---
title: Interactions of Photons and Neutrons
module: Radiation and Applications
moduleNumber: 11
lessonNumber: 2
order: 1102
summary: >
  Photons are removed from a beam in single events, so their intensity falls
  exponentially with a linear attenuation coefficient built from three processes:
  the photoelectric effect at low energy, Compton scattering at intermediate
  energy, and pair production above twice the electron rest energy, each with its
  own atomic-number and energy dependence. Neutrons carry no charge and interact
  only with nuclei, moderating by elastic scattering and being captured with a
  cross section that rises as one over speed away from resonances.
topics: [Radiation and Applications]
sources:
  - book: Krane
    ref: "Ch. 7 — Detecting Nuclear Radiations; §7.1 Interactions of Radiation with Matter (photons and neutrons)"
  - book: Wong
    ref: "Ch. 4 — Nuclear Collective Motion; §4-1 Interaction of Radiation with Matter"
draft: false
---

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.[^krane-photons]

## Exponential attenuation

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

$$
I(x) = I_0\, e^{-\mu x}, \qquad \mu = n\sigma = \frac{N_A \rho}{A}\,\sigma.
$$

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**

$$
x_{1/2} = \frac{\ln 2}{\mu}.
$$

$$
% caption: A collimated photon beam is attenuated exponentially; each half-value
% layer removes half of the photons that reach it, so intensity falls by a
% constant factor per equal thickness.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.8,0) node[right, black!70] {thickness $x$};
  \draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {$I(x)$};
  \draw[acc, very thick] (0,4.1)
    .. controls (0.9,2.9) and (1.6,2.35) .. (2.3,2.05)
    .. controls (3.3,1.65) and (3.9,1.35) .. (4.5,1.02)
    .. controls (5.6,0.72) and (6.4,0.6) .. (7.2,0.5);
  \node[anchor=east, black!70] at (0,4.1) {$I_0$};
  \draw[black, dashed] (0,2.05) -- (2.3,2.05) -- (2.3,0);
  \node[anchor=east, black, font=\scriptsize] at (0,2.05) {half $I_0$};
  \node[anchor=north, black!70, font=\scriptsize] at (2.3,0) {$x_{\frac{1}{2}}$};
  \draw[black, dashed] (0,1.02) -- (4.5,1.02) -- (4.5,0);
  \node[anchor=east, black, font=\scriptsize] at (0,1.02) {quarter $I_0$};
  \node[anchor=north, black!70, font=\scriptsize] at (4.5,0) {$2\,x_{\frac{1}{2}}$};
\end{tikzpicture}
$$

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,

$$
\tau_\mathrm{pe} \propto \frac{Z^n}{E_\gamma^{7/2}}, \qquad n \approx 4\text{–}5,
$$

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

$$
\lambda' - \lambda = \frac{h}{m_e c}\,(1 - \cos\theta),
\qquad \frac{h}{m_e c} = 2.43\ \mathrm{pm},
$$

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

$$
E'_\gamma = \frac{E_\gamma}{1 + (E_\gamma/m_e c^2)(1 - \cos\theta)}.
$$

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 $E_\gamma = 2 m_e c^2 = 1.022\ \mathrm{MeV}$ 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,

$$
E_\gamma = 2 m_e c^2 + T_{e^-} + T_{e^+}.
$$

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.

$$
% caption: Which process dominates depends on photon energy and absorber atomic
% number: the photoelectric effect at low energy and high Z, Compton scattering
% across the middle, and pair production at high energy and high Z; the boundary
% curves mark equal cross sections.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.2,0) node[right, black!70] {photon energy (log)};
  \draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {atomic number $Z$};
  % lower boundary: photoelectric vs Compton (rising then bending right)
  \draw[very thick]
    (0.6,0.2) .. controls (1.6,1.4) and (2.2,2.6) .. (2.9,4.3);
  % upper boundary: Compton vs pair (falling to the right at high energy)
  \draw[very thick]
    (5.2,0.2) .. controls (5.7,1.6) and (5.9,3.0) .. (6.1,4.3);
  % region labels
  \node[black!70, font=\scriptsize, align=center] at (1.3,3.2) {photo\\electric};
  \node[black!70, font=\scriptsize] at (4.0,2.2) {Compton};
  \node[black!70, font=\scriptsize, align=center] at (7.1,3.2) {pair\\production};
  \node[black, font=\scriptsize, anchor=south] at (2.7,4.35) {equal};
\end{tikzpicture}
$$

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.

