15 Neutron multiplication factor
Designing a nuclear reactor is about controlling the neutron chain reaction. The underlying neutron physics allows this control.
The neutron multiplication factor ([latex]k[/latex]) describes the chain reaction. It is defined as the ratio of neutrons in generation [latex](n + 1)[/latex] to the number of neutrons in generation [latex]n[/latex].
Then, if [latex]k \lt 1[/latex], the chain reaction decreases in time. If [latex]k \gt 1[/latex], the chain reaction is increasing. These are defined as subcritical and supercritical, respectively. However, if [latex]k = 1[/latex], then the chain reaction is self-sustaining. This state is called critical. In terms of reactor operation, [latex]k[/latex] is manipulated to obtain a state of criticality at a designated power level.
The four factor formula can be calculated to obtain the neutron multiplication factor for an ‘infinite reactor’. This means that the reactor is theoretically large such that none of the neutrons ‘leak’. [latex]k[/latex] then can be applied to determine the size of the critical reactor, based on fuel type, coolant, geometry, etc.
Four factor formula
[latex]k \equiv \eta f \epsilon p[/latex]
- [latex]\eta \equiv \;[/latex] neutron reproduction factor
- [latex]f \equiv \;[/latex] fuel utilization factor
- [latex]\epsilon \equiv \;[/latex] fast fission factor
- [latex]p \equiv \;[/latex] resonance escape probability
Remember, if the goal is for [latex]k = 1[/latex], then think of the range of values for each parameter when studying the factors below.
Neutron reproduction factor
[latex]\eta \equiv \frac{\nu\Sigma_F}{\Sigma_A}[/latex]
- [latex]\nu \equiv \;[/latex] average number of neutrons released per fission (dependent on fissonable material)
- [latex]\Sigma_F \equiv \;[/latex] macroscopic fission cross section
- [latex]\Sigma_A \equiv \;[/latex] macroscopic absorption cross section
The neutron reproduction factor gives a ratio of the ‘usable’ neutrons to neutrons absorbed per birth of neutrons.
Fuel utilization factor
[latex]f \equiv \frac{\Sigma^{fuel}_{A}}{\Sigma^{fuel}_{A} + \Sigma^{mod}_{A}}[/latex]
- [latex]\Sigma^{fuel}_{A} \equiv \;[/latex] macroscopic absorption cross section in the fuel
- [latex]\Sigma^{mod}_{A} \equiv \;[/latex] macroscopic absorption cross section in the moderator
The fuel utilization factor basically shows the ‘miles per gallon’ for the core. Neutrons absorbed in the moderator will not produce any more fissions.
Resonance escape probability
Resonance escape ([latex]p[/latex]) is the probability neutron is not absorbed in the resonance region. Most neutrons are absorbed by [latex]^{238}U[/latex] when slowing down in commercial reactors. Empirical results are typically used because it is extremely difficult to compute.
Fast fission factor
The fast fission factor is ratio of the total number of fast and thermal neutrons produced to the number produced by just thermal fission. Empirical relationships are again used because it is a really difficult parameter to calculate.
Lastly, neutron leakage has to be considered. These are the neutrons that completely leave the system (reactor vessel). There is leakage of fast neutrons and thermal neutrons. The parameter of interest then is the ‘non leakage’ probability; that is, the probability of the neutrons that do not leak. This value is needed to compute [latex]k[/latex] for the critical reactor. Leakage is largely a function of geometry, but the choice of reflector is also important.
The following image illustrates how [latex]k[/latex] can be calculated.
Additional notes