5.4 The ideal gas of bosons or fermions: beyond the classical approximation

Take-home message: Where the number of available states approaches the number of particles in the system, the properties of the gas will depend on whether multiple occupancy is allowed (bosons) or not (fermions).

Bose and Fermi gases are the subject of an entire third year course PC3151. This is just a very brief taster.

When we derived the properties of the ideal gas previously, our results were only valid if the number of available single-particle levels greatly exceeded the number of particles in the gas ($n_Q\gg n$). This was because we knew that we were not treating states with more than one particle in them correctly. Now we know that if the gas particles are fermions, that isn't even possible, so we need a new approach. What we do is lift the restriction that the number of particles in the gas is fixed, and use the Gibbs distribution instead of Boltzmann. We then find that rather than focus on a single particle in the gas, it is easier to focus on what is happening in a single energy level. Then we can write

$${\cal Z}={\cal Z}_1{\cal Z}_2{\cal Z}_3\ldots=\prod_r{\cal Z}_r$$

where $r$ labels the energy level, not the particle. Since the log of a product is the sum of logs of the individual terms, the grand potential $\Phi_G$, the energy, the particle number and the entropy all consist of sums of the contributions from each level: $\left\langle N \right\rangle =\sum_r\left\langle N_r \right\rangle$, $\left\langle E \right\rangle =\sum_r\left\langle N_r \right\rangle \varepsilon_r$ etc.

Furthermore we have already found the single-level grand partition functions ${\cal Z}_r$ and the average occupancies $\left\langle N_r \right\rangle$: for fermions, which obey the Pauli exclusion principle:

$${\cal Z}_r=1+e^{(\mu-\varepsilon_r)\beta}\qquad\qquad \left\langle N_r \right\rangle ={1\over e^{(\varepsilon_r-\mu)\beta}+1}$$

and for bosons, which don't:

$${\cal Z}_r ={1\over 1-e^{(\mu-\varepsilon_r)\beta}}\qquad\qq... ...le N_r \right\rangle ={1\over e^{(\varepsilon_r-\mu)\beta}-1}.$$

(see here.)

For a gas the sum over discrete energy levels is replaced by an integral over the wavenumber $k$, weighted by the density of states:

$$\left\langle N \right\rangle =\int_0^\infty\!D(k)N(k){\rm d}... ...\right\rangle =\int_0^\infty\!D(k)\varepsilon(k) N(k){\rm d}k$$

where $\varepsilon(k)=\hbar^2k^2/2m$ and

$$N(k)={1\over e^{(\varepsilon(k)-\mu)\beta}\pm1}\qquad\hbox{for $\left\{ {\hbox{fermions}\atop \hbox{bosons}}\right\}$}$$

Note that for bosons, $\mu$ must be less than the energy of the lowest level (zero for most purposes) but for fermions $\mu$ can be (and usually will be) greater than 0.

References

• Mandl 11.2,11.5
• Bowley and Sánchez 10.2-3