Corresponding author: Anatoly G. Yuferov (

Academic editor: Yury Kazansky

The article considers methodological issues related to the conceptual and terminological apparatus of the dynamics of nuclear reactors. Based on an elementary analysis of the standard point reactor kinetics equations, the author shows that it is necessary to clarify the physical meaning of the parameter β included in the equations, which is traditionally interpreted as the “effective delayed neutrons fraction” (

the statement “if the delayed neutron fraction is β, then the prompt neutron fraction is equal to 1 – β”, used in the problems of analyzing the nuclear reactor dynamics as a starting position, cannot be considered applicable to any reactor conditions;

an increase in the β parameter by multiplying it by the “delayed neutron efficiency factor” leads, contrary to traditional interpretations, not to an increase but to a decrease in neutron reproduction in a supercritical reactor.

The proposed clarifications are appropriate both in terms of more adequate descriptions of processes in nuclear reactors and in relation to the formulations of nuclear safety requirements.

The “effective delayed neutrons fraction” (

According to the generally accepted interpretation of the

However, a detailed consideration of the equations of nuclear reactor dynamics reveals that the parameter β, which appears in these equations as the

Obviously, it is necessary to distinguish between the fractions in single fission events, in reactor steady states and in transient processes. Of interest are both the fractions of the current number of delayed and prompt neutrons relative to the total volume of the neutron population, and the fractions of generation and loss of these particles, i.e., the ratio of the process rates. An elementary analysis of the nuclear reactor dynamics equations (both point and distributed ones) shows that these quantities, as applied to delayed neutrons, are by no means always expressed in terms of the parameter β. Therefore, the historically formed interpretation of the parameter β as the “effective delayed neutrons fraction” requires both substantive and terminological clarifications.

In the equations of nuclear reactor point dynamics

(where all the notations are commonly-accepted: _{j}

is the difference between the generation rate of prompt neutrons

We will call the value

According to Eqs. (2), the term

is the total rate of generation of delayed neutron precursors or, in relation to Eq. (1), the rate of consumption of available neutrons per _{0} = β/Λ has the meaning of the relative rate, i.e., the share of consumption (per unit time) of these neutrons per

Accordingly, _{j} is the share of prompt neutron consumption per

Thus, the parameter β should be considered as a constant that characterizes a single fission event and relates the number of

with the corresponding number of prompt neutrons ν_{pn}. It is in this guise that the parameter β appears in the first works on the nuclear reactor neutron dynamics (

The interpretation of the parameter β in Eq. (1), which took root later in many works as an “effective quantity”, that in some way specifies the delayed neutrons fraction in the total yield per fission, is erroneous both in terms of content and terminology.

Firstly, the parameter β has the meaning of a “fraction” (as a characteristic of some part of a set that is homogeneous in terms of a certain attribute) when it characterizes the prompt neutron consumption per

Secondly, although the decay of the precursor nucleus gives one delayed neutron, i.e., ν_{dn} = ν_{dpn}, but the fraction of delayed neutrons in the total yield per fission

Of course, it is possible to use the parameter

Distributed neutron transport models are structurally similar to Eqs. (1), (2). In particular, the generation of prompt and delayed neutrons is described by the following terms (see, for example, (

However, it is important to note that the parameter ν(^{*}) here should be interpreted as the prompt neutron yield per fission. Then the meaning of the parameter β(^{*}) will be similar to that defined by Eqs. (4), (5). If, however, as is sometimes done (^{*}) is understood as the total yield, including delayed neutrons, then the interpretation β(^{*}) as =

_{0})∙

That is, in the case of criticality (

The parameter β can be omitted explicitly and only the rate constant _{0} = β/Λ can be used, i.e., the fraction of the neutron population that goes into the

Here, the neutron population reproduction rate v(

is equal to the current total reproduction rate of the

_{0}_{j}c_{j}

Equation (8) written as

v(

shows that the model under consideration describes the dynamics of the neutron population with the reflection of only two processes: (1) the prompt neutron reproduction at a rate of

This is due to the fact that equations (1), (2), (8) describe only the change in the neutron population. In this case, to reflect the increased contribution of delayed neutrons, it would be necessary to correct the term Σλ_{j}c_{j}, the delayed neutron generation rate. However, in fact, models (1), (2), (8) do not require such refinement. Firstly, these models operate on population size as such, without any qualitative subdivision of neutrons. Secondly, the balance of replenishment-consumption rates the population is determined only by the current values of the generation time Λ and the lifetime _{j}c_{j}.

