last update March 9, 1997

Sooner or later nearly all reactor texts identify beta as "the delayed neutron fraction". Many then use beta as a weighting factor in defining neutron generation time. In so doing, beta is taken to be the fraction of the total neutron population that consists of delayed neutrons. This is not correct.

The immediate mass products of a fission event are two large fission fragments, which are remnants of the original U-235 nucleus, plus, on average, two or three free neutrons, called "prompt" neutrons. Later, a fission fragment occasionally, but very rarely, undergoes a stage of radioactive decay that yields an additional neutron, called a "delayed" neutron. These neutron emitting fission fragments are called delayed neutron precursor atoms. Each precursor atom emits a single delayed neutron. If all prompt neutrons and precursor atoms resulting from a set of fissions are summed, the the fraction of this total that is precursor atoms is 0.0064. For example, if the total of prompt neutrons plus precursor atoms formed is 10,000, then the number of precursor atoms is 64. Strictly speaking the fraction 0.0064 is "the precursor yield fraction". More importantly, it is beta.

Beta is a physical property (constant) of U-235. And, it has no correlation to the fraction of the reactor neutron population that is delayed neutrons, which is not a constant value.

The important distinction between prompt and delayed neutrons is that while prompt neutrons appear at the instant of fission, delayed neutrons appear at some later time, extending from the immediately after the fission to well in excess of 100 seconds following the fission event. If the neutron population is changing with time, then it is not possible that both the precursor atoms and the identical number of delayed neutrons make up the same fraction, beta, of different neutron populations.

Misapplication of beta can be avoided by designating the delayed neutron fraction, i.e. the fraction of the total neutron population that consists of delayed neutrons, as DNF. So that:

Thus, if by the time the precursor atoms in the above example decay to emit delayed neutrons, the total neutron population has increased to 20,000, then DNF is 64/20000 = 0.0032, or significantly less than the value of beta. Obviously, DNF and beta represent different entities.

In the Delayed-Critical region, i.e. at power levels sufficiently high to render the non-fission neutron source negligible, the DNF value for operational type transients is always (beta - rho), where rho is reactivity. This definition applies to both transient and steady state power. And, consistent with the above example, DNF, as (beta - rho), is smaller than beta for positive reactivity and greater than beta for negative reactivity. At the steady state condition of criticality, with rho = 0, (beta - rho) is equivalent to beta.

In the Sub-Critical region, i.e. at low power levels where the non-fission neutrons are a significant part of the total neutron population, there is no similar analytic expression for the transient DNF. However, the total source neutron fraction, the total source being composed of non-fission plus delayed neutrons, is always (beta - rho). Computer programs exist that can easily track the DNF for complex transients in the Sub-Critical region, or for transitions between the Sub-Critical and Delayed-Critical regions, as does The Reactor Trainer. At steady state in the Sub-Critical region, i.e. at equilibrium subcritical multiplication, DNF = beta. This is because, at equilibrium multiplication, the non-fission source exactly makes up for neutron losses represented by the negative reactivity.

To summarize, the total neutron source strength is always equal to (beta - rho), whether in the Sub-Critical or Delayed-Critical region. But, since the non-fission neutrons are negligible in the Delayed-Critical region, the DNF is equal to (beta - rho). At steady-state, whether criticality or equilibrium subcritical multiplication, the DNF is equivalent to (beta). The delayed neutron population fraction (DNF) as a function of reactivity is illustrated on the following graphic:

The vertical axis is the delayed neutron population fraction, DNF, while the horizontal axis is reactivity. The linear portion of the line extending downward from the upper left to the lower right corner of the graphic is (beta - rho). The horizontal line at DNF = 0.0064, extending from rho = -0.0100 to rho = 0 is beta. Thus, it is readily apparent that, even over this limited reactivity range, the fraction of the neutron population that is made up of delayed neutrons varies greatly. In subsequent NUKEFACTS, the DNF behavior during actual power transients will be illustrated.

Is all this fuss about the real meaning of beta important? Absolutely! Understanding reactor behavior, or anything else, is not possible when the underlying terminology is sloppy, especially when it is then used to concoct phoney concepts like neutron generation time, which will be discussed as a future topic. But more importantly, the total neutron source fraction is crucial to the determination of the reactor power level. See NUKEFACT #3 on Source multiplication.

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