Module Code: PHYS231001
Module Title: Physics 4 © UNIVERSITY OF LEEDS
School of Physics and Astronomy August Resit 2020/2021
This is an open book assessment. You may consult any of your own notes. You must
provide an explanation for all your answers in your own words. This will vary from a
few words to two to three sentences depending on the material. This will be to demonstrate
your understanding of the course.
Do not just repeat answers from your notes without this explanation. Make sure you
method of calculation is clearly shown.
If you make use of websites or textbooks to answer specific questions, you must list
them at the end of the relevant answer.
Assessment information:
• This assessment is made up of 7 pages.
• You must upload your answers via GradeScope to Minerva within 48 hours of the
assessment being released. You are advised to allow up to four hours to photograph
your answers, and upload as a PDF to GradeScope. The upload link will be found in
the Assessment section for each module on Minerva and will be available throughout
the period of the assessment.
• Although the upload is open for the full period of the assessment, you are advised
that the assessment should only require 2 hours to complete.
• Late submission of answers is not possible.
• You must answer all of the questions in this assessment.
• You should cross out any work you do not want to be marked.
• You should indicate the final answer to each question by underlining it.
• As part of the process of submitted through GradeScope you must identify which
questions are answered on which uploaded pages. You must also check that you
have uploaded all the work you wish to be marked as part of this assessment and
that the answers uploaded are clearly legible. Failure to do so may result in your work
not being marked.
• This is a formal University assessment. You must not share or discuss any aspect
of this assessment, your answers or the module more generally with anyone,
whether a student or not, during the period the assessment is open.
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Module Code: PHYS231001
Approximate values of some constants
Speed of light in a vacuum, c 2.998× 108 m s−1
Electron Charge, e 1.602× 10−19 C
Electron rest mass, me 9.11× 10−31 kg = 0.511 MeV c−2
Proton rest mass, mp 1.673× 10−27 kg = 938.3 MeVc−2
Unified atomic mass unit, u 1.661× 10−27 kg = XXXXXXXXXXMeVc−2
Fine structure constant, α 1/137.036
Planck constant, h 6.626× 10−34 J s
Boltzmann constant, kB 1.381× 10−23 J K−1 = 8.617× 10−5 eV K−1
Coulomb constant, k = 1/4π�0 8.987× 109 N m2 C−2
Rydberg constant, R XXXXXXXXXX× 107 m−1
Avogadro constant, NA 6.022× 1023 mol−1
Gas constant, R 8.314 J K−1 mol−1
Stefan Boltzmann constant, σ 5.670× 10−8 W m−2 K−4
Bohr magneton, µB 9.274× 10−24 J T−1
Gravitational constant, G 6.673× 10−11 m3 kg−1 s−2
Acceleration due to gravity, g 9.806 m s−2
Permeability of free space, µ0 4π × 10−7 H m−1
Permittivity of free space, �0 8.854× 10−12 F m−1
1 Parsec, pc 3.086× 1016 m
Solar mass, M� 1.99× 1030 kg
Magnetic flux quantum, Φ XXXXXXXXXX× 10−15 W
Some SI prefixes
Multiple Prefix Symbol Multiple Prefix Symbol
10−18 atto a 10−9 nano n
10−15 femto f 109 giga G
10−12 pico p 1012 tera T
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Module Code: PHYS231001
SECTION A
• You must answer all the questions from this section.
• This section is worth 20 marks.
• You are advised to spend 30 minutes on this section.
A1. Electrons with a velocity of 1.3× 10−6 m/s are incident on a ba
ier of height 11.5 eV
and width 5 Å. Using the approximate expression for the transmission probability, find
the probability that the electrons will pass through the ba
ier. What minimum energy
would incident electrons need to have so that resonant tunnelling would occur?
[4 Marks]
A2. In a system of two electrons explain why the parallel a
angement of the two electron
spins has a lower energy than the antiparallel a
angement. How is this connected to
the occupation of electron levels in atoms?
[4 Marks]
A3. The valence band of a hypothetical one-dimensional semiconductor has the disper-
sion relation E = A
2
(cos ka− 1). Derive an expression for the constant A for the case
where holes close to the top of the band have an effective mass of −me.
[4 Marks]
A4. Show that the total magnetic moment J of a 4f o
ital containing n electrons is given
y n(6− n)/2 for small enough n. Up to what value of n is this result valid and why is
this the case?
[4 Marks]
A5. Find the Q value for the α decay of 84 Be. What is the relevance of the sign of Q? Use
your value to find the momentum of the emitted α particle if a 84 Be nucleus decays at
est. You may use the following values: M(84 Be) = 8.0053u, M(
4
2 He) = 4.0026u.
[4 Marks]
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Module Code: PHYS231001
SECTION B
• You must answer all the questions from this section.
• This section is worth 75 marks.
• You are advised to spend 90 minutes on this section.
B1. A particle is moving on a unit circle in the x−y plane. The only coordinate of impor-
tance is the polar angle φ which can vary from 0 to 2π as the particle goes around the
circle. We are interested in measurements of the angular momentum L of the particle
about the perpendicular z-axis. The angular momentum operator of such a particle is
given by
L̂ = −i~ d
dφ
.
(a) Suppose that the state of the particle is described by the wavefunction
ψ1(φ) = N e−iφ,
where N is the normalization constant. Determine N .
[4]
(b) What values could we find when we measure the angular momentum of the par-
ticle in the state ψ1? If more than one value is possible, what is the probability of
obtaining each result? What is the expectation value of the angular momentum?
[5]
(c) Does the particle have a definite energy in the state ψ1? Explain your answer.
[3]
(d) Now suppose that the state of the particle is described by the normalised wave-
function
ψ2(φ) = N
(√
3
4
e−iφ − i
2
e2iφ
)
.
When we measure the angular momentum of the particle, what value(s) could
we find? If more than one value is possible, what is the probability of obtaining
each result? [3]
[15 Marks]
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Module Code: PHYS231001
B2. For the n = 1 harmonic oscillator eigenfunction given by(
2
√
π
) 1
2
(2bx)e
−b2x2
2 where b =
(mκ
~2
) 1
4
(a) Why are the expectation values 〈p〉 and 〈x〉 equal to zero? [2]
(b) Show that the expectation value for 〈p2〉 = 3m~ω
2
. [6]
(c) Show that the expectation value for 〈x2〉 = 3~
2mω
. [4]
(d) Hence show that these results are consistent with the uncertainty principle. [3]
You may find the following integrals useful:∫ ∞
−∞
xe−ax
2
=
√
π
a
;
∫ ∞
−∞
x2e−ax
2
=
1
2a
√
π
a
;
∫ ∞
−∞
x4e−ax
2
=
3
√
π
4a5/2
;
[15 Marks]
B3. (a) Draw sketches of the energy dependence of the density of states for electrons
for each of a simple alkali metal, an insulator, a transition metal, and an intrinsic
semiconductor. Mark the Fermi level on each sketch. Draw the sketches in orde
of the typical room temperature electrical conductivity of these four classes of
materials. Explain the reasons for your choice of ordering. [8]
(b) Calculate the density of states for electrons at the Fermi level for a hypotheti-
cal body centred cubic monovalent free electron-like metal with lattice constant
0.25 nm. [3]
(c) By what factor must the molecular field exceed the magnetisation for this hypo-
thetical metal to become an itinerant fe
omagnet? Comment on your answer.
[4]
[15 Marks]
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Module Code: PHYS231001
B4. (a) Show that the Drude conductivity σ = ne
2τ
m
can be written as σ = 1
3
v2Fg(EF)e
2τ
for the case of free electrons. [5]
(b) Explain why it is reasonable to assume that the interaction between a conduction
electron and an impurity atom is propotional to Ze2, where Z is the difference in
atomic number between the impurity and the host metal. [2]
(c) Use this assumption to make dimensional arguments that show that the scatter-
ing cross-section of an electron interacting with an impurity is given by
Σi ≈
Z2e4
(4π�0)2E2F
.
[3]
(d) Compare the predictions of this formula with the experimental data shown in
Fig. 1, where the increase in residual resistivity ρ0 is proportional to the concen-
tration of impurities ci according to ρ0 = αci.
Some relevant data for Cu: EF ≈ 7 eV; vF ≈ 106 m/s; assume Cu to be mono-
valent in all cases. [5]
Figure 1: Residual resistivity contributions of selected impurities in Cu. Data from [F. Pawlek
and K. Riecher, Z. Metallkunde 47, XXXXXXXXXX)].
[15 Marks]
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Module Code: PHYS231001
B5. (a) Sketch the potential experienced by an α particle in the presence of a large
nucleus, as a function of distance from the centre of the nucleus. Label you
axes appropriately with typical values and explain the shape of the potential.
[5]
(b) Use your diagram to explain why the Q value for the α decay of a particula
nuclide has a large effect on the half-life of that decay.
[6]
(c) With some exceptions, typical half-lives for β decay are on the order of 10−3 s,
while typical half-lives for electron capture are on the order of several days. Ex-
plain these observations.
[4]
[15 Marks]
Page 7 of 7 End.
Module Code: PHYS2310 MID-TERM RESIT PAPER
Module Title: Physics 4 ©UNIVERSITY OF LEEDS
School of Physics and Astronomy August Resit 202021
• This is an open book assessment. You may consult any of your own notes.
• You must provide an explanation for all your answers in your own words. This
will vary from a few words to two to three sentences depending on the material. This
will be to demonstrate your understanding of the course. Do not just repeat answers
from your notes without this explanation.
• Make sure your method of calculation is clearly shown. If you make use of websites
or textbooks to answer specific questions, you must list them at the end of
the relevant answer.
Assessment information
• This assessment is made up of 6 pages.
• You must upload your answers