Chapter 8: Quantum Statistical Mechanics | Lectures on Thermodynamics and Statistical Mechanics

8. Quantum Statistical Mechanics

There are two kinds of particles from the point of view of statistics, bosons and fermions. The corresponding statistical distributions are called the Bose-Einstein distribution and the Fermi-Dirac distribution. Bosons have the property that one can have any number of particles in a given quantum state, while fermions obey the Pauli exclusion principle which allows a maximum of only one particle per quantum state. Any species of particles can be put into one of these two categories. The natural question is, of course, how do we know which category a given species belongs to; is there an a priori way to know this? The answer to this question is yes, and it constitutes one of the deep theorems in quantum field theory, the so-called spin-statistics theorem. The essence of the theorem, although this is not the precise statement, is given as follows.

Theorem 8.0.1 — Spin-Statistics Theorem.

Identical particles with integer values for spin (or intrinsic angular momentum) are bosons, they obey Bose-Einstein statistics. Identical particles with spin (or intrinsic angular momentum) = , where 𝓃 is an integer, obey the Pauli exclusion principle and hence they are fermions and obey Fermi-Dirac statistics.

Thus, among familiar examples of particles, photons (which have spin = 1), phonons (quan- tized version of elastic vibrations in a solid), atoms of , etc. are bosons, while the electron (spin = ), proton (also spin- ), atoms of , etc. are fermions. In all cases, particles of the same species are identical.

8.1 Bose-Einstein distribution

We will now consider the derivation of the distribution function for free bosons carrying out the counting of states along the lines of what we did for the Maxwell-Boltzmann distribution. Let us start by considering how we obtained the binomial distribution. We considered a number of particles and how they can be distributed among, say, boxes. As the simplest case of this, consider two particles and two boxes. The ways in which we can distribute them are as shown below. The boxes may correspond to different values of momenta, say and , which have

the same energy. There is one way to get the first or last arrangement, with both particles in one box; this corresponds to

(8.1)

There are two ways to get the arrangement of one particle in each box, corresponding to

(8.2)

The generalization of this counting is what led us to the Maxwell-Boltzmann statistics. But if particles are identical with no intrinsic attributes distinguishing the particles we have labeled 1 and 2, this result is incorrect. The possible arrangements are just

Counting the arrangement of one particle per box twice is incorrect; the correct counting should give and , giving a total of 3 distinct arrangements or states. To generalize this, we first consider 𝓃 particles to be distributed in 2 boxes. The possible arrangements are 𝓃 in box 1, 0 in box 2; in box 1, 1 in box 2; · · · ; 0 in box 1, n in box 2, giving distinct arrangements or states in all. If we have 𝓃 particles and 3 boxes, we can take particles in the first two boxes (with possible states) and particles in the third box. But can be anything from zero to , so that the total number of states is

(8.3)

We may arrive at this in another way as well. Represent the particles as dots and use 3 − 1 = 2 partitions to separate them into three groups. Any arrangement then looks like

Clearly any permutation of the entities (dots or partitions) gives an acceptable arrangement. There are such permutations. However, permutations of the two partitions does not change the arrangement, neither does the permutations of the dots among themselves. Thus the number of distinct arrangements or states is

(8.4)

Generalizing this argument, for boxes and 𝓃 particles, the number of distinct arrangements is given by

(8.5)

We now consider particles, with of them having energy each, of them with energy ϵ₂ each, etc. Further, let be the number of states with energy , the number of states with energy ϵ₂, etc. The degeneracy may be due to different values of momentum (e.g., different directions of with the same energy or other quantum numbers, such as spin. The total number of distinct arrangements for this configuration is

(8.6)

The corresponding entropy is and we must maximize this subject to and . With the Stirling formula for the factorials, the function to be extremized is thus

(8.7)

The extremization condition is

(8.8)

with the solution

(8.9)

where we have approximated , since the use of the Stirling formula needs large numbers. As the occupation number per state, we can take the result as

(8.10)

with the degeneracy factors arising from the summation over states of the same energy. This is the Bose-Einstein distribution. For a free particle, the number of states in terms of the momenta can be taken as the single-particle , which, from (7.55), is

(8.11)

Thus the normalization conditions on the Bose-Einstein distribution are

(8.12)

The remaining sum in this formula is over the internal states of the particle, such as spin states.

It is also useful to write down the partition function . Notice that we may write the occupation number n in (8.10) as

(8.13)

This result holds for each state for which we are calculating 𝓃. But recall that the total number should be given by

(8.14)

Therefore, we expect the partition function to be given by

(8.15)

For each state with fixed quantum numbers, we can write

(8.16)

This shows that the partition function is the sum over states of possible occupation numbers of the Boltzmann factor . This is obtained in a clearer way in the full quantum theory where

(8.17)

where and are the Hamiltonian operator and the number operator, respectively, and Tr denotes the trace over a complete set of states of the system.

