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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 渭for T>Tc.

(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/Tc,c=渭/kTc,andx=系/kTc. Express the integral that defines 渭, equation N=鈭�0鈭�g(系)1e(系-渭)/kT-1d系, in terms of these variables, you should obtain the equation

2.315=鈭�0鈭�xdxe(x-c)/t-1

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

(c) Now vary 渭, holding Tfixed, to find the precise value of 渭for T=2Tc. Repeat for values ofT/Tcranging from 1.2up to 3.0, in increments of 0.2. Plot a graph of 渭as a function of temperature.

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Most popular questions from this chapter

Consider the electromagnetic radiation inside a kiln, with a volume of V= I m3 and a temperature of 1500 K.

(a) What is the total energy of this radiation?

(b) Sketch the spectrum of the radiation as a function of photon energy.

(c) What fraction of all the energy is in the visible portion of the spectrum, with wavelengths between 400 nm and 700 nm?

The results of the previous problem can be used to explain why the current temperature of the cosmic neutrino background (Problem 7.48) is 1.95 K rather than 2.73 K. Originally the temperatures of the photons and the neutrinos would have been equal, but as the universe expanded and cooled, the interactions of neutrinos with other particles soon became negligibly weak. Shortly thereafter, the temperature dropped to the point where kT/c2 was no longer much greater than the electron mass. As the electrons and positrons disappeared during the next few minutes, they "heated" the photon radiation but not the neutrino radiation.

(a) Imagine that the universe has some finite total volume V, but that V is increasing with time. Write down a formula for the total entropy of the electrons, positrons, and photons as a function of V and T, using the auxiliary functions u(T) and f(T) introduced in the previous problem. Argue that this total entropy would have ben conserved in the early universe, assuming that no other species of particles interacted with these.

(b) The entropy of the neutrino radiation would have been separately conserved during this time period, because the neutrinos were unable to interact with anything. Use this fact to show that the neutrino temperature Tv and the photon temperature T are related by

TT谓32蟺445+u(T)+f(T)=constant

as the universe expands and cools. Evaluate the constant by assuming that T=Tv when the temperatures are very high.

(c) Calculate the ratio T/Tv, in the limit of low temperature, to confirm that the present neutrino temperature should be 1.95 K.

(d) Use a computer to plot the ratio T/Tv, as a function of T, for kT/mc2ranging from 0 to 3.*

For a system of fermions at room temperature, compute the probability of a single-particle state being occupied if its energy is

(a) 1eVless than 渭

(b) 0.01eVless than 渭

(c) equal to 渭

(d) 0.01eVgreater than 渭

(e) 1eVgreater than渭

Figure 7.37 shows the heat capacity of a Bose gas as a function of temperature. In this problem you will calculate the shape of this unusual graph.

(a) Write down an expression for the total energy of a gas of Nbosons confined to a volume V, in terms of an integral (analogous to equation 7.122).

(b) For T<Tcyou can set 渭=0. Evaluate the integral numerically in this case, then differentiate the result with respect to Tto obtain the heat capacity. Compare to Figure 7.37.

(c) Explain why the heat capacity must approach 32Nkin the high- Tlimit.

(d) For T>Tcyou can evaluate the integral using the values of 渭calculated in Problem 7.69. Do this to obtain the energy as a function of temperature, then numerically differentiate the result to obtain the heat capacity. Plot the heat capacity, and check that your graph agrees with Figure 7.37.

Figure 7.37. Heat capacity of an ideal Bose gas in a three-dimensional box.

An atomic nucleus can be crudely modeled as a gas of nucleons with a number density of 0.18fm-3(where 1fm=10-15m). Because nucleons come in two different types (protons and neutrons), each with spin 1/2, each spatial wavefunction can hold four nucleons. Calculate the Fermi energy of this system, in MeV. Also calculate the Fermi temperature, and comment on the result.
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