NUKEFACT #4

CHAIN REACTIONS ARE NOT SELF-SUSTAINING AT CRITICALITY

last update February 28, 1997

The condition of criticality is the predominant mode of reactor operation. An understanding of this most important condition is usually based on the following:

Description and Definition: Criticality is a steady-state condition. Power and neutron population are constant (dn/dt = 0 in the neutron balance equation); precursor inventory is constant with time (dC/dt = 0 in the precursor balance equation); and, no significant extraneous neutron source is present. The chain reactions are said to be self-sustaining, which to most probably infers that the chains are infinite in length.
Equation: The principal equation defining criticality is:

where:
k_eff = the effective multiplication factor
k_inf = the infinite multiplication factor
B² = core buckling
Tau = Fermi age
L = diffusion length of themal neutrons
Model: A chain reaction model comparable to that for equilibrium subcritical multiplication, is not given; rather, the definition of k_eff, based on neutron behavior during successive neutron generation times, implies a model:

As it turns out a more realistic model, one that treats prompt and delayed neutrons separately, is useful for understanding the physical process at criticality. Solving the neutron balance equation for steady state power in the Delayed-Critical region gives:

where:
P = n/(nu x H), power in watts
n = thermal neutrons causing fission, sec^-1
nu = 2.5 neutrons/fission
H = 3.1x10¹⁰ (fissions/second)/watt
Lambda = single precursor group effective decay constant, seconds^-1
C-bar = C/(nu x H)
C = core precursor inventory, atoms
beta = precursor yield fraction = 0.0065

This equation derives from the neutron balance equation but is also supported by a model similar to that for equilibrium multiplication. This "criticality model" is as follows:

where:
C' = C x l_p
l_p = the prompt neutron lifetime

Thus, the product (Lambda x C') represents the number of delayed neutrons introduced into each prompt lifetime interval. These delayed neutrons act as source neutrons, initiating chain reactions at the start of each time unit. The chains are propagated by prompt neutrons, whose lifetime in commercial reactors is of the order of 1x10^-4 seconds. Since delayed neutrons initiate the chains, they cannot also be counted in the propagation of the chains. Hence, the effective multiplication factor is multiplied by (1 - beta), the fraction of fission neutron production that is prompt, to remove the delayed neutrons from the propagation. This bracketed factor, [k_eff(1 - beta)], is known as the prompt multiplication factor, k_p. For criticality, with k_eff = 1.0000 and beta = 0.0065, k_p = 0.9935. And, since k_p < 1, each set of chains is finite, i.e. each set of chains contracts from lifetime to lifetime, and eventually expires. The individual chain reactions are not self-sustaining at criticality. The overall process is self-sustaining on fission neutrons alone (prompts + delays) because new chains initiated by delayed neutrons exactly make up for the expiring chains.

At criticality, the prompt multiplication factor is defined as:

So that, after 1000 lifetimes we have a chain remnant that is 0.9935¹⁰⁰⁰ = 0.0015. This means that after 0.1 second only one in a thousand prompt neutrons remain from a set of chain reactions that were initiated in a particular lifetime interval.

The common feature in the two multiplication factor definitions, effective versus prompt, is that both are a ratio of neutrons in successive time units. The difference is that k_eff counts all delayed neutrons from precursors produced in time unit-one as appearing in time unit-2, while k_p counts only prompt neutrons in time unit-2. The generation time, as used in the definition of k_eff, is the average lifetime of the fission neutron, i.e. the weighted lifetime of the prompt and delayed neutrons. For commercial reactors, l_g is of the order of 0.1 second. But, since delayed neutrons are emitted from precursors having mean lives extending up to 80 seconds, that do not recognize any averaging, it is not very likely that many appear in the 0.1 second interval of generation-2. In short, neutron generation time, with a so called average neutron, is flawed concept and will be addressed in a future NUKEFACT. Conversely, in the definition of k_p, the time unit is the actual lifetime of the prompt neutrons, meaning that all prompt neutrons produced by fissions in lifetime-1 appear in lifetime-2.

Also, consider the condition of prompt criticality. Conventionally, when k_eff = 1.0065, the reactor is said to be at prompt criticality, where the chain reaction can be maintained by means of the prompt neutrons alone. The criticality model is consistent with this definition because k_p is then equal to 1.0000. At prompt criticality the chains are self sustaining on prompt neutrons, which are the propagators of the chain. Once initiated, these chains are infinite in length. But, for any k_eff value less than 1.0065, which includes all operational k_eff, the reactor is subcritical on prompt neutrons. It is impossible for delayed neutrons to participate in successive life cycles of any model. The chains are not "self-sustaining", or unending, at criticality ... they are finite.

Does operational evidence exist to support the proposition that the chain reactions are not individually self-sustaining at criticality? Certainly! It is the phenomenon of prompt jump in power, which will be discussed in a future NUKEFACT.

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