In a realistic reactor of finite size, some of the fast neutrons leak out of the boundaries of the reactor core before they begin the slowing down process. The fast non-leakage probability () is defined as the ratio of the number of fast neutrons that do not leak from the reactor core to the number of fast neutrons produced by all fissions. This ratio is stated as follows.

Thermal Non-Leakage Probability ()

Neutrons can also leak out of a finite reactor core after they reach thermal energies. The thermal non-leakage probability (Sft) is defined as the ratio of the number of thermal neutrons that do not leak from the reactor core to the number of neutrons that reach thermal energies. The thermal non-leakage probability is represented by the following.

number of thermal neutrons that do not leak from reactor

number of neutrons that reach thermal energies

The fast non-leakage probability () and the thermal non-leakage probability () may be combined into one term that gives the fraction of all neutrons that do not leak out of the reactor core. This term is called the total non-leakage probability and is given the symbol , where , are both effected by a change in coolant temperature in a heterogeneous water-cooled, water-moderated reactor. As coolant temperature rises, the coolant expands. The density of the moderator is lower; therefore, neutrons must travel farther while slowing down. This effect increases the probability of leakage and thus decreases the non-leakage probability. Consequently, the temperature coefficient (defined later) for the non-leakage probabilities is negative, because as temperature rises, decrease.

Six Factor Formula

With the inclusion of these last two factors it is possible to determine the fraction of neutrons that remain after every possible process in a nuclear reactor. The effective multiplication factor (k_eff ) can then be determined by the product of six terms.

Equation (3-3) is called the six factor formula. Using this six factor formula, it is possible to trace the entire neutron life cycle from production by fission to the initiation of subsequent fissions. Figure 1 illustrates a neutron life cycle with nominal values provided for each of the six factors. Refer to Figure 1 for the remainder of the discussion on the neutron life cycle and sample calculations. The generation begins with 1000 neutrons. This initial number is represented by . The first process is fast fission and the population has been increased by the neutrons from this fast fission process. From the definition of the fast fission factor it is possible to calculate its value based on the number of neutrons before and after fast fission occur.

The total number of fast neutrons produced by thermal and fast fission is represented by the quantity .

Next, it can be seen that 140 neutrons leak from the core before reaching the thermal energy range. The fast non-leakage probability is calculated from its definition, as shown below.

The number of neutrons that remain in the core during the slowing down process is represented by the quantity .

Figure 1 Neutron Life Cycle with

The next step in the analysis is to consider the number of neutrons that are absorbed in the intermediate energy level. The probability of escaping this resonance absorption (p) is stated as follows.

The number of neutrons entering the thermal energy range is now represented by the quantity .

After reaching thermal energies, 100 neutrons leak from the core. The value for can be calculated by substitution of the known values in the definition as shown below.

The number of thermal neutrons available for absorption anywhere in the core is represented by the quantity .

Figure 1 indicates that 125 neutrons were absorbed in non-fuel materials. Since a total of 620 thermal neutrons were absorbed, the number absorbed by the fuel equals 620 - 125 = 495. Therefore, the thermal utilization factor can be calculated as follows.

The final factor numerically describes the production of fission neutrons resulting from thermal neutrons being absorbed in the fuel. This factor is called the reproduction factor (Tj). The value for the reproduction factor can be determined as shown below.

The number of fission neutrons that exist at the end of the life cycle which are available to start a new generation and cycle is represented by the quantity .

In the example illustrated in Figure 1, k_eff is equal to one. available to start the next generation.

Therefore, 1000 neutrons are

Example:

10,000 neutrons exist at the beginning of a generation. The values for each factor of the six factor formula are listed below. Calculate the number of neutrons that exist at the points in the neutron life cycle listed below.

1) Number of neutrons that exist after fast fission.

2) Number of neutrons that start to slow down in the reactor.

3) Number of neutrons that reach thermal energies.

4) Number of thermal neutrons that are absorbed in the reactor.

5) Number of thermal neutrons absorbed in the fuel.

6) Number of neutrons produced from thermal fission.

Solution: