Kramers' law

The formula for Kramers' law is usually given as the distribution of intensity (photon count) ${\displaystyle I}$ against the wavelength ${\displaystyle \lambda }$ of the emitted radiation: $${\displaystyle I(\lambda )d\lambda =K\left(\frac{\lambda }{{\lambda }_{\text{min}}}-1\right)\frac{1}{{\lambda }^2}d\lambda }$$

The constant K is proportional to the atomic number of the target element, and ${\displaystyle {\lambda }_{\text{min}}}$ is the minimum wavelength given by the Duane–Hunt law. The maximum intensity is ${\displaystyle \frac{K}{4{\lambda }_{\text{min}}^2}}$ at ${\displaystyle 2{\lambda }_{\text{min}}}$.

The intensity described above is a particle flux and not an energy flux as can be seen by the fact that the integral over values from ${\displaystyle {\lambda }_{min}}$ to ${\displaystyle \mathrm{\infty }}$ is infinite. However, the integral of the energy flux is finite.

To obtain a simple expression for the energy flux, first change variables from ${\displaystyle \lambda }$ (the wavelength) to ${\displaystyle \omega }$ (the angular frequency) using ${\displaystyle \lambda =2\pi c/\omega }$ and also using ${\displaystyle \tilde{I}(\omega )=I(\lambda )\frac{-d\lambda }{d\omega }}$. Now ${\displaystyle \tilde{I}(\omega )}$ is that quantity which is integrated over ${\displaystyle \omega }$ from 0 to ${\displaystyle {\omega }_{\text{max}}}$ to get the total number (still infinite) of photons, where ${\displaystyle {\omega }_{\text{max}}=2\pi c/{\lambda }_{\text{min}}}$: $${\displaystyle \tilde{I}(\omega )=\frac{K}{2\pi c}\left(\frac{{\omega }_{\text{max}}}{\omega }-1\right)}$$

The energy flux, which we will call ${\displaystyle \psi (\omega )}$ (but which may also be referred to as the "intensity" in conflict with the above name of ${\displaystyle I(\lambda )}$) is obtained by multiplying the above ${\displaystyle \tilde{I}}$ by the energy ${\displaystyle \hslash \omega }$: $${\displaystyle \psi (\omega )=\frac{K}{2\pi c}(\hslash {\omega }_{\text{max}}-\hslash \omega )}$$ for ${\displaystyle \omega \le {\omega }_{\text{max}}}$ $${\displaystyle \psi (\omega )=0}$$ for ${\displaystyle \omega \ge {\omega }_{\text{max}}}$.

It is a linear function that is zero at the maximum energy ${\displaystyle \hslash {\omega }_{\text{max}}}$.