Annals of Nuclear Energy 63 (2014) 276–284

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Annals of Nuclear Energy

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Alpha eigenvalue calculations with TRIPOLI-4�

0306-4549/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.anucene.2013.07.018

⇑ Corresponding author. Tel.: +33 (0)1 69 08 95 44.E-mail address: andrea.zoia@cea.fr (A. Zoia).

Andrea Zoia ⇑, Emeric Brun, Fausto MalvagiCEA/Saclay, DEN/DM2S/SERMA/LTSD, 91191 Gif-sur-Yvette Cedex, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 2 April 2013Received in revised form 12 July 2013Accepted 15 July 2013Available online 26 August 2013

Keywords:Alpha eigenvaluesAlpha static algorithmMonte CarloTime-dependent transportTRIPOLI-4

�

Characterizing the time behavior of neutron transport is crucial in several technological issues, such as forinstance reactor start-up analysis, reactivity measurements, or kinetics studies of accelerator-drivensystems. Basically, this amounts to computing the spectrum of the Boltzmann operator: analytical tech-niques, though quite ingenuous, are often restricted to simple geometries and one-speed transport, sothat numerical solutions become mandatory to this aim. In this paper we develop a Monte Carlo methodfor determining the dominant eigenvalue of the Boltzmann operator and the associated fundamentalmode for arbitrary geometries, materials, and boundary conditions; this is done in view of adding suchcapabilities to the Monte Carlo code TRIPOLI-4

�. Extensive verification and validation tests are performedon multiplying and moderating media, and a critique of the algorithm is proposed.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The path of a neutron traveling in a host medium and collidingwith the surrounding nuclei can be thought of as a random walk ofa particle evolving in the position-velocity {r,v} phase space, as afunction of time. Such a random walk can be entirely characterizedby assigning a streaming operator T and a collision operator C: thestreaming operator is T = �v � rr � v R, and describes the freeexponentially-distributed displacements of the neutrons betweenrandom collisions with the constituents of the traversed medium.Here R = R(r,v) is the macroscopic total cross section, andv = jvj. The collision operator can be generally written asC ¼

Rc ðv0 ! vjrÞdv0, and redistributes particle velocities at colli-

sion events that occur at r. The kernel c depends on the mediumcross sections and keeps into account particle multiplicity due todistinct nuclear reactions (as given for instance by fission or(n,2n) reactions, if any). Then, the linear integral–differential trans-port operator A = T + C

A ¼ �v � rr � vRþZ

cðv0 ! vjrÞdv0 ð1Þ

describes the rate of variation of the neutron density w(r,v, t) inphase space, namely,

@

@twðr;v; tÞ ¼ Awðr;v; tÞ: ð2Þ

Eq. (2) is a general form of the linear Boltzmann equation governingthe (prompt) time-dependent neutron transport in arbitrary

geometries, when delayed neutron contributions are neglected(Duderstadt and Martin, 1979; Bell and Glasstone, 1970; Wing,1962).

Determining the spectrum r[A] of the Boltzmann operator A isan essential prerequisite for the analysis of time-dependent trans-port problems. Moreover, knowing the whole spectrum r[A] (i.e.,the set of eigenvalues a and associated eigenfunctions Wa)amounts to characterizing the operator A itself, and this in turn al-lows for eigenfunction expansions of the full time-dependent solu-tion, starting from an assigned initial distribution w(r,v,0). Theseissues are crucial in many technological problems associated withneutron transport (Betzler et al., 2012), such as for instancestart-up of commercial reactors (Pfeiffer et al., 1974), analysis ofaccelerator-driven systems (Persson, 2008), material control andaccountability in critical assemblies (Sanchez and Jaegers, 1998),and pulsed neutron reactivity measurements (Cao and Lee, 2008),just to name a few.

Unfortunately, the operator A is in general not self-adjoint, notcompact, and possibly unbounded, which makes its analysis highlynon trivial (Duderstadt and Martin, 1979; Greenberg et al., 1987).Only a few general properties of the spectrum of A are known sofar, since the pioneering works on one-speed isotropic transportin slab geometries (Lehner and Wing, 1955; Lehner and Wing,1956; Pimbley, 1959). Some surprising features have emerged,such as for instance the possible presence of a continuous spec-trum in addition to a point (discrete) spectrum, which do not al-ways have a counterpart in diffusion theory (Nelkin, 1963;Corngold, 1964). In most applications, one is naturally led to con-sider bodies of finite size, convex, and with vacuum boundary con-ditions (particles cannot re-enter once they have left the system):even with this restriction, the properties of the spectrum depend

2 The word ‘static’ refers to the fact that this algorithm is used to determine thefundamental eigenvalue of a stationary equation for the flux; ‘dynamic’ alpha

A. Zoia et al. / Annals of Nuclear Energy 63 (2014) 276–284 277

on the specific details of the streaming and collision operators.1

The literature concerning the properties of r[A] as a function ofgeometry, scattering kernels, and boundary conditions is consider-able: see for instance (Greenberg et al., 1987; Duderstadt and Martin,1979; Sahni and Sjöstrand, 1990; Sahni, 1996) and referencestherein.