$$
% caption: The total mass attenuation coefficient is the sum of a steeply falling
% photoelectric term, a slowly falling Compton term, and a rising pair-production
% term above about one MeV; their sum has a broad minimum near a few MeV.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.2,0) node[right, black!70] {photon energy (log)};
  \draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {mass attenuation (log)};
  % photoelectric: steep fall
  \draw[black, densely dotted, thick]
    (0.4,4.3) .. controls (1.2,2.6) and (1.9,1.2) .. (2.9,0.35);
  \node[black, font=\scriptsize, anchor=west] at (1.4,3.0) {photoelectric};
  % compton: slow fall
  \draw[black, densely dashed, thick]
    (0.6,3.1) .. controls (2.8,2.5) and (5.0,1.9) .. (7.8,1.3);
  \node[black, font=\scriptsize, anchor=south] at (5.8,1.75) {Compton};
  % pair: rise after threshold
  \draw[black, thick]
    (4.6,0.2) .. controls (5.6,0.9) and (6.6,1.9) .. (7.8,2.7);
  \node[black, font=\scriptsize, anchor=north] at (6.9,2.0) {pair};
  % total: sum with broad minimum
  \draw[acc, very thick]
    (0.4,4.45) .. controls (1.6,3.0) and (2.6,2.1) .. (3.8,1.85)
    .. controls (5.0,1.65) and (5.6,1.7) .. (6.4,2.05)
    .. controls (7.0,2.35) and (7.4,2.6) .. (7.9,2.9);
  \node[acc, font=\scriptsize, anchor=south west] at (3.6,1.9) {total};
\end{tikzpicture}
$$

| Process | Energy regime | $Z$ dependence | Energy dependence |
| --- | --- | --- | --- |
| Photoelectric | low | $Z^{4\text{–}5}$ | $\propto E^{-7/2}$ |
| Compton | intermediate | $Z$ (per electron) | slow, $\sim E^{-1}$ |
| Pair production | above $1.022\,\mathrm{MeV}$ | $Z^2$ | rises, $\sim\ln E$ |

## 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

$$
\frac{\Delta E_\mathrm{max}}{E} = \frac{4A}{(1+A)^2}
$$

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,

$$
\xi = \left\langle \ln\frac{E_\mathrm{before}}{E_\mathrm{after}} \right\rangle
\approx \frac{2}{A + 2/3},
$$

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,

$$
\sigma_\mathrm{abs}(v) \propto \frac{1}{v} \quad\Longleftrightarrow\quad
\sigma_\mathrm{abs}(E) \propto \frac{1}{\sqrt{E}}.
$$

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.

$$
% caption: The neutron absorption cross section follows a one-over-speed straight
% line on log axes at low energy, broken by sharp Breit-Wigner resonances where a
% compound-nucleus level matches the neutron energy, then flattens at high energy.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.2,0) node[right, black!70] {neutron energy (log)};
  \draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {cross section (log)};
  % 1/v line (falling with energy) then resonances then flat
  \draw[acc, very thick] (0.4,4.1) -- (4.2,1.35);
  \node[black, font=\scriptsize, anchor=south west] at (1.4,2.9) {$\frac{1}{v}$ region};
  % resonance spikes
  \draw[acc, very thick] (4.2,1.35)
    -- (4.6,1.2) -- (4.75,3.5) -- (4.9,1.1)
    -- (5.4,1.0) -- (5.55,3.0) -- (5.7,0.95)
    -- (6.2,0.9) -- (6.35,2.6) -- (6.5,0.85);
  \node[black, font=\scriptsize, anchor=south] at (5.5,3.1) {resonances};
  % high-energy smooth tail
  \draw[acc, very thick] (6.5,0.85) .. controls (7.2,0.7) and (7.6,0.65) .. (7.9,0.62);
  \node[black, font=\scriptsize, anchor=north] at (7.4,0.7) {fast region};
\end{tikzpicture}
$$

> **Result.** Uncharged radiations are attenuated exponentially, $I = I_0 e^{-\mu x}$,
> with $\mu$ a sum over independent processes. For photons those are the
> photoelectric effect (low energy, $Z^{4\text{–}5}$), Compton scattering
> (intermediate, $\propto Z$), and pair production (above $1.022\ \mathrm{MeV}$,
> $\propto Z^2$). For neutrons they are elastic moderation, most efficient on light
> nuclei, and $1/v$ absorption punctuated by resonances. These cross sections set
> both the shielding required against a source and the response of the detectors
> built to register the radiation, taken up
> [next](/nuclear-physics/radiation-matter-applications/radiation-detectors).

[^krane-photons]: **Krane**, _Introductory Nuclear Physics_, §7.1, and **Wong**,
_Introductory Nuclear Physics_, §4-1. Exponential attenuation and the mass
attenuation coefficient; the photoelectric $Z^n/E^{7/2}$ law and absorption edges;
the Compton wavelength shift and Klein-Nishina scaling; the pair-production
threshold and $Z^2$ dependence; and neutron moderation with the logarithmic
decrement $\xi$ and the $1/v$ absorption law. Photon attenuation coefficients are
tabulated in the NIST XCOM database, [https://physics.nist.gov/](https://physics.nist.gov/);
evaluated neutron cross sections are served by the IAEA Nuclear Data Services,
[https://www-nds.iaea.org/](https://www-nds.iaea.org/).