8.2 Fermi-Dirac distribution

The counting of distinct arrangements for fermions is even simpler than for the Bose-Einstein case, since each state can have an occupation number of either zero or 1. Thus consider states with 𝓃 particles to be distributed among them. There are n states which are singly occupied and these can be chosen in ways. The total number of distinct arrangements is thus given by

(8.18)

The function to be maximized to identify the equilibrium distribution is therefore given by

(8.19)

The extremization condition reads

(8.20)

with the solution

(8.21)

So, for fermions in equilibrium, we can take the occupation number to be given by

(8.22)

with the degeneracy factor arising from summation over states of the same energy. This is the Fermi-Dirac distribution. The normalization conditions are again,

(8.23)

As in the case of the Bose-Einstein distribution, we can write down the partition function for free fermions as

(8.24)

Notice that, here too, the partition function for each state is ;it is just that, in the present case, n can only be zero or 1.

8.3 Applications of the Bose-Einstein distribution

We shall now consider some simple applications of quantum statistics, focusing in this section on the Bose-Einstein distribution.

8.3.1 The Planck distribution for black body radiation

Any material body at a finite nonzero temperature emits electromagnetic radiation, or photons in the language of the quantum theory. The detailed features of this radiation will depend on the nature of the source, its atomic composition, emissivity, etc. However, if the source has a sufficiently complex structure, the spectrum of radiation is essentially universal. We want to derive this universal distribution, which is also known as the Planck distribution.

Since a black body absorbs all radiation falling on it, treating all wavelengths the same, a black body may be taken as a perfect absorber. (Black bodies in reality do this only for a small part of the spectrum, but here we are considering the idealized case.) By the same token, black bodies are also perfect emitters and hence the formula for the universal thermal radiation is called the black body radiation formula.

The black body radiation formula was obtained by Max Planck by fitting to the observed spectrum. He also spelled out some of the theoretical assumptions needed to derive such a result and this was, as is well known, the beginning of the quantum theory. Planck’s derivation of this formula is fairly simple once certain assumptions, radical for his time, are made; from the modern point of view it is even simpler. Photons are particles of zero mass, the energy and momentum of a photon are given as

(8.25)

where the frequency of the radiation and the wave number are related to each other in the usual way, . Further photons are spin-1 particles, so we know from the spin-statistics connection that they are bosons. Because they are massless, they have only two polarization states, even though they have spin equal to 1. (For a massive particle we should expect polarization states for a spin-1 particle.) We can apply the Bose-Einstein distribution (8.10) directly, with one caveat. The number of photons is not a well defined concept. Since long wavelength photons carry very little energy, the number of photons for a state of given energy could have an ambiguity of a large number of soft or long wavelength photons. This is also seen more theoretically; there is no conservation law in electromagnetic theory beyond the usual ones of energy and momentum. This means that we should not have a chemical potential which is used to fix the average number of photons. Thus the Bose-Einstein distribution simplifies to

(8.26)

Figure 8.1: The Planck distribution as a function of frequency for three sample values of

temperature, with ; units are arbitrary

function of momentum is given by

(8.27)

where the factor of 2 is from the two polarization states. Using (8.25), for the energy density, we find

(8.28)

This is Planck’s radiation formula. If we use and carry out the integration over angular directions of , it reduces to

(8.29)

This distribution function vanishes at and as . It peaks at a certain value which is a function of the temperature. In Fig. 8.1, we show the distribution for some sample values of temperature. Note that the value of 𝓌 at the maximum increases with temperature; in addition, the total amount of radiation (corresponding to the area under the curve) also increases with temperature.

If we integrate (8.29) over all frequencies, the total energy density comes out to be

(8.30)

where we have used the result

(8.31)

Rate of radiation from a black body

We can convert the formula for the energy density to the intensity of the radiation by considering the conservation of energy in electrodynamics. The energy density of the electromagnetic field is given by

(8.32)

Using the Maxwell equations in free space, we find

(8.33)

Integrating over a volume , we find

(8.34)

Thus the energy flux per unit area or the intensity is given by the Poynting vector . For electromagnetic waves,, ~E and ~B are orthogonal to each other and both are orthogonal to , the wave vector which gives the direction of propagation, i.e., the direction of propagation of the photon. In this case we find

(8.35)

Using the Planck formula (8.28), the magnitude of the intensity of blackbody radiation is given by

(8.36)

We have considered radiation in a box of volume in equilibrium. To get the rate of

radiation per unit area of a blackbody, note that, because of equilibrium, the radiation rate

from the body must equal the energy flux falling on the area under consideration (which is all taken to be absorbed since it is a blackbody); thus emission rate equals absorption rate as expected for equilibrium. The flux is given by

(8.37)

where is the normal to the surface and is the angle between and . Further, in the equilibrium situation, there are photons going to and away from the surface under consideration, so we must only consider positive values of or . Thus the radiation rate over all wavelengths per unit area of the emitter is given by

(8.38)

This result is known as the Stefan-Boltzmann law.

Radiation pressure

Another interesting result concerning thermal radiation is the pressure of radiation. For this, it is convenient to use one of the relations in (6.30), namely,

(8.39)

From (8.30), we have

(8.40)

Equations (8.39) and (8.40) immediately lead to

(8.41)

Radiation pressure is significant and important in astrophysics. Stars can be viewed as a gas or fluid held together by gravity. The gas has pressure and the pressure gradient between the interior of the star and the exterior region tends to create a radial outflow of the material. This is counteracted by gravity which tends to contract or collapse the material. The hydrostatic balance in the star is thus between gravity and pressure gradients. The normal fluid pressure is not adequate to prevent collapse. The radiation produced by nuclear fusion in the interior creates an outward pressure and this is a significant component in the hydrostatic equilibrium of the star. Without this pressure a normal star would rapidly collapse.

Maximum of Planck distribution

We have seen that the Planck distribution has a maximum at a certain value of 𝓌. It is interesting to consider the wavelength at which the distribution has a maximum. This can be done in terms of frequency or wavelength, but we will use the wavelength here as this is more appropriate for the application we consider later. (The peak for frequency and wavelength occur at different places since these variables are not linearly related, but rather are reciprocally related.) Using , we can write down the Planck distribution (8.36) in terms of the wavelength as

(8.42)

(The minus sign in only serves to show that when the intensity increases with frequency, it should decrease with and vice versa. So we have dropped the minus sign. is the solid angle for the angular directions.) Extremization with respect to gives the condition

(8.43)

Where . The solution of this transcendental equation is

(8.44)

This relation is extremely useful in determining the temperature of the outer layer of stars, called the photosphere, from which we receive radiation. By spectroscopically resolving the radiation and working out the distribution as a function of wavelength, we can see where the maximum is, and this gives, via (8.44), the temperature of the photosphere. Notice that higher temperatures correspond to smaller wavelengths; thus blue stars are hotter than red stars. For the Sun, the temperature of the photosphere is about , corresponding to a wavelength . Thus the maximum for radiation from the Sun is in the visible region, around the color green.

Another case of the importance in which the radiation pressure and the we calculated are important is in the early history of the universe. Shortly after the Big Bang, the universe was in a very hot phase with all particles having an average energy so high that their masses could be neglected. The radiation pressure from all these particles, including the photon, is an important ingredient in solving the Einstein equations for gravity to work out how the universe was expanding. As the universe cooled by expansion, the unstable massive particles decayed away, since there was not enough average energy in collisions to sustain the reverse process. Photons continued to dominate the evolution of the universe. This phase of the universe is referred to as the radiation dominated era.

Later, the universe cooled enough for electrons and nuclei to combine to form neutral atoms, a phase known as the recombination era. Once this happened, since neutral particles couple only weakly (through dipole and higher multipole moments) to radiation, the existing radiation decoupled and continued to cool down independently of matter. This is the matter dominated era in which we now live. The radiation obeyed the Planck spectrum at the time of recombination, and apart from cooling would continue to do so in the expanding universe. Thus the existence of this background relic radiation is evidence for the Big Bang theory. This cosmic microwave background radiation was predicted to be a consequence of the Big Bang theory, by Gamow, Dicke and others in the 1940s. The temperature was estimated in calculations by Alpher and Herman and by Gamow in the 1940s and 1950s. The radiation was observed by Penzias and Wilson in 1964. The temperature of this background can be measured in the same way as for stars, by comparing the maximum of the distribution with the formula (8.44). It is found to be approximately . (Actually this has been measured with great accuracy by now, the latest value being .) The corresponding is in the microwave region, which is why this is called the cosmic microwave background.

8.3.2 Bose-Einstein condensation

We will now work out some features of an ideal gas of bosons with a conserved particle number; in this case we do have a chemical potential. There are many atoms which are bosons and, if we can neglect the interatomic forces as a first approximation, this discussion can apply to gases made of such atoms. The partition function for a gas of bosons was given in (8.15). Since is related to pressure as in (7.73), this gives immediately

(8.45)

where is the fugacity and denotes the polylogarithm defined by

(8.46)

The total number of particles is given by the normalization condition (8.12) and works out to

(8.47)

We have defined the thermal wavelength by

(8.48)

Apart from some numerical factors of order 1, this is the de Broglie wavelength for a particle of energy .

If we eliminate 𝒵 in favor of from this equation and use it in (8.45), we get the equation of state for the ideal gas of bosons. For high temperatures, this can be done by keeping the terms up to order in the polylogarithms. This gives

(8.49)

This equation shows that even the perfect gas of bosons does not follow the classical ideal gas law. In fact, we may read off the second virial coefficient as

(8.50)

The thermal wavelength is small for large , so this correction is small at high temperatures, which is why the ideal gas was a good approximation for many of the early experiments in thermal physics. If we compare this with the second virial coefficient of a classical gas with interatomic potential as given in (7.84), namely,

(8.51)

we see that we can mimic (8.50) by an attractive interatomic potential. Thus bosons exhibit a tendency to cluster together.

