By Herbert Uberall
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Extra resources for Electron Scattering from Complex Nuclei, Part A
2-12a) if recoil is taken into account. ,/^ = 0, and Eq. (2-54c) holds, we find from Eq. ) Actually, Mott had derived Eq. (2-70a) not in first Born approximation, but by using a phase-shift method, and had only later expanded in powers of Za (Mott 29, 49). He also had obtained the next higher term in this 48 BASIC ELECTRON SCATTERING THEORY [Ch. 2 expansion, of order (Za)3, but with a wrong coefficient (Dalitz 51). We shall come back to this later; for now, we shall take the nonrelativistic limit of Eq.
The result is easily shown to be (Olsen 58; Überall 60): Vin out(*) = [(exp t'ke · r) + (ία · V — ßme — Ee)ITin out (r, K)]u(K) (2-76a) this being the solution that satisfies Eq. (2-75a) to first order in V(r)> and where //. -(r, k, - J -fa Λ 7 ^ ^ η τ , ) "«">exp * · '' (2"76b) The limit η —► + 0 is understood. If the integral over ds is evaluated so that the poles at s = ±k are encircled in the way prescribed by r\y one sees that the plus (minus) sign in the denominator corresponds to in coming (outgoing) spherical waves at infinity, caused by the scattering from the potential.
The first Born approximation alone is of re stricted validity, nominally determined by the criterion (Mott49): Za\ß<^\ (2-51) (at least for the Coulomb interaction), where ß = vjc> the electron velocity (in units of light velocity). Significantly, this shows [see also Parzen (50)] that the approximation does not improve with energy for the relativistic electrons which we shall mainly consider here, since, for them, ß ^ 1 already. It is expected to hold better for lighter nuclei, but it is now becoming increasingly obvious that even for the lightest nuclei, the lowest-order Born approximation is no longer sufficient to describe the nuclear properties to that accuracy which the most modern experimental facilities have become capable of achieving.
Electron Scattering from Complex Nuclei, Part A by Herbert Uberall