Exercises

1. Demonstrate that the geometric series
   $$S_n = \sum_{k=0,n-1}a r^{ k}$$
   can be summed to give
   $$S_n = a\left(\frac{1-r^{ n}}{1-r}\right).$$
   Here, $r\neq 1$ . Hence, deduce that
   $$S_\infty = \frac{a}{1-r},$$
   assuming that $0<r<1$ .

2. Let
   $$I(n) = \int_0^\infty x^{ n} {\rm e}^{-\alpha x^{ 2}} dx.$$
   Demonstrate that
   $$I(n)=-\frac{\partial I(n-2)}{\partial \alpha}.$$
   Furthermore, show that
   $$I(0) = \frac{\sqrt{\pi}}{2 \alpha^{ 1/2}}$$
   (see Exercise 2), and
   $$I(1)=\frac{1}{2 \alpha}.$$
   Hence, deduce that

   $$I(2) =\frac{\sqrt{\pi}}{4 \alpha^{ 3/2}},$$

   $$I(3) = \frac{1}{2 \alpha^{ 2}},$$

   $$I(4) = \frac{3\sqrt{\pi}}{8 \alpha^{ 5/2}},$$

   $$I(5) =\frac{1}{\alpha^{ 3}}.$$

   
   

3. A sample of mineral oil is placed in an external magnetic field $B$ . Each proton has spin $1/2$ , and a magnetic moment $\mu$ . It can, therefore, have two possible energies, $\epsilon=\mp \mu B$ , corresponding to the two possible orientations of its spin. An applied radio-frequency field can induce transitions between these two energy levels if its frequency $\nu$ satisfies the Bohr condition $h \nu = 2 \mu B$ . The power absorbed from this radiation field is then proportional to the difference in the number of nuclei in these two energy levels. Assume that the protons in the mineral oil are in thermal equilibrium at a temperature $T$ that is sufficiently high that $\mu B \ll k T$ . How does the absorbed power depend on the temperature, $T$ , of the sample?

4. Consider an assembly of $N_0$ weakly-interacting magnetic atoms per unit volume, held at temperature $T$ . According to classical physics, each atomic magnetic moment, $\mu$ , can be orientated so as to make an arbitrary angle $\theta$ with respect to the $z$ -direction (say). In the absence of an external magnetic field, the probability that the angle lies between $\theta$ and $\theta+d\theta$ is simply proportional to the solid angle, $2\pi \sin\theta d\theta$ , enclosed in this range. In the presence of a magnetic field of strength $B_z$ , directed parallel to the $z$ -axis, this probability is further proportional to the Boltzmann factor, $\exp(-\beta E)$ , where $\beta = 1/(k T)$ , and $E=-$$\mu$ $\cdot {\bf B}$ is the magnetic energy of the atom.
   1. Show that the classical mean magnetization is
      $$\overline{M_z} = N_0 \mu {\cal L}(x),$$
      where $x=\mu B_z/(k T)$ , and
      $${\cal L}(x)=\coth x-\frac{1}{x}$$
      is known as the Langevin function.

   2. Demonstrate that the corresponding quantum mechanical expression for a collection of atoms with overall angular momentum $J \hbar$ is
      $$\overline{M_z} = N_0 \mu {\cal B}_J(x),$$
      where $x=\mu B_z/(k T)$ , $\mu=g J \mu_B$ , and
      $${\cal B}_J(x)= \left[\left(\frac{2 J+1}{2 J}\right)\coth\left(\... ... J+1}{2 J} x\right)-\frac{1}{2 J} \coth\left(\frac{x}{2 J}\right)\right]$$
      is the Brillouin function.

   3. Show, finally, that the previous two expressions are identical in the classical limit $J\gg 1$ . (This is the classical limit because the spacing between adjacent magnetic energy levels, $E_m = \mu B_z m/J$ , where $m$ is an integer lying between $-J$ and $+J$ , is ${\mit\Delta} E_m=\mu B_z/J$ , which tends to zero as $J\rightarrow \infty$ .)

5. Consider a spin-1/2 (i.e., $J=1/2$ and $g=2$ ) paramagnetic substance containing $N$ non-interacting atoms.
   1. Show that the overall magnetic partition function, $Z$ , is such that
      $$\ln Z = N \ln\left[2 \cosh\left(\frac{\mu_B B_z}{k T}\right)\right],$$
      where $\mu_B$ is the Bohr magneton, $B_z$ the magnetic field-strength, and $T$ the absolute temperature.

