1. Where Drude and Sommerfield Fail A metal’s compressibility is pretty well predicted. Drude was 100s off from c v and thermoelectric, f FD fixed Wiedemann-Franz  /  good at high/low temps only R H depends on temp and field (sign Al), alkalis close Why does DC conductivity depend on T? (have to add  ) Current density isn’t always parallel to E field. Why? Optical properties seem much more complex. Color? Why does heat capacity go as T 3 at low temperature?

2. Fundamental Questions Remaining What determines the number of conduction electrons per atom? Some elements (like iron) have multiple possible valences. Why aren’t boron, bismuth and antimony good conductors? [Xe] 4f 14 5d 10 6s 2 6p 3Xe [Kr] 4d 10 5s 2 5p 3Kr [He] 2s 2 2p 1He

3. Limitations of the Drude Model—and Beyond The Drude model, augmented by quantum mechanics, was extremely successful in accounting for many of the properties of metals. Some flawed assumptions behind the FEG model: 1.The free-electron approximation The positive ions act only as scattering centers and is assumed to have no effect on the motion of electrons between collisions. 2. The independent electron approximation Interactions between electrons are ignored. Considerable progress comes from abandoning only the free- electron approximation in order to take into account the effect of the lattice on the conduction electrons.

4. What is crystallography? The branch of science that deals with the geometric description of crystals and their internal arrangement. Platinum Platinum surface Crystal lattice and structure of Platinum (scanning tunneling microscope)

5. Structure of Solids Objectives By the end of this section you should be able to: Use correct notation for directions/planes/families Find the distance between planes (when angles 90  ) Identify a unit cell in a symmetrical pattern Identify a crystal structure Define cubic, tetragonal, orthorhombic and hexagonal unit cell shapes

6. Crystal Direction Notation Figure shows [111] direction Choose one lattice point on the line as an origin (point O). Choice of origin is completely arbitrary, since every lattice point is identical. Then choose the lattice vector joining O to any point on the line, say point T. This vector can be written as; R = N 1 a 1 + N 2 a 2 + N 3 a 3 a 1, a 2, a 3 often written as a, b, c or even x, y, z To distinguish a lattice direction from a lattice point (x,y,z), the triplet is enclosed in square brackets and use no comas. Example: [n 1 n 2 n 3 ] [ n 1 n 2 n 3 ] is the smallest integer of the same relative ratios. Example: [222] would not be used instead of [111]. Negative directions can be written as Also sometimes [-1-1-1]

7. X = -1, Y = -1, Z = 0 [110] X = 1, Y = 0, Z = 0 [1 0 0] Group: Determine the crystal directions X = 1, Y = ½, Z = 0 [1 ½ 0] [2 1 0] X = ½, Y = ½, Z = 1 [½ ½ 1] [1 1 2] [210]

8. Group: Determine the Crystal Direction X =-1, Y = 1, Z = -1/6 [-1 1 -1/6] [6 6 1] We can move vectors to the origin as long as don’t change direction or magnitude. Now let’s do one that’s a little harder.

9. Crystal Planes In Chapter 5, but useful to know now. Within a crystal lattice it is possible to identify sets of equally spaced parallel planes, called lattice planes. The density of lattice points on each plane of a set is the same. b a b a A couple sets of planes in a 2D lattice.

10. Why are planes in a lattice important? (A) Determining crystal structure * Diffraction methods measure the distance between parallel lattice planes of atoms to determine the lattice parameters, etc. (B) Plastic deformation * Plastic deformation in metals occurs by the slip of atoms past each other. * This slip tends to occur preferentially along specific crystal-dependent planes. (C) Transport Properties * In certain materials, atomic structure in some planes causes the transport of electrons and/or heat to be particularly rapid in that plane, and relatively slow not in the plane. Example: Graphite: heat conduction is more in sp 2 -bonded plane.

11. Miller Indices ( h k l ) Miller Indices are a vector representation for the orientation of an a plane in a crystal lattice and are defined as the reciprocals of the fractional intercepts which the plane makes with the crystallographic axes. To determine Miller indices of a plane, take the following steps: 1) Determine the intercepts of the plane along each of the three crystallographic directions 2) Take the reciprocals of the intercepts 3) If fractions result, multiply each by the denominator of the smallest fraction (multiply again if needed)

12. Crystal Structure12 Axis XYZ Intercept points 1 ∞∞ Reciprocals 1/11/ ∞ Smallest Ratio 100 Miller İndices (100) Example-1 (1,0,0)

13. Crystal Structure13 Axis XYZ Intercept points 11 ∞ Reciprocals 1/1 1/ ∞ Smallest Ratio 110 Miller İndices (110) Example-2 (1,0,0) (0,1,0)

