## Physics Friday 32: Blackbody Radiation (Part 1)

Consider a cubical cavity with side length a, filled with radiation in equilibrium, with the cavity large compared to the radiation wavelength (). To confine radiation, the conductivity of the cavity walls must be nonzero. For equilibrium, we need the electric field strenth at the walls to be zero, as  would be lossy at conducting walls. So, at equilibrium, we have only standing waves with nodes at the walls.

Consider standing waves in the x direction. The requirement that  leads to the condition , with n any positive integer. Considering the similar conditions for the other axes, and using some trig, we gain the condition that
, where nx, ny, and nz are positive integers. Using , we can find the frequencies of the allowed modes:
.
We can treat the allowed modes as a lattice of ponts in a three-dimensional space (nx, ny, nz). Since these points have unit spacing, a volume in that space encloses a number of points equal to the volume (to integer approximation). Choosing a radial coordinate r in this space (), we see
. Next, we must note that for each value of (nx, ny, nz), there are two radiation modes, one of each polarization.

Consider now the number of modes with frequencies in the range ν to ν+, which we shall call N(ν) (so that N(ν) is a sort of density function of mode number/frequency range).
Note that the number of modes between r and r+dr is double (again, because of polarization) the volume of the first octant portion of the spherical shell between r and r+dr, which for small dr is , and so .
Now, rewriting the right hand side in terms of frequency, we use , and thus  and , so we get
.

Now, if each mode has, in thermal equilibrium at temperature T, an average energy , then the energy density of radiation in the frequency range ν to ν+ is
.

Now, we can use the fact that blackbody radiation is isotropic to find from this the specific intensity (the radiated power per unit area per unit solid angle per unit frequency) Iν:
.

If we assume that the radiation is classical (able to have any positive energy), then it obeys the Boltzmann probability distribution: , where k is the Boltzmann constant.
This gives
.
The average energy per mode is thus
,
giving us
,
which is known as the Rayleigh-Jeans law, and which describes blackbody radiation at low frequencies (long wavelengths). However, it diverges toward infinity at high frequencies; this is the “ultraviolet catastrophe,” which, contrary to many textbooks, was not the motivation for Planck’s work (see here).

Next week, I will give a derivation of the solution to this discrepancy.