1. ESO 214: Nature and Properties of Materials
Ashish Garg, Dept. of MME, IITK

2. ESO 214: Nature and Properties of Materials
Points in Space
Illustrate the concept of point lattice
Lack of regular arrangement
No periodicity
Non-identical neighbourhood
Regular arrangement
Periodicity
Identical neighbourhood of each point
Ashish Garg, Dept. of MME, IITK

3. ESO 214: Nature and Properties of Materials
Point Lattice a b 
Periodic array of points in space with each point having identical neighborhood
Ashish Garg, Dept. of MME, IITK

4. Example: Point Lattice
ESO 214: Nature and Properties of Materials
Example: Point Lattice A B A B
Not A Point Lattice!
Is A Point lattice
A and B dont have identical surroundings
A and B have identical surroundings
Ashish Garg, Dept. of MME, IITK

5. Motif + Point Lattice = Crystal Structure
ESO 214: Nature and Properties of Materials
Motif + Point Lattice = Crystal Structure
Motif can be defined as a unit of pattern. For a crystal, it is an atom, an ion or a group of atoms or ions or a formula unit or formula units. When motif replaces points in a periodic point lattice, it gives rise to what is called as a crystal with a defined structure.
Ashish Garg, Dept. of MME, IITK

6. Primitive/Non-primitive lattices
ESO 214: Nature and Properties of Materials
Primitive/Non-primitive lattices
Primitive unit-cell: Consists of one lattice point
Non-primitive cell: More than one lattice points / unit cells
Volume of NP Cell = No of Motifs x Volume of Primitive Unit Cell
Ashish Garg, Dept. of MME, IITK

7. ESO 214: Nature and Properties of Materials
Unit Cells
Smallest repeatable unit
Many unit cells can be formed by As shown by cell vectors
Choice of cell is not unique
Lattice vector R can be represented as
Lattice parameters are usually defined as
Lattice Translations a, b, and c
Angle ,, and 
Ashish Garg, Dept. of MME, IITK

8. ESO 214: Nature and Properties of Materials
Summary
Delete this Line
Ashish Garg, Dept. of MME, IITK

9. ESO 214: Nature and Properties of Materials
Symmetry in Crystals
Symmetry
An operation which brings the object back to its original confiscation.
Symmetry elements underlying a point lattice (see the figures)
Reflection: reflection across a mirror plane
Rotation: rotation around a crystallographic axis by an angle such as 360˚/ is an integer of value 1, 2, 3, 4 and 6.
Inversion: a point at x,y,z becomes its equivalent at (–x,-y,-z)
Rotation-Inversion: Rotation followed by inversion OR Rotation-Reflection: Rotation followed by reflection.
Ashish Garg, Dept. of MME, IITK

10. ESO 214: Nature and Properties of Materials
(b) Rotation: for four fold axis, A1 goes to A2, A3 and then A4; for three fold axis A1 goes to A3; two fold axis, A1 goes to A4
(a) Reflection about a plane: point A1 reflects to A2
(d) Rotation-Inversion center: A1 becomes A’1 due to four fold rotation and then inversion takes A’1 to A2
(c) Inversion center
Ashish Garg, Dept. of MME, IITK

11. ESO 214: Nature and Properties of Materials
Crystal Systems
Based on the symmetry considerations, for various primitive lattice shapes, unit-cells can be divided into seven crystal systems.
Crystal system and lattice parameters
Minimum symmetry elements:
Cubic; a=b=c, ===90 1.
Four 3-fold rotation axes
Tetragonal; a=bc
===90 2.
One 4-fold rotation (or rotation-inversion) axis 3.
Orthorhombic; abc
===90
Three perpendicular 2-fold rotation (or rotation-inversion) axis
Ashish Garg, Dept. of MME, IITK

12. ESO 214: Nature and Properties of Materials
Crystal system and lattice parameters
Minimum symmetry elements: 4.
Rhombohedral;
a=b=c, ==90
One 3-fold rotation (or rotation-inversion) axis
Hexagonal;
a=bc
==90 =120
One 6-fold rotation (or rotation-inversion) axis 5.
Ashish Garg, Dept. of MME, IITK

13. ESO 214: Nature and Properties of Materials
Crystal system and lattice parameters
Minimum symmetry elements:
Monoclinic; abc
==90 6.
One 2-fold rotation (or rotation-inversion) axis
Triclinic
abc
90 7.
None
Ashish Garg, Dept. of MME, IITK

14. ESO 214: Nature and Properties of Materials
Further, seven crystal systems can be categorized in 14 Bravais lattices including both primitive and non-primitive .types.
1. Cubic; a=b=c, ===90
5. Hexagonal;
a=bc, ==90, =120
Crystal Systems
&
Bravais Lattices
(Plz put it page title)
4. Rhombohedral;
a=b=c, ==90
2. Tetragonal; a=bc, ===90
6. Monoclinic; abc, ==90
7. Triclinic
abc; 90
3. Orthorhombic; abc, ===90
Ashish Garg, Dept. of MME, IITK

15. ESO 214: Nature and Properties of Materials
Planes and Directions
Essential for the completeness of crystal structures
Millers Indices (in the names of William Hallowes Miller)
Crystallographic Planes
Identification of various faces seen on the crystal
(h,k,l) for a plane or {h,k,l} for identical set of planes
h, k, l are integers
Directions
Atomic directions in the crystal
[u,v,w] for a direction or <u,v,w> for identical set of directions
u, v, w are integers
Ashish Garg, Dept. of MME, IITK

16. Determination of a Crystal Plane
ESO 214: Nature and Properties of Materials
Determination of a Crystal Plane
A crystallographic plane in a crystal satisfies following equation
h/a, k/b, and c/l are the intercepts of the plane on x, y, and z axes.
a,b,c are the unit cell lengths
h, k, l are the integers called as Miller indices and the plane is represented as (h, k, l)
Ashish Garg, Dept. of MME, IITK

17. ESO 214: Nature and Properties of Materials
Unit Cell Parameters
4A, 8A and 3A
Fractional intercepts: 2A/4A, 6A/8A, 3A/3A
Reciprocal of fractional intercepts: 2, 4/3, 1
Convert to smallest set of integers: (6, 4, 3)
Ashish Garg, Dept. of MME, IITK

18. ESO 214: Nature and Properties of Materials
Crystal Planes
Interplanar angle is given by (cubic only)
Interplanar spacing is given by (Cubic)
Ashish Garg, Dept. of MME, IITK

19. ESO 214: Nature and Properties of Materials
Ashish Garg, Dept. of MME, IITK

20. ESO 214: Nature and Properties of Materials
Directions
Denoted as [u,v,w]
Vector components of the direction resolved along each of the crystal axis reduced to smallest set of integers
Ashish Garg, Dept. of MME, IITK

21. ESO 214: Nature and Properties of Materials
Crystal Directions
How to locate a direction:
Example: [231] direction would be
1/3 intercept on cell a-length
1/2 intercept on cell b-length and
1/6 intercept on cell c-length
Directions are always denoted with [uvw] with square brackets and family of directions in the form <uvw>
Ashish Garg, Dept. of MME, IITK

22. ESO 214: Nature and Properties of Materials
Weiss Zone Law
For a direction [u v w] lying in a plane (h k l)
h.u + k.v + l.w = 0
Ashish Garg, Dept. of MME, IITK

23. ESO 214: Nature and Properties of Materials
Summary
Point lattice is regular arrangement of points in space with identical neighbourhood.
Motif (Unit or basis) replaces the point to create crystal structure
Unit cell: smallest repeatable unit.
Unit-cell containing one lattice point is called as primitive whereas unit-cell containing more than one lattice points is called as non-primitive.
There are a total of seven crystal systems and fourteen Bravais lattices.
Unit-cell can be defined by crystal planes and directions in terms of their Miller indices.
Ashish Garg, Dept. of MME, IITK