Differential Cross-Section: What is it and Why Does it Matter?

[AxiomOfChoice]

Can someone *please* explain what a "differential (scattering) cross-section" is?

I've tried and tried and tried to wrap my head around what this really is, and to develop some physical intuition about it, but I just can't do it. Can someone please explain what the "differential cross-section," or "scattering cross-section," or "differential scattering cross-section" is? It's usually abbreviated in textbooks by $\frac{d\sigma}{d\Omega}$. What the hell does it TELL us about the physical situation we're considering? Why does it really MATTER?

Thanks!

 

[QuantumPion]

A differential scattering cross section is the probability that a neutron scattering event will cause a neutron to go from energy E to E' and angle $$\Omega$$->$$\Omega$$'.

To get a feel for what this represents, think about the integral of the differential scattering cross section. If you integrate the differential cross section over all energies and angles, the result will be the total scattering cross section. E.g., the sum of the probabilities of a neutron to scatter from ANY energy TO any energy, and FROM any angle TO any angle, is the total scattering cross section. The differential cross section is the probability a neutron will scatter from SOME PARTICULAR energy TO some other particular energy, and some particular angle to some other particular angle.

The above description above would actually be $$\frac{d2\sigma}{d\Omega dE}$$. Therefore $$\frac{d\sigma}{d\Omega}$$ would refer to the same thing, but independent of energy. E.g. the probability a neutron would scatter from $$\Omega$$->$$\Omega$$' for all energies.

 

[QuantumPion]

Oh yes, I forgot to add why this is important. If you want to know whether a fast neutron will be able to bounce around a few nuclei to slow down and cause another fission, you need to know how that neutron interacts with all those nuclei, how far it will travel, how much energy it loses in each collision, how many collisions it will make before being absorbed, etc. These things are dependent on how the neutron interacts with those nuclei for any particular energy and change in angle. E.g. will a fast neutron in a medium make a few grazing hits and escape the reactor like a rock skipping across a pond? Or will it smack head on to another nuclei and stop like a football to the groin? Will it lose enough energy in a collision to slow down, or will it be absorbed in the resonance region, etc. :)

 

[vanesch]

To add to what QP said, a differential cross section is the "normalized result of an experiment (or the calculation of it)" that has to do with 2-particle collisions.
It describes the probability that after the collision, the particle will be going out under such and such an angle, and with such and such an energy.

The normalisation has to do with the following hypotheses:
- we assume that the NUMBER of particles going out under a certain angle and with a certain direction is proportional to the incoming flux density and the number of target particles.
- we assume that there are no interference effects between interactions with different incoming or target particles.

In other words, quantum-mechanically, we assume that we can consider a SINGLE 2-particle collision, and consider that for a setup with many particles, the outgoing state is a statistical mixture of all the single "pair" collisions without interference.

The last statement is for instance not true in the case of crystal diffraction, but is usually true with amorphe targets (unless you become nitpicking and look at small angles...).

So, making the above assumption, you can say that differential cross sections are:
1) resumes of experimental results, where people normalised over what is considered just factors of proportionality (beam flux intensity and target size)
2) can be the result of a quantum-mechanical calculation which tries to model a scattering experiment, starting out with a single incoming particle and a single target particle, and "doing the collision interaction".

In a way, you can think of a table of cross sections as say, the "data sheet" of the kind of collision you're occupied with.

Once you have it, you can use it to do calculations (but beware of coherence effects!) of more complicated systems, to do radiation transport, calculate nuclear reactors, ... or, on a more fundamental scientific level, compare the result of experiment with a calculation.