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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 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 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. Page 1 of 7 Turn the page over 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 = 931.494 MeVc−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 1.09373× 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, Φ0 2.0679× 10−15 Wb 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 Page 2 of 7 Turn the page over 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 barrier 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 barrier. 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 arrangement of the two electron spins has a lower energy than the antiparallel arrangement. 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 orbital containing n electrons is given by 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 rest. You may use the following values: M(84 Be) = 8.0053u, M( 4 2 He) = 4.0026u. [4 Marks] Page 3 of 7 Turn the page over 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] Page 4 of 7 Turn the page over Module Code: PHYS231001 B2. For the n = 1 harmonic oscillator eigenfunction given by( b 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 order 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 ferromagnet? Comment on your answer. [4] [15 Marks] Page 5 of 7 Turn the page over 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, 347 (1956)]. [15 Marks] Page 6 of 7 Turn the page over 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 your 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 particular 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