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Chapter 7: Q. 7.66 (page 323)
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Number of photons in a photon gas.
(a) Show that the number of photons in equilibrium in a box of volume V at temperature T is
The integral cannot be done analytically; either look it up in a table or evaluate it numerically.
(b) How does this result compare to the formula derived in the text for the entropy of a photon gas? (What is the entropy per photon, in terms of k?)
(c) Calculate the number of photons per cubic meter at the following temperatures: 300 K; 1500 K (a typical kiln); 2.73 K (the cosmic background radiation).
In Section 6.5 I derived the useful relation between the Helmholtz free energy and the ordinary partition function. Use analogous argument to prove that , where is the grand partition function and is the grand free energy introduced in Problem 5.23.
Consider a system consisting of a single impurity atom/ion in a semiconductor. Suppose that the impurity atom has one "extra" electron compared to the neighboring atoms, as would a phosphorus atom occupying a lattice site in a silicon crystal. The extra electron is then easily removed, leaving behind a positively charged ion. The ionized electron is called a conduction electron, because it is free to move through the material; the impurity atom is called a donor, because it can "donate" a conduction electron. This system is analogous to the hydrogen atom considered in the previous two problems except that the ionization energy is much less, mainly due to the screening of the ionic charge by the dielectric behavior of the medium.
For a system of fermions at room temperature, compute the probability of a single-particle state being occupied if its energy is
(a) less than
(b) less than
(c) equal to
(d) greater than
(e) greater than
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 (), 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: . Furthermore, neutrinos and antineutrinos can be produced and annihilated in pairs by the reaction
(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?
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