Statistical Mechanics/Thermal Radiation

Planck Distribution Function For thermal radiation we know the following equation: εn=s&hbar;ωn which we can apply our previously made Partition-function 'Infrastructure' to: Z = Σs=0 exp(-s&hbar;ωn/T) By algebra: = 1/(1 - exp(-&hbar;ωn/T)) Therefore, we can also find the probability: P(s) = exp(-s&hbar;ωn/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(-s&hbar;ωn/T) Which if we carry out the mathematics of the sum: <s>=1/(exp(&hbar;ωn/T) - 1) Stefan-Boltzmann Law Remember that for a mode: εn=s&hbar;ωn Average it: <εn> = <s&hbar;ωn> = <s>&hbar;ωs From the previous section: = &hbar;ωn/(exp(&hbar;ωn/T) - 1) Thus, if we sum up over all the modes: U = Σn &hbar;ωn/(exp(&hbar;ωn/T) - 1) Note that ωn = nπc/L, now because &hbar; 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 = π&hbar;cn/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=&hbar;ωn, and f(ε)=(exp(&hbar;ωn/T) - 1)-1 U = (L3T4/π2&hbar;3c3) ∫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 = π2/15π2&hbar;3c3 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 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 = &hbar;/π2c3 ω3/(exp(&hbar;ω/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: JU = cU(T)/4V (the extra 4 is a geometrical factor) If we take and apply the Stefan-Boltzmann law to this: JU = π2T4/60&hbar;3c2 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.