1. 1 Chapter 3 Crystal Geometry and Structure Determination

2.  Crystal  Unit cell, building blocks of crystal, shape filling  Lattice parameters a, b, c and interaxial angles , ,  to characterize the size and shape of the unit cell  Lattice  Primitive and non primitive unit cells  14 Bravais lattices, 7 crystal systems Recap

3. Crystal SystemBravais Lattices 1.CubicPIF 2.TetragonalPI 3.OrthorhombicPIFC 4.HexagonalP 5.TrigonalP 6.MonoclinicPC 7.TriclinicP P: Primitive; I: body-centred; F: Face-centred; C: End-centred *The notations comes from Germans 7 Crystal Systems and 14 Bravais Lattices

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

5. Lattices are classified on the basis of their symmetry Crystal class is defined by certain minimum symmetry (defining symmetry) What is the basis for classification of lattices into 7 crystal systems and 14 Bravais lattices?

6. 6/87 7 crystal Systems Cubic Defining Crystal system Conventional symmetry unit cell 4 A single 3 1 none Tetragonal Orthorhombic Hexagonal Rhombohedral Triclinic Monoclinic a=b=c,  =  =  =90  a=b  c,  =  =  =90  a  b  c,  =  =  =90  a=b  c,  =  = 90 ,  =120  a=b=c,  =  =  90  a  b  c,  =  =90  a  b  c, 

7. 7 Cubic symmetry 4 triads: 4 body diagonals

8. 8 Tetragonal symmetry 1 tetrad

9. Similarly, you can check for other crystal systems Courtesy: H Bhadhesia

10. A 3D translationally periodic arrangement of atoms Crystal A 3D translationally periodic arrangement of points Lattice

11. 11 How would you create a crystal structure from lattice?? Crystal structure means you now have to place something at each lattice point

12. What is the relation between the two? Crystal = Lattice + Motif Motif or basis: an atom or a group of atoms associated with each lattice point

13. Crystal=lattice+basis Lattice: the underlying periodicity of the crystal, Basis: atom or group of atoms associated with each lattice points Lattice: how to repeat Motif: what to repeat

14. 14 Create the crystal structure of brass Cubic P Each of these points are lattice points

15. 1/2 Crystal Structure Motif Coordinates of Cu and Zn atoms Structure of brass Courtesy: H Bhadhesia

16. 16 Courtesy: H Bhadhesia

17. lattice + motif = structure primitive cubic lattice motif = Cu at 0,0,0 Zn at 1/2, 1/2, 1/2 Courtesy: H Bhadhesia

18. 18 Create some complicated crystal structure: Structure of diamond Face-centred cubic Cubic F

19. 1/4 3/4 Lattice: face-centred cubic Motif: C at 0,0,0 C at 1/4,1/4,1/4 Courtesy: H Bhadhesia

20. Structure of Diamond All the C atoms are tetrahedrally bonded by covalent bond Courtesy: H Bhadhesia

21. How many C atoms per unit cell?? You know about total no. of lattice points in cubic F 4 How many C atoms you are putting per lattice point? 2 So total no. of C atoms per unit cell would be 8

22. 1/4 3/4 Structure of ZnS Courtesy: H Bhadhesia

23. 1/4 3/4 Lattice: face-centred cubic Motif: Zn at 0,0,0 S at 1/4,1/4,1/4 Structure of ZnS Courtesy: H Bhadhesia

25. Miller Indices of directions and planes William Hallowes Miller (1801 – 1880) University of Cambridge

26. 1. Choose a point on the direction as the origin. 2. Choose a coordinate system with axes parallel to the unit cell edges. x y3. Find the coordinates of another point on the direction in terms of a, b and c 4. Reduce the coordinates to smallest integers. 5. Put in square brackets Miller Indices of Directions [100] 1a+0b+0c z 1, 0, 0 Miller Indices 2

27. y z Miller indices of a direction represents only the orientation of the line corresponding to the direction and not its position or sense All parallel directions have the same Miller indices [100] x Miller Indices 3

28. x y z O A 1/2, 1/2, 1 [1 1 2] OA=1/2 a + 1/2 b + 1 c P Q x y z PQ = -1 a -1 b + 1 c -1, -1, 1 Miller Indices of Directions (contd.) [ 1 1 1 ] __ -ve steps are shown as bar over the number

29. 29 Courtesy: H Bhadhesia

31. Miller indices of a family of symmetry related directions [100] [001] [010] = [uvw] and all other directions related to [uvw] by the symmetry of the crystal = [100], [010], [001] = [100], [010] Cubic Tetragonal [010] [100] Miller Indices 4