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

Corrections to the Sommerfeld formula for the LII-LIII 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 e² i.e., of relative order 1/Z, compared to the Sommerfeld splitting. The result is applied to experimental data for elements varying from $Z=60$ to $Z=92$, to obtain a direct experimental value for the fine structure constant. By the method of least squares, we find 1α=ce²=136.93 with an estimated limit of error of 0.18.