The Mass Absorption Coefficient of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>K</mml:mi></mml:math>Shell According to the Dirac Relativistic Theory of the Electron

Abstract

Taking as model an atom containing two non-interacting electrons and a fixed nucleus with charge $\mathrm{Ze}$, the mass absorption coefficient is calculated by use of the proper functions of the Dirac relativistic equation. $Z$ is determined so as to make the lowest energy level agree with the experimental value determined from the K absorption edge. The numerical calculation presented difficulty because of lack of tables of complex gamma functions. The relativistic coefficient is found to be from 0 to 40 percent smaller than the non-relativistic coefficient calculated by Nishina and Rabi, the greatest difference occurring for the heavy elements and short wave-lengths; it agrees slightly worse with experiment than the non-relativistic coefficient. The difference between theory and experiment is least for the heavy atoms, as would be expected, since for the heavy atoms (large $Z$) the neglected electronic interaction-field is small in comparison with the nuclear field. The variation of the relativistic coefficient with wave-length is complicated, but in the range $\frac{1}{2}{\ensuremath{\lambda}}_{k}$ to ${\ensuremath{\lambda}}_{k}$ (${\ensuremath{\lambda}}_{k}=$ wave-length of $K$ absorption edge) it is more nearly linear with ${\ensuremath{\lambda}}^{3}$ than the non-relativistic coefficient. The importance of using the relativistic equation for heavy atoms and short x-ray wavelengths is emphasized by these results, which also show that the model chosen is too approximate, even for the heavy elements.The general normalizing factors for the discrete and continuous spectrum proper functions of a hydrogen-like atom are given.