Structural Aspects of Neutron Survival Probabilities

Abstract

The neutron survival probability (and related quantities including probabilities of extinction and initiation) is a central element of the broader stochastic theory of neutron populations and finds application in fields including reactor start-up, analysis of reactor power bursts and criticality accidents, and safeguards. In a full neutron transport formulation, the equation governing the single-neutron survival probability is a backward or adjoint-like integro-partial differential equation with the added complexity of being highly nonlinear. Analogous formulations of this equation exist in the context of many approximate theories of neutron transport, with the point kinetics formulation having received significant theoretical attention since the 1940s. This work continues this tradition by providing a novel analysis of the single-neutron survival probability equation using the tools of boundary layer theory. The analysis reveals that the “fully dynamic” solution of the single-neutron survival probability equation—and some key probability distributions derived from it—may be cast as a singular perturbation around the underlying quasi-static single-neutron probability of initiation. In this perturbation solution, the expansion parameter is the ratio of the neutron generation time to a macroscopic time scale characterizing the overall system evolution; this interpretation illuminates some of the fundamental structural aspects of neutron survival phenomena.

1. Introduction

At the level of single particles and nuclei, the interaction of neutrons with matter is a random process (i.e., is not predictable using deterministic methods). Consequently, the behavior of a neutron population in a reactor or “system” is always stochastic. In practice, the degree to which this stochasticity is consequential or even observable depends on many factors, notably including the size of the neutron population itself. Most commonly these behaviors manifest in modest-population applications (e.g., reactor start-up, power bursts, criticality accidents, and safeguards applications) [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17].

Beyond capturing mean behaviors, even the highest fidelity neutron transport equations, as classically formulated, do not account for the random fluctuations inherent to the evolution of neutron populations. Indeed, while various formulations of neutron transport equations have long been known and used to characterize only the mean behavior of neutron populations, they are in fact always underwritten by more fundamental entities that describe probability distributions of the neutron population or number within some system. In practice, while the derivation of mean-behavior neutron transport equations is almost always performed classically, these structures may also be obtained from their underlying probability distributions.

In broadening the scope of neutron transport beyond mean behaviors, some assumptions are often made regarding the underlying physics. These include:

• The neutron number states (with attached probabilities of realization) are discrete. In this case, the number of neutrons potentially present in a system assumes the integer values 0, 1, 2, and so forth.
• The evolutionary process is Markovian in nature. A Markov process is defined as being without memory in the sense that all information about the present population is independent of the state at earlier times.
• Transfer between states is characterized in terms of rates that, when multiplied against an arbitrarily small collision interval (such that only a single state transition may occur), yield single-event probabilities within that same span.
• The variation in time, space, angle, and energy is considered continuous (though this assumption is not necessary in general).

In view of these assumptions, a general probability distribution in the neutron number may be shown to obey a backward or adjoint-like master equation, which essentially encodes all mutually exclusive probabilistic pathways a system may experience in a transition between initial and final states. In the space-angle-energy-time phase space formulation of the Boltzmann neutron transport equation, the relevant backward master equation is given by

$$\begin{array}{c}\left[-{\displaystyle \frac{1}{\mathrm{v}\left(E\right)}}{\displaystyle \frac{\partial }{\partial t}}-\widehat{\Omega }·\nabla +{\Sigma }_{\mathrm{t}}\left(\stackrel{⃑}{r},E,t\right)\right]p_n\left(\mathrm{R},t_{\mathrm{f}};\stackrel{⃑}{r},\widehat{\Omega },E,t\right)\hfill \\ ={\int }_0^{\infty }d{E}^{\prime }{\int }_{4\pi }{\displaystyle \frac{d{\Omega }^{\prime }}{4\pi }}{\Sigma }_s\left(\stackrel{⃑}{r},\widehat{\Omega }·{\widehat{\Omega }}^{\prime },E\to E^{\prime },t\right)p_n\left(\mathrm{R},t_{\mathrm{f}};\stackrel{⃑}{r},\widehat{\Omega }\prime ,E\prime ,t\right)+{\Sigma }_{\mathrm{c}}\left(\stackrel{⃑}{r},E,t\right){\delta }_{n,0}\hfill \\ +{\Sigma }_{\mathrm{f}}\left(\stackrel{⃑}{r},E,t\right){\displaystyle \sum _{j=0}^J}{q}_j{\displaystyle \sum _{n_1+\dots +n_j=n}}{\displaystyle \prod _{k=1}^j}{\int }_0^{\infty }d{E}^{\prime }{\int }_{4\pi }{\displaystyle \frac{d{\Omega }^{\prime }}{4\pi }}\chi \left(E\prime \right)p_{n_k}\left(\mathrm{R},t_{\mathrm{f}};\stackrel{⃑}{r},\widehat{\Omega }\prime ,E\prime ,t\right),\hfill \end{array}$$

(1)

where

$\left(\stackrel{⃑}{r},\widehat{\Omega },E,t\right)$: space-angle-energy-time independent phase space variables,

t_f: final time (i.e., t_f > t),

R: terminal phase space region,

p_n = probability of n neutrons present in R at t_f due to a single neutron present at t < t_f in phase space element $\left(\stackrel{⃑}{r},\widehat{\Omega },E\right)$,

v(E): neutron speed,

Σ_i: macroscopic total (t), capture (c), differential scattering (s), and fission (f) cross sections,

δ_n,0: Kronecker delta function (associated with the probability that the system is in state n = 0 at time t),

q_j: fission multiplicity distribution or probability of j (0, 1, 2, …, J) neutrons emitted in a fission event (and where J is the maximum),

χ: fission neutron energy distribution,

and subject to the boundary and final conditions

$$p_n\left(\mathrm{R},t_{\mathrm{f}};{\stackrel{⃑}{r}}_{\mathrm{b}},\widehat{\Omega },E,t\right)=0 \mathrm{f}\mathrm{o}\mathrm{r} \widehat{\Omega }·\widehat{\mathrm{n}}>0,$$

(2)

and

$$p_n\left(\mathrm{R},t_{\mathrm{f}};\mathrm{R},t_{\mathrm{f}}\right)={\delta }_{n,1}.$$

(3)

Equation (1) is also known as the Pál-Bell equation owing to independent derivations of forms of it obtained by L. Pál and G. Bell in the late 1950s and early 1960s [18,19,20] (see also Williams, Lewins and collaborators, Munoz-Cobo and Verdu, Saxby and collaborators, Pázsit and Pál, and Horton and Kyprianou [21,22,23,24,25,26,27,28]). As formulated, it is an open set of coupled integro-differential-difference equations for the population probabilities p_n, where n may assume integer values ranging from zero to, for many practical scenarios, exceptionally large and effectively infinite numbers (e.g., the neutron population within a typical nuclear reactor may reach values of 10¹⁵ or higher depending on the power level).

Equation (2) is a typical non-reentrant boundary condition for convex geometry (with surface location ${\stackrel{⃑}{r}}_{\mathrm{b}}$ and outward normal $\widehat{\mathrm{n}}$) in the adjoint setting. Equation (3) amounts to a self-consistency statement: the probability that a single neutron introduced into the system at final time t_f in phase space element R certainly results in the presence of a single neutron at that same time and phase space location.

While highly general in nature, Equations (1)–(3) are still subject to several limitations; for example, they do not include the effects of delayed neutron precursors, and their strictly Markovian nature significantly complicates the additional inclusion of thermal feedback processes [29,30]. Consequently, among the aforementioned practical settings, forms of Equations (1)–(3) typically enjoy the most applicability in low-power, fast-transient scenarios (i.e., reactor bursts or criticality accidents); see, for example, Hansen, Hankins, Méchitoua, Ramsey and Hutchens, or Gregson [3,10,12,31,32].

Moreover, despite their fundamental importance, Equations (1)–(3) are seldom treated directly due to their practically infinite size and rather form the starting point for the construction of daughter equations in more widespread use. For example, equations characterizing the behavior of certain members of the neutron number probability distribution may also be obtained directly from Equation (1), such as that governing the zero-population probability p₀. The complement of p₀, namely

$$p_{\mathrm{s}}\left(\mathrm{R},t_{\mathrm{f}};\stackrel{⃑}{r},\widehat{\Omega },E,t\right)=1-p_0\left(\mathrm{R},t_{\mathrm{f}};\stackrel{⃑}{r},\widehat{\Omega },E,t\right),$$

(4)

may then be shown to satisfy the backward nonlinear integro-differential transport equation

$$\begin{array}{c}\left[-{\displaystyle \frac{1}{\mathrm{v}\left(E\right)}}{\displaystyle \frac{\partial }{\partial t}}-\widehat{\Omega }·\nabla +{\Sigma }_{\mathrm{t}}\left(\stackrel{⃑}{r},E,t\right)\right]p_{\mathrm{s}}\left(\mathrm{R},t_{\mathrm{f}};\stackrel{⃑}{r},\widehat{\Omega },E,t\right)\hfill \\ ={\int }_0^{\infty }d{E}^{\prime }{\int }_{4\pi }{\displaystyle \frac{d{\Omega }^{\prime }}{4\pi }}{\Sigma }_s\left(\stackrel{⃑}{r},\widehat{\Omega }·{\widehat{\Omega }}^{\prime },E\to E^{\prime },t\right)p_{\mathrm{s}}\left(\mathrm{R},t_{\mathrm{f}};\stackrel{⃑}{r},\widehat{\Omega }\prime ,E\prime ,t\right)\hfill \\ +\overline{\nu }{\Sigma }_{\mathrm{f}}\left(\stackrel{⃑}{r},E,t\right){\int }_0^{\infty }d{E}^{\prime }{\int }_{4\pi }{\displaystyle \frac{d{\Omega }^{\prime }}{4\pi }}\chi \left(E^{\prime }\right)p_{\mathrm{s}}\left(\mathrm{R},t_{\mathrm{f}};\stackrel{⃑}{r},\widehat{\Omega }\prime ,E\prime ,t\right)\hfill \\ -{\Sigma }_{\mathrm{f}}\left(\stackrel{⃑}{r},E,t\right){\displaystyle \sum _{j=2}^J}{\left(-1\right)}^j{\displaystyle \frac{{\chi }_j}{j!}}{\left[{\int }_0^{\infty }d{E}^{\prime }{\int }_{4\pi }{\displaystyle \frac{d{\Omega }^{\prime }}{4\pi }}\chi \left(E^{\prime }\right)p_{\mathrm{s}}\left(\mathrm{R},t_{\mathrm{f}};\stackrel{⃑}{r},{\widehat{\Omega }}^{\prime },E^{\prime },t\right)\right]}^j,\hfill \end{array}$$

(5)

subject to

$$p_{\mathrm{s}}\left(\mathrm{R},t_{\mathrm{f}};{\stackrel{⃑}{r}}_{\mathrm{b}},\widehat{\Omega },E,t\right)=0 \mathrm{f}\mathrm{o}\mathrm{r} \widehat{\Omega }·\widehat{\mathrm{n}}>0,$$

(6)

and

$$p_{\mathrm{s}}\left(\mathrm{R},t_{\mathrm{f}};\mathrm{R},t_{\mathrm{f}}\right)=1.$$

(7)

The nomenclature appearing in Equations (5)–(7) is identical to that appearing in Equations (1)–(3), with the additional definitions:

p_s: probability that a single neutron present in the system at some time t in the phase space element $\left(\stackrel{⃑}{r},\widehat{\Omega },E\right)$ results in the presence of a non-zero (but otherwise unspecified) neutron population at some later time t_f > t in phase space element R,

$\overline{\nu }$: mean number of neutrons released in an induced fission event,

χ_j: order-j fission factorial moment, or the mean number of neutron doublets, triplets, and j-tuplets released in an induced fission event.

As such, p_s is typically referred to as the single-neutron survival probability. Similarly, Equation (5) itself is known as the single-neutron survival probability equation, or sometimes the Bell-Lee equation owing to the derivation and exercise of a form of it by G. Bell and C. Lee in the 1960s and 1970s [33] (see also Baker [34]).

Equations (5)–(7) are straightforward to computationally treat [33,34,35,36,37,38,39,40,41] compared to the far more general formulation afforded by Equations (1)–(3) but still encode useful information. Indeed, in the time since its derivation, forms of the Bell-Lee equation have found widespread use in the analysis of many of the aforementioned low-population applications. Consequently, further analysis of the structure of this important equation—or, more precisely, surrogate representations thereof—is the principal focus of the remainder of this work.

In support of this program of study, Section 2 includes motivations and derivations surrounding the analogous point kinetics instantiation of Equations (1)–(7), including additional discussions surrounding notions of extinction, survival, and initiation. Section 3 provides an analysis of the resulting mathematical model, including considerations based on dimensional analysis and singular perturbation/boundary layer theory. A simple illustrative example of these concepts is furnished in Section 4, and some conclusions and recommendations for future study are provided in Section 5.

2. Point Kinetics

On a fundamental level, much of classical neutron transport theory may be traced back to equations formulated in the “full transport” phase space of independent variables; that is, space-angle-energy-time as appearing throughout Section 1. However, in many practical circumstances, certain approximations to neutron transport equations enjoy extensive usage; for example, P_N, S_N, or diffusion theories, slowing-down theories, multigroup theories, point kinetics theories, or some suitable combinations thereof, to name a few. The utility of each or any of these approximations to a full transport formulation certainly depends on both the physical regime under investigation and the necessary degree of informational fidelity demanded by the analyst.

With this idea in mind, analogs of Equations (1)–(3) and (5)–(7) for the Pál-Bell and Bell-Lee equations, respectively, may be devised within any of the approximate theories typically associated with neutron transport. In some cases these formulations are already achieved: see, for example, work beginning with Frankel, Feynman, and Bell (see also Hill) in the context of point kinetics theory [1,2,42,43], Ramsey and Hutchens, Williams, and Saxby et al. within certain diffusion theories [12,44,45], Stacey within a space and energy-dependent model [7,8], Saxby et al. within slowing-down theory [25,26], Lewins, O’Rourke and Prinja, and Gregson and Prinja within various monoenergetic scattering theories [23,41,46], and Salmi and Lewins and Myers within some multigroup kinetics theories [47,48,49,50].

Among these formulations, the most attention by far is devoted to the monoenergetic point kinetics setting, which dispenses with all independent variable dependences except time. As observed by Ott and Neuhold [51],

“… point kinetics equations … allow meaningful insight into kinetics concepts and provide the basis for … ‘exact’ treatment … It should be clearly understood, however, that the preliminary formulations are too inaccurate for practical applications. This holds true particularly for the one-group formulations, which are merely included to illustrate the meaning of … otherwise more complicated formulas.”

Indicating that this setting sacrifices the physical fidelity inherent to its full transport counterpart, albeit set against the advantage of broad preservation of essential behaviors within an analytically tractable mathematical setting. To this point, as noted by Sachdev and Janna [52],

“… the search for exact solutions is now motivated by the desire to understand the mathematical structure of the solutions, and hence, a deeper understanding of the physical phenomena described by them. Analysis, computation, and not insignificantly, intuition all pave the way to their discovery…”

Thus rendering the exact-solution-rich point kinetics formulation as a low-fidelity but potentially high-value setting in which to perform analysis, and any conclusions obtained therein enable cogent exploration in higher-fidelity (and inevitably computational) settings.

With this motivation in mind, the point kinetics analog of Equations (1)–(3) may again be constructed using a probability balance including all mutually exclusive events associated with a population of n neutrons (which again may assume the values 0, 1, 2, …) within a system at some time t_f owing to the introduction of a single neutron at some earlier time t < t_f. In the point kinetics formalism, the relevant processes include:

(1)

No event occurs,

(2)

A removal (i.e., capture or leakage) event occurs,

(3)

A fission event featuring the emission of j secondary neutrons occurs.

Given these processes,

$$-{\displaystyle \frac{d{p}_n}{dt}}=-{\lambda }_{\mathrm{t}}{p}_n+{\lambda }_{\mathrm{r}}{\delta }_{n,0}+{\lambda }_{\mathrm{f}}\sum _{j=0}^J{q}_j\sum _{n_1+\dots +n_j=n}\prod _{j^{\prime }=1}^j{p}_{n_{j^{\prime }}},$$

(8)

and

$$p_n\left(t_{\mathrm{f}};t=t_{\mathrm{f}}\right)={\delta }_{n,1},$$

(9)

are the backward master equation-final condition pairing inherent to the point kinetics setting [53,54]. In Equations (8) and (9),

p_n(t_f;t): probability of there being n (0, 1, 2, …) neutrons within the system at time t_f given the presence of a single neutron at an earlier time t < t_f,

λ_i(t) (i = t, r, f): total (t; removal plus fission), removal (r; capture plus leakage), and fission (f) reaction rates. These reaction rates may be regarded as functions of time as indicated,

and t_f, q_j, and δ_n,0 retain their definitions as provided in Section 1. Equation (9) again indicates that a single neutron introduced into the system at time t_f by definition results in p₁ = 1 at that same time.

The three terms appearing on the right-hand side of Equation (8) encode the three processes included in the point kinetics formulation. Already Equation (8) is perhaps more intuitive than its full-transport analog given in Equation (1): in Equation (8), the immediately adjacent states in n are connected more simply than in the expanded phase space of Equation (1). In a sense there are far fewer pathways or degrees of freedom available for connection to the neutron number state at time t.

Like Equation (1), for many practical purposes, Equation (8) is essentially an infinite coupled, open system of nonlinear ordinary differential equations (ODEs) for the probabilities p_n. However, this structure may be cast in a still simpler form via the introduction of a probability generating function G defined by

$$G\left(z,t_{\mathrm{f}};t\right)=\sum _{n=0}^{\infty }{z}^n{p}_n,$$

(10)

where the transform variable z lies in the range 0 ≤ z ≤ 1. With Equation (10) and its complement

$$H\left(z,t_{\mathrm{f}};t\right)=1-G,$$

(11)

Equations (8) and (9) may then be rendered as, after some algebra (see, for example, O’Rourke et al. [54]), the equivalent ODE-final condition pairing

$${\displaystyle \frac{dH}{dt}}={\lambda }_{\mathrm{t}}H+{\lambda }_{\mathrm{f}}\left[-1+\sum _{j=0}^J{\displaystyle \frac{{\left(-1\right)}^j}{j!}}{\chi }_j{H}^j\right],$$

(12)

and

$$H\left(z,t_{\mathrm{f}};t=t_{\mathrm{f}}\right)=1-z,$$

(13)

where the fission factorial moments defined by

$${\chi }_j=\sum _{i=j}^J{\displaystyle \frac{i!}{\left(i-j\right)!}}{q}_i,$$

(14)

are again associated with the moments of the fission multiplicity distribution q_i; for example, this definition yields the normalization condition χ₀ = 1 and ${\chi }_1=\overline{\nu }$, the mean number of neutrons released per fission. Accordingly, Equation (12) may be further simplified by expanding the first two terms within its summation to yield:

$$l{\displaystyle \frac{dH}{dt}}=-\rho H+\sum _{j=2}^J{\left(-1\right)}^j{\beta }_j{H}^j,$$

(15)

where the reactivity is defined as [55]

$$\rho \left(t\right)=1-{\displaystyle \frac{{\lambda }_{\mathrm{t}}}{\overline{\nu }{\lambda }_{\mathrm{f}}}},$$

(16)

the neutron generation time is defined as [55]

$$l\left(t\right)={\displaystyle \frac{1}{\overline{\nu }{\lambda }_{\mathrm{f}}}},$$

(17)

and each of the β_j (j = 2, 3, …) is defined by

$${\beta }_j={\displaystyle \frac{{\chi }_j}{j!\overline{\nu }}},$$

(18)

and are typically taken as independent of time.

Equation (15) for H(z,t_f;t), subject to Equation (13), may be viewed as a simplification over the equivalent backward master equation. This is the case due to the now-parametric appearance of the transform variable z, which, via Equation (10), encodes the infinite nonlinear coupling between the probabilities p_n appearing in Equation (8). Otherwise, Equation (15) is a first-order nonlinear ODE with variable coefficients; moreover, as expected within the adjoint-like formulation, it is solved backward in time from the final condition at t = t_f to earlier times t.

With a solution of Equations (13) and (15) for H, the associated probability distribution p_n may be directly recovered using

$$p_n=-{\displaystyle \frac{1}{n!}}{{\displaystyle \frac{{\partial }^{n}H}{\partial z^n}}|}_{z=0},$$

(19)

as may be inferred from Equations (10) and (11). The construction associated with the introduction of Equations (10), (11) and (19) is thus seen to “exchange” the complexity inherent to Equation (8) with another form of complexity inherent to Equation (15): a large number of coupled ODEs with almost uniformly empty final conditions have been cast as an equivalent single ODE featuring a non-trivial final condition, followed by the necessity of extracting successively higher derivatives.

For many of the practical applications outlined in Section 1, knowledge of the full probability distribution p_n is not strictly necessary, and recourse may be made to even simpler entities. Among these is the single-neutron survival probability defined in the full transport formulation via Equations (5)–(7). The point kinetics analog of this formalism is explored further in Section 2.1.