We can now consider what happens when we lower the temperature. It is useful to calculate a typical value of . Putting in the constants,

(8.52)

( is the mass of the proton the mass of the hydrogen atom.) Thus for hydrogen at room temperature, is of atomic size. Since is approximately the free volume available to a molecule, we find from (8.47) that 𝒵 must be very small under normal conditions. The function starts from zero at and rises to about at , see Fig.8.2. Beyond that, even though the function can be defined by analytic continuation, it is imaginary. In fact there is a branch cut from to . Thus for , we can solve (8.47) for 𝒵 in terms of . As the temperature is lowered, decreases and eventually we get to the point where . This happens at a temperature

(8.53)

If we lower the temperature further, it becomes impossible to satisfy (8.47). We can see the problem at more clearly by considering the partition function, where we separate the contribution due to the zero energy state,

(8.54)

We see that the partition function has a singularity at . This is indicative of a phase transition. The system avoids the singularity by having a large number of particles making a Transition to the state of zero energy and momentum. Recall that the factor may be viewed as , as a sum over difference possibly occupation numbers for the ground state. The idea here is that, instead of various possible occupation number sf or the ground state, what happens below is that there is a certain occupation number for the ground state, say, , so that the partition function should read

(8.55)

Thus, rather than having different probabilities for the occupation numbers for the ground state, with correspondingly different probabilities as given by the Boltzmann factor, we have a single multiparticle quantum state, with occupation number , for the ground state. The normalization condition (8.47) is then changed to

(8.56)

Below , this equation is satisfied, with , and with N0 compensating for the second term on the right hand side as increases. This means that a macroscopically large number of particles have to be in the ground state. This is known as Bose-Einstein condensation. In terms of , we can rewrite (8.56) as

(8.57)

which gives the fraction of particles which are in the ground state.

Since for temperatures below , we have . This is then reminiscent of the case of photons where we do not have a conserved particle number. The proper treatment of this condensation effect requires quantum field theory, using the concept of spontaneous symmetry

breaking. In such a description, it will be seen that the particle number is still a conserved operator but that the condensed state cannot be an eigenstate of the particle number.

There are many other properties of the condensation phenomenon we can calculate. Here we will focus on just the specific heat. The internal energy for the gas is given by

(8.58)

At high temperatures, Z is small and and (8.47) gives . Thus in agreement with the classical ideal gas. This gives .

For low temperatures below , and we can set Li5/2(z) = Li5/2(1) ≈ 1.3415. The

specific heat becomes

(8.59)

We see that the specific heat goes to zero at absolute zero, in agreement with the third law of thermodynamics. It rises to a value which is somewhat above at . Above , we must solve for 𝒵 in terms of and substitute back into the formula for . But qualitatively, we can see that the specific heat has to decrease for reaching the ideal gas value of at very high temperatures. A plot of the specific heat is shown in Fig. 8.3.

There are many examples of Bose-Einstein condensation by now. The formula for the thermal wavelength (8.52) shows that smaller atomic masses will have larger and one may expect them to undergo condensation at higher temperatures. While molecular hydrogen (which is a boson) may seem to be the best candidate, it turns to a solid at around . The best candidate is thus liquid Helium. The atoms of the isotope are bosons. Helium becomes a liquid below and it has a density of about (under normal atmospheric pressure) and if this value is used in the formula (8.53), we find to be about . What is

remarkable is that liquid Helium undergoes a phase change at . Below this temperature, it becomes a superfluid, exhibiting essentially zero viscosity. ( atoms are fermions, there is superfluidity here too, at a much lower temperature, and the mechanism is very different.) This transition can be considered as an example of Bose-Einstein condensation. Helium is not an ideal gas of bosons, interatomic forces (particularly a short range repulsion) are important and this may explain the discrepancy in the value of . The specific heat of liquid is shown in Fig. 8.4. There is a clear transition point, with the specific heat showing a discontinuity in addition to the peaking at this point. Because of the similarity of the graph to the Greek letter , this is often referred to as the -transition. The graph is very similar, in a broad qualitative sense, to the behavior we found for Bose-Einstein condensation in Fig. 8.3; however, the Bose-Einstein condensation in the noninteracting gas is a first order transition, while the -transition is a second order transition, so there are differences with the Bose-Einstein condensation of perfect gas of bosons.

The treatment of superfluid Helium along the lines we have used for a perfect gas is very inadequate. A more sophisticated treatment has to take account of interatomic forces and incorporate the idea of spontaneous symmetry breaking. By now, there is a fairly comprehensive theory of liquid Helium.

Recently, Bose-Einstein condensation has been achieved in many other atomic systems such as a gas of atoms, atoms, and a number of others, mostly alkaline and alkaline earth elements.

8.3.3 Specific heats of solids

We now turn to the specific heats of solids, along the lines of work done by Einstein and Debye. In a solid, atoms are not free to move around and hence we do not have the usual translational degrees of freedom. Hence the natural question which arises is: When a solid is heated, what are the degrees of freedom in which the energy which is supplied can be stored? As a first approximation, atoms in a solid may be taken to be at the sites of a regular lattice. Interatomic

forces keep each atom at its site, but some oscillation around the lattice site is possible. This is the dynamics behind the elasticity of the material. These oscillations, called lattice vibrations, constitute the degrees of freedom which can be excited by the supplied energy and are thus the primary agents for the specific heat capacity of solids. In a conductor, translational motion of electrons is also possible. There is thus an electronic contribution to the specific heat as well. This will be taken up later; here we concentrate on the contribution from the lattice vibrations. In an amorphous solid, a regular lattice structure is not obtained throughout the solid, but domains with regular structure exist, and so, the elastic modes of interest are still present.

