hirth, and R. W. Balluffi: ‘On grain boundary dislocations and ledges’,
Acta Metallurgica, 1973, 21, 929–942.
21. D. G. Brandon: ‘The structure of high-angle grain boundaries’, Acta Metallur-
gica, 1966, 14, 1479–1484.
22. A. H. King, and D. A. Smith: ‘The effects on grain-boundary processes of
the steps in the boundary plane associated with the cores of grain-boundary
dislocations’, Acta Crystallographica A, 1980, 36, 335–343.
23. R. C. Pond, and D. A. Smith: ‘On the role of grain boundary dislocations in
plastic deformation’, In: J. L. Walter, J. H. Westbrook, and D. A. Woodford.,
eds. 4th Bolton Landing Conference on Grain Boundaries in Eng. Materials.
Baton Rouge, USA: Claitors Publishing Division, 1975:309–315.
24. H. Grimmer, W. T. Bollmann, and D. H. Warrington: ‘Coincidence-site lattices
and complete pattern-shift in cubic crystals’, Acta Crystallographica A, 1974,
30, 197–207.
25. D. H. Warrington, and H. Grimmer: ‘Dislocation burgers vectors for cubic
metal grain boundaries’, Philosophical Magazine, 1974, 30, 461–468.
26. W. Bollmann, B. Michaut, and G. Sainfort: ‘Pseudo-subgrain-boundaries in
stainless steel’, Physica Status Solidi A, 1972, 13, 637–649.

27. A. P. Sutton, and V. Vitek: ‘On the structure of tilt grain boundaries in cubic
metals I. symmetrical tilt boundaries’, Philosophical Transactions of the Royal
Society A, 1983, 309, 1–36.
Appendices

259
A
Matrix methods

A.1 Vectors
Quantities (such as force, displacement) which are characterised by both magnitude
and direction are called vectors; scalar quantities (such as time) only have magnitude.
A vector is represented by an arrow pointing in a particular direction, and can be
identified by underlining the lower–case vector symbol (e.g. u). The magnitude of
u (or |u|) is given by its length, a scalar quantity. Vectors u and v are only equal if
they both point in the same direction, and if |u| = |v|. The parallelism of u and v is
indicated by writing u ∥ v. If x = −u, then x points in the opposite direction to u,
although |x| = |u|.
Vectors can be added or removed to give new vectors, and the order in which
these operations are carried out is not important. Vectors u and x can be added by
placing the initial point of x in contact with the final point of u; the initial point of
the resultant vector u + x is then the initial point of u and its final point corresponds
to the final point of x. The vector mu points in the direction of u, but |mu|/|u| = m,
m being a scalar quantity. A unit vector has a magnitude of unity; dividing a vector
u by its own magnitude u gives a unit vector parallel to u.
It is useful to refer vectors to a fixed frame of reference; an arbitrary reference
frame would consist of three non-coplanar basis vectors a1 , a2 and a3 . The vector u
could then be described by means of its components u1 , u2 and u3 along these basis
vectors, respectively, such that

ui = |u| cos θi /|ai |

where I = 1, 2, 3 and θi = angle between u and ai .

If the basis vectors ai form an orthonormal set (i.e. they are mutually perpendic-
ular and each of unit magnitude), then the magnitude of u is:

|u|2 = u21 + u22 + u23

If the basis vectors ai form an orthogonal set (i.e. they are mutually perpendicu-
lar) then the magnitude of u is:

|u|2 = (u1 |a1 |)2 + (u2 |a2 |)2 + (u3 |a3 |)2

A dot or scalar product between two vectors u and x (order of multiplication not

261
262 A Matrix methods

important) is given by u.x = |u| × |x| cos θ, θ being the angle between u and x. If x
is a unit vector then u.x gives the projection of u in the direction x.
The cross or vector product is written u ∧ x = |u| × |x| sin θy, where y is a unit
vector perpendicular to both u and x, with u,x and y forming a right–handed set. A
right–handed set u, x, y implies that a right–handed screw rotated through an angle
less than 180◦ from u to x advances in the direction y. The magnitude of u ∧ x gives
the area enclosed by a parallelogram whose sides are the vectors u and x; the vector
y is normal to this parallelogram. Clearly, u ∧ x ̸= x ∧ u.
If u, x and z form a right–handed set of three non-coplanar vectors then u ∧ x.z
gives the volume of the parallelepiped formed by u, x and z. It follows that u ∧ x.z =
u.x ∧ z = z ∧ u.x.
The following relations should be noted:

u ∧ x = −x ∧ u
u.(x ∧ y) = x.(y ∧ u) = y.(u ∧ x)
u ∧ (x ∧ y) ̸
= (u ∧ x) ∧ y
u ∧ (x ∧ y) = (u.y)x − (u.x)y (A.1)

