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Chapter 7: Q 7.38 (page 293)

It's not obvious from Figure 7.19 how the Planck spectrum changes as a function of temperature. To examine the temperature dependence, make a quantitative plot of the functionu(系) for T = 3000 K and T = 6000 K (both on the same graph). Label the horizontal axis in electron-volts.

Short Answer

Expert verified

The function is:

u(系)=8蟺(hc)3系3e系/kT-1

Step by step solution

01

Given information

The Planck spectrum changes as a function of temperature. To examine the temperature dependence, make a quantitative plot of the functionu(系) for T = 3000 K and T = 6000 K

02

Explanation

The photon's Planck spectrum is given as:

u(系)=8蟺(hc)3系3e系/kT-1(1)

This function must be plotted at temperatures of T= 3000 K and T = 6000 K, with the constants in eV supplied by:

h=4.136脳10-15eV路sk=8.62脳10-5eV/Kc=3.00脳108m/s

Using python to plot the function and the code is:

The graph is:

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Most popular questions from this chapter

Consider any two internal states, s1 and s2, of an atom. Let s2 be the higher-energy state, so that Es2-Es1=系 for some positive constant. If the atom is currently in state s2, then there is a certain probability per unit time for it to spontaneously decay down to state s1, emitting a photon with energy e. This probability per unit time is called the Einstein A coefficient:

A = probability of spontaneous decay per unit time.

On the other hand, if the atom is currently in state s1 and we shine light on it with frequency f=系/h, then there is a chance that it will absorb photon, jumping into state s2. The probability for this to occur is proportional not only to the amount of time elapsed but also to the intensity of the light, or more precisely, the energy density of the light per unit frequency, u(f). (This is the function which, when integrated over any frequency interval, gives the energy per unit volume within that frequency interval. For our atomic transition, all that matters is the value of u(f)atf=系/h) The probability of absorbing a photon, per unit time per unit intensity, is called the Einstein B coefficient:

B=probability of absorption per unit timeu(f)

Finally, it is also possible for the atom to make a stimulated transition from s2down to s1, again with a probability that is proportional to the intensity of light at frequency f. (Stimulated emission is the fundamental mechanism of the laser: Light Amplification by Stimulated Emission of Radiation.) Thus we define a third coefficient, B, that is analogous to B:

B'=probability of stimulated emission per unit timeu(f)

As Einstein showed in 1917, knowing any one of these three coefficients is as good as knowing them all.

(a) Imagine a collection of many of these atoms, such that N1 of them are in state s1 and N2 are in state s2. Write down a formula for dN1/dt in terms of A, B, B', N1, N2, and u(f).

(b) Einstein's trick is to imagine that these atoms are bathed in thermal radiation, so that u(f) is the Planck spectral function. At equilibrium, N1and N2 should be constant in time, with their ratio given by a simple Boltzmann factor. Show, then, that the coefficients must be related by

B'=BandAB=8蟺hf3c3

Consider a two-dimensional solid, such as a stretched drumhead or a layer of mica or graphite. Find an expression (in terms of an integral) for the thermal energy of a square chunk of this material of area , and evaluate the result approximately for very low and very high temperatures. Also, find an expression for the heat capacity, and use a computer or a calculator to plot the heat capacity as a function of temperature. Assume that the material can only vibrate perpendicular to its own plane, i.e., that there is only one "polarization."

In a real semiconductor, the density of states at the bottom of the conduction band will differ from the model used in the previous problem by a numerical factor, which can be small or large depending on the material. Let us, therefore, write for the conduction band g(系)=g0c系-系cwhere g0cis a new normalization constant that differs from g0by some fudge factor. Similarly, write g鈭�at the top of the valence band in terms of a new normalization constant g0v.

(a) Explain why, if g0v鈮�g0c, the chemical potential will now vary with temperature. When will it increase, and when will it decrease?

(b) Write down an expression for the number of conduction electrons, in terms of T,渭,鈭�candg0cSimplify this expression as much as possible, assuming 系c-渭鈮�kT.

(c) An empty state in the valence band is called a hole. In analogy to part (b), write down an expression for the number of holes, and simplify it in the limit 渭-系v鈮�kT.

(d) Combine the results of parts (b) and (c) to find an expression for the chemical potential as a function of temperature.

(e) For silicon, g0cg0=1.09andg0vg0=0.44*.Calculate the shift in碌 for silicon at room temperature.

In this problem you will model helium-3 as a non-interacting Fermi gas. Although He3liquefies at low temperatures, the liquid has an unusually low density and behaves in many ways like a gas because the forces between the atoms are so weak. Helium-3 atoms are spin-1/2 fermions, because of the unpaired neutron in the nucleus.

(a) Pretending that liquid 3He is a non-interacting Fermi gas, calculate the Fermi energy and the Fermi temperature. The molar volume (at low pressures) is 37cm3鈥�

(b)Calculate the heat capacity for T<<Tf, and compare to the experimental result CV=(2.8K-1)NkT(in the low-temperature limit). (Don't expect perfect agreement.)

(c)The entropy of solid H3ebelow 1 K is almost entirely due to its multiplicity of nuclear spin alignments. Sketch a graph S vs. T for liquid and solid H3eat low temperature, and estimate the temperature at which the liquid and solid have the same entropy. Discuss the shape of the solid-liquid phase boundary shown in Figure 5.13.

Use the results of this section to estimate the contribution of conduction electrons to the heat capacity of one mole of copper at room temperature. How does this contribution compare to that of lattice vibrations, assuming that these are not frozen out? (The electronic contribution has been measured at low temperatures, and turns out to be about40% more than predicted by the free electron model used here.)
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