Where Drude and Sommerfield Fail

•A metal’s compressibility is pretty well predicted.•Drude was 100s off from cv and thermoelectric, fFD fixed

•Wiedemann-Franz / good at high/low temps only•RH 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 T3 at low temperature?

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] 4f14 5d10 6s2 6p3

[Kr] 4d10 5s2 5p3

[He] 2s2 2p1

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.

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)

Structure of SolidsObjectives

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

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 = N1 a1 + N2 a2 + N3 a3

• a1, a2, a3 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: [n1n2n3]

• [n1n2n3] is the smallest integer of the same relative

ratios. Example: [222] would not be used instead of [111].

• Negative directions can be written as ][ 321 nnn Also sometimes

[-1-1-1]

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]

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.

Crystal PlanesIn 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.

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 sp2-bonded plane.

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)

Crystal Structure 12

Axis X Y Z

Intercept points 1 ∞ ∞

Reciprocals 1/1 1/ ∞ 1/ ∞Smallest

Ratio 1 0 0

Miller İndices (100)

Example-1

(1,0,0)

Crystal Structure 13

Axis X Y Z

Intercept points 1 1 ∞

Reciprocals 1/1 1/1 1/ ∞Smallest

Ratio 1 1 0

Miller İndices (110)

Example-2

(1,0,0)

(0,1,0)

Crystal Structure 14

Axis X Y Z

Intercept points 1 1 1

Reciprocals 1/1 1/1 1/1Smallest

Ratio 1 1 1

Miller İndices (111)(1,0,0)

(0,1,0)

(0,0,1)

Example-3

Crystal Structure 15

Axis X Y Z

Intercept points 1/2 1 ∞

Reciprocals 1/(½) 1/1 1/ ∞Smallest

Ratio 2 1 0

Miller İndices (210)(1/2, 0, 0)

(0,1,0)

Example-4

Note change of axis

orientation

Axis a b c

Intercept points 1 ∞ ½

Reciprocals 1/1 1/ ∞ 1/(½)

Smallest Ratio 1 0 2

Miller İndices (102)

Group: Example-5

Can always shift the plane(note doesn’t make a difference)

Axis a b c

Intercept points -1 ∞ ½

Reciprocals 1/-1 1/ ∞ 1/(½)

Smallest Ratio -1 0 2

Miller İndices (102)

Group: Example-6Yes, I know it’s difficult to visualize. That’s actually part of the point of

doing this one.

(102)

What are the Miller Indices (h k l) of this plane and the direction perpendicular to it?

Reciprocal numbers are: 2

1 ,

2

1 ,

3

1Plane intercepts axes at cba 2 ,2 ,3

Indices of the plane (Miller): (2 3 3)

Indices of the direction: [2 3 3]a

3

2

2

bc

[2,3,3]

Miller indices still apply for a non-cubic system (even if angles are not at 90 degrees)

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

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?

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:

)111(),111(),111(),111(),111(),111(),111(),111(}111{

)001(),100(),010(),001(),010(),100(}100{

]001[],100[],010[],001[],010[],100[100

• 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?

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”

Crystal Structure 24

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

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 = n1 a1 + n2 a2

locates an exactly equivalent point, i.e. a point with the same environment. This is translational symmetry.

The vectors a1 and a2 are known as lattice vectors and (n1, n2) 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)

a2

a1A

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

Sb

S

2D Unit Cell example -(NaCl)

We define lattice points ; these are points with identical environments

Can the box be a unit cell?

Crystal Structure 28

Is this the minimum unit cell size?

Crystal Structure 29

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

Crystal Structure 30

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

Crystal Structure 31

- or if you don’t start from an atom

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

Crystal Structure 33

Why can't the blue triangle be a unit cell?

Lattice Vectors – 3D(same as the directions we already discussed)

A three dimensional crystal is described by 3 fundamental translation vectors a1, a2 and a3.

  R = n1 a1 + n2 a2 + n3 a3 (book)

or r = n1 a + n2 b + n3 c

  (figure)

Remember any direction [n1 n2 n3] is perpendicular to the plane (n1 n2 n3).

Sometimes people will use [h k l] instead of n’s for direction too.