Procedure to determine miller indices of plane: (1) Choose origin in such a way which lies outside the plane of interest (choice of origin is arbitrary) (2) Find the intercepts of the plane on the three co – ordinate axes (3) Take reciprocals (4) Convert into smallest integers in the same ratio (5) Enclose them in parentheses without camos eg : (h k l)
( ) denotes plane, { } denotes family of planes
Useful aspects about miller indices for planes (1) A plane and its negative are identical eg: (0 1 0) = (0 -1 0) (2) Planes and their multiples are not identical as planar densities and packing factors are different (3) Family of planes will have same type of atomic packing but not all members of family are parallel to one another
Miller indices of direction Directions in crystal are specified in a shorthand vector notation. Let a vector r represents a direction in a crystal. The miller indices are simply the vector components of direction resolved along each of the co – ordinate axes and reduce to smallest integers i.e the components of the vector along 3 axes are determined as multiples of the unit vector corresponding to each direction.
[ ] denotes direction, < > denotes family of directions
Useful aspects about miller indices for directions (1) A direction and its negative are not identical, [1 0 0] ≠ [-1 0 0] same line but opposite direction (2) A direction and its multiple are identical, [1 1 0] = [2 2 0], but should not be reduced to lowest integers (3) Crystal directions of family are not necessary parallel to one another (4) Crystal plane and a crystal direction normal to it have same indices i.e [1 1 1] ﬩ (1 1 1)
Miller – bravias indices For hexagonal crystals a four digit notation h k i l known as Miller – bravias indices is used. The use of such a notation enables crystallographically equivalent planes or directions in hexagonal crystals to be denoted by the same set of indices. 3 axes a1, a2, a3 are coplanar and lie on the basal plane of the hexagonal prism with a 120° angle between them. The fourth axis is the c axis perpendicular to the basal plane. The indices of plane or direction are calculated similar to that in miller indices. For 3 coplanar vectors h+k = -i
Inter planar spacing The distance or spacing between the plane and a parallel plane passing through the origin. In case of cubic system it is given by
Angle between planes or directions
For cubic crystals, the angle between two planes (h1 k1 l1) and (h2 k2 l2) or two directions [h1 k1 l1] and [h2 k2 l2] is
Line of intersection [h k l] of two planes (h1 k1 l1) and (h2 k2 l2) is cross product of two planes.
h = k1l2 – l1k2 k = l1h2 – h1l2 l = h1k2 – k1h2
The direction [h1 k1 l1] lies in the plane (h2 k2 l2) if h1h2 + k1k2 + l1l2 = 0.
Interstitial Voids The empty space that is unoccupied by the atoms in a closed packed system.
Types: (1) tetrahedral (2) octahedral
Tetrahedral void
· This forms when an atom is put in the valley formed by three spheres of a closed – packed plane
· The named has been derived because regular tetrahedron is formed when he centers of 4 atoms are joined
Octahedral void
· It is surrounded by 6 solvent atoms situated at 6 corners of a regular octahedron
· The name has been derived because of 8 equal faces of equilateral triangles
· 4 atoms in a plane (square based) and one on top and other at bottom