5.4 Ideal Bose gas

For bosons the mean population number of a state is

$$\langle f_{\vec{p}} \rangle \equiv \frac{f_{i}^{*}}{g_{i}} = \frac{1}{z^{-1}e^{\beta E_{i}} - 1}$$ (5.12)

This function looks a bit like the Boltzmann factor $\propto e^{-E_{\vec{p}}/kT}$ but is everywhere larger than the latter. For small $\mu$ and large $T$ we again find that $f_{B} \approx f_{Bm}$.

Figure 5.3: Mean population number of states in a Bose-Einstein system with $\mu =0$. For comparison we include a graph of the classical density $\exp \left[ -E / kT\right]$ at $kT=5$.

PHOTONS IN A BOX
A popular example for this type of system is a ``gas'' of photons. A box with an internal coating of reflecting material in which a photon gas is held in thermodynamic equilibrium is often called a ``black body''. Originally this name refers to a grain of black material that might be placed in the box to allow for absorption and re-emission of photons, thus enabling energy exchange and equilibration. In actual fact the internal walls of the box are never perfectly reflecting, rendering the insertion of a black body grain unnecessary.

Since the total energy in the box is conserved and the photons may change their energy upon absorption and reemission, the total number of photons is not conserved. This is tantamount to assuming $\mu =0$.

To determine the energy spectrum of the photons we first calculate the number $d\Sigma$ of states within a small energy interval. These states will then be populated according to equ. 5.12. A simple geometric consideration - how many lattice points are lying within a spherical shell - leads to

$$d \Sigma = \frac{8 \pi V p^{2}dp}{h^{3}} = \frac{8 \pi V \nu^{2}d\nu}{c^{3}}$$ (5.13)

where $p=h \nu / c$ is the momentum pertaining to the energy $E=h \nu$. The total number of photons in the system is thus

$$N = \int \langle f_{\vec{p}} \rangle d \Sigma = \frac{8 \pi V}{c^{3}} \int \frac{\nu^{2}}{e^{h \nu / kT}-1} d \nu$$ (5.14)

and accordingly the number of photons in a frequency interval $[ \nu, d \nu ]$ is

$$n(\nu) d \nu = \frac{8 \pi V}{c^{3}} \frac{\nu^{2}}{e^{h \nu / kT}-1} d \nu$$ (5.15)

The amount of energy carried by these photons is the spectral density of black body radiation; it is given by

$$I(\nu) d\nu \equiv \frac{dE(\nu)}{d \nu} = E(\nu) n(\nu) d \... ...\frac{8 \pi V}{c^{3}} \frac{h \nu^{3}}{e^{h \nu / kT}-1} d \nu$$ (5.16)

For pressure and internal energy of the photon gas we find - in contrast to the classical and Fermi cases - the relation $PV = U/3$.

Providing an explanation for the experimentally measured spectrum of black body radiation was one of the decisive achievements in theoretical physics around 1900. Earlier attempts based on classical assumptions had failed, and it was Max Planck who, by the ad hoc assumption of a quantization of energy, could reproduce the correct shape of the spectrum. The subsequent efforts to understand the physical implications of that assumption eventually lead up to quantum physics.