Unit cell, types of crystal structure, Miller indices, X- Ray diffiraction

1. Unit-III
Crystallography
Prof. Arvind D. Kanwate
SVC, Dept. of Physics Deulgaon Raja

2. Crystallography
 Introduction
 Crystalline and Amorphous materials
 Unit Cell
 Millar Indices
 Types of Lattices
 Coordination Number
 Diffraction of X-ray by crystals
 Bragg’s law
 Determination of lattice parameter of crystal
 Reciprocal lattice
 Defects in solid

3. What is Crystallography?
 It is a branch of science in which the geometry,
internal structure and physical properties of
crystalline materials are studied.
 Structures should be classified into different types
according to the symmetries they possess.

4. What is Matter

5. SOLIDS
 Solid material is classified into two categories
1) Crystalline Solids
2) Amorphous Solids

6. Crystalline Solids
 Crystalline solids are those in which the
atoms (ions or molecules) are arranged in a
periodic manner in all the three directions.
 The majority of all solids are crystalline.

7. 1) Amorphous Solids
 Amorphous Solids are those in which atoms (or molecules are
arranged in random manner. There is no regularity or periodicity in
the arrangement of atoms in space.
 Amorphous silicon can be used in solar cells and thin film
transistors.

8. Difference Between Crystalline Solids And Amorphous Solids
Crystalline Solids
 Atoms or molecules have regular,
periodic arrangements
 They exhibit different magnitudes
of physical properties in different
directions.
 They are anisotropic in nature.
 They exhibit directional
properties.
 They have sharp melting points.
 Crystal breaks along regular
crystal planes and hence the
crystal pieces have regular shape
 Ex: Copper, Silver, Aluminium etc
Amorphous Solids
 Atoms or molecules are not
arranged in a regular, periodic
manner. They have random
arrangement.
 They exhibit same magnitudes of
physical properties in different
directions
 They are isotropic in nature.
 They do not exhibit directional
properties.
 They do not possess sharp melting
points
 Amorphous solids breaks into
irregular shape due to lack of
crystal plane.
 Ex: Glass, Plastic, rubber, etc.

9. Unit Cell
• Unit cell is defined as the smallest volume of a crystal from
which the complete crystal can be constructed by translational
repetition in three dimensions.
• The smallest block or geometrical figure from which the
crystal is build up by repetition in three dimensions, is called
unit cell.
• The complete crystal is found to consist of identical blocks or
unit cells.

10. Unit Cell in 2D
 The smallest component of the crystal (group of atoms,
ions or molecules), which when stacked together with
pure translational repetition reproduces the whole
crystal.
10
S
a
b
S
S
S
S
S
S
S
S
S
S
S
S
S
S

13. • Crystallographic Axes: These are the lines drawn parallel to the lines
of intersection of any three faces of the unit cell which do not lie in the
same plane. ox,oy oz.
• Primitives: The three sides of unit cell are called Primitives. They are
denoted by a,b,c. They are also known as lattice constants.
• Interfacial angles: The angles between three crystallographic axes of
the unit cell are called interfacial angles.
• The angle b/w Y and Z axis is α
• Z and X axes is β
• X and Y axes is γ
• Lattice parameter defines actual shape and size of crystal.

15. Crystal Structure
 Crystal structures can be obtained by attaching atoms, groups
of atoms or molecules which are called basis (motif) to the
lattice sides of the lattice point.
15
Crystal Structure = Crystal Lattice + Basis

16. A two-dimensional Bravais lattice
with different choices for the basis

17. Five Bravais Lattices in 2D 17

20. Unit Cell in 3D
Crystal Structure
20

21. Simple Cubic Crystal Structure (SCC)
(P)
 All atoms are placed at corners of unit cells.
 The co-ordination number is 6.

22. Body Centered Cubic Crystal
Structure (BCC) (I)
 Atoms are placed at corners and center of unit cell
 The co-ordination number is 8.

23. Face Centered Cubic Crystal
Structure (FCC) (F)
Atoms are placed corners as well as faces of unit cells.
The co-ordination number is 12.

