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When neutrons interact with matter scattering, what direction to they scatter? Is it random, or some probability distribution of directions?

I'm trying to understand why neutron reflectors reflect neutrons, instead of allowing them to go through. Is it because of the angles that neutrons scatter, or because neutrons travel shorter paths before scattering again and so between interface of reflective material and other "stuff", neutrons are more probable to end up scattering back rather than going through?

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  • $\begingroup$ As it turns out, neutron reflectors aren't like optical mirrors, reflecting almost all of the neutrons. Yes, it is a stochastic response from many single neutron scattering events. Each neutron-atom interaction has an angle-dependent scattering cross section. $\endgroup$ – Jon Custer Jun 12 at 13:23
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Your question (and my answer) relates in large parts to How does neutron reflection work?, which I am quoting without specific notice.

Neutrons interact with a material mainly in three possible ways depending on energy:

  1. High-energy neutrons undergo inelastic scattering with nuclei. When they scatter of the target, their incident angle is almost unchanged, that is why high-energy neutrons will not reflect at all. However, there is a chance for spallation or the production of evaporation neutrons, where the nucleus gets excited for a short time and "evaporates" fast (Mev) neutrons. This process is isotropic, so you'll find good reflection of incident neutrons for example from metals or dry soil surfaces.

  2. Fast (MeV range) neutrons undergo elastic scattering. Following the Born approximation, wavefunctions of the scattered particles are s-waves with a spherical symmetry. I.e., the scattered neutron is equally likely to be found in all directions. Elastic scattering also follows the principle of elastic collision, where the energy transfer between neutron and particle is lower the higher the target's mass. That's why heavy atoms are used as neutron reflectors: neutrons only change the direction without losing much energy. In a dense material with heavy atoms, it only takes a couple of scatters before any information about the neutron's incident direction is completely lost, which effectively leads to an isotropic angular distribution. Hence the best possible reflector can only reflect 50 % of the neutrons.

  3. Thermal neutrons get captured/absorbed. A material with a high absorption cross section hinders neutrons from escaping, hence it would be a horrible reflector.

The whole elastic scattering process (2.) works like light scattering off ice crystals in snow. Crystals appear transparent for photons, but they make up a great (statistically isotropic) refractor / reflector. Hence, snow appears bright on top and darker the deeper you dig. In the neutron particle world ice corresponds to a dense material with very low absorption cross section and heavy atoms.

In summary, the best neutron reflectors (for the specific energy-range of interest) should have:

  • low scattering cross section (==low energy loss per collision, roughly corresponds to high mass)
  • low absorption cross section (you don't want to lose neutrons)
  • high material density (leads to short mean free paths)

This can be understood by looking at the mean free path $\lambda$ of a neutron between two collisions. It can be described by $\lambda=1/n\sigma$, where $n$ is the number density and $\sigma$ is the scattering cross section of the material. At a depth $\lambda$ from the the surface, the likelyhood is 50% that the incident neutron has eventually turned around and reflected back out.

A few example materials for neutrons reflectors are:

  • graphite and beryllium, an order of magnitude heavier than a neutron, and very low absorption cs
  • dense heavy metals such as lead (low cost) or tungsten (extremly dense but expensive)

Side note: There are fancy-pancy techniques to control neutron scattering at very small angles, see grazing-incidence small-angle neutron scattering, GISANS.

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