Introduction

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

where F(q2), in literature, isknown as the form factor. This accounts for the spatial extent of thescatterer. F(q2) not only tellsabout the distribution of the charge in space but using it, one can estimatethe size of the target particle as well as its charge distribution anddensity of magnetization. Thus, for an extended charge distribution, theprobability amplitude for a point-like scatterer is modified by a formfactor.

Physical Significance of the Form Factor

Consider the elastic scattering of a “spinless” electron from astatic “spinless” point object having charge. In the Bornapproximation, where the perturbation is assumed to be weak, the scatteringamplitude is written as

where and are the wave functions of the initial and final electron withmomentum and, respectively. These waves are assumed to be plane waves suchthat

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

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

Assuming, which leads to,

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

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