In general, one is interested in using the knowledge on r[A] inorder to derive the eigenfunction expansion of the initial valueproblem in Eq. (2). In view of the functional form of Eq. (2), it isnatural to postulate for w a time dependence of exponential kind,namely w(r,v, t) = Wa(r,v)eat, which leads to

�v � rrWa � vRWa þZ

v0W0acðv0 ! vjrÞdv0 ¼ aWa; ð3Þ

where W0a ¼ Waðr;v0Þ. Due to the possible presence of a continuousspectrum, the general form of the eigenvalue expansion will be

wðr;v; tÞ ¼Xai

aiWaieai t þ

ZgðaÞWaeatda; ð4Þ

where the former term on the right hand side is the eigenfunctionexpansion pertaining to the discrete component of r[A], with (gen-erally complex) discrete eigenvalues ai and weight coefficients ai

depending on the initial conditions, and the latter is the contribu-tion due to the continuous portion of the spectrum, with densityg(a) (Duderstadt and Martin, 1979; Albertoni and Montagnini,1966; Wing, 1962; Van Norton, 1960). A more rigorous proof canbe given by taking the Laplace transform of Eq. (2). Then, seekingthe formal inverse transform (the Bromwich integral) and applyingCauchy’s residual theorem precisely yields Eq. (4) (Duderstadt andMartin, 1979).

Very broad conditions for the well-posedness of the eigenvalueEq. (3) for reasonable scattering kernels and bounded domainshave been thoroughly discussed (Larsen and Zweifel, 1974; Larsen,1979; Duderstadt and Martin, 1979). In particular, it has beenshown that under mild assumptions a dominant discrete eigen-value a0 exists, which is simple, real, larger than the real parts ofall the other a, and whose associated eigenfunction Wa0 ðr;vÞ isnon-negative (Larsen and Zweifel, 1974; Larsen, 1979). This en-sures that, after a transient, the neutron population will grow intime as w ’ ea0t: when a0 > 0, the system is supercritical and thepopulation diverges exponentially fast; when a0 < 0, the systemis subcritical and the population dies out exponentially fast; finally,when a0 = 0 the system is critical and the population is exactlyconstant in time. However, it has been argued that the discretespectrum may not exist (for instance when the size of the systemis below some critical threshold): in this case, the time behaviorwould be imposed by the continuous portion of r[A], and theasymptotic decay of w would be non-exponential (Nelkin, 1963;Corngold, 1964; Duderstadt and Martin, 1979; Larsen and Zweifel,1974; Larsen, 1979).

Exact calculations leading to a0 are hardly feasible but for thevery simplest configurations, and the situation is even worse forhigher eigenvalues (Montagnini and Pierpaoli, 1971): that is thereason why, in parallel with theoretical progress on r[A], efficientnumerical methods have been developed so as to solve Eq. (3) forassigned geometries and materials. Such numerical algorithmsare essentially based on either Monte Carlo or deterministic meth-ods: for an overview, see, e.g., (Brockway et al., 1985; Kornreich andParsons, 2005; Betzler et al., 2012; Cullen et al., 2003; Sahni and Sjö-strand, 1990; Sahni, 1996; Singh et al., 2009; Hill, 1983). The goal ofour paper is to present a Monte Carlo algorithm for the computation

1 For instance, in the case of isotropic scattering the spectrum is purely discretewhen the minimum neutron velocity, say v0, is bounded away from zero, whereas acontinuous spectrum can arise in the region Rfag < �min½vRðr;vÞ� when v0 isallowed to vanish (Jörgens, 1958; Van Norton, 1960).

algorithms also exist, which can be used to compute the full time evolution of the fluxstarting from an initial configuration; see, e.g., the extensive discussion in (Akcasuet al., 1971; Cullen et al., 2003).

3 See for instance the algorithms discussed in (Duderstadt and Hamilton, 1976; Rieand Kschwendt, 1967).

of the fundamental eigenvalue a0, in view of adding alpha static cal-culation capabilities to the code TRIPOLI-4

� (Tripoli-4 Project Team,2008). This work is organized as follows: in Section 2 we first brieflyrecall the basic Monte Carlo method that has been historically pro-posed to compute a0 and discuss some possible improvements.Then, is Section 3 we apply such algorithm to verification and val-idations case studies, and the performances of the method are illus-trated. Perspectives are finally discussed in Section 4.

2. Monte Carlo alpha static algorithms

We begin by observing that the eigenvalue Eq. (3) can be recastin the equivalent form

x � r/þ R/þ av / ¼

Zcsðv0 ! vÞ/0dv0 þ

Zcf ðv0 ! vÞ/0dv0; ð5Þ

where the neutron flux /(r,v,t) is simply related to the particle den-sity by / = vw, x = v/v, and the collision operator C has been explic-itly split into scattering cs and fission cf parts. Then, Eq. (5) isformally identical to a stationary equation for the flux /, wherethe modified total cross section R⁄ = R + Ra depends on the un-known parameter a, with Ra = a/v.

The Monte Carlo alpha static method2 was first developed forpositive a, so that R⁄ > R (Brockway et al., 1985). In this case, the to-tal cross section is augmented by a quantity Ra = a/v > 0, which canbe conceptually seen as an additional absorber with /1/v cross sec-tion (not depending on spatial position), hence also the name ‘timeabsorption’ (Cullen et al., 2003; Nolen et al., 2012). The fundamentaleigenvalue a0 can then be estimated via usual Monte Carlo methodsby adapting the well-known power iteration algorithm3 for k staticcriticality calculations (Brockway et al., 1985; Goad and Johnston,1959). By introducing a fictitious parameter k dividing the fissionterm, we write

x � r/þ R/þ av / ¼

Zcsðv0 ! vÞ/0dv0 þ 1

k

Zcf ðv0 ! vÞ/0dv0;

ð6Þ

which becomes a standard k eigenvalue equation, the parameter abeing though unknown. The basic strategy is to seek then the valuea for which k = 1, i.e., the system is exactly critical. Hence, startingfrom a guess value a0 for a and /0 (the source) for /, Eq. (5) is solvedonce (a ‘cycle’ of the power iteration algorithm) to get the stationaryflux /j and the corresponding effective multiplication coefficient kj

kj ¼ productionsj

source particlesjð7Þ

at each cycle j, where productions are the sum of all neutron births.One needs then to update aj+1 for the fundamental eigenvalue, onthe basis of the value kj: this root finding procedure can be achievedin several ways (Brockway et al., 1985; Nolen et al., 2012), the sim-plest being the linear extrapolation

ajþ1 ¼ ajkj: ð8Þ

This procedure is then iterated over several cycles, until kn atsome later nth iteration converges to 1: the corresponding an con-verges to the fundamental eigenvalue, and /n to the fundamentalmode. When no prior knowledge on the system dynamics is avail-

f

5

278 A. Zoia et al. / Annals of Nuclear Energy 63 (2014) 276–284

able, the choice of the guess value a0 can be possibly improved byfirst running a k static criticality calculation (Hill, 1983; Brockwayet al., 1985).

When a is negative, the positivity of the term R⁄ cannot beensured, so that Eq. (5) is rather rearranged as

x � r/þ R/ ¼Z

csðv0 ! vÞ/0dv0 þZ

cf ðv0 ! vÞ/0dv0 � av /: ð9Þ

By using the identity

av / ¼

Zav 0 dðv

0 � vÞ/0dv0; ð10Þ

the last term at the right hand side of Eq. (9) can be interpreted as acreation term with section Ra = �a/v > 0 and degenerate delta spec-trum: for this reason, in this case the algorithm takes the name of‘time production’ (Cullen et al., 2003; Nolen et al., 2012). Then,Eq. (9) can be solved via the power iteration method exactly as donefor the time absorption case.4

This power iteration algorithm has been implemented withminor modifications in most commercial Monte Carlo codesallowing for alpha static calculations, such as for instance MCNP(X-5 Monte Carlo Team, 2003), TART (Cullen, 2003), or SERPENT(Leppänen, 2012). An alternative scheme based on weight correc-tion has been also proposed (Yamamoto and Miyoshi, 2003). Ineither case, it is known that for a < 0 (i.e., for subcritical systems)these methods may become unstable and lead to anomalous termi-nation (Nolen et al., 2012; Yamamoto and Miyoshi, 2003; Ye et al.,2006). The same problem has been reported also for deterministicsolvers (Hill, 1983). To overcome this limitation, in (Ye et al., 2006)it has been suggested to rewrite Eq. (9) as

x � r/þ R� kav

� �/�

Zcsðv0 ! vÞ/0dv0

¼Z

cf ðv0 ! vÞ/0dv0 �Zð1þ kÞ av 0 dðv

0 � vÞ/0dv0 ð11Þ

for negative a, where k > 0 is an arbitrary parameter (to be fixedbefore running a simulation) that acts as an additional absorptionterm. Use of Eq. (11) within the power iteration scheme (Ye et al.,2006) or the weight correction scheme (Yamamoto, 2011) has beenrecently shown to improve the stability of the alpha static algo-rithm for subcritical systems where the standard power iterationtechnique failed.

In view of adding alpha static calculations capabilities to theMonte Carlo code TRIPOLI-4

�, we have decided to resort to the stan-dard power iteration algorithm (with time absorption or produc-tion) and to keep into account the additional parameter k as in(Ye et al., 2006) for subcritical systems. When a > 0, we simplyresort to the unmodified Eq. (5). When a < 0, we propose to slightlyrearrange Eq. (11) as follows

x � r/þ R� kav

� �/�

Zcsðv0 ! vÞ/0dv0

¼Z

cf ðv0 ! vÞ/0dv0 �Z

�mkkav 0 dðv

0 � vÞ/0dv0; ð12Þ

where we have set

�mk ¼1þ k

k> 0: ð13Þ

The main advantage of Eq. (12) is that the cross section termRa,k = �k a/v > 0 now appears on both sides of the equation, i.e.,summed to the total cross section R on the left hand side and asa creation term on the right hand side. This means that for a < 0we have (mathematically) added a distributed absorber (with /

4 In both cases, the Ra acts as a population control mechanism that allows neutronnumber to be kept constant throughout iterations (Cullen et al., 2003).

1/v section) emitting �mk new particles in the system: all standardMonte Carlo variance reduction techniques, such as Russian rou-lette, particle splitting or implicit capture, can be easily integratedinto the algorithm without any special treatment. Attention shouldhowever be paid to the fact that Eq. (12) introduces correlationsbetween consecutive estimates aj, because of the parameter k: asa consequence, the measured variance on aj might be underesti-mated (Yamamoto, 2011). In most cases tested below, a valuek ’ 1 proved satisfactory for achieving convergence.

3. Verification and validation tests

Before implementing the alpha static method for a futurerelease of TRIPOLI-4

�, we have decided to test its performances onsome case studies, within the development version of the code. Avalidation of the alpha static algorithm would require experimen-tally assessing the fundamental eigenvalue, for instance by Rossialpha or pulsed neutron measures (Orndoff, 1957; Hansen,1985); these are unfortunately available only for a limited set ofconfigurations, e.g., research reactors or critical assemblies closeto delayed criticality. However, even in the absence of experimen-tal data, the alpha static algorithm can be validated for arbitrarygeometries and material compositions as follows. First, an analog(fixed source) Monte Carlo simulation is run, where neutron pathsare tracked in time, starting from an initial arbitrary source. As weare interested in determining the prompt alpha eigenvalues, onlyprompt fission neutrons (if any) are simulated. The energy- andgeometry-integrated neutron flux is then computed, and its behav-ior as a function of time is recorded. After an initial transient thatdepends on the source distribution, the neutron flux / is assumedto asymptotically behave as / ’ exp (a0t): therefore, by fitting theresulting curve of the neutron flux as a function of time one canestimate the fundamental eigenvalue a0.5

