Lattice Planes and Miller Indices

1. Lattice Planes and Miller Indices Lattice Planes : - Aggregate of set of parallel equidistant planes passing through the lattice points Miller Indices: - The reciprocals of the intercept’s made by the plane on the three rectangular axis and written as (h, k, l) The Miller indices are defined by X, Y, Z are the intersections of one plane with on a, b, c respectively Note - plane // to axis, intercept = ∞ and 1/∞ = 0 family of lattice planes parallel to

2. z 1a+1b+0c The Rule to obtain Miler Indices … It is more useful to specify the orientation of a plane by the following rules: Set up Co-ordinate axis along the edge of the unit cell and then note where the plane to be indexed intercepts the axis. Record the resulting normalized intercept sent in the order X, Y, Z Invert the intercept values (i.e. 1/intercepts) Using an appropriate multiplier, convert the 1/intercept set to the smallest possible set of whole numbers Enclose the whole-number set in the in curly brackets around the indices {h, k, l}. y x [110]

3. Miller Indices for planes z y x 1.Select an origin not on the plane 2.select a crystallographic coordinate system (0,0,1) 3. Find intercepts along axes 2 3 1 (0,3,0) 4. Take reciprocal 1/2 1/3 1/1 5. Convert to smallest integers in the same ratio : 3,2,6 (2,0,0) 6. Enclose in parenthesis : (3,2,6)

4. Examples of Crystallographic Planes c c b b a a c 0.5 b a (100) (111) (212)

5. dhkl 2D d'h’k’l’ Inter-Planar Spacing, dhkl, and Miller Indices • The inter-planar spacing (dhkl) between crystallographic planes belonging to the same family (h,k,l) is denoted (dhkl) • Distances between planes defined by the same set of Miller indices are unique for each material Inter-planar spacing’s can be measured by x-ray diffraction (Bragg’s Law)

6. INTER-PLANNER SPACING Let us consider simple cubic lattice and OX, OY, OZ are co-ordinate axis. Reference Plane passing through origin. Plane A,B, C defined by miller indices (h, k, l) and cuts intercepts a/h, b/k, c/l with coordinate axis. Normal ON = d is normal to the plane A, B, C from origin. Because ON d = normal to the plane A, B, C we can write The Law of direction Cosine is

7. Lattice parameters a, b, c of a unit cell can then be calculated The relationship between d and the lattice parameters can be determined geometrically and depends on the crystal system The expressions for the remaining crystal systems are more complex

8. Concepts to remember • Unit cell, unit vector, and lattice parameters. • Bravais Lattices. • Counting number of atoms for a given unit cell. • Coordination number = number of nearest neighbor atoms. • Atomic Packing Factor (APF) = Volume of atoms in a unit cell/Volume of unit cell. • Close-packing • FCC, BCC, HCP • Crystallographic coordinates, directions, and planes. • Densities • X-ray diffraction

9. References • Solid State Physics by A. J. Dekker –Macmillan India Ltd. • Introduction to Solid State Physics by Charles Kittel- Wiley Eastern Ltd. • Solid State Physics by R. L. Singhal – Kedar Nath Ram Nath and Co. Meerut. • Solid State Physics by Neil W. Ashcroft and N. David Mermin – Thomson Books/COLE • Unified PHYSICS (B Sc Final Year) by R. P. Goyal – Shiv Lal Agrawala and Co. Indore