- Print publication year: 2020
- Online publication date: May 2020

- Publisher: Cambridge University Press
- DOI: https://doi.org/10.1017/9781108489065.011
- pp 385-424

Introduction

In the last chapter, we have discussed how to calculate the cross sections for the scattering of two point-like particles. Now the question arises, what happens when an electron interacts with a charge which is distributed in space like the one shown in Figure 10.1. The standard technique to measure the charge distribution and get information about the structure of the hadron is to measure the differential/total scattering cross sections of electron with a hadron and compare it with the cross section of electron scattering with a spinless (J = 0) point target (known as Mott scattering cross section). The ratio of these two is generally expressed as

where F(q2), in literature, is known as the form factor. This accounts for the spatial extent of the scatterer. F(q2) not only tells about the distribution of the charge in space but using it, one can estimate the size of the target particle as well as its charge distribution and density of magnetization. Thus, for an extended charge distribution, the probability amplitude for a point-like scatterer is modified by a form factor.

Physical Significance of the Form Factor

Consider the elastic scattering of a “spinless” electron from a static “spinless” point object having charge. In the Born approximation, where the perturbation is assumed to be weak, the scattering amplitude is written as

where and are the wave functions of the initial and final electron with momentum and, respectively. These waves are assumed to be plane waves such that

Instead of a point charge distribution, if we assume an extended charge distribution with normalization, then the potential felt by the electron located at is given by

where is the maximum range of the charge distribution. The scattering amplitude modifies to

Assuming, which leads to,

The term in the square brackets on the right-hand side of Eq. (10.6) is known as the form factor, which is nothing but the Fourier transform of the charge density distribution, given as

In field theory, if we consider the scattering of a spin electron from an external electromagnetic field (shown in Figure 10.2), the electromagnetic field in the momentum space is written as

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