Chapter 4- Answers to selected exercises

1. Consider a model of $N$ localized magnetic ions, given by the spin Hamiltonian

$$\mathcal{H}=D\dsum\limits_{j=1}^{N}S_{j}^{2},$$

where the spin variable $S_{j}$ may assume the values $-1,$ $0,$ or $+1,$ for all $j$ (see exercise 8 of Chapter 2). Given the total energy $E$, use the expression for the number of accessible microstates, $\Omega \left( E,N\right) ,$ to obtain the entropy per particle, $s=s\left( u\right)$, where $u=E/N$. Obtain an expression for the specific heat $c$ in terms of the temperature $T$. Sketch a graph of $c$ versus $T$. Check the existence of a broad maximum associated with the Schottky effect. Write an expression for the entropy as a function of temperature. What are the limiting values of the entropy for $T\rightarrow 0$ and $T\rightarrow \infty$? The entropy per particle is given by

$$\frac{1}{k_{B}}s=\frac{u}{D}\ln 2-\left( 1-\frac{u}{D}\right) \ln \left( 1- \frac{u}{D}\right) -\frac{u}{D}\ln \frac{u}{D},$$

where $u=E/N$. Thus, we have the equation of state

$$\frac{1}{k_{B}T}=\frac{1}{D}\ln \frac{2\left( 1-u/D\right) }{u/D},$$

from which we have

$$u=\frac{2D\exp \left( -\beta D\right) }{1+2D\exp \left( -\beta D\right) },$$

where $\beta =1/\left( k_{B}T\right)$. The specific heat is given by the derivative of $s$ with respect to $T$. For $T\rightarrow 0$, $s\rightarrow 0$; for $T\rightarrow \infty$, $s\rightarrow k_{B}\ln 2$.

2. In the solid of Einstein, we may introduce a volume coordinate if we make the phenomenological assumption that the fundamental frequency $\omega$ as a function of $v=V/N$ is given by =

$$\omega =\omega \left( v\right) =\omega _{o}-A\ln \left( \frac{v}{v_{o}} \right) ,$$

where $\omega _{o},$ $A$, and $v_{o}$ are positive constants. Obtain expressions for the expansion coefficient and the isothermal compressibility of this model system. As $\omega =\omega \left( v\right)$, the entropy of Einstein's solid can be written as a function of energy and volume, $s=s\left( u,v\right)$. From the equations of state, it is straightforward to obtain the expansion coefficient and the compressibility.

3. Consider the semiclassical model of $N$ particles with two energy levels ( 0 and $\epsilon >0$). As in the previous exercise, suppose that the volume of the gas may be introduced by the assumption that the energy of the excited level depends on $v=V/N$,

$$\epsilon =\epsilon \left( v\right) =\frac{a}{v^{\gamma }},$$

where $a$ and $\gamma$ are positive constants. Obtain an equation of state for the pressure, $p=p\left( T,v\right)$, and an expression for the isothermal compressibility (note that the constant $\gamma$ plays the role of the Grüneisen parameter of the solid). Again, as $\epsilon =\epsilon \left( v\right)$, we can write $s=s\left( u,v\right)$. From the equations of state, it is easy to write the isothermal compressibility.

4. The total number of the accessible microscopic states of the Boltzmann gas, with energy $E$ and number of particles $N$, may be written as

$$\Omega \left( E,N\right) =\dsum\limits_{N_{1},N_{2},...}\frac{N!}{ N_{1}!N_{2}!\cdots },$$

with the restrictions

$$\dsum\limits_{j}N_{j}=N \dsum\limits_{j}\epsilon _{j}N_{j}=E.$$

Except for an additive constant, show that the entropy per particle is given by

$$s=-k_{B}\sum\limits_{j}\left( \frac{\tilde{N}_{j}}{N}\right) \ln \left( \frac{\tilde{N}_{j}}{N}\right) ,$$

where $\left\{ \tilde{N}_{j}\right\}$ is the set of occupation numbers at equilibrium. Using the continuum limit of the Boltzmann gas, show that the entropy depends on temperature according to a term of the form $-k_{B}T\ln T$ .

5. Consider a lattice gas of $N$ particles distributed among $V$ cells (with $N\leq V$). Suppose that each cell may be either empty or occupied by a single particle. The number of microscopic states of this system will be given by

$$\Omega \left( V,N\right) =\frac{V!}{N!\left( V-N\right) !}.$$

Obtain an expression for the entropy per particle, $s=s\left( v\right)$, where $v=V/N$. From this fundamental equation, obtain an expression for the equation of state $p/T$. Write an expansion of $p/T$ in terms of the density $\rho =1/v$. Show that the first term of this expansion gives the Boyle law of the ideal gases. Sketch a graph of $\mu /T$, where $\mu$ is the chemical potential, in terms of the density $\rho$. What is the behavior of the chemical potential in the limits $\rho \rightarrow 0$ and $\rho \rightarrow 1$? The entropy particle is given by

$$\frac{1}{k_{B}}s=v\ln v-\left( v-1\right) \ln \left( v-1\right) .$$

Thus, we have the equation of state

$$\frac{p}{k_{B}T}=-\ln \left( 1-\frac{1}{v}\right) =\frac{1}{v}+\frac{1}{ 2v^{2}}+\frac{1}{3v^{3}}+...$$

Note that Bolyles's law is already given by the first term is this expansion. To find $\mu /T$, we write $S=Ns=S\left( V,N\right)$, and take the derivative with respect to $N$.