Consider a gas of $n$ identical spin-0 bosons confined by an isotropic three-dimensional harmonic oscillator potential. (In the rubidium experiment discussed above, the confining potential was actually harmonic, though not isotropic.) The energy levels in this potential are $\epsilon =nhf$, where $n$ is any nonnegative integer and $f$ is the classical oscillation frequency. The degeneracy of level $n\text{is}(n+1)(n+2)/2$.

(a) Find a formula for the density of states, $g\left(\epsilon \right)$, for an atom confined by this potential. (You may assume $n>>1$.)

(b) Find a formula for the condensation temperature of this system, in terms of the oscillation frequency $f$.

(c) This potential effectively confines particles inside a volume of roughly the cube of the oscillation amplitude. The oscillation amplitude, in turn, can be estimated by setting the particle's total energy (of order $kT$ ) equal to the potential energy of the "spring." Making these associations, and neglecting all factors of 2 and $\mathrm{\pi }$ and so on, show that your answer to part (b) is roughly equivalent to the formula derived in the text for the condensation temperature of bosons confined inside a box with rigid walls.

Most spin-1/2 fermions, including electrons and helium-3 atoms, have nonzero magnetic moments. A gas of such particles is therefore paramagnetic. Consider, for example, a gas of free electrons, confined inside a three-dimensional box. The z component of the magnetic moment of each electron is ±µa. In the presence of a magnetic field B pointing in the z direction, each "up" state acquires an additional energy of $-{\mu }_{\mathrm{B}}B$, while each "down" state acquires an additional energy of $+{\mu }_{\mathrm{B}}B$

(a) Explain why you would expect the magnetization of a degenerate electron gas to be substantially less than that of the electronic paramagnets studied in Chapters 3 and 6, for a given number of particles at a given field strength.

(b) Write down a formula for the density of states of this system in the presence of a magnetic field B, and interpret your formula graphically.

(c) The magnetization of this system is ${\mu }_{\mathrm{B}}\left(N_{\uparrow }-N_{\downarrow }\right.$, where Nr and N1 are the numbers of electrons with up and down magnetic moments, respectively. Find a formula for the magnetization of this system at $T=0$, in terms of N, µa, B, and the Fermi energy.

(d) Find the first temperature-dependent correction to your answer to part (c), in the limit $T\ll T_{\mathrm{F}}$. You may assume that ${\mu }_{\mathrm{B}}B\ll kT$; this implies that the presence of the magnetic field has negligible effect on the chemical potential $\mu$. (To avoid confusing µB with µ, I suggest using an abbreviation such as o for the quantity µaB.)

In Section 6.5 I derived the useful relation $F=-kT\ln \left(Z\right)$ between the Helmholtz free energy and the ordinary partition function. Use analogous argument to prove that $\varphi =-kT\times \ln \left(\hat{Z}\right)$, where $\hat{Z}$ is the grand partition function and $\varphi$ is the grand free energy introduced in Problem 5.23.

A ferromagnet is a material (like iron) that magnetizes spontaneously, even in the absence of an externally applied magnetic field. This happens because each elementary dipole has a strong tendency to align parallel to its neighbors. At $t=0$ the magnetization of a ferromagnet has the maximum possible value, with all dipoles perfectly lined up; if there are $N$ atoms, the total magnetization is typically$~2\mu eN$, where µa is the Bohr magneton. At somewhat higher temperatures, the excitations take the form of spin waves, which can be visualized classically as shown in Figure $7.30$. Like sound waves, spin waves are quantized: Each wave mode can have only integer multiples of a basic energy unit. In analogy with phonons, we think of the energy units as particles, called magnons. Each magnon reduces the total spin of the system by one unit of $\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$21r$}\right.$and therefore reduces the magnetization by $~2\mu e$. However, whereas the frequency of a sound wave is inversely proportional to its wavelength, the frequency of a spin-wave is proportional to the square of $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.$.. (in the limit of long wavelengths). Therefore, since$\in =hf$ and $p=\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.$.. for any "particle," the energy of a magnon is proportional

In the ground state of a ferromagnet, all the elementary dipoles point in the same direction. The lowest-energy excitations above the ground state are spin waves, in which the dipoles precess in a conical motion. A long-wavelength spin wave carries very little energy because the difference in direction between neighboring dipoles is very small.

to the square of its momentum. In analogy with the energy-momentum relation for an ordinary nonrelativistic particle, we can write $\in =\raisebox{1ex}{$p^2$}\!\left/ \!\raisebox{-1ex}{$2pm$}\right.$*, where$m$* is a constant related to the spin-spin interaction energy and the atomic spacing. For iron, m* turns out to equal $1.24\times {10}^{29}kg$, about$14$ times the mass of an electron. Another difference between magnons and phonons is that each magnon ( or spin-wave mode) has only one possible polarization.

(a) Show that at low temperatures, the number of magnons per unit volume in a three-dimensional ferromagnet is given by

$\frac{Nm}{V}=2\pi {\left(\frac{2m\times kT}{h^2}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\int }_0^{\infty }\frac{\sqrt{x}}{e^x-1}dx.$

Evaluate the integral numerically.

(b) Use the result of part ($a$) to find an expression for the fractional reduction in magnetization, $\left(M\right(O)-M(T\left)\right)/M\left(O\right).$ Write your answer in the form ${(T/To)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$, and estimate the constant$T_0$ for iron.

(c) Calculate the heat capacity due to magnetic excitations in a ferromagnet at low temperature. You should find $Cv/Nk={(T/Ti)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ , where $Ti$ differs from To only by a numerical constant. Estimate$Ti$ for iron, and compare the magnon and phonon contributions to the heat capacity. (The Debye temperature of iron is $470k$.)

(d) Consider a two-dimensional array of magnetic dipoles at low temperature. Assume that each elementary dipole can still point in any (threedimensional) direction, so spin waves are still possible. Show that the integral for the total number of magnons diverge in this case. (This result is an indication that there can be no spontaneous magnetization in such a two-dimensional system. However, in Section $8.2$ we will consider a different two-dimensional model in which magnetization does occur.)

Suppose that the concentration of infrared-absorbing gases in earth's atmosphere were to double, effectively creating a second "blanket" to warm the surface. Estimate the equilibrium surface temperature of the earth that would result from this catastrophe. (Hint: First show that the lower atmospheric blanket is warmer than the upper one by a factor of $2^{1/4}$. The surface is warmer than the lower blanket by a smaller factor.)

If you have a computer system that can do numerical integrals, it's not particularly difficult to evaluate $\mu$ for $T>T_c$.

(a) As usual when solving a problem on a computer, it's best to start by putting everything in terms dimensionless variables. So define $t=T/T_c,c=\mu /k{T}_c,\text{and}x=ϵ/k{T}_c$. Express the integral that defines $\mu$, equation $N={\int }_0^{\infty }g\left(ϵ\right)\frac{1}{e^{(ϵ-\mu )/kT}-1}dϵ$, in terms of these variables, you should obtain the equation

$2.315={\int }_0^{\infty }\frac{\sqrt{x}dx}{e^{(x-c)/t}-1}$

(b) According to given figure , the correct value of $c$ when $T=2T_c$ , is approximately $-0.8$. Plug in these values and check that the equation above is approximately satisfied.

(c) Now vary $\mu$, holding $T$ fixed, to find the precise value of $\mu$ for $T=2T_c$ . Repeat for values of$T/T_c$ ranging from $1.2$ up to $3.0$, in increments of $0.2$. Plot a graph of $\mu$ as a function of temperature.