Turning to the details of the lattice vibrations, for atoms on a lattice, we expect modes, since each atom can oscillate along any of the three dimensions. Since the atoms are like beads on an elastic string, the oscillations can be transferred from one atom to the next and so we get traveling waves. We may characterize these by a frequency w and a wave number . The dispersion relation between 𝓌 and can be obtained by solving the equations of motion for coupled particles. There are distinct modes corresponding to different relations; the typical qualitative behavior is shown in Fig. 8.5. There are three acoustic modes for which , for low , cs being the speed of sound in the material. The three polarizations correspond to oscillations in the three possible directions. The long wavelength part of these modes can also be obtained by solving for elastic waves (in terms of the elastic moduli) in the continuum approximation to the lattice. They are basically sound waves, hence the name acoustic modes. The highest value for is limited by the fact that we do not really have a continuum; the shortest wavelength is of the order of the lattice spacing.

There are also the so-called optical modes for which for any . The minimal energy needed to excite these is typically in the range of or so; in terms of a photon energy this corresponds to the infrared and visible optical frequencies, hence the name. Since , the optical modes are not important for the specific heat at low temperatures.

Just as electromagnetic waves, upon quantization, can be viewed as particles, the photons, the elastic waves in the solid can be described as particles in the quantum theory. These particles are called phonons and obey the expected energy and momentum relations

(8.60)

The relation between w and may be approximated for the two cases rather well by

(8.61)

where is a constant independent of . If there are several optical modes, the corresponding ’s may be different. Here we consider just one for simplicity. The polarizations correspond to the three Cartesian axes and hence they transform as vectors under rotations; i.e., they have spin = 1 and hence are bosons. The thermodynamics of these can now be worked out easily.

First consider the acoustic modes. The total internal energy due to these modes is

(8.62)

The factor of is for the three polarizations. For most of the region of integration which contributes significantly, we are considering modes of wavelengths long compared to the lattice spacing and so we can assume isotropy and carry out the angular integration. For high , the specific crystal structure and anisotropy will matter, but the corresponding 𝓌’s are high and the factor will diminish their contributions to the integral. Thus

(8.63)

Here is the Debye frequency which is the highest frequency possible for the acoustic modes. The value of this frequency will depend on the solid under consideration. We also define a Debye temperature by . We then find

(8.64)

For low temperatures is so large that one can effectively replace it by in a first approximation to the integral. For high , we can expand the integrand in powers of u to carry out the integration. This way we find

(8.65)

The internal energy for

(8.66)

The specific heat at low temperatures is thus given by

(8.67)

We can relate this to the total number of atoms in the material as follows. Recall that the total number of vibrational modes for atoms is . Thus

(8.68)

If we ignore the optical modes, we get

(8.69)

This formula will hold even with optical modes if is interpreted as the number of unit cells rather than the number of atoms. In terms of , we get, for ,

(8.70)

The expression for in (8.67) or (8.70) is the famous law for specific heats of solids at low temperatures derived by Debye in 1912. There is a universality to it. The derivation relies only on having modes with at low . There are always three such modes for any elastic solid. These are the sound waves in the solid. (The existence of these modes can also be understood from the point of view of spontaneous symmetry breaking, but that is another matter.) The power is of course related to the fact that we have three spatial dimensions. So any elastic solid will exhibit this behavior for the contribution from the lattice vibrations. As we shall see shortly, the optical modes will not alter this result. Some sample values of the Debye temperature are given in Table 8.1. This will give an idea of when the low temperature approximation is applicable.

For , we find

(8.71)

The specific heat is then given by

(8.72)

Turning to the optical modes, we note that the frequency 𝓌 is almost independent of , for the whole range of . So it is a good approximation to consider just one frequency , for each optical mode. Let be the total number of degrees of freedom in the optical mode of frequency . Then the corresponding internal energy is given by

(8.73)

The specific heat contribution is given by

(8.74)

These results were derived by Einstein a few years before Debye’s work.

Both contributions to the specific heat, the acoustic contribution given by Debye’s -law and the optical contribution given in (8.74), vanish as . This is in accordance with the third law of thermodynamics. We see once again how the quantum statistics leads to the third law. Further, the optical contribution is exponentially small at low temperatures. Thus the inclusion of the optical modes cannot invalidate Debye’s -law. Notice that even if we include the slight variation of 𝓌 with , the low temperature value will be as given in (8.74) if is interpreted as the the lowest possible value of 𝓌.

8.4 Applications of the Fermi-Dirac distribution

We now consider some applications of the Fermi-Dirac distribution (8.22). It is useful to start by examining the behavior of this function as the temperature goes to zero. This is given by

(8.75)

Thus all states below a certain value, which is the zero-temperature value of the chemical potential, are filled with one fermion each. All states above this value are empty. This is a highly quantum state. The value of ϵ for the highest filled state is called the Fermi level. Given the behavior (8.75) it is easy to calculate the Fermi level in terms of the number of particles. Let correspond to the magnitude of the momentum of the highest filled level. Then

(8.76)

where is the number of polarizations for spin, . Denoting , the Fermi level is thus given by

(8.77)

The ground state energy is given by

(8.78)

(8.79)

The pressure is then easily calculated as

(8.80)

(Since depends on 𝓃, it is easier to use (8.78) for this.) The multiparticle state here is said to be highly degenerate as particles try to go to the single quantum state of the lowest energy possible subject to the constraints of the exclusion principle. The pressure (8.80) is referred to as the degeneracy pressure. Since fermions try to exclude each other, it is as if there is some repulsion between them and this is the reason for this pressure. It is entirely quantum mechanical in origin, due to the needed correlation between the electrons. As we will see, it plays an important role in astrophysics.

