Neutron interactions

Lina Morales from radiation protection joins the sixth session with a question from the field. A survey of the Clinch River construction laydown area found nothing unusual, but she wants to know how neutrons from the future reactor will behave in the concrete, steel and water around the core, and why neutron shielding looks so different from gamma shielding. Paul answers that neutrons are unusual particles: they carry no charge, so they ignore electrons entirely and interact only with nuclei, and the type of nucleus matters far more than its density.

Why neutrons are different

Charged particles and photons lose energy mainly by interacting with atomic electrons, so the material's electron density sets how fast they stop. Neutrons have no charge and only a tiny magnetic moment, so they pass through electron clouds untouched. They interact only when they come within a few femtometers of a nucleus, through the strong nuclear force. Because nuclei occupy about one part in \(10^{12}\) of the volume of matter, neutrons travel centimeters between collisions even in dense solids.

Two consequences follow. Neutrons are highly penetrating, and their interactions depend on individual nuclides rather than bulk properties such as density or atomic number. Lead, an excellent gamma shield, is nearly useless for slowing neutrons, while water and polyethylene, which are poor gamma shields per centimeter, are excellent at slowing them. Neutron shields therefore combine light materials to slow neutrons, absorbers such as boron to capture them, and heavy materials to stop the gamma rays produced along the way.

The five main interactions

Neutron interactions fall into two families: scattering, in which a neutron comes out again, and absorption, in which it does not.

Elastic scattering

In elastic scattering, the neutron and nucleus behave like billiard balls: total kinetic energy and momentum are both conserved. In a head-on collision with a nucleus of mass number \(A\), the neutron loses the most energy, leaving with \(\alpha E\), where

For hydrogen, \(A = 1\), so \(\alpha = 0\): a neutron can lose all its energy in one collision, just as a cue ball stops dead when it hits another ball squarely. For carbon, \(\alpha = 0.716\), so the neutron keeps at least 72 percent of its energy; for uranium, \(\alpha = 0.983\), so it barely slows. For scattering that is isotropic in the center-of-mass frame, which is a good approximation for neutrons below about 1 MeV on light nuclei, the energy after a collision is uniformly distributed between \(\alpha E\) and \(E\), and the average energy after collision is \((1+\alpha)E/2\).

# Neutron interactions: elastic-scattering kinematics and reaction branching.
import math

def alpha(A):
    """Smallest fraction of its energy a neutron keeps after one elastic collision."""
    return ((A - 1) / (A + 1)) ** 2

print(f"{'target':10s} {'A':>4s} {'alpha':>7s} {'max loss':>9s} {'avg E after (from 2 MeV)':>26s}")
for name, A in (('hydrogen', 1), ('deuterium', 2), ('carbon', 12), ('oxygen', 16), ('iron', 56), ('uranium', 238)):
    a = alpha(A)
    avg = 2.0 * (1 + a) / 2          # isotropic in the center-of-mass frame: mean of E and alpha*E
    print(f"{name:10s} {A:4d} {a:7.4f} {1 - a:9.1%} {avg:21.3f} MeV")

# Which reaction happens? Branching fractions at thermal energy (approximate cross sections, barns)
for name, sig_s, sig_g in (('H-1', 20.5, 0.332), ('C-12', 4.75, 0.0035), ('Fe-56', 12.4, 2.59), ('U-238', 9.3, 2.68)):
    tot = sig_s + sig_g
    print(f"{name:6s}: scatter {sig_s / tot:6.1%}, capture {sig_g / tot:6.1%} of thermal collisions")

# Neutron speed from kinetic energy (non-relativistic)
mn = 1.674927e-27; eV = 1.602177e-19
for E in (0.0253, 1.0, 1e3, 2e6):
    v = math.sqrt(2 * E * eV / mn)
    print(f"E = {E:>10.4g} eV  speed {v:12,.0f} m/s")

target        A   alpha  max loss   avg E after (from 2 MeV)
hydrogen      1  0.0000    100.0%                 1.000 MeV
deuterium     2  0.1111     88.9%                 1.111 MeV
carbon       12  0.7160     28.4%                 1.716 MeV
oxygen       16  0.7785     22.1%                 1.779 MeV
iron         56  0.9311      6.9%                 1.931 MeV
uranium     238  0.9833      1.7%                 1.983 MeV
H-1   : scatter  98.4%, capture   1.6% of thermal collisions
C-12  : scatter  99.9%, capture   0.1% of thermal collisions
Fe-56 : scatter  82.7%, capture  17.3% of thermal collisions
U-238 : scatter  77.6%, capture  22.4% of thermal collisions
E =     0.0253 eV  speed        2,200 m/s
E =          1 eV  speed       13,832 m/s
E =       1000 eV  speed      437,393 m/s
E =      2e+06 eV  speed   19,560,831 m/s

The first table in the output is the core of moderator design. Light nuclei take large bites out of the neutron's energy, heavy nuclei take tiny ones. Lesson 11 turns this into the number of collisions needed to slow a fission neutron from 2 MeV to thermal energy, and shows why water, heavy water and graphite are the moderators of choice.

