# Different Formulations of Planck's Law

- 1. Introductory Remarks
- 1.1. Radiometric Quantities
- 1.2. Spectral variables
- 1.3. Spectral Distribution
- 1.3.1. Change of the Radiometric Quantity
- 1.3.2. Change of the Spectral Variable
- 2. Overview of Different Forms
- 2.1. In Terms of Wavelength
- 2.2. In Terms of Frequency
- 2.3. In Terms of Angular Frequency

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 |

### 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} |
$k = \frac{2\pi}{\lambda}$ |

^{[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] | A First Course in Atmospheric Radiation Sundog Publishing 2006 (ch. 2.7.1) |

[2] | Introduction to Radiometry SPIE Press 1998 (ch. 2.5) |

[3] | Physik - Der Studienbegleiter Springer 2012 (p. 428) |

[4] | Environmental Physics Wiley 2011 (p. 9) |

[5] | Radiation Heat Transfer Wiley 2002 (ch. 2.16) |

[6] | Physik - Der Studienbegleiter Springer 2012 (p. 428) |

[7] | An Introduction to Atmoshperic Physics Cambridge University Press 2000 (p. 58) |

[8] | Physik - Der Studienbegleiter Springer 2012 (p. 428) |