A.2 Matrices
A.2.1 Definition, addition, scalar multiplication
A matrix is a rectangular array of numbers, having m rows and n columns, and is
said to have an order m by n. A square matrix J of order 3 by 3 may be written as
⎛ ⎞ ⎛ ⎞
J11 J12 J13 J11 J21 J31
⎜ ⎟ ⎜ ⎟
J=⎜ ⎝J21 J22 J23 ⎠
⎟ and its transpose J′ = ⎜⎝J12 J22 J32 ⎠
⎟

J31 J32 J33 J13 J23 J33

(A.2)
where each number Jij (i = 1, 2, 3 and j = 1, 2, 3) is an element of J. J’ is called the
transpose of the matrix J. An identity matrix (I) has the diagonal elements J11 , J22
& J33 equal to unity, all the other elements being zero. The trace of a matrix is the
sum of all its diagonal elements J11 + J22 + J33 . If matrices J and K are of the same
order, they are said to be equal when Jij = Kij for all i, j. Multiplying a matrix by a
constant involves the multiplication of every element of that matrix by that constant.
Matrices of the same order may be added or subtracted, so that if L = J + K, it
follows that Lij = Jij + Kij .
A.2 Matrices 263

A.2.2 Einstein summation convention

In order to simplify more complex matrix operations we now introduce the summa-
tion convention. The expression
u1 a1 + u2 a2 + ...un an
can be shortened by writing
3
.
u i ai
i=1
A further economy in writing is achieved by adopting the convention that the repeti-
tion of a subscript or superscript index in a given term implies summation over that
index from 1 to n. Using this summation convention, the above sum can be written
ui ai . Similarly,
xi = yj zij for i = 1, 2 j = 1, 2 implies that
x1 = y1 z11 + y2 z12
x2 = y1 z21 + y2 z22

A.2.3 Multiplication and Inversion

The matrices J and K can be multiplied in that order to give a third matrix L if
the number of columns (m) of J equals the number of rows of K (J is said to be
conformable to K). L is given by
Lst = Jsr Krt
where s ranges from 1 to the total number of rows in J and t ranges from 1 to the
total number of columns in K. If J and K are both of order 3 × 3 then, for example,
L11 = J11 K11 + J12 K21 + J13 + K31
Note that the product JK does not in general equal KJ.
Considering a n × n square matrix J, it is possible to define a number ∆ which
is the determinant (of order n) of J. A minor of any element Jij is obtained by
forming a new determinant of order (n − 1), of the matrix obtained by removing all
the elements in the ith row and the jth column of J. For example, if J is a 2 × 2
matrix, the minor of J11 is simply J22 . If J is a 3 × 3 matrix, the minor of J11 is:
: :
: :
:J22 J23 :
: : = J22 J33 − J23 J32 (A.3)
:J J :
32 33

where the vertical lines imply a determinant. The cofactor jij of the element Jij is
then given by multiplying the minor of Jij by (−1)i+j . The determinant (∆) of J is
thus
.n
det J = J1j j1j withj = 1, 2, 3 (A.4)
j=1
264 A Matrix methods

Hence, when J is a 3 × 3 matrix, its determinant ∆ is given by:

∆ = J11 j11 + J12 j12 + J13 j13
= J11 (J22 J33 − J23 J32 ) + J12 (J23 J31 − J21 J33 ) + J13 (J21 J32 − J22 J31 )
The inverse of J is written J−1 and is defined such that
J.J−1 = I
−1
The elements of J−1 are Jij such that:
−1
Jij = jji /det J
Hence, if L is the inverse of J, and if detJ = ∆, then:
L11 = (J22 J33 − J23 J32 )/∆
L12 = (J32 J13 − J33 J12 )/∆
L13 = (J12 J23 − J13 J22 )/∆
L21 = (J23 J31 − J21 J33 )/∆
L22 = (J33 J11 − J31 J13 )/∆
L23 = (J13 J21 − J11 J23 )/∆
L31 = (J21 J32 − J22 J31 )/∆
L32 = (J31 J12 − J32 J11 )/∆
L33 = (J11 J22 − J12 J21 )/∆
If the determinant of a matrix is zero, the matrix is singular and does not have an
inverse.
The following matrix equations are noteworthy:
(JK)L = J(KL)
J(K + L) = JK + JL
(J + K)′ = J′ + K′
(JK)′ = K′ J′
(JK)−1 = K−1 J−1
(J−1 )′ = (J′ )−1

A.2.4 Orthogonal matrices

A square matrix J is said to be orthogonal if J−1 = J′ . It the columns or rows
of a real orthogonal matrix are taken to be components of column or row vectors
respectively, then these vectors are all unit vectors. The set of column vectors (or row
vectors) form an orthonormal basis; if this basis is right–handed, the determinant of
the matrix is unity, but if it is left–handed then ∆ = −1.
Orthogonal matrices arise in co-ordinate transformations between orthonormal
bases and where rigid body rotations are represented in a single orthonormal basis.
A.2 Matrices 265

Example A.1: Simple operations

⎛ ⎞ ⎛ ⎞
2 0 2 2 3 5
⎜ ⎟ ⎜ ⎟
A=⎜
⎝3 4 5⎠ ,
⎟ A′ = ⎜
⎝0 4 6⎠ ,
⎟ det A = −8
5 6 7 2 5 7
⎛ ⎞
0.25 −1.5 1
⎜ ⎟
A−1 =⎜ ⎟
⎝−0.5 −0.5 0.5⎠
0.25 1.5 −1

= > = > = >

2 3 2 1 0.8 −0.6
B= , B′ = , det B = 5, B−1 =
1 4 3 4 −0.2 0.4

= > = >
2 3 ′ 2 1
A= A =
1 4 3 4
= >
−1 0.8 −0.6
det A = 5 A =
−0.2 0.4
B
General rotation matrix

The simple matrix representing a right-handed rotation of θ = 45◦ about [001] was
written on page 111 as: ⎛ ⎞
m n 0
⎜ ⎟
R=⎜
⎝−n m
⎟
0⎠
0 0 1
where m = cos θ and n = sin θ. The matrix was derived by inspection of Fig. 7.5.
This was possible because a rotation about one of the basis vectors is easy to visu-
alise. At the same time, it was stated without derivation that a general rotation about
an arbitrary unit vector [u1 u2 u3 ] can be described by the matrix g R = JRJ′ , where
the rotation matrix J represents the new coordinate system:
⎛ ⎞
u1 u1 (1 − m) + m u1 u2 (1 − m) + u3 n u1 u3 (1 − m) − u2 n
g
⎜ ⎟
R=⎜ ⎝ u1 u2 (1 − m) − u3 n u2 u2 (1 − m) + m u2 u3 (1 − m) + u1 n ⎠
⎟

u1 u3 (1 − m) + u2 n u2 u3 (1 − m) − u1 n u3 u3 (1 − m) + m
(B.1)
The right-handed angle of rotation can be obtained from the fact that

J11 + J22 + J33 = 1 + 2 cos θ (B.2)

and the components of the vector u along the axis of rotation are given by

u1 = (J23 − J32 )/2 sin θ

u2 = (J31 − J13 )/2 sin θ (B.3)
u3 = (J12 − J21 )/2 sin θ

We shall now go on to prove all this by doing a similarity transformation of the

simple rotation matrix into the more general form, i.e. transforming the rotation into
a new coordinate system where the rotation axis is not necessarily parallel to a basis

267
268 13 General rotation matrix

vector. This can be done as follows:

