Elastic Coulomb Scatter from an Unscreened Point Charge

Abstract

This chapter begins Part II of this book with an exploration of the physics of a process which will be fundamental throughout our future discussions, that of the Coulomb interaction between a moving charged particle and a stationary charged body or, equivalently, a Coulomb potential. The result is scatter, or the deviation of the moving particle from its original trajectory. Energy can be transferred through the recoil of the target. Here, the target is assumed to be a bare (unscreened) point charge. We begin with two independent classical descriptions of the problem: one based upon our development of the scattering integral and the other based upon the Laplace–Runge–Lenz vector. Both yield the same result in that the Coulomb scatter differential cross section with solid angle is proportional to \( \frac{1}{{\mathrm{ si}{{\mathrm{ n}}^4}\left( {\frac{\theta }{2}} \right)}} \), where θ is the angle through which the projectile is scattered from its original trajectory. The validity conditions for such a classical description are then examined. We then proceed to derivations of this differential cross section through quantum-mechanical means. We first use the first term of the Born expansion to yield an expression for the scattering amplitude and, thus, the differential cross section which is in exact agreement with that derived classically. But as the Coulomb potential does not satisfy the requirements of this approximation, we extend the Born expansion to second order for the Coulomb potential and determine that this second-order term will diverge, but can be made convergent for a screened charged target. An exact nonrelativistic quantum-mechanical solution for the Coulomb scattering differential cross section is then derived from the Schrodinger equation and compared to the classical results and that based upon the Born expansion. The problem of electron (Mott) scattering, in which the intrinsic spin of the electron is not neglected, is examined and a differential cross section derived from Dirac theory. Finally, the case of electron–electron (Moller) scatter and the inherent need to account for spin statistics and the indistinguishability between the electrons in the final state is derived.