Deterministic methods

These methods are essentially more or less elaborate approximations of the Boltzmann equation. The most widely used approximation is the multigroup diffusion theory which we outline here, as an example. The different groups correspond to energy bands E_i<E<E_i+1. The set of multigroup equations reads

$$D_{i}\Delta\varphi_{i}(\overrightarrow{r} )-\Sigma_{t,i}\varp... ...erset{j}{\sum}\nu\Sigma_{f,j}\varphi _{j}(\overrightarrow{r} )$$ (4.65)

where i(j) denotes the i(j)th group. $\Sigma_{r,j\rightarrow i}$ is the cross-section for a jump from group j to group i. $\Sigma_{t,i} =\Sigma_{a,i}+\underset{j}{\sum}\Sigma_{r,j\rightarrow i}$ is the cross-section for removing neutrons from group i. $\Sigma_{f,j}$ is the fission cross-section in group j. $\chi_{i}$ is the fraction of the fission neutrons which have energies within group i. The cross-sections should be computed as averages over the group energy domain by

$$\Sigma_{i}=\frac{\overset{E_{i+1}}{\underset{E_{i}}{\int}}\Si... ...E}{\overset{E_{i+1}}{\underset{E_{i}}{\int}}\varphi _{i}(E)dE}$$ (4.66)

which means that equations 3.68 are, in fact, a set of complicated integro-differential equations. In particular, in the resonance regions the flux has a complicated structure due to its depletion at energy in the vicinity of resonance energy. Thus, approximations are made on the calculation of the group cross-sections 3.69. In particular, in heteregeneous reactors one, first, computes the cross-sections, with a large number of groups, for the cells, with simplifying assumptions on the shape of the flux, and, possibly, correction factors. In a second step one computes the flux on the cell network. In practice, experiments are needed to validate the group cross-sections for each type of reactors.