Texture Analysis by Pole Figure
Method
Dr. Debalay Chakrabarti
How to Measure Texture
• X-ray diffraction; pole figures; measures average
texture at a surface (µms penetration); projection (2
angles).
• Neutron diffraction; type of data depends on
neutron source; measures average texture in bulk
(cms penetration in most materials) ; projection (2 angles).
• Electron [back scatter] diffraction; easiest [to
automate] in scanning electron microscopy (SEM);
local surface texture (nms penetration in most
materials); complete orientation (3 angles).
• Optical microscopy: optical activity (plane of
polarization); limited information (one angle).
Texture: Quantitative Description
• Three (3) parameters needed to describe the
orientation [of a crystal relative to the embedding
body or its environment].
• Most common: 3 [rotation] Euler angles.
• Most experimental methods [X-ray and neutron
pole figures included] do not measure all 3 angles,
so orientation distribution must be calculated.
• Best mathematical representation for graphing,
illustraitng symmetry: Rodrigues-Frank vectors.
• Best mathematical representation for calculations:
quaternions.
X-ray Pole Figures
• X-ray pole figures are the most common source of texture information;
cheapest, easiest to perform. They have the advantage of providing an
average texture over a reasonably large surface area (~1mm2), compared
to EBSD. For a grain size finer than about 100 µm, this means that
thousands of grains are included in the measurement, which ensures
statistical viability.
• Pole figure:= variation in diffracted intensity with respect to direction in the
specimen.
• Representation:= map in projection of diffracted intensity.
• Each PF is equivalent to a geographic map of a hemisphere (North pole in
the center).
• Map of the density of a specific crystal direction w.r.t. sample reference
frame. More concretely, it is the frequency of occurrence of a given crystal
plane normal per unit spherical area. Think of a (spherical) pin cushion
with each pin representing the normal to {hkl}.
PF apparatus
• From Wenk’s chapter in
Kocks book.
• Fig. 20: showing path
difference between
adjacent planes leading to
destructive or constructive
interference. The path
length condition for
constructive interference
is the basis for the Bragg
equation:
2 d sinθ = n λ
• Fig. 21: pole figure
goniometer for use with x-
ray sources.
Crystal Directions on the Sphere
• Uses the inclination of the
normal to the crystallographic
plane: the points are the
intersection of each crystal
direction with a (unit radius)
sphere.
• This is an orthographic
projection to illustrate the
physical directions, not a
stereographic projection.
Obj/notation AxisTransformation Matrix EulerAngles Components
Miller indices of a pole
Miller indices are a convenient way to represent a direction or a plane normal in a
crystal, based on integer multiples of the repeat distance parallel to each axis of the
unit cell of the crystal lattice. This is simple to understand for cubic systems with
equiaxed Cartesian coordinate systems but is more complicated for systems with lower
crystal symmetry. Directions are simply defined by the set of multiples of lattice repeats
in each direction. Plane normals are defined in terms of reciprocal intercepts on each
axis of the unit cell. In cubic materials only, plane normals are parallel to directions with
the same Miller indices.
When a plane is written with
parentheses, (hkl), this
indicates a particular plane
normal: by contrast when it is
written with curly braces, {hkl},
this denotes a the family of
planes related by the crystal
symmetry. Similarly a direction
written as [uvw] with square
brackets indicates a particular
direction whereas writing within
angle brackets , <uvw>
indicates the family of directions
related by the crystal symmetry.
Projection from Sphere to Plane
• The measured pole figure exists on the
surface of a (hemi-)sphere. To make
figures for publication one must project
the information onto a flat page. This is
a traditional problem in cartography.
We exploit just two of the many possible
projection methods.