2.1. Extinction and Survival

Conspicuous in the resolution of Equation (8) for n = 0, namely (after some algebra),

$$-{\displaystyle \frac{d{p}_0}{dt}}=-{\lambda }_{\mathrm{t}}{p}_0+{\lambda }_r+{\lambda }_{\mathrm{f}}\sum _{j=0}^J{\displaystyle \frac{{\left(-1\right)}^j}{j!}}{\chi }_j{\left(1-p_0\right)}^j,$$

(20)

is the appearance of an equation with no coupling to the remainder of the probability distribution; that is, Equation (20) features no terms involving p₁, p₂, or any other probabilities. In the context of Equation (15), the equivalent result may be inferred under the parameterization z = 0, such that Equation (10) yields

$$G\left(0,t_{\mathrm{f}};t\right)=p_0\left(t_{\mathrm{f}};t\right),$$

(21)

where the zero-population probability p₀ introduced in Section 1 is also referred to as the single-neutron extinction probability (see Figure 1): this is the probability that a single neutron present in the system at time t culminates in a zero-neutron state at the later time t_f > t. Accordingly, with Equation (11),

$$H\left(0,t_{\mathrm{f}};t\right)=1-p_0\left(t_{\mathrm{f}};t\right)=p_{\mathrm{s}}\left(t_{\mathrm{f}};t\right),$$

(22)

again defining a single-neutron survival probability analogous to that appearing in Equation (4) (see Figure 1): in the point kinetics setting, this is the probability that a single neutron present in the system at time t culminates in a non-zero (but otherwise ambiguous) neutron population at the later time t_f > t. This quantity is alternatively interpreted as the probability that the initial neutron instigates a fission chain persisting to t_f > t.

Given the parametric realization of the transform variable z appearing in Equations (13) and (15), with z = 0 an explicit equation for the survival probability may be immediately obtained as

$$l{\displaystyle \frac{d{p}_{\mathrm{s}}}{dt}}=-\rho p_{\mathrm{s}}+\sum _{j=2}^J{\left(-1\right)}^j{\beta }_j{p}_{\mathrm{s}}^j,$$

(23)

subject to

$$p_{\mathrm{s}}\left(t_{\mathrm{f}};t=t_{\mathrm{f}}\right)=1.$$

(24)

Unlike Equations (13) and (15), Equations (23) and (24) nowhere contain z, even parametrically. Given, via Equation (10), the transform variable z in some sense encodes the coupling between the discrete elements of the probability distribution p_n, its complete absence from Equations (23) and (24) is thus analogous to the complete absence from Equation (20) of the non-p₀ elements of the probability distribution, and in turn the essential independence of p₀ (or p_s) (see, for example, O’Rourke et al. or Prinja and Souto [54,56] for further discussion of this topic).

This property established, like Equation (15), Equation (23) is a first-order, nonlinear ODE with variable coefficients, again solved backward in time from the final condition given by Equation (24). A solution of this equation may be used to define additional quantities in widespread use.

2.2. External Neutron Sources

The single-neutron survival probability may be convolved with an external neutron source rate function (i.e., a neutron source not owing to induced fission) s = s(t) to construct a neutron survival probability associated with that source. To see this, following Hansen [3], the quantity

$$s\left(t\right)p_{\mathrm{s}}\left(t_{\mathrm{f}};t\right) dt,$$

(25)

is simply the probability that a fission chain surviving to t_f is sponsored by s at time t in the infinitesimal interval dt. If sp_s is essentially constant during this interval, the sponsoring by s of independent surviving fission chains may be assumed to follow a continuous geometric or exponential distribution [57], wherein

$$P_0\left(t_{\mathrm{f}};t\right)=\exp \left[-{\int }_{-\infty }^{t}s\left(t\prime \right)p_{\mathrm{s}}\left(t_{\mathrm{f}};t\prime \right) dt\prime \right],$$

(26)

is the probability that no such events occur at or before t (see Figure 1). The survival probability density function (see Figure 1) given by

$$\mathrm{s}\mathrm{p}\mathrm{d}\mathrm{f}\left(t_{\mathrm{f}};t\right) dt=\exp \left[-{\int }_{-\infty }^{t}s\left(t\prime \right)p_{\mathrm{s}}\left(t_{\mathrm{f}};t\prime \right) dt\prime \right]s\left(t\right)p_{\mathrm{s}}\left(t_{\mathrm{f}};t\right) dt,$$

(27)

thus denotes the probability that the first surviving fission chain is sponsored by s at time t in dt, and the complement of Equation (26) (see Figure 1), namely

$$\mathrm{s}\mathrm{c}\mathrm{d}\mathrm{f}\left(t_{\mathrm{f}};t\right)=1-\exp \left[-{\int }_{-\infty }^{t}s\left(t\prime \right)p_{\mathrm{s}}\left(t_{\mathrm{f}};t\prime \right) dt\prime \right],$$

(28)

is then the survival cumulative distribution function representing the probability that a fission chain surviving to t_f is sponsored by s at or before time t. The scdf given in Equation (28) is thus sometimes referred to as the source-neutron survival probability.

Equations (25)–(28) reflect the underlying assumption that s represents a singlet-emitting source; extensions of these relations to sources capable of multiplet-emitting behaviors are given by numerous authors including Bell, Hill, and O’Rourke et al. [42,43,54].

3. Dimensional Analysis

The existing literature on the extinction and survival of stochastic neutron populations emphasizes forms of Equations (23) and (24) featuring constant coefficients, and hence time-independent or static systems [43,58,59,60,61]; some exceptions include the studies by Hansen, Ramsey and Hutchens, Hill, Prinja and collaborators, and O’Rourke et al. [3,11,43,53,54,56].

In the interest of further generalizing this body of investigations, dimensional considerations suggest that the arbitrary time-dependence appearing in Equation (23) via both the neutron generation time l and reactivity ρ is more realistically expressed as

$$l\left(t\right)\to l_{0}L\left({\displaystyle \frac{t}{t_0}}\right),$$

(29)

and

$$\rho \left(t\right)\to \rho \left({\displaystyle \frac{t}{t_0}}\right),$$

(30)

where l₀ and t₀ are constants with dimensions of time, and L and ρ are dimensionless functions of the indicated dimensionless argument.

Given that l and ρ feature the physical dimensions of time and 1 (dimensionless), respectively, a formulation analogous to Equations (29) and (30) will almost inevitably appear even in cases initially featuring the more generic but ultimately naive parameterization l = l(t) and ρ = ρ(t). Put another way, in Equations (29) and (30), the characteristic time scales l₀ and t₀ feature the same dimensions as l and t, respectively, and thus explicitly parameterize the time-dependence of l and ρ in a dimensionally consistent but still suitably arbitrary manner that is representative of a wide variety of practical circumstances.

Otherwise, like the specific functional forms of L and ρ themselves, more precise definitions of l₀ and t₀ are scenario dependent. However, some general guidance is afforded via reference to the classical point kinetics setting, wherein scenarios featuring parameterizations akin to Equation (30) are commonplace. For example, authors such as Ash, Ott and Neuhold, and Hetrick [9,51,55] consider various “reactor excursion” models ranging from prompt-jumps to ramp-inputs to more complicated reactivity insertion-shutdown mechanisms: these models generally include dimensional quantities such as insertion rate constants, heat capacities, delay times, or other feedback coefficients. Inherent to each of these mechanisms is, through a suitable dimensional analysis, the emergence of a “macro-physics” time scale representing the forcing, drive, or evolution of the entire system.

On the other hand, parameterizations of the form given by Equation (29) appear to be much less common in the classical point kinetics literature, wherein, during excursion analysis, the neutron generation time is either taken to be constant or somehow bundled up with the time-variation in the reactivity function. A notable exception appears in Hill’s work involving survival probabilities [43], wherein the analysis of criticality excursions allows for simultaneous, independent variation in both l and ρ (via a multiplication factor k) in the style of Equations (29) and (30); see Figure 2.

In this case, t₀ features a macro-interpretation common to Equations (29) and (30). However, Equation (29) also features the additional parameter l₀, which arises from the inherent dimensionality of the neutron generation time. From Figure 2, this additional time scale may be interpreted as a minimum, maximum, or other fiducial governing the scale of the ordinate axis. Furthermore, given the definition of l appearing in Equation (17), l₀ also represents the “micro-physics” time scale associated with the instantaneous neutronic evolution embedded within the macro-physics excursion.

The parameters l₀ and t₀ suitably interpreted, a dimensionless time variable τ is naturally defined as

$$\tau ={\displaystyle \frac{t}{t_0}},$$

(31)

so that Equation (23) may be written as

$$\mathcal{r}L\left(\tau \right){\displaystyle \frac{d{p}_{\mathrm{s}}}{d\tau }}=-\rho \left(\tau \right) p_{\mathrm{s}}+\sum _{j=2}^J{\left(-1\right)}^j{\beta }_j{p}_{\mathrm{s}}^j,$$

(32)

where now p_s = p_s(τ_f;τ), subject to

$$p_{\mathrm{s}}\left({\tau }_{\mathrm{f}};\tau ={\tau }_{\mathrm{f}}\right)=1,$$

(33)

and

$$\mathcal{r}={\displaystyle \frac{l_0}{t_0}},$$

(34)

$${\tau }_{\mathrm{f}}={\displaystyle \frac{t_{\mathrm{f}}}{t_0}},$$

(35)

are dimensionless numbers.

Equation (32) may be further simplified by introducing the differential transformation

$$d\tilde{\tau }={\displaystyle \frac{d\tau }{L\left(\tau \right)}},$$

(36)

thus defining, in view of Equation (17), and in analogy to the optical depth pervasive in the radiation transport literature, $\tilde{\tau }$ as an “optical time” or effective number of fission neutrons evolved up to time τ according to

$$\tilde{\tau }={\int }_{-\infty }^{\tau }{\displaystyle \frac{1}{L\left(\tau \prime \right)}}d{\tau }^{\prime }.$$

(37)

With Equations (36) and (37), Equations (32) and (33) become

$$\mathcal{r}{\displaystyle \frac{d{p}_{\mathrm{s}}}{d\tilde{\tau }}}=-\rho \left(\tilde{\tau }\right) p_{\mathrm{s}}+\sum _{j=2}^J{\left(-1\right)}^j{\beta }_j{p}_{\mathrm{s}}^j,$$

(38)

and

$$p_{\mathrm{s}}\left({\tilde{\tau }}_{\mathrm{f}};\tilde{\tau }={\tilde{\tau }}_{\mathrm{f}}\right)=1,$$

(39)

respectively, where

$${\tilde{\tau }}_{\mathrm{f}}={\int }_{-\infty }^{{\tau }_{\mathrm{f}}}{\displaystyle \frac{1}{L\left(\tau \prime \right)}}d{\tau }^{\prime },$$

(40)

is the effective number of fission neutrons evolved up to the final time τ_f. In further analysis of Equations (38) and (39) (i.e., during the remainder of this study), the tildes will be suppressed in the interest of notational brevity.

