During the scattering reaction, a fraction of the neutron’s kinetic energy is transferred to the nucleus
. Using the laws of conservation of momentum and energy
and the analogy of collisions of billiard balls for elastic scattering, it is possible to derive the following equation for the mass of target or moderator
nucleus (M), energy of incident neutron
) and the energy of scattered neutron (Es
where A is the atomic mass number.
In case of the hydrogen (A = 1) as the target nucleus, the incident neutron can be completely stopped. But this works when the direction of the neutron is completely reversed (i.e. scattered at 180°). In reality, the direction of scattering ranges from 0 to 180 ° and the energy transferred also ranges from 0% to maximum. Therefore, the average energy of scattered neutron is taken as the average of energies with scattering angle 0 and 180°.
Moreover, it is useful to work with logarithmic quantities and therefore one defines the logarithmic energy decrement per collision (ξ) as a key material constant describing energy transfers during a neutron slowing down. ξ is not dependent on energy, only on A and is defined as follows:
For heavy target nuclei, ξ may be approximated by following formula:
From these equations it is easy to determine the number of collisions required to slow down a neutron from, for example from 2 MeV to 1 eV.
Determine the number of collisions required for thermalization for the 2 MeV neutron in the carbon.
ξCARBON = 0.158
N(2MeV → 1eV) = ln 2⋅106/ξ =14.5/0.158 = 92
Table of average logarithmic energy decrement for some elements.
For a mixture of isotopes: