Quantum Statistics

Near the cells where oxygen is used, its chemical potential is significantly lower than near the lungs. Even though there is no gaseous oxygen near these cells, it is customary to express the abundance of oxygen in terms of the partial pressure of gaseous oxygen that would be in equilibrium with the blood. Using the independent-site model just presented, with only oxygen present, calculate and plot the fraction of occupied heme sites as a function of the partial pressure of oxygen. This curve is called the Langmuir adsorption isotherm ("isotherm" because it's for a fixed temperature). Experiments show that adsorption by myosin follows the shape of this curve quite accurately.

In a real hemoglobin molecule, the tendency of oxygen to bind to a heme site increases as the other three heme sites become occupied. To model this effect in a simple way, imagine that a hemoglobin molecule has just two sites, either or both of which can be occupied. This system has four possible states (with only oxygen present). Take the energy of the unoccupied state to be zero, the energies of the two singly occupied states to be $-0.55 eV$, and the energy of the doubly occupied state to be $-1.3 eV$ (so the change in energy upon binding the second oxygen is $-0.75 eV$ ). As in the previous problem, calculate and plot the fraction of occupied sites as a function of the effective partial pressure of oxygen. Compare to the graph from the previous problem (for independent sites). Can you think of why this behavior is preferable for the function of hemoglobin?

Consider a system consisting of a single hydrogen atom/ion, which has two possible states: unoccupied (i.e., no electron present) and occupied (i.e., one electron present, in the ground state). Calculate the ratio of the probabilities of these two states, to obtain the Saha equation, already derived in Section 5.6 Treat the electrons as a monotonic ideal gas, for the purpose of determining $\mu$. Neglect the fact that an electron has two independent spin states.

Repeat the previous problem, taking into account the two independent spin states of the electron. Now the system has two "occupied" states, one with the electron in each spin configuration. However, the chemical potential of the electron gas is also slightly different. Show that the ratio of probabilities is the same as before: The spin degeneracy cancels out of the saha equation.

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.

Repeat the previous problem, taking into account the two independent spin states of the electron. Now the system has two "occupied" states, one with the electron in each spin configuration. However, the chemical potential of the electron gas is also slightly different. Show that the ratio of probabilities is the same as before: The spin degeneracy cancels out of the Saha equation.

 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. 

(b) Assuming that the conduction electrons behave like an ordinary ideal gas (with two spin states per particle), write their chemical potential in terms of the number of conduction electrons per unit volume,$\raisebox{1ex}{$N_c$}\!\left/ \!\raisebox{-1ex}{$V$}\right.$ .

(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 $N_c$ in terms of the number of donor atoms $\left(N_d\right)$, eliminatingµ. Solve for $N_c$ using the quadratic formula. (Hint: It's helpful to introduce some abbreviations for dimensionless quantities. Try$x=\raisebox{1ex}{$N_c$}\!\left/ \!\raisebox{-1ex}{$N_d$}\right.$,$t=\raisebox{1ex}{$kT$}\!\left/ \!\raisebox{-1ex}{$l$}\right.$ and so on.)

(d) For phosphorus in silicon, the ionization energy is $0.044eV$. Suppose that there are $1017 p$ atoms per cubic centimeter. Using these numbers, calculate and plot the fraction of ionized donors as a function of temperature. Discuss the results.

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

$\stackrel{—}{N}=\frac{kT}{Z}\frac{\partial Z}{\partial \mu }$

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

$\overline{N^2}=\frac{{\left(kT\right)}^2}{Z}\frac{{\partial }^{2}Z}{\partial {\mu }^2}$

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

${\sigma }_N=\sqrt{kT\left(\partial \stackrel{—}{N}/\partial \mu \right)},$

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

In Section 6.5 I derived the useful relation $F=-kT \ln \left(Z\right)$ between the Helmholtz free energy and the ordinary partition function. Use analogous argument to prove that $\varphi =-kT\times \ln \left(\hat{Z}\right)$, where $\hat{Z}$  is the grand partition function and $\varphi$ is the grand free energy introduced in Problem 5.23.

Suppose you have a "box" in which each particle may occupy any of $10$ single-particle states.  For simplicity, assume that each of these states has energy zero.

(a)  What is the partition function of this system if the box contains only one particle?

(b)  What is the partition function of this system if the box contains two distinguishable particles?

(c)  What is the partition function if the box contains two identical bosons?

(d)  What is the partition function if the box contains two identical fermions?

(e)  What would be the partition function of this system according to equation $7.16$?

(f)  What is the probability of finding both particles in the same single particle state, for the three cases of distinguishable particles, identical bosom, and identical fermions?

Compute the quantum volume for an $N_2$ molecule at room temperature, and argue that a gas of such molecules at atmospheric pressure can be

treated using Boltzmann statistics.  At about what temperature would quantum statistics become relevant for this system (keeping the density constant and pretending that the gas does not liquefy)?

Consider a system of five particles, inside a container where the allowed energy levels are nondegenerate and evenly spaced. For instance, the particles could be trapped in a one-dimensional harmonic oscillator potential. In this problem you will consider the allowed states for this system, depending on whether the particles are identical fermions, identical bosons, or distinguishable particles.

(a) Describe the ground state of this system, for each of these three cases.

(b) Suppose that the system has one unit of energy (above the ground state). Describe the allowed states of the system, for each of the three cases. How many possible system states are there in each case?

(c) Repeat part (b) for two units of energy and for three units of energy.

(d) Suppose that the temperature of this system is low, so that the total energy is low (though not necessarily zero). In what way will the behavior of the bosonic system differ from that of the system of distinguishable particles? Discuss.

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

(a) $1 eV$ less than $\mu$

(b) $0.01 eV$ less than $\mu$

(c) equal to $\mu$

(d) $0.01 eV$ greater than $\mu$

(e) $1 eV$ greater than $\mu$

Consider two single-particle states, A and B, in a system of fermions, where ${ϵ}_A=\mu -x$ and ${ϵ}_B=\mu +x;$ that is, level A lies below $\mu$ by the same amount that level B lies above $\mu$. Prove that the probability of level B being occupied is the same as the probability of level A being unoccupied. In other words, the Fermi-Dirac distribution is "symmetrical" about the point where $ϵ=\mu$.

For a system of bosons at room temperature, compute the average occupancy of a single-particle state and the probability of the state containing $0,1,2,3$ bosons, if the energy of the state is 

(a) $0.001\mathrm{eV}$ greater than $\mu$

(b) $0.01\mathrm{eV}$ greater than $\mu$

(c)  $0.1\mathrm{eV}$ greater than $\mu$

 (d) $1\mathrm{eV}$greater than $\mu$

For a system of particles at room temperature, how large must $ϵ-\mu$ 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 system obeying Boltzmann statistics, we know what $\mu$ is from Chapter 6. Suppose, though, that you knew the distribution function (equation $7.31$ ) but didn't know $\mu$. You could still determine $\mu$ by requiring that the total number of particles, summed over all single-particle states, equal N. Carry out this calculation, to rederive the formula $\mu =-kT\ln \left(Z_1/N\right)$. (This is normally how $\mu$ is determined in quantum statistics, although the math is usually more difficult.)

Consider an isolated system of $N$ identical fermions, inside a container where the allowed energy levels are nondegenerate and evenly spaced.* For instance, the fermions could be trapped in a one-dimensional harmonic oscillator potential. For simplicity, neglect the fact that fermions can have multiple spin orientations (or assume that they are all forced to have the same spin orientation). Then each energy level is either occupied or unoccupied, and any allowed system state can be represented by a column of dots, with a filled dot representing an occupied level and a hollow dot representing an unoccupied level. The lowest-energy system state has all levels below a certain point occupied, and all levels above that point unoccupied. Let $\eta$ be the spacing between energy levels, and let  be the number of energy units (each of size $11$) in excess of the ground-state energy. Assume that$q<N$. Figure 7 .8 shows all system states up to $\mathrm{q}=3$.

(a) Draw dot diagrams, as in the figure, for all allowed system states with $q=4,q=5,\text{ and }q=6$. (b) According to the fundamental assumption, all allowed system states with a given value of q are equally probable. Compute the probability of each energy level being occupied, for $q=6$. Draw a graph of this probability as a function of the energy of the level. ( c) In the thermodynamic limit where $q$ is large, the probability of a level being occupied should be given by the Fermi-Dirac distribution. Even though 6 is not a large number, estimate the values of $\mu$ and T that you would have to plug into the Fermi-Dirac distribution to best fit the graph you drew in part (b).

A representation of  the system states of a fermionic sytern with evenly spaced, nondegen erate energy levels. A filled dot rep-  resents an occupied single-particle  state, while a hollow dot represents an unoccupied single-particle state . {d) Calculate the entropy of this system for each value of q from $0 to 6$, and draw a graph of entropy vs. energy. Make a rough estimate of the slope of this graph near $q=6$, to obtain another estimate of the temperature of this system at that point. Check that it is in rough agreement with your answer to part ( c).

Imagine that there exists a third type of particle, which can share a single-particle state with one other particle of the same type but no more. Thus the number of these particles in any state can be $0,1$ or $2$ . Derive the distribution function for the average occupancy of a state by particles of this type, and plot the occupancy as a function of the state's energy, for several different temperatures.

In analogy with the previous problem, consider a system of identical spin$‐0bosons$ trapped in a region where the energy levels are evenly spaced. Assume that $N$is a large number, and again let $q$ be the number of energy units.

(a) Draw diagrams representing all allowed system states from $q=0$ up to $q=6$.Instead of using dots as in the previous problem, use numbers to indicate the number of bosons occupying each level.

(b) Compute the occupancy of each energy level, for $q=6$. Draw a graph of the occupancy as a function of the energy at each level.

(c) Estimate values of $\mu$ and $T$ that you would have to plug into the Bose-Einstein distribution to best fit the graph of part(b).

(d) As in part (d) of the previous problem, draw a graph of entropy vs energy and estimate the temperature at $q=6$ from this graph. 

Consider a degenerate electron gas in which essentially all of the electrons are highly relativistic $\left(ϵ\gg m{c}^2\right)$ so that their energies are $ϵ=pc$ (where p is the magnitude of the momentum vector).

(a) Modify the derivation given above to show that for a relativistic electron gas at zero temperature, the chemical potential (or Fermi energy) is given by =

$\mu =hc(3N/8\pi V{)}^{1/3}$ 

(b) Find a formula for the total energy of this system in terms of N and $\mu$.

An atomic nucleus can be crudely modeled as a gas of nucleons with a number density of $0.18{\mathrm{fm}}^{-3}$ (where $1\mathrm{fm}={10}^{-15} \mathrm{m}$ ). 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.

At the center of the sun, the temperature is approximately ${10}^7 \mathrm{K}$ and the concentration of electrons is approximately ${10}^{32}$ per cubic meter. Would it be (approximately) valid to treat these electrons as a "classical" ideal gas (using Boltzmann statistics), or as a degenerate Fermi gas (with $T\approx 0$ ), or neither?

Each atom in a chunk of copper contributes one conduction electron. Look up the density and atomic mass of copper, and calculate the Fermi energy, the Fermi temperature, the degeneracy pressure, and the contribution of the degeneracy pressure to the bulk modulus. Is room temperature sufficiently low to treat this system as a degenerate electron gas? 

A white dwarf star (see Figure 7.12) is essentially a degenerate electron gas, with a bunch of nuclei mixed in to balance the charge and to provide the gravitational attraction that holds the star together. In this problem you will derive a relation between the mass and the radius of a white dwarf star, modeling the star as a uniform-density sphere. White dwarf stars tend to be extremely hot by our standards; nevertheless, it is an excellent approximation in this problem to set T=0.