The Fermi energy determines what temperatures can be considered as high or low. For electrons in a metal, is of the order of , corresponding to temperatures around . Thus, for most of the physics considerations, electrons in a metal are at low temperatures. For atomic gases, the Fermi level is much smaller due to the factor in (8.77), and room temperature is high compared to . We will first consider the high temperature case, where we expect small deviations from the classical physics.

The expression for , given by the normalization condition (8.23) is

(8.81)

Where is the thermal wavelength, defined as before, by The partition function , from (8.24) is given by

(8.82)

This being , the equation of state is given by

(8.83)

At low densities and high temperatures, we see from the power series expansion of the polylogarithms that it is consistent to take 𝒵 to be small. Keeping terms up to the quadratic order in 𝒵, we get

(8.84)

So, as in the bosonic case, we are not far from the ideal gas law. The correction may be identified in terms of the second virial coefficient as

(8.85)

This is positive; so, unlike the bosonic case, we would need a repulsive potential between classical particles to mimic this effect via the classical expression (8.51) for .(8.85)

8.4.1 Electrons in a metal

Consider a two-state system in quantum mechanics and, to begin with, we take the states to be degenerate. Thus the Hamiltonian is just a diagonal matrix,

(8.86)

If we consider a perturbation to this system such that the Hamiltonian becomes

(8.87)

then the degeneracy between the two eigenstates of is lifted and we have two eigenstates with eigenvalues

(8.88)

Now consider a system with N states, with the Hamiltonian as an matrix. Starting with all states degenerate, a perturbation would split the levels by an amount depending on the perturbing term. We would still have eigenstates, of different energies which will be close to each other if the perturbation is not large. As becomes very large, the eigenvalues will be almost continuous; we get a band of states as the new eigenstates. This is basically what happens in a solid. Consider atoms on a lattice. The electronic states, for each atom by itself, is identical to the electronic states of any other atom by itself. Thus we have a Hamiltonian with a very large degeneracy for any of the atomic levels. The interatomic forces act as a perturbation to these levels. The result is that, instead of each atomic level, the solid has a band of energy levels corresponding to each unperturbed single-atom state. Since typically the Avogadro number, it is a very good approximation to treat the band as having continuous energy eigenvalues between two fixed values. There are gaps between different bands, reflecting the energy gaps in the single-atom case. Thus the structure of electronic states in a solid is a series of well-separated bands with the energy levels within each band so close together as to be practically continuous. Many of these eigenstates will have wave functions localized around individual nuclei. These correspond to the original single- atom energy states which are not perturbed very much by the neighboring atoms. Typically, inner shell electrons in a multi-electron atom would reside in such states. However, for the outer shell electrons, the perturbations can be significant enough that they can hop from one atomic nucleus to a neighbor, to another neighbor and so on, giving essentially free electrons subject to a periodic potential due to the nuclei. In fact, for the calculation of these bands, it is a better approximation to start from free electrons in a periodic potential rather than perturbing individual atomic states. These nonlocalized bands are crucial for the electrical conductivity. The actual calculation of the band structure of a solid is a formidable problem, but for understanding many physical phenomena, we only need the general structure.

Consider now a solid with the electronic states being a set of bands. We then consider filling in these bands with the available electrons. Assume that the number of electrons is such that a certain number of bands are completely filled, at zero temperature. Such a material is an insulator, because if an electric field is applied, then the electrons cannot respond to the field because of the exclusion principle, as there are no unoccupied states of nearby energy.

The only available unoccupied states are in the next higher band separated by an energy gap. As a result the electrical conductivity is zero. If the field is strong enough to overcome the gap, then, of course, there can be conduction; this amounts to a dielectric breakdown.

However, when all the available electrons have been assigned to states, if there is a band of nonlocalized states which is not entirely filled, it would mean that there are unoccupied states very close to the occupied ones. Electrons can move into these when an electric field is applied, even if the amount of energy given by the potential is very small. This will lead to nonzero electrical conductivity. This is the case for conductors; they have bands which are not fully filled. Such bands are called conducting bands, while the filled ones are called valence bands.

The nonlocalized states of the conducting band can be labeled by the electron momentum with energy . The latter is, in general, not a simple function like , because of interactions with the lattice of atoms and between electrons. In general it is not isotropic either but will depend on the crystalline symmetry of the lattice. But for most metals, we can approximate it by the simple form

(8.89)

The effect of interactions can be absorbed into an effective electron mass . (Showing that this can actually be done is a fairly complicated task; it goes by the name of fermi liquid theory, originally guessed, with supporting arguments, by Landau and proved to some extent by Migdal, Luttinger and others. We will not consider it here.) At zero temperature, when we have a partially filled band, the highest occupied energy level within the band is the Fermi level . The value of can be calculated from (8.77), knowing 𝓃^→, the number of electrons (not bound to sites); this is shown in Table 8.2. Since 1 is of the order in terms of temperature, we see that, for phenomena at normal temperatures, we must consider the low temperature regime of the Fermi-Dirac distribution. The fugacity is very large and we need a large fugacity asymptotic expansion for various averages. This is done using a method due to Sommerfeld. Consider the expression for 𝓃^→, from (8.81), which we can write as