The second table shows something less obvious. Even hydrogen, whose thermal capture cross section is only 0.33 barns, captures a neutron in about 1.6 percent of thermal collisions. That is enough to make light water reactors need enriched fuel, while heavy water, whose deuterium captures far less, lets CANDU reactors, such as the existing units at Darlington and Pickering in Ontario, run on natural uranium. In iron and uranium-238, capture competes strongly with scattering, so thermal neutrons that wander into steel structures or fuel are likely to be absorbed.

Inelastic scattering

If the neutron has enough energy, it can leave the target nucleus in an excited state. The neutron emerges with less kinetic energy than elastic scattering would leave, and the nucleus then emits one or more gamma rays. The threshold is slightly above the energy of the first excited level: about 45 keV for uranium-238, 0.85 MeV for iron-56, 4.4 MeV for carbon-12 and over 6 MeV for oxygen-16.

Inelastic scattering matters for fast neutrons in heavy materials. In a reactor core, fission neutrons with energies of a few MeV lose energy rapidly through inelastic collisions with uranium-238 before elastic scattering in the moderator takes over. In shields, steel and iron take fast neutrons below the inelastic thresholds quickly, which is why many shield designs place steel near the source and hydrogenous material farther out. The gamma rays produced by inelastic scattering, together with capture gammas, are a major source of secondary radiation that shields must stop.

Radiative capture

In radiative capture, the neutron is absorbed and the compound nucleus releases its excitation, about 5 to 8 MeV for most nuclei, as one or more gamma rays. Capture cross sections are largest for slow neutrons, following the one-over-v law at low energies and showing resonances higher up (Lesson 5).

Capture appears throughout reactor engineering. It breeds new fuel: uranium-238 captures a neutron to become uranium-239, which beta decays through neptunium-239 to plutonium-239, so a third or more of the energy late in a fuel cycle comes from fissioning plutonium made in the reactor. It poisons the reactor: xenon-135 and samarium-149 capture neutrons that would otherwise cause fission. It activates materials (Lesson 4). And it produces high-energy capture gamma rays that dominate the dose outside thick concrete shields; hydrogen's 2.22 MeV capture gamma ray and iron's 7.6 MeV gamma rays are the ones shield designers watch.

Charged-particle reactions

Some light nuclei absorb neutrons and emit a proton or alpha particle instead of a gamma ray. Lesson 4 covered boron-10, lithium-6 and helium-3 for slow neutrons, and the fast-neutron reactions that produce nitrogen-16 from oxygen-16. Boron-10's huge thermal cross section, 3840 barns, makes it the standard neutron absorber: boric acid dissolved in the coolant controls reactivity in pressurized water reactors, boron carbide fills control blades in boiling water reactors such as the BWRX-300 planned at Clinch River, and borated concrete and polyethylene line neutron shields. Because the products are charged particles that stop within micrometers, boron absorbers produce very little penetrating secondary radiation, a major advantage over cadmium, whose capture releases high-energy gamma rays.

Fission as an interaction

Fission is an absorption reaction in which the compound nucleus splits. For neutron cross-section bookkeeping, it is one more partial cross section: \(\sigma_a = \sigma_\gamma + \sigma_f\) for a fissile nuclide. The ratio \(\sigma_\gamma/\sigma_f\), written \(\alpha\) in reactor physics (a different alpha from the scattering parameter), tells how many absorptions are wasted on capture. For uranium-235 at thermal energies it is about 0.17, so about 85 percent of absorptions cause fission. Lesson 7 takes up the physics of the split itself.

Neutron energy ranges

Engineers sort neutrons by kinetic energy, because each range behaves differently. The boundaries are conventions rather than sharp physics, and different texts draw them differently, but the following are typical.

The program's speed table shows the scale. A thermal neutron at 0.0253 eV moves at 2200 m/s, about the speed of a rifle bullet. A 2 MeV fission neutron moves at nearly 20 000 km/s, about 7 percent of the speed of light. Slowing a neutron from fission energy to thermal energy reduces its speed by a factor of nearly 10 000 and its energy by a factor of about 80 million.

Thermal neutrons and the moderator

Once neutrons have slowed to energies comparable with the thermal motion of the moderator atoms, they can gain as well as lose energy in collisions. They come into rough equilibrium with the moderator, and their energies follow approximately a Maxwell-Boltzmann distribution characteristic of the moderator temperature. The most probable energy is \(kT\), which is 0.0253 eV at 20 °C. In a hot reactor moderator at about 300 °C, the distribution is shifted to higher energies, about 0.05 eV, and absorption by neutrons that do not follow 1/v exactly must be averaged over the distribution. Reactor physics codes handle this with temperature-dependent scattering laws for bound hydrogen in water and for carbon in graphite.

Neutron sources

Away from reactors, neutrons come from a handful of sources used in industry and research. Radioisotope sources mix an alpha emitter with beryllium, whose \((\alpha,n)\) reaction releases neutrons; americium-beryllium sources are used in moisture gauges and well logging. Californium-252 fissions spontaneously, producing about \(2.3 \times 10^{12}\) neutrons per second per gram, and serves as a compact source for startup and calibration. Accelerator generators fuse deuterium with tritium to give 14.1 MeV neutrons. Spallation sources fire high-energy protons at heavy targets to produce intense neutron beams for science. For reactor startup, the the Tennessee Valley Authority's Clinch River project design includes a neutron source and the boron-lined detectors discussed in Lesson 4, so that operators can see the neutron population rising long before the reactor produces measurable power.