Cross sections

The formalism for treating a scattering process including QM is fairly sophisticated but the main ingredients and the overview are not too complicated.  The first point that we need to understand is the notion of asymptotic states. We encountered a similar idea in the kaon system, there are two appropriate sets of states, the states that are produced via the interaction of strongly interacting particles and the states that are suited to describe the time evolution when the weak interaction is present. These are the eigenstates for different Hamiltonians .For a scattering problem we define:

The quantum evolution of the system is described by the amplitude for a transition from and incoming state to an outgoing state.

The scattering amplitude is based on developing an operator that determines how as the interaction becomes important it changes the in states so that they overlap with certain out states. For example, how a beam of particles moving along the z direction is scattered to states where the particle moves along a direction at an angle  wrt the z-axis.

Classical Cross section

Most discussions start with the definition of the differential cross section.  The scattering process is depicted as a beam of particles that scatters into range of angles, . The cross section then represents the relationship between the beam and the scattered or outgoing particles. This leads to the natural set of descriptive variables b, . The cross section can be generalized to describe a process where a reasonable set of variables is used to characterize the before and after situation.

Classically a beam is described by the perpendicular distance from the z-axis or the impact parameter, b.  If I assume a beam spread uniformly over the x-y plane and moving along the z-axis I can determine the number of particles incident per unit impact parameter value by defining b, db and generating an area

This little portion of the beam will be scattered into an annulus

Requiring that for a given dA1 the same rate of particles emerges through dA2 [angular range] and defining the ratio of chosen areas to be the differential cross section we find

There is a relationship based on the interaction between b and  so that  can be computed for a given problem. [The above equations also assume symmetry about the z axis.]

 http://en.wikipedia.org/wiki/Image:ScatteringDiagram.svg

In general for some range of parameters that define the outgoing particles, . Where can involve an energy range or some combination of ranges of parameters

The key in general is to define the scattering intensities with respect a generalized differential. Normally there is no attempt to change the incident flux because most results are obtained from experiments with a beam that is best characterized by area and impact parameter. In the standard development the incident and scattered intensities are

[If Ib is independent of I can change variables and integrate over .]

per unit area and per unit solid angle respectively but when more variables are added [e.g. the outgoing energy] this redefines the outgoing flux definition.

This is also the case for the integration over . This just defines the relevant outgoing flux as per angular area  rather than .

So you can define the outgoing flux per unit angle or per unit solid angle in the normal fashion. The resulting measured outgoing flux must be either per unit angle or per unit solid angle.

Therefore,

Crossection is defined as the area of the incoming beam that is scattered.

[units of area]

Now if the scattering depends on something I can label the cross section by that variable.

is the part of the total cross section that specifically is scattered by an angle  into .

Since this is a continuous and not a discrete label we need to describe this process as the area of the beam that is scattered into a range of values. [ is the symbol commonly used to denote solid angle. The standard definition of  in spherical coordinates allows one to define an area on the surface of a sphere . The angular part is obtained by dividing by r2 ; .]

Take the map of the area of the incoming variable b to the outgoing variable .

when one integrates over . In many cases this integral can be performed because the interaction is independent of .]

The standard definition of cross section:

 differential cross section is the ratio of the beam area to the scattered solid angle. One determines the probability to find a final state particle with certain measured values or quantum numbers in a given solid angle when a beam of particles of an intesity Io [FLUX] which determines a  impinges on a target through an area dA.   The above definition does not account for the fact that there may be multiple targets. Scattering off a proton is normally accomplished using  a hydrogen target. The target will consist of millions of hydrogen atoms and therefore millions of protons so one need to normalize by target density (ρt), the length of the beam-target interaction region (lbt). In addtion practical measurement considerations such as ththe geometrical "size" of detector ((ΔΩ)d), and the "counting" efficiency of the detector (fd) must also be included. Note that the product of denstiy and target length yields the number of targets/area. wikipedia

FLUXES

It may help to imagine how one can quantify the number of particles that are moving through a system. In our case we will want to know the beam flux and the scattering flux.

Imagine a surface that is emitting particles or light. Each area of surface has a certain number N of particles, energy [n.b. energy is proportional to N] per unit area dA and these particles can be heading in different directions [] so that you can define a solid angle region .  Thus you can define the number of particles that come from the surface dA and head into an angular realm. To characterizing the experimental environment specifically to evaluate how busy a detector might be the luminosity is defined. If you know the cross section for a given process then the number of interactions may be calculated as . To increase luminosity one can either increase the beam intensity or increase the target length. So luminosity characterizes the rate of interactions for an experimental setup normalized to a unit cross-sectional strength.

 Scattered Flux using solid angle rather than dxdy number of particles scattered into a detector at angle   per unit solid angle.

 number  [N] number of particles (proportional to Energy) power [Po] e.g. Watts or J/s Beam Intensity [I] Here the energy or number of particles is measured per unit area. This is equivalent to the common definition of FLUX. Light intensity [I] (Slightly different definition than above often used with light) Radiant intensity è Power/Ω; power per unit solid angle. [watts per steradian (W·sr-1) ] (Particularly relevant for a point source.) Flux number of particles through/from an area dA in a time dt. luminosity [L] number of incident particles per unit area per unit time times the fraction of the beam scattered from  the target per unit cross sectional area. Characterizes the beam target configuration. General flux of particles or photons where different directions vary from the same point. Number of particles from an area that head into an angular bin per unit time. Light reflected off a white surface. Each point emits light in all directions. So the any area has light that covers all angles so one can talk about the light confined in a range

Further quantum considerations

There are basically two approaches to the cross section

1. measure
2. predict

The measurement follows the classical approach in that you simply put a detector in location and measure the scattering intensity and compare that to the beam intensity. Whether it is quantum or classical is irrelevant because you are dealing with the number of particle detected.

The prediction or theoretical calculation is very different.  Indeed if you look at the scattering of a plane water wave from an object

The outgoing wave can be a spherical wave that therefore has waves propagating in all directions. The quantum nature then has incoming states that evolve into outgoing states with probabilities for a range of angles.

The method for making theoretical predictions is to calculate how the interaction evolves the in states into the out states

This accomplished using perturbation theory. Feynman diagrams are a symbolic representation of the mathematical process that is used to evolve the system.  Once the amplitude is found the probability is determined by squaring the amplitude.  On then must sum up all configurations for final state particles that lead to the included out states. This is referred to as the allowable phase space. In finding configurations of the final state total energy and total momentum should be conserved.  The details are beyond our interest.