Supplement 2.2: Planck's Radiation Law    (2/2)

Energy of waves

According to the Law of Equipartition of Energy in thermodynamics, the mean kinetic temperature of an atom or another microparticle is determined solely by the absolute temperature T and given by kT. k=1.38 10 23 J/K is the Boltzmann constant (it is of no relevance right now, that this is a simplified exposition).

This expression meets the mean energy of a wave in a cavity as well. The mean value of the total energy of all waves in a frequency interval df can be obtained by multiplying the mean energy of the single waves by dN, the total number of waves in that frequency interval:

kTdN=8π f 2 c 3 VkTdf

Relating the total energy to the volume, we obtain the energy density of the waves within the frequency interval df:

dU= u f df= kT V dN=8π f 2 c 3 kTdf

The equation of spectral energy density

u f =8π f 2 c 3 kT

is the Rayleigh-Jeans radiation law. It has been derived by Lord Rayleigh and Sir James Hopwood Jeans in 1900, within a narrow time frame to Planck's radiation law. It describes the radiation of black bodies at low frequencies, shorter than the radiation spectrum's maximum (or: at long wavelengths, longer than the maximum) quite reliably. However, the equation does not have a maximum. Instead, its values increase quadratically towards higher frequencies (or: lower wavelengths), which has been named the 'ultraviolet catastrophe' of this law.

Max Planck by contrast chose a different formulation than kT to approach the waves' energy. He considered the oscillating atoms at the surface of the cavity and set their kinetic energy equal to

E n =nhf ,

with n=0,1,2,3,.... The quantum number n of the atomic oscillators corresponds to the mode number of the electromagnetic waves. In effect, Planck set the energy of an oscillator equal to the energy of the wave linked to it. This represents a quantization of the radiation, what caused the photon model to evolve.

Classical oscillators
Quantum mechanical oscillators

The probability pn of an oscillator being in the state n with the energy E n =nhf is named occupation probability of the state and it is described by the Boltzmann distribution:

p n = 1 Z exp( nhf kT )

The occupation probability decreases exponentially to an increase in energy. Temperature plays an important role in this case as well: high energy states are more likely occupied at higher temperatures.

 

The quantity Z in the denominator of the occupation probability is the so-called state sum:

Z= n=0 exp( nhf kT )

The state sum is describable as a geometric series. The result is:

Z= 1 1exp( hf / kT )
Prove

Z limits the occupation probabilities in a manner, so that its sum over all energy values equals 1 (or: 100%):

n=0 p n =1

The thermic mean value of the quantum number n of the cavity radiation is:

n = n=0 n p n = 1 Z n=0 nexp( nhf / kT )

For easier legibility, we set hf / kT =α , so that the sum expression becomes:

n=0 nexp( nα ) = d dα n=0 exp( nα )

A conversion into the derivation with respect to α simplifies the calculation notably, since the sum has already been calculated on the right-hand side at the beginning. It follows:

d dα n=0 exp( nα ) = d dα ( 1 1exp( α ) )= exp( α ) ( 1exp( α ) ) 2

Application to the equation for n provides the result for Z:

n = exp( hf / kT ) 1exp( hf / kT ) = 1 exp( hf / kT )1

The mean thermal energy of the cavity radiation is then:

E = n hf= hf exp( hf / kT )1

This expression replaces the mean kinetic energy kT mentioned at the beginning of this page. Considering this for the equation of spectral energy density, one obtains Planck's radiation equation:

u f = 8π f 2 c 3 hf exp( hf / kT )1
Transition to classical physics