Physics for Non-Physicists: Cross Sections

I’ve decided to start adding explanations of physics topics every once in a while. I’ll make a page for these so they’re easy to find. The idea is that I can compile a series of explanations for non-physicists.

My first post is on cross sections. I chose this because this is important in understanding nearly all measurements in high energy and nuclear physics.

Cross Sections in Classical Physics

The cross section in classical physics is closely related to the informal definition of a cross section.

Consider a three dimensional object, such as a sphere or a cylinder. If I rotate the object in some arbitrary way and look at it, I see that it covers a certain fraction of my field of vision. The effective size of that object as projected onto the plane perpendicular to my line of sight is its cross section. For a circle, this is always just πr2 where r is the radius of the sphere. For a cylinder, the size depends on how it’s been rotated. This kind of cross section is the informal definition.

The total cross section described above only gives a very rough description of the size of the object. A cross section can also be used to glean information about the shape of the object. In physics, this is the differential cross section.

Again imagine your field of vision as a plane with the object at some rotation with respect to this plane. Now shoot tiny particles uniformly from that plane (and perpendicular to it) toward the object. Mathematically, I will quantify this as the intensity, I, which is just the number of particles passing through the plane per unit area per unit time.

If the particles are too far from the center of the object (assuming it is finite in size) they will miss. If they hit the object, they will be deflected at some angle related to the orientation of the surface of the object at the intersection between the object and the path of the particle. Thus, the position at which I fire the particle and its deflection angle tell us about the shape of the object’s surface. This is described mathematically by the differential cross section:

\frac{d\sigma(\theta,\phi)}{d\Omega} = \frac{\rm Number\ of\ particles\ deflected\ into\ a\ region\ d\Omega\ around\ (\theta,\phi)}{Id\Omega}.

The d in dΩ indicates that we look at an infinitesimally small region of scattering direction space, although we can get a good estimate as long as the region is small enough for whatever we’re trying to measure. The total cross section is then the sum over all scattering directions,

\sigma = \int d\Omega \frac{d\sigma}{d\Omega}.

Now add in some long range force like gravity. Gravity pulls particles toward the object, so their paths are deflected. Some particles will get pulled into the object, so clearly the effective cross section has increased. Particles with more momentum will be harder to deflect, so this effective cross section will also depend on the properties of the incoming particles; a generic cross cross section is a function of properties such as particle mass, particle momentum, magnetic moment, etc.

What if I leave in gravity but reduce the physical size of my object to nothing? The incoming particles can no longer fall into the object, but they will still be deflected. But remember that the differential cross section is defined only by the properties of the scattered particles (and maybe the orientation of the object) – not on anything related to the physical size of the object. In fact, we could even model an object with a physical size as simply being a point object that happens to have an infinitely strong repulsive force at what would normally be called the object’s surface. So, we can retain our original definition of the differential cross section but then define the total cross section as the integral of the differential cross section (hence the term “differential”). This gives us a definition of cross section based on the object’s interactions with particles. This definition is also a convenient one for comparing to experimental data.

Cross Sections in Quantum Physics

Quantum mechanics and quantum field theory (QFT – basically relativistic quantum mechanics) have different definitions of the cross section, although they should reduce to the classical one in the classical limit of the theories.

In quantum theories, the scattering process described by a cross section occurs in a probabilistic manner. Our classical definition of the differential cross section is basically probabilistic too. Ignoring the overall normalization giving the total cross section it describes the probability that a scattered particle will scatter in a particular way. However, the classical definition does not describe a probabilistic process. If we shoot a particle at our classical object, we can calculate exactly what it would do. In quantum physics, we can only calculate the probability that the given type of scattering will happen. Additionally, we can scatter particles off of a field localized in some region of space or scatter particles off of each other; objects as we know them won’t really exist at the scales of quantum theories.

The cross section in quantum theories basically just tells us the probability that a given process (what given outgoing particles come out of scattering of some incoming particles with known properties)  will happen. The differential cross section tells us the details (what directions, orientations, momenta, etc. the particles in the final state end up with). More formally,

R = \mathcal{L}\sigma.

The rate of interactions for the cross section σ that we want to measure is the product of the cross section and the luminosity – the analog of the intensity used in QFT calculations. (The quantum mechanics definition is based on waves rather than particles so  I won’t discuss here). The luminosity just quantifies the strength of the “beam” (or beams for a collider). This definition tells us how to get the cross section from a measurement: we know the properties of our beams (or beam and target) and we measure an event rate. The differential cross section just requires us to determine the rate of events with the given kinematics instead of the total rate.

By defining everything through probabilities, it is also easy to treat inelastic scattering, where the outgoing particle(s) (final state) are different from the incoming particle(s) (initial state). Even if there are dozens of particles in the final state, the cross section is really just the probability that that final state will occur (with an extra normalization factor making it an area).

Some Concluding Thoughts

Cross sections are a fundamental piece of much of modern physics, particularly high energy (particle) and nuclear physics. Event rates or numbers of events are very basic kinds of measurements, and these can be turned into cross sections by relating to the expected or measured luminosity.

In some cases just measuring the existence (or nonexistence) of a process is interesting. It could show us the existence of a new particle or a new interaction. Quantum field theories such as the Standard Model predict cross sections (total and differential) for many kinds of interactions and disallow others. Comparing measured cross sections to theoretical models lets us test the validity of those models.

There are many examples of cross sections in measurements that have been in the news over the past few years.

In dark matter direct detection, experiments seek to place upper bounds on the possible interaction cross sections between dark matter particles and Standard Model particles. This requires theoretical modeling of the dark matter distribution near Earth to determine the correct “luminosity” since the “beam” is just particles flying around in space and through detectors on Earth.

At the LHC, physicists are trying to determine whether or not the particle believed to be the Higgs boson really is a Standard Model Higgs by studying its interactions. The cross sections (often ratios of cross sections rather than absolute cross sections to reduce uncertainties) for many initial and final states are being compared to Standard Model predictions (and even non-Standard Model ones to test other models) to see if they are consistent.