X-Ray Diffraction and TEM
- Dr. SANKET A. MEHAR
Course content:
Unit-1
Introduction to crystallography, Symmetry – Bravais
lattices, Pole figures (Stereographic projection and their
applications), point group/space groups, Reciprocal
lattice
Unit-2
X‐ray diffraction and analysis: Production and properties
of X‐rays, X‐rays absorption, filter and detectors. Bragg’s
law, Diffraction Methods, Structure factor and intensity
calculations.
Course content:
Unit-3
Determination of crystal structure, sources of error in
measurements. Chemical Analysis by X‐ray techniques, Effect of
texture, grain size, plastic deformation, micro strain, residual
stresses etc. on diffraction lines.
Unit 4:
TEM: Principle and operation, sample preparation
techniques, detectors and imaging modes, Introduction to
HRTEM, diffraction patterns, Indexing of selected area
diffraction patterns.
Reference Books:
➢B.D. Cullity, Elements of X‐ray Diffraction by (II edition),
Addison‐Wesley Publishing Co Inc., Reading, USA, 1978
➢P.J. Goodhew and F.J. Humphreys, Electron Microscopy and
Analysis by Taylor and Francis, London,
2001(ISBN‐0‐7484‐0968‐8).
➢Basics of X-Ray Diffraction and Its Applications by K.
Ramakanth Hebbar
Evaluation
➢Mid semester exam
➢End Semester exam
➢Class tests
➢Assignments/ Short Presentation/ Internal
bonus/Attendance
Characterization using XRD
➢A fundamental process without which no scientific understanding
of engineering materials could be established.
Crystallography
➢A crystal may be defined as a solid composed of atoms
arranged in a pattern - periodic in three dimensions.
➢As such, crystals differ in a fundamental way from gases and
liquids because the atomic arrangements in the latter do not
possess the essential requirement of periodicity.
Crystallography
▪ Arrangement of atoms in solids.
▪ A Crystal is any solid material in which the component atoms are
arranged in a definite pattern and whose surface regularity reflects its
internal symmetry.
Crystal Faces
➢Unit cell is the smallest unit of volume that permits identical
cells to be stacked together to fill all space.
➢By repeating the pattern of the unit cell over and over in all
directions, the entire crystal lattice can be constructed.
➢3 D network of imaginary lines connecting the atoms – Space Lattice
or Bravais Lattices.
Atomic Hard sphere model Lattice points
➢A point lattice, which is defined as an array of
points in space so arranged that each point has
identical surroundings.
➢“Identical surroundings' mean-
that the lattice of points, when viewed in a
particular direction from one lattice point,
would have exactly the same appearance when
viewed in the same direction from any other
lattice point.
Unit cells need to be able to “Stack” them
to fill all space !!
• This puts restrictions on Unit Cell
Shapes
• Cubes Work!
• Pentagons Don’t!
Different types of unit cell are possible and they are classified based on their level
of symmetry
There are 14 possible types of space lattices (Bravais Lattice) and they fall into 7
crystal systems.
Where Can I Put the Lattice Points?
•The French scientist August Bravais, demonstrated in 1850 that only
these 14 types of unit cells are compatible with the orderly
arrangements of atoms found in crystals.
• These three-dimensional configurations of points used to describe
the orderly arrangement of atoms in a crystal.
• Each point represents one or more atoms in the actual crystal, and if
the points are connected by lines, a crystal lattice is formed.
Ref: School of Chemistry, Univ. of Bristol
➢We must first distinguish between simple, or
primitive, cells (symbol P or R) and
nonprimitive cells (any other symbol):
primitive cells have only one lattice point per
cell while nonprimitive have more than one.
➢A lattice point in the interior of a cell
"belongs" to that cell, while one in a cell face is
shared by two cells and one at a corner is
shared by eight.
Symmetry operations
➢Bravais lattices and the real crystals - exhibit various kinds of symmetry.
➢A body or structure is said to be symmetrical when its component parts are
arranged in such balance, that certain operations can be performed on the
body which will bring it into coincidence with itself. These are
termed symmetry operations.
➢There are in all four macroscopic symmetry operations or elements:
reflection, rotation, inversion, and rotation-inversion.
➢A body has n-fold rotational symmetry about an axis if a rotation of
360 /n brings it into self-coincidence.
6- Fold
5 fold, 7 fold, 8 fold or higher :
Does not exist in crystal because it cannot fill all the space.
An object has reflection symmetry if it can be
divided in half by one or more lines or axes of
symmetry
Inversion
➢A body has an inversion center if corresponding points of the body are located
at equal distances from the center on a line drawn through the center.
➢A molecule with an inversion center can only have ONE center of inversion.