(a) Use dimensional analysis to argue that the gravitational potential energy of a uniform-density sphere (mass M, radius R) must equal 

${\mathrm{U}}_{\mathrm{grav}}=-\left(\mathrm{constant}\right)\frac{{\mathrm{GM}}^2}{\mathrm{R}}$

where (constant) is some numerical constant. Be sure to explain the minus sign. The constant turns out to equal 3/5; you can derive it by calculating the (negative) work needed to assemble the sphere, shell by shell, from the inside out. 

(b) Assuming that the star contains one proton and one neutron for each electron, and that the electrons are nonrelativistic, show that the total (kinetic) energy of the degenerate electrons equals 

${\mathrm{U}}_{\mathrm{kinetic}}= (0.0086) \frac{{\mathrm{h}}^2{\mathrm{M}}^{{\displaystyle \frac{5}{3}}}}{{\mathrm{m}}_{\mathrm{e}}{\mathrm{m}}_{\mathrm{p}}^{{\displaystyle \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{\mathrm{R}}^2}$

Figure 7.12. The double star system Sirius A and B. Sirius A (greatly overexposed in the photo) is the brightest star in our night sky. Its companion, Sirius B, is hotter but very faint, indicating that it must be extremely small-a white dwarf. From the orbital motion of the pair we know that Sirius B has about the same mass as our sun. (UCO /Lick Observatory photo.) 

( c) The equilibrium radius of the white dwarf is that which minimizes the total energy ${\mathrm{U}}_{\mathrm{gravity}}+{\mathrm{U}}_{\mathrm{kinetic}}$· Sketch the total energy as a function of R, and find a formula for the equilibrium radius in terms of the mass. As the mass increases, does the radius increase or decrease? Does this make sense?

( d) Evaluate the equilibrium radius for $\mathrm{M}=2\times {10}^{30} \mathrm{kg}$, the mass of the sun. Also evaluate the density. How does the density compare to that of water?

( e) Calculate the Fermi energy and the Fermi temperature, for the case considered in part (d). Discuss whether the approximation T = 0 is valid. 

(f) Suppose instead that the electrons in the white dwarf star are highly relativistic. Using the result of the previous problem, show that the total kinetic energy of the electrons is now proportional to 1 / R instead of $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\mathrm{R}}^2$}\right.$• Argue that there is no stable equilibrium radius for such a star. 

(g) The transition from the nonrelativistic regime to the ultra relativistic regime occurs approximately where the average kinetic energy of an electron is equal to its rest energy, ${\mathrm{mc}}^2$ Is the nonrelativistic approximation valid for a one-solar-mass white dwarf? Above what mass would you expect a white dwarf to become relativistic and hence unstable?

A star that is too heavy to stabilize as a white dwarf can collapse further to form a neutron star: a star made entirely of neutrons, supported against gravitational collapse by degenerate neutron pressure. Repeat the steps of the previous problem for a neutron star, to determine the following: the mass radius relation; the radius, density, Fermi energy, and Fermi temperature of a one-solar-mass neutron star; and the critical mass above which a neutron star becomes relativistic and hence unstable to further collapse. 

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 about$40\%$ more than predicted by the free electron model used here.)

In this problem you will model helium-3 as a non-interacting Fermi gas. Although ${}^3\mathrm{He}$ liquefies 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 $37 {\mathrm{cm}}^3$•

(b)Calculate the heat capacity for $\mathrm{T}<<{\mathrm{T}}_{\mathrm{f}}$, and compare to the experimental result ${\mathrm{C}}_{\mathrm{V}} = (2.8 {\mathrm{K}}^{-1})\mathrm{NkT}$ (in the low-temperature limit). (Don't expect perfect agreement.)

(c)The entropy of solid ${}^{3}He$ below 1 K is almost entirely due to its multiplicity of nuclear spin alignments. Sketch a graph S vs. T for liquid and solid ${}^{3}He$ at 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.

The argument given above for why ${\mathrm{C}}_{\mathrm{v}} \propto \mathrm{T}$ does not depend on the details of the energy levels available to the fermions, so it should also apply to the model considered in Problem 7.16: a gas of fermions trapped in such a way that the energy levels are evenly spaced and non-degenerate.

(a) Show that, in this model, the number of possible system states for a given value of q is equal to the number of distinct ways of writing q as a sum of positive integers. (For example, there are three system states for q = 3, corresponding to the sums 3, 2 + 1, and 1 + 1 + 1. Note that 2 + 1 and 1 + 2 are not counted separately.) This combinatorial function is called the number of unrestricted partitions of q, denoted p(q). For example, p(3) = 3. 

(b) By enumerating the partitions explicitly, compute p(7) and p(8). 

(c) Make a table of p(q) for values of q up to 100, by either looking up the values in a mathematical reference book, or using a software package that can compute them, or writing your own program to compute them. From this table, compute the entropy, temperature, and heat capacity of this system, using the same methods as in Section 3.3. Plot the heat capacity as a function of temperature, and note that it is approximately linear. 

(d) Ramanujan and Hardy (two famous mathematicians) have shown that when q is large, the number of unrestricted partitions of q is given approximately by

$\mathrm{p}\left(\mathrm{q}\right)\approx \frac{{\mathrm{e}}^{\mathrm{\pi }\sqrt{{\displaystyle \raisebox{1ex}{$2\mathrm{q}$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}}{4\sqrt{3}\mathrm{q}}$

Check the accuracy of this formula for q = 10 and for q = 100. Working in this approximation, calculate the entropy, temperature, and heat capacity of this system.  Express the heat. capacity as a series in decreasing powers of $\mathrm{kT}/\mathrm{\eta }$, assuming that this ratio is large and keeping the two largest terms. Compare to the numerical results you obtained in part (c). Why is the heat capacity of this system independent of N, unlike that of the three dimensional box of fermions discussed in the text?

Consider a free Fermi gas in two dimensions, confined to a square area $A=L^2$•

(a) Find the Fermi energy (in terms of $N$ and $A$), and show that the average energy of the particles is $\raisebox{1ex}{${\in }_F$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$.

(b) Derive a formula for the density of states. You should find that it is a constant, independent of $\in$.

(c) Explain how the chemical potential of this system should behave as a function of temperature, both when $kT\ll {\in }_F$ and when $T$ is much higher.

(d) Because $g\left(\in \right)$ is a constant for this system, it is possible to carry out the integral 7.53 for the number of particles analytically. Do so, and solve for $\mu$ as a function of $N$. Show that the resulting formula has the expected qualitative behavior.

(e) Show that in the high-temperature limit, $kT\gg {\in }_F$, the chemical potential of this system is the same as that of an ordinary ideal gas. 

Although the integrals ($7.53$and $7.54$) for$N$and $U$ cannot be

carried out analytically for all T, it's not difficult to evaluate them numerically

using a computer. This calculation has little relevance for electrons in metals (for

which the limit $kT<<EF$ is always sufficient), but it is needed for liquid ${}^{3}He$ and

for astrophysical systems like the electrons at the center of the sun.

(a) As a warm-up exercise, evaluate the$N$ integral ($7.53$) for the case$kT={\epsilon }_F$

and $\mu =0$, and check that your answer is consistent with the graph shown

above. (Hint: As always when solving a problem on a computer, it's best to

first put everything in terms of dimensionless variables. So let $t=\raisebox{1ex}{$kT$}\!\left/ \!\raisebox{-1ex}{${\epsilon }_F$}\right.$ $, c=\raisebox{1ex}{$\mu $}\!\left/ \!\raisebox{-1ex}{${\epsilon }_F$}\right.$

, and $x=\raisebox{1ex}{$\epsilon $}\!\left/ \!\raisebox{-1ex}{$kT$}\right.$. Rewrite everything in terms of these variables,

and then put it on the computer.)

(b) The next step is to vary$\mu$ holding$T$ fixed, until the integral works out to

the desired value,$N$. Do this for values of  $\raisebox{1ex}{$kT$}\!\left/ \!\raisebox{-1ex}{${\epsilon }_F$}\right.$ranging from $0.1$ up to $2$,

and plot the results to reproduce Figure$7.16$. (It's probably not a good idea

to try to use numerical methods when $\raisebox{1ex}{$kT$}\!\left/ \!\raisebox{-1ex}{${\epsilon }_F$}\right.$ is much smaller than $0.1$, since

you can start getting overflow errors from exponentiating large numbers.

But this is the region where we've already solved the problem analytically.)

(c) Plug your calculated values ofµ into the energy integral ($7.54$), and evaluate

that integral numerically to obtain the energy as a function of temperature

for$kT$up to $2{\epsilon }_F$ Plot the results, and evaluate the slope to obtain the

heat capacity. Check that the heat capacity has the expected behavior at

both low and high temperatures.

Carry out the Sommerfeld expansion for the energy integral $(7.54)$, to obtain equation $7.67$. Then plug in the expansion for $\mu$to obtain the final answer, equation $7.68$.

In Problem $7.28$ you found the density of states and the chemical potential for a two-dimensional Fermi gas. Calculate the heat capacity of this gas in the limit $kT \ll {\epsilon }_F$· Also show that the heat capacity has the expected behavior when $kT \gg {\epsilon }_F$. Sketch the heat capacity as a function of temperature.

The Sommerfeld expansion is an expansion in powers of $\raisebox{1ex}{$kT$}\!\left/ \!\raisebox{-1ex}{${\epsilon }_F$}\right.$, which is assumed to be small. In this section I kept all terms through order ${\left(\raisebox{1ex}{$kT$}\!\left/ \!\raisebox{-1ex}{${\epsilon }_F$}\right.\right)}^2$, omitting higher-order terms. Show at each relevant step that the term proportional to $T^3$ is zero, so that the next nonvanishing terms in the expansions for$\mu$ and $U$are proportional to $T^4$. (If you enjoy such things, you might try evaluating the $T^4$terms, possibly with the aid of a computer algebra program.)

When the attractive forces of the ions in a crystal are taken into account, the allowed electron energies are no longer given by the simple formula 7.36; instead, the allowed energies are grouped into bands, separated by gaps where there are no allowed energies. In a conductor the Fermi energy lies within one of the bands; in this section we have treated the electrons in this band as "free" particles confined to a fixed volume. In an insulator, on the other hand, the Fermi energy lies within a gap, so that at T = 0 the band below the gap is completely occupied while the band above the gap is unoccupied. Because there are no empty states close in energy to those that are occupied, the electrons are "stuck in place" and the material does not conduct electricity. A semiconductor is an insulator in which the gap is narrow enough for a few electrons to jump across it at room temperature. Figure 7 .17 shows the density of states in the vicinity of the Fermi energy for an idealized semiconductor, and defines some terminology and notation to be used in this problem. 

(a) As a first approximation, let us model the density of states near the bottom of the conduction band using the same function as for a free Fermi gas, with an appropriate zero-point: $g\left(ϵ\right)=g0\sqrt{ϵ-{ϵ}_c}$, where go is the same constant as in equation 7.51. Let us also model the density of states near the top

Figure 7.17. The periodic potential of a crystal lattice results in a densityof-states function consisting of "bands" (with many states) and "gaps" (with no states). For an insulator or a semiconductor, the Fermi energy lies in the middle of a gap so that at T = 0, the "valence band" is completely full while the-"conduction band" is completely empty. of the valence band as a mirror image of this function. Explain why, in this approximation, the chemical potential must always lie precisely in the middle of the gap, regardless of temperature.

(b) Normally the width of the gap is much greater than kT. Working in this limit, derive an expression for the number of conduction electrons per unit volume, in terms of the temperature and the width of the gap.

 (c) For silicon near room temperature, the gap between the valence and conduction bands is approximately 1.11 eV. Roughly how many conduction electrons are there in a cubic centimeter of silicon at room temperature? How does this compare to the number of conduction electrons in a similar amount of copper? 

( d) Explain why a semiconductor conducts electricity much better at higher temperatures. Back up your explanation with some numbers. (Ordinary conductors like copper, on the other hand, conduct better at low temperatures.) (e) Very roughly, how wide would the gap between the valence and conduction bands have to be in order to consider a material an insulator rather than a semiconductor? 

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 $g\left(ϵ\right)=g_{0c}\sqrt{ϵ-{ϵ}_c}$ where $g_{0c}$is a new normalization constant that differs from $g_0$ by some fudge factor. Similarly, write $g\left(\in \right)$ at the top of the valence band in terms of a new normalization constant $g_{0v}$.

(a) Explain why, if $g_{0v}\ne g_{0c}$, 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 $T, \mu , {\in }_c and g_{0c}$ Simplify this expression as much as possible, assuming ${ϵ}_c-\mu \gg kT$.

(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 $\mu -{ϵ}_v\gg kT$.

(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, $\raisebox{1ex}{$g_{0c}$}\!\left/ \!\raisebox{-1ex}{$g_{0 }=1.09 and {\displaystyle \raisebox{1ex}{$g_{0v}$}\!\left/ \!\raisebox{-1ex}{$g_0=0.{44}^{*}.$}\right.}$}\right.$Calculate the shift inµ for silicon at room temperature. 

The previous two problems dealt with pure semiconductors, also called intrinsic semiconductors. Useful semiconductor devices are instead made from doped semiconductors, which contain substantial numbers of impurity atoms. One example of a doped semiconductor was treated in Problem 7.5. Let us now consider that system again. (Note that in Problem 7.5 we measured all energies relative to the bottom of the conduction band, Ee. We also neglected the distinction between $g_0$and $g_{0c}$; this simplification happens to be ok for conduction electrons in silicon.)

(a) Calculate and plot the chemical potential as a function of temperature, for silicon doped with ${10}^{17}$ phosphorus atoms per $c{m}^3$ (as in Problem 7.5). Continue to assume that the conduction electrons can be treated as an ordinary ideal gas.

(b) Discuss whether it is legitimate to assume for this system that the conduction electrons can be treated as an ordinary ideal gas, as opposed to a Fermi gas. Give some numerical examples.

(c)Estimate the temperature at which the number of valence electrons excited to the conduction band would become comparable to the number of conduction electrons from donor impurities. Which source of conduction electrons is more important at room temperature?

Most spin-1/2 fermions, including electrons and helium-3 atoms, have nonzero magnetic moments. A gas of such particles is therefore paramagnetic. Consider, for example, a gas of free electrons, confined inside a three-dimensional box. The z component of the magnetic moment of each electron is ±µa. In the presence of a magnetic field B pointing in the z direction, each "up" state acquires an additional energy of $-{\mu }_{\mathrm{B}}B$, while each "down" state acquires an additional energy of $+{\mu }_{\mathrm{B}}B$ 

(a) Explain why you would expect the magnetization of a degenerate electron gas to be substantially less than that of the electronic paramagnets studied in Chapters 3 and 6, for a given number of particles at a given field strength. 

(b) Write down a formula for the density of states of this system in the presence of a magnetic field B, and interpret your formula graphically. 

(c) The magnetization of this system is ${\mu }_{\mathrm{B}}\left(N_{\uparrow }-N_{\downarrow }\right.$, where Nr and N1 are the numbers of electrons with up and down magnetic moments, respectively. Find a formula for the magnetization of this system at $T=0$, in terms of N, µa, B, and the Fermi energy.

(d) Find the first temperature-dependent correction to your answer to part (c), in the limit $T\ll T_{\mathrm{F}}$. You may assume that ${\mu }_{\mathrm{B}}B\ll kT$; this implies that the presence of the magnetic field has negligible effect on the chemical potential $\mu$. (To avoid confusing µB with µ, I suggest using an abbreviation such as o for the quantity µaB.)

 Prove that the peak of the Planck spectrum is at x = 2.82.

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 function $u\left(ϵ\right)$ for T = 3000 K and T = 6000 K (both on the same graph). Label the horizontal axis in electron-volts.

Change variables in equation 7.83 to $\lambda =hc/ϵ$ and thus derive a formula for the photon spectrum as a function of wavelength. Plot this spectrum, and find a numerical formula for the wavelength where the spectrum peaks, in terms of hc/kT. Explain why the peak does not occur at hc/(2.82kT).

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.

Consider any two internal states, s₁ and s₂, of an atom. Let s₂ be the higher-energy state, so that $E\left(s_2\right)-E\left(s_1\right)=ϵ$ 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 $f=ϵ/h$, 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 $u\left(f\right)\text{ at }f=ϵ/h$) The probability of absorbing a photon, per unit time per unit intensity, is called the Einstein B coefficient:

$B=\frac{\text{ probability of absorption per unit time }}{u\left(f\right)}$

Finally, it is also possible for the atom to make a stimulated transition from s₂ down 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:

$B^{\text{'}}=\frac{\text{ probability of stimulated emission per unit time }}{u\left(f\right)}$

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 $d{N}_1/dt$ 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, N₁ and N₂ should be constant in time, with their ratio given by a simple Boltzmann factor. Show, then, that the coefficients must be related by

$B^{\text{'}}=B \text{ and } \frac{A}{B}=\frac{8\pi h{f}^3}{c^3}$

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?

At the surface of the sun, the temperature is approximately 5800 K.

(a) How much energy is contained in the electromagnetic radiation filling a cubic meter of space at the sun's surface?

(b) Sketch the spectrum of this radiation as a function of photon energy. Mark the region of the spectrum that corresponds to visible wavelengths, between 400 nm and 700 nm.

(c) What fraction of the energy is in the visible portion of the spectrum? (Hint: Do the integral numerically.)

Use the formula $P=-(\partial U/\partial V{)}_{S,N}$ to show that the pressure of a photon gas is 1/3 times the energy density (U/V). Compute the pressure exerted by the radiation inside a kiln at 1500 K, and compare to the ordinary gas pressure exerted by the air. Then compute the pressure of the radiation at the centre of the sun, where the temperature is 15 million K. Compare to the gas pressure of the ionised hydrogen, whose density is approximately 10⁵ kg/m³.

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

$N=8\pi V{\left(\frac{kT}{hc}\right)}^3{\int }_0^{\infty }\frac{x^2}{e^x-1}dx$

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).

 Sometimes it is useful to know the free energy of a photon gas.

(a) Calculate the (Helmholtz) free energy directly from the definition

 (Express the answer in terms of T' and V.)

(b) Check the formula $S=-(\partial F/\partial T{)}_V$ for this system.

(c) Differentiate F with respect to V to obtain the pressure of a photon gas. Check that your result agrees with that of the previous problem.

(d) A more interesting way to calculate F is to apply the formula $F=-kT\ln Z$ separately to each mode (that is, each effective oscillator), then sum over all modes. Carry out this calculation, to obtain

$F=8\pi V\frac{(kT{)}^4}{(hc{)}^3}{\int }_0^{\infty }{x}^2\ln \left(1-e^{-x}\right)dx$

Integrate by parts, and check that your answer agrees with part (a).

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.)

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 ($\stackrel{-}{v}$), 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: ${\mu }_{\nu }={\mu }_{\overline{\nu }}$. Furthermore, neutrinos and antineutrinos can be produced and annihilated in pairs by the reaction

$\nu +\overline{\nu }\leftrightarrow 2\gamma$

(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?