In view of the interpretation of t₀ furnished above, the dimensionless number τ_f appearing in Equation (39) simply indexes the placement of the final time t_f against the characteristic time scale of an underlying criticality excursion. When τ_f >> 1 the final time is taken as occurring beyond the relevant duration of the excursion, and Equations (38) and (39) therefore interrogate for neutron survival through it. While this condition is assumed throughout the remainder of this study, Section 5 discusses some circumstances wherein alternate final conditions might instead be employed.

Finally, the dimensionless number $\mathcal{r}$ may be interpreted as the ratio of micro-to-macro physics time scales. Given for many practical scenarios of interest (e.g., Figure 2),

l₀ ≤ 10⁻⁷ s,

t₀ ≥ 10⁻⁴ s,

and hence $\mathcal{r}$ << 1. Given the interpretation of $\mathcal{r}$, these scenarios feature neutron collision time scales significantly smaller than those associated with the enveloping macro-physics evolution. Put another way, in these scenarios neutron collisions occur much more rapidly than the comparatively plodding reactivity insertion-feedback processes.

In view of this likely outcome, the placement of $\mathcal{r}$ within Equation (38) is conspicuous and ultimately suggestive of a productive approach to further analysis.

3.1. Boundary Layer Theory

Equations (38) and (39) have no known general closed-form solution. In the existing literature, considerable attention has been devoted to devising approximate solutions of forms of these equations under a variety of further simplifying assumptions, including:

• Equation (38) is considerably simplified under the assumption J = 2, which is referred to throughout the existing literature as the “quadratic approximation” [42,43,53,54,56,59,62]. The quadratic approximation involves the truncation of the factorial moment summation given in Equation (14), and not the fission multiplicity distribution defined following Equation (1). The limitations and consequences of this approximation are considered at length by authors including Hill and Ramsey and Hutchens [43,60]; in short, the quadratic approximation is typically viewed as being more viable for uranium-based systems (e.g., β₂ = 0.89, β₃ = 0.43, β₄ = 0.12, … for U-235 at 1 MeV [60]) than their plutonium-based counterparts (e.g., β₂ = 1.13, β₃ = 0.69, β₄ = 0.24, … for Pu-239 at 1 MeV [60]). Under this assumption, Equation (38) indeed attains only quadratic nonlinearity and is hence both a homogeneous Riccati equation and an index-2 Bernoulli equation [63,64,65]. For arbitrary ρ(τ), this equation features a solution in terms of quadratures; by far the most analysis is thus devoted to the closed-form special case where ρ = const [43,53,54,56,58,59,62,66,67,68].
• For J ≥ 3, Equations (38) and (39) feature implicit solutions for certain parameterizations of ρ(τ) (the constant parameterization again being noteworthy). The Abel equation [64] resulting from the J = 3 “cubic approximation” is among the most hopeful of this sort, but, except for some mention by Lewins and collaborators [59,66] and Ramsey and collaborators [60,68], it is not widely explored.

These activities are broadly useful in exposing some of the fundamental properties of Equations (38) and (39), and hence Equations (5)–(7) by association. However, the structure of Equation (38) may be even more thoroughly explored using another approximation technique.

As noted in Section 3, many practical circumstances wherein Equations (38) and (39) find application feature $\mathcal{r}$ << 1. This outcome naturally suggests a perturbative approach to further analysis of Equations (38) and (39), though in practice, this matter is somewhat complicated by the conspicuous placement of $\mathcal{r}$ therein. Indeed, as $\mathcal{r}\to 0$ Equation (38) nominally becomes an order-J polynomial,

$$-\rho \left(\tau \right){ p}_{\mathrm{s}}+\sum _{j=2}^J{\left(-1\right)}^j{\beta }_j{p}_{\mathrm{s}}^j=0,$$

(41)

the differential order of which is abruptly changed with respect to Equation (38); hence, the solution of Equation (41) cannot, in general, satisfy Equation (39). This result in turn suggests the existence of a narrow region within the solution of Equation (38) near where Equation (39) is satisfied, wherein

$$\left|\mathcal{r}{\displaystyle \frac{d{p}_{\mathrm{s}}}{d\tau }}\right|\nless 1,$$

(42)

implying that the derivative term appearing in Equation (38) cannot be neglected within this “boundary layer”.

Closely related situations arise throughout many branches of mathematical physics, including quantum mechanics and, perhaps most notably, boundary layer theory in the context of fluid mechanics. The common theme occurring in this and analogous scenarios—including Equations (38) and (39)—is the multiplication of a small parameter against the highest-order derivative appearing within the salient differential equation. These scenarios are broadly called boundary layer problems by, for example, Bender and Orszag [65], and within them any attempted perturbation analyses must be of the singular variety.

A singular perturbation/boundary layer analysis performed in the context of Equations (38) and (39) results in the decomposition of their perturbative solution into two elements:

(1)

An “outer region” in some sense removed from the boundary layer wherein a classical perturbation solution of Equation (38) may be attempted,

(2)

An “inner region” interior to the boundary layer wherein Equation (42) holds, and solution of a distinct limiting form of Equations (38) and (39) may be attempted,

(3)

With the outer and inner solutions so devised, asymptotic matching is typically used to resolve any remaining constants of integration, resulting in the construction of an approximate global solution.

However, in the case of Equations (38) and (39), the outer component associated with the boundary layer solution necessarily involves the solution of relations such as Equation (41). As an algebraic equation, the solution of Equation (41) involves no constants of integration, thus complicating the asymptotic matching procedures typical to boundary layer theory. Among certain alternatives [e.g., with recourse to an equivalent, 2nd order linear ODE structure obtained using the Riccati transformation [63,64,65] of Equations (38) and (39)], throughout the remainder of Section 3.1, a simple envelope solution is employed, the consequences of which are discussed as both a matter of course and within Section 5.

The Outer Solution

Construction of the outer solution of Equation (38) involves the perturbation expansion to arbitrary order M,

$$p_{\mathrm{s}}^{\left(\mathrm{o}\mathrm{u}\mathrm{t}\right)}\left({\tau }_{\mathrm{f}};\tau \right)=\sum _{i=0}^M{\mathcal{r}}^i{p}_{\mathrm{s}}^{\left(i\right)}\left({\tau }_{\mathrm{f}};\tau \right),$$

(43)

which may be substituted into Equation (38) to yield

$$\sum _{i=0}^M{\mathcal{r}}^{i+1}{\displaystyle \frac{{dp}_{\mathrm{s}}^{\left(i\right)}}{d\tau }}=-\rho \sum _{i=0}^M{\mathcal{r}}^i{p}_{\mathrm{s}}^{\left(i\right)}+\sum _{j=2}^J{\left(-1\right)}^j{\beta }_j{\left(\sum _{i=0}^M{\mathcal{r}}^i{p}_{\mathrm{s}}^{\left(i\right)}\right)}^j.$$

(44)

The nonlinear term appearing in Equation (44) may be resolved using the multinomial theorem [69], namely

$${\left(p_{\mathrm{s}}^{\left(0\right)}+\mathcal{r}{p}_{\mathrm{s}}^{\left(1\right)}+\dots +{\mathcal{r}}^M{p}_{\mathrm{s}}^{\left(M\right)}\right)}^j=\sum _{k_0+k_1+\dots +k_M=n}\left(\begin{array}{c}j\\ k_0,k_1,\dots ,k_M\end{array}\right)\prod _{i=0}^M{\left({\mathcal{r}}^i{p}_{\mathrm{s}}^{\left(i\right)}\right)}^{k_i},$$

(45)

where all the k_i ≥ 0, and the multinomial coefficients are given by

$$\left(\begin{array}{c}j\\ k_0,k_1,\dots ,k_M\end{array}\right)={\displaystyle \frac{j!}{k_0!k_1!\dots k_M!}}.$$

(46)

With Equation (45), Equation (44) becomes

$$\begin{array}{c}{\displaystyle \sum _{i=0}^{\infty }}{\mathcal{r}}^{i+1}{\displaystyle \frac{{dp}_{\mathrm{s}}^{\left(i\right)}}{d\tau }}=-\rho {\displaystyle \sum _{i=0}^{\infty }}{\mathcal{r}}^i{p}_{\mathrm{s}}^{\left(i\right)}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +{\displaystyle \sum _{j=2}^J}{\left(-1\right)}^j{\beta }_j{\displaystyle \sum _{k_0+k_1+\dots +k_M=n}}\left(\begin{array}{c}j\\ k_0,k_1,\dots ,k_M\end{array}\right){\displaystyle \prod _{i=0}^M}{\left({\mathcal{r}}^i{p}_{\mathrm{s}}^{\left(i\right)}\right)}^{k_i},\end{array}$$

(47)

which may now be collected against the powers of $\mathcal{r}$, again under the assumption that the outer solution holds far from the boundary layer and is thus not rapidly varying [i.e., Equation (42) does not hold]. The first two terms in this construction are

$${\mathcal{r}}^0: \rho p_{\mathrm{s}}^{\left(0\right)}-\sum _{j=2}^J{\left(-1\right)}^j{\beta }_j{\left(p_{\mathrm{s}}^{\left(0\right)}\right)}^j=0,$$

(48)

$${\mathcal{r}}^1: {\displaystyle \frac{d{p}_{\mathrm{s}}^{\left(0\right)}}{d\tau }}+\rho p_{\mathrm{s}}^{\left(1\right)}-p_{\mathrm{s}}^{\left(1\right)}\sum _{j=2}^J{\left(-1\right)}^{j}j{\beta }_j{\left(p_s^{\left(0\right)}\right)}^{j-1}=0,$$

(49)

and so forth, explicitly realizing a one-way coupled system of equations that may be solved in sequence for p_s⁽⁰⁾, p_s⁽¹⁾, and so forth. As is now evident, this system requires no differential equation solutions. Consequently, and as anticipated, none of the solutions of these equations nor their sum can generally satisfy Equation (39), thus necessitating eventual supplementation via an inner solution.

