Statistical Mechanics/Thermal Radiation
Planck Distribution Function
editFor 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
editRemember 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
editNow, 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
editSay 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.