Photon Laws for Black Body Radiation: Starting with a relation showing the temperature volume changes for black body radiation which is undergoing an adiabatic change, it is shown how photon laws which correspond one by one with known radiant energy laws may be derived. In all instances, excepting two, namely λ_{m}′T=const and ν_{m}′T=const, the photon relations differ from the corresponding radiant energy relations chiefly by having a power of T decreased by one, a power of λ increased by one, or a power of ν decreased by one. The entropy of black body radiation is strictly proportional to the number of photons involved. Table of Black Body Radiation Constants: A table includes values in common units for the collective groups of constants entering the various photon and radiant energy equations mentioned in the paper.

Guilian Shi, Fei Han, Lin Wang, Chengwen Liang, and Kaiyang Li Biomed. Opt. Express 6(4) 1109-1123 (2015)

References

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These experimentally determined values taken together form the basis for the other, computed constants. Unfortunately, the values for σ and c_{2} vary somewhat from those obtained with the aid of generally accepted formulae using other observed data. While in the table the number of places used in expressing the values for the bracketed constants are those to be expected, were the values given for c_{2} and σ accepted, it is understood that they possess actual uncertainties of the order of
${\scriptstyle \frac{1}{2}}$ to 1 percent. How to shift from the values reported here to others based on other values for σ and c_{2} will be apparent to the user.

Tables (2)

Table I

Symbols, quantities and units.

Symbol

Quantity

Common Unit

λ

Wave-length

μ

ν

Frequency

vibration/sec.

T

Temperature

K°

h

Planck’s constant

erg sec.

k

Boltzmann gas constant for a single molecule

erg/K°

u

Radiant energy density

erg/cm^{3}

u_{λ}

Radiant energy density per unit wave-length interval

erg/(cm^{3}μ)

u_{ν}

Radiant energy density per unit frequency interval

erg/(cm^{3} vib./sec.)

C

Photon concentration

photon/cm^{3}

C_{λ}

Photon concentration per unit wave-length interval

photon/(cm^{3}μ)

C_{ν}

Photon concentration per unit frequency interval

photon/(cm^{3} vib./sec.)

s

Entropy density

erg/(cm^{3} K°)

σ

Stefan-Boltzmann constant for radiant energy

erg/(cm^{2} sec. K° ^{4})

σ′

Stefan-Boltzmann constant for photons

photon/(cm^{2} sec. K° ^{3})

a

Constant of the fourth power law relating to radiant energy density

erg/(cm^{3} K° ^{4})

a′

Constant of the third power law relating to photon concentration

photon/(cm^{3} K° ^{3})

ℛ

Radiancy

erg/(cm^{2} sec.)

ℛ_{λ}

Radiancy per unit wave-length interval

erg/(cm^{2} sec. μ)

ℛ_{ν}

Radiancy per unit frequency interval

erg/(cm^{2} sec. vib./sec.)

N

Photon emission rate per unit area

photon/(cm^{2} sec.)

N_{λ}

Photon emission rate per unit area and unit wave-length interval

photon/(cm^{2} sec. μ)

N_{ν}

Photon emission rate per unit area and unit frequency interval

photon/(cm^{2} sec. vib./sec.)

c_{2}

Second radiation constant

μ K°

λ_{m}

Wave-length at which, for a given T, ℛ_{λ} is a maximum

μ

λ_{m}′

Wave-length at which, for a given T, N_{λ} is a maximum

μ

ν_{m}

Frequency at which, for a given T, ℛ_{ν} is a maximum

vib./sec.

ν_{m}′

Frequency at which, for a given T, N_{ν} is a maximum

vib./sec.

T_{m}

Temperature for the maximum spectral efficiency of production of radiant energy at wave-length λ

K°

T_{m}′

Temperature for the maximum spectral efficiency of photons production at wave-length λ

K°

λ_{e}

Effective wave-length for the total radiation

μ

ν_{e}

Effective frequency for the total radiation

vib./sec.

Table II

Equations and values for the collective groups of constants enclosed in [ ].

These experimentally determined values taken together form the basis for the other, computed constants. Unfortunately, the values for σ and c_{2} vary somewhat from those obtained with the aid of generally accepted formulae using other observed data. While in the table the number of places used in expressing the values for the bracketed constants are those to be expected, were the values given for c_{2} and σ accepted, it is understood that they possess actual uncertainties of the order of
${\scriptstyle \frac{1}{2}}$ to 1 percent. How to shift from the values reported here to others based on other values for σ and c_{2} will be apparent to the user.