• Collection of symmetry operations which when applied about a
lattice point leaves the lattice/ crystal structure invariant, are
called as POINT GROUPS. ( No change in the surroundings)
• Rotation + Reflection = 2D
• Rotation + Reflection+ Inversion = 3D
• Reflection
• Rotation
POINT OPERATIONS
• Inversion SPACE OPERATIONS
• Translation
POINT GROUPS
➢In 2D, 10 different point group exists in a crystal
➢1) 1 : Rotation of 360֯ only, will bring it to same position
1m - mirror plane – reflection
symmetry
2 – two fold axis – two fold rotation
2mm –
2 - two fold axis – two fold rotation
m - mirror plane parallel to rotation axis
m - mirror plane perpendicular to
rotation axis
3D Point groups
➢Hermann- Mauguin Notations (H-M symbols):
➢Numbers 1,2,3.. Axis of rotation (Rotation axis)
➢Letter m- mirror plane
➢Letter mm- parallel mirror planes
➢Bar – Inversion
➢/m – mirror plane normal to the rotation axis
➢32 point groups in 3D
Stereographic Projection
➢For determining -Angle between two different planes/ two
different directions
➢Difficult in 3D – so resolve it in 2D
➢Geographical globe/ atlas – area true projection
➢Stereographic Projection- angle true projection
(100) Family of planes
➢Next step- draw perpendiculars to these 6 faces of unit
cell
➢Wherever these normal or perpendiculars cut the
spheres – that point is called as a POLE
➢That is, a perpendicular from (100) plane, if cuts a
sphere at point A, then that point A is called as pole of
(100) plane.
➢In doing so, we have reduced down the 6 planes of a unit
cell in 6 points – lying on a surface of a reference sphere.
4 poles are shown, 2 other poles are the one
which comes out of the board/ front and back.
➢Now, put a piece of paper/ hold a paper perpendicular
to the vertical axis
➢Put a source of light at the top ( at 001)
➢Allow the light to pass through all the poles and collect
it on the paper.
➢If we do this, whole of a reference sphere will be
projected as a circle – called as basic circle.
➢Mark these poles on a basic circle, that is on a paper
which we have held.
This is nothing but a projection of poles of 6 planes-
This diagram is called as Stereographic Projection - Angle true projection
110 family of planes
All (110) type of poles lie
midway of all (100) type of
poles
111 family of
planes also
included
WULFF NET: To calculate angles
Reciprocal Lattice
➢X-Ray diffraction – from sets of parallel lattice planes in a crystal
➢But in a crystal- there exists many sets of parallel planes with different
orientations and interplanar spacings which can diffract given beam of x-rays
➢Visualization and to obtain information of these planes – difficult
➢Therefore- Reciprocal lattice – given by Ewald
➢Two lattices associated with crystal- direct or real lattice and Reciprocal
lattice.
➢Using reciprocal lattice concept, it becomes easy to find interplanar spacings.
Reciprocal Lattice
➢When we observe diffraction pattern, the diffraction spots form a picture of
crystal lattice. This periodic structure like is not direct picture of crystal
➢Inverse image – Reciprocal lattice
➢Reciprocal lattice points are inverse of actual lattice points.
➢Thus the distance in reciprocal lattice system is 1/ distance corresponding to
actual distance d in a direct crystal lattice.
➢Such a space – is reciprocal space or Fourier space.
➢In direct lattice – lattice vectors - denoted by a1, a2, a3 Reciprocal lattice
vectors – b1, b2, b3
To construct a lattice from a direct lattice, steps-
➢Take a point in the direct lattice as origin
➢From this origin, draw normal to every set of planes in the direct lattice.
➢Set the length of each normal equal to reciprocal of interplanar spacing (d)
for its particular plane of parallel planes.
➢Place a point at the end of each normal
➢The collection of points so obtained is called as reciprocal lattice
➢Reciprocal lattice is a mathematical concept. It is a Fourier space, however, the
distance between two given points in this space is equal to the inverse of
corresponding inter-planar d-spacing in the direct lattice (Bravais lattice). The word
lattice indicates a set of mathematical points in the direct space which satisfy
translational symmetry.
➢The vector that connects two points in the reciprocal space is the reciprocal lattice
vector ‘G’
➢The reciprocal lattice can be constructed for each direct crystal lattice. The indices of
points in a reciprocal lattice represent the Miller indices of planes in the
corresponding direct crystal lattice.
➢As per quantum physics electrons can be considered as waves associated with a wave
vector k = 2П/λ
➢Here, the wave vector k has dimensions as similar as the reciprocal lattice vector G.
Therefore, electrons present in crystals can be treated mathematically as waves in
reciprocal space.
Primitive lattice translation vector
➢The primitive lattice translation vectors specify unit cell of
smallest volume.
➢A lattice translation operator is defined as a displacement of a
crystal with a crystal translation operator : T = u1a1 + u2a2 +
u3a3
➢To describe a crystal, it is necessary to specify three things:
1. What is the lattice 2. What are the lattice translation vectors 3.
What is the basis
• To Find the Reciprocal lattices of Simple cubic, BCC and
FCC lattices.
