 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:
where E_{0} 
= original neutron energy, and 
E 
= neutron energy after collision 
 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 precollision 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. H_{2}0).
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} = N_{0} × s_{S}
where N_{0} = atoms per unit volume and s_{S}
= microscopic cross section.
Moderator ratio (MR)  Compromise between too
much moderation and good slowing down power.