⎛ ⎞⎛ ⎞⎛ ⎞
J11 J12 J13 m n 0 J11 J21 J31
g
⎜ ⎟⎜ ⎟⎜ ⎟
R = ⎜ ⎟⎜
⎝J21 J22 J23 ⎠ ⎝−n m
⎟⎜
0⎠ ⎝J12 J22
⎟
J32 ⎠
J31 J32 J33 0 0 1 J13 J23 J33
⎛ ⎞⎛ ⎞
J11 m − J12 n J11 n + J12 m J13 J11 J21 J31
⎜ ⎟⎜ ⎟
≡ ⎜ ⎟⎜ ⎟
⎝J21 m − J22 n J21 n + J22 m J23 ⎠ ⎝J12 J22 J32 ⎠
J31 m − J32 n J31 n + J32 m J33 J13 J23 J33
⎛ ⎞
2 2 2
[m(J11 + J12 ) + J13 ] [m(J11 J21 + J12 J22 ) + n(J11 J22 − J12 J21 ) + J13 J23 ] . . .
⎜ ⎟
≡ ⎜
⎝ ... ... . . .⎠
⎟

... ... ...

(B.4)

where for clarity, only the first two elements of g R are listed.
• The components of the rotation axis in the basis R are [001], which in the co-
ordinates of J become [J13 J23 J33 ] ≡ [u1 u2 u3 ]. And since each row and each
2 2 2
column of a rotation matrix is a unit vector, J11 + J12 = 1 − J13 . If follows that
g
the element R11 becomes u1 u1 (1 − m) + m as in equation B.1.
• Since JJ′ = I, the identity matrix, if follows that J11 J21 + J12 J22 + J13 J23 = 0
and since J13 = u1 and J23 = u2 , the term m(J11 J21 + J12 J22 ) = −u1 u2 m.
• The determinant of J is

J33 (J11 J22 − J12 J21 ) + J23 (J31 J12 − J11 J32 ) + J13 (J21 J32 − J22 J31 ) = 1
u3 (J11 J22 − J12 J21 ) + u2 (J31 J12 − J11 J32 ) + u1 (J21 J32 − J22 J31 ) = 1

Since u21 + u22 + u23 = 1, it follows that J11 J22 − J12 J21 = u3 etc. so considering
the last two listed-items together, g R12 = u1 u2 (1 − m) + u3n as in equation B.1.
The remaining terms can be proven similarly.
Finally, the Euler angles described on page 99 can be converted into a rotation
matrix. An individual rotation matrix is created for each of the three operations,
followed by taking their product in the correct sequence, to give an overall rotation
matrix (and therefore an axis-angle pair) for the equivalent orientation relation-
ship.
13.0 INDEX 269

Index

Affine transformations, 151 intensity, 79

Austenite neutrons, 86
fcc cell, 7 ordered crystals, 82
primitive cell, 7 structure factor, 79
triclinic cell, 185 thin crystals, 85
Dimensions of crystals, 3
Bain strain, 119, 152, 154 Displacive transformation, 167
alternative correspondence, 184 DSC lattice, 251
undistorted lines, 154
Basis vectors, 6 Eigenvalues, 156
Brass, 19 Eigenvectors, 156
Bravais lattices, 8 Euler angles, 99
Burgers circuit, 232
Glide plane, 17, 62
Camera constant, 82 axial glide, 62
Cementite, 130 diagonal glide, 62
Bagaryatski orientation relation- diamond glide, 62
ship, 130 Grain deformation
growth direction, 57 topology, 162
Pitsch orientation relationships, Graphene, 3, 22
130
space group, 68, 130 Homogeneous deformation, 151
Centre of symmetry, 17 complex, 166
Coincidence site lattice, 107, 239 grain topology, 162
Σ value, 239 multi-directional, 166
fcc-hcp, 241 plane strain, 164
secondary dislocations, 249 pure strain, 154
Copper, 20 sequential, 165
Correspondence matrix, 182 Homogeneous deformations
Crystal structure, 19 eigenvectors and eigenvalues, 156
ε-carbide, 31 interfaces, 161
brass, 19 stretch and rotation, 158
copper, 20
diamond, 20 Interfaces, 105, 231
graphene, 22 α/β brass, 237, 248
Cuprite, 63 asymmetrical tilt, 236
Burgers circuit, 232
Definition of crystal, 3 Burgers vector content, 232
Diamond, 20, 69 coincidence site lattice, 239
Diffraction, 78 coincidence site lattices, 107
Bragg equation, 78 deformation, 161
electron diffraction, 80 difficulties, 253
Ewald sphere, 78 dislocation interactions, 188
270 13 General rotation matrix

