3.6 Scattering in three dimensions
Sophisticated methods have been developed to analyse scattering in three-dimensions. The complexity of these methods makes them unsuitable for inclusion in this course but it is appropriate to say something about the basic quantities involved.
In three dimensions, we are obliged to think in terms of scattering at a given angle, rather than in terms of one-dimensional reflection or transmission. We distinguish between the incident particles (some of which may be unaffected by the target) and the scattered particles which are affected by the target in some way (changing their direction of motion, energy or state of internal excitation). The detectors for the scattered particles are placed far from the target, well outside the range of interaction of the incident beam and the target, so the scattering process is complete by the time the particles are detected. The incident beam is assumed to be uniform and broad enough to cover all the regions in which the beam particles interact with the target. The incident beam is characterised by its flux; this is the rate of flow of particles per unit time per unit area perpendicular to the beam.
If we consider a particular scattering experiment (electron-proton scattering, for example), one of the main quantities of interest is the total cross-section, σ. This is the total rate at which scattered particles emerge from the target, per unit time per unit incident flux. The total cross-section has the dimensions of area. Very loosely, you can think of it as representing the ‘effective’ area that the target presents to an incident projectile, but you should not give too much weight to this classical interpretation, as most total cross-sections vary markedly with the energy of the incident particles. An acceptable SI unit for the measurement of total cross-sections would be m2, but measured cross-sections are generally so small that physicists prefer to use a much smaller unit called the barn, defined by the relation 1 barn = 1 × 10−28 m2. The name is intended as a joke, 1 barn being such a large cross-section in particle and nuclear physics that it can be considered to be ‘as big as a barn door’. Many cross-sections are measured in millibarn (mb), microbarn (μb) or even nanobarn (nb).
Scattering processes that conserve the total kinetic energy of the colliding particles are said to be examples of elastic scattering. They may be contrasted with cases of inelastic scattering where the particles may change their internal state of excitation or be absorbed; particles may even be created or destroyed, especially at very high energies. In reality, total cross-sections often contain both elastic and inelastic contributions.
Scattering experiments are often analysed in great detail. The total cross-section arises as a sum of contributions from particles scattered in different directions. For each direction, we can define a quantity called the differential cross-section, which tells us the rate of scattering in a small cone of angles around the given direction. The integral of the differential cross-section, taken over all directions, is equal to the total cross-section. We can also vary the energy of the incident beam. Both the total cross-section and the differential cross-section depend on the energies of the incident particles. There is therefore a wealth of experimental information to collect, interpret and explain.
In exploring the microscopic world of atoms, nuclei and elementary particles, physicists have few options, other than to carry out a scattering experiment. This process has been compared with that of trying to find out how a finely crafted watch works by the expedient of hurling stones at it and seeing what bits come flying out. It is not a delicate business, but by collecting all the data that a scattering experiment provides, and by comparing these data with the predictions of quantum physics, physicists have learnt an amazing amount about matter on the scale of atoms and below. One early discovery in the scattering of electrons from noble gas atoms (such as xenon) was a sharp dip in the measured cross-section at an energy of about 1 eV. The experimental discovery of this Ramsauer-Townsend effect in the early 1920s was an early indication from elastic scattering that some new theoretical insight was needed that would take physics beyond the classical domain. The effect is now recognised as a three-dimensional analogue of the transmission resonance we met earlier.
At the much higher collision energies made available by modern particle accelerators, such as those at the CERN laboratory in Geneva, total cross-sections become dominated by inelastic effects, as new particles are produced. As an example, Figure 17 shows some data concerning the scattering of K− mesons by protons. The upper curve shows the variation of the total cross-section over a very wide range of energies, up to several GeV (1 GeV = 109 eV). The lower curve shows the contribution from elastic scattering alone. As the collision energy increases the contribution from elastic scattering becomes less and less important as inelastic processes become more common.