   2. Demonstrate that the mean magnetic energy of the system is
      $$\overline{E} =-N \epsilon \tanh\left(\frac{\epsilon}{k T}\right),$$
      where $\epsilon=\mu_B B_z$ . Show that $\overline{E}\rightarrow -N \epsilon$ as $T\rightarrow 0$ , and $\overline{E}\rightarrow 0$ as $T\rightarrow\infty$ . Plot $\overline{E}/(N \epsilon)$ versus $k T/\epsilon$ .

   3. Demonstrate that the magnetic contribution to the specific heat of the substance is
      $$C= N k\left(\frac{\epsilon}{k T}\right)^2{\rm sech}^2\left(\frac{\epsilon}{k T}\right),$$
      and that
      $$C\simeq N k \left(\frac{2 \epsilon}{k T}\right)^2\exp\left(-\frac{2 \epsilon}{k T}\right)$$
      when $k T\ll \epsilon$ , whereas
      $$C\simeq N k \left(\frac{2 \epsilon}{k T}\right)^2$$
      when $k T\gg \epsilon$ . Plot $C/(N k)$ versus $k T/\epsilon$ . The sharp peak that is evident when $k T\sim\epsilon$ is known as the Schottky anomaly.

   4. Show that the magnetic contribution to the entropy of the substance is
      $$S=k N\left\{\ln\left[2 \cosh\left(\frac{\epsilon}{k T}\right)\... ...(\frac{\epsilon}{k T}\right)\tanh\left( \frac{\epsilon}{k T}\right)\right\},$$
      and demonsrate that $S\rightarrow 0$ as $T\rightarrow 0$ and $S\rightarrow N k \ln 2$ as $T\rightarrow\infty$ . Plot $S/(N k)$ versus $k T/\epsilon$ .

6. The nuclei of atoms in a certain crystalline solid have spin one. According to quantum theory, each nucleus can therefore be in any one of three quantum states labeled by the quantum number $m$ , where $m=1$ , 0, or $-1$ . This quantum number measures the projection of the nuclear spin along a crystal axis of the solid. Because the electric charge distribution in the nucleus is not spherically symmetric, but ellipsoidal, the energy of a nucleus depends on its spin orientation with respect to the internal electric field existing at its location. Thus a nucleus has the same energy $E=\epsilon$ in the state $m=1$ and the state $m=-1$ , compared with energy $E=0$ in the state $m=0$ .
   1. Find an expression, as a function of absolute temperature, $T$ , of the nuclear contribution to the molar internal energy of the solid.
   2. Find an expression, as a function of $T$ , of the nuclear contribution to the molar entropy of the solid.
   3. By directly counting the total number of accessible states, calculate the nuclear contribution to the molar entropy of the solid at very low temperatures. Calculate it also at high temperatures. Show that the expression in part (b) reduces properly to these values as $T\rightarrow 0$ and $T\rightarrow\infty$ .
   4. Make a qualitative graph showing the temperature dependence of the nuclear contribution to the molar heat capacity of the solid. Calculate the temperature dependence explicitly. What is the temperature dependence for large values of $T$ ?

7. A dilute solution of a macromolecule (large molecules of biological interest) at temperature $T$ is placed in an ultracentrifuge rotating with angular velocity $\omega$ . The centripetal acceleration $\omega^{ 2} r$ acting on a particle of mass $m$ may then be replaced by an equivalent centrifugal force $m \omega^{ 2} r$ in the rotating frame of reference.
   1. Find how the relative density, $\rho(r)$ , of molecules varies with their radial distance, $r$ , from the axis of rotation.
   2. Show qualitatively how the molecular weight of the macromolecules can be determined if the density ratio $\rho_1/\rho_2$ at the radii $r_1$ and $r_2$ is measured by optical means.

8. Consider a homogeneous mixture of inert monatomic ideal gases at absolute temperature $T$ in a container of volume $V$ . Let there be $\nu_1$ moles of gas 1, $\nu_2$ moles of gas 2, ..., and $\nu_k$ moles of gas $k$ .
   1. By considering the classical partition function of this system, derive its equation of state. In other words, find an expression for its total mean pressure, $\overline{p}$ .
   2. How is this total pressure, $\overline{p}$ , of the gas related to the so-called partial pressure, $\overline{p}_i$ , that the $i$ th gas would produce if it alone occupied the entire volume at this temperature?

9. Monatomic molecules adsorbed on a surface are free to move on this surface, and can be treated as a classical ideal two-dimensional gas. At absolute temperature $T$ , what is the heat capacity per mole of molecules thus adsorbed on a surface of fixed size?

