Ride or Die

Question 1

structure factor

Answer 1

factor within diffraction scattering intensity equation that accounts for scattering power of basis atoms within conventional unit cell

Question 2

what effect does structure factor have on diffraction pattern?

Answer 2

structure factor alters relative intensity of scattering from different reciprocal lattice vector

Question 3

assumptions of tight binding theory of atomic bonding? (3)

Answer 3

1. surrounding atoms only weakly perturb electrons of each atom (electrons are tightly bound to atomic nucleus)
2. electron states in a solid are well approximated by sums of atomic orbitals and the wavefunction has the periodicity of lattice + satisfies Bloch’s equation
3. coulomb interaction between electrons is ignored

Question 4

Born-Oppenhaimer approximation

Answer 4

the fixing of nuclei positions in a crystal with respect to the motion of electrons

Question 5

why is Born-Oppenheimer a good approximation?

Answer 5

electrons are 2000x smaller than a proton and 50x slower

Question 6

Bloch’s theorem states…

Answer 6

eigenfunctions of a Hamiltonian = plane wave x periodic function (translational periodicity)

Question 7

dispersion relation

Answer 7

relation between wavevector and associated energy of a particle

Question 8

Fermi energy

Answer 8

chemical potential at T=0. energy level at which occupied energy states and unoccupied energy states are divided

Question 9

Fermi-surface

Answer 9

surface in reciprocal k-space for which all points on the surface have magnitude equal to that of the Fermi wavevector

Question 10

allowed k-state

Answer 10

value for wavevector in reciprocal space that satisfies the constraints places on the system

Question 11

why does x-ray diffraction measured from a crystalline material in powder form produce conical shaped beams?

Answer 11

crystalline powers contains polycrystalline randomly oriented crystal grains so they can be oriented in any possible direction. there is always some angle at which the crystal can be oriented. the resulting diffraction pattern = combination of diffraction for all possible orientations of a single crystal. multiple conical beams are observed where each corresponds to diffraction from a single Bragg reflection from a single crystal

Question 12

crystal properties that can be determined from X-ray diffraction?

Answer 12

reciprocal lattice vector length, crystal structure (in cubic case), temperature and pressure dependence

Question 13

energy of free electron

Answer 13

E = hbar^2k^2/2m_e

Question 14

assumptions for free electron (jellium) model? (3)

Answer 14

1. QM obeyed
2. electrons are free to move inside the material
3. each electron only feels an average potential from the others (that’s included in total potential energy)

Question 15

electron mobility

Answer 15

drift velocity per unit applied electric field = parametrises how fast electrons travel in an electric field

Question 16

electron conductivity

Answer 16

electric field inside a material per unit current density

Question 17

steady state condition?

Answer 17

m dv_d/dt = 0

Question 18

how many atoms in unit cell for FCC lattice?

Answer 18

4

Question 19

how to find electron number density knowing type of lattice and side length of unit cell?

Answer 19

n = atoms in unit cell/(length of unit cell)^3

Question 20

paramagnetism

Answer 20

magnetic phenomenon where already existing magnetic moments align parallel to B field. can arise from atoms/ions with permanent magnetic moments or free electrons = magnetic response linear with B field at low magnetic fields with a positive susceptibility

Question 21

equation describing paramagnetism

Answer 21

mu_0M = XB

Question 22

ferromagnetism

Answer 22

significant observable magnetic permeability allowing the material to form a permanent magnet = permanent magnetic dipoles even in absence of external B field for T<Tc. above Tc = material is paramagnetic with susceptibility that doesn’t follow Curie’s law

Question 23

magnetic susceptibility

Answer 23

= X = parametrises magnetisation M due to an applied magnetic field B

Question 24

what are 2 physical mechanisms described by:
Be = B0 + λM ?

Answer 24

1. mean field approximation to the exchange interaction between electrons
2. also describes magnetic moment due to all other magnetic moments in the material in form of Be

Question 25

Curie-Weiss law =

Answer 25

X = C/(T-Tc)

Question 26

how to find electron number density knowing type of lattice and lattice constant, a in Armstrong?

Answer 26

n = number of atoms in unit cell/ (a x 10^-10)^2

Question 27

where does free electron model breakdown?

Answer 27

it does not allow for electronic band gap

Question 28

example of free electron model breakdown with Copper

Answer 28

plasma frequency found is bigger than visible light frequency, but we know copper has a reddish colour. copper has interbond transitions in visible range of EM spectrum which results in absorption of light at these frequencies

Question 29

quasi-particle for charge oscillations

Answer 29

plasmon. has energy hbar*omega_p

Question 30

why do we use quasiparticles to describe excitations in solids? (2)

Answer 30

1. allows description of collective behaviour of a material as though it's due to a single particle 2. can assign QM values of real particles to quasi-particles

Question 31

quasi-particles examples

Answer 31

phonon -> vibration of atoms in solids electron hole -> nearly full electronic band electron-quasi-particle -> electron exciton -> bound electron-hole

Question 32

how is a covalent bond formed between two neutral atoms?

Answer 32

electrons are equally shared between the two atoms. nucleus of atom A attracts electrons of atom B and the atoms are bound together by Coulomb forces (electrons and nuclei repel each other)#

Question 33

bonding orbital

Answer 33

sum of linear combinations. maximises charge density between nuclei and forms a stable bound state

Question 34

antibonding orbital

Answer 34

difference of linear combinations. zero charge density at midpoint

Question 35

why do we get bonding and antibonding orbitals?

Answer 35

when electron distributions around 2 neutral atoms overlap we need new orbitals to describe those states

Question 36

diamagnetism

Answer 36

repelled by a B field. no unpaired spins or permanent moments in structure. distortion of electron orbits produces weak negative susceptibility

Question 37

Bloch wavevector

Answer 37

describes a phase variation of wavefunction from cell to cell

Question 38

crystal momentum =

Answer 38

ħk

Question 39

crystal momentum

Answer 39

behaves like momentum of an excitation state modified by diffraction of the lattice

Question 40

why can all possible Bloch states be described by wavevectors within first Brillouin zone?

Answer 40

k and k+G describe the same phase variation from cell to cell. exp(i(k+G).R) = exp(ikRn)exp(iGRn)= exp(ik.R) x 1. all distinct values of k are found in 1 unit cell of reciprocal lattice

Question 41

how does structure factor affect directions of scattered beams?

Answer 41

no effect

Question 42

how does structure factor affect intensities of scattered beams?

Answer 42

controls distribution of intensity between different order G

Question 43

when do 'missing order' appear in a diffraction pattern?

Answer 43

when there are several identical atoms in the atomic basis. they result from destructive interference of waves scattered out of phase from planes of atoms midway between lattice points

Question 44

layers of atoms in square, lattice constant a. primitive translation vectors of the space lattice?

Answer 44

a1 = ax, a2 = ay

Question 45

layers of atoms in square, lattice constant a. primitive translation vectors of the reciprocal lattice?

Answer 45

a1* = 2π/a x, a2* = 2π/a y

Question 46

what's the electronic ground state of a 2D free electron fermi gas like?

Answer 46

1. due to Pauli exclusion principle, 1 electron can occupy a single particle orbital and 2 electrons can occupy each k-state (spin-1/2) 2. allowed k-states form a rectangular lattice (2π)^-2 per unit area k-space per unit area box 3. ground state of fermi gas fill up single electron states in order of increasing energy

Question 47

what happens to Fermi function at T=0?

Answer 47

it becomes a step function

Question 48

when does a Fermi surface just touch 1st Brillouin zone?

Answer 48

1 bisector of primitive reciprocal lattice vector = Brillouin zone boundary

Question 49

how will diffraction by atom affect the shape of fermi surface?

Answer 49

fermi surface is drawn out into nucleus to track BZ boundaries (dispersion relation affected by diffraction close to boundary)

Question 50

condition for a crystal to form a diffracted beam

Answer 50

wavevector of diffracted radiation changes by an amount equal to a reciprocal lattice translation vector of crystal

Question 51

condition for a crystal to forma diffracted beam =

Answer 51

for some integer p, G.Rn = 2πp

Question 52

why are neutrons rather than X-rays used to investigate magnetic ordering of crystals and to investigate lattice vibrations? (2)

Answer 52

1. magnetic moment of neutrons interacts differently with up and down spins 2. at a given energy, neutrons has a much larger k-vector than X-ray (as it has mass) so allows for inelastic scattering and probing vibrational quarks across BZ

Question 53

what's the defining property of a semiconductor?

Answer 53

dispersion relation has an energy gap at the BZ boundary, separating filled and unfilled energy bands in ground state. Eg > kT

Question 54

intrinsic semiconductor

Answer 54

no defect/impurity state concentrations = crystal is insulating

Question 55

extrinsic semiconductor

Answer 55

defect states are thermally ionised = crystal is conducting

Question 56

n-type semiconductor

Answer 56

defects use donors; levels below conduction band edge

Question 57

p-type semiconductor

Answer 57

defects use acceptors; levels above valence edge

Question 58

why are minority carriers always present in an extrinsic semiconductor?

Answer 58

electron-hole pairs are thermally generated and recombined. they balance in equilibrium. product of electron and hole concentrations = constant at given T. even in presence of strong doping and large electron/hole concentration, minority population persists

Question 59

why do extrinsic carriers in a semiconductor doped with acceptor impurities appear to be positively charged?

Answer 59

1. ionised acceptor creates a vacant state in otherwise filled valence band 2. E field applied in x-direction accelerates all band electrons to states of higher k_x 3. sum of filled band k = 0. properties of band with unfilled state at k_x are the same as properties of unpaired state at -k_x 4. unpaired state accelerated in k-space in the opposite direction to the unfilled state