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TSP Problem Sheet 3
Thermal and Statistical Physics
University of Cambridge
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University of Cambridge Cavendish Laboratory
TSP-2020/21 — Thermal and Statistical Physics (Part II)
Problem sheet 3: Quantum statistics
Point defect A point defect in a solid may be occupied by 0, 1 (spin up or down) or 2 electrons, and the solid provides a reservoir of electrons at chemical potential μ. The energy for occupation by a single electron is , and that for 2 electrons is 2+U , where U is the Coulomb repulsion energy between the electrons. Obtain an expression for the average electron occupancy of the defect and produce a sketch of average occupancy vs. μ − for low temperatures T << U/kB.
Plasma Show that the equilibrium constant KN for the ionisation reaction He ⇀↽ He+ + e− is to a good approximation
KN =
1
4 V
(
2 πh ̄ 2 mekB T
) 3 / 2
eeφ/kB T
where φ is the first ionisation potential of He, which is 24 V. Find the proportion of He that is ionised at 10 4 K (i) at atmospheric pressure, and (ii) at 10 − 2 Nm− 2. What is the cause of the change in the equilibrium constant? This effect is important for spectral lines from interstellar gases, one finds a surprisingly large intensity corresponding to spectral lines of ionised atoms.
Trap Atoms can be held in a spherical trap with potential V (r) = ar. Calculate the partition function ZN of a gas of N indistinguishable non-interacting atoms in this trap at temper- ature T in the classical limit. Sketch the temperature dependence of the entropy of the classical gas in this trap for two different values of a and demonstrate that by decreasing a adiabatically, the gas can be cooled reversibly. A trap contains a variable number of atoms. Find an expression for the chemical potential of the system. Estimate the number of atoms required for quantum statistics to become important.
Degenerate or non-degenerate The temperature at the centre of the sun is T = 1. 6 × 107 K, and plasma at the centre of the sun consists of hydrogen at a density of ρH = 6 × 104 kg m− 3 and helium at a density of ρHe = 1 × 105 kg m− 3. (a) Calculate the thermal wavelengths of the electrons, protons and He nuclei. (b) Determine whether the electrons, protons and He nuclei are degenerate or non-degenerate under these conditions. (c) Estimate the pressure at the centre of the sun due to these particles and that due to the radiation pressure. (d) Is it the pressure due to the particles or the radiation which prevents gravitational collapse of the sun?
Helium- At temperatures below 0 K, a dilute solution of 3 He in liquid 4 He behaves like a gas of 3 He atoms moving freely in vacuo except that the effective mass of each 3 He atom is enhanced by a factor of about 2. The concentration of 3 He is 5 atomic percent and the density of the solution is 140 kg/m 3. Sketch the temperature dependence of the heat capacity per 3 He atom at low temperatures. Calculate the Fermi temperature, TF , and
TSP-2020/21 Problem sheet 3 1 Michaelmas Term
TSP Problem Sheet 3
Module: Thermal and Statistical Physics
University: University of Cambridge
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