Kramers' law explained

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 λ of the emitted radiation:^[2] $$I(\backslash lambda)d\backslash lambda\; =\; K\; \backslash left(\backslash frac\; -\; 1\; \backslash right)\; \backslash frac\; d\backslash lambda$$

The constant K is proportional to the atomic number of the target element, and

λ _{min} is the minimum wavelength given by the Duane–Hunt law. The maximum intensity is (K)/(4λ ^{2} _{min}) at 2 λ _{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

λ _{min} to inf ty is infinite. However, theintegral of the energy flux is finite.

To obtain a simple expression for the energy flux, first change variables from

λ (the wavelength) to \omega (the angular frequency) using λ = 2 \pi c/\omega and also using \tilde I(\omega) = I( λ ) (- d λ)/(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 / λ _{min} :$$\backslash tilde\; I(\backslash omega)\; =\; \backslash frac\; \backslash left(\backslash frac\; -\; 1\backslash right)$$

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( λ ) ) is obtained by multiplying the above \tilde I by the energy \hbar\omega :$$\backslash psi(\backslash omega)\; =\; \backslash frac(\backslash hbar\backslash omega\_\backslash text-\backslash hbar\backslash omega)$$for \omega \le \omega _{max} $$\backslash psi(\backslash omega)=0$$for \omega\ge \omega _{max} .

It is a linear function that is zero at the maximum energy

\hbar\omega _{max} .

Notes and References

1. On the theory of X-ray absorption and of the continuous X-ray spectrum . Kramers. H.A.. 1923. Phil. Mag.. 46. 836. 10.1080/14786442308565244.
2. 10.1002/xrs.1300060409. 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. 1977XRS.....6..201L.