(8.90)

Where and . The idea is to change the variable of integration to . The lower limit of integration will then be , which may be replaced by as a first approximation. But in doing so,we need to ensure that the integrand vanishes at . For this one needs to do a partial integration first. Explicitly, we rewrite (8.90) as

(8.91)

In the first line we have done a partial integration of the expression from (8.90); in the second line we replaced the lower limit by . The discrepancy in doing this is at least of order due to the in the denominator of the integrand. This is why we needed a partial integration. We can now expand in powers of ω; the contribution from large values of will be small because the denominator ensures the integrand is sharply peaked around . Odd powers of 𝓌give zero since integrand would be odd under . Thus

(8.92)

This gives us the equation for as

(8.93)

By writing ,we can solve this to first order as

(8.94)

where we have used the expression for in terms of 𝓃^→. As expected, the value of at zero temperature is . Turning to the internal energy, by a similar procedure, we find

(8.95)

Using the result (8.94) for , this becomes

(8.96)

The most interesting result of this calculation is that there is an electronic contribution to the specific heat which, at low temperatures, is given by

(8.97)

As expected from the third law, this too vanishes as .

8.4.2 White dwarf stars

A gas of fermions which is degenerate is also important in many other physical phenomena, including astrophysics. Here we will briefly consider its role in white dwarf stars.

The absolute magnitude of a star which is proportional to its luminosity or total output of energy per unit time is related to its spectral characteristic , which is in turn related to the temperature of its photosphere. Thus a plot of luminosity versus spectral classification, known as a Hertsprung-Russell diagram, is a useful guide to classifying stars. Generally, bluer stars or hotter stars have a higher luminosity compared to stars in the red part of the spectrum. They roughly fall into a fairly well defined curve. Stars in this category are called main sequence stars. Our own star, the Sun, is a main sequence star. There are two main exceptions, white dwarfs which tend to have low luminosity even though they are white and red giants which have a higher luminosity than expected for the red part of the spectrum. White dwarfs have lower luminosity because they have basically run of hydrogen for fusion and usually are not massive enough to pass the threshold for fusion of higher nuclei. They are thus mostly made of helium. The radiation is primarily from gravitational contraction. (Red giants are rather low mass stars which have exhausted the hydrogen in their cores. But then the cores contract, hydrogen from outer layers get pulled in somewhat and compressed enough to sustain fusion outside the core. Because the star has a large radius, the total output is very high even though the photosphere is not very hot, only around .)

Returning to white dwarfs, what keeps them from completely collapsing is the degeneracy pressure due to electrons. The stars are hot enough for most of the helium to be ionized and so there is a gas of electrons. The Fermi level is around or so, while the temperature in the core is of the order of . Thus the electron gas is degenerate and the pressure due to this is important in maintaining equilibrium. Electron mass being , the gas is relativistic. If we use the extreme relativistic formula, the energy-momentum relation is

(8.98)

Calculating the Fermi level and energy density, we find

(8.99)

For the condition for hydrostatic equilibrium, consider a spherical shell of material in the star, of thickness at a radius 𝓇 from the center. If the density (which is a function of the radius) is , then the mass of this shell is . The attractive force pulling this towards the center is , where is the mass enclosed inside the sphere of radius 𝓇. The pressure difference between the inside and outside of the shell under consideration is , with an outward force Thus equilibrium requires

(8.100)

Further, the mass enclosed can be written as

(8.101)

These two equations, along with the equation of state, gives a second order equation for . The radius of the star is defined by .

What contributes to the pressure? This is the key issue in solving these equations. For a main sequence star which is still burning hydrogen, the kinetic pressure (due to the random movements of the material particles) and the radiation pressure contribute. For a white dwarf, it is basically the degeneracy pressure. Thus we must solve these equations, using the pressure from (8.99). The result is then striking. If the mass of the star is beyond a certain value, then the electron degeneracy pressure is not enough to counterbalance it, and hence the star cannot continue as a white dwarf. This upper limit on the mass of a white dwarf is approximately times the mass of the Sun. This limit is known as the Chandrasekhar limit.

What happens to white dwarfs with higher masses? They can collapse and ignite other fusion processes, usually resulting in a supernova. They could end up as a neutron star, where the electrons, despite the degeneracy pressure, have been squeezed to a point where they combine with the protons and we get a star made of neutrons. This (very dense) star is held up by neutron degeneracy pressure. (The remnant from the Crab Nebula supernova explosion is such a neutron star.) There is an upper limit to the mass of neutron stars as well, by reasoning very similar to what led to the Chandrasekhar limit; this is known as the Tolman-Oppenheimer-Volkov limit. What happens for higher masses? They may become quark stars, and for even higher masses, beyond the stability limit of quark stars, they may completely collapse to form a black hole.

8.4.3 Diamagnetism and paramagnetism

Diamagnetism and paramagnetism refer to the response of a material system to an external magnetic field . To quantify this, we look at the internal energy of the material, considered as a function of the magnetic field. The magnetization of the material is then defined by

(8.102)

The magnetization is the average magnetic dipole moment (per unit volume) which the material develops in response to the field and, in general, is itself a function of the field . For ferromagnetic materials, the magnetization can be nonzero even when we turn off the external field, but for other materials, for small values of the field, we can expect a series expansion in powers of , so that

(8.103)

is the magnetic susceptibility of the material. In cases where the linear approximation (8.103) is not adequate, we define

(8.104)

In general is a tensor, but for materials which are isotropic to a good approximation, we can take , defining a scalar susceptibility . Materials for which are said to be diamagnetic while materials for which are said to be paramagnetic. The field , which appears in the Maxwell equation which has the free current as the source, is related to the field by ; is the magnetic permeability.

Regarding magnetization and susceptibility, there is a theorem which is very simple but deep in its implications. It is originally due to Niels Bohr and later rediscovered by . van Leeuwen. The theorem can be rephrased as follows.

Theorem 8.4.1 - Bohr-van Leeuwen theorem. The equilibrium partition function of a system of charged particles obeying classical statistics in an external magnetic field is independent of the magnetic field

It is very easy to prove this theorem. Consider the Hamiltonian of a system of N charged particles in an external magnetic field. It is given by

(8.105)

where α refers to the particle, as usual, and is the potential energy. It could include the electrostatic potential energy for the particles as well as the contribution from any other source. is the vector potential which is evaluated at the position of the α^-th particle. The classical canonical partition function is given by

(8.106)

The strategy is to change the variables of integration to

(8.107)

The Hamiltonian becomes

(8.108)

Although this eliminates the external potential from the Hamiltonian, we have to be careful about the Jacobian of the transformation. But in this case, we can see that the Jacobian is 1. For the phase space variables of one particle, we find

(8.109)

The determinant of the matrix in this equation is easily verified to be the identity and the argument generalizes to particles. Hence

(8.110)

We see that has disappeared from the integral, proving the theorem. Notice that any mutual binding of the particles via electrostatic interactions, which is contained in , does not change this conclusion. The argument extends to the grand canonical partition since it is

For the cognoscenti, what we are saying is that one can describe the dynamics of charged particles in a magnetic field in two ways. We can use the Hamiltonian (8.105) with the symplectic form or one can use the Hamiltonian (8.108) with the symplectic from . The equations of motion will be identical. But in the second form, the Hamiltonian does not involve the vector potential. The phase volume defined by is also independent of . Thus the partition function is independent of

This theorem shows that the explanation for diamagnetism and paramagnetism must come from the quantum theory. We will consider these briefly, starting with diamagnetism. The full treatment for an actual material has to take account of the proper wave functions of the charged particles involved, for both the localized states and the extended states. We will consider a gas of charged particles, each of charge ℯ and mass , for simplicity. We take the magnetic field to be along the third axis. The energy eigenstates of a charged particle in an external uniform magnetic field are the so-called Landau levels, and these are labeled by . The energy eigenvalues are

(8.111)

where . P is the momentum along the third axis and labels the Landau level. Each of these levels has a degeneracy equal to

(8.112)

The states with the same energy eigenvalue are labeled by . The particles are fermions (electrons) and hence the occupation number of each state can be zero or one. Thus the partition function is given by

(8.113)

Where 𝒵 is the fugacity as usual. For high temperature, we can consider a small 𝒵-expansion. Retaining only the leading term,

(8.114)

For high temperatures, we can also use a small 𝒳-expansion,

(8.115)

This leads to

(8.116)

The definition (8.102) is equivalent to .From

, we have ,so that

(8.117)

Which shows that

(8.118)

Using (8.102), (8.118), we see that

(8.119)

Further, the average particle number is given by

(8.120)

Using this to eliminate 𝒵, we find from (8.119),

(8.121)

The diamagnetic susceptibility is thus

(8.122)

Although the quantum mechanical formula for the energy levels is important in this derivation, we have not really used the Fermi-Dirac distribution, since only the high temperature case was considered. At low temperatures, the Fermi-Dirac distribution will be important. The problem also becomes closely tied in with the quantum Hall effect, which is somewhat outside the scope of what we want to discuss here. So we will not consider the low temperature case for diamagnetism here. Instead we shall turn to a discussion of paramagnetism.

Paramagnetism can arise for the spin magnetic moment of the electron. Thus this is also very much a quantum effect. The Hamiltonian for a charged point particle including the spin-magnetic field coupling is

(8.123)

Here is the spin vector and is the gyromagnetic ratio. For the electron , being the Pauli matrices, and is very close to ; we will take

. Since we want to show how a positive can arise from the spin magnetic moment, we will, for this argument, ignore the vector potential A in the first term of the Hamiltonian. The energy eigenvalues are thus

(8.124)

The partition function is thus given by

(8.125)

Where with

(8.126)

From , we get

(8.127)

By taking , we see that the number density of electrons for both spin states together is . For high temperatures, we can approximate the integral in (8.127) by

(8.128)

Using this, the magnetization becomes,

(8.129)

The susceptibility at high temperatures is thus given as

(8.130)

Turning to low temperatures, notice that we have already obtained the required expansion for the integral in (8.127); this is what we have done following (8.90), so we can use the formula (8.94) for , along with (8.77) for the Fermi level in terms of $\overset{\rightarrow}{\mathcal{n}}$, applied in the present case to separately. Thus