Of these equations, only forms of Equation (48) appear in the existing literature, the treatment of which is largely as extensive as that of Equations (38) and (39). This is because, when ρ = const., both Hill and Prinja [43,53] show that Equation (48) is satisfied by the probability that a single neutron present in the system at time τ survives (or its progeny survive) indefinitely. Moreover, Prinja and Souto [56] (see also O’Rourke et al. [54]) explore the circumstances under which the solution of Equation (48) may also be interpreted as

$$p_{\mathrm{s}}^{\left(0\right)}=p_{\infty },$$

(50)

or the probability that the indefinitely persistent fission chain is simultaneously the infinite population probability. That is, as noted by Prinja and Souto [56],

“… given one initiating neutron, the neutron population in a … supercritical system will eventually (as t → ∞) either become extinct with probability P₀(∞) or it will diverge with probability 1 − P₀(∞). The finite-state (0 < n < ∞) occupation probabilities are all zero…”

For this reason, the solution of Equation (48) given by p_∞ is also called the single-neutron divergent chain probability or single-neutron probability of initiation (POI). In view of this interpretation, Equation (48) features several notable properties:

• Equation (48) features the trivial solution p_∞ = 0 for any constant value of ρ.
• When constant ρ < 0, Equation (48) features only negative or imaginary non-trivial solutions, which are discarded in favor of the trivial solution. Physically, there is zero probability that the progeny of a single neutron introduced into a static subcritical system can survive indefinitely or approach an infinite number.
• When constant ρ > 0, Equation (48) features a single non-trivial and physically relevant solution 0 < p_∞ < 1. Physically, there is a finite probability that the progeny of a single neutron introduced into a static supercritical system can survive indefinitely, approaching an infinite number.

These outcomes are further explored by O’Rourke et al. [70] using modern analysis techniques for high-order polynomial equations. Furthermore, the special cases of:

• J = 2 (again corresponding to the quadratic approximation) is treated in numerous references [43,53,54,56,58,59,62,66,67,68],
• J = 3 (the cubic approximation) is treated by Lewins and collaborators [59,66] and Ramsey and collaborators [60,68],
• J = 4 and 5 solutions can be obtained in closed form, but are algebraically cumbersome,
• Solutions for J ≥ 6 are associated with high-order polynomial equations, the analysis of which remains a longstanding problem of interest within the pure mathematics community. In this case, Ramsey and Hutchens have devised approximate solutions using A-hypergeometric series [71]; additional investigations using group theory techniques or more recent advances featuring hyper-Catalan numbers [72] remain a matter for future study.

The fundamental difference between these existing studies and Equation (48) is, in the latter case, the residual time-dependence still encoded in ρ. However, given that this time-dependence may be regarded as parametric, the resulting solution of Equation (48) is taken to feature the same behaviors as its ρ = const. counterpart. Consequently, for ρ = ρ(τ), the solution of Equation (48) is labeled parametrically as p_∞(τ) and is referred to as the quasi-static single-neutron POI. Put another way, p_∞ determined at time τ assumes the underlying ρ realized at that same τ is fixed for all time. The time-parameterized solution of Equation (48) is thus effectively a series of “static snapshots” of these fixed p_∞ values, with τ serving as an index.

In this paradigm, the instantaneous nature of Equation (48) is rectified by inclusion of the corrections encoded in Equation (49) and any higher-order members. The corrections resulting from p_s⁽¹⁾ and so forth approximate the action of the time-derivative appearing in Equation (38), partially restoring the truly dynamical character of the time-evolving system. In view of Equation (43), these amendments to purely quasi-static behavior are ordered via the indicated power series in $\mathcal{r}$, reflecting the correlation between their magnitude, the time-scale ratio defined in Equation (34), and the presence of a non-zero l₀ parameter.

Through this interpretation, the quasi-static single-neutron POI is indeed seen to be the l₀ → 0 limit of a higher-order single-neutron POI-like structure (i.e., the outer solution) that in some sense includes knowledge of the past and future evolutionary state of ρ. From a physical standpoint, the quasi-static single-neutron POI features the assumption that the overall system macro-evolution encoded in t₀ appears infinitely long (and hence the associated dynamics fixed) in comparison to the neutron population micro-evolution encoded in l₀, in turn realizing p_∞(τ) as a series of static snapshots as extracted from Equation (51).

In reality $\mathcal{r}$ > 0 (however small it may be), as via Equation (17), there always exists a finite time scale inherent to neutron generation. The associated corrections to the quasi-static single-neutron POI indicated in Equation (43) are thus associated with finite neutron generation time effects. Further investigation of these effects and their consequences on various neutron survival probabilities is laid out in Section 4.

The Inner Solution

The outer solution associated with Equation (43) in hand, the complementary inner solution is devised in view of Equation (42). Where applicable, Equation (42) suggests an alternate approximation to Equation (38) wherein the time-derivative term is explicitly retained. In turn, the retention of the time-derivative term allows the inner solution to non-trivially satisfy Equation (39); that is

$$p_{\mathrm{s}}^{\left(\mathrm{i}\mathrm{n}\right)}\left({\tau }_{\mathrm{f}};\tau ={\tau }_{\mathrm{f}}\right)=1.$$

(51)

The inner solution is thus expected to be valid in a boundary layer near τ = τ_f. Given p_s⁽ⁱⁿ⁾~1 within this layer, and the typical realization β_j+1 << β_j (j = 2, 3, …) as may be inferred from Equation (18), when

$$\left|\rho \right|\ll {\beta }_2,$$

(52)

the inner solution satisfies the homogeneous Riccati or index-2 Bernoulli equation approximation to Equation (38) given by

$$\mathcal{r}{\displaystyle \frac{d{p}_{\mathrm{s}}^{\left(\mathrm{i}\mathrm{n}\right)}}{d\tilde{\tau }}}-{\beta }_2{\left(p_{\mathrm{s}}^{\left(\mathrm{i}\mathrm{n}\right)}\right)}^2=0,$$

(53)

the solution of which is subject to Equation (51)

$$p_{\mathrm{s}}^{\left(\mathrm{i}\mathrm{n}\right)}={\displaystyle \frac{1}{1+\frac{{\beta }_2}{\mathcal{r}}\left({\tau }_{\mathrm{f}}-\tau \right)}}.$$

(54)

Equation (53) features the notable property that no perturbation expansion is necessary in its construction; hence, the exact result appears in Equation (54). Unlike Equation (43), this result is correct to all orders in $\mathcal{r}$.

Moreover, for $\mathcal{r}\ll 1$ Equation (54) approaches zero as (τ − τ_f)⁻¹ near τ = τ_f, as expected, and consistent with the interpretation of p_s⁽ⁱⁿ⁾ as being valid in a boundary layer near this same location. Following Bender and Orszag [65], the time-duration or “thickness” δ of this boundary layer may be approximated according to

$$\delta \sim O\left(\mathcal{r}\right),$$

(55)

so that in the case of Equation (54),

$$\delta \sim {\displaystyle \frac{\mathcal{r}}{{\beta }_2\left({\tau }_{\mathrm{f}}-\tau \right)}}.$$

(56)

Equation (56) indicates that the time-duration of the boundary layer is linearly proportional to $\mathcal{r}$ and hence, via Equation (34), the finite neutron generation time scale l₀. As with the outer solution, near-τ_f corrections to the quasi-static single-neutron POI encoded in Equation (54) are thus directly associated with a non-zero neutron generation time effect. This effect arises due to the inclusion of a finite survival window but diminishes as $\mathcal{r}\to 0$ (i.e., even a finite survival window “appears infinite” to neutrons featuring ostensibly infinitesimal inter-reaction times).

Moreover, Equation (54) also features two essentially equivalent limits:

$$\underset{\mathcal{r}\to 0}{\lim }{p}_{\mathrm{s}}^{\left(\mathrm{i}\mathrm{n}\right)}=0,$$

(57)

which is an inherent property of a boundary layer, and

$$\underset{{\tau }_{\mathrm{f}}⟶\infty }{\lim }{p}_{\mathrm{s}}^{\left(\mathrm{i}\mathrm{n}\right)}=0,$$

(58)

a suggestive result that is most appropriately considered in tandem with the associated outer solution.

The Global Solution

A simple singularly perturbed/boundary layer approximation to the solution of Equations (38) and (39) is given by the envelope

$$p_{\mathrm{s}}\left({\tau }_{\mathrm{f}};\tau \right)=p_{\mathrm{s}}^{\left(\mathrm{o}\mathrm{u}\mathrm{t}\right)}\left(\tau \right)+p_{\mathrm{s}}^{\left(\mathrm{i}\mathrm{n}\right)}\left({\tau }_{\mathrm{f}};\tau \right),$$

(59)

or, with Equations (43) and (50),

$$p_{\mathrm{s}}\left({\tau }_{\mathrm{f}};\tau \right)=p_{\infty }\left(\tau \right)+\sum _{i=1}^M{\mathcal{r}}^i{p}_{\mathrm{s}}^{\left(i\right)}\left(\tau \right)+p_{\mathrm{s}}^{\left(\mathrm{i}\mathrm{n}\right)}\left({\tau }_{\mathrm{f}};\tau \right),$$

(60)

where p_∞ and the p_s⁽ⁱ⁾ are the solutions of Equations (48) and (49), and any additional higher-order members, and p_s⁽ⁱⁿ⁾ is given by Equation (54).

Equation (60) indicates that the solution of Equations (38) and (39) for the “fully dynamic” single-neutron survival probability may be approximated by a singular perturbation expansion around the associated quasi-static single-neutron POI, where the expansion parameter $\mathcal{r}$ is defined via Equation (34) as the ratio of the micro-to-macro physics evolutionary time scales. In this construction order-${\mathcal{r}}^i$ corrections to static snapshot behavior are encoded in the outer solution, and satisfaction of the final condition is ensured solely by the presence of the inner solution, which is exact in $\mathcal{r}$.

Equations (57) and (58) may now be revisited in view of Equation (60), thus illuminating some additional implications. Specifically,

$$\underset{\mathcal{r}\to 0}{\lim }{p}_{\mathrm{s}}=p_{\infty },$$

(61)

confirming that in the l₀ → 0 (or t₀ → ∞) limit the single-neutron survival probability indeed collapses to the quasi-static single-neutron POI.

As viewed from the standpoint of the inner solution, Equations (57) and (58) are functionally equivalent. However, when extended to Equation (60), the τ_f → ∞ limit instead yields

$$\underset{{\tau }_{\mathrm{f}}\to \infty }{\lim }{p}_{\mathrm{s}}=p_{\infty }+\sum _{i=1}^M{\mathcal{r}}^i{p}_{\mathrm{s}}^{\left(i\right)}.$$

(62)

Unlike Equation (61), in this limit, the single-neutron survival probability does not collapse to the quasi-static single-neutron POI. From a physical standpoint, although in this limit the dependence on the final time τ_f is explicitly removed, finite neutron generation time effects are still present and encoded in the summation appearing in Equation (62).

Overall, Equation (60) is a useful result in that it explicitly resolves phenomena surrounding both finite neutron generation time and final time effects inherent to the single-neutron survival probability. This boundary layer approach has the added advantage of essentially being built on the foundation of quasi-static single-neutron POI, which, as previously noted, is subject to thorough analysis within the existing literature. With Equation (60), the abundance of existing knowledge pertaining to solutions of Equation (48) is then readily extended to an associated dynamic formulation.

