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PX3142, PXT111
Section A
PX3142, PXT111
Section A
Answer all questions
1. Within the limits of the Drude model.
(a) Sketch how the resistivity of a copper:gold alloy would change with increasing
gold content.
Discuss how this model breaks down when compared to experimental measure-
ments. [3]
The Drude-Lorentz formula ((ω)) for the dielectric constant of a solid is:
(ω) = 1 +
ω2p
(ω20 − ω2)− iτω−1
where the plasma frequency ω2p is given by (Nq
2)/(0m), with charge q, electron
mass m, 0 the vacuum permittivity and the concentration of free electrons N
is 8.4 x 1028 m−3. τ is the scattering time of the electron and the resonant
frequency ω0 is the energy gap for the intraband transition.
(b) Assuming that Ohm’s Law applies and the polarisation current density, J =
dP/dt where P is the polarisation calculate the complex frequency dependent
conduction (σ(ω)). [7]
2. (a) Two techniques to measure the Fermi surface are Compton scattering and de
Hass-von Alphen oscillations. Discuss the relative strengths and weaknesses of
the two techniques. [3]
(b) Explain how Landau levels can be used to measure the Fermi Surface. [4]
3 Turn over
PX3142, PXT111
(c) A measurement is made of the magnetisation with the field applied along the
[111] direction. The results (with unmarked axes) are shown below resulting in
a belly (Γ point) and neck (L point) orbit. What is the ratio of the extremal
areas of the surface perpendicular to the Γ−L direction? How should the x and
y axis be labelled? [3]
4
PX3142, PXT111
3. (a) On a simple cubic lattice sketch the Miller planes (300) and (022). [4]
(b) The structure factor (Fhkl) can be expressed as
Fhkl =
n∑
j=1
fje
iGhkl·rj
where (fj) is the atomic form factor, Ghkl is the reciprocal lattice vector and rj
is the real space lattice vector. Consider the following orthorhombic crystal.
Calculate if there is scattering intensity from the (001) and the (100) plane. [6]
5 Turn over
PX3142, PXT111
4. Consider the two-dimensional reciprocal lattice along kx and ky, with real lattice
spacing a and 2a respectively. The Γ,M and X points are also shown on the diagram.
(a) Sketch the first Brillouin zone in the (kx,ky) plane, labelling all axis and inter-
cepts appropriately. [4]
(b) The kinetic energy of a free electron at the corner of the first Brillouin zone
(ΓM) is larger by a factor of α than that of a free electron at the midpoint of
the side of the face of the first Brillouin zone (ΓX). Calculate α. [3]
(c) Consider a one dimensional energy band with a dispersion relation
E(k) = E0 + E1cos
2(
ka
2
)
Calculate the effective mass at k = 0 and k = pi/a. [3]
6
PX3142, PXT111
Section B
Answer two questions
5. (a) Sketch the (111) plane on a face centred cubic (fcc) lattice. [2]
(b) Assuming the Laue condition holds, show that an incoming wave vector k is a
perpendicular bisector of the reciprocal lattice vector G, such that 2k ·G = G2.
[4]
(c) Consider a one dimensional line of alternating atoms ABAB.....AB, with a A-
B bond length of a/2. The form factors are fA and fB for atoms A and B
respectively. The incident beam of x-rays with wavelength λ, is perpendicular
to the line of atoms.
(i) Using an appropriately labelled sketch, show that the condition for
constructive interference is nλ = a cosθ where θ is the angle between the
diffracted beam and the line of atoms and n is an integer. [3]
(ii) Show that the intensity of the diffracted beam is proportional to
∣∣∣fA− fB∣∣∣2
for n odd and
∣∣∣fA + fB∣∣∣2 for n even. [9]
(iii) Explain what happens if fA = fB. [2]
7 Turn over
PX3142, PXT111
6. (a) Consider the Drude model. A material has a scattering lifetime τ , of 3x10−16 s
and the electron velocity is 1x108 cm s−1. Calculate the mean free path in the
material and state whether your answer indicates that the material is a good
metal. [4]
(b) Consider the following diagram for the resistivity, ρ of a sample as a function of
temperature, T .
The resistivity of the sample can be accounted for by considering contributions
from both the impurities (ρimpurity) and the phonons (ρphonons). The number
of electron collisions with phonons is proportional to the number density (ns)
of phonons and is given by:
ns ∝ 1
x3
∫ ∞
0
y2dy
ey − 1
where x = ΘD/T with Θ the Debye temperature and T the temperature and
y=~ωDkbT , where ωD is the Debye frequency.
(i) From the figure explain whether the material is a metal or an insulator. [2]
(ii) Explain the temperature dependence of the impurity resistivity and also
estimate the Debye temperature. [3]
(iii) By considering electron scattering off phonons only, show that:
ρ ∝ T 5
The electron momentum change when scattered off a phonon is small such
that the small angle approximation 1 − cosθ ≈ θ2/2 can be used. Also
assume that
∫∞
0
y2dy
ey−1 = K where K is a constant. [9]
(c) Consider a quantum solid, where the dominant repulsive energy is the zero
point motion (essentially a quantum fluctuation) of atoms. By considering a
1-dimensional lattice of He4 atoms, separated by length L, find the zero point
kinetic energy per particle as a function of L. [2]
8
PX3142, PXT111
7. The dielectric susceptibility due to oscillators of charge q, density N , resonant fre-
quency ω0 and damping γ is:
χ(ω) =
Nq2
0m
(ω20 − ω2 − iγω)−1
(a) Using this write the relative permittivity (ω) given that the background relative
permittivity is b. [2]
(b) Make a labelled sketch of the real and imaginary part of (ω) with and without
damping. [4]
(c) To what value does (ω) tend at frequencies less than the resonant frequency?
[2]
(d) For longitudinal oscillations, what is the resonant frequency without damping?
[2]
(e) The frequency range between the transversal and longitudinal frequency is
highly reflective. Explain why this is so, considering that the intensity reflection
coefficient is given by:
R =
∣∣∣ n˜− 1
n˜+ 1
∣∣∣2
where n˜ is the complex refractive index. [2]
(f) Calculate the wavelength range of high reflectivity created by the optical phonon
resonance in NaCl, which has a lattice constant of a = 0.565 nm, and an fcc
lattice. Assume that the transverse phonon frequency is ω0 = 3 x10
13 Hz, the
oscillator mass is given by the reduced mass of the relative Na-Cl motion, and
a charge of q = e . The mass of Na is 23 a.u. and that of Cl is 35 a.u. and b=
2. [8]
9X Turn over
Section A
Answer all questions
1. Within the limits of the Drude model.
(a) Sketch how the resistivity of a copper:gold alloy would change with increasing
gold content.
Discuss how this model breaks down when compared to experimental measure-
ments. [3]
The Drude-Lorentz formula ((ω)) for the dielectric constant of a solid is:
(ω) = 1 +
ω2p
(ω20 − ω2)− iτω−1
where the plasma frequency ω2p is given by (Nq
2)/(0m), with charge q, electron
mass m, 0 the vacuum permittivity and the concentration of free electrons N
is 8.4 x 1028 m−3. τ is the scattering time of the electron and the resonant
frequency ω0 is the energy gap for the intraband transition.
(b) Assuming that Ohm’s Law applies and the polarisation current density, J =
dP/dt where P is the polarisation calculate the complex frequency dependent
conduction (σ(ω)). [7]
2. (a) Two techniques to measure the Fermi surface are Compton scattering and de
Hass-von Alphen oscillations. Discuss the relative strengths and weaknesses of
the two techniques. [3]
(b) Explain how Landau levels can be used to measure the Fermi Surface. [4]
3 Turn over
PX3142, PXT111
(c) A measurement is made of the magnetisation with the field applied along the
[111] direction. The results (with unmarked axes) are shown below resulting in
a belly (Γ point) and neck (L point) orbit. What is the ratio of the extremal
areas of the surface perpendicular to the Γ−L direction? How should the x and
y axis be labelled? [3]
4
PX3142, PXT111
3. (a) On a simple cubic lattice sketch the Miller planes (300) and (022). [4]
(b) The structure factor (Fhkl) can be expressed as
Fhkl =
n∑
j=1
fje
iGhkl·rj
where (fj) is the atomic form factor, Ghkl is the reciprocal lattice vector and rj
is the real space lattice vector. Consider the following orthorhombic crystal.
Calculate if there is scattering intensity from the (001) and the (100) plane. [6]
5 Turn over
PX3142, PXT111
4. Consider the two-dimensional reciprocal lattice along kx and ky, with real lattice
spacing a and 2a respectively. The Γ,M and X points are also shown on the diagram.
(a) Sketch the first Brillouin zone in the (kx,ky) plane, labelling all axis and inter-
cepts appropriately. [4]
(b) The kinetic energy of a free electron at the corner of the first Brillouin zone
(ΓM) is larger by a factor of α than that of a free electron at the midpoint of
the side of the face of the first Brillouin zone (ΓX). Calculate α. [3]
(c) Consider a one dimensional energy band with a dispersion relation
E(k) = E0 + E1cos
2(
ka
2
)
Calculate the effective mass at k = 0 and k = pi/a. [3]
6
PX3142, PXT111
Section B
Answer two questions
5. (a) Sketch the (111) plane on a face centred cubic (fcc) lattice. [2]
(b) Assuming the Laue condition holds, show that an incoming wave vector k is a
perpendicular bisector of the reciprocal lattice vector G, such that 2k ·G = G2.
[4]
(c) Consider a one dimensional line of alternating atoms ABAB.....AB, with a A-
B bond length of a/2. The form factors are fA and fB for atoms A and B
respectively. The incident beam of x-rays with wavelength λ, is perpendicular
to the line of atoms.
(i) Using an appropriately labelled sketch, show that the condition for
constructive interference is nλ = a cosθ where θ is the angle between the
diffracted beam and the line of atoms and n is an integer. [3]
(ii) Show that the intensity of the diffracted beam is proportional to
∣∣∣fA− fB∣∣∣2
for n odd and
∣∣∣fA + fB∣∣∣2 for n even. [9]
(iii) Explain what happens if fA = fB. [2]
7 Turn over
PX3142, PXT111
6. (a) Consider the Drude model. A material has a scattering lifetime τ , of 3x10−16 s
and the electron velocity is 1x108 cm s−1. Calculate the mean free path in the
material and state whether your answer indicates that the material is a good
metal. [4]
(b) Consider the following diagram for the resistivity, ρ of a sample as a function of
temperature, T .
The resistivity of the sample can be accounted for by considering contributions
from both the impurities (ρimpurity) and the phonons (ρphonons). The number
of electron collisions with phonons is proportional to the number density (ns)
of phonons and is given by:
ns ∝ 1
x3
∫ ∞
0
y2dy
ey − 1
where x = ΘD/T with Θ the Debye temperature and T the temperature and
y=~ωDkbT , where ωD is the Debye frequency.
(i) From the figure explain whether the material is a metal or an insulator. [2]
(ii) Explain the temperature dependence of the impurity resistivity and also
estimate the Debye temperature. [3]
(iii) By considering electron scattering off phonons only, show that:
ρ ∝ T 5
The electron momentum change when scattered off a phonon is small such
that the small angle approximation 1 − cosθ ≈ θ2/2 can be used. Also
assume that
∫∞
0
y2dy
ey−1 = K where K is a constant. [9]
(c) Consider a quantum solid, where the dominant repulsive energy is the zero
point motion (essentially a quantum fluctuation) of atoms. By considering a
1-dimensional lattice of He4 atoms, separated by length L, find the zero point
kinetic energy per particle as a function of L. [2]
8
PX3142, PXT111
7. The dielectric susceptibility due to oscillators of charge q, density N , resonant fre-
quency ω0 and damping γ is:
χ(ω) =
Nq2
0m
(ω20 − ω2 − iγω)−1
(a) Using this write the relative permittivity (ω) given that the background relative
permittivity is b. [2]
(b) Make a labelled sketch of the real and imaginary part of (ω) with and without
damping. [4]
(c) To what value does (ω) tend at frequencies less than the resonant frequency?
[2]
(d) For longitudinal oscillations, what is the resonant frequency without damping?
[2]
(e) The frequency range between the transversal and longitudinal frequency is
highly reflective. Explain why this is so, considering that the intensity reflection
coefficient is given by:
R =
∣∣∣ n˜− 1
n˜+ 1
∣∣∣2
where n˜ is the complex refractive index. [2]
(f) Calculate the wavelength range of high reflectivity created by the optical phonon
resonance in NaCl, which has a lattice constant of a = 0.565 nm, and an fcc
lattice. Assume that the transverse phonon frequency is ω0 = 3 x10
13 Hz, the
oscillator mass is given by the reduced mass of the relative Na-Cl motion, and
a charge of q = e . The mass of Na is 23 a.u. and that of Cl is 35 a.u. and b=
2. [8]
9X Turn over
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