DSC lattice, 251 twin, 141

glissile, 117, 236
jogs, 236 Partial dislocation, 17
misfit, 232 Phenomenological theory, 121
secondary dislocations, 249 Point groups, 23
stepped, 186 Celestite, 25
symmetrical tilt, 234 chess pieces, 24
symmetrical tilt boundary, 105 epsomite, 25
twist boundary, 236 gypsum, 25
Invariant-line strain, 117 notation, 23
invariant normal, 213 precipitate shape, 70
Invariant-plane strain, 171, 198 quartz, 25
conjugate, 197 water molecule, 23
inverse, 175 Polar point group, 43
two combined, 198 Polonium, 3
Primitive unit cell, 6
Kurdjumov-Sachs orientation, 135 Principal deformation, 154
Pure strain, 154
Lattice, 6 Pyrite, 4
Lattice correspondence, 182
Lattice-invariant deformation, 16 Quasicrystals, 19

Martensite Reciprocal lattice, 75

Bain strain, 118 Rotation matrix
characteristics, 117 general, 112
crystallographic theory, 121 Rotoinversion, 17
habit plane, 117
hexagonal, 122, 161, 172, 191, Screw axis, 17, 62
241 Secondary dislocations, 249
interfacial structure, 206 Shape, 4
shape deformation, 117, 203 Shape deformation, 203
Mechanical twinning, 176 Shape deformations, 172
Mechanical twins, 179 Shuffles, 167, 194
Metric tensor, 142 fcc→hcp, 195
Miller indices, 14 Similarity transformation, 158, 172
Motif, 19 Single-crystal deformation, 91, 175
duplex slip, 178
O-lattice, 246 Space groups, 61
α/β brass, 248 F m3m, 70
Orientation distribution function, 99 F mmm, 70
Orientation relationships, 129 P bnm, 131
Bagaryatski, 130 P n3m, 64
coincidence site lattices, 107 P nma, 131
Kurdjumov-Sachs, 135 Space-time crystals, 3
Pitsch, 130 Superdislocation, 83
representation, 108 Superlattice reflections, 84
13.0 INDEX 271

Symmetry, 16
axis-angle pairs, 242

Tetrakaidecahedron, 162
Texture, 96
displacive transformations, 222
Euler angles, 99
inverse pole figures, 96
orientation distribution, 99
pole figures, 96
representation, 96
Topology, 162
Transformation twinning, 221
Translations, 23
Turbine blade, 4
Twinning
deformation, 179
double twinning, 141
orientation relationship, 141
slip across twins, 186
transformation twins, 221

Unit cell
primitive, 6

Vector cross product, 77

Vector product, 146

Weiss zone law, 76

Zinc sulphide, 69
 Organized into a two-part structure aimed at readers of
differing experience levels, Geometiy of Crystals, Poly
crystals, and Phase Transformations is accessible to both
newcomers and advanced researchers within the field of
crystallography. The rst part of the text covers what any
reader in the materials sciences, physics, chemistry, earth
sciences "and natural sciences in general“ should know
about crystallography. lt is intentionally concise and covers
sufficient material to form a firm foundation. The second
part is aimed at researchers and discusses phase transfor- ‘X

mations, deformations, and interface crystallography in

depth. The phase transformations are limited to those dom §.

inated by crystallography. The entire book contains worked

examples and uniquely deals not just with crystals but
aggregates of crystals and solid-state transformations
between crystals.

Provides a firm foundation for any natural scientist to

appreciate the tools of crystallography and a taste of
op
what is possible at an advanced level es;

Contains worked examples to communicate difficult E

.,.

concepts T

YKX
Supported by slides, video lectures, and other
,»?<
..,¢
electronic ancillaries freely accessible at: 5
4??

http.//www.phase trans.msm.cam.ac.uk/teaching.html
§¢
,<k

32> 1;’:
9?.
at
gi
we Q
gs
0/