Loading
$ \renewcommand{\D}[2][]{\,\text{d}^{#1} {#2}} $ $\DeclareMathOperator{\Tr}{Tr}$

Last edited on 19. October 2015

Different Formulations of Planck's Law

For a given object at temperature $T$ and in thermal equilibrium with its environment, Planck's law gives an upper limit for the spectral distribution of the emitted thermal radiation. It is completely determined by the object's temperature $T$ and independent of it's size or shape.[1]

In the literature one finds many different formulations of Planck's law. That can be very confusing and one might wonder, which one is the right one or what is the difference? This article is intended to create clarity in this issue.

Introductory Remarks

Radiometric Quantities

The emitted radiation is isotropic (the same for all directions) and can be expressed in terms of different radiometric quantities:

Any radiometric quantity which is capable of charaterising the radiation field locally can serve as the function value in Planck's law. The following ones are perhaps most common:

quantity label units meaning
radiance $L$ [W m-2 sr-1] spatial+directional power density
flux density $M = \pi L$ [W m-2] spatial power density
energy density \begin{align*} u &= \frac{4\pi}{c} L \\ &= \frac{4}{c} M \end{align*} [J m-3] spatial energy density
Overview of different radiometric quantities that can be used to characterise black body radiation. In the case of isotropic radiance, these are equivalent and differ only by a geometric factor ($c$ denotes the propagation velocity of the radiation, sr is the unit of a solid angle). You can find the derivation of the relationship for the energy density and for the radiant exitance in other articles.

Spectral variables

Likewise, different spectral variables can be used. Here is an overview:

Name Symbol SI-units Relation
frequency $\nu$ Hz
wavelength $\lambda$ m $\lambda = \frac{c}{f}$
spectroscopic wavenumber $\widetilde{\nu}$ m-1 $\widetilde{\nu} = \frac{1}{\lambda}$
angular frequency $\omega$ rad s-1 $\omega = 2\pi \nu$
wavenumber $k$ rad m-1 $k = \frac{2\pi}{\lambda}$
Overview over different spectral variables. The frequency $\nu$ is intrinsic for photons and remains unchanged, the wavelength may however change due to changes in the speed of light $c$.[2]

Spectral Distribution

For the spectral distribution of any of the radiometric quantities mentioned above (for instance the radiance $L$), the corresponding spectral density (say $L_\lambda := \frac{\partial L}{\partial\lambda}$) is plotted against the spectral variable (here $\lambda$).

If you are wondering why there is the partial derivative in the definition of the spectral density, I recommend you to read the article about the spectral distribution of radiometric quantities.

Change of the Radiometric Quantity

When changing the radiometric quantity, the Planck's law transforms as shown in fig. 1: \begin{align} L \overset{\cdot\pi}{\longrightarrow} M \overset{\cdot\frac{4}{c}}{\longrightarrow} u \end{align}

Change of the Spectral Variable

When the spectral variable is changed, all instances must be substituted and additionally the equation must be multiplied with the partial derivative of the old variable with respect to the new one in order to take the differential nature of the spectral distribution into account: \begin{align} L_\nu = \frac{\partial L}{\partial \nu} = \frac{\partial L}{\partial\lambda} \cdot \left| \frac{\partial\lambda}{\partial\nu} \right| = L_\lambda \cdot \left|\frac{\partial\lambda}{\partial\nu}\right| \end{align} You find a more detailed background of this in the article on the spectral distribution of radiometric quantities.

Overview of Different Forms

In Terms of Wavelength

With the wavelength $\lambda$ as the independent variable Planck's law reads: \begin{align} L_\lambda (\lambda) &= \frac{2hc^2}{\lambda^5} \cdot \frac{1}{\exp(\frac{hc}{\lambda k T})-1} \\ M_\lambda (\lambda) &= \frac{2\pi hc^2}{\lambda^5} \cdot \frac{1}{\exp (\frac{hc}{\lambda k T})-1} \\ u_\lambda (\lambda) &= \frac{8\pi hc}{\lambda^5} \cdot \frac{1}{\exp (\frac{hc}{\lambda k T})-1} \end{align} Here $h$ denotes Planck's constant, $c$ the speed of light, $k$ the Boltzmann constant and $T$ the black body temperature. [3] [4] [5]

In Terms of Frequency

When changing to the spectral variable frequency $\nu$, all $\lambda$'s have to be substituted by $\lambda = \frac{c}{\nu}$ and a factor $\left| \frac{\partial \lambda}{\partial\nu} \right| = \frac{c}{\nu^2}$ is added: \begin{align} L_\nu (\nu) &= \frac{2h\nu^3}{c^2} \cdot \frac{1}{\exp (\frac{h\nu}{k T})-1} \\ M_\nu (\nu) &= \frac{2\pi h\nu^3}{c^2} \cdot \frac{1}{\exp (\frac{h\nu}{k T})-1} \\ u_\nu (\nu) &= \frac{8\pi h \nu^3}{c^3} \cdot \frac{1}{\exp (\frac{h\nu}{k T})-1} \end{align} Here $h$ denotes Planck's constant, $c$ the speed of light, $k$ the Boltzmann constant and $T$ the black body temperature. [6] [7]

In Terms of Angular Frequency

When changing to the spectral variable angular frequency $\omega$, all $\nu$'s must be substituted by $\nu = \frac{\omega}{2\pi}$ and a factor $\left| \frac{\partial \nu}{\partial\omega} \right| = \frac{1}{2\pi}$ is added: \begin{align} L_\omega (\omega) &= \frac{2\hbar\omega^3}{(2\pi)^3 c^2} \cdot \frac{1}{\exp (\frac{\hbar\omega}{k T})-1} \\ M_\omega (\omega) &= \frac{\hbar\omega^3}{(2\pi)^2 c^2} \cdot \frac{1}{\exp (\frac{\hbar\omega}{k T})-1} \\ u_\omega (\omega) &= \frac{\hbar\omega^3}{\pi^2 c^3 } \cdot \frac{1}{\exp (\frac{\hbar\omega}{k T})-1} \end{align} Here $h$ denotes Planck's constant, $c$ the speed of light, $k$ the Boltzmann constant and $T$ the black body temperature.[8]

References

[1] G. W. Petty A First Course in Atmospheric Radiation Sundog Publishing 2006 (ch. 2.7.1)
[2] W. L. Wolfe Introduction to Radiometry SPIE Press 1998 (ch. 2.5)
[3] Lowry A. Kirkby Physik - Der Studienbegleiter Springer 2012 (p. 428)
[4] E. Boeker, R. van Grondelle Environmental Physics Wiley 2011 (p. 9)
[5] J. R. Mahan Radiation Heat Transfer Wiley 2002 (ch. 2.16)
[6] Lowry A. Kirkby Physik - Der Studienbegleiter Springer 2012 (p. 428)
[7] David G. Andrews An Introduction to Atmoshperic Physics Cambridge University Press 2000 (p. 58)
[8] Lowry A. Kirkby Physik - Der Studienbegleiter Springer 2012 (p. 428)