• Projection of spherical information onto
a flat surface
– Equal area projection, or,
Schmid projection
– Equiangular projection, or,
Wulff projection,
more common in crystallography
[Cullity]
Obj/notation AxisTransformation Matrix EulerAngles Components
Stereographic Projections
• Connect a line from the South pole to the point on the surface of the
sphere. The intersection of the line with the equatorial plane defines the
project point. The equatorial plane is the projection plane. The radius
from the origin (center) of the sphere, r, where R is the radius of the
sphere, and α is the angle from the North Pole vector to the point to be
projected (co-latitude), is given by:
r = R tan(α/2)
• Given spherical coordinates (α,ψ), where the longitude is ψ (as before),
the Cartesian coordinates on the projection are therefore:
(x,y) = r(cosψ, sinψ) = R tan(α/2)(cosψ, sinψ)
• To obtain the spherical angles from [uvw], we calculate the co-latitude
and longitude angles as:
cosα = w
tanψ = v/u !Careful: Use ATAN2(v,u)!
Concept Params. Euler Normalize Vol.Frac. Cartesian Polar Components
Standard Stereographic Projections
• Pole figures are familiar diagrams. Standard
Stereographic projections provide maps of low index
directions and planes.
• PFs of single crystals can be derived from SSTs by
deleting all except one Miller index.
• Construct {100}, {110} and {111} PFs for cube
component.
Typical X-ray Diffractometer
X-ray Diffractometer (Schematic)
Pole Figure measurement
• PF measured with 5-axis goniometer.
• 2 axes used to set Bragg angle (choose a specific crystallographic plane
with θ/2θ), which determines the Miller indices associated with the PF.
These settings remain constant during the measurement of a given pole
figure.
• Third axis tilts specimen plane w.r.t. the focusing plane (co-latitude angle
in the PF, i.e. distance from North Pole). Although this angle can be as
large as 90°, no diffracted intensity will be measured with the plane of the
beams parallel to the surface: this limits the maximum tilt angle at which
PFs can be measured in reflection to about 80°.
• Fourth axis spins the specimen about its normal (longitude angle in the
PF).
• Fifth axis (optional) oscillates the Specimen under the beam in order to
maximize the number of grains included in the measurement.
• For texture calculation, at least 2 PFs required and 3 are preferable even
for materials with high crystal symmetry.
• N.B. deviations of relative intensities in a standard θ/2θ scan from powder
file indicate texture but only on a qualitative basis.
Standard (001) Projection
Pole Figure Example
• If the goniometer is set for {100} reflections, then
all directions in the sample that are parallel to
<100> directions will exhibit diffraction.
Texture Component → Pole Figure
• To calculate where a texture component shows up in a pole figure, there are various operations that must
be performed.
• The key concept is that of thinking of the pole figure as a set of crystal plane normals (e.g. {100}, or {111}) in
the reference configuration (“cube component”) and applying the orientation as a transformation to that pole
(or set of poles) to find its position with respect to the sample frame.
• Step 1: write the crystallographic pole (plane normal) of interest as a unit vector; e.g. (111) = 1/√3(1,1,1) = h.
In general, you will repeat this for all symmetrically equivalent poles (so for cubics, one would also calculate
{-1,1,1}, {1,-1,1} etc.)
• Step 2: apply the inverse transformation (passive rotation), g-1, to obtain the coordinates of the pole in the
pole figure:
h’ = g-1h
(pre-multiply the vector by, e.g. the transpose of the orientation matrix, g, that represents the orientation;
Rodrigues vectors or quaternions can also be used).
• Step 3: convert the rotated pole into spherical angles (to help visualize the result, and to simplify Step 4)
where Θ is the co-latitude and φ is the longitude:
Θ = cos-1(h’z), φ = tan-1(h’y/h’x).
Remember - use ATAN2(h’y,h’x) in your program or spreadsheet and be careful about the order of the
arguments!
• Step 4: project the pole onto a point, p, in the plane (stereographic or equal-area):
px = tan(Θ/2) cosφ; py = tan(Θ/2) sinφ. [corrected sine and cosine for py and px components 25 i 08]
The previous slide explains where this formula comes from.
• Note: why do we use the inverse transformation (passive rotation)?! One way to understand this is to recall
that the orientation is, by convention (in materials science), written as an axis transformation from sample
axes to crystal axes. The inverse of this description can also be used to describe a vector rotation of the
crystal, all within the sample reference frame, from the reference position to the actual crystal orientation.
