Thermalization of Neutrons Role of Moderators Kinematic Relations
As explained in Section 7.1.1, fission in natural uranium will not result in a chain reaction if the neutrons interact primarily at energies close to those at which they are emitted, in the vicinity of 1 or 2 MeV. The neutrons are reduced in energy from this region to the more favorable thermal energy region by elastic collisions with the nuclei of the 'moderator.
The energy transfer in an elastic nuclear collision depends on the angle at which the incident neutron is scattered. The energy of the scattered neutron Es can be calculated from the kinematics of "billiard ball" collisions (i.e., by the application of conservation of energy and momentum). If a neutron of initial energy Ei is scattered at 0° by a nucleus of the moderator material— namely it continues in the forward direction—then the energy of the scattered neutron will be Es = Ei. In brief, nothing is changed. If the neutron is scattered at 180°, then the energy of the scattered neutron can be shown to be where M is the atomic mass of the moderator, mn is the mass of the neutron, and the approximation has been made that M = A and mn = 1 (both in atomic mass units, u). Equation (7.6) describes the condition for minimum neutron energy or maximum energy loss. If we define S to be the ratio of average scattered neutron energy to Ei and assume this average to be the
mean of the 0° and 180° cases, it follows that
For example, S = 0.50 for hydrogen (A =1), S = 0.86 for carbon (A = 12), and S = 0.99 for uranium (A = 238).
Obviously, Eq. (7.7) could not give a correct average if the scattering events were either predominantly forward or predominantly backward. Less obviously, it gives an exactly correct result if the scattering is isotropic, namely an equal number of neutrons scattered in all directions.7 This isotropy condition is well satisfied for neutrons incident on light elements such as hydrogen and carbon, at the energies relevant to fission; therefore, Eq. (7.7) is a good approximation.
An effective moderator (i.e., one that reduces the neutron energy with relatively few collisions) requires a low value of the parameter S. As seen from Eq. (7.7) and illustrated by the above examples, this means that the moderator must have a relatively low atomic mass number A. This rules out using uranium as its own moderator. Instead, hydrogen in water (H2O), deuterium in heavy water (D2O), or carbon in graphite (C) are commonly used as moderators.
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