The parameters Λ and _{j}c_{j}

After zero reactivity is established, equation (8), which in this case takes the form v(_{j}

Further, in the case of v(_{0}_{j}c_{j}, i.e., the neutron consumption rate for the _{0} (or, which is the same, the parameter β) by some “efficiency factor”, we, contrary to traditional interpretations, will not increase but decrease the reproduction of the population. This fact unequivocally points to the fallacy of the usual interpretations of the “effective” parameter β.

An analysis of the nuclear reactor dynamics often begins with the statement “if the delayed neutron fraction is β, then the prompt neutron fraction is equal to 1 – β”, assuming it is applicable to any reactor states. On this basis, the expression for the rates of processes on prompt neutrons is presented in the following form (see, for example, (

From here, taking into account expressions (3) for

The above thesis is specified, for example, as follows (

In a population of _{j}_{j}

In the critical state, the generation rates of _{j}_{j}, so that β/Λ = 6.4 delayed neutrons are generated per population neutron every second, i.e., 1006.4 neutrons are produced every second. In this case, the fraction of DNs in the total output (PN+DN) is obviously equal to 6.4/1006.4 = 0.00636 ≠ β. Therefore, the wording of the above example should be changed as follows: “out of every 100,000 fission neutrons in the critical reactor, 99,364 are prompt and 636 are delayed ones.”

As we can see, in order to clarify the content of the

According to equation (1), the parameter

only in the reactor steady state, when the _{0} expresses the population renewal rate, i.e., the relative delayed neutron generation rate is equal to _{0} = Σλ_{j}c_{j}/_{j}_{j}

In unsteady states, the balance that determines the relative neutron population reproduction rate α(

Representing here the relative delayed neutron generation rate as

we will obtain the following expression for the delayed neutron generation fraction determined by relation (11) (i.e.,

For slow transients in the reactor operating conditions, when the terms α and

and shows that under the specified conditions (α = 0,

At ρ = β, relation (12) takes (at

indicating the equality of the neutron population reproduction rate and the delayed neutron generation rate. The equality ρ = β also means that the neutron consumption rate per

The given interpretations of the parameter

A fairly obvious analysis of the reactor dynamics equations performed in this paper allows us, in particular, to state the following:

The parameter β, which appears in the above equations as the “effective delayed neutron fraction”, is generally equal to the prompt neutron fraction consumed for the

Only in the reactor steady state, the delayed neutron yield per prompt neutron is equal to β. At the same time, nothing can be said about the delayed neutron fraction relative to the total neutron population within the framework of models (1), (2).

In transient reactor processes, the delayed neutron generation fraction relative to prompt

Multiplying the parameter β by the “efficiency factor” means, according to equation (1), an increase in the neutron consumption per

Thus, the usual definition of the parameter β as the “effective delayed neutron fraction” does not fully correspond to the physical content of this parameter and should be used only with appropriate reservations. In particular, it is necessary to distinguish between the fractions in single fission events, in reactor steady states and in transient processes. In the last two cases, we can talk about fractions of a relative total neutron population or about ratios of the process rates.

It seems that the proposed clarifications are appropriate both in terms of more adequate descriptions of processes in nuclear reactors and in relation to the formulations of nuclear safety requirements.

Russian text published: Izvestiya vuzov. Yadernaya Energetika (ISSN 0204-3327), 2022, n. 2, pp. 174–182.