Percolation properties of the neutron population in nuclear reactors Benjamin Dechenauxand Thomas Delcambre Institut de Radioprotection et de S uret e Nucl eaire IRSN

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Percolation properties of the neutron population in nuclear reactors

Benjamin Dechenaux∗and Thomas Delcambre

Institut de Radioprotection et de Sˆuret´e Nucl´eaire (IRSN)

PSN-RES/SNC/LN, F-92260, Fontenay-aux-Roses, France.

Eric Dumonteil

Institut de Recherche sur les Lois Fondamentales de l’Univers

CEA, Universit´e Paris-Saclay, 91191 Gif-sur-Yvette, France

(Dated: October 6, 2022)

Abstract: Reactor physics aims at studying the neutron population in a reactor core under the inﬂuence of

feedback mechanisms, such as the Doppler temperature eﬀect. Numerical schemes to calculate macroscopic

properties emerging from such coupled stochastic systems however require to deﬁne intermediate quantities (e.g.

the temperature ﬁeld), which are bridging the gap between the stochastic neutron ﬁeld and the deterministic

feedback. By interpreting the branching random walk of neutrons in ﬁssile media under the inﬂuence of a

feedback mechanism as a directed percolation process and by leveraging on the statistical ﬁeld theory of birth

death processes, we will build a stochastic model of neutron transport theory and of reactor physics. The critical

exponents of this model, combined to the analysis of the resulting ﬁeld equation involving a fractional Laplacian

will show that the critical diﬀusion equation cannot adequately describe the spatial distribution of the neutron

population and shifts instead to a critical super-diﬀusion equation. The analysis of this equation will reveal

that non-negligible departure from mean ﬁeld behavior might develop in reactor cores, questioning the attainable

accuracy of the numerical schemes currently used by the nuclear industry.

I. INTRODUCTION

At their inception in the 30’s, Monte Carlo algorithms

intended to describe the neutron transport in ﬁssile me-

dia [1, 2] in support of the ﬁrst applications of nu-

clear power, might it be for the design of the ﬁrst nu-

clear weapons or in support of nuclear energy produc-

tion. Since then, the nuclear energy industry has used

the so-called Monte Carlo criticality codes as reference

numerical schemes to solve the linear Boltzmann equa-

tion. Nuclear safety demonstrations in particular rely

on a Monte Carlo solving of this equation with few, if

any, hypotheses, while the use of deterministic codes is

preferred for an approximate but fast -and even online-

solving of a simpliﬁed version of this equation: the two

(energy) group critical diﬀusion equation. In both cases,

the aim of such neutronics codes [3] is to calculate the

neutron spatial distribution in the context of mean-ﬁeld

hypotheses. Indeed, even at the startup of a nuclear re-

actor (i.e. when the neutron densities are small), such

mean-ﬁeld equations are believed to accurately describe

the average behavior of the stochastic regime. Though, in

the power regime of the reactor, stabilizing eﬀects come

into action. For example, local power excursions and po-

tential sudden rise of local neutron densities are tempered

by the Doppler broadening of the neutron cross-sections

(giving the probabilities that a neutron at a given energy

induce a particular reaction on a nuclei): the quantum

resonances of the neutron-nuclei system are broadened

by the thermal agitation of the nuclei from which re-

sults a modiﬁcation of the neutron capture cross section.

The study of the properties of neutron transport in ﬁssile

∗benjamin.dechenaux@irsn.fr

media subject to feedback mechanisms is called ’reactor

physics’. Reactor physics numerical solvers hence cou-

ple the mean-ﬁeld equations of neutronics to thermal or

thermal-hydraulics solvers via the deﬁnition of interme-

diate quantities (such as the temperature scalar ﬁeld) to

extract macroscopic measurable quantities characteriz-

ing the neutron population (such as local neutron ﬂuxes

measured by ﬁssion rate chambers for example).

The ﬂuctuations of the neutron population were also

largely investigated in the past ﬁfty years [4–6], while

being assumed to be of marginal concern for nuclear

safety, since they vanish as the core power increases.

However, a few years ago, this paradigm was questioned

[7–9] by the study of a neutronics toy-model based on

the celebrated branching Brownian motion [10], that

couples the diﬀusive random walk of neutrons [11],

mimicking their transport in the Brownian regime,

to a Galton-Watson birth-death process, reproducing

the induced ﬁssion reaction with variable number of

outgoing neutrons. Indeed, the ﬁrst moments of the

master equation of this model revealed that the spatial

correlations within the neutron population might in

some cases diverge, invalidating the use of the mean-ﬁeld

equation. From a physical point of view, this clustering

phenomenon (already characterized at the time in

the theory of population ecology [12–14]) causes the

emergence of spatial patterns in the neutron gas. While

always present and increasing with time in 1 and 2

dimensional systems, the spatial correlations however

saturate in 3 dimensions. A dedicated experiment

taking place at the Rensselaer Poytechnic Institute

was designed and took place in 2018 to characterize

such spatio-temporal correlations aﬀecting the neutron

distributions. This experiment revealed the existence

of such correlations and triggered questions relative to

arXiv:2210.02413v1 [cond-mat.stat-mech] 5 Oct 2022

their persistence in the power regime of reactor physics

where feedbacks occur [15].

