Chapter 5- Answers to selected exercises

1. The energy of a system of $N$ localized magnetic ions, at temperature $T$ , in the presence of a field $H$, may be written as

$$\mathcal{H}=D\dsum\limits_{i=1}^{N}S_{i}^{2}-\mu _{o}H\dsum\limits_{i=1}^{N}S_{i},$$

where the parameters $D$, $\mu _{o}$, and $H$ are positive, and $S_{i}=+1,$ $0,$ or $-1$, for all sites $i$. Obtain expressions for the internal energy, the entropy, and the magnetization per site. In zero field ($H=0$), sketch graphs of the internal energy, the entropy, and the specific heat versus temperature. Indicate the behavior of these quantities in the limits $T\rightarrow 0$ and $T\rightarrow \infty$. Calculate the expected value of the ``quadrupole moment,''

$$Q=\frac{1}{N}\left\langle \dsum\limits_{i=1}^{N}S_{i}^{2}\right\rangle ,$$

as a function of field and temperature. The canonical partition function is given by

$$Z=\dsum\limits_{\left\{ S_{i}\right\} }\exp \left[ -\beta D\dsum... ...exp \left( -\beta D\right) \cosh \left( \beta \mu _{o}H\right) +1\right] ^{N}.$$

The internal energy per ion comes from

$$u=-\frac{1}{N}\frac{\partial }{\partial \beta }\ln Z.$$

The magnetization per spin is given by

$$m=\frac{1}{\beta N}\frac{\partial }{\partial H}\ln Z,$$

and the ``quadrupolar moment'' by

$$Q=-\frac{1}{\beta D}\frac{\partial }{\partial D}\ln Z.$$

In zero field, we have $u\left( T=0\right) =0$, $u\left( T\rightarrow \infty \right) =2D/3$, $s\left( T=0\right) =0$, and $s\left( T\rightarrow \infty \right) =k_{B}\ln 3$.

2. Consider a one-dimensional magnetic system of $N$ localized spins, at temperature $T$, associated with the energy

$$\mathcal{H}=-J\dsum_{i=1,3,5,\ldots ,N-1}\sigma _{i}\sigma _{i+1}-\mu _{o}H\dsum\limits_{i=1}^{N}\sigma _{i},$$

where the parameters $J$, $\mu _{o}$, and $H$ are positive, and $\sigma _{i}=\pm 1$ for all sites $i$. Assume that $N$ is an even number, and note that the first sum is over odd integers. (a) Obtain an expression for the canonical partition function and calculate the internal energy per spin, $u=u\left( T,H\right)$. Sketch a graph of $u\left( T,H=0\right)$ versus temperature $T$. Obtain an expression for the entropy per spin, $s=s\left( T,H\right)$. Sketch a graph of $s\left( T,H=0\right)$ versus $T$. (b) Obtain expressions for the magnetization per particle, $_{{}}$

$$m=m\left( T,H\right) =\frac{1}{N}\left\langle \mu _{o}\dsum\limits_{i=1}^{N}\sigma _{i}\right\rangle ,$$

and for the magnetic susceptibility,

$$\chi =\chi \left( T,H\right) =\left( \frac{\partial m}{\partial H}\right) _{T}.$$

Sketch a graph of $\chi \left( T,H=0\right)$ versus temperature. The canonical partition function is given by

$$Z=\left[ 2\exp \left( \beta J\right) \cosh \left( 2\beta \mu _{o}H\right) +2\exp \left( -\beta J\right) \right] ^{N/2}.$$

The magnetization per particle is given by

$$m=\frac{1}{N\beta }\frac{\partial }{\partial H}\ln Z=\mu _{o}\fra... ...right) }{\cosh \left( 2\beta \mu _{o}H\right) +\exp \left( -2\beta J\right) },$$

whose derivative with respect to $H$ yields the susceptibility.

3. Consider a system of $N$ classical and noninteracting particles in contact with a thermal reservoir at temperature $T$. Each particle may have energies 0, $\epsilon >0$, or $3\epsilon$. Obtain an expression for the canonical partition function, and calculate the internal energy per particle, $u=u\left( T\right)$. Sketch a graph of $u$ versus $T$ (indicate the values of $u$ in the limits $T\rightarrow 0$ and $T\rightarrow \infty$ ). Calculate the entropy per particle, $s=s\left( T\right)$, and sketch a graph of $s$ versus $T$. Sketch a graph of the specific heat versus temperature. The canonical partition function is given by

$$Z=\left[ 1+\exp \left( -\beta \epsilon \right) +\exp \left( -3\beta \epsilon \right) \right] ^{N}.$$

For $T\rightarrow \infty$, we have $u=4\epsilon /3$ and $s=k_{B}\ln 3$.

4. A system of $N$ localized and independent quantum oscillators is in contact with a thermal reservoir at temperature $T$. The energy levels of each oscillator are given by

$$\epsilon _{n}=\hbar \omega _{o}\left( n+\frac{1}{2}\right) , n=1,3,5,7,....$$

Note that $n$ is an odd integer. (a) Obtain an expression for the internal energy $u$ per oscillator as a function of temperature $T$. What is the form of $u$ in the classical limit ( $\hbar \omega _{o}<<k_{B}T$)? (b) Obtain an expression for the entropy per oscillator as a function of temperature. Sketch a graph of entropy $_{{}}$versus temperature. What is the expression of the entropy in the classical limit? (c) What is the expression of the specific heat in the classical limit? The answers come from the expression

$$Z_{1}=\frac{\exp \left( -\frac{3}{2}\beta \hbar \omega _{o}\right) }{1-\exp \left( -2\beta \hbar \omega _{o}\right) }.$$

The internal energy is given by

$$u=-\frac{\partial }{\partial \beta }\ln Z_{1}=\frac{3}{2}\hbar \o... ...o}+ \frac{2\hbar \omega _{o}}{\exp \left( 2\beta \hbar \omega _{o}\right) -1}.$$

Note that $u\rightarrow k_{B}T$ in the classical limit (the specific heat is just $k_{B}$).

5. Consider a system of $N$ noninteracting classical particles. The single-particle states have energies $\epsilon _{n}=n\epsilon$, and are $n$ times degenerate ( $\epsilon >0$; $n=1,2,3,...$). Calculate the canonical partition function and the entropy of this system. Obtain expressions for the internal energy and the entropy as a function of temperature. What are the expressions for the entropy and the specific heat in the limit of high temperatures? The thermodynamic functions come from canonical partition function, given by $Z=Z_{1}^{N}$, where

$$Z_{1}=\dsum\limits_{n=1,2,3,...}n\exp \left( -\beta n\epsilon \ri... ... \epsilon \right) }{\left[ \exp \left( \beta \epsilon \right) -1\right] ^{2}}.$$

6. A set of $N$ classical oscillators in one dimension is given by the Hamiltonian

$$\mathcal{H}=\dsum\limits_{i=1}^{N}\left( \frac{1}{2m}p_{i}^{2}+\frac{1}{2} m\omega ^{2}q_{i}^{2}\right) .$$

Using the formalism of the canonical ensemble in classical phase space, obtain expressions for the partition function, the energy per oscillator, the entropy per oscillator, and the specific heat. Compare with the results from the classical limit of the quantum oscillator. Calculate an expression for the quadratic deviation of the energy as a function of temperature. The thermodynamic functions come from canonical partition function, given by $Z=Z_{1}^{N}$, where

$$Z_{1}=\dint\limits_{-\infty }^{+\infty }\dint\limits_{-\infty }^{... ...}p^{2}-\frac{\beta m\omega ^{2}}{2}q^{2} \right] =\frac{2\pi }{\beta \omega }.$$

