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An Introduction to Thermal Physics

Daniel V. Schroeder

Physics

1st Edition · 356 pages

ISBN: 9780201380279

Chapter 7: Q. 7.69 (page 323)

If you have a computer system that can do numerical integrals, it's not particularly difficult to evaluate $μ$ 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 $t = T / T_{c}, c = μ / k T_{c}, and x = ϵ / k T_{c}$ . Express the integral that defines $μ$ , equation $N = \int_{0}^{\infty} g (ϵ) \frac{1}{e^{(ϵ - μ) / k T} - 1} d ϵ$ , in terms of these variables, you should obtain the equation
$2.315 = \int_{0}^{\infty} \frac{\sqrt{x} d x}{e^{(x - c) / t} - 1}$
(b) According to given figure , the correct value of $c$ when $T = 2 T_{c}$ , is approximately $- 0.8$ . Plug in these values and check that the equation above is approximately satisfied.
(c) Now vary $μ$ , holding $T$ fixed, to find the precise value of $μ$ for $T = 2 T_{c}$ . Repeat for values of $T / T_{c}$ ranging from $1.2$ up to $3.0$ , in increments of $0.2$ . Plot a graph of $μ$ as a function of temperature.

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Most popular questions from this chapter

For a system of particles at room temperature, how large must $ϵ - μ$ be before the Fermi-Dirac, Bose-Einstein, and Boltzmann distributions agree within $1 %$ ? Is this condition ever violated for the gases in our atmosphere? Explain.

For a brief time in the early universe, the temperature was hot enough to produce large numbers of electron-positron pairs. These pairs then constituted a third type of "background radiation," in addition to the photons and neutrinos (see Figure 7.21). Like neutrinos, electrons and positrons are fermions. Unlike neutrinos, electrons and positrons are known to be massive (ea.ch with the same mass), and each has two independent polarization states. During the time period of interest, the densities of electrons and positrons were approximately equal, so it is a good approximation to set the chemical potentials equal to zero as in Figure 7.21. When the temperature was greater than the electron mass times $\frac{c^{2}}{k}$ , the universe was filled with three types of radiation: electrons and positrons (solid arrows); neutrinos (dashed); and photons (wavy). Bathed in this radiation were a few protons and neutrons, roughly one for every billion radiation particles. the previous problem. Recall from special relativity that the energy of a massive particle is $ϵ = \sqrt{(p c)^{2} + {(m c^{2})}^{2}}$ .

(a) Show that the energy density of electrons and positrons at temperature $T$ is given by

$u (T) = \int_{0}^{\infty} \frac{x^{2} \sqrt{x^{2} + {(m c^{2} / k T)}^{2}}}{e^{\sqrt{x^{2} + {(m c^{2} / k T)}^{2}}} + 1} d x; w h e r e u (T) = \int_{0}^{\infty} \frac{x^{2} \sqrt{x^{2} + {(m c^{2} / k T)}^{2}}}{e^{\sqrt{x^{2} + {(m c^{2} / k T)}^{2}}} + 1} d x$

(b) Show that $u (T)$ goes to zero when $k T ≪ m c^{2}$ , and explain why this is a

reasonable result.

( c) Evaluate $u (T)$ in the limit $k T ≫ m c^{2}$ , and compare to the result of the

the previous problem for the neutrino radiation.

(d) Use a computer to calculate and plot $u (T)$ at intermediate temperatures.

(e) Use the method of Problem 7.46, part (d), to show that the free energy

density of the electron-positron radiation is

$\frac{F}{V} = - \frac{16 π (k T)^{4}}{(h c)^{3}} f (T); w h e r e f (T) = \int_{0}^{\infty} x^{2} \ln (1 + e^{- \sqrt{x^{2} + {(m c^{2} / k T)}^{2}}}) d x$

Evaluate $f (T)$ in both limits, and use a computer to calculate and plot $f (T)$ at intermediate

temperatures.

(f) Write the entropy of the electron-positron radiation in terms of the functions

$u (T)$ and $f (T)$ . Evaluate the entropy explicitly in the high-T limit.

