The fourth factor *p’ measures the efficiency of the design in ‘slowing down’ the fast neutrons to thermal energies. During the slowing down process many of the neutrons will diffuse back into the fuel and be captured in the resonance capture peaks of U-238, see Fig 1.6 (b). [In the numerical example of Fig 1,13, of the initial 1190 fast neutrons 180 neutrons are captured and 1010 thermalised.] The main interest, however, is in how many neutrons escape resonance capture; hence the resonance escape probability:

number of thermal neutrons emerging from
the slowing down process

15 ~ number of fast neutrons starting the slowing down process

Thus p^fni neutrons are successfully thermalised [0 85 x 1190 = 1010 thermal neutrons for p = 0.85].

The more fuel there is in the reactor relative to the moderator the more likely is resonance capture of the neutrons and the lower the value of p; ultimately p = 0 for 100% fuel. Conversely the more modera — [or there is the greater the value of p; ultimately p _ I for 100% moderator. Hence we may conclude that the value of p increases progressively from zero towards unity as more and more moderator is added to a given amount of fuel. This is in exact contrast to the thermal utilisation factor T, Section 6.2.1 of this chapter.

6.3 к * = peijf

In Fig 1.13 and the preceding sections the assump­tion was made of having an initial П| [1000] thermal neutrons in the moderator. These neutrons were all subsequently absorbed and replaced a neutron life cycle later by perjfni [1010] thermal neutrons in the moderator. Hence

peqfni
ni

к 00 = pei/f

This is known as the four factor formula In the numerical example here:

кос = 0.85 x 1.03 x 1.33 x 0.87 = 1.01

This represents a supercritical reactor in which the number of neutrons increase by a multiple of 1.01 per neutron life cycle. If the values of the four factors had been such that their product was unity then of course this would be an exactly critical reactor. A subcritical reactor implies a product less than unity.

The usefulness of the four factor formula is in giving an understanding of how the value of k® may change, or be changed, by for example altering the geometry or the operating characteristics of the re­actor. Thus the operator, by insertion or withdrawal of the control rods, changes T and hence k®. Again, the isotopic content of the fuel will change during irradiation as the U-235 is consumed and plutonium created by the neutron capture in U-238; V will change and hence k®.

Referring back to the reactor design considerations of Section 6.1 of this chapter we may now see that, for given moderator and fuel materials, k® is largely determined by the fuel enrichment and by the quan­tity of moderator relative to fuel. For a given amount of fuel T decreases and ‘p’ increases as the quantity of moderator is increased.

The task of the designer is to calculate the amount of moderator that will give maximum value for k® and this value can be increased further, if required, by enrichment. Assuming a simple rod design for the fuel the optimum lattice pitch and fuel rod diameter may then be easily determined.