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Deviations from the Stefan Boltzmann law at low temperatures

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The total radiation energy in a cavity is studied in the limit of small volume and temperature. The validity of refined high-temperature expansions is examined. For the cube-shaped cavity with edge lengthL complete results covering the range 0≦LT<∞ are presented.

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k :

Boltzmann constant

T :

absolute temperature

h :

Planck's constant

h :

Dirac's symbolh/2π

c :

velocity of light

V :

volume of cavity

K :

degree Kelvin



v i :

frequency of thei-th cavity mode

g i :

weight ofv i

\(\bar D(v)\) :

averaged mode density of cavity radiation

L :

edge length of cube shaped cavity, length of cylinder shaped cavity


order of


circumference of cylindrical cavity

γ i :

i-th smooth piece of γ

α j :

j-th corner angle of γ

A :

effective length in second order correction of mode density

K :

curvature of γ i

R :

radius of cylinder shaped cavity

E :

exact total radiation energy

E i :

i-th term in the high-temperature expansion ofE

E i * :

i-th term in the low-temperature expansion ofE

p :

diameter-to-length ratio=2R/L for cylindrical cavities

j s, l :

s-th zero of Bessel functionJ l(x)

j's, l :

s-th zero of derivative of Bessel function,J'l(x)


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Baltes, H.P. Deviations from the Stefan Boltzmann law at low temperatures. Appl. Phys. 1, 39–43 (1973). https://doi.org/10.1007/BF00886803

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Index Headings

  • Radiometry
  • Stefan Boltzmann formula
  • Black body radiation