Chapter 9 - Answers to selected exercises

1. What is the compressibility of a gas of free fermions at zero temperature? Obtain the numerical value for electrons with the density of conduction electrons in metallic sodium. Compare your results with experimental data for sodium at room temperature.

2. An ideal gas of fermions, with mass $m$ and Fermi energy $\epsilon _{F}$, is at rest at zero temperature. Find expressions for the expected values $\left\langle v_{x}\right\rangle$ and $\left\langle v_{x}^{2}\right\rangle$ , where $\vec{v}$ is the velocity of a fermion.

3. Consider a gas of free electrons, in a $d$-dimensional space, within a hypercubic container of side $L$. Sketch graphs of the density of states $D\left( \epsilon \right)$ versus energy $\epsilon$ for dimensions $d=1$ and $d=2$. What is the expression of the Fermi energy in terms of the particle density for $d=1$ and $d=2$?

4. Show that the chemical potential of an ideal classical gas of $N$ monatomic particles, in a container of volume $V$, at temperature $T$, may be written as

$$\mu =k_{B}T\ln \left( \frac{\lambda ^{3}}{v}\right) ,$$

where $v=V/N$ is the volume per particle, and $\lambda =h/\sqrt{2\pi mk_{B}T}$ is the thermal wavelength. Sketch a graph of $\mu /k_{B}T$ versus $T$. Obtain the first quantum correction to this result. That is, show that the chemical potential of the ideal quantum gas may be written as the expansion

$$\frac{\mu }{k_{B}T}-\ln \left( \frac{\lambda ^{3}}{v}\right) =A\l... ...ambda ^{3}}{v}\right) +B\ln \left( \frac{\lambda ^{3}}{v}\right) ^{2}+\ldots ,$$

and obtain explicit expressions for the prefactor $A$ for fermions and bosons. Sketch a graph of $\mu /k_{B}T$ versus $\lambda ^{-2}$ (that is, versus the temperature in convenient units) for fermions, bosons, and classical particles.

5. Obtain an asymptotic form, in the limit $T<<T_{F}$, for the specific heat of a gas of $N$ free fermions adsorbed on a surface of area $A$, at a given temperature $T$.

6. Consider a gas of $N$ free electrons, in a region of volume $V$, in the ultrarelativistic regime. The energy spectrum is given by

$$\epsilon =\left[ p^{2}c^{2}+m^{2}c^{4}\right] ^{1/2}\approx pc,$$

where $\vec{p}$ is the linear momentum. (a) Calculate the Fermi energy of this system. (b) What is the total energy in the ground state? (c) Obtain an asymptotic form for the specific heat at constant volume in the limit $T<<T_{F}$.

7. At low temperatures, the internal energy of a system of free electrons may be written as an expansion,

$$U=\frac{3}{5}N\epsilon _{F}\left\{ 1+\frac{5\pi ^{2}}{12}\left( \frac{T}{ T_{F}}\right) ^{2}-A\left( \frac{T}{T_{F}}\right) ^{4}+\ldots \right\} .$$

Obtain the value of the constant $A$, and indicate the order of magnitude of the terms that have been discarded.

8. Consider a system of free fermions in $d$ dimensions, with the energy spectrum

$$\epsilon _{\vec{k},\sigma }=c\left\vert \vec{k}\right\vert ^{a},$$

where $c>0$ and $a>1$. (a) Calculate the prefactor $A$ of the relation $pV=AU$. (b) Calculate the Fermi energy as a function of volume $V$ and number of particles $N$. (c) Calculate an asymptotic expression, in the limit $T<<T_{F}$, for the specific heat at constant volume.

9. Consider again the gas of $N$ ultrarelativistic free electrons, within a container of volume $V$, at temperature $T$, in the presence of a magnetic field $\vec{H}$. If we neglect the effects of orbital magnetism, the energy spectrum is given by

$$\epsilon _{\vec{p},\sigma }=cp-\mu _{B}H\sigma ,$$

where $\mu _{B}$ is the Bohr magneton and $\sigma =\pm 1$. (a) Show that the Fermi energy of this system may be written as

$$\epsilon _{F}=A+BH^{2}+O\left( H^{4}\right) .$$

Obtain expressions for the prefactors $A$ and $B$. (b) Show that the magnetization in the ground state can be written in the form

$$M=CH+O\left( H^{3}\right) .$$

Obtain an expression for the constant $C$. (c) Calculate the susceptibility of the ground state in zero field.

10. In the classical paramagnetic theory of Langevin, proposed before the advent of quantum statistics, we assume a classical Hamiltonian, given by

$$\mathcal{H}=-\dsum\limits_{i=1}^{N}\vec{\mu}_{i}\cdot \vec{H} =-\dsum\limits_{i=1}^{N}\mu H\cos \theta _{i},$$

where $\vec{\mu}_{i}$ is the magnetic moment of a localized ion. (a) Show that the canonical partition function of this system is given by

$$Z=Z_{1}^{N}=\left\{ \int d\Omega \exp \left( \beta \mu H\cos \theta \right) \right\} ^{N},$$

where $d\Omega$ is the elementary solid angle of integration. (b) Show that the magnetization (along the direction of the field) is given by

$$M=N\left\langle \mu \cos \theta \right\rangle =N\mu \mathcal{L}\left( \beta \mu H\right) ,$$

where

$$\mathcal{L}\left( x\right) =\coth \left( x\right) -\frac{1}{x}$$

is the Langevin function. (c) Show that the susceptibility in zero field is given by the Curie law,

$$\chi _{o}=\frac{N\mu ^{2}}{3kT}.$$

11. Obtain an expression for the magnetic susceptibility associated with the orbital motion of free electrons in the presence of a uniform magnetic field $H$, under conditions of strong degeneracy, $T<<T_{F}$, and very weak fields, $\mu _{B}H<<k_{B}T$. To simplify the expression of $\Xi$, you may use Euler's sum rule,

$$\dsum\limits_{n=0}^{\infty }f\left( n+\frac{1}{2}\right) \approx ... ...mits_{0}^{\infty }f\left( x\right) dx+\frac{1}{24}f^{\prime }\left( 0\right) .$$