In this paper, leveraging on the seminal approach of

Janssen and De Dominicis [16, 17], we will build a sim-

pliﬁed model of reactor physics to investigate the validity

of mean ﬁeld equations in the power regime of nuclear

reactors. In that aim, a ﬁeld theoretic formulation of

neutron physics will be built in Section 2. While being

0 dimensional, this formulation will then be exploited in

Section 3 so as to take into account the principal features

of neutronics such as the calculation of ﬂuctuations and

correlations of the neutron ﬁeld or the eﬀects of an ex-

ternal neutron source. It will also be extended in order

to add the eﬀect of delayed neutrons in Section 4 and to

take into account spatial phenomena in Section 5. Along

the path, we will highlight the close relationship between

the response functional formalism of Janssen and De Do-

minics with the Doi and Peliti ﬁeld theory [18, 19] which

lend itself to an exact microscopic interpretation, thus

strengthening the use of both approaches in neutronics.

Finally, in Section 6, we will add a simpliﬁed model of

thermal feedback, showing that the neutron gas in a nu-

clear reactor operated at criticality in the power regime

can be described as a phase transition belonging to the

directed percolation universality class. The calculation of

the critical exponents at the percolation threshold using

the renormalization group will in return allow to specify

the mean ﬁeld equation, formulated in terms of fractional

Laplacian. We will in particular show that in the directed

percolation model, the critical diﬀusion equation shifts to

a critical super-diﬀusion equation, whose stochastic gen-

erators are L´evy ﬂights, ultimately dynamically modify-

ing the medium properties in which neutrons are prop-

agating. The analysis of the fundamental mode of the

associated fractional Laplacian using an approximate for-

mula will allow to quantify the diﬀerences between this

formal approach and classical coupling schemes, thus set-

ting an accuracy limit on all actual numerical solvers of

reactor physics.

II. A FIELD THEORETIC FORMULATION OF

NEUTRONICS

Branching processes are among the simplest models

capable of characterizing the phenomenology of neutrons

evolving in ﬁssile materials [5, 6]. In their simplest ex-

pression, these models approximate the path of neutrons

in matter as a stochastic process, in which they are sub-

ject to random extinction or reproduction events, occur-

ring at constant average rates. They are also known to

display a second order phase transition between an active

and an inactive state [10]. This transition is at the core

root of nuclear reactors operations and corresponds, in

this context, to the appearance of self-sustained chain re-

actions: the ’percolation threshold’ where neutron chains

survive indeﬁnitely is precisely the point at which nuclear

reactors do operate.

Field theoretic method have long been used in statis-

tical and condensed matter physics for their eﬃciency

in the study of the intricate behavior of systems under-

going phase transitions. They indeed prove particularly

adapted to unravel the universal properties and scaling

behaviors in the vicinity of the phase transition’s criti-

cal point. Along this line of thought, the present paper

ought to demonstrate that ﬁeld theory also provides an

adapted framework for neutronics, allowing notably to

unravel highly non-trivial behaviors of the neutron pop-

ulation close to criticality.

The derivation of a ﬁeld-theoretic version of neutron-

ics followed in this paper is based on the seminal work

of Janssen [16] and De Dominicis [17]. Starting from

the mean ﬁeld equation of the problem, to which is sup-

plemented a random noise source term (as is commonly

done to study the stability of nuclear reactors [4]), the

neutron population evolving in a nuclear reactor can be

decoupled into two components:

•a smoothly varying neutron ﬁeld N(t). It corre-

sponds to the instantaneous number of neutrons

that can be measured in a reactor (proportional

to the operating power for instance) and it is only

sensitive to the global reactivity ρof the reactor;

•a random noise source η(t), arising from various

perturbations (such as the stochasticity of the in-

duced ﬁssion reactions that have a random number

of outgoing neutrons, or such as assembly vibra-

tions) which are generally driven by unknown or at

least unspeciﬁed phenomena.

In the simplest model of the branching process asso-

ciated to neutrons evolving in an idealized, inﬁnite and

homogeneous reactor, such a decoupling transforms the

mean ﬁeld equation into a Stochastic Diﬀerential Equa-

tion (SDE), whose form is given by

dN(t)

dt =ρN(t) + η(t).(1)

Any macroscopic observable quantity O[N] is insen-

sitive to the rapidly varying and low lying noise term.

