Particle scattering

Particle scattering, specifically in the context of particle physics, is scattering that occurs between particles. Particles are considered to be quantum objects, which makes this a form of quantum scattering, and often they are also considered to be relativistic. As such, rigorously solving particle scattering behavior can be incredibly complex. Moreover, partly due to quantum physics and partly due to the minuscule scale, particle scattering is a largely random phenomenon, speaking of the Probability of scattering more so than the certainty of it. Finally, due to wave-particle duality, the participating particles can also be considered waves.

The typical particle scattering process (or reaction) looks like this:

The letters represent the particles, with $a$ and $b$ being the incident particles and/or the target and $c$ and $d$ being the reaction products. $c$ and $d$ may very well be the same as $a$ and $b$, albeit in a different state. This is just an example of a two-particle interaction resulting in just as many particles. It's absolutely possible for there to be more inputs and outputs to the reaction, and for the number of inputs and outputs to differ from one another.

Particle scattering is ubiquitous. It's the foundation of many physical phenomena that we know, such as the color of the sky (Rayleigh scattering), and it is a fantastic experimental tool to probe the structure of particles, both as a whole and their internals. High-energy particle scattering is artificially created using a particle accelerator, specifically a collider. In colliders specifically, you don't have a single incident particle. Rather, it's an entire particle beam being aimed at a target, with each particles in the beam being referred to as a projectile. The target maybe stationary or moving, with the most common moving targeting being another particle beam going in the opposite direction and made to collide.

Particles involved in scattering may be elementary particles like electrons, composite particles like protons or even entire nuclei, atoms or ions.

Particle scattering is often explained through Feynman diagrams, which are standardized schematic representations of particle scattering. Being entire diagrams, they can carry much more information than a simple mathematical representation, which is sorely needed given the complexity of many-particle interactions.

Elastic scattering#

Particle scattering is elastic when kinetic energy is conserved (but may be redistributed) and the incident particles do not change in the process. A typical elastic process reads

where $a'$ and $b'$ are primed just for visual clarity, and are otherwise identical to $a$ and $b$. $b$ is assumed to remain in its ground state, but nevertheless absorb some momentum through its recoil.

In the context of using scattering for probing the structure of particles, the wavelength of the incident wave-particle must be smaller than the length scale of the target. We can determine a range by invoking the Heisenberg inequality

The momentum is provided by the de Broglie formula

where $\bar{\lambda}$ is the reduced de Broglie wavelength. If we take the error on momentum to be the same scale as the momentum itself ($p=\Delta p$, not unreasonable for such a small scale), we get

So the reduced wavelength must be in the order of the spatial uncertainty. Recalling that the energy of a wave $E=pc$, we can also write

If we take a $\Delta x$ of a few femtometers (typical radius of the atomic nucleus), we get something between 10 and 100 MeV. If we go further and take the charge radius of an individual proton (0.84 fm) we get just over 100 MeV. To go even deeper and probe the internals of single protons, the energy needs to go even higher, in the order several GeV¹.

Inelastic scattering#

Particle scattering becomes inelastic when the kinetic energy transferred from $a$ to $b$ results in $b$ transitioning into an excited state $b^{*}$. $b^{*}$ then decays into its ground state, emitting a particle or splitting into two or more new particles. Here it splits into $c$ and $d$. In symbols:

In inelastic scattering, it's also possible for the projectile $a$ to be absorbed by the target $b$ in order to fuse into one or more new particles, such as this:

This is especially common in beam-on-beam collisions in particle colliders (it's also the end goal, generally speaking).

In an experimental setting, if the original projectile $a'$ is the only particle that's detected from the scattering, then it's said to be inclusive. If all of the scattering and decay products are detected, it's said to be exclusive. Of course, if the projectile is absorbed, the scattering can't be inclusive².