Kramers' law

This article is about X-ray emission spectra. For other uses, see Kramers' law (disambiguation).

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 $\omega \leq \omega _{{max}}$

\psi (\omega )=0

for $\omega \geq \omega _{{max}}$ .

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

References

↑ 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.
↑ 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.

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