25. Base Centered lattice (C)
 In this lattice along with
the corner atoms, the base
and opposite face will
have centre atoms.

27. 3D Bravais Lattice
 A three dimensional space lattice is generated by
repeated translation of three translational vectors a,
b and c.
 Crystals are grouped under seven systems on the
basis of the shape of the unit cell.

28. CUBIC
Simple Body-Centered Face-Centered
Cubic
a = b = c, a = b = g = 90º

29. TETRAGONAL
Simple Body-Centered
Tetragonal
a = b ≠ c, a = b = g = 90º

30. ORTHORHOMBIC
Simple Base-
Centered
Body-CenteredFace-
Centered
Orthorhombic
a ≠ b ≠ c, a = b = g = 90º

31. RHOMBOHEDRAL or TRIGONAL
a = b = c, a ≠ b ≠ g

32. HEXAGONAL
a = b ≠ c, a = b = 90º g = 120º

33. MONOCLINIC
Simple Base Centered
a ≠ b ≠ c, a = g = 90º ≠ b

34. TRICLINIC
a ≠ b ≠ c, a ≠ b ≠ g

35. Crystal Structure 35

38. Miller Indices
 Miller Indices are a symbolic vector representation for the
orientation of an atomic plane in a crystal lattice and are defined
as the reciprocals of the fractional intercepts which the plane
makes with the crystallographic axes.
 To determine Miller indices of a plane, we use the following steps
1) Determine the intercepts of the plane along each of the three crystallographic
directions
2) Take the reciprocals of the intercepts
3) If fractions result, multiply each by the denominator of the smallest fraction
4) The result is written in paranthesis. This is called the `Miller Indices’ of the
plane in the form (h k l).

39. IMPORTANT HINTS:
 Miller indices denotes the crystal planes and crystal directions.
 All equally spaced family of parallel planes have the same miller
indices.
 When a plane is parallel to any axis, the intercept of the plane on that
axis is infinity. So, the Miller index for that axis is Zero
 A bar is put on the Miller index when the intercept of a plane on any
axis is negative
 The normal drawn to a plane(h,k,l) gives the direction [h,k,l]

40. Example-1

41. Example-2

42. Example-3

44. Example-4

45. Example-5

48. X-ray Diffraction
 X-ray crystallography, also called X-ray diffraction, is used to
determine crystal structures by interpreting the diffraction
patterns formed when X-rays are scattered by the electrons of
atoms in crystalline solids. X-rays are sent through a crystal to
reveal the pattern in which the molecules and atoms contained
within the crystal are arranged.
 This x-ray crystallography was developed by physicists William
Lawrence Bragg and his father William Henry Bragg. In 1912-
1913, the younger Bragg developed Bragg’s law, which connects
the observed scattering with reflections from evenly spaced
planes within the crystal.

50. Bragg’s Law

52. Bragg’s Law : 2dsinθ = nλ

54. Sodium Chloride Structure
 Sodium chloride also crystallizes
in a cubic lattice, but with a
different unit cell.
 Sodium chloride structure consists
of equal numbers of sodium and
chlorine ions placed at alternate
points of a simple cubic lattice.
 Each ion has six of the other kind
of ions as its nearest neighbours.
54

55. Defects in Solids
Real crystals are never perfect: there are always defects!
distortion
of planes
POINT DEFECTS
a) Vacancies: This defect is formed when
an atom is missing from a position that
ought to be filled in the crystal, creating a
vacancy.
b) Self-Interstitials:
An extra atoms positioned between atomic sites.
distortion
of planesc) Impurity Atoms:

56. Line Defects
Deformation of ductile materials occurs when a line defect
(dislocation) moves (slip) through the material

57. EDGE DISLOCATION
Edge dislocation centers around the edge dislocation line that
is defined along the end of the extra half-plane of atoms
 extra half-plane of atoms
inserted in a crystal structure.

58. Screw dislocation
spiral planar ramp resulting from shear deformations.