Condensed Matter

Question 1

What is a crystal

Answer 1

A crystal is a solid material whose constituent atoms feature long range structure.

Question 2

What is a lattice?

Answer 2

A lattice is an infinite array of points in space, each having identical surroundings to the other points.

Question 3

How is a lattice defined?

Answer 3

By primitive unit vectors, 2 needed in 2D and 3 in 3D.

They are not unique
Their lengths are called lattice constants or lattice parameters
Can also be specified by the angle between them.

Question 4

What is a unit cell, what is the difference between a primitive unit cell and a conventional unit cell?

Answer 4

A unit cell is a region in space which can be repeatedly stacked to assemble the lattice without gaps ( can tessellate ).

A primitive unit cell is a unit cell which contains only 1 lattice point.

A conventional unit cell is a cell chosen in preference to the primitive cell eg, because it is more symmetric.

Question 5

What is a wigner-Seitz primitive unit cell?

Answer 5

Contains all the space closer to one lattice point than any other

Question 6

What is a Bravais lattice?

Answer 6

Any of the 14 possible 3D configuration of points used to describe orderly arrangement of atoms in crystals.

In 2D any of the 5 symmetrically distinct points of configuration used to describe orderly the arrangement of atoms in crystals.

Question 7

Given a regularly repeating arrangement of atoms, how can you determine where the lattice points describing the crystal are?

Answer 7

Pick points with translational symmetry, i.e moving at that point, the immediate surroundings look identical to another lattice point.

Question 8

What is a basis and how do you specify one?

Answer 8

The basis is the identical assembly of atoms associated with each lattice point.

The basis is specified as fractions of the unit cell vectors, you pick out the set of atoms for the basis by looking at all the atoms which are not translationally symmetric within the primitive unit cell.

Question 9

How do you pick primitive lattice vectors, and how are these different to ordinary lattice vectors?

Answer 9

A primitive lattice vector is simply a vector which moves from one lattice point to another.

The lattice vectors can always be expressed as a linear combination of primitive lattice vectors.

Question 10

How do you know how many atoms are needed to be specified in the basis?

Answer 10

The number of whole atoms within the unit cell being used is the number of atoms needed by the basis

Question 11

How are directions specified in crystals

Answer 11

Along vectors R = h A + k B + l C (capitals are vectors), where h k l are the miller indices.

Standard notation :
R = [h k l]
Set of symmetric directions , eg
<1 0 0> = [1 0 0 ][1(bar) 0 0 ][0 1 0] [0 1bar 0] [0 0 1] [ 0 0 1bar].

Question 12

How are atomic planes specified?

Give any relevant formula for the indices given.

Answer 12

By miller indices, (h k l).
h=alpha a / p, k = alpha b / q and l = alpha c / r, where p q r are the axis intercepts of a, b and c respectively.
Alpha is a scale factor so there is no common denominator (i.e integer values for the indices)

Question 13

What are the notational differences relating to the miller indices describing directions and the miller indices describing planes?

Answer 13

Directions:
R=[ h k l ]
Family of symmetric directions given by < h l k> .
Planes
( h k l )
Family of symmetric planes given by {h k l}.

Question 14

What is meant by the packing fraction

Answer 14

Fraction of space filled by touching spheres.

Question 15

What is Bragg’s law, explain any symbols used.

Answer 15

nlamda = 2dsintheta

n is the diffraction order
Lambda is the wavelength of light
d is the distance between planes
Bragg angle of diffraction

Question 16

How many sets of diffraction lines do you need to identify what lattice structure the crystal corresponds too?

Answer 16

As many terms as are needed till there is a difference in the series of ratios.

Question 17

Explain why for a CsCl crystal, radiation scattering off a plane is only in phase for even n, and out of phase for odd n?

Answer 17

As the path difference between 2 planes of Cl atoms is exactly half the path difference between the planes of the Cs atoms, if there is an odd number of wavelengths as the path difference between the planes of Cs atoms, then Cl will have a path difference of ??? If odd number of wavelengths then wont there be destructive interference anyway, no wave to leave and interference with the Cs???

Question 18

Explain why is it hard to distinguish the presence of hydrogen using the scattering spectra of compounds

Answer 18

The strength of the Bragg’s law effect is proportional to the number of electrons in the atom for x ray diffraction. Hydrogen only have 1 electron and so if in a compound with a heavy element, the x ray diffraction spectra of the compound will look identical to the x ray diffraction spectra of the heavy element alone, with the same lattice structure.

Question 19

Explain why KCl and KBr have different diffraction spectra but the same lattice structure

Answer 19

KCl ionic compound is formed by the exchange of electrons. As the K ions have the same number of electrons as in the Cl ions, the strength of the diffracted x rays will be the same, so when the diffracted x rays are out of phase no signal will be detected, whereas, in the KBr, when the signals are out of phase,they don’t completely cancel since the ions have different numbers of electrons.
This means you will see different spectra for the compounds, despite them having the same lattice structure.

Question 20

Explain the benefits of neutron scattering

Answer 20

The neutron form factors do not change by much between heavy and light elements and so the effect of neutron diffraction will be as pronounced for hydrogen , as it would be for a heavier elements. Making it easier to detect lighter elements in compounds.

Question 21

What are form factors?

Answer 21

Atomic scattering angles amplitudes

Question 22

How can you determine the basis from diffraction?

Answer 22

Looking at the relative intensities of the diffraction lines tells us what Elements can be present

Question 23

Explain how the Bragg condition of diffraction is identical to the laue condition.

Answer 23

Bragg = > n lambda = 2d Sin theta

Question 24

What are the laue conditions?

Answer 24

They are the set of equations the scattering vector must satisfy in order for constructive interference at the detector.

Question 25

What is the reciprocal lattice and how does a diffraction experiment measure it?

Answer 25

The reciprocal lattice is the k space equivalent of a real lattice. It is a useful construct because every scattering vector that results in a diffraction spot is a reciprocal lattice vector. Consequently the ?

Question 26

Prove that the reciprocal lattice of a simple cubic structure is still simple cubic.

Answer 26

Use the equations for A B and C, and use a = ai , b = aj and c = ak. Will end up with another set of orthogonal axis vectors, just with a different scaling.

Question 27

What the key assumptions made in the drude model?

Answer 27

Between collisions, electron interactions with other electrons and the ions are neglected. These are the independent and free electron assumptions.

Collisions are instantaneous - abrupt changes in velocity.

Electrons collide with a probability of 1/tau per unit time.

Electrons are always in local thermal eq.

Question 28

What does the drude model successfully model and where does it break down.

Answer 28

Can predict ohms law.
Predicts wiedemann franz law

L’s
Fails to predict the mean free path measured at low T’s

Fails to predict that heat capacities are independent of no of valence electrons

Hall coefficients dependence of sample preparation and temperature, the sign change of the hall coefficient for certain species.

Off by a factor of 2 for the Lorenz number

Properties of alloys, such as alloy resistivity.

Magnetic susceptibility

Question 29

What are the basic assumptions of the sommerfield model?

Answer 29

• Free electrons are valence electrons of composing atoms
• A valence electron in metal finds itself in the field of all ions and that of other electrons. The mutual repulsion between the electrons is neglected and the potential filed representing the attractive interaction of ions is assumed to be completely uniform every where inside the solid.
• Distribution of energy in an electron gas obeys the Fermi-Dirac quantum statistics.
• Only electrons close to the Fermi energy are scattered.
  The energy levels are filled in accordance with the Pauli’s exclusion principle according to which an energy level can accommodate at the most two electrons, one with spin up and one with spin down.

Question 30

How does the Pauli exclusion principle explain why valence electrons do not contribute to a crystals specific heat capacity?

Answer 30

Because electrons can not occupy the same quantum states. Only electrons that have an energy within KT of the Fermi energy can be excited to higher states with the new thermal energy. However the fraction of such electrons is T/Tf, this is tiny. Thus, as not all the electrons are excited to the higher energy, the specific heat capacity will be much lower.

Question 31

Explain why the allowed k vectors for a free electron can be represented through a simple cubic lattice?

Answer 31

Solving the Schrödinger equation with periodic boundary conditions gives the quantised k vectors. The separation between subsequent k vectors is constant and so a simple cubic lattice, where the lattice points represent available k vectors can model the available states.

Question 32

What is the fermi sphere?

Answer 32

Taking a simple cubic lattice to represent all the available k states of an electron, filling states from the lowest energy level with N electrons fills the simple lattice from the origin outwards. The radius of the sphere is described by the energy available at T=0, the Fermi energy. The most energetic electron will have the Fermi energy and be at the fermi radius away from the origin.

Question 33

Explain the concept of a density of states

Answer 33

The number of states available per unit energy, at a given energy.

The number of states available with an energy between e and e+de = g(e)de

Question 34

Explain the concept of electrical conduction and scattering using fermi spheres.

Answer 34

F_ext=-eE

F_ext = dp/dt = d hbar k /dt
Solve for k: k = -eE t/ h bar
K increases in the direction opposite to the applied E field.

As average K in the fermi sphere, once shifted away from the origin, is no longer 0, there is a net movement of charge. This is a current. For a steady state current to exist, there must be scattering to opposite this movement. These processes will return the sphere back to the origin when the applied field is turned off.

Question 35

If a fermi sphere shows more thermally excited electrons on the right side, what does this imply about the metal sample?

Answer 35

The electrons are therefore on average moving to the right, this means the left side is hotter.

Question 36

Explain the quantum free electrons gas models explanation on the effect of sample preparation on resistivity

Answer 36

As there are different scattering processes, there should be different scattering rates. The scattering rates do depend on the purity of the samples, so the resitivities, dependence is explained.

Question 37

Explain bloch’s theorem

Answer 37

Due to the symmetry associated with crystals, psi (x+a)^2 = psi(x)^2, ‘a’ being the lattice parameter. This gives Psi (x) a form= u(x)e^ikx, where u(x) = u(x+a).

Question 38

What is a brillouin zone?

Answer 38

locus of points in reciprocal space that are closer to the origin of the reciprocal lattice than they are to any other reciprocal lattice points.

Question 39

Why can each band only hold 2 electrons from each unit cell?

Answer 39

In each brillouin zone there are N available k states, where N is the number of unit cells. Each k state can take 2 electrons. Therefore if each unit cell provides 2 electrons, there is 1 to 1 ratio between number of available states in each brillouin zone and the number of electrons.

Question 40

What is the effect of imposing a periodic potential on our free electron gas model?

Answer 40

Energy bands

Question 41

Why do we use periodic boundary conditions when solving for the electrons wave function?

Answer 41

To get travelling wave solutions, rather than standing wave ones.
These solutions allow for transport properties to be modelled.

Question 42

Describe the effect of impurities on a samples resistivity.

Answer 42

It translates the resistivity function by an amount equal to the difference in scattering times between the pure sample and and impure sample.

Question 43

What does the free electron model solve over the drude model?

Answer 43

+
Explains the lack of heat capacities dependence on number of valence electrons

Mean free path in pure samples

Lorenz number

-
AT LOW T, T dep on resistivity.

Properties of alloys
Optical properties
Effects of orientation.

Question 44

Explain the concept of negative effective mass

Answer 44

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