# Group Diffusion Equation

As mentioned earlier, the energy ranges are divided into a finite number of discrete energy groups, which results in a multigroup equation. The division of the neutron energy range into *G* groups is shown in Figure 7.1. The maximum and minimum energies in the range of interest are denoted by *E _{0}* and

*E*respectively. Here, the first group represents the group with the highest energy, that is

_{G},*g*= 1. The

*g*value increases with the decrease in neutron energy. Consider a neutron which is introduced into a group

*g'*by fission or scattering. Then, the neutron will pass during moderation into a group

*g*(i.e. at lower energy), where

*g*> g', whereas a certain amount of up-scattering may also occur (i.e.

*g*< g') in the thermal energy region.

In steady state, the neutron balance in any group *g* can be represented as follows:

The parameters *D _{g}* (diffusion coefficient) and

*~L*(for absorption and scattering) are introduced here to derive the expression Equation 7.1 for the rates of these processes. The actual rates of neutron interactions within the group may be obtained by combining these parameters with the group flux ф

_{g}_{г}

Take a small volume element which is located at a point x. Here, for simplicity, a single coordinate is used but the results are applicable to any coordinate system. According to the diffusion theory, the leakage from group *g* (term 1) in the neutron balance Equation 7.1 can be represented by

where *D _{g}(x)* and ф^х) are the group

*g*diffusion coefficient and the neutron flux, respectively, at the point x in the system.

Absorption in group *g* (term 2) and scattering out of *g* (term 3) together represent the total rate of neutron interaction, which are absorption and scattering of neutrons in group g. The sum of these two terms is represented as

where Sg = S g + Sg is the total interaction cross section for neutrons in group g. Scattering into group *g* (term 4) is defined as

FIGURE 7.1

Divisions of neutron into * G* groups.

where *Sg® _{g}* denotes the cross section for the scattering of neutrons from any group g' into g. This term also includes the scattering in which the neutron remains in the same group (when g' = g). The summation over values g' from 1 to

*G*allows for scattering from all groups into group

*g*.

Finally, there is a production in group *g* (term 5), which represents the rate at which neutrons are produced in the group. The term 5 can be written in the following way:

Term 5 is equal to the rate at which neutrons with energies in group *g* are generated as a result of fission by the neutrons of all energies. Now, considering all the terms together, the steady-state neutron balance equation for group *g* can be written as

Finally, the complete set of multigroup diffusion equations (7.6) consists of *G* equations with *g*=1, 2, ... , G. These equations are coupled due to the flux ф_{g} in group *g* is dependent on the values of in other groups.