4.1. The neutron transport equation

The behaviour of neutrons in any medium can be described by the transport
equation, first used by Boltzmann for the description of the behaviour of gas molecules.

Neutrons can indeed be thought as forming a kind of a gas whose treatment is simplified by the fact that neutrons do not interact with each other. The general time, position and angle-dependent form of the transport equation is

mean:

neutron velocity corresponding to energy E, neutron angular density,

total neutron cross-sections (generally function of r and E), neutron source, space coordinate,

unit vector in the direction of the neutron motion,

energy,

time,

scattering cross-section from E’, ft’ into E, ft.

This equation represents simply a neutron balance in a volume element dV for the neutrons having energy between E and E + dE and flight direction in the solid angle dfl around ft.

The first term on the right side is the leakage out of dV, the second term represents the loss due to absorption and scattering, the third gives the sources due to scattering from other directions and energies and the fourth is the source term (including fission and external sources). We do not repeat here the well-known difinitions of neutron angular density, cross-sections, etc. (see refs. 1 and 2).

Defining the neutron angular flux

Ф(г, E, ft, t) = vN(r, E, ft, t),

eqn. (4.1) becomes

— дф(г’ f; a—~ = — ftVt/r(r, £, ft, t) — Х, ф(г, E, ft, О v at

+ J Xs (E’ -* E, ft’ -* Г1)ф(г, E’, ft’, 1) dCT + S(r, E, ft, t).

(4.2)

The boundary conditions for this equation are continuity at the interface between two media;

at the outer free surface ф{г, Е,£1, t) = 0 for directions entering the system; ф(г, £, ft, t) = 0 for energies greater than the maximum source or fission neutron energy. For reactors this limit corresponds to 10-15 MeV.

The transport equation (4.2) without external sources takes the form:
j дф(г, Е,(1, t) = _nv^( E () — х, ф(г, E, ft, t)

V ot

+ JXs (E’ -*E, ft’-* П)ф(г, E’, ft’, t) dE’ d(l’

+ J Х/(Е’)ф(г, E’, ft’, t)v(E’) dE’ dft’, (4.3)

here the last term represents the fission source which is supposed to be isotropic;

x(E) is the fission spectrum, probability that a fission neutron is generated with energy E

£ x(E)dE=l;

v(E) is the average number of neutrons generated by a fission induced by a neutron having energy E.

If the leakage and absorption terms are equal to the production terms (fission and slowing down) the time derivative of the left-hand side of eqn. (4.3) vanishes.

In this case the reactor is said to be critical and the neutron population does not change with time.

For a given material composition all terms on the right-hand side of eqn. (4.3) are fixed except the first one. This means that criticality can only be reached for a particular value of the term

ftVt//(r, E, ft)

which represents the neutron leakage out of the systems and depends on the reactor geometry. In this way the critical geometry is defined. For a given reactor criticality can be obtained changing X, by means of insertion or extraction of neutron absorbers. Static calculations of non-critical systems are usually performed artificially dividing the term v(E) of eqn. (4.3) by a factor keff so that the equation is always satisfied without considering the time dependence of the neutron population.