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L A N C A S T E R U N I V E R S I T Y
2012 EXAMINATIONS
Part II
PHYSICS - Paper 3.B
- Candidates should answer all those sections identified with the modules for which they are registered.
- The time allocated is 60 minutes per section.
- An indication of mark weighting is given by the numbers in square brackets following each part.
- In each section attempted, candidates should answer question 1 (30 marks) and either question 2 or question 3 (30 Marks).
- Use a separate answer book for each section.
PHYSICAL CONSTANTS
Planck’s constant h = 6. 63 × 10 −^34 J s ℏ = 1. 05 × 10 −^34 J s Boltzmann’s constant kB = 1. 38 × 10 −^23 J K−^1 Mass of electron me = 9. 11 × 10 −^31 kg Mass of proton mp = 1. 67 × 10 −^27 kg Electronic charge e = 1. 60 × 10 −^19 C Speed of light c = 3. 00 × 108 m s−^1 Avogadro’s number NA = 6. 02 × 1023 mol−^1 Permittivity of the vacuum ≤ 0 = 8. 85 × 10 −^12 F m−^1 Permeability of the vacuum μ 0 = 4 π × 10 −^7 H m−^1 Gravitational constant G = 6. 67 × 10 −^11 N m^2 kg−^2 Bohr magneton μB = 9. 27 × 10 −^24 J T−^1 (or A m^2 ) Bohr radius a 0 = 5. 29 × 10 −^11 m Gas constant R = 8 .31 J K−^1 mol−^1 Acceleration due to gravity g = 9.81 m s−^2 1 standard atmosphere = 1. 01 × 105 N m−^2 Mass of Earth = 5. 97 × 1024 kg Radius of Earth = 6. 38 × 106 m
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Section A: Module 321: Atomic Physics (The time allocated for this section is 60 minutes. Candidates should answer question A1 and either question A2 or question A3.)
Compulsory question:
A1. (a) (i) State the main assumptions of the Hartree theory of an atom and explain why it is called self-consistent. (ii) Explain what is meant by the shell penetration effect with the help of a diagram showing the radial distribution of electron density in different shells. (iii) In the independent electron approximation, the Li atom (Z=3) has an 8-fold degenerate ground level. Show schematically the splitting of the ground level of Li due to the shell penetration effect and indicate the de- generacy of each resulting sublevel. [12] (b) (i) Define the term selection rule. (ii) State the selection rules for the dipole optical transitions in an atom. [6] (c) (i) Explain why the noble gases such as He (Z=2) or Ne (Z=10) are chemically inert. (ii) Consider a hypothetical universe in which an electron does not have spin but obeys the Pauli principle. State the atomic numbers of the two lightest noble gases in this universe’s periodic table. (iii) The difference between the first ionization energies of Ne (Z=10) and Na (Z=11) is significantly larger than the difference between the first ion- ization energies of F (Z=9) and Ne (Z=10). Explain this using the shell model of the many-electron atom. [12]
Section B: Module 322: Statistical Physics (The time allocated for this section is 60 minutes. Candidates should answer question B1 and either question B2 or question B3.)
Compulsory question:
B1. (a) Explain as used in statistical physics the difference between the following pairs of terms; (i) macrostate and microstate, (ii) distinguishable and indistinguishable particles, (iii) fermions and bosons. [9] (b) (i) Show that ni identical fermions can be distributed amongst a group of gi states in (^) n gi! i!(gi −^ ni)!
ways. (ii) Write down the Fermi-Dirac distribution f as a function of energy of the particle ε, the Fermi energy εF and the temperature of the system T. (iii) Derive an expression in terms of εF and T , for the energy of a state in a degenerate electron gas in which the number of particles per state is 23.8% of the number of particles per state at the Fermi energy. [11] (c) (i) Write down Boltzmann’s expression for statistical entropy defining all sym- bols used. (ii) Consider a system of N distinguishable magnetic atoms with spin 3/2 and four equidistant energy levels. Show that the entropy for such a system tends to zero as the temperature approaches T =0. Qualitatively sketch the behaviour of entropy for this system as a function of temperature. (iii) What is the value of entropy of such a system at T → ∞? Explain your reasoning. [10]
Answer one of the following two questions:
B2. (a) (a) Define the terms density of states in energy g(ε) and spin factor G, as used in statistical physics. [6] (b) Consider a gas of phonons in a crystal of volume V using the approximation of constant speed of sound cs for all sound waves propagating in the crystal. (i) Show that the density of states function g(ε) for phonons can be expressed as g(ε) =
3 V ε^2 2 π^2 (ℏcs)^3 [9] (ii) Given that (^) ∫ ∞
0
x^3 ex^ − 1
dx =
π^4 15 calculate the total energy of a gas of thermal phonons in the low temper- ature limit. [8] (iii) Find the ratio R of the energy of phonons in a crystal of volume V to the energy of photons in a box of the same volume at T → 0. [7]
B3. (a) Define the terms Fermi energy and Fermi temperature as used in statistical physics. [6] (b) Consider an electron moving in a two dimensional free electron gas of area A. (i) Show that the density of states g(k) in k-space for such an electron is
g(k)dk =
A
π
kdk
(ii) Hence calculate the density of states g(ε) for a two-dimensional electron gas with area density n 2 D = N/A confined to a graphene sheet, and derive expression for the Fermi energy εF of such a system. [14] (c) Derive an approximate expression for electron heat capacity C for a degenerate two-dimensional gas of electrons as a function of area density n 2 D = N/A explaining all assumptions used. [10]
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Answer one of the following two questions:
C2. (a) (i) State Newton’s law of viscosity and define all of the symbols used. (ii) Give the two mathematical conditions that velocity has to satisfy in the steady flow of an incompressible fluid. [12] (b) An incompressible viscous fluid flows steadily through a pipe of radius a. The velocity field v has the form
v =
4 η 0
dp dz
(r^2 − a^2 ) ˆz
where the unit vector ˆz points along the pipe, r is the distance from the pipe axis and dp/dz is constant. (i) Sketch the profile of the velocity field v across the pipe’s cross-section. (ii) Determine the volumetric discharge rate by integrating the velocity across the cross-section of the pipe. (^) [6] (c) A viscous fluid flows between the inner and outer walls of a coaxial pipe with inner radius a and outer radius b.
2 b 2 a ˆ z
The velocity field v = vz (r) ˆz, where the unit vector ˆz points along the pipe, depends only on the radial distance r from the pipe’s axis of symmetry. In cylindrical polar coordinates (r, ϕ, z), the Navier-Stokes equation can be writ- ten as η 0
r
d dr
r
dvz dr
= A 0
where A 0 is a constant. (i) State the appropriate boundary condition on the fluid at the walls of the pipe. (ii) Use this condition to determine vz (r). [12]
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C3. (a) (i) Give the definitions of each of the following terms: ideal fluid, potential flow, steady flow, drag. (ii) State d’Alembert’s paradox. [10] (b) (i) The dispersion relation for gravity waves on deep water is
ω =
g 0 k
where ω is angular frequency, k is wavelength and g 0 is the gravitational acceleration. Determine the phase velocity and group velocity of such waves and use your answer to explain the concept of a wave packet. [8] (ii) Consider a sinusoidal gravity wave with amplitude a propagating in the x direction across a sea of constant depth d. The velocity potential is
Ψ = f (z) cos(kx − ωt) where f (z) = A exp(kz) + B exp(−kz)
and A and B are constants. Determine the dispersion relation using the (linearized) open surface (z=0) boundary conditions
f
∣z=0 = −g^0 a ω
df dz
z=
= −aω
and the no-through-flow boundary condition at the sea bed z = −d. [12]
End of Paper
