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STATISTICAL MECHANICS

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Paramagnetic susceptibility oi an imperfect Fermi gas with repulsive interactions. The model used is well founded only for kT/f F 1.

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The function (TX/C) rises linearly at T

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It reaches a maximum value, which is greater than unity, at kT/ f. F Z 1. Then it approaches unity as T --+ 00. A qualitative plot of TX/C is shown in Fig. 11.19. If we calculate X for an ideal Fermi gas endowed with the same magnetic moment, we find the slope

a (TX) 3 (ideal Fermi gas) (11.151) aT C T~O 2 The imperfect gas has a steeper slope, as (11.150) shows, which is again a reflection of the enhancement of spin alignment by the repulsive interaction. The result is sometimes described by saying that imperfect gas behaves like an ideal gas with a higher Fermi energy. *

PROBLEMS

Give numerical estimates for the Fermi energy of (a) electrons in a typical metal; ( b) nucleons in a heavy nucleus; (c) He 3 atoms in liquid He 3 (atomic volume = 46.2 P2 /atom). Treat all the mentioned particles as free particles.

*See, however, Problem

FERMI SYSTEMS

11.2 Show that for the ideal Fermi gas the Helmholtz free energy per particle at low temperatures is given by

11.3 A collection of free nucleons is enclosed in a box of volume V. The energy of a single nucleon of momentum p is

f p =

where me = 1000 MeV. (a) Pretending that there is no conservation law for the number of nucleons, calculate the partition function of a system of nucleons (which obey Fermi statistics) at temperature T. (b) Calculate the average energy density. (e) Calculate the average particle density. (d) Discuss the necessity for a conservation law for the number of nucleons, in the light of the foregoing calculations.

11.4 (a) What is the heat capacity C v of a three-dimensional cubic lattice of atoms at room temperature Assume each atom to be bound to its equilibrium position by Hooke's law forces. (b) Assuming that a metal can be r~prc"'~t1kd by such a lattice of atoms plus freely moving electrons, compare the specific heat due to the electrons with that due to the lattice, at room temperature. 11.15 A cylinder is separated into two compartments by a free sliding piston. Two ideal Fermi gases are placed into the two compartments, numbered 1 and 2. The particles in compartment 1 have spin ~, while those in compartment 2 have spin ~. They all have the same mass. Find the equilibrium relative density of the two gases at T = 0 and at T ---> 00. 11.6 Consider a two-dimensional electron gas in a magnetic field strong enough so that all particles can be accommodated in the lowest Landau level. Taking into account both orbital and spin paramagnetism, find the magnetization at absolute zero.

11.7 (a) Show that for the imperfect Fermi gas discussed in Section 11.6 the specific heat at constant volume is given by

a 32 - T [( ar -2k- [I(r)T 2 -ar] + - 'TTan aT aT

)2 + ra r] 2

where

(b) Show that when there is no spontaneous magnetization

C V )ideal gas

and hence the interpretation that the imperfect gas behaves like an ideal gas with a higher Fermi energy cannot be consistently maintained.

BOSE SYSTEMS

The dominant characteristic of a system of bosons is a "statistical" attraction between the particles. In contradistinction to the case of fermions, the particles like to have the same quantum numbers. When the particle number is conserved. this attraction leads to the Bose-Einstein condensation, which is the basis of superfluidity. In this chapter we illustrate various bose systems, discuss the Bose-Einstein condensation, and introduce the notion of the superfluid order parameter.

12.1 PHOTONS Consider the equilibrium properties of electromagnetic radiation enclosed in a volume Vat temperature T, a system known as a "blackbody cavity." It can be experimentally produced by making a cavity in any material, evacuating the cavity completely, and then heating the material to a given temperature. The atoms in the walls of this cavity will constantly emit and absorb electromagnetic radiation, so that in equilibrium there will be a certain amount of electromagnetic radiation in the cavity, and nothing else. If the cavity is sufficiently large, the thermodynamic properties of the radiation in the cavity should be independent of the nature of the wall. Accordingly we can impose on the radiation field any boundary condition that is convenient. The Hamiltonian for a free electromagnetic field can be written as a sum of terms, each having the form of a Hamiltonian for a harmonic oscillator of some frequency. This corresponds to the possibility of regarding any radiation field as a linear superposition of plane waves of various frequencies. In quantum theory each harmonic oscillator of frequency w can only have the energies (n + Dhw, where n = 0, 1, 2, .... This fact leads to the concept of photons as quanta of the electromagnetic field. A state of the free electromagnetic field is specified by the number n for each of the oscillators. In other words, it is specified by enumerating the number of photons present for each frequency.