Statistical Mechanics/Thermal Radiation

Planck Distribution Function

For thermal radiation we know the following equation:

ε_n=sℏω_n

which we can apply our previously made Partition-function 'Infrastructure' to:

Z = Σ_s=0 exp(-sℏω_n/T)

By algebra:

= 1/(1 - exp(-ℏω_n/T))

Therefore, we can also find the probability:

P(s) = exp(-sℏω_n/T)/Z

Now, we can start calculating some interesting thermodynamic quantities. Let's start with the thermal average of s, the average mode of thermal radiation given a certain temperature:

<s> = Σ_s=0 sP(s) = Z^-1 Σsexp(-sℏω_n/T)

Which if we carry out the mathematics of the sum:

<s>=1/(exp(ℏω_n/T) - 1)

Stefan-Boltzmann Law

Remember that for a mode:

ε_n=sℏω_n

Average it:

<ε_n> = <sℏω_n>

= <s>ℏω_s

From the previous section:

= ℏω_n/(exp(ℏω_n/T) - 1)

Thus, if we sum up over all the modes:

U = Σ_n ℏω_n/(exp(ℏω_n/T) - 1)

Note that ω_n = nπc/L, now because ℏ is so small, we can approximate this sum to an integral. In the process we will change the coordinates of the integral over n in spherical coordinates, and we will let x = πℏcn/LT (an extra 1/8 comes in because we are integrating over only positive values of n, and an extra 2 due to two independent set of cavity modes of frequencies):

Note: actually, this is a density of states problem with D(n) = 4n² because of the spherical shell * 1/8 * 2 = n², ε=ℏω_n, and f(ε)=(exp(ℏω_n/T) - 1)^-1

U = (L³T⁴/π²ℏ³c³) ∫₀^∞ x³/(exp(x) - 1) dx

The integral has a definite value found in an integral table, L³=V, and thus we come upon the Stefan-Boltzmann law of radiation:

U/V = π²/15π²ℏ³c³ T⁴

Planck Radiation Law

Now, in our previous derivation, instead of integrating in terms of dn, say we left it as dω, there would be something of the form:

U/V = ∫dω u_ω

Carrying along the comparison with statistical properties, this is like a density, to be specific, a spectral density, if we carry the math out:

u_ω = ℏ/π²c³ ω³/(exp(ℏω/T) - 1)

And this is known as Planck's radiation Law.

Kirchhoff's Law

Say we are concerned with the radiant flux density, by the definition of flux density:

J_U = cU(T)/4V (the extra 4 is a geometrical factor)

If we take and apply the Stefan-Boltzmann law to this:

J_U = π²T⁴/60ℏ³c²

The only difference between this and Kirchhoff's law is an extra constant thrown in known as the absorption/emissivity constant, dependent on the material.