1. Chapter 3 Crystal Geometry and Structure Determination

2. Crystals: long range periodicity, Anisotropic
Amorphous: Homogeneous, isotropic
Courtesy: H Bhadhesia

3. Crystal ?
A 3D translationaly periodic arrangement of atoms in space is called a crystal.
2D crystal

4. Translational Periodicity
Unit cell description : 1
Translational Periodicity
Crystal
One can select a small volume of the crystal which by periodic repetition generates the entire crystal (without overlaps or gaps)
Unit Cell
Space filling
Building block of
crystal

5. 2D crystal

6. Unit cell is the imaginary, it doesn't really
exist: We use them to understand the
crystallography
It should be space filling, no gaps, no overlaps
We tend to choose unit cells with angles close
to 90° and shortest unit cell edge length

7. The most common shape of a unit cell is a parallelopiped.
3D UNIT CELL:
Unit cell description : 2
The most common shape of a unit cell is a parallelopiped.

8. The description of a unit cell requires:
Unit cell description : 3
The description of a unit cell requires:
1. Its Size and shape (lattice parameters)
2. Its atomic content (fractional coordinates)

9. Lets just think about size and shape
first!!

10. Size and shape of the unit cell:
Unit cell description : 4
Size and shape of the unit cell:
1. A corner as origin
2. Three edge vectors {a, b, c} from the origin define a CRSYTALLOGRAPHIC COORDINATE SYSTEM a b c   
3. The three lengths a, b, c and the three interaxial angles , ,  are called the LATTICE PARAMETERS

11. Crystal
Unit cell
Characterize the size and shape of a unit cell
Lattice

12. Lattice?
A 3D translationally periodic arrangement of points in space is called a lattice.

13. Lattice
A 3D translationally periodic arrangement of points
Each lattice point in a lattice has identical neighbourhood of other lattice points.

14. A unit cell of a lattice is NOT unique.
UNIT CELLS OF A LATTICE
If the lattice points are only at the corners, the unit cell is primitive otherwise non-primitive
Non-primitive cell
Primitive cell
A unit cell of a lattice is NOT unique.
Primitive cell

15. Can we select a triangular unit cell? Since it can give a very small
UNIT CELLS OF A LATTICE
Can we select a triangular unit cell? Since it can give a very small
repeat unit
Unit cell is a small volume of the crystal which by periodic repetition generates the entire crystal (without overlaps or gaps)
Primitive cell

16. In 2D there are only 5 possible ways of arranging points
which are regular in space
A 3D space lattice can be generated by repeated
translation of three vectors a, b and c
It turns out there are 14 distinguishable ways of arranging
points in 3 dimensional space such that each arrangement
conforms to the definition of a space lattice
These 14 space lattices are known as Bravais lattices,
named after their originator

17. Think about 2D crystal which is making big news??
Carbon nanotube: Graphene sheet
A layer of C atoms
in hexagonal arrangement
Cylindrical crystal
In general we mostly deal with 3 dimensional crystals

18. Classification of lattice
The Seven Crystal System
And
The Fourteen Bravais Lattices

19. 7 Crystal Systems and 14 Bravais Lattices
Crystal System Bravais Lattices
Cubic P I F
Tetragonal P I
Orthorhombic P I F C
Hexagonal P
Trigonal P
Monoclinic P C
Triclinic P
P: Primitive; I: body-centred; F: Face-centred; C: End-centred
*The notations comes from Germans

20. Cubic Crystals
a=b=c; ===90

21. The three cubic Bravais lattices
Crystal system Bravais lattices
Cubic P I F
Simple cubic Primitive cubic Cubic P
Body-centred cubic Cubic I
Face-centred cubic Cubic F

22. Orthorhombic C End-centred orthorhombic Base-centred orthorhombic

23. Unit cell parameters for different crystal systems
Courtesy: H Bhadhesia

24. Trinclinc Crystal

25. Courtesy: H Bhadhesia

26. Courtesy: H Bhadhesia

27. Why half the boxes are empty?
Crystal System Bravais Lattices
Cubic P I F
Tetragonal P I
Orthorhombic P I F C
Hexagonal P
Trigonal P
Monoclinic P C
Triclinic P ?
E.g. Why cubic C is absent?

28. End-centred cubic not in the Bravais list ?
End-centred cubic = Simple Tetragonal

29. 14 Bravais lattices divided into seven crystal systems
Crystal system Bravais lattices
Cubic P I F C
Tetragonal P I
Orthorhombic P I F C
Hexagonal P
Trigonal P
Monoclinic P C
Triclinic P

30. Similarly, answer why face centred tetragonal
is not in the list?
Face-centred tetragonal = Body-centred Tetragonal

31. What is the basis for classification of lattices into 7 crystal systems and 14 Bravais lattices?

32. Lattices are classified on the basis of their symmetry
Crystal class is defined by certain minimum symmetry (defining symmetry)

33. Symmetry?
If an object is brought into self-coincidence after some operation it said to possess symmetry with respect to that operation.

34. Translational symmetry
Lattices also have translational symmetry
In fact this is the defining symmetry of a lattice

35. Rotation Axis
If an object come into self-coincidence through smallest non-zero rotation angle of  then it is said to have an n-fold rotation axis where
=180
n=2
2-fold rotation axis
=90
n=4
4-fold rotation axis

36. Rotational SymmetriesZ
Angles:
180
120
90
72
60
45
Fold: 2 3 4 5 6 8
Graphic symbols

37. Symmetry of lattices
Lattices have
Translational symmetry
Rotational symmetry
Reflection symmetry