Assessing the fundamental eigenvalue via analog Monte Carlosimulation can be thought of as a way of ‘experimentally’ measur-ing a0 when benchmark results or reference data are not available.As such, this technique has been successfully applied, e.g., in(Yamamoto and Miyoshi, 2003; Yamamoto, 2011; Ye et al., 2006)to make up for the lack of experimental measures. This procedurehas however two main shortcomings: for subcritical configura-tions, the number of particles is rapidly decreasing, so that it is of-ten necessary to simulate a huge number of neutrons in order tohave proper statistics at long times; for supercritical configura-tions, a time cutoff is mandatory, so as to prevent the neutron pop-ulation from growing unbounded and eventually saturatingcomputer memory. Morevoer, it should be mentioned that the fit-ting procedure of the exponential decay is often sensitive to con-tamination from higher modes and to statistical fluctuations(Spriggs et al., 1997; Hansen, 1985). To prevent this problem, onehas to choose the observation time T so as to ensure that the fun-damental mode has set: a convenient criterion, e.g., consists inwaiting until T� 1/a0 (Keepin, 1965). Once the fundamentaleigenvalue has been estimated by analog Monte Carlo simulation,the value a0 thus obtained (together with the associated asymp-totic flux shape at long times) can be used as a reference for thevalidation of the alpha static calculations.

As case studies, we have in particular analysed (i) a simpleone-dimensional, one-speed transport problem, the so-called rodmodel (Wing, 1962; Montagnini and Pierpaoli, 1971), where thefundamental eigenvalue a0 can be explicitly computed (Montagniniand Pierpaoli, 1971), (ii) the ‘Godiva-like’ criticality problems

Actually, virtually any physical observable associated to the system, such as forstance the leakage current or the total neutron number in the geometry, will

symptotically display the same time behavior, and can in principle be used to infer0. Here, we focus on the neutron flux for the sake of simplicity.

inaa

Fig. 1. Alpha eigenvalues of the rod model. The fundamental root v0 = a0/(vR) + 1 ofEq. (14) is displayed as a function of the dimensionless rod size z = LR. Solid linescorrespond to numerical solutions for �m ¼ 1:5 (blue), �m ¼ 1:0 (red), and �m ¼ 0:5(green). The results of Monte Carlo alpha static algorithm are displayed assymbols for a few values of z. The critical dimensionless rod sizezc ¼ LcR ¼ 2 tan�1 ð

ffiffiffiffiffiffiffiffiffiffiffiffi�m� 1p

Þ=ffiffiffiffiffiffiffiffiffiffiffiffi�m� 1p

is also shown for the curve �m ¼ 1:5. Dashedlines correspond to the asymptotic values v10 ¼ �m. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)

A. Zoia et al. / Annals of Nuclear Energy 63 (2014) 276–284 279

discussed in (Cullen et al., 2003), (iii) a few configurations from theRossi alpha validation benchmark (Mosteller and Kiedrowski,2011), and (iv) some moderating non-fissile materials. In the fol-lowing, we will discuss each item in detail.

3.1. The rod model

The rod model is possibly the simplest example of a transportproblem: particles move at constant speed v along a line (therod) and undergo collision events at a rate vR, whereupon q newprompt particles are re-emitted, whose number distribution isP(q). Because of the geometric constraints, only two directions offlight are allowed, namely forward (x+) and backward (x�); here,we furthermore assume that scattering is isotropic, i.e., that direc-tions are sampled with equal probability. When vacuum boundaryconditions are imposed, the viable space is a segment [0,L], and thefollowing explicit dispersion relation can be derived

cosh zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivðv� �mÞ

p� �þ

v� �m2

� �sinh z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivðv� �mÞ

p� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivðv� �mÞ

p ¼ 0; ð14Þ

which relates the dimensionless variable v = a/(vR) + 1 to thedimensionless length z = LR and to the average number of second-ary particles per collision �m ¼

PqqPðqÞ (Montagnini and Pierpaoli,

1971). For an interpretation of this formula in terms of survivalprobability for exponential flights in bounded domains, see for in-stance (Zoia et al., 2012a,b).

The zeros of Eq. (14), solved as a function of a, form the spec-trum of the Boltzmann operator for this problem: it has beenshown that there exists a finite number of real eigenvalues, plusa countable infinity of complex eigenvalues associated to oscillat-ing modes (Montagnini and Pierpaoli, 1971). When the dominanteigenvalue a0 ¼ a0ðz; �mÞ < 0, the neutron chain reaction will dieout because of leakages and possibly absorptions; when a0 > 0,the chain reaction will diverge, as particles born from fission arenot sufficiently compensated by leakages and absorptions; thecrossover between these two regimes is reached for a0 = 0, i.e., atcriticality. When �m 6 1, all eigenvalues a have negative real part.When �m > 1, imposing a0 = 0 in Eq. (14) allows determining the(dimensionless) critical segment length zc = LcR at which the neu-tron population will stay constant, namely,

zc ¼ 2tan�1 1ffiffiffiffiffiffi

�m�1p� �ffiffiffiffiffiffiffiffiffiffiffiffi�m� 1p : ð15Þ

It is not difficult to show that a10 ¼ limz!1a0ðz; �mÞ ¼ vRð�m� 1Þfor very large rod sizes, i.e., in the absence of spatial leakages;hence, also v10 ¼ a10 =ðvRÞ þ 1 ¼ �m. As the dominant zero of Eq.(14) can be easily computed numerically as a function of the sys-tem parameters z and �m, the rod model provides a simple, yet non-trivial example for the verification of the alpha static method. Tothis aim, we prepared specially modified nuclear data files for nuc-lides with constant scattering cross section Rs, absorption crosssection Ra, and fission cross section Rf, and constant number ofsecondary neutrons per fission �mf . The cross sections are chosenso as to impose the parameters R = Rs + Ra + Rf and�m ¼ ðRs þ �mf Rf Þ=R. The scattering routines in TRIPOLI-4

� are easilymodified to reproduce the needed one-speed isotropic scatteringlaw where the neutron direction after scattering is preserved or re-versed with equal probability 1/2.

