Interactions of Photons and Neutrons

Photons and neutrons carry no charge, so neither ionizes continuously the way a proton or electron does. Each travels through matter until a single event either absorbs it or scatters it out of the beam. A collimated beam is therefore attenuated exponentially rather than stopped at a definite range, and the physics is captured by a cross section for each competing process. Photons interact with the atomic electrons and the Coulomb field of the nucleus; neutrons interact only with the nuclei themselves, through the strong force.¹

Exponential attenuation

Let a narrow beam of $N$ photons per second pass through a slab. In a thickness $\mathbf{d}x$ the probability that a given photon is removed is $n\sigma \mathbf{d}x$, where $n$ is the number density of target atoms and $\sigma$ the total cross section per atom. The beam loses $\mathbf{d}N=-Nn\sigma \mathbf{d}x$, which integrates to

The linear attenuation coefficient $\mu$ has units of inverse length; its reciprocal is the mean free path ${\lambda }_{\mathrm{mfp}}=1/\mu$, the average distance between interactions. Because $\mu$ scales with density, the tabulated quantity is the mass attenuation coefficient $\mu /\rho$, which depends on the element and photon energy but not on the physical state of the material. The thickness that halves the beam is the half-value layer

The total cross section is a sum over the independent processes, $\sigma ={\tau }_{\mathrm{pe}}+{\sigma }_{\mathrm{C}}+{\kappa }_{\mathrm{pair}}$, so the attenuation coefficient decomposes the same way. Which term dominates depends on the photon energy and the atomic number of the absorber.

The photoelectric effect

At low energy the photon is absorbed entirely by a bound atomic electron, which is ejected with kinetic energy $T_e=E_{\gamma }-E_b$, the photon energy less the electron binding energy. Momentum conservation requires the nucleus to absorb recoil, so a free electron cannot photoabsorb; the process needs a bound electron, and its cross section is largest for the most tightly bound (K-shell) electrons. The atomic cross section rises steeply with atomic number and falls steeply with energy,

away from the absorption edges. The strong $Z^n$ dependence is why high-$Z$ materials such as lead are efficient gamma shields and why photographic contrast in radiography traces atomic number. The plot of ${\tau }_{\mathrm{pe}}$ versus energy shows sharp absorption edges where the photon energy crosses a shell binding energy and a new group of electrons becomes available.

Compton scattering

At intermediate energies the photon scatters off an electron loosely bound compared to the photon energy, transferring part of its energy and continuing at a reduced frequency. Treating the electron as free and at rest, energy-momentum conservation gives the wavelength shift

where $\theta$ is the photon scattering angle and $h/m_{e}c$ is the Compton wavelength. In terms of the photon energy $E_{\gamma }=hc/\lambda$,

The maximum energy transfer, at $\theta =180^{\circ }$ backscatter, leaves the scattered photon with the least energy and gives the recoil electron its greatest energy, the Compton edge of the electron spectrum. The differential cross section per electron is the Klein-Nishina formula; integrated, the Compton cross section per atom scales as $Z$ (one contribution per electron) and falls gradually with energy, roughly as $1/E_{\gamma }$ in the relevant range. Because it grows only as $Z$, Compton scattering dominates the mid-energy region for all materials.

Pair production

Above the threshold $E_{\gamma }=2m_e{c}^2=1.022\mathrm{MeV}$ a photon can convert into an electron-positron pair in the Coulomb field of a nucleus, which absorbs the recoil momentum that makes the conversion possible. The excess energy becomes kinetic energy of the pair,

The cross section rises from threshold and, well above it, grows logarithmically with energy while scaling as $Z^2$ because the process couples to the nuclear Coulomb field. The positron subsequently annihilates with an electron, producing two $0.511\mathrm{MeV}$ photons that carry energy away and, in a detector, appear as characteristic escape peaks. Pair production dominates the high-energy region, and its $Z^2$ dependence again favors heavy absorbers.

The total mass attenuation coefficient is the sum of the three contributions, and plotting them together shows the photoelectric term dominating at the lowest energies, Compton taking over through a broad minimum, and pair production rising after $1\mathrm{MeV}$. The minimum in the total near a few MeV is why photons of that energy are the most penetrating.

Neutron interactions

A neutron has no charge and does not interact with atomic electrons. It reaches a nucleus without any Coulomb barrier and interacts through the strong force, so its mean free path can be centimeters even at low energy. The two dominant channels are elastic scattering, which slows the neutron, and absorption, which removes it.

Elastic moderation. A neutron scattering elastically from a nucleus of mass number $A$ loses, on a head-on collision, the fraction

of its energy. For hydrogen ($A=1$) this is unity: a single collision can stop the neutron. The efficiency of slowing is measured by the average logarithmic energy decrement per collision,

which is independent of energy, so the number of collisions to thermalize a fast neutron is $\ln (E_0/E_{\mathrm{th}})/\xi$. Light nuclei are the best moderators: about $18$ collisions in hydrogen versus more than $2000$ in ${}^{238}\mathrm{U}$. The optimal moderator combines a large $\xi$ with a small absorption cross section, which is why ordinary and heavy water, and graphite, are used.

Absorption and the $1/v$ law. Radiative capture $(n,\gamma )$ and other absorption reactions have a cross section that, away from resonances, rises as the neutron slows. The time a slow neutron spends within range of a nucleus scales as $1/v$, and the reaction probability follows,

This $1/v$ law is why thermal (slow) neutrons are captured far more readily than fast ones, and why a moderator that thermalizes fission neutrons multiplies their capture rate. Superimposed on the $1/v$ background are sharp resonances where the compound nucleus formed by neutron plus target has an excited level at the available energy; each resonance follows the Breit-Wigner shape, and the resonance region separates the thermal $1/v$ behavior at low energy from the smooth, small cross sections at high energy.