10. Consider a system in thermal equilibrium with a heat bath held at absolute temperature $T$ . The probability of observing the system in some state $r$ of energy $E_r$ is is given by the canonical probability distribution:
    $$P_r = \frac{\exp(-\beta E_r)}{Z},$$
    where $\beta = 1/(k T)$ , and
    $$Z = \sum_r \exp(-\beta E_r)$$
    is the partition function.
    1. Demonstrate that the entropy can be written
       $$S=-k\sum_r P_r \ln P_r.$$

    2. Demonstrate that the mean Helmholtz free energy is related to the partition function according to
       $$Z =\exp\left(-\beta \overline{F}\right).$$

11. Show that the logarithm of the classical partition function of an ideal gas consisting of $N$ identical molecules of mass $m$ , held in a container of volume $V$ , and in thermal equilibrium with a heat bath held at absolute temperature $T$ , is
    $$\ln Z = N\left[\ln\left(\frac{V}{N}\right)-\frac{3}{2} \ln\beta +\sigma\right],$$
    where
    $$\sigma =\frac{3}{2}\ln\left(\frac{2\pi m}{h_0^{ 2}}\right)+1.$$
    Here, $\beta=k T$ , and $h_0$ parameterizes how finely classical phase-space is partitioned. Demonstrate that:

    1. $$\overline{E} = \frac{3}{2} \nu R T.$$

    2. $$\overline{H} = \frac{5}{2} \nu R T.$$

    3. $$\overline{F} = -\nu R T\left[\ln\left(\frac{V}{N}\right)-\frac{3}{2} \ln\beta +\sigma\right].$$

    4. $$\overline{G} = -\nu R T\left[\ln\left(\frac{V}{N}\right)-\frac{3}{2} \ln\beta +\sigma-1\right].$$

    5. $$\frac{{\mit\Delta}^\ast E}{\overline{E}}= \left(\frac{2}{3 N}\right)^{1/2}.$$

12. Use the Debye approximation to calculate the contribution of lattice vibrations to the thermodynamic functions of a solid.
    1. To be more specific, show that

       $$\ln Z = -\frac{9}{8} N \frac{\theta_D}{T}-3 N \ln\left(1-{\rm e}^{-\theta_D/T}\right)+ N D\left(\frac{\theta_D}{T}\right),$$

       $$\overline{E} = \frac{9}{8} N k \theta_D + 3 N k T D\left(\frac{\theta_D}{T}\right),$$

       $$S = N k\left[-3 \ln\left(1-{\rm e}^{-\theta_D/T}\right)+4 D\left(\frac{\theta_D}{T}\right)\right].$$

       
       
       Here, $N$ is the number of atoms in the solid, $T$ the absolute temperature, $Z$ the partition function, $\overline{E}$ the mean energy, $S$ the entropy, $D(y)\equiv(3/y^{ 3})\int_0^y x^{ 3} dx/({\rm e}^{ x}-1)$ , $\theta_D= \hbar \omega_D/k$ , and $\omega_D$ is the Debye frequency.

    2. Show that in the limit $T\ll \theta_D$ ,

       $$\frac{\ln Z}{N} \simeq -\frac{9}{8}\left(\frac{\theta_D}{T}\right) + \frac{\pi^{ 4}}{5}\left(\frac{T}{\theta_D}\right)^3,$$

       $$\frac{\overline{E}}{N k T} \simeq \frac{9}{8}\left(\frac{\theta_D}{T}\right) + \frac{3 \pi^{ 4}}{5}\left(\frac{T}{\theta_D}\right)^3,$$

       $$\frac{S}{N k} \simeq \frac{4\pi^{ 4}}{5}\left(\frac{T}{\theta_D}\right)^3.$$

       
       
       [Hint: $\int_0^y x^{ 3} dx/({\rm e}^{ x}-1)=\pi^{ 4}/15$ .]

