In 1911, Ernest Rutherford and his students and coworkers were studying the scattering of what Rutherford called alpha particles, emitted by a radioactive source, from a thin gold foil. To the astonishment of everyone, the alpha particles, which had a mass of 4 times the hydrogen atom and a positive charge of 2 times the magnitude of the electron charge, scattered back as if they were interacting with a point source of charge. Rutherford had discovered the atomic nucleus. To describe the probability of an alpha particle scattering to a given angle θ relative to its incident direction, Rutherford assumed the alpha particles were moving along classical Newtonian trajectories describing the hyperbolic motion of one positive charge relative to another fixed positive charge. That assumption gave an excellent description of the data for relative number of alpha particles scattered as a function of scattering angle.  Calculations are normally done in the center of mass system of projectile and target nucleus, so the scattering angle is measured in that system.  When Rutherford's classical calculation was redone using quantum physics in the mid-1920s, the same equation for the differential cross section resulted.  This agreement between classical and quantum approaches happens because the range of the Coulomb force is infinite.

See the text for the detailed definitions of cross section, and differential cross section.

The standard unit for the total cross section σ, a measure of the probability of a given process, which has units of area, is the millibarn, which is 10^-31 square meters. The standard unit of distance for nuclear processes is the fermi (fm) which is 10^-15 meters. Thus a millibarn (mb) is 0.1 fm².  [The total cross section for Rutherford scattering is infinite, because the Coulomb force has an infinite range.  For the nuclear and the weak fundamental forces of nature, which have very short ranges, the total cross section is well-defined and finite.] The so-called differential cross section, dσ/dΩ, has units of mb/sr, where an sr is a steradian. To avoid using this concept, the author of the textbook expresses the differential cross section as dσ/d(cosθ), where θ is the scattering angle. By the 1940s ,it was well known and understood that atomic nuclei have radii of several fm, and consist of two kinds of particles, protons and neutrons, which themselves have radii of about 0.8 fm, and essentially the same mass. With extremely relativistic beams of electrons, or relativistic protons, and using relativistic quantum physics to analyze the results of the scattering of these electrons and protons from nuclei, it was found that nuclei have a radius given by R = r₀A^1/3 where the constant r₀ is 1.2 fm and A is the total number of protons and neutrons in the nucleus. With even higher energies, so that the probability wavelength of the incident particles was much less than 1 fm, it was found that both protons and neutrons have a charge distribution, even though the total charge of the neutron is zero.  [How do you know when a kinetic energy is relativistic?  Relativity must be used when the kinetic energy of a particle is comparable to its mass.  For example, an electron has a mass of 0.5 MeV, so a 100 MeV electron beam is ultra-relativistic. A proton has a mass of about 940 MeV, so a 300 MeV proton is relativistic and a 100,000 MeV proton is ultra-relativistic.]

In the 1950s, ultra-relativistic electron beams were used to probe the distribution of charge inside various atomic nuclei. It was found that all nuclei have about the same interior charge (proton) density, no matter how many neutrons and protons are present. The half-radius of the distribution is given very accurately by R = r0A1/3 with A = N + Z, the total number of nucleons present, and r0 = 1.2 fm. To learn about the density distribution of neutrons in the nucleus, proton beams of energies 200 to 1000 MeV were used. Note that the observed proton differential cross sections exhibit the quantum version of Fresnel diffraction, because of the fairly sharp “edge” of most nuclei.

Optical examples of Fresnel diffraction from sharp edges.

History repeated itself in the 1970s when physicists used the technique of deep inelastic scattering to observe directly the fundamental pointlike particles that the proton is composed of. Previous studies of submicroscopic systems had used elastic scattering, in which the outgoing particle after scattering has the same energy as before the scattering. In these new experiments, the incoming particle, an electron in most studies, interacts with a single constituent inside the proton, transferring a large amount of momentum and energy to it before coming back out of the proton. This makes it easy to measure the initial momentum of the struck particle inside the proton! Results are often displayed as a function of x, which is defined as the fraction of the proton's total momentum that was carried by the particle which was struck. The bosons of the strong interaction, which holds the proton and neutron together, are called gluons. They are massless virtual particles. We would expect the proton to be full of gluons, and it is. It is also full of virtual quark-antiquark pairs, in addition to the three real ”valence” quarks of which the proton is made. One of the biggest problems in understanding the strong force, which is mediated by gluons, is that the strong interaction field couples to color... and gluons carry color so gluons can interact with one another! Just as bad is the fact that the strong interaction is scale invariant, which means that if you picture the internal state of the proton down to some specific level of complexity, and then zoom in on a very tiny part of that state, it exhibits the same complexity and number of processes as the much larger volume you originally described!

The interactions in these deep-inelastic scattering studies can either involve the photon, γ, which is of course the boson of the electromagnetic interaction, or one of the three bosons of the so-called weak interaction, Z⁰ and W^±.  The four different quarks referred to in the diagram are u, d, s and c.  A bar over the symbol means it denotes an antiquark.  The valence quarks are denoted by a v subscript.  The proton is made of two u valence quarks, and one d valence quark.  The experiments allow the measurement of the charge, intrinsic spin, and many other properties of the particles that are struck.  All the particles are pointlike.

Here Q or q is the momentum transfer in the process, and the form factor F(q2) is proportional to the cross section dσ/dΩ. Also, this form factor is basically just the Fourier transform of the scattering potential,  or the matter density ρ(r) of the target system.

A general, simple quantum view of scattering. The beam, considered as particles in a momentum and energy eigenstate, is depicted as a plane probability wave incident on the scattering center. The scattered probability waves are outgoing spherical waves, still momentum and energy eigenstates, and they have an amplitude that varies as a function of the scattering angle. It is this variation that is measured as the differential cross section dσ/dΩ, by the detector of the scattered particles.