The neutron density n in a reactor is in general a function of position r, the energy E of the neutrons, the direction Й in which they are trav­elling, and time t. The neutron density n obeys a linear version of the Boltzmann equation called the neutron transport equation, the deriv­ation of which is given in many standard works on neutron transport (e. g. Davison and Sykes, 1957; Duderstadt and Hamilton, 1976). It can be written in a simplified form as

д П

= — v& • grad n — v^n + T + F + S. дt (1.1)

1 2 3 4 5

If each term in equation 1.1 is multiplied by a small element dV dE dfi the terms on the right-hand side can be thought of as the contributions to neutrons appearing in the volume dV and the energy interval dE and travelling in a small solid angle dfi surrounding the direction Й as follows.

1. This is the rate at which neutrons already in dEd ^ move into dV across its boundary, v being the neutron speed corresponding to E, so that v2 = 2E/m where m is the mass of a neutron. (Relativistic effects are not important: even a 4 MeV neutron is travelling at only a tenth of the speed of light.)

2. This is the rate at which the neutrons in dV dEd ^ interact with the nuclei in dV. It is assumed that any interaction removes the neutron from dEd ^. £r is the total macroscopic cross-section in dVdE and is independent of Й.

3. This is the rate at which neutrons already in dV are scattered (elast­ically or inelastically) from other energies and directions dE7 d&7 into dE d&.

4. This is the rate at which new neutrons from fission appear in dV dEd ^. Delayed neutrons are ignored in equation 1.1 for simplicity.

5. This is an additional source of neutrons in dV dEd ^ (from spon­taneous fission, for example, or from a spallation source driven by an accelerator).

T is given by

T = d&’ dE’v’£s (E’ ^ E, Й ^ to)n(i, E’, ft7, t). (1.2)

4n 0

£s is the macroscopic scattering cross-section in dV (including both elastic and inelastic components). It is a function of E’, E and ft — ft7. Up-scattering in energy is not important in a fast reactor so £s is zero for E’ < E.

F is given by

TO

F = dE’n(E’ )v’£ f (E’ )v(E’ )x(E )/4n. (1.3)

0

£ f is the macroscopic fission cross-section in dV and v is the average number of neutrons generated in each fission event. x is the fission spectrum that is assumed to be independent of the energy of the neut­ron causing the fission. It is also assumed that fission neutrons are generated isotropically.

S is usually assumed to be zero in a critical reactor because in an operating power reactor spontaneous fission is negligible as a source of neutrons.

If S = 0 all the terms in equation 1.1 are linear in vn and it is usual to work in terms of the neutron flux ф defined by

ф(г, E, Й, t) = v(E)n(r, E, Й, t). (1.4)

v is about 14000 m/s at 1 eV and 14000 km/s at 1 MeV.