Chapter VIII
An elementary approach to the quantum theory of scattering by a potential

1. A Introduction
   1. A-1 Importance of collision phenomena
   2. A-2 Scattering by a potential
   3. A-3 Definition of the scattering cross section
   4. A-4 Organization of this chapter
2. B Stationary scattering states. Calculation of the cross section
   1. B-1 Definition of stationary scattering states
   2. B-2 Calculation of the cross section using probability currents
   3. B-3 Integral scattering equation
   4. B-4 The Born approximation
3. C Scattering by a central potential. Method of partial waves
   1. C-1 Principle of the method of partial waves
   2. C-2 Stationary states of a free particle
   3. C-3 Partial waves in the potential V(r)
   4. C-4 Expression of the cross section in terms of phase shifts

A. Introduction

A-1. Importance of collision phenomena

Many experiments in physics, especially in high energy physics, consist of directing a beam of particles (1) (produced for example by an accelerator) onto a target composed of particles (2), and studying the resulting collisions: the various particles1 constituting the final state of the system - that is, the state after the collision (cf. Fig. 1) - are detected and their characteristics (direction of emission, energy, etc.) are measured. Obviously, the aim of such a study is to determine the interactions that occur between the various particles entering into the collision.

Figure 1: Diagram of a collision experiment involving the particles (1) of an incident beam and the particles (2) of a target. The two detectors represented in the figure measure the number of particles scattered through angles ?1 and ?2 with respect to the incident beam.

The phenomena observed are sometimes very complex. For example, if particles (1) and (2) are in fact composed of more elementary components (protons and neutrons in the case of nuclei), the latter can, during the collision, redistribute themselves amongst two or several final composite particles which are different from the initial particles; in this case, one speaks of "rearrangement collisions". Moreover, at high energies, the relativistic possibility of the "materialization" of part of the energy appears: new particles are then created and the final state can include a great number of them (the higher the energy of the incident beam, the greater the number). Broadly speaking, one says that collisions give rise to reactions, which are described most often as in chemistry:

Amongst all the reactions possible2 under given conditions, scattering reactions are defined as those in which the final state and the initial state are composed of the same particles (1) and (2). In addition, a scattering reaction is said to be elastic when none of the particles' internal states change during the collision.

A-2. Scattering by a potential

We shall confine ourselves in this chapter to the study of the elastic scattering of the incident particles (1) by the target particles (2). If the laws of classical mechanics were applicable, solving this problem would involve determining the deviations in the incident particles' trajectories due to the forces exerted by particles (2). For processes occurring on an atomic or nuclear scale, it is clearly out of the question to use classical mechanics to resolve the problem; we must study the evolution of the wave function associated with the incident particles under the influence of their interactions with the target particles [which is why we speak of the "scattering" of particles (1) by particles (2)]. Rather than attack this question in its most general form, we shall introduce the following simplifying hypotheses:

1. We shall suppose that particles (1) and (2) have no spin. This simplifies the theory considerably but should not be taken to imply that the spin of particles is unimportant in scattering phenomena.
2. We shall not take into account the possible internal structure of particles (1) and (2). The following arguments are therefore not applicable to "inelastic" scattering phenomena, where part of the kinetic energy of (1) is absorbed in the final state by the internal degrees of freedom of (1) and (2) (cf. for example, the experiment of Franck and Hertz). We shall confine ourselves to the case of elastic scattering, which does not affect the internal structure of the particles.
3. We shall assume that the target is thin enough to enable us to neglect multiple scattering processes; that is, processes during which a particular incident particle is scattered several times before leaving the target.
4. We shall neglect any possibility of coherence between the waves scattered by the different particles which make up the target. This simplification is justified when the spread of the wave packets associated with particles (1) is small compared to the average distance between particles (2). Therefore we shall concern ourselves only with the elementary process of the scattering of a particle (1) of the beam by a particle (2) of the target. This excludes a certain number of phenomena which are nevertheless very interesting, such as coherent scattering by a crystal (Bragg diffraction) or scattering of slow neutrons by the phonons of a solid, which provide valuable information about the structure and dynamics of crystal lattices. When these coherence effects can be neglected, the flux of particles detected is simply the sum of the fluxes scattered by each of the N target particles, that is, N times the flux scattered by any one of them (the exact position of the scattering particle inside the target is unimportant since the target dimensions are much smaller than the distance between the target and the detector).
5. We shall assume that the interactions between particles (1) and (2) can be described by a potential energy V(r1 - r2), which depends only on the relative position r = r1 - r2 of the particles. If we follow the reasoning of § B, Chapter VII, then, in the center-of-mass reference frame3 of the two particles (1) and (2), the problem reduces to the study of the scattering of a single particle by the potential V (r). The mass µ of this "relative particle" is related to the masses m1 and m2 of (1) and (2) by the formula: (A-2)

A-3. Definition of the scattering cross section

Let Oz be the direction of the incident particles of mass µ (fig. 2). The potential V (r) is localized around the origin O of the coordinate system [which is in fact the center of mass of the two real particles (1) and (2)]. We shall designate by Fi the flux of particles in the incident beam, that is, the number of particles per unit time which traverse a unit surface perpendicular to Oz in the region where z takes on very large negative values. (The flux Fi is assumed to be weak enough to allow us to neglect interactions between different particles of the incident beam.)

We place a detector far from the region under the influence of the potential and in the direction fixed by the polar angles ? and f, with an opening facing O and subtending the solid angle dO (the detector is situated at a distance from O which is large compared to the linear dimensions of the potential's zone of influence). We can thus count the number dn of particles scattered per unit time into the solid angle dO about the direction (?, f). The differential dn is obviously proportional to dO and to the incident flux Fi. We shall define s (?, f) to be the coefficient of proportionality between dn and Fi dO:

The dimensions of dn and Fi are, respectively, T-1 and (L2T)-1,s (?, f) therefore has the dimensions of a surface; it is called the differential scattering cross section in the direction (?, f). Cross sections are frequently measured in barns and submultiples of barns:

The definition (A-3) can be interpreted in the following way: the number of particles per unit time which reach the detector is equal to the number of particles which would cross a surface s (?, f) dO placed perpendicular to Oz in the incident beam.

Similarly, the total scattering cross section s is defined by the formula:

Comments:

1. Definition (A-3), in which dn is proportional to dO, implies that only the scattered particles are taken into consideration. The flux of these particles reaching a given detector D [of fixed surface and placed in the direction (?, f)] is...