Kramers' law

Kramers' law is a formula for the spectral distribution of X-rays produced by an electron hitting a solid target. The formula concerns only bremsstrahlung radiation, not the element specific characteristic radiation. It is named after its discoverer, the Dutch physicist Hendrik Anthony Kramers.^[1]

The formula for Kramers' law is usually given as the distribution of intensity (photon count) $I$ against the wavelength $\lambda$ of the emitted radiation:^[2]

$I(\lambda) d\lambda = K \left( \frac{\lambda}{\lambda_{min}} - 1 \right)\frac{1}{\lambda^2} d\lambda$

The constant K is proportional to the atomic number of the target element, and $\lambda_{min}$ is the minimum wavelength given by the Duane–Hunt law.

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 $\lambda_{min}$ to $\infty$ is infinite. However, the integral of the energy flux is finite.

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

$\tilde I(\omega)=\frac{K}{2\pi c}\left( \frac{\omega_{max}}{\omega}-1\right)$

The energy flux, which we will call $\psi(\omega)$ (but which may also be referred to as the "intensity" in conflict with the above name of $I(\lambda)$) is obtained by multiplying the above $\tilde I$ by the energy $\hbar\omega$:

$\psi(\omega)=\frac{K}{2\pi c}(\hbar\omega_{max}-\hbar\omega)$

for $\omega \le \omega_{max}$

$\psi(\omega)=0$

for $\omega\ge \omega_{max}$.

It is a linear function that is zero at the maximum energy $\hbar\omega_{max}$.

References

1. Kramers, H.A. (1923). "On the theory of X-ray absorption and of the continuous X-ray spectrum". Phil. Mag. 46: 836. doi:10.1080/14786442308565244.
2. Laguitton, Daniel; William Parrish (1977). "Experimental Spectral Distribution versus Kramers' Law for Quantitative X-ray Fluorescence by the Fundamental Parameters Method". X-ray Spectrometry 6 (4): 201. doi:10.1002/xrs.1300060409.