In addition to the cosmic background radiation of photons, the universe is thought to be permeated with a background radiation of neutrinos (v) and antineutrinos ( $\bar{v}$ ), currently at an effective temperature of 1.95 K. There are three species of neutrinos, each of which has an antiparticle, with only one allowed polarisation state for each particle or antiparticle. For parts (a) through (c) below, assume that all three species are exactly massless

(a) It is reasonable to assume that for each species, the concentration of neutrinos equals the concentration of antineutrinos, so that their chemical potentials are equal: $μ_{ν} = μ_{\bar{ν}}$ . Furthermore, neutrinos and antineutrinos can be produced and annihilated in pairs by the reaction

$ν + \bar{ν} \leftrightarrow 2 γ$

(where y is a photon). Assuming that this reaction is at equilibrium (as it would have been in the very early universe), prove that u =0 for both the neutrinos and the antineutrinos.

(b) If neutrinos are massless, they must be highly relativistic. They are also fermions: They obey the exclusion principle. Use these facts to derive a formula for the total energy density (energy per unit volume) of the neutrino-antineutrino background radiation. differences between this "neutrino gas" and a photon gas. Antiparticles still have positive energy, so to include the antineutrinos all you need is a factor of 2. To account for the three species, just multiply by 3.) To evaluate the final integral, first change to a dimensionless variable and then use a computer or look it up in a table or consult Appendix B. (Hint: There are very few

(c) Derive a formula for the number of neutrinos per unit volume in the neutrino background radiation. Evaluate your result numerically for the present neutrino temperature of 1.95 K.

d) It is possible that neutrinos have very small, but nonzero, masses. This wouldn't have affected the production of neutrinos in the early universe, when me would have been negligible compared to typical thermal energies. But today, the total mass of all the background neutrinos could be significant. Suppose, then, that just one of the three species of neutrinos (and the corresponding antineutrino) has a nonzero mass m. What would mc² have to be (in eV), in order for the total mass of neutrinos in the universe to be comparable to the total mass of ordinary matter?

Show that when a system is in thermal and diffusive equilibrium with a reservoir, the average number of particles in the system is

$\overset{—}{N} = \frac{k T}{Z} \frac{\partial Z}{\partial μ}$

where the partial derivative is taken at fixed temperature and volume. Show also that the mean square number of particles is

$\bar{N^{2}} = \frac{{(k T)}^{2}}{Z} \frac{\partial^{2} Z}{\partial μ^{2}}$

Use these results to show that the standard deviation of $N$ is

$σ_{N} = \sqrt{k T (\partial \overset{—}{N} / \partial μ)},$

in analogy with Problem $6.18$ Finally, apply this formula to an ideal gas, to obtain a simple expression for $σ_{N}$ in terms of $\bar{N}$ Discuss your result briefly.

Consider a gas of noninteracting spin-0 bosons at high temperatures, when $T ≫ T_{c}$ . (Note that “high” in this sense can still mean below 1 K.)

Show that, in this limit, the Bose-Einstein function can be written approximately as
${\bar{n}}_{B E} = e^{- (\in - μ) / k T} [1 + e^{- (\in - μ) / k T} + \dots]$ .
Keeping only the terms shown above, plug this result into equation 7.122 to derive the first quantum correction to the chemical potential for gas of bosons.
Use the properties of the grand free energy (Problems 5.23 and 7.7) to show that the pressure of any system is given by In $P = (k T / V)$ , where $Z$ is the grand partition function. Argue that, for gas of noninteracting particles, In $Z$ can be computed as the sum over all modes (or single-particle states) of In $Z_{i}$ , where $Z_{i}$ ; is the grand partition function for the $i th$ mode.
Continuing with the result of part (c), write the sum over modes as an integral over energy, using the density of states. Evaluate this integral explicitly for gas of noninteracting bosons in the high-temperature limit, using the result of part (b) for the chemical potential and expanding the logarithm as appropriate. When the smoke clears, you should find
$p = \frac{N k T}{V} (1 - \frac{N v_{Q}}{4 \sqrt{2} V})$ ,
again neglecting higher-order terms. Thus, quantum statistics results in a lowering of the pressure of a boson gas, as one might expect.
Write the result of part (d) in the form of the virial expansion introduced in Problem 1.17, and read off the second virial coefficient, $B (T)$ . Plot the predicted $B (T)$ for a hypothetical gas of noninteracting helium-4 atoms.
Repeat this entire problem for gas of spin-1/2 fermions. (Very few modifications are necessary.) Discuss the results, and plot the predicted virial coefficient for a hypothetical gas of noninteracting helium-3 atoms.

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