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 = |
|
|
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.
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