Cube Component = {001}<100>
{100}
{111}
{110}
Think of the θ-2θ setting as acting as a filter on the
standard stereographic projection,
Practical Aspects
• Typical to measure three PFs for the 3 lowest values of Miller
indices (smallest available angles of Bragg peaks).
• Why?
– Small Bragg angles correspond to normals coincident with
symmetry elements of the crystal, which means fewer
symmetry-related poles, and, consequently, greater
dynamic range of intensity (peak to valley).
– A single PF does not uniquely determine orientation(s),
texture components because only the plane normal is
measured, but not directions in the plane (2 out of 3
parameters).
– Multiple PFs required for calculation of Orientation
Distribution.
Corrections to Measured Data
• Random texture [=uniform dispersion of orientations] means same
intensity in all directions.
• Background count must be subtracted, just as in conventional x-ray
diffraction analysis.
• X-ray beam becomes defocused at large tilt angles (> ~60°); measured
intensity even from a sample with random texture decreases towards
edge of PF.
• Defocusing correction required to increase the intensity towards the
edge of the PF. (Despite the uncertainty associated with this correction,
it is better to measure in reflection out to as large a tilt as possible, in
preference to trying to combine reflection and transmission figures.)
• After these corrections have been applied, the dataset must be
normalized in order that the average intensity is equal to unity (similar to,
although not the same as, making sure that a probability distribution has
unit area under the curve).
• Units: multiples of a random density (MRD). To be explained …
Defocussing
• The combination of the θ−2θ
setting and the tilt of the
specimen face out of the
focusing plane spreads out
the beam on the specimen
surface.
• Above a certain spread, not all
the diffracted beam enters the
detector.
• Therefore, at large tilt angles,
the intensity decreases for
purely geometrical reasons.
• This loss of intensity must be
compensated for, using the
defocussing correction.
Defocusing Correction
• Defocusing correction more important with decreasing 2θ and
narrower receiving slit.
• Best procedure involves measuring the intensity from a
reference sample with random texture.
• If such a reference sample is not available, one may have to
correct the available defocusing curves in order to optimize the
correction. This will be explained again in the context of using
popLA.
[Kocks]
popLA and the Defocussing Correction
Values for
Values for correcting
correcting data background
demo (from Cu1S40, smoothed a bit: UFK)
0 100.00
111
5 100.00
0 1000.00 10 100.00
5 999. 15 100.00
10
15
999.
999.
Tilt 20
25
100.00
100.00
Tilt 20
25 999.
Angles
30
35
100.00
100.00
999.
Angles
30 40 100.00
35 999. 45 100.00
At each tilt
40 999. 50 99.00
45 55 96.00
999.
50 60 92.00
55
60
999.
982.94 angle, the 65
70
83.00
72.00
65
70
939.04
870.59 data is 75
80
54.00
32.00
75
multiplied
759.37 85 13.00
80 90 .00
85 650.83
90 505.65
344.92 by If you change the
163.37
2.19 1000/value DFB file, always plot
the curves to check
them visually!
Area Element, Volume Element
• Spherical coordinates
result in an area element,
dA, whose magnitude Θ
depends on the declinationdA
(or co-latitude): dθ
dψ
dA = sinΘ dΘ dψ
[Kocks]
Concept Params. Euler Normalize Vol.Frac. Cartesian Polar Components
Normalization
• Normalization is the operation that ensures that “random” is equivalent
to an intensity of one.
• This is achieved by integrating the un-normalized intensity, f’(θ,ψ), over
the full area of the pole figure, and dividing each value by the result,
taking account of the solid area. Thus, the normalized intensity,
•
f(θ,ψ), must satisfy the following equation, where the 2π accounts for
the area of a hemisphere:
1
2π
∫ f (Θ,ψ )sinΘdΘdψ = 1
Note that in popLA files, intensity levels are represented by i4 integers, so
the random level = 100. Also, in .EPF data sets, the outer ring (typically,
Θ > 80°) is empty because it is unmeasurable; therefore the integration for
normalization excludes this empty outer ring.
Stereographic vs Equal Area
Projection
Stereographic
Equal Area
* Many texts, e.g. Cullity, show the
plane touching the sphere at N: this
changes the magnification factor for
the projection, but not its geometry. [Kocks]
Concept Params. Euler Normalize Vol.Frac. Cartesian Polar Components
Equal Area Projection
• Connect a line from the North Pole to the point to be
projected. Rotate that line onto the plane tangent to the
North Pole (which is the projection plane). The radius, r,
of the projected point from the North Pole, where R is the
radius of the sphere, and α is the angle from the North
Pole vector (co-latitude) to the point to be projected, is
given by:
r = 2R sin(α/2)
• Given spherical coordinates (α,ψ), where the longitude is
ψ (as before), the Cartesian coordinates on the
projection are therefore:
(x,y) = r(cosψ, sinψ) = 2R sin(α/2)(cosψ, sinψ)
Concept Params. Euler Normalize Vol.Frac. Cartesian Polar Components
Inverse Pole Figure - Procedure
• To calculate where a sample direction appears in an inverse pole figure, there are
various operations that must be performed.
• The key concept is that of thinking of the inverse pole figure as a set of sample
directions (e.g. RD, or ND) in the reference configuration and applying the orientation
as a transformation to that direction (here one only needs to deal with a single
direction, in contrast to the Pole Figure case) to find its position with respect to the
sample frame.
• Step 1: write the sample direction of interest as a unit vector; e.g. ND≡[001] = h.
• Step 2: apply the transformation (passive rotation), g (not g-1), to obtain the
coordinates of the direction in the inverse pole figure:
h’ = gh
(pre-multiply the vector by, e.g. the orientation matrix, g, that represents the
orientation; Rodrigues vectors or quaternions can also be used).
• Step 3: convert the rotated direction into spherical angles (to help visualize the result,
and to simplify Step 4) where Θ is the co-latitude and φ is the longitude:
Θ = cos-1(h’z), φ = tan-1(h’y/h’x).
Remember - use ATAN2(h’y,h’x) in your program or spreadsheet and be careful about
the order of the arguments!
• Step 4: project the direction onto a point, p, in the plane (stereographic or equal-
area):
px = tan(Θ/2) cosφ; py = tan(Θ/2) sinφ. [corrected sine and cosine for py and px components 25 i
08]
The previous slide explains where this formula comes from. The axes of the inverse
pole figure are x=100 and y=010. (Caution - this is simple and obvious for cubics. For
low symmetry crystals, these are Cartesian x and y, which may or may not
Summary
• Microstructure contains far more than qualitative
descriptions (images) of cross-sections of
materials.
• Most properties are anisotropic which means
that it is critically important for quantitative
characterization to include orientation
information (texture).
• Many properties can be modeled with simple
relationships, although numerical
implementations are (almost) always necessary.
Supplemental Slides
• The following slides contain revision
material about Miller indices from the first
two lectures.
Miller Indices
• Cubic system: directions, [uvw], are
equivalent to planes, (hkl).
• Miller indices for a plane specify reciprocals
of intercepts on each axis.
Miller <-> vectors
• Miller indices [integer representation of
direction cosines] can be converted to a
unit vector, n: {similar for [uvw]}.
) (h, k,l)
n= 2 2 2
h + k +l
Miller Index Definition of Texture
Component
• The commonest method for specifying a
texture component is the plane-direction.
• Specify the crystallographic plane normal
that is parallel to the specimen normal
(e.g. the ND) and a crystallographic
direction that is parallel to the long
direction (e.g. the RD).
(hkl) || ND, [uvw] || RD, or (hkl)[uvw]
Direction Cosines
• Definition of direction cosines:
• The components of a unit vector are equal
to the cosines of the angle between the
vector and each (orthogonal, Cartesian)
reference axis.
• We can use axis transformations to
describe vectors in different reference
frames (room, specimen, crystal, slip
system….)