Several configurations have been tested, with one or severalnuclides, fissile and non-fissile materials, and various segmentlengths, so as to explore sub- and super-critical regimes. All stan-dard variance reduction techniques, e.g., implicit capture, Russianroulette and particle splitting, were activated during these tests.The results of Monte Carlo simulations always converge to the

exact values that can be numerically computed based on Eq.(14). The modification proposed in Eq. (12) turns our to be manda-tory for subcritical configurations, for which the standard time pro-duction algorithm would make the computation diverge after afew iterations. The obtained results are illustrated in Fig. 1: thenumerical roots v0 = a0/(vR) + 1 of Eq. (14), corresponding to thefundamental eigenvalues a0, are compared to the Monte Carlo al-pha static results as a function of the dimensionless rod sizez = LR for a few values of �m. An excellent agreement is found.

3.2. The ‘Godiva-like’ criticality problems

In (Cullen et al., 2003), a set of five ‘Godiva-like’ criticality testshas been proposed, with the aim of comparing alpha static, dynamicand k static calculations in the context of the continuous-energyMonte Carlo code TART (Cullen, 2003) for a variety of sub- andsupercritical configurations. In particular, it has been shown that al-pha static calculations are equivalent to dynamic alpha calculationsinsofar as only the stationary behavior of the system is sought,whereas k static calculations always differ from the previous (un-less the system is close to prompt criticality). These configurationsprovide a natural framework so as to test the performances of thealpha static algorithm implemented in TRIPOLI-4

�. Since only theprompt behavior of the system is concerned, delayed neutrons arenot simulated. All calculations are performed by resorting toENDF/B-VII.0 nuclear data (at room temperature), includingprobability tables in the unresolved resonance range. Within alphastatic calculations, the removal time sR of the neutrons in the sys-tem can be also computed by tracking the rate at which particlesleave the system from the boundaries or disappear by absorption:the reciprocal of this rate defines precisely sR (Cullen et al., 2003).

The computed values for a0 and sR are summarized in Table 1for problem I–V. The fundamental eigenvalue a0 estimated by thealpha static method is in good agreement with the exponentialfit af.s. of the flux decay obtained from fixed source simulations.Moreover, both a0 and sR lie remarkably close to the respective val-ues given in (Cullen et al., 2003) by using TART, with the possibleexception of problem I (see Section 3.2.1 for details). Precise geom-etry and composition specifications for each problem can be found

Table 1‘Godiva-like’ criticality tests as in Cullen et al. (2003). Fundamental eigenvalue a0 andremoval time sR as computed by TRIPOLI-4

� (alpha static algorithm (a.s.) and fixedsource (f.s.) calculations), and the corresponding values computed by TART, as takenfrom Cullen et al. (2003).

Problem: I II III IV V

a0 T4-a.s. (1/ls) �1.09 144.9 146.9 0.671 �0.979af.s. T4 (1/ls) �1.09 146 148 0.689 �1.0a0 TART (1/ls) �0.739 144.7 146.6 0.653 �1.048sR T4-a.s. (ns) 6.24 4.57 9.35 1110 235sR TART (ns) 6.124 4.559 9.203 1108 226

Fig. 3. Problem II: neutron flux spectra /(E) as a function of the incident neutronenergy E, as computed by TRIPOLI-4

�. For comparison, fluxes have been normalizedand are expressed in arbitrary units. Blue triangles: fundamental mode corre-sponding to k static calculations; red squares: fundamental mode corresponding toalpha static calculations. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

280 A. Zoia et al. / Annals of Nuclear Energy 63 (2014) 276–284

in (Cullen et al., 2003). In the following, we just sketch somedetails.

3.2.1. Problem IProblem I corresponds to a bare sphere of highly enriched ura-

nium, with kprompt = 0.99349. The system is close to prompt criti-cality, which is precisely the specification for the simplifiedGodiva benchmark in (OECD NEA, 2010); see also Section 3.3.The system is spatially homogeneous, and the neutron spectrumis fast. The fundamental mode of alpha static calculations is com-pared to the fundamental mode of k static calculations in Fig. 2:for this case, the spectra are quite similar, as expected from keff

being close to 1. There is a discrepancy between the a0 value com-puted by TART and those computed by TRIPOLI-4

�, which we mightperhaps attribute to differences in the nuclear data libraries (forcomparison, the measured value for Godiva is a ’ �1.1 [1/ls](OECD NEA, 2010)).

3.2.2. Problem IIProblem II corresponds to a bare sphere of highly enriched ura-

nium, with a density twice as big as in Problem I, which makes thisconfiguration very super critical (kprompt = 1.58455). The system isspatially homogeneous, and the neutron spectrum is fast. The fun-damental mode of alpha static calculations is compared to the fun-damental mode of k static calculations in Fig. 3: for this case, thetwo spectra are quite different; in particular, the k static methodyields a larger flux shape at lower energies, whereas the situationis reversed at higher energies.

