In the thermal reactor neutrons should be slowed down before they are captured. The survival probability was expressed in equation (3.61). The survival integral reads

(3.85)

with

This integral is only valid for small absorption. For E lying between fission neutron energy and thermal energy a parametric expression of the survival probability p can be used [55]

■]1Q ТПЛ

with the values of a and c for U and Th (which are the main resonance absorbers) shown in table 3.5. cabs is the atomic concentration of the absorber nuclei.

For heavy water £ = 0.509 while Ss = 0.452 cm-1. Taking the case of 232Th and requiring that p should be equal to 0.95, we get the concentration of 232Th atoms.

and, at equilibrium

and the value of kn

The diffusion length has to take into account the slowing down stage. This amounts to adding half the age of the neutron given by equation (3.56) to the diffusion length Lc

a2 (r : E) = —2 ln 0 = 2r.

V 2 3£S2 E

For thermal final energies the slowing down term reads, in the case studied,

with L = JL2 + a2(r : Eth) = 19 cm. The critical mass of fuel is around 1.2 tons. The lower critical mass for thermal systems is, obviously, a consequence of the higher capture and fission cross-sections.

The above calculations of critical masses neglected absorption in the structural elements, and, most important, by fission products.