Fine-structure correction in the x-ray-edge problem

The effect of the fine structure of the one-electron propagator is considered in the edge problem of x-ray spectra of metals. Until now this problem was treated neglecting the logarithmic terms and the fine structure was simply time averaged. However the detailed fine structure may be important at the instant the ion potential varies suddenly. We show that it contributes by a factor $1+A {\left|\epsilon \right|}^{\frac{1+2{\delta }_l}{\pi }}\ln \left|\epsilon \right|$ varying strongly at the edge frequency ($\epsilon =0\mathrm{}$).