KAHIBARO
Table of Contents
Basic idea
Elastic scattering of neutrons is one of the most important ways neutrons interact with matter. In this process, a neutron collides with a nucleus and bounces away, while the nucleus recoils. The key feature is that the total kinetic energy of the neutron plus nucleus is conserved. Energy is not used to excite the nucleus or change its internal state.
Because neutrons have no electric charge, they do not lose energy mainly through electromagnetic forces like charged particles do. Instead, they interact by getting very close to nuclei. Elastic scattering is therefore a major mechanism by which neutrons change direction and gradually lose energy in materials.
In elastic scattering, the nucleus remains in its ground state and the total kinetic energy before and after the collision is conserved.
What happens in a collision
Imagine a neutron moving toward a nucleus at rest in the material. After the collision, both particles move away. The neutron usually changes direction, and the nucleus gains some kinetic energy, called recoil energy.
If the nucleus is light, the neutron can lose a large fraction of its energy in a single collision. If the nucleus is heavy, the neutron usually loses only a small fraction.
This is why light materials, such as hydrogen-rich substances, are very effective for slowing down neutrons.
Conservation laws
Elastic scattering follows the conservation of momentum and kinetic energy. If a neutron of mass $m_n$ collides with a nucleus of mass $M$, then
$$
\vec{p}_{\text{before}} = \vec{p}_{\text{after}}
$$
and
$$
K_{\text{before}} = K_{\text{after}}
$$
where $K$ is total kinetic energy.
These two conservation laws determine how much energy the neutron keeps and how much is transferred to the nucleus.
For elastic scattering, both momentum and total kinetic energy are conserved.
Energy transfer
The amount of energy lost by the neutron depends strongly on the target nucleus mass. It is convenient to describe the nucleus mass by the mass number $A$, so that approximately
$$
M \approx A m_n
$$
for many purposes.
In a head on collision, the neutron can lose its maximum possible fraction of energy. The neutron's final energy can be as low as
$$
E'_{\min} = \alpha E
$$
where $E$ is the initial neutron energy, $E'$ is the final neutron energy, and
$$
\alpha = \left( \frac{A-1}{A+1} \right)^2
$$
The larger the value of $\alpha$, the less energy the neutron can lose in one collision.
For hydrogen, $A=1$, so
$$
\alpha = 0
$$
This means that in a head on collision with hydrogen, a neutron can in principle lose all of its kinetic energy to the proton.
For heavy nuclei, $A$ is large, and $\alpha$ is close to 1, so only a small fraction of energy is lost per collision.
The minimum possible neutron energy after one elastic collision is
$$
E'_{\min} = \alpha E, \qquad \alpha = \left( \frac{A-1}{A+1} \right)^2
$$
Light nuclei are much better at slowing neutrons than heavy nuclei.
Why light nuclei are effective moderators
A moderator is a material used to slow down neutrons. Elastic scattering is the main slowing process in many moderators. Since energy transfer is greatest when the target nucleus mass is similar to the neutron mass, materials containing light nuclei are especially effective.
Hydrogen is the best case for energy transfer. Deuterium and carbon are also useful because they can slow neutrons with relatively small absorption compared with some other materials.
| Material nucleus | Approximate $A$ | $\alpha = \left(\frac{A-1}{A+1}\right)^2$ | Slowing ability per collision |
|---|---|---|---|
| Hydrogen | 1 | 0 | Very high |
| Deuterium | 2 | $\frac{1}{9}$ | High |
| Carbon | 12 | $\left(\frac{11}{13}\right)^2$ | Moderate |
| Lead | 207 | close to 1 | Very low |
This is why water, heavy water, and graphite are widely used in neutron moderation.
Scattering angle
After collision, the neutron usually emerges at some angle. The direction changes from one collision to the next, so repeated elastic scatterings make neutron motion look like a random walk through matter.
The scattering angle affects the energy transfer. Large deflections often correspond to larger energy transfer, while small deflections usually mean the neutron keeps more of its energy.
In many beginner treatments, the exact angular formulas are not required. The main physical idea is that the neutron's path becomes more irregular and its energy gradually decreases through repeated collisions.
Microscopic description
The likelihood of elastic scattering is described by the elastic scattering cross section. A larger cross section means elastic collisions are more probable.
If a beam of neutrons passes through a material, some neutrons scatter elastically, some may undergo other interactions, and some may pass through without interacting. The cross section is therefore a measure of interaction probability, not a literal geometric size of the nucleus.
For a given material, the importance of elastic scattering depends on neutron energy. For many nuclei, the elastic scattering cross section changes with energy, sometimes strongly.
A cross section represents probability of interaction. A larger elastic scattering cross section means elastic scattering is more likely.
Role in neutron slowing down
Elastic scattering is the foundation of neutron moderation. Fast neutrons produced in nuclear reactions can be slowed by many successive elastic collisions until they become thermal neutrons.
A simple picture is this. A fast neutron enters a moderator, collides with nuclei, changes direction, gives up some energy in each collision, and eventually reaches much lower energies.
Special case of hydrogen
Hydrogen is especially important because its nucleus is a single proton, with a mass very close to the neutron mass. In an elastic collision between two nearly equal masses, energy transfer can be very large.
This makes hydrogen-rich materials, such as ordinary water or polyethylene, very useful for reducing neutron energy. However, in practical systems one must also consider neutron absorption, which is discussed separately under other topics.
Elastic scattering compared with other neutron interactions
Elastic scattering changes the neutron's energy and direction, but it does not alter the internal state of the nucleus. This distinguishes it from inelastic scattering, where some kinetic energy goes into nuclear excitation, and from neutron capture, where the neutron is absorbed.
So, the main effect of elastic scattering is moderation and redirection, not removal of the neutron from the beam.
Elastic scattering does not absorb the neutron and does not excite the nucleus. It mainly changes neutron direction and transfers kinetic energy to the recoiling nucleus.
Summary
Elastic scattering is a collision between a neutron and a nucleus in which total kinetic energy is conserved. The neutron changes direction and usually loses some energy, while the nucleus recoils. Light nuclei are much more effective than heavy nuclei at taking energy from neutrons, which is why elastic scattering in hydrogen, deuterium, and carbon is central to neutron moderation. The probability of this interaction is described by the elastic scattering cross section, and repeated elastic collisions are the main way fast neutrons are slowed in many materials.