# Planck Black Body Irradiation Equation Derivation

An oscillator has energy <E> = kT with the mean temperature T. The energy emission density, u(ν)dν, multiplied by frequency range [ν, ν+dν] is proportional to the oscillator amount in a volume unit in this region, dN(ν), multiplied by kT and it is as follows:

u(ν)dν = <E> dN(ν) = kT dN(ν)

u(ν)dν = Emission energy in range [ν,ν+dn]/Volume

dN(ν) was calculated by Rayleigh and Jeans: dN(ν) = 8pn²/c³ dν

u(ν) = 8pn²/c³ kT | The Rayleigh-Jeans law is confirmed by experiments in IR region !

1900 Max Planck draw a conclusion that energy doesn't emit continuously but by small fractions (energy quanta). The probability that normal vibration has energy nhν is as follows (Boltzmann distribution):

Wn = e-nhν/kT/Q

The quantity Q (state sum) is for probability normalization on 1:

S¥n=0 Wn = 1/Q S¥n=0 (e−hν/kT)n = 1/Q .1/1 − e−hν/kT = 1

where the sum gives geometrical series. We obtain:

1/Q = 1 − e−hν/kT

The mean energy <E> is as follows

<E> = Σ Wn nhν = (1 − e−hν/kT) Σ nhν e−nhν/kT = (1 − e−hν/kT) Σ-d/d(1/kT) e-nhν/kT =

- (1 - e−hν/kT) d/d(1/kT)Σ e−nhν/kT = − (1 − e−hν/kT) d/d(1(kT) 1/1 − e−hν/kT = hν e−hν/kT/(1 − e−hν/kT)

<E> = hν/e+hν/kT− 1

We haven't yet obtained the degrees of freedom which can be obtained after the Rayleigh-Jeans law: U(ν) dν = <E> dN = 8pn²/c²<E> dν

One can easily obtain the spectral energy density u(ν)dν = <E> · dN(ν) in the frequency range [ν, ν+dν]:

| <E> | dN
u(ν)dν = | hν/e+hν/kT− 1 | . 8pn ²/c³ dν

And finally we can write down the Planck emission formula:

u(ν) = 8πhν³/c³.1/ehν/kT− 1

that is completely coincide with an experiment.