The Boltzmann equation expresses the variation with time of the number of neutrons present in an elementary volume V of surface S. We can write this as:

We define each of the terms of the right-hand side of equations (3.16) and (3.17):

entering — exiting

div(J(r, v, t)) d3r

V

expresses the total current entering the volume if the normal is directed outwards;

S(r, v, t) is the external neutron source. In the second term of the right-hand side, the fission source r/f(v), is the velocity spectrum of the fission neutrons, ni is the number of neutrons per fission of species i, and the macroscopic cross-sections are assumed to be time independent; i indexes fissioning nuclei

inscattered =

where j indexes all species of scattering nuclei.

outscattered+absorbed

‘(r, v, t) E.(T](r, v) d3r

where XT = Xs + Xa.

In the above expressions we have, for the sake of simplicity, neglected the fi dependence of the cross-sections and integrated over fi. The Boltz­mann equation is obtained as

div(J(r, v, t)) + S(r, v, t)

‘(r, v, t)J2 чФ№)(Ц1)(г, v’) + sij)(r, v! v)) dv

hJ

— ‘(r, v, oX) v)

j

where we made use of ‘(r, v, t) = vn(r, v, t).