3.2. Correction Factors

The developments of Section 3 and Section 3.1 are also readily integrated into the various source-neutron survival probabilities defined in Section 2.1. Analogous to the realistic interpretations of l = l(t) and ρ = ρ(t) furnished in Equations (29) and (30),

$$s\left(t\right)\to s\left({\displaystyle \frac{t}{t_0}}\right),$$

(63)

so that with Equation (31) and the definition of the dimensionless source function S given by

$$S\left(\tau \right)=t_{0}s\left(\tau \right),$$

(64)

Equation (60) may now also be incorporated into the source-neutron survival probabilities defined in Section 2.1. With the various redefinitions of the time variable appearing throughout Section 3 and including Equations (60) and (64), Equations (27) and (28) become

$$\begin{array}{c}\mathrm{s}\mathrm{p}\mathrm{d}\mathrm{f}\left({\tau }_{\mathrm{f}};\tau \right) d\tau =\exp \left[-{\int }_{-\infty }^{\tau }S\left({\tau }^{\prime }\right)p_{\infty }\left({\tau }^{\prime }\right) d{\tau }^{\prime }\right]\\ \times \exp \left[-{\int }_{-\infty }^{\tau }S\left({\tau }^{\prime }\right){\displaystyle \sum _{i=1}^M}{\mathcal{r}}^i{p}_{\mathrm{s}}^{\left(i\right)}\left({\tau }^{\prime }\right) d{\tau }^{\prime }\right]\\ \times \exp \left[-{\int }_{-\infty }^{\tau }S\left(\tau \prime \right)p_{\mathrm{s}}^{\left(\mathrm{i}\mathrm{n}\right)}\left({\tau }_{\mathrm{f}};\tau \prime \right) d\tau \prime \right]S\left(\tau \right)\left[p_{\infty }\left(\tau \right)+{\displaystyle \sum _{i=1}^M}{\mathcal{r}}^i{p}_{\mathrm{s}}^{\left(i\right)}\left(\tau \right)+p_{\mathrm{s}}^{\left(\mathrm{i}\mathrm{n}\right)}\left({\tau }_{\mathrm{f}};\tau \right)\right] d\tau ,\end{array}$$

(65)

and

$$\begin{array}{c}\mathrm{s}\mathrm{c}\mathrm{d}\mathrm{f}\left({\tau }_{\mathrm{f}};\tau \right)=1-\exp \left[-{\int }_{-\infty }^{\tau }S\left({\tau }^{\prime }\right)p_{\infty }\left({\tau }^{\prime }\right) d{\tau }^{\prime }\right]\\ \times \exp \left[-{\int }_{-\infty }^{\tau }S\left({\tau }^{\prime }\right){\displaystyle \sum _{i=1}^M}{\mathcal{r}}^i{p}_{\mathrm{s}}^{\left(i\right)}\left({\tau }^{\prime }\right) d{\tau }^{\prime }\right]\\ \times \exp \left[-{\int }_{-\infty }^{\tau }S\left(\tau \prime \right)p_{\mathrm{s}}^{\left(\mathrm{i}\mathrm{n}\right)}\left({\tau }_{\mathrm{f}};\tau \prime \right) d\tau \prime \right],\end{array}$$

(66)

for the survival probability density function and survival cumulative distribution function, respectively.

The essential behavior of Equations (65) and (66) is further illuminated by the definitions

$$c^{\left(\mathrm{out}\right)}=\exp \left[-{\int }_{-\infty }^{\tau }S\left({\tau }^{\prime }\right)\sum _{i=1}^M{\mathcal{r}}^i{p}_{\mathrm{s}}^{\left(i\right)}\left({\tau }^{\prime }\right) d{\tau }^{\prime }\right],$$

(67)

$$c^{\left(\mathrm{in}\right)}=\exp \left[-{\int }_{-\infty }^{\tau }S\left(\tau \prime \right)p_{\mathrm{s}}^{\left(\mathrm{i}\mathrm{n}\right)}\left({\tau }_{\mathrm{f}};\tau \prime \right) d\tau \prime \right],$$

(68)

so that Equations (65) and (66) may be written as

$$\begin{array}{c}\mathrm{s}\mathrm{p}\mathrm{d}\mathrm{f}\left({\tau }_{\mathrm{f}};\tau \right) d\tau =c^{\left(\mathrm{i}\mathrm{n}\right)}{c}^{\left(\mathrm{o}\mathrm{u}\mathrm{t}\right)}\exp \left[-{\int }_{-\infty }^{\tau }S\left({\tau }^{\prime }\right)p_{\infty }\left({\tau }^{\prime }\right) d{\tau }^{\prime }\right]S\left(\tau \right)[p_{\infty }\left(\tau \right)\\ +{\displaystyle \sum _{i=1}^M}{\mathcal{r}}^i{p}_{\mathrm{s}}^{\left(i\right)}\left(\tau \right)+p_{\mathrm{s}}^{\left(\mathrm{i}\mathrm{n}\right)}\left({\tau }_{\mathrm{f}};\tau \right)] d\tau ,\end{array}$$

(69)

and

$$\mathrm{s}\mathrm{c}\mathrm{d}\mathrm{f}\left({\tau }_{\mathrm{f}};\tau \right)=1-c^{\left(\mathrm{i}\mathrm{n}\right)}{c}^{\left(\mathrm{o}\mathrm{u}\mathrm{t}\right)}\exp \left[-{\int }_{-\infty }^{\tau }S\left({\tau }^{\prime }\right)p_{\infty }\left({\tau }^{\prime }\right) d{\tau }^{\prime }\right] ,$$

(70)

respectively. Equations (69) and (70) reveal that c⁽ⁱⁿ⁾ and c^(out) may be interpreted as “correction factors” applied to

$$P_0^{\left(0\right)}\left(\tau \right)=\exp \left[-{\int }_{-\infty }^{\tau }S\left({\tau }^{\prime }\right)p_{\infty }\left({\tau }^{\prime }\right) d{\tau }^{\prime }\right],$$

(71)

which, in reference to Equations (26) and (28), may be interpreted as the quasi-static probability that no indefinitely persistent fission chains are sponsored by S at or before τ, or the quasi-static source-neutron extinction probability. This outcome is analogous to Equation (60): here, c⁽ⁱⁿ⁾ and c^(out) are associated with singular perturbative corrections to Equation (71) owing to both finite neutron generation time and final time effects. In the case of Equations (69) and (70), these corrections are seen to be multiplicative, owing to the presence of summation terms in the arguments of the relevant exponentials.

In view of their definitions given in Equations (67) and (68), c⁽ⁱⁿ⁾ and c^(out) fall in the range 0 < c^(in,out) ≤ 1. Consequently, compared to the quasi-static kernel, these corrections reduce the quasi-static source-neutron extinction probability given in Equation (71). Conversely, owing to these corrections, the survival cumulative distribution function given by Equation (70) assumes larger values than its purely quasi-static counterpart. This behavior is explored in more detail in Section 4.

Finally, Equations (69) and (70) may be evaluated in the τ_f → ∞ limit to obtain probabilities associated with indefinite survival. In this limit, Equation (58) holds so that Equation (68) becomes

$$\underset{{\tau }_{\mathrm{f}}\to \infty }{\lim }{c}^{\left(\mathrm{i}\mathrm{n}\right)}=1,$$

(72)

and Equation (69) becomes

$$\mathrm{s}\mathrm{p}\mathrm{d}\mathrm{f}\left(\tau \right) d\tau =c^{\left(\mathrm{o}\mathrm{u}\mathrm{t}\right)}\exp \left[-{\int }_{-\infty }^{\tau }S\left({\tau }^{\prime }\right)p_{\infty }\left({\tau }^{\prime }\right) d{\tau }^{\prime }\right]S\left(\tau \right)\left[p_{\infty }\left(\tau \right)+\sum _{i=1}^M{\mathcal{r}}^i{p}_{\mathrm{s}}^{\left(i\right)}\left(\tau \right)\right] d\tau ,$$

(73)

while Equation (70) becomes

$$\mathrm{s}\mathrm{c}\mathrm{d}\mathrm{f}\left(\tau \right)=1-c^{\left(\mathrm{o}\mathrm{u}\mathrm{t}\right)}\exp \left[-{\int }_{-\infty }^{\tau }S\left({\tau }^{\prime }\right)p_{\infty }\left({\tau }^{\prime }\right) d{\tau }^{\prime }\right].$$

(74)

In this case, Equations (73) and (74) obviously retain c^(out), the correction factor associated solely with finite neutron generation time effects.

4. Example

The fundamental properties of Equations (38) and (39) as inferred from boundary layer theory may be even further illuminated via exploration of a representative analytical example. One such example may be obtained through the prescription of both the reactivity function ρ = ρ(τ) and the maximum factorial moment index J.

For example, selecting

$$\rho ={\rho }_0\left({\displaystyle \frac{1}{1+{\tau }^2}}-\epsilon \right),$$

(75)

for ρ₀ > 0, captures a simple but broadly representative power burst model: when 0 < ε << 1, the excursion proceeds from a slightly subcritical state as τ → −∞ to criticality (ρ = 0) at

$${\tau }_{\mathrm{c}}=-\sqrt{{\displaystyle \frac{1}{\epsilon }}-1},$$

(76)

and subsequently attaining a maximum ρ~ρ₀ at τ = 0. The second half of the excursion during τ > 0 then mirrors the first half (see Figure 3).

