# Calculation of the Effective Delayed Neutron Fraction by Deterministic and Monte Carlo Methods

Authors: M. Carta; S. Dulla; V. Peluso; P. Ravetto; G. Bianchini
Journal: Science and Technology of Nuclear Installations
 (2011)

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## Abstract

The studies on Accelerator-Driven Systems (ADSs) have renewed the interest in the theoretical and computational evaluation of the main integral parameters characterizing subcritical systems (e.g., reactivity, effective delayed neutron fractionβeff, and mean prompt neutron generation time). In particular, some kinetic parameters, as the effective delayed neutron fraction, are evaluated in Monte Carlo codes by formulations which do not require the calculation of the adjoint flux. This paper is focused on a theoretical and computational analysis about how the different βeff definitions are connected and which are the approximations inherent to the Monte Carlo definition with respect to the standard definition involving weighted integrals. By means of a refined transport computational analysis carried out in a coherent and consistent way, that is, using the same deterministic code and neutron data library for the βeff evaluation in different ways, the theoretical analysis is numerically confirmed. Both theoretical and numerical results confirm the effectiveness of the Monte Carlo βeff evaluation, at least in cases where spectral differences between total and prompt fluxes are negligible with respect to the value of the functionals entering the classical βeff formulation.


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## Body

## 1. Introduction

The studies on Accelerator-Driven Systems (ADSs) have renewed the interest in the theoretical and computational evaluation of the main integral parameters characterizing subcritical systems (e.g., reactivity, effective delayed neutron fraction, and mean neutron generation time [1]). In particular, the extensive use of Monte Carlo codes for the analysis of the ADS neutronic behaviour is challenging the deterministic codes, used in the past for the analysis of critical systems, for what concerns the capability to reproduce Monte Carlo results, usually taken as reference [2–4]. Nevertheless, some particular parameters, as the effective delayed neutron fraction, are evaluated in Monte Carlo codes by formulations which do not require the calculation of the adjoint flux. The assessment of the various formulations of the effective delayed neutron fraction is crucial for the system evaluation, since it plays an important role in determining its dynamic characteristics [5]. This paper is focused on a theoretical and computational analysis on the connections among the different βeff definitions, with special attention to the approximations inherent to the Monte Carlo definition with respect to the standard definition involving weighted integrals. Theoretical results show how the Monte Carlo formulation of βeff may be related to the classical definition, interpreting the classical one through a reactivity evaluation based on an “improved” first-order approach of perturbation theory. The computational analysis is carried out in a coherent and consistent way, using the same deterministic code and neutron data library for the βeff evaluations. A simplified system allows to investigate the features of the various procedures and to use the results to obtain a physical insight. The GUINEVERE system is then selected as a relevant test case for ADS technology. The GUINEVERE experience [6, 7], mainly devoted to the issues concerning online reactivity monitoring in ADS, is analysed by using a modified layout of the VENUS critical facility located at the SCK•CEN site in Mol, Belgium, coupling the subcritical core facility to a deuteron accelerator delivering 14 MeV neutrons by deuterium-tritium reactions, by a continuous or pulsed beam. For the GUINEVERE experience the ERANOS system [8] and neutron data library JEFF 3.1 [9] have been used to perform transport calculations with 49 energy groups for a cylindrical schematization of the GUINEVERE start-up (at critical) configuration. The analysis of the results allows to draw some conclusions on the merits and the limits of the various formulations.


## 2. Theoretical Analysis

### 2.1. The General Case

Let us take asreference system the following eigenvalue problem:(1)Lφ=ωFφ,
where L is the loss operator, F the fission operator, and ω=1/k, inverse of the classical multiplication eigenvalue. Then, a perturbation δF is introduced in the system. As a consequence, the eigenvalue problem for the perturbed system is(2)Lφ*=[ω+δω][F+δF]φ*,
with φ* as the perturbed flux. Following the exact approach of the Perturbation Theory (PT) [10], that is, considering the adjoint problem corresponding to the reference system(3)L+φ+=ωF+φ+,
one obtains (the symbol 〈⋯〉 denotes integration over the full phase space, while, in the following, 〈⋯〉x is denoting integration only over the specified variable x):(4)δωω+δω=-〈φ+δFφ*〉〈φ+Fφ*〉
or equivalently(5)δωω=-〈φ+δFφ*〉〈φ+(F+δF)φ*〉.
If ω+δω=1/k*, we can write (4) in terms of the k eigenvalue:(6)1-k*k=-〈φ+δFφ*〉〈φ+Fφ*〉.
If we write (2) as(7)L[φ+δφ]=[ω+δω][F+δF][φ+δφ]
and follow the first-order PT approach, while retaining the second-order termδωδFφ, we obtain the “improved” first-order formulation(8)δωω+δω≅-〈φ+δFφ〉〈φ+Fφ〉
or, in terms of the k eigenvalue,(9)1-k*k≅-〈φ+δFφ〉〈φ+Fφ〉.


