Coulomb scattering of dirac particles

The continuum Coulomb wave function in the Johnson and Deck form—a matrix operator acting on a plane wave spinor—is discussed. General relations for the elastic differential cross section, the asymmetry parameters, and the precession and nutation of polarization are developed. An expansion of the asymptotic wave function correct to third order in λ = αZ for all Born parameters $\text{ν =}\text{αZ}\text{β}$ is obtained. This expansion involves a two parameter function T(θ, ν) which in turn may be expanded for small ν to give results previously obtained by Johnson, Weber, and Mullin. On the other hand, a large ν asymptotic expansion of T(θ, ν) valid for |2πν| > 1 permits a discussion of the non-relativistic limit. Both the small and large approximations substantially agree for $\text{|ν| ⋍}\text{1}\text{2}$. Consequently, these two approximations may be used to span the whole range of Born parameters.The asymptotic Born parameter approximation predicts markedly different scattering behavior for electrons and positrons. A striking feature is that functions pertaining to electron scattering are oscillatory in the scattering angle θ while the positron functions are not. In particular, the oscillations in the asymmetry parameter for electron scattering obtained numerically by Sherman are contained in the analytical approximate form. Also, the approach to the Rutherford cross section is different than that quoted by Mott and Massey.