The further prescription J = 2 (the quadratic approximation) affords some simplification of Equations (38) and (39) subject to Equation (75); while this approximation underestimates the contributions to fission chains of fissions releasing comparatively large numbers of neutrons, it has the benefit of retaining much of the phenomenological structure of Equations (38) and (39) in an analytically tractable manner [43,53,54,56,58,59,62,66,67,68]. Nonetheless, in this case, even the J = 2 approximation of Equations (38) and (39) subject to Equation (75) features a solution only in terms of quadratures. However, an approximate boundary layer solution for this model may be written in closed form. For example, with ρ given by Equation (75) and J = 2, the nontrivial solution of Equation (48) is given by

$$p_{\infty }={\displaystyle \frac{{\rho }_0}{{\beta }_2}}\left({\displaystyle \frac{1}{1+{\tau }^2}}-\epsilon \right).$$

(77)

Equation (77) features two zeroes, namely

$${\tau }_{\pm }^{\left(0\right)}=\pm \sqrt{{\displaystyle \frac{1}{\epsilon }}-1},$$

(78)

which are seen to be the critical times associated with Equation (75); Equation (77) thus predicts p_∞ < 0 before and after these time stamps. Therefore, leveraging the trivial solution of Equation (48) the quasi-static single-neutron POI is instead taken as

$$p_{\infty }=\{\begin{array}{cc}0,& \tau <{\tau }_{-}^{\left(0\right)}\\ {\displaystyle \frac{{\rho }_0}{{\beta }_2}}\left({\displaystyle \frac{1}{1+{\tau }^2}}-\epsilon \right),& {\tau }_{-}^{\left(0\right)}\le \tau \le {\tau }_{+}^{\left(0\right)}\\ 0,& \tau >{\tau }_{+}^{\left(0\right)}\end{array}.$$

(79)

With ρ given by Equation (75) and J = 2, the first-order correction to Equation (79) is given by the solution of Equation (49),

$$p_{\mathrm{s}}^{\left(1\right)}={\displaystyle \frac{2\tau }{{\beta }_2\left(1+{\tau }^2\right)\left(\epsilon {\tau }^2+\epsilon -1\right)}},$$

(80)

or, since ε << 1,

$$p_{\mathrm{s}}^{\left(1\right)}=-{\displaystyle \frac{2\tau }{{\beta }_2\left(1+{\tau }^2\right)}}-{\displaystyle \frac{2\tau }{{\beta }_2}}\epsilon -{\displaystyle \frac{2\tau \left(1+{\tau }^2\right)}{{\beta }_2}}{\epsilon }^2+\dots ,$$

(81)

whence, to first order (i.e., in both $\mathcal{r}$ and ε, suppressing cross-terms), the outer solution is thus given by

$$p_{\mathrm{s}}^{\left(\mathrm{o}\mathrm{u}\mathrm{t}\right)}={\displaystyle \frac{{\rho }_0}{{\beta }_2}}\left({\displaystyle \frac{1}{1+{\tau }^2}}-\epsilon \right)-{\displaystyle \frac{2\tau \mathcal{r}}{{\beta }_2\left(1+{\tau }^2\right)}}.$$

(82)

Equation (82) again features two zeroes, in this case given by

$${\tau }_{\pm }^{\left(1\right)}={\displaystyle \frac{-\mathcal{r}\pm \sqrt{{\mathcal{r}}^2+\epsilon \left(1-\epsilon \right){\rho }_0^2}}{\epsilon {\rho }_0}}.$$

(83)

Comparison of Equations (78) and (83) indicates τ₋⁽¹⁾ < τ₋⁽⁰⁾ and τ₊⁽¹⁾ < τ₊⁽⁰⁾ for $\mathcal{r}$ > 0, ρ₀ > 0, and ε > 0. Moreover, Equation (83) again indicates p_s^(out) < 0 before and after these time stamps; hence the outer solution is taken as

$$p_{\mathrm{s}}^{\left(\mathrm{o}\mathrm{u}\mathrm{t}\right)}=\{\begin{array}{cc}0,& \tau <{\tau }_{-}^{\left(1\right)}\\ {\displaystyle \frac{{\rho }_0}{{\beta }_2}}\left({\displaystyle \frac{1}{1+{\tau }^2}}-\epsilon \right)-{\displaystyle \frac{2\tau \mathcal{r}}{{\beta }_2\left(1+{\tau }^2\right)}},& {\tau }_{-}^{\left(1\right)}\le \tau \le {\tau }_{+}^{\left(1\right)}\\ 0,& \tau >{\tau }_{+}^{\left(1\right)}\end{array}.$$

(84)

Finally, with Equations (54) and (84), a first-order boundary layer approximation to the solution of Equations (38) and (39) is given by

$$p_{\mathrm{s}}=\{\begin{array}{cc}0,& \tau <{\tau }_{-}^{\left(1\right)}\\ {\displaystyle \frac{{\rho }_0}{{\beta }_2}}\left({\displaystyle \frac{1}{1+{\tau }^2}}-\epsilon \right)-{\displaystyle \frac{2\tau \mathcal{r}}{{\beta }_2\left(1+{\tau }^2\right)}}+{\displaystyle \frac{1}{1+\frac{{\beta }_2}{\mathcal{r}}\left({\tau }_{\mathrm{f}}-\tau \right)}},& {\tau }_{-}^{\left(1\right)}\le \tau \le {\tau }_{+}^{\left(1\right)}\\ {\displaystyle \frac{1}{1+\frac{{\beta }_2}{\mathcal{r}}\left({\tau }_{\mathrm{f}}-\tau \right)}},& {\tau }_{+}^{\left(1\right)}<\tau \le {\tau }_{\mathrm{f}}\end{array}.$$

(85)

Equations (79), (84) and (85), and (for reference) a numerical solution of Equations (38) and (39) are depicted in Figure 4a–c. The numerical solution of Equations (38) and (39) depicted in Figure 4a–c (and figures to follow based on these results) is obtained using the Wolfram Mathematica 14.2 default numerical solution package for first-order ODEs, which employs a multistep method with adaptive switching between nonstiff (Adams) and stiff (Gear) solvers; additional details on this solution package are provided in Reference [73]. These figures starkly illustrate a variety of trends:

• The quasi-static single-neutron POI exactly mirrors the shape of the time-dependent reactivity depicted in Figure 3 and thus encodes static snapshot (i.e., indefinite fission chain persistence) behavior.
• In the proximity of the quasi-static curve (i.e., within the outer solution), higher-order or finite neutron generation time effects induce both early-time and late-time probability “tails”. This phenomenon is also observed by both Ramsey and Hutchens and Hill [12,43] and is associated with a neutron “waiting” or “dwell” time that is itself a manifestation of the finite window between successive neutron generation events. In a sense, the finite neutron generation time (i.e., l₀) enables neutrons to “anticipate” the onset of both higher and lower reactivity states, with a commensurate effect on fission chain persistence indexed at those times.
• Consequently, given that the quasi-static single-neutron POI does not account for this finite neutron generation time effect, it underestimates the survival probability associated with the early-time tail and overestimates the survival probability associated with the late-time tail.
• The self-consistency statement of the final condition given in Equation (39) is encoded within the inner solution. Any earlier-time effects associated with this condition manifest exclusively within the inner solution and diminish outside of the indicated boundary layer.

With one of Equations (79), (84), or (85) in hand, the various source-neutron probabilities appearing in Section 3.2 may also be resolved. For example, given the source function parameterized by

$$S\left(\tau \right)=S_0=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}. ,$$

(86)

for dimensionless source amplitude S₀, the quasi-static source-neutron extinction probability given in Equation (71) becomes

$$P_0^{\left(0\right)}=\{\begin{array}{c}\begin{array}{cc}1,& \tau <{\tau }_{-}^{\left(0\right)}\\ \exp \left({\displaystyle \frac{{\rho }_0{S}_0}{{\beta }_2}}{\left[{\tan }^{-1}\tau \prime -\epsilon \tau \prime \right]}_{{\tau }_{-}^{\left(0\right)}}^{\tau }\right),& {\tau }_{-}^{\left(0\right)}\le \tau \le {\tau }_{+}^{\left(0\right)}\end{array}\\ \begin{array}{cc}\exp \left({\displaystyle \frac{{\rho }_0{S}_0}{{\beta }_2}}{\left[{\tan }^{-1}\tau \prime -\epsilon \tau \prime \right]}_{{\tau }_{-}^{\left(0\right)}}^{{\tau }_{+}^{\left(0\right)}}\right),& \tau >{\tau }_{+}^{\left(0\right)}\end{array}\end{array} ,$$

(87)

and hence Equations (69) and (70) yield the associated quasi-static survival probability density function and survival cumulative distribution function, respectively. However, given that these probabilities are formed using only the quasi-static single-neutron POI p_∞, they may also be referred to as the quasi-static initiation density function and source-neutron POI, respectively.

Similarly, with Equations (85) and (86), Equations (67), (68), and (71) may be evaluated to yield

$$P_0^{\left(1\right)}=\{\begin{array}{cc}1,& \tau <{\tau }_{-}^{\left(1\right)}\\ c^{\left(\mathrm{o}\mathrm{u}\mathrm{t}\right)}{c}^{\left(\mathrm{i}\mathrm{n}\right)}\exp \left({\displaystyle \frac{{\rho }_0{S}_0}{{\beta }_2}}{\left[{\tan }^{-1}\tau \prime -\epsilon \tau \prime \right]}_{{\tau }_{-}^{\left(1\right)}}^{\tau }\right),& {\tau }_{-}^{\left(1\right)}\le \tau \le {\tau }_{+}^{\left(1\right)}\\ c^{\left(\mathrm{o}\mathrm{u}\mathrm{t}\right)}{c}^{\left(\mathrm{i}\mathrm{n}\right)}\exp \left({\displaystyle \frac{{\rho }_0{S}_0}{{\beta }_2}}{\left[{\tan }^{-1}\tau \prime -\epsilon \tau \prime \right]}_{{\tau }_{-}^{\left(1\right)}}^{{\tau }_{+}^{\left(1\right)}}\right),& \tau >{\tau }_{+}^{\left(1\right)}\end{array} ,$$

(88)

where

$$c^{\left(\mathrm{o}\mathrm{u}\mathrm{t}\right)}=\{\begin{array}{cc}{\left[{\displaystyle \frac{1+{\tau }^2}{1+{\left({\tau }_{-}^{\left(1\right)}\right)}^2}}\right]}^{{\displaystyle \frac{\mathcal{r}{S}_0}{{\beta }_2}}},& {\tau }_{-}^{\left(1\right)}\le \tau \le {\tau }_{+}^{\left(1\right)}\\ {\left[{\displaystyle \frac{1+{\left({\tau }_{+}^{\left(1\right)}\right)}^2}{1+{\left({\tau }_{-}^{\left(1\right)}\right)}^2}}\right]}^{{\displaystyle \frac{\mathcal{r}{S}_0}{{\beta }_2}}},& \tau >{\tau }_{+}^{\left(1\right)}\end{array},$$

(89)

$$c^{\left(\mathrm{i}\mathrm{n}\right)}={\left({\displaystyle \frac{\frac{\mathcal{r}}{{\beta }_2}+{\tau }_{\mathrm{f}}-\tau }{\frac{\mathcal{r}}{{\beta }_2}+{\tau }_{\mathrm{f}}+{\tau }_{-}^{\left(1\right)}}}\right)}^{{\displaystyle \frac{\mathcal{r}{S}_0}{{\beta }_2}}}.$$

(90)

whence Equations (69) and (70) again yield the associated quasi-static survival probability density function and survival cumulative distribution function, respectively. As noted throughout Section 3.2, these probabilities feature finite neutron generation time and final time corrections to their quasi-static counterparts obtained using Equation (87).

In the τ_f → ∞ limit Equation (90) indeed realizes Equation (72). However, in this limit, Equations (87) and (89) remain unchanged and may thus be used to once again construct the quasi-static survival probability density function and survival cumulative distribution function using Equations (69) and (70), respectively. These results are plotted in Figure 5a,b and Figure 6a,b, which reveal a variety of additional trends:

• Both the quasi-static initiation probability density function and quasi-static source-neutron POI are zero for τ < τ_c, the critical time as defined in Equation (76). For the same reasons as delineated for the quasi-static single-neutron POI, within this approximation there is zero probability that the progeny of source neutrons introduced into a static subcritical system can survive indefinitely or approach an infinite number.
• In the proximity of the quasi-static curve (i.e., within the outer solution) higher-order effects again drag the probability fields leftward, most notably resulting in a non-zero probability tail during late-time subcriticality. This tail extends from τ₋⁽¹⁾ ≤ τ ≤ τ_c, and its magnitude within this range is proportional to c^(out) [and hence, via Equation (89), both $\mathcal{r}$ and S₀]. During this interval, the (zero) quasi-static initiation probability density function and quasi-static source-neutron POI underestimate the probabilities owing to the finite dwell time between neutron generation events.

Given the shape of the single-neutron survival probability curve as observed in Figure 4a–c, the survival cumulative distribution function saturates at τ = τ₊⁽¹⁾, attaining the value 1 − P₀⁽¹⁾ extracted from Equation (88) with c⁽ⁱⁿ⁾ = 1. From Equation (88) it may be inferred that the survival cumulative distribution function may, in some cases, effectively saturate near unity for τ < τ₊⁽¹⁾, principally depending on ρ₀ and S₀ as indicated.

5. Summary

Unlike analogous entities encountered in the context of deterministic neutron transport theories, the neutron survival probability equations [e.g., Equations (5)–(7) or (38) and (39)] endemic to stochastic neutronics are inherently nonlinear. While this unfamiliar property often induces significant challenges for the neutron transport theorist, it also provides an opportunity for the deployment of analysis techniques more commonly encountered in fields traditionally dominated by nonlinear equations. Two such examples appearing in Section 3 are:

• Dimensional analysis: Broadly speaking, dimensional analysis of a differential equation structure or broader mathematical model results in the identification of both dimensionless variables and dimensionless numbers that are somehow key to the formulation, solution, and interpretation of the underlying physics. In the case of Equations (38) and (39), the relevant dimensionless numbers $\mathcal{r}$ and τ_f are given by Equations (34) and (35), respectively.
• Boundary layer theory: The likely magnitude of the time-scale ratio $\mathcal{r}$ appearing in Equation (38) suggests a perturbative approach to its approximate solution, which, given the placement of $\mathcal{r}$ therein, must be of the singular variety. The boundary layer approach pursued in Section 3.1 is one such manifestation of a singular perturbation theory, which employs $\mathcal{r}$ as an expansion parameter within an outer solution region. In this construction, the boundary layer—wherein the associated final condition must be satisfied—is a comparatively narrow region joined to the outer solution and satisfies a distinct limiting form of the single-neutron survival probability equation.

Synthesis of these two outcomes results in Equation (60). The benefit of this novel approach is the construction of a perturbative solution wherein the quasi-static single-neutron POI appears as the lowest-order member: both the dimensionless numbers $\mathcal{r}$ and τ_f appear in corrections to this fundamental and well-known structure. In this result $\mathcal{r}$ is revealed to represent both finite neutron generation time effects associated with reactivity transients and the width of the boundary layer, and τ_f the location of the boundary layer.

Given the definitions of $\mathcal{r}$ and τ_f appearing in Equations (34) and (35), respectively, a clear connection may be made to the guiding physical mechanisms underlying neutron survival phenomena. Indeed, both governing parameters are time-scale ratios: $\mathcal{r}$ being the ratio of micro- to macro-physics evolution occurring within a reactor transient scenario, and hence its dynamical evolution; and τ_f being the ratio of the final survival time to the macro-physics evolution of that same scenario, and hence its terminal characteristics. Together, via the incorporation of finite neutron generation time and survival time effects, these two ratios lend additional physical structure to the (in some sense) naïve interpretation of fission chain persistence afforded by the quasi-static POI interpretation.

Indeed, in view of Equations (38) and (39) [or, more broadly, Equations (5)–(7)], the “fully dynamic” single-neutron survival probability equation—and both the finite neutron generation time and final condition effects it encodes—is inherently of higher fidelity than its analogous quasi-static counterpart. Consequently, and in addition to the clear physical interpretation afforded by its approximate representations disseminated in Section 3, the approximate solution of Equations (38) and (39) produced via boundary layer theory may be interpreted as a higher-fidelity representation of neutron survival phenomena than that encoded in the quasi-static single-neutron POI and its associated probabilities. This outcome is likely of consequence in a variety of practical settings, including the small neutron population scenarios disseminated in Section 1. In these contexts, higher-fidelity instantiations of neutron survival phenomena have the potential of more accurately influence disparate applications, including detector response phenomena, safety margins, and reactor startup procedures, to name a few.

Recommendations for Future Study

In the space of singular perturbation problems (see, for example, Bender and Orszag [65]), Equations (38) and (39) are relatively simple in structure; consequently, their approximate solution using the boundary layer method is subject to a variety of nuances worthy of further exploration. For example,

• The outer solution given by Equation (43) features no differential equation solutions and hence potentially manifests some pathologies. For example, the solution of Equation (79) to infinite-order in ε [i.e., Equation (80)] is singular when $\tau =\pm {\tau }_{\mathrm{c}}$. This behavior is essentially the cost of the infinite-order accuracy in ε as encoded in Equation (80); in addition to being patently inconsistent with the finite-order accuracy associated with the additional power expansion in $\mathcal{r}$, this behavior is undesirable on the grounds of physical realism and suggests the possibility of an alternate solution strategy that is uniformly valid over [−τ_c, τ_c].

The Taylor series representation of Equation (80) given by Equation (81), valid for ε < 1, represents one such uniformly valid approximate solution of Equation (49); in particular, Equation (82) is self-consistent in that it is valid to first order in both $\mathcal{r}$ and ε. Within this paradigm, higher-order corrections in ε should be accompanied by commensurate higher-order corrections in $\mathcal{r}$, which will yield a series of consistent, uniformly valid approximate solutions of Equation (38). While coming at the cost of infinite-order accuracy in ε, these approximate outer solutions have the obvious benefit of remaining non-singular, so that the error associated with them remains bounded over [−τ_c, τ_c].

• The contrived properties of the solutions of Equations (48) and (49), and any higher-order members notwithstanding, the outer solution as formulated cannot feature arbitrary integration constants. Therefore, there is no opportunity for asymptotic matching to the inner solution given by Equation (54).

Indeed, among other candidates, the simple global solution given by Equation (60) is the sum or envelope, wherein the outer and inner solutions are fully decoupled. In contrast to the variety of boundary layer solutions disseminated by Bender and Orszag [65], Equation (60) is thus prone to “double counting” in solution regions wherein both Equations (43) and (54) are non-negligible; see, for example, Figure 4a–c. However, in view of Equation (85), both the extent and magnitude of the double-count effect are clearly controlled by the parameters $\mathcal{r}$ and τ_f; notably, for either $\mathcal{r}\to 0$ or ${\tau }_{\mathrm{f}}\to \infty$ the overlap, while non-zero, decreases.

The envelope solution given by Equation (85) features the important property of allowing for the explicit quantification of the overlap in terms of both $\mathcal{r}$ and τ_f. For example, for a scenario-dependent selection of τ_f, the error incurred within the overlap region is solely controlled by the $\mathcal{r}$ parameter, thus setting requirements on the outer solution expansion order employed in Equation (49). Alternatively, given the essential decoupling between Equations (43) and (54), as observed in Equations (62), (73), and (74) the τ_f → ∞ limit formally eliminates the overlap and hence yields no effect on the outer solution.

These outcomes collectively suggest the deeper analysis of Equations (38) and (39) as a singular perturbation problem. Beyond any attempts at alternate boundary layer approximations, multiple-scale analysis or renormalization group techniques [65,74,75] may prove useful in obtaining both asymptotically matched and uniformly valid approximate solutions of Equations (38) and (39). Further still, generalizing beyond the dimensional analysis provided in Section 3, Equations (38) and (39) might also be made subject to a formal program of Lie symmetry analysis [76,77,78,79] in the interest of identifying additional geometric and structural properties. Recent work along all these lines has been performed for closely related nonlinear ODEs appearing in both the theory of oscillation and penetration mechanics [80,81].

Another important theme arising from these considerations is the behavior of solutions of Equations (38) and (39) with respect to the final time τ_f. The approximate boundary layer solution disseminated in Section 3 (and via example in Section 4) is valid under a specific set of circumstances, namely τ_f >> 1. In view of Equation (35), in this case, the final time t_f occurs far beyond the termination of the supercritical reactivity excursion parameterized by t₀, and Equation (52) holds. Equations (51) and (53) thus manifest as the distinct limiting form of Equations (38) and (39) within the boundary layer and give rise to the archetypal single-neutron survival probability solution structure depicted in, for example, Figure 4a–c (and as computationally exhibited by Trahan and Jia et al. [39,82]; this fundamental structure may serve as a basis for code verification activities using Monte Carlo or other high-fidelity computational tools).

This construction–and its consequences in the τ_f → ∞ limit, namely Equations (62), (73) and (74)–is by no means unique. Indeed, the cogent, scenario-dependent placement of the final time (e.g., with respect to short-duration transients or in the context of pulsed reactors) and the consequent sensitivity of solutions of Equations (5)–(7) represent a significant outstanding problem in neutron survival phenomena. Further investigation of these matters using a combined program of analytical and computational study remains an important avenue of future exploration.

Whether conducted using boundary layer theory or using some other strategy, an analysis analogous to that conducted in Section 3 may be extended from the single-neutron survival probability equation to closely related entities including the generating function equation given in Equation (15) or neutron population moments equations extracted from it (see, for example, Bell, Hill, or O’Rourke et al. [42,43,54]). Moreover, the texts by O’Malley and Johnson [83,84] on the deployment of singular perturbation methods within systems of ODEs or even partial differential equations suggest potentially viable extensions of the work presented herein to higher-fidelity mathematical models, including diffusion theories, multi-group formulations, and variations in any such models featuring delayed neutron precursor effects. Combined with an analysis like that performed by Prinja and Souto [85] (see also O’Rourke et al. [54]), any such program of study will ultimately provide a similar level of theoretical insight into the broader neutron number probability distribution itself, thus setting the stage for complementary analysis in the context of higher-fidelity neutron transport settings.