One reason for expressing core nuclear status in terms of reactivity is because rho shares a common denominator with beta, which allows the difference (beta - rho) to appear in both the General Equation for Reactor Power and the Reactor Rate Equation. This common denominator is “fission neutrons produced.” The reason that (beta - (delta-k)) does not appear is because it would be like adding apples and oranges; beta and delta-k do not possess a common denominator.
In order to identify the underlying difference between delta-k and rho it is necessary to utilize a term that represents the fraction of fast neutrons that reach thermal energy and cause fission. This important term, k/nu, is not widely recognized. It can be obtained from the definition of k-effective that ratios fast neutrons in generation-2 (N2) to fast neutrons in generation-1 (N1), as follows:
|
25.1 |
where:
N2 = nu x n1
nu = 2.5 neutrons/fission
n1 = thermal neutrons causing fission in generation-1
The product (nu x n1) is the fast neutrons, N2 , appearing in generation-2. Then dividing both sides of Equation 25.1 by nu gives:
 |
25.2 |
Thus, k/nu is the fraction of fast neutrons in a generation that produce thermal fission. At a subcritical k = 0.8, this fraction is 0.32. At criticality, with k = 1, this fraction is 0.40. As criticality is approached, the fraction of fast neutrons that reach thermal energy and cause fission increases.
Early in the study of reactor behavior, subcritical source multiplication is developed as:
 |
25.3 |
where:
N = the total number of fast neutrons in a generation (non-fission source and fission neutrons)
S = the non-fission neutron source strength in neutrons/generation
k = k-effective (subcritical)
delta-k = keff - 1
N , on the left-hand-side of Equation 25.3, represents fast neutrons because the non-fission source on the right-hand-side emits only fast neutrons and the multiplying fission neutrons are fast at birth. If both sides of the equation are multiplied by k/nu to obtain thermal neutrons causing fission (n) , we have:
 |
25.4 |
In this process, the k/nu term on the right-hand-side of Equation 25.4 is actually applied to the non-fission source strength, S, so as to obtain the number of source neutrons that cause thermal fission. However, in practice this equation is expressed as:
 |
25.5 |
Instead of being reduced by a factor of k/nu the non-fission source strength, S, is reduced by the lesser factor of 1/nu. Since k is always less than 1, the number of source neutrons that reach thermal energy and cause fission is overstated as S/nu. The errant k is carried into the denominator as a divisor to delta-k, which is then called reactivity, rho . Again since k is always less than 1, reactivity is of larger magnitude than delta-k. The consequence of this mathematical manipulation is a wash ... the value of “n” is unchanged and remains correct. The reason is that the overstatement of the source neutrons causing fission is compensated by an equal overstatement of the nuclear status, rho. However, as criticality is approached the overstatement of the source strength and reactivity diminish, until finally being eliminated at criticality. The net effect is that, with constant delta(delta-k) increments, the delta-rho increments become smaller as criticality is approached. Thus when subcritical delta-k is halved, repeating that same delta(delta-k) addition will achieve criticality, but will exceed criticality if the second reactivity increment equals the first. In NUKEFACT #24 the reactivity increase associated with the second delta(delta-k) addition of +0.0500 is delta-rho = +0.0526. This is the reactivity addition required to attain criticality.
Thus, Equation 25.3 represents multiplication of fast source neutrons and employs delta-k as a measure of the nuclear status. Equation 25.5 represents multiplication of thermal source neutrons causing fission, which is directly proportional to power, and employs rho as a measure of the nuclear status. As the equations for reactor behavior are developed, delta-k applies to fast neutrons and rho applies to thermal neutrons causing fission.
As long as these two forms of nuclear status are retained within their respective frameworks, no difficulty arises. However, delta-k and reactivity are sometimes used interchangeably in other applications. And as was illustrated in NUKEFACT #24, this creates contradictions and apparent discrepancies. Even worse, it leaves the student in confusion, with uncertainty and doubt that he really understands what should be straightforward.
So what is the solution? It seems that the best approach is to set delta-k aside after the early development of non-fission source multiplication (Equation 25.3). Who cares about fast neutrons? Thermal neutrons causing fission are where the power is produced (Equation 25.5). In a future issue we will discuss “power doubling” as applied to reactor startup. It’s power doubling ... not fast neutron doubling. But in addition, reactivity is by far the more common means of expressing the nuclear status, in commercial operational environments. All this is supported by the fact that both the General Equation Reactor Power and the Reactor Rate Equation contain reactivity as a term, not delta-k. We need to refrain from using delta-k and reactivity interchangeably ... especially as is done in the INPO exam bank. It serves no useful purpose, but does create confusion.