Statistical Mechanics/Thermal Radiation

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Planck Distribution Function[edit]

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[edit]

Remember that for a mode:

εn=s&hbar;ωn

Average it:

n> = <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 approxiamate 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 becaues 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 = (L3T42&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[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 = &hbar;/π2c3 ω3/(exp(&hbar;ω/T) - 1)

And this is known as Planck's radiation Law.

Kirchoff'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 = π2T4/60&hbar;3c2

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