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计算代写-PXT111

时间：2021-01-19

PX3142, PXT111

CARDIFF UNIVERSITY

EXAMINATION PAPER

Academic year: 2020-21

Examination Period: Practise

Examination Paper Number: PX3142, PXT111

Examination Paper Title: Condensed Matter Physics

Duration: 2 hours

Additional time: 1 hour to scan and submit your solutions

Structure of Examination Paper:

There are six pages.

There are SEVEN questions in total.

This examination paper is divided into two sections.

There are no appendices.

The maximum mark for the examination paper is 80 (80% of the module mark)

The mark obtainable for a question or part of a question is shown in brackets alongside

the question.

Instructions for Completing the Assessment:

• Answer ALL questions from Section A, and TWO questions from Section B. If addi-

tional Section B questions are attempted be clear which should be marked, otherwise

only the first two written answers will be marked. You are advised to spend equal

amounts of time on Section A and B.

• Use black/blue ink and dark pencils, on white paper. You may use graph paper where

necessary.

• Before the end of the exam duration, allow time to use Microsoft-Lens, Adobe PDF

scan (or equivalent software) to photograph your answers and convert into a PDF to

upload.

• Make sure the pages of your answers are in the correct order and orientation.

• Include your Student Number clearly on the first page, and in the filename of your

PDF.

• Submit your PDF answer file before the end of the duration by:

uploading as assignment on PX3142 Learning Central module,

1

PX3142, PXT111

or emailing to physics-exams@cardiff.ac.uk if you think the upload did not work.

Notice to Students:

• This is an “open book” assessment, you may make use of any resource.

• The use of calculators / computers is permitted.

• You are expected to work without communication with any other person.

• By submitting your answers, you declare that your submission (and all the material

contained in it), is your own work, and you have not shared it with anyone else.

• The examiners reserve the right to further question you, if necessary, by pre-arranged

telephone calls, so you can further explain the answers you have provided in your

submission. The University’s plagiarism detection software may also be used to check

originality of, and any collusion in, the submitted work.

• By submitting your answers, you declare you understand that deceiving, or attempt-

ing to deceive the examiners by passing off the work of another as your own is

plagiarism. You also declare that you understand that plagiarising another’s work,

or knowingly allowing another student to plagiarise from your work, is against Uni-

versity Regulations and that doing so will result in loss of marks and disciplinary

proceedings.

2

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

CARDIFF UNIVERSITY

EXAMINATION PAPER

Academic year: 2020-21

Examination Period: Practise

Examination Paper Number: PX3142, PXT111

Examination Paper Title: Condensed Matter Physics

Duration: 2 hours

Additional time: 1 hour to scan and submit your solutions

Structure of Examination Paper:

There are six pages.

There are SEVEN questions in total.

This examination paper is divided into two sections.

There are no appendices.

The maximum mark for the examination paper is 80 (80% of the module mark)

The mark obtainable for a question or part of a question is shown in brackets alongside

the question.

Instructions for Completing the Assessment:

• Answer ALL questions from Section A, and TWO questions from Section B. If addi-

tional Section B questions are attempted be clear which should be marked, otherwise

only the first two written answers will be marked. You are advised to spend equal

amounts of time on Section A and B.

• Use black/blue ink and dark pencils, on white paper. You may use graph paper where

necessary.

• Before the end of the exam duration, allow time to use Microsoft-Lens, Adobe PDF

scan (or equivalent software) to photograph your answers and convert into a PDF to

upload.

• Make sure the pages of your answers are in the correct order and orientation.

• Include your Student Number clearly on the first page, and in the filename of your

PDF.

• Submit your PDF answer file before the end of the duration by:

uploading as assignment on PX3142 Learning Central module,

1

PX3142, PXT111

or emailing to physics-exams@cardiff.ac.uk if you think the upload did not work.

Notice to Students:

• This is an “open book” assessment, you may make use of any resource.

• The use of calculators / computers is permitted.

• You are expected to work without communication with any other person.

• By submitting your answers, you declare that your submission (and all the material

contained in it), is your own work, and you have not shared it with anyone else.

• The examiners reserve the right to further question you, if necessary, by pre-arranged

telephone calls, so you can further explain the answers you have provided in your

submission. The University’s plagiarism detection software may also be used to check

originality of, and any collusion in, the submitted work.

• By submitting your answers, you declare you understand that deceiving, or attempt-

ing to deceive the examiners by passing off the work of another as your own is

plagiarism. You also declare that you understand that plagiarising another’s work,

or knowingly allowing another student to plagiarise from your work, is against Uni-

versity Regulations and that doing so will result in loss of marks and disciplinary

proceedings.

2

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