$^{\text{*}}$7. Consider again the preceding problem. The canonical partition function can be written as an integral form,

$$Z\left( \beta \right) =\dint\limits_{0}^{\infty }\Omega \left( E\right) \exp \left( -\beta E\right) dE,$$

where $\Omega \left( E\right)$ is the number of accessible microscopic states of the system with energy $E$. Note that, in the expressions for $Z\left( \beta \right)$ and $\Omega \left( E\right)$, we are omitting the dependence on the number $N$ of oscillators. Using the expression for $Z\left( \beta \right)$ obtained in the last exercise, perform a reverse Laplace transformation to obtain an asymptotic form (in the thermodynamic limit) for $\Omega \left( E\right)$. Compare with the expression calculated in the framework of the microcanonical ensemble. First, we use an integral representation of the $\delta$-function (see Appendix) to see that

$$\frac{1}{2\pi }\dint\limits_{-i\infty }^{+i\infty }Z\left( \beta ... ...\left( E^{\prime }-E\right) \right] d\beta =\Omega \left( E^{\prime }\right) .$$

Using the result of the previous exercise, we have

$$\Omega \left( E\right) =\frac{1}{2\pi }\dint\limits_{-i\infty }^{... ...ft( \frac{2\pi }{\beta \omega }\right) ^{N}\exp \left( \beta E\right) d\beta ,$$

which can be written in the form of a saddle-point integration (see Appendix),

$$\Omega \left( E\right) =\frac{1}{2\pi }\dint\limits_{-i\infty }^{... ...eft( \frac{2\pi }{\omega }\right) -\ln \beta +u\beta \right] \right\} d\beta ,$$

where $u=E/N$. Using the asymptotic integration techniques of the Appendix, we locate the saddle point at $\beta =1/u$ and write the asymptotic form (for $N\rightarrow \infty$),

$$\Omega \left( E\right) \sim \left( 2\pi u^{2}\right) ^{-1/2}\exp ... ...\{ N \left[ \ln \left( \frac{2\pi }{\omega }\right) +\ln u+1\right] \right\} .$$

Therefore, we have the entropy per oscillator,

$$\frac{1}{k_{B}}s=\ln \left( \frac{2\pi }{\omega }\right) +\ln u+1,$$

which should be compared with the well-known result for the classical one-dimensional harmonic oscillator in the microcanonical ensemble.

8. A system of $N$ one-dimensional localized oscillators, at a given temperature $T$, is associated with the Hamiltonian

$$\mathcal{H}=\dsum\limits_{i=1}^{N}\left[ \frac{1}{2m}p_{i}^{2}+V\left( q_{i}\right) \right] ,$$

where

$$V\left( q\right) =\left\{ \begin{array}{ll} \frac{1}{2}m\omega... ...\frac{1}{2}m\omega ^{2}q^{2}+\epsilon ; & \text{for }q<0, \end{array} \right.$$

with $\epsilon >0$. (a) Obtain the canonical partition function of this classical system. Calculate the internal energy per oscillator, $u=u\left( T\right)$. What is the form of $u\left( T\right)$ in the limits $\epsilon \rightarrow 0$ and $\epsilon \rightarrow \infty$? (b) Consider now the quantum analog of this model in the limit $\epsilon \rightarrow \infty$. Obtain an expression for the canonical partition function. What is the internal energy per oscillator of this quantum analog? The classical partition function is given by $Z=Z_{1}^{N}$, where

$$Z_{1}=\frac{\pi }{\beta \omega }\left[ \exp \left( -\beta \epsilon \right) +1 \right] .$$

Note that we have $u\rightarrow k_{B}T$ for both limits, $\epsilon \rightarrow 0$ and $\epsilon \rightarrow \infty$. In order to write the quantum partition function, in the $\epsilon \rightarrow \infty$ limit, we should consider odd values of $n$ only (note that even values of $n$ are associated with wave functions that do not vanish at the origin).