As such, physical observables must be averaged over all

possible conﬁgurations of the noise. This is formally

achieved by performing a functional integration over the

noise probability functional P[η(t)],

hO[N]i ∝ ZDηO[N]P[η].(2)

The development of the ﬁeld-theoretic model starts by

noting that the Nappearing under the integral sign of

Eq. 2 is constrained to be a solution of Eq. 1. The con-

straint (valid for all t),

C[N] = dN(t)

dt −ρ N(t)−η(t)= 0,(3)

can be enforced with the help of a functional version of

the resolution of the identity [20],

1 = ZDNY

δ(C[N]) .(4)

Performing a (functional) Fourier transform, one can

write

1 = ZD[ie

N]ZDN e−Rdt e

N(t)C[N]

=ZD[ie

N]ZDN e−Rdt e

N(t)(dN(t)

dt −ρ N(t)−η(t)),(5)

where the purely imaginary auxiliary ﬁeld e

N(t), known

as the Martin-Siggia-Rose response ﬁeld [21], has been

introduced. Inserting this Fourier transformed identity

back into Eq. 2, one obtains

hO[N]i ∝ ZD[ie

N]ZDNne−Re

N(t)(d

dt −ρ)N(t)dt

× O[N]×ZDη eRe

N(t)η(t)dtP[η].(6)

To go further, one needs to integrate over ηand thus

specify the probability functional to which it is associ-

ated. In the following, it will be supposed that at any

given moment in time, the noise follows a Gaussian prob-

ability distribution

P[η]∝e−1

4Rη(t)Γ−1η(t)dt (7)

with zero mean hη(t)i= 0 and a covariance given by,

hη(t)η(t0)i= 2Γ(N)δ(t−t0).(8)

Note that an explicit dependence of the variance Γ(N)

on the global power level (i.e. N(t)) is to be expected on

physical ground : the expected noise in a reactor indeed

depends on the operating power level. The integral over

the noise can now be straightforwardly evaluated as

ZDη eRe

N(t)η(t)dtP[η]∝eΓe

N2.(9)

Inserting this result back into Eq. 6, one ﬁnally arrives

at an expression that solely depends on the variance of

the noise probability distribution

hO[N]i=NZD[ie

N]ZDNO[N]e−S[N, e

N],(10)

where both a normalisation factor Nand the so-called

response functional have been introduced, this last func-

tional being deﬁned through

S[N, e

N] = Ze

N(t)d

dt −ρN(t)−e

N(t)Γ e

N(t)dt.

(11)

It often reveals convenient to include in Eq. 3 an initial

condition of the type N(t0) = N0. The initial condition

is then included directly into the response functional and

it will be taken into account naturally in the further cal-

culation of observables. One can show that the inclusion

of an initial condition in the response functional gives

S[N, e

N] = Ze

N(t)d

dt −ρN(t)

−e

N(t)δ(t−t0)N0−e

N(t)Γ e

N(t)odt. (12)

It can then be demonstrated that the net eﬀect of

this supplementary term appearing in the response

functional is to multiply each quantity evaluated at the

initial time by the factor N0. In the remainder of the

paper, only the latter requirement will be retained and

the terms corresponding to the initial conditions will be

systematically omitted in all of the response functionals

that will be considered.

Equations 10–12 form the ﬁeld-theoretic version of the

branching process. It presents itself under the form of

a (euclidean) path integral. The further calculation of

observable quantities can now be addressed thanks to the

powerful machinery developed in Quantum Field Theory

(QFT) and statistical mechanics.

III. CALCULATION OF OBSERVABLES

It can generally safely be assumed that in a nuclear

reactor, the random power ﬂuctuations driven by random

noise are much weaker than the actual operating power

level, i.e. the noise term is small. This suggests a strategy

to extract sensible results from the response functional

formalism. One can expand the function Γ(N) into the

Taylor series

Γ(N) = λ0+λ1N+λ2N2+..., (13)

with small λi’s. Under this form, the noise term appear-

ing in the response functional can be treated perturba-

tively, just as one does with an interaction potential in

quantum mechanics. This perturbative treatment and

the calculation of observables of the type given in Eq. 10

is most conveniently represented under the form of Feyn-

man diagrams [22, 23].

The observables derived from the response functional

will all depend on the parameters λi, which play here the

same role as the interaction coupling constants in con-

ventional QFT. The precise meaning of these parameters

and in particular their microscopic origin is, however,

left unspeciﬁed. In this respect, the response functional

formalism can be viewed as a purely phenomenological

approach: the precise interpretation of the coupling pa-

rameters is to be sought either a posteriori or from an

external knowledge of the problem.

One might thus rightly be worried that the calcula-

tion of observables relies upon an inﬁnite power series

expansion, whose structure and convergence depends on

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PercolationpropertiesoftheneutronpopulationinnuclearreactorsBenjaminDechenauxandThomasDelcambreInstitutdeRadioprotectionetdeS^ureteNucleaire(IRSN)PSN-RES/SNC/LN,F-92260,Fontenay-aux-Roses,France.EricDumonteilInstitutdeRecherchesurlesLoisFondamentalesdel'UniversCEA,UniversiteParis-Saclay,91191Gif...

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Percolation properties of the neutron population in nuclear reactors Benjamin Dechenauxand Thomas Delcambre Institut de Radioprotection et de S uret e Nucl eaire IRSN

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