Prolog

1. Prolog Text Book: C.Kittel, "Introduction to Solid State Physics", 8th ed.,Wiley (2005) Website: http://ckw.phys.ncku.edu.tw Homework submission: class@ckw.phys.ncku.edu.tw Grades: Exercises: 50% MidTerm: 20% Final: 30%

2. 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

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

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

5. Basis:

6. Primitive Lattice Cell Recapitulation: Parallelepiped defined by axes ai 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 ai 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.

7. Fundamental Types of Lattices Only rotations Cn 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.

8. 2-D Bravais Lattices

9. Γ = P = primary , Γb = C = base centered, Γv = I = body centered, Γf = F = face centerd

10. Cubic Lattices

12. Index System for Crystal Planes Miller indices of a crystal plane: Express the intercepts of the plane with the crystal axes in units of lattice constants a1 , a2 , a3. Take the reciprocal of these numbers. Reduce them to integers of the same ratio: (h,k,l). Intercepts at 3a1, 2a2, and 2a3. Reciprocals are (1/3, 1/2, 1/2). Miller indices = (233).

13. Simple Crystal Structures Sodium Chloride (NaCl)

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

15. Diamond Structure Cubic Zinc Sulfide Structure

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

17. 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.

18. Crystal Structure Data