    3. Show that in the limit $T\gg \theta_D$ ,

       $$\frac{\ln Z}{N} \simeq -\frac{9}{8}\left(\frac{\theta_D}{T}\right) -3 \ln\left(\frac{\theta_D}{T}\right)+1,$$

       $$\frac{\overline{E}}{N k T} \simeq \frac{9}{8}\left(\frac{\theta_D}{T}\right) +3,$$

       $$\frac{S}{N k} \simeq -3 \ln\left(\frac{\theta_D}{T}\right)+4.$$

       
       

    4. Further, show that
       $$\bar{p} = \frac{\gamma \overline{E}}{V},$$
       where $\bar{p}$ is the mean pressure, $V$ the volume, and
       $$\gamma= -\frac{d\ln\theta_D}{d\ln V} = \frac{1}{3}.$$

13. For the quantized lattice waves (phonons) in the Debye theory of specific heats, the frequency, $\omega$ , of a propagating wave is related to its wavevector, ${\bf k}$ , by the dispersion relation $\omega=c_s k$ , where $c_s$ is the velocity of sound. On the other hand, in a ferromagnetic solid at low temperatures, quantized waves of magnetization (spin waves) have their frequencies, $\omega$ , related to their wavevectors, ${\bf k}$ , according to the dispersion relation $\omega=A k^{ 2}$ , where $A$ is a constant. Show that, at low temperatures, the contribution of spin waves to the heat capacity of the ferromagnet varies as $T^{ 3/2}$ .

14. Verify directly that
    $$\overline{v^{ 2}} = \frac{3 k T}{m}$$
    for a Maxwellian velocity distribution. Here, $m$ is the molecular mass, and $T$ the absolute temperature.

15. Show that the mean speed of molecules effusing through a small hole in a gas-filled container is $3\pi/8=1.18$ times larger than the mean speed of the molecules within the container.

16. A vessel is closed off by a porous partition through which gases can pass by effusion and then be pumped off to some collecting chamber. The vessel is filled with dilute gas containing two types of molecule which differ because they contain different atomic isotopes, and thus have the different masses, $m_1$ and $m_2$ . The concentrations of these molecules are $c_1$ and $c_2$ , respectively, and are maintained constant within the vessel by constantly replenishing the supply of gas in it.
    1. Let $c_1'$ and $c_2'$ be the concentrations of the two types of molecule in the collecting chamber. What is the ratio $c_1'/c_2'$ ?
    2. By using the gas ${\rm UF}_6$ , one can attempt to separate ${\rm U}^{235}$ from ${\rm U}^{237}$ , the first of these isotopes being the one that undergoes nuclear fission reactions. The molecules in the vessel are then ${\rm U}^{235}{\rm F}_6^{ 19}$ and ${\rm U}^{238}{\rm F}_6^{ 19}$ . The concentrations of these molecules in the vessel corresponds to the natural abundances of the two isotopes: $c_{238}=99.3$ percent, and $c_{235}=0.7$ percent. What is the ratio, $c_{235}'/c_{238}'$ , of the two isotopic concentrations in the gas collected after effusion, compared to the original concentration ratio, $c_{235}/c_{238}$ ?

17. Show that the mean force per unit area exerted on the walls of a container enclosing a Maxwellian gas is
    $$\bar{p}= \int_{v_z>0}2 m v_z {\mit\Phi}({\bf v}) d^{ 3}{\bf v} =\int_{v_z>0}2 m v_z^{ 2} f(v) d^{ 3}{\bf v},$$
    where $m$ is the molecular mass, and the outward normal to the wall element is directed in the $z$ -direction. Hence, deduce that
    $$\bar{p}= \frac{1}{3} m n \overline{v^{ 2}}= n k T,$$
    where $n$ is the molecular concentration, and $T$ the absolute gas temperature.

18. Consider a spin-1/2 ferromagnetic material consisting of $N$ identical atoms with $S=1/2$ and $g=2$ . Let each atom have$n$ nearest neighbors.
    1. Show that
       $${\cal B}_{1/2}(\eta)=\tanh\left(\frac{\eta}{2}\right),$$
       where ${\cal B}_S(\eta)$ is a Brillouin function.
    2. Use the molecular field approximation to demonstrate that

       $$m = \tanh\left[\frac{\beta}{\beta_c}\left(m+\beta_c \mu_B B_z\right)\right],$$

       
       
       where $m=\overline{M}_z/(N \mu_B)$ , $\beta_c=1/(k T_c)$ , and $k T_c=J n/2$ .
    3. Show that for $T$ slightly less than $T_c$ , and in the absence of an external magnetic field,

       $$m\simeq \sqrt{3}\left(1-\frac{T}{T_c}\right)^{1/2},$$

       
       

    4. Demonstrate that exactly at the critical temperature,
       $$m \simeq \left(\frac{3 \mu_B B_z}{k T_c}\right)^{1/3}.$$

    5. Finally, show that for $T$ slightly larger than $T_c$ ,
       $$m\simeq \frac{\mu_B B_z}{k (T-T_c)}.$$

    [Hint: At small arguments $\tanh(z)\simeq z-z^{ 3}/3$ .]