Both delta-k and reactivity are commonly used to describe the nuclear status of a reactor, as alternates to using k-effective. Typically, both delta-k and rho are defined as measures of the deviation from criticality. Reactivity is also frequently identified as the fractional change in the neutron population over a generation time. Perhaps the advantage of using delta-k or reactivity is that the status is expressed with reference to the normal operational condition of criticality, from which deviations tend to be small, making delta-k and rho more meaningful representations of nuclear status than k-effective.
The following are the conventional equation definitions:
|
24.1 |
 |
24.2 |
where:
k = k-effective
delta-k = keff - 1
rho = reactivity
Both equations apply to one specific nuclear condition having a given k-effective. Two less commonly recognized equations for change in nuclear status are:
 |
24.3 |
 |
24.4 |
where:
delta(delta-k)= the change in delta-k between two nuclear conditions = delta-k2 - delta-k1
delta-k2 = delta-k at the final condition
delta-k1 = delta-k at the initial condition
delta-rho = the change in reactivity between two nuclear conditions = rho2 - rho1
rho2 = reactivity at the final condition
rho1 = reactivity at the initial condition
When moving from one nuclear condition to another, these two equations provide a measure of the change in nuclear status. With this as background, let’s try some real numbers.
Given two k-effectives, determine delta-k, reactivity, and the respective changes:
................... k1 = 0.9000
................... k2 = 0.9500
Applying the above equations, we have:
| delta-k1 = -0.1000 | ... rho1 = -0.1111 |
| delta-k2 = -0.0500 | ... rho2 = -0.0526 |
| delta(delta-k) = -0.0500 - (-0.1000) | ... delta-rho = -0.0526 - (-0.1111) |
| ............ = +0.0500 | ............ = +0.0585 |
Now consider the following. In terms of delta-k, the change in nuclear status moves the reactor half way toward criticality. The initial subcritical delta-k is halved, from -0.1000 to -0.0500, by adding a delta-delta-k increment of +0.0500. Addition of a second delta-delta-k increment of the same magnitude, brings the reactor to exact criticality. However, for the same two nuclear conditions, the change in reactivity is +0.0585, or more than half way toward criticality. Addition of a second delta-rho of the same magnitude as the first (+0.0585) brings the reactor supercritical because rho2 is only -0.0526. Sure, it is well known that the relation between delta-k and reactivity is not linear. But how can this be? The cause of this apparent contradiction will be discussed in the next NUKEFACT.
-----------------------------------------------
In NUKEFACT #23 we presented excerpts from news releases about recent goings on at Chernobyl Unit 4 ... specifically concerning a detected increase in neutron level in proximity to the solidified core melt. The original reactor, an RBMK, was overmoderated, as evidenced by a positive void coefficient. However, it is not likely that much of the graphite moderator remains in the solidified core melt, meaning that it is probably significantly undermoderated. Then if, as reported, rainwater were leaking into the sarcophagus and reaching the solidified core melt, it is quite possible that the moderation would increase k-effective and source multiplication. This can be illustrated by an equation for equilibrium subcritical multiplication, as previously presented in NUKEFACT #2, namely:
 |
24.5 |
where:
P = core power, watts
S-bar = S/(3.1x1010x 2.46)
S = non-fission source strength, neutrons/sec
rho = subcritical reactivity
With the inverse relation between power and reactivity, and with a constant non-fission source strength, power (and neutron level) can increase by a factor-of-F only if the reactivity is reduced (divided by) a factor-of -F, as follows:
 |
24.6 |
Viktor Baryakhtar, vice-president of Ukraine’s Academy of Sciences, stated that readings had increased by as much as a factor-of-110. If the increased neutron level was strictly due to a reactivity change, then reactivity was reduced by a factor-of-110. If we assume the initial reactivity were rho = - 1.0000 (keff = 0.5000), then the altered reactivity is rho = -1.0000/110 = -0.0091. Not only is this a huge reduction in subcritical reactivity, the final nuclear status is then uncomfortably close to criticality.