Amplitude reconstruction in charged particles scattering

Abstract

It is shown how to obtain, in modulus and phase, the forward short-range amplitude in the elastic channel for a collision between charged particles. The problem The reconstruction ofthe short range or, as often called, the nuclear amplitude f_N(θ) □ f_N will be discussed in the following sections and some new data presented. One consider two spinless, non identical, particles with charges Z₁e,Z₂e (e electron charge).If these charges are localized in points the pure Coulomb amplitude f_C(θ) □ f_c can be obtained in a closed form /1/ as well as the corresponding Rutherford cross-section dσ _C/dω = |f_C|². The amplitude f_C is governed by the Sommerfeld parameter n = Z₁Z₂e²/v,where v is the relative velocity of the two particles. When the long range potential is due to charges spread in space superimposed to short-range interactions, new elastic amplitude f(θ)□ f and cross-section dσ/dω = |f|² are computed. The short range or nuclear amplitude is by definition f_N = f − f_C. f and f_C are not functions but distributions /2/ and it is always assumed here that

$$f_N \in c^2 [0,\pi ]$$

((1))

A non-linear equation between f_N and the two above cross-sections is easily derived as

$$d\sigma /d\Omega - d\sigma _C /d\Omega = |f_N |^2 + 2\operatorname{Re} f_C^ * f_N$$

((2))

In (2) the LHS can be obtained from measurements in the elastic channel from some angle θ₀ up to π .Equation (2) is very difficult to analyze and the author is not aware of any formal discussion of it as it is the case for standard amplitude reconstruction /3/. The experimentalists interested by these problems have given methods appropriate to find the solution(s) f_N of (2) at a given energy E taking into account errors of measurement (for references see /4/). The new results concerning f_N are not obtained from (2) but from an other equation derived from it.

Laboratoire associé au C.N.R.S.