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Chapter 7: Q 7.40 (page 293)

Starting from equation 7.83, derive a formula for the density of states of a photon gas (or any other gas of ultra relativistic particles having two polarisation states). Sketch this function.

Short Answer

Expert verified

Hence, the formula for density of states of a photon gas isg(ϵ)=8πVϵ2(hc)3

Step by step solution

01

Given information

The equation 7.83 is

UV=8π(hc)3∫0∞ϵ3eϵ/kT-1dϵ

02

Explanation

The equation 7.83 is:

UV=8π(hc)3∫0∞ϵ3eϵ/kT-1dϵ

We can write the equation as:

localid="1647752992962">U=∫0∞ϵ8πVϵ2(hc)31eϵ/kT-1dϵ(1)

Distribution function for Planck's constant is given as:

n¯Pl=1eϵ/kT-1

Substituting this into (1)

U=∫0∞ϵ8πVϵ2(hc)3n¯Pldϵ

Hence the energy density for Planck's constant is

g(ϵ)=8πVϵ2(hc)3

Using Python to solve this function, the code is:

The graph is:

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Most popular questions from this chapter

The sun is the only star whose size we can easily measure directly; astronomers therefore estimate the sizes of other stars using Stefan's law.

(a) The spectrum of Sirius A, plotted as a function of energy, peaks at a photon energy of2.4eV, while Sirius A is approximately 24times as luminous as the sun. How does the radius of Sirius A compare to the sun's radius?

(b) Sirius B, the companion of Sirius A (see Figure 7.12), is only role="math" localid="1647765883396" 3%as luminous as the sun. Its spectrum, plotted as a function of energy, peaks at about7eV. How does its radius compare to that of the sun?

(c) The spectrum of the star Betelgeuse, plotted as a function of energy, peaks at a photon energy of 0.8eV, while Betelgeuse is approximately10,000times as luminous as the sun. How does the radius of Betelgeuse compare to the sun's radius? Why is Betelgeuse called a "red supergiant"?

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=0the magnetization of a ferromagnet has the maximum possible value, with all dipoles perfectly lined up; if there are Natoms, the total magnetization is typically~2μeN, 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 h21rand therefore reduces the magnetization by ~2μe. 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 1λ.. (in the limit of long wavelengths). Therefore, since∈=hfand p=hλ.. 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 ∈=p22pm*, wherem* is a constant related to the spin-spin interaction energy and the atomic spacing. For iron, m* turns out to equal 1.24×1029kg, about14times 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

NmV=2π2m×kTh232∫0∞xex-1dx.

Evaluate the integral numerically.

(b) Use the result of part (a) to find an expression for the fractional reduction in magnetization, (M(O)-M(T))/M(O).Write your answer in the form (T/To)32, and estimate the constantT0for iron.

(c) Calculate the heat capacity due to magnetic excitations in a ferromagnet at low temperature. You should find Cv/Nk=(T/Ti)32, where Tidiffers from To only by a numerical constant. EstimateTifor 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 8.2we will consider a different two-dimensional model in which magnetization does occur.)

In the text I claimed that the universe was filled with ionised gas until its temperature cooled to about 3000 K. To see why, assume that the universe contains only photons and hydrogen atoms, with a constant ratio of 109 photons per hydrogen atom. Calculate and plot the fraction of atoms that were ionised as a function of temperature, for temperatures between 0 and 6000 K. How does the result change if the ratio of photons to atoms is 108 or 1010? (Hint: Write everything in terms of dimensionless variables such as t = kT/I, where I is the ionisation energy of hydrogen.)

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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.

(a) Write down a formula for the probability of a single donor atom being ionized. Do not neglect the fact that the electron, if present, can have two independent spin states. Express your formula in terms of the temperature, the ionization energy I, and the chemical potential of the "gas" of ionized electrons.

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(c) Now assume that every conduction electron comes from an ionized donor atom. In this case the number of conduction electrons is equal to the number of donors that are ionized. Use this condition to derive a quadratic equation for Ncin terms of the number of donor atoms Nd, eliminatingµ. Solve for Ncusing the quadratic formula. (Hint: It's helpful to introduce some abbreviations for dimensionless quantities. Tryx=NcNd,t=kTland so on.)

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