Fig. 2. Problem I: neutron flux spectra /(E) as a function of the incident neutronenergy E, as computed by TRIPOLI-4

�. For comparison, fluxes have been normalizedand are expressed in arbitrary units. Blue triangles: fundamental mode corre-sponding to k static calculations; red squares: fundamental mode corresponding toalpha static calculations. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

3.2.3. Problem IIIProblem III corresponds to a sphere of highly enriched uranium

surrounded by a thick water reflector, resulting in a very supercrit-ical configuration (kprompt = 1.66782). The system is spatially heter-ogeneous, and the neutron spectrum is fast. The fundamentalmode of alpha static calculations is compared to the fundamentalmode of k static calculations in Fig. 4: for this case, the k staticmethod yields a dramatically larger flux shape at lower energies.A nice explanation for this fact is provided in (Cullen et al.,2003): the time scale of transport in the uranium sphere is muchshorter than in water. As a consequence, the alpha static method(via the a/v time absorption) basically says that thermal neutronsescaping to water and eventually coming back to the fissile spheregive no contribution to the flux: by the time they can get back tothe sphere, the neutron flux has already climbed up to a very large

10−10

10−5

100

10 −12

10 −10

10 −8

10 −6

10 −4

10 −2

10 0

E [MeV]

Fig. 4. Problem III: neutron flux spectra /(E) as a function of the incident neutronenergy E, as computed by TRIPOLI-4

�. For comparison, fluxes have been normalizedand are expressed in arbitrary units. Blue triangles: fundamental mode corre-sponding to k static calculations; red squares: fundamental mode corresponding toalpha static calculations. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

10−6 10−4 10−2 100

10−10

10−8

10−6

10−4

10−2

100

E [MeV]

Fig. 6. Problem V: neutron flux spectra /(E) as a function of the incident neutronenergy E, as computed by TRIPOLI-4

�. For comparison, fluxes have been normalizedand are expressed in arbitrary units. Blue triangles: fundamental mode corre-sponding to k static calculations; red squares: fundamental mode corresponding toalpha static calculations. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

A. Zoia et al. / Annals of Nuclear Energy 63 (2014) 276–284 281

value (a being positive). Conversely, the k static method does nottake time into account and results into a largely enhanced contri-bution from thermal neutrons.

3.2.4. Problem IVProblem IV corresponds to a sphere of highly enriched uranium

homogeneously mixed with water, resulting in a very supercriticalconfiguration (kprompt = 1.79282). The system is spatially homoge-neous, and the neutron spectrum is thermal due to the presenceof admixed hydrogen nuclei. The time scale of neutron dynamicsis much slower than in the previous three problems. The funda-mental mode of alpha static calculations is compared to the funda-mental mode of k static calculations in Fig. 5: with respect toproblem III, the flux shapes are almost identical, although differ-ences are observable especially in the thermal region.

3.2.5. Problem VProblem V corresponds to a sphere of highly enriched uranium

at low density (half the density of problem I), resulting in a verysubcritical configuration (kprompt = 0.52542). The system is spatiallyhomogeneous, and the neutron spectrum is fast. The fundamentalmode of alpha static calculations is compared to the fundamentalmode of k static calculations in Fig. 6: the alpha static fundamentalmode shows a relevant contribution at thermal energies, which isnot apparent in k static mode. The large spike close to the eV regionin the alpha static flux is rather puzzling: in view of analogousobservations made for moderating materials (De Saussure, 1962;Conn and Corngold, 1969), we are led to conjecture that the spikemight be possibly due to the so-called ‘neutron trapping effect’ in-duced by the strong variations in the elastic cross section in theepithermal region. This effect surely deserves further investiga-tions, which go beyond the scope of the present work.

3.3. The Rossi alpha criticality benchmark

Recently, a benchmark for criticality calculations has been pro-posed, under the name of Rossi alpha validation suite (Mostellerand Kiedrowski, 2011). For a series of multiplying systems takenfrom (OECD NEA, 2010), it is proposed to compare the Rossi alphavalue at delayed criticality adc

p as obtained from Monte Carlo

Fig. 5. Problem IV: neutron flux spectra /(E) as a function of the incident neutronenergy E, as computed by TRIPOLI-4

�. For comparison, fluxes have been normalizedand are expressed in arbitrary units. Blue triangles: fundamental mode corre-sponding to k static calculations; red squares: fundamental mode corresponding toalpha static calculations. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

simulations to the experimental values available from literature.The experimental adc

p is estimated by operating each system at var-ious reactivities q = (keff � 1)/keff (below prompt criticality condi-tions), and measuring the corresponding prompt Rossi alphaap = ap(q) coming from fission chain coincidence counting at somedetector (Keepin, 1965; Orndoff, 1957). Close to delayed criticality,we have

apðqÞ ¼ adcp ½1� qð$Þ�; ð16Þ

where adcp is the prompt alpha value at q = 0 (delayed criticality),

and q($) is the reactivity expressed in dollars, i.e., units of beff, theeffective delayed neutrons fraction (Keepin, 1965). The value adc

p

is then obtained by extrapolating the measures of ap(q) to q = 0.When a multiplying system is at delayed criticality, the prompt al-pha is given by

adcp ¼ �

beff

Keff; ð17Þ

where beff is the effective fraction of delayed neutrons and Keff is theeffective (prompt) mean generation time (Bell and Glasstone, 1970;Hansen, 1985; Orndoff, 1957; Keepin, 1965). Since the kineticsparameters beff and Keff can be estimated in a regular k static calcu-lation (and are not very sensitive to reactivity variations close todelayed criticality), comparing their ratio to the experimentalextrapolated measure allows validating the accuracy of a MonteCarlo code and the associated nuclear data library for criticalitycalculations (Mosteller and Kiedrowski, 2011).