### 2.2. The “Prompt” Perturbation

If the perturbation is assumed to be equal to the delayed neutrons fission operatorFd, hence δF=-Fd, the perturbed system given in (2) can be written as(10)Lφp=ωp[F-Fd]φp,
with the perturbed flux given by the prompt flux φp and  ωp=1/kp. Then (6) becomes(11)1-kpk=〈φ+Fdφp〉〈φ+Fφp〉
and (9) takes the following form:(12)1-kpk≅〈φ+Fdφ〉〈φ+Fφ〉=βeff.
Thus, the effective delayed neutron fraction βeff is an “improved” PT first-order formulation of the relationship 1-(kp/k), widely used in Monte Carlo codes as βeff estimator. The pure PT first-order formulation provides:(13)δωω=kkp-1≅-〈φ+δFφ〉〈φ+Fφ〉=〈φ+Fdφ〉〈φ+Fφ〉=βeff.
To pass from (11) to (12), it is sufficient to replace φp by φ.


## 2.1. The General Case

Let us take asreference system the following eigenvalue problem:(1)Lφ=ωFφ,
where L is the loss operator, F the fission operator, and ω=1/k, inverse of the classical multiplication eigenvalue. Then, a perturbation δF is introduced in the system. As a consequence, the eigenvalue problem for the perturbed system is(2)Lφ*=[ω+δω][F+δF]φ*,
with φ* as the perturbed flux. Following the exact approach of the Perturbation Theory (PT) [10], that is, considering the adjoint problem corresponding to the reference system(3)L+φ+=ωF+φ+,
one obtains (the symbol 〈⋯〉 denotes integration over the full phase space, while, in the following, 〈⋯〉x is denoting integration only over the specified variable x):(4)δωω+δω=-〈φ+δFφ*〉〈φ+Fφ*〉
or equivalently(5)δωω=-〈φ+δFφ*〉〈φ+(F+δF)φ*〉.
If ω+δω=1/k*, we can write (4) in terms of the k eigenvalue:(6)1-k*k=-〈φ+δFφ*〉〈φ+Fφ*〉.
If we write (2) as(7)L[φ+δφ]=[ω+δω][F+δF][φ+δφ]
and follow the first-order PT approach, while retaining the second-order termδωδFφ, we obtain the “improved” first-order formulation(8)δωω+δω≅-〈φ+δFφ〉〈φ+Fφ〉
or, in terms of the k eigenvalue,(9)1-k*k≅-〈φ+δFφ〉〈φ+Fφ〉.


## 2.2. The “Prompt” Perturbation

If the perturbation is assumed to be equal to the delayed neutrons fission operatorFd, hence δF=-Fd, the perturbed system given in (2) can be written as(10)Lφp=ωp[F-Fd]φp,
with the perturbed flux given by the prompt flux φp and  ωp=1/kp. Then (6) becomes(11)1-kpk=〈φ+Fdφp〉〈φ+Fφp〉
and (9) takes the following form:(12)1-kpk≅〈φ+Fdφ〉〈φ+Fφ〉=βeff.
Thus, the effective delayed neutron fraction βeff is an “improved” PT first-order formulation of the relationship 1-(kp/k), widely used in Monte Carlo codes as βeff estimator. The pure PT first-order formulation provides:(13)δωω=kkp-1≅-〈φ+δFφ〉〈φ+Fφ〉=〈φ+Fdφ〉〈φ+Fφ〉=βeff.
To pass from (11) to (12), it is sufficient to replace φp by φ.


## 3. Standard and Monte Carloβ Formulations

The effectiveness of delayed neutrons is normally obtained by one of the following two relationships:(14)βeff=∑m∑iν̅d,i(m)〈〈Φ+(r,E)χd,i(m)(E)〈Σf(m)(r,E′)Φ(r,E′)〉E′〉E〉r∑m〈〈Φ+(r,E)χ(m)(E)〈ν(m)(E′)Σf(m)(r,E