14. Crystal Structure14 Axis XYZ Intercept points 111 Reciprocals 1/1 Smallest Ratio 111 Miller İndices (111) (1,0,0) (0,1,0) (0,0,1) Example-3

15. Crystal Structure15 Axis XYZ Intercept points 1/21 ∞ Reciprocals 1/(½)1/11/ ∞ Smallest Ratio 210 Miller İndices (210) (1/2, 0, 0) (0,1,0) Example-4

16. Note change of axis orientation Axis abc Intercept points 1 ∞ ½ Reciprocals 1/11/ ∞ 1/(½) Smallest Ratio 102 Miller İndices (102) Group: Example-5 Can always shift the plane (note doesn’t make a difference)

17. Axis abc Intercept points ∞ ½ Reciprocals 1/-11/ ∞ 1/(½) Smallest Ratio 02 Miller İndices (102) Group: Example-6 Yes, I know it’s difficult to visualize. That’s actually part of the point of doing this one. (102)

18. What are the Miller Indices ( h k l ) of this plane and the direction perpendicular to it? Reciprocal numbers are: Plane intercepts axes at Indices of the plane (Miller): (2 3 3) Indices of the direction: [2 3 3] 3 2 2 [2,3,3] Miller indices still apply for a non-cubic system (even if angles are not at 90 degrees)

19. Miller Indices ( h k l ), Lattice directions (a, b, c)=(x,y,z) If you do have 90 degree angles, use this formula for distance between planes

20. What is the distance between the (111) planes on a cubic lattice of lattice parameter a? Find the distance between (1 2 3) in a cubic lattice?

21. Indices of a Family or Form Sometimes several nonparallel planes may be equivalent by virtue of symmetry, in which case it is convenient to lump all these planes in the same Miller Indices, but with curly brackets. Thus indices {h,k,l} represent all the planes equivalent to the plane (hkl) through rotational symmetry. Similarly, families of crystallographic directions are written as:

22. Crystal Lattice = an infinite array of points in space Each lattice point has identical surroundings. Arrays are arranged exactly in a periodic manner. Could the centers of both Na and Cl be lattice points at the same time?

23. Crystal Structure =Lattice +Basis Crystal structure can be obtained by attaching atoms, groups of atoms or molecules, which are called the basis (AKA motif) to the lattice sides of the lattice point. AKA means “also known as”

24. Crystal Structure24 Crystal structure Don't mix up atoms with lattice points! Lattice points are infinitesimal points in space Atoms can lie at positions other than lattice points Crystal Structure = Crystal Lattice + Basis

25. Translational Lattice Vectors – 2D A Bravais lattice is a set of points such that a translation from any point in the lattice by a vector; R = n 1 a 1 + n 2 a 2 locates an exactly equivalent point, i.e. a point with the same environment. This is translational symmetry. The vectors a 1 and a 2 are known as lattice vectors and (n 1, n 2 ) is a pair of integers whose values depend on the lattice point. What are the lattice points (integers) for points D, F and P, where point A is the origin? P Point D (n1, n2) = (0,2) Point F (n1, n2) = (0,-1) Point P (n1, n2) = (3,2) a2a2 a1a1 A

26. 26 Unit Cell in 2D The smallest component of the crystal (group of atoms, ions or molecules), which when stacked together with pure translational repetition reproduces the whole crystal. S S The choice of unit cell is not unique. a S b S

27. 2D Unit Cell example -(NaCl) We define lattice points ; these are points with identical environments Can the box be a unit cell?

28. Crystal Structure28 Is this the minimum unit cell size?

29. Crystal Structure29 Choice of origin is arbitrary - lattice points need not be atoms - but unit cell size should always be the same.

30. Crystal Structure30 This is also a unit cell - it doesn’t matter if you start from Na or Cl

31. Crystal Structure31 - or if you don’t start from an atom

32. Bravais Lattices in 2D In 2D there are five ways to order atoms in a lattice Primitive unit cell: contains only one atom (but 4 points?) Are the dotted lattices primitive? Non-primitive unit cells sometimes useful if orthogonal coordinate system can be used Special case where angles go to 90  a=b Special case where point halfway a=b

33. Crystal Structure33 Why can't the blue triangle be a unit cell?

34. Lattice Vectors – 3D (same as the directions we already discussed) A three dimensional crystal is described by 3 fundamental translation vectors a 1, a 2 and a 3. R = n 1 a 1 + n 2 a 2 + n 3 a 3 (book) or r = n 1 a + n 2 b + n 3 c (figure) Remember any direction [n 1 n 2 n 3 ] is perpendicular to the plane (n 1 n 2 n 3 ). Sometimes people will use [h k l] instead of n’s for direction too.