Small-angle scattering and quasiclassical approximation beyond leading order

TL;DR

This paper assesses the accuracy of the quasiclassical approximation in small-angle electron scattering, deriving precise formulas for the cross section and Sherman function, and highlighting the importance of non-quasiclassical contributions for higher-order corrections.

Contribution

It provides the first derivation of next-to-leading order corrections to the quasiclassical approximation for small-angle electron scattering, including non-quasiclassical effects.

Findings

The correction to the cross section is of order θ^2.

The correction to the Sherman function is of order θ^1.

Non-quasiclassical contributions are essential for higher-order accuracy.

Abstract

In the present paper we examine the accuracy of the quasiclassical approach on the example of small-angle electron elastic scattering. Using the quasiclassical approach, we derive the differential cross section and the Sherman function for arbitrary localized potential at high energy. These results are exact in the atomic charge number and correspond to the leading and the next-to-leading high-energy small-angle asymptotics for the scattering amplitude. Using the small-angle expansion of the exact amplitude of electron elastic scattering in the Coulomb field, we derive the cross section and the Sherman function with a relative accuracy $\theta^2$ and $\theta^1$, respectively ($\theta$ is the scattering angle). We show that the correction of relative order $\theta^2$ to the cross section, as well as that of relative order $\theta^1$ to the Sherman function, originates not only from the contribution of large angular momenta $l\gg 1$, but also from that of $l\sim 1$. This means that, in general, it is not possible to go beyond the accuracy of the next-to-leading quasiclassical approximation without taking into account the non-quasiclassical terms.