• It has been seen that slow or thermal neutrons are very efficient at inducing nuclear reactions - hence fast neutrons sometimes need to be moderated.
• Slow neutrons are thus a product of many elastic collisions with nuclei.  In all cases the conservation of momentum applies:

S (mass × velocity)_before = S (mass × velocity)_after

• Therefore if you take the mass of a neutron to be 1, and the mass of the target nucleus = A:

• The average change in the 'ln' of energy is called x and is proportional to the slowing down power and moderator ratio.
• Each neutron has a unique history and overall we can only deduce the average behaviour, hence only the average cosines (cos q) of the angles scattered. The average scattering parameter arises as the average loss of energy is proportional to the pre-collision energy.

x (Greek 'Xi') - As the neutrons go slower, the energy loss per collision becomes smaller.

where (A = atomic mass)

This shows why light elements are used as moderators (e.g. H₂0).  For hydrogen, A=1 and a=0, so the neutron loses all its energy.

Slowing down power (SDP) - A large x is of little importance unless there is a high probability of scattering reactions taking place.

SDP = x × S_S

where S_S = macroscopic scattering cross section of the moderator for epithermal neutrons.

S_S = N₀ × s_S 

where N₀ = atoms per unit volume and s_S = microscopic cross section.

Moderator ratio (MR) - Compromise between too much moderation and good slowing down power.