Why Study Solid State Physics?

1. Why Study Solid State Physics?

2. Ideal Crystal • An ideal crystal is a periodic array of structural units, such as atoms or molecules. • It can be constructed by the infinite repetition of these identical structural units in space. • Structure can be described in terms of a lattice, with a group of atoms attached to each lattice point. The group of atoms is the basis.

3. Bravais Lattice • An infinite array of discrete points with an arrangement and orientation that appears exactly the same, from any of the points the array is viewed from. • A three dimensional Bravais lattice consists of all points with position vectors R that can be written as a linear combination of primitive vectors. The expansion coefficients must be integers.

4. Crystal lattice: Proteins

5. Crystal Structure

6. Honeycomb: NOT Bravais

7. Honeycomb net: Bravais lattice with two point basis

8. Crystal structure: basis

9. Translation Vector T

10. Translation(a1,a2), Nontranslation Vectors(a1’’’,a2’’’)

11. Primitive Unit Cell • A primitive cell or primitive unit cell is a volume of space that when translated through all the vectors in a Bravais lattice just fills all of space without either overlapping itself or leaving voids. • A primitive cell must contain precisely one lattice point.

13. Fundamental Types of Lattices • Crystal lattices can be mapped into themselves by the lattice translations T and by various other symmetry operations. • A typical symmetry operation is that of rotation about an axis that passes through a lattice point. Allowed rotations of : 2 π, 2π/2, 2π/3,2π/4, 2π/6 • (Note: lattices do not have rotation axes for 1/5, 1/7 …) times 2π

14. Five fold axis of symmetry cannot exist

15. Two Dimensional Lattices • There is an unlimited number of possible lattices, since there is no restriction on the lengths of the lattice translation vectors or on the angle between them. An oblique lattice has arbitrary a1 and a2 and is invariant only under rotation of π and 2 π about any lattice point.

16. Oblique lattice: invariant only under rotation of pi and 2 pi

17. Two Dimensional Lattices

18. Three Dimensional Lattice Types

19. Wigner-Seitz Primitive Cell: Full symmetry of Bravais Lattice

20. Conventional Cells

21. Cubic space lattices

22. Cubic lattices

23. BCC Structure

24. BCC Crystal

25. BCC Lattice

26. Primitive vectors BCC

27. Elements with BCC Structure

28. Summary: Bravais Lattices (Nets) in Two Dimensions

29. Escher loved two dimensional structures too

30. Summary: Fourteen Bravais Lattices in Three Dimensions

31. Fourteen Bravais Lattices …

32. FCC Structure

33. FCC lattice

34. Primitive Cell: FCC Lattice

35. FCC: Conventional Cell With Basis • We can also view the FCC lattice in terms of a conventional unit cell with a four point basis. • Similarly, we can view the BCC lattice in terms of a conventional unit cell with a two point basis.

36. Elements That Have FCC Structure

37. Simple Hexagonal Bravais Lattice

38. Primitive Cell: Hexagonal System

39. HCP Crystal

40. Hexagonal Close Packing

41. HexagonalClosePacked HCP lattice is not a Bravais lattice, because orientation of the environment Of a point varies from layer to layer along the c-axis.

42. HCP: Simple Hexagonal Bravais With Basis of Two Atoms Per Point

43. Miller indices of lattice plane • The indices of a crystal plane (h,k,l) are defined to be a set of integers with no common factors, inversely proportional to the intercepts of the crystal plane along the crystal axes:

44. Indices of Crystal Plane

45. Indices of Planes: Cubic Crystal

46. 001 Plane

47. 110 Planes

48. 111 Planes

49. Simple Crystal Structures • There are several crystal structures of common interest: sodium chloride, cesium chloride, hexagonal close-packed, diamond and cubic zinc sulfide. • Each of these structures have many different realizations.

50. NaCl Structure