Having in mind the validation of the alpha static algorithm, wehave chosen some of the configurations proposed in the bench-mark, and performed the following calculations. First, we havecomputed beff, Keff and q in a regular k static simulation involvingboth prompt and delayed neutrons, which allows deriving ap(q) asdefined in Eq. (16) for the chosen configuration.6 The obtainedap(q) is then compared to the value a0 coming from an alpha staticalgorithm (without delayed neutrons) for the same configuration.This analysis is further supported by extensive analog fixed source

6 Computing beff and Keff formally demands weighting by the adjoint neutron flux(Keepin, 1965). In TRIPOLI-4

�, beff and Keff are currently estimated by resorting toNauchy’s method (Nauchy and Kameyama, 2005).

282 A. Zoia et al. / Annals of Nuclear Energy 63 (2014) 276–284

Monte Carlo simulations without delayed neutrons, so as to ascer-tain the asymptotic time behavior of the neutron population, usedas a reference. All calculations are performed by resorting toENDF/B-VII.0 nuclear data (at room temperature).

3.3.1. Bare spheresWe begin our validation by considering the three bare spheres

Godiva (HEU-MET-FAST-001), Jezebel (PU-MET-FAST-001) andJezebel-233 (U233-MET-FAST-001) operated in the 1950s close todelayed criticality conditions at the Los Alamos Scientific Labora-tory (now LANL). Godiva is a bare homogeneous sphere of highlyenriched uranium (keff = 0.99999), Jezebel is a bare homogeneoussphere of plutonium (keff = 0.99994), and Jezebel-233 is a barehomogeneous sphere of 233U (keff = 0.99977). Benchmark specifica-tions closely follow the details reported in (Mosteller and Kiedrow-ski, 2011; OECD NEA, 2010). A good agreement is found betweenthe value ap(q) computed from k static method and the fundamen-tal eigenvalue a0 obtained from alpha static method. Results ofcalculations are summarized in Table 2. The exponential fit af.s. ob-tained by fixed source simulations is also displayed for reference. Inparticular, the case of Jezebel is illustrated in Fig. 7 (top), where thetotal neutron flux within the sphere is displayed as a function oftime, revealing an exponential decay.

3.3.2. Stacy-30Stacy-30 (LEU-SOL-THERM-007) is an unreflected stainless steel

cylinder containing a solution of uranium nitrate at intermediateenrichment (keff = 0.99771). It is based on an experiment

Table 2Comparison between Rossi alpha value ap(q) as computed by k static method and a0

as computed by alpha static method in TRIPOLI-4�. The exponential fit obtained by fixed

source (f.s.) simulations is also displayed for reference. For the case of Thor, twoexponential fits are reported for fixed source simulations.

ap(q) (1/ls) a0 (1/ls) af.s. (1/ls)

Godiva �1.126 �1.149 �1.142Jezebel �0.634 �0.639 �0.638Jezebel-233 �1.131 �1.137 �1.131Stacy-30 �1.66 � 10�4 �1.70 � 10�4 �1.69 � 10�4

Thor �0.429 �0.051 (i) �0.430; (ii) �0.055

Fig. 7. Top. The time behavior of the flux for the Jezebel configuration (bluetriangles). Flux is given in arbitrary units. The exponential fit is also displayed as ared dashed line. Bottom. The time behavior of the flux for the Thor configuration(blue triangles). Flux is given in arbitrary units. The exponential fits for the twoexponential decay modes are also displayed as red dashed lines. (For interpretationof the references to color in this figure legend, the reader is referred to the webversion of this article.)

performed at the Japan Atomic Energy Research Institute in 1995(Mosteller and Kiedrowski, 2011; OECD NEA, 2010). A good agree-ment is found between the Rossi alpha value ap(q) computed fromk static method and the fundamental eigenvalue a0obtained fromalpha static method. Calculations are summarized in Table 2. Theexponential fit af.s. obtained by fixed source simulations is alsodisplayed for reference.

3.3.3. ThorThe Thor benchmark (PU-MET-FAST-008) consists of a sphere of

plutonium enclosed in the center of a right circular cylinder of tho-rium (keff = 0.99735), based on a configuration operated at the LosAlamos Scientific Laboratory in 1961–1962 (Mosteller and Kied-rowski, 2011; OECD NEA, 2010). An interesting feature emergesin this configuration: in fixed source simulations, two exponentialdecay modes are apparent (see Fig. 7, bottom). The former (rapidlydecaying) mode can be intuitively associated to the transit time ofthe neutrons in the core, whereas the latter (slowly decaying) tothe (much longer) time that the neutrons spend wandering aroundin the reflector. A similar behavior has been reported, e.g., in (Yeet al., 2006) for a polyethylene-reflected plutonium sphere: for abare sphere, a single exponential decay is observed for the neutronflux; then, by increasing the thickness of the reflector, the singleexponential decay evolves towards a double exponential decay.For the case of Thor, the Rossi alpha value ap(q) is in agreementwith the exponential fit of the first slope (fast decay at short times),whereas the fundamental eigenvalue a0 is in agreement with theexponential fit of the second slope (slow decay at long times), asshown in Table 2.

The appearance of multiple (usually two) prompt decay modesfor slightly sub-critical reflected systems close to delayed critical-ity has been long studied, and cannot be interpreted within theframework of Rossi alpha formulas with a single exponential decay(Hansen, 1985). Basically, this is due to the fact that Eq. (16) as-sumes that a single space- and energy-mode adequately capturesthe system behavior (point-kinetics), whereas this hypothesis be-comes suspect in presence of strong heterogeneities (in the spaceas well as in the energy domain). Several techniques have beenproposed in literature so as to take into account such multiple de-cay modes, by resorting to either simplified two-region models(Spriggs et al., 1997), and/or sophisticated corrections based onthe theory of branching processes (Muñoz-Cobo et al., 2011; Pázsitand Pál, 2008). Despite these improvements, a question concerningthe practical possibility of measuring the slower decay mode actu-ally arises, because of the rapid die-away of the neutron flux asso-ciated to the faster decay mode: in most experiences, the dominant(slower) mode will be concealed under the unavoidable measurenoise (Spriggs et al., 1997).

