Prolog Text Book: C.Kittel, "Introduction to Solid State Physics", 8th ed.,Wiley (2005) Website: Homework Grades: Exercises: 50% MidTerm:20% Final:30%

1. Crystal Structure Periodic Arrays of Atoms Fundamental Types of Lattices Index System for Crystal Planes Simple Crystal Structures Direct Imaging of Atomic Structure Crystal Structure Data

Periodic Arrays of Atoms Experimental evidence of periodic structures: Integral idex numbers. X-ray diffraction (Laue’s theory). Crystal building from identical blocks.

Lattice Translation Vectors Crystal = Lattice + Basis Lattice = set of points given by d = dimension of latticea i = translation vectors If every pair of equivalent points in a crystal are related by (1), then a i are called primitive translation vectors. (1) Volume bounded by a i is called a cell. A crystal is invariant under the translation Cell bounded by primitive translation vectors is called a primitive cell.

Basis:

Primitive Lattice Cell Recapitulation: Parallelepiped defined by axes a i is called a cell. A cell must fill all space when subject to all possible lattice translations of the crystal. Parallelepiped defined by primitive axes a i is called a primitive cell. A crystal with 1 atom in its primitive cell is called a Bravais crystal. Characteristics of a primitive cell : Cell volume is minimal. Number of basis atoms is minimal. Contains exactly 1 lattice point. Wigner-Seitz cell Primitive cell centered at a lattice point and bounded by planes normal to and bisecting the lines joining the lattice point to its neighboring points.

Fundamental Types of Lattices Only rotations C n with n = 2, 3, 4 and 6 are compatible with the translational symmetry. There’re 32 crystallographic point groups (classes). Lattices with the same maximal point group are said to belong to the same crystal systems. There’re only 7 crystal systems in 3-D. Besides the primitive lattice (denoted by P or R ), some crystal systems may allow other centered lattices (denoted by C, A, F, or I ). → There’re 14 Bravais lattices (lattice types) in 3-D and 5 in 2-D.

2-D Bravais Lattices

Γ = P = primary, Γ b = C = base centered, Γ v = I = body centered, Γ f = F = face centerd

Cubic Lattices

Index System for Crystal Planes Miller indices of a crystal plane: 1.Express the intercepts of the plane with the crystal axes in units of lattice constants a 1, a 2, a 3. 2.Take the reciprocal of these numbers. 3.Reduce them to integers of the same ratio: (h,k,l). Intercepts at 3a 1, 2a 2, and 2a 3. Reciprocals are (1/3, 1/2, 1/2). Miller indices = (233).

Simple Crystal Structures Sodium Chloride (NaCl)

Cesium Chloride Structure Close-Packed Structure Hexagonal Closed-Packed ABCABC… → fcc ABABAB… → hcp

Diamond Structure Cubic Zinc Sulfide Structure

Direct Imaging of Atomic Structure Scanning Tunneling Microscopy (Chap 19) (111) surface of fcc Pt at 4K. Nearest neighbor distance = 2.78A.

Non-Ideal Crystal Structures Closed packing: ABCABC… → fcc ABABAB… → hcp Random Stacking Polytypism: stacking with long period. E.g., ZnS has >150 polytypes; longest period =360 layers. SiC has >45 polytypes; longest period =594 layers. Cause: spiral steps due to dislocations in growth nucleus.

Crystal Structure Data