Fermi and Bose Gases

Lets now look more closely at ideal gases.An ideal gas is a gas of non-interacting atoms in the limit of low concentration.The limit is defined in terms of the thermal average value of the number of particles, <N>, that occupy an orbital.We will usually call <N> the distribution function.An orbital is a state of the Schrodinger equation for only one particle.The orbital model gives an exact solution of the N?particle problem only if there are no interactions between the particles.

From quantum mechanics we know that there are two different types of particles in the universe, fermions and bosons.Fermions are particles with half integer spin. Examples of fermions are electrons and quarks. In terms of fundamental particles fermions are physical matter in the universe.Bosons are particles with integer spins. Examples of bosons are photons and phonons.In terms of fundamental particles bosons are associated with the forces in the universe.If an object is a composite of fundamental particles, its total spin is just the sum of the spins of the fundamental particles.For example, He⁴ has two electrons, two protons and two neutrons, each with a spin of ½.So the overall spin of the He atom is an integer and He₄ acts like a boson. He³ on the other hand has two electrons, two protons and one neutron, for a total of five particles.This results in a half-integer spin and He³ acts like a fermion.The effect of the difference in particle types shows up in the orbital model of non-interacting particles as occupancy rules:

1. An orbital can be occupied by any integral number of bosons of the same species, including zero.
2. An orbital can be occupied by 0 or 1 fermions of the same species.

The second rule is one of many different forms of the Pauli exclusion principle.It is because of the exclusion principle that the elements form the periodic table the way they do.The different occupancy rules have profound effects on many different parts of physics, especially on thermodynamics.The grand sum associated with fermions is very different from that associated with bosons.Only in the limit <N> << 1 do the two types converge and act the same. We call this limit the classical regime.

Fermions

Let's look first at fermions.We can treat each orbital as a system.

From the Pauli exclusion principle, we see that for one orbital the grand sum becomes

(13.1)

The distribution function is then

(13.2)

This is known as the Fermi-Dirac distribution function.The value of <N> always lies between zero and one.A graph of the distribution function looks like

At t = 0, we define m(t=O) = m(0) = e_F.e_F is called the Fermi energy.In solid state physics, it is also called the Fermi level.At temperature t = 0, all of the orbitals with energy below the Fermi energy are occupied by exactly one fermion each, and all orbitals with higher energy are empty.m(t), which is also called the Fermi level, is the temperature dependent chemical potential.For a free electron gas, m(t) can be approximated by e_F with negligible deviation.

Bosons

What about bosons?Since any integral number of bosons can occupy an orbital, then the energy associated with each orbital can be written as Ne, where e is the energy associated with one particle.Then the grand sum for one orbital is

So

(13.3)

This is known as the Bose-Einstein distribution function.We can write the two distribution functions together with the shorthand

where (+) indicates Fermi?Dirac statistics and (?) indicates Bose?Einstein statistics.Notice that when t << (e?m) the exponential dominates the denominator and the two distribution functions converge.

This is what we mean by the classical regime.In this regime, the distribution function for both fermions and bosons becomes

(13.4)

Chemical Potential

What is the chemical potential of photons and phonons?Recall that for photons

butfor photons and phonons.

What is the chemical potential of a system? In the classical regime,

Notice that if there are no interactions between the particles, the summation is the partition function Z₁, for a single particle in a volume V, so

<N> = lZ₁,

Earlier we saw that Z₁ = n_qV, so n/n_q = e^m/t and

m = t ln (n/n_q)

But this is the chemical potential of an ideal gas!Thus, we see that if we assume no interactions, the classical distribution function is the distribution function of an ideal gas. There are various ways in which this expression for the chemical potential can be changed.If the zero of the energy scale is shifted by an amount D so that the zero of the kinetic energy of the orbital falls at e₀ = D instead of e₀ = 0, then m = D + t ln(n/n_q). Notice that if this shift is applied to all of our systems, then it is not observable physically since we are only interested in differences of chemical potential.Another way of shifting the chemical potential is when we include the spin of atoms.In this case the number of orbitals in the grand sum is multiplied by the spin multiplicity 2s + l.This will increase the value of Z_lby 2s + l, and lower the chemical potential by a factor of t ln(2s+1).

Free Energy

What about the free energy? Remember that

so

(13.5)

Once we have the free energy, we can again determine the pressure, energy and entropy of an ideal gas.The pressure is given by

or

pV = Nt

The energy is found from

To get the entropy, we use the definition

Finally, we can derive the heat capacity at constant volume from

(13.6)

In a manner similar to that used for the heat capacity at constant volume, we can define the heat capacity at constant pressure.In terms of the entropy, the heat capacity at constant pressure is

(13.7)

Using the thermodynamic identity at constant number, tds = dU + p dV, this can be rewritten as

(13.8)

For an ideal gas, the energy depends only on the temperature, so , with .Similarly, from the ideal gas law, we have that V = Nt/p, so C_p becomes

Cp = CV + N

(13.9)

Substituting in the results for C_V, we see that

The ratio C_p/C_V is denoted as g.For an ideal gas we have g = 5/3.