Statistical Mechanics/Thermal Radiation
Planck Distribution Function edit
For thermal radiation we know the following equation:
εn=sℏωn
which we can apply our previously made Partition-function 'Infrastructure' to:
Z = Σs=0 exp(-sℏωn/T)
By algebra:
= 1/(1 - exp(-ℏωn/T))
Therefore, we can also find the probability:
P(s) = exp(-sℏωn/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> = Σs=0 sP(s) = Z-1 Σsexp(-sℏωn/T)
Which if we carry out the mathematics of the sum:
<s>=1/(exp(ℏωn/T) - 1)
Stefan-Boltzmann Law edit
Remember that for a mode:
εn=sℏωn
Average it:
<εn> = <sℏωn>
= <s>ℏωs
From the previous section:
= ℏωn/(exp(ℏωn/T) - 1)
Thus, if we sum up over all the modes:
U = Σn ℏωn/(exp(ℏωn/T) - 1)
Note that ωn = 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=ℏωn, and f(ε)=(exp(ℏωn/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 edit
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 uω
Carrying along the comparison with statistical properties, this is like a density, to be specific, a spectral density, if we carry the math out:
uω = ℏ/π2c3 ω3/(exp(ℏω/T) - 1)
And this is known as Planck's radiation Law.
Kirchhoff's Law edit
Say we are concerned with the radiant flux density, by the definition of flux density:
JU = cU(T)/4V (the extra 4 is a geometrical factor)
If we take and apply the Stefan-Boltzmann law to this:
JU = 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.