Therefore, utmost care should be taken when comparing thevalues of ap(q) coming from experimental measures with those ob-tained from alpha static calculations: while the Monte Carlo alphastatic algorithm converges to the fundamental a0 (the dominanteigenvalue of the system), experimental measures might singleout the faster decay mode (Spriggs et al., 1997; Hansen, 1985). Inthis respect, Monte Carlo analog fixed source simulations can beseen as an ideal (numerical) experimental technique (Ye et al.,2006), capable of following the full time behavior of the system,insofar as simulations are performed with sufficient statistics(which might require large memory occupation and long runningtimes), and thus providing a reliable tool for the validation of thealpha static method.

3.4. Moderating materials

We conclude the validation tests by computing thefundamental a0 eigenvalue for non-multiplying media. In view of

Table 3Fundamental eigenvalue a0 and (inverse) removal time sR as computed by TRIPOLI-4

�

by means of alpha static algorithm (a.s.) and fixed source (f.s.) calculations.

C-graphite H2O water

a0 T4-a.s. (1/s) �265.3 �4912(sR)�1 T4-a.s. (1/s) �265.1 �4907af.s. T4 (1/s) �265.3 �4900

A. Zoia et al. / Annals of Nuclear Energy 63 (2014) 276–284 283

determining thermal neutron diffusion parameters, pulsed neutronexperiments are usually performed, where a burst of neutrons isinjected into a moderating (non-multiplying) material, and the re-sponse at a given detector is recorded as a function of time (Belland Glasstone, 1970; Keepin, 1965). It is known that the thermalflux typically has an asymptotic relaxation of the kind / ’ exp(�aDt), where the constant aD is related to the physical propertiesof the medium by

aD ¼ hvRai þ D0B2 þ CB4 þ � � � ; ð18Þ

Ra being the absorption cross section, D0 the diffusion coefficient, B2

the geometrical buckling, and C < 0 the so-called diffusion-coolingconstant (Nelkin, 1960). In view of the previous discussion, the fun-damental eigenvalue a0 computed by alpha static eigenvalue calcu-lations on moderating materials should yield a fairly accurateestimate of the relaxation parameter aD; this, in turn, provides auseful tool for the assessment of the physical properties of moder-ating materials. In this case, a0 < 0 and sR = �1/a0 allows estimatingthe characteristic flight time of the neutrons in the moderator, frombirth to absorption or leakage. Besides, this analysis allows testingthe behavior of the alpha static algorithm when the system doesnot contain any fissile isotopes (i.e., keff = 0).

Similarly as done above, the values of a0 estimated by resortingto the alpha static method are compared to those obtained by fixedsource calculations, which can be thought of as idealized pulsedneutron experiments. The case of graphite and water have beenseparately considered, both at room temperature and nominaldensities, for an 1 m-radius homogeneous sphere. In both cases,the values a0 are in excellent agreement with the correspondingexponential fits af.s. coming from fixed source calculations and liewell within the interval of experimental values aD reported in(Keepin, 1965). Results are recalled in Table 3. All calculationsare performed by resorting to ENDF/B-VII.0 nuclear data, includ-ing S(a,b) thermal sections.

4. Conclusions

In this work we have proposed a Monte Carlo method aimed atassessing the fundamental time eigenvalue a0 and the associatedfundamental mode of a time-dependent transport problem. Whena > 0 (time absorption), we have closely followed the prescriptionsof the power iteration algorithm. Conversely, when a < 0 (time pro-duction), the power iteration algorithm has been slightly modifiedso as to ensure convergence even for strongly subcritical configura-tions, where the standard power iteration has been reported to fail.

Several verification and validation tests have been performed,including comparisons with analytical results for the rod model,the modified ‘Godiva-like’ criticality problems proposed in (Cullenet al., 2003), some delayed-critical configurations from the Rossialpha benchmark suite (Mosteller and Kiedrowski, 2011), and timerelaxation in moderating materials. In all such cases, the algorithmhas given satisfactory results, and a good agreement has beenfound as compared to analytical results and experimental mea-sures (when available) and/or fixed source analog Monte Carlosimulations (used as a reference).

A few words of caution are in order: as opposed to regular k sta-tic calculations, alpha static method requires longer settle cyclesbefore converging to the fundamental eigenvalue, and convergencecan be especially difficult close to prompt criticality. Care musttherefore be taken to avoid false convergence.

Future work will be directed to a twofold aim. On one hand, thealgorithm that we have proposed to implement in TRIPOLI-4

� cancurrently compute prompt alpha eigenvalues alone: it wouldtherefore be interesting to add the possibility of computing alsoso-called delayed alpha eigenvalues, associated to delayed neu-trons. This could be achieved by following the strategies recentlyproposed, e.g., in (Hoogenboom, 2002; Singh et al., 2011; Betzleret al., 2012), which might in principle be adapted to both deter-ministic and Monte Carlo methods. On the other hand, severalimprovements to the algorithm could be explored: for instance,keeping into account the effects of batch-to-batch correlations in-duced by the parameter k in subcritical calculations, or accelerat-ing the convergence for subcritical cases by sampling the initialdistribution on a guess of the fundamental mode.

Acknowledgment

The authors wish to thank AREVA and Electricité de France(EDF) for partial financial support.

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