So far the diffusion equation has been discussed in a non-multiplying medium with an external neutron source. Here, fission neutron production (fission source) in a reactor core is applied to the diffusion equation. Assuming a volumetric fission reaction rate If ф and an average number of neutrons released per fission v in a unit cubic volume (1 cm3) leads to a volumetric fission neutron production rate by multiplying both ones. Hence, substituting the fission source vIf ф for the external source (S) in Eq. (2.41) gives the one-group diffusion equation at the steady-state reactor core as the following.

DV20 — Іаф + vZf(f) = 0 (2.66)

This equation represents a complete balance between the number of neutrons produced by fission and the number of neutrons lost due to leakage and absorption, in an arbitrary unit volume at the critical condition of the reactor. In actual practice, however, it is hard to completely meet such a balance in the design calculation step. A minor transformation of Eq. (2.66) can be made as

(2.67)

Keff

where keff is an inherent constant characterizing the system, called the eigen­value. If keff = 1.0, Eq. (2.67) becomes identical to Eq. (2.66). The equation may be solved for the eigenvalue kf

(2.68)

where L, A, and P denote neutron leakage, absorption, and production, respec­tively. If the space is extended to the whole reactor core (i. e., each term is integrated over the whole reactor core), keff can be interpreted as follows. The numerator is the number of neutrons that will be born in the reactor core in the next generation, whereas the denominator represents those that are lost from the current generation. Hence, the condition of the reactor core depends on the keff value as given next.

(keff > 1 : Supercritical

(2.69)

[keff < 1 : Subcritical

Another interpretation can be given for kef in Eq. (2.67). As mentioned above, it is highly unlikely to hit on the exact neutron balance of L + A = P in a practical core design calculation. Neither the supercritical (L + A < P) nor subcritical (L + A > P) condition gives a steady-state solution to Eq. (2.66). Hence, if keff is introduced into the equation as an adjustment parameter like Eq. (2.67), the neutron balance can be forced to maintain (P! P/kejf) and the equation will always have a steady-state solution. In a supercritical condition (keff > 1), for example, the neutron balance can be kept by adjusting down the neutron production term (vZfp! vZfp/keff).

A problem expressed as an equation in the form of Eq. (2.67) is referred to as an eigenvalue problem. By contrast, a problem with an external neutron source such as Eq. (2.41) is referred to as a fixed-source problem. The big difference between both equations is that the eigenvalue problem equation has an infinite set of solutions. For example, if p is a solution of Eq. (2.67), it is self-evident
that 2ф and 3ф are also other solutions. Consequently, the solutions of the eigenvalue problem are not absolute values of ф but the eigenvalue kef and a relative spatial distribution of ф. Meanwhile, the fixed-source problem has a spatial distribution of absolute ф which is proportional to the intensity of the external neutron source.