Precise Determination of the Fine Structure Constant from X-Ray Spin Doublet Splitting

Corrections to the Sommerfeld formula for the $L_{\mathrm{II}}-L_{\mathrm{III}}$ x-ray spin doublet splitting, to take into account departures from a pure Coulomb field for heavy elements caused by interactions between electrons, are calculated by using Dirac wave functions for a Coulomb field and calculating, with all relativistic corrections, the terms in the electron interaction of order $e^2$ i.e., of relative order $\frac{1}{Z}$, compared to the Sommerfeld splitting. The result is applied to experimental data for elements varying from $Z=60\mathrm{}$ to $Z=92\mathrm{}$, to obtain a direct experimental value for the fine structure constant. By the method of least squares, we find $\frac{1}{\alpha }=\frac{\hslash c}{e^2}=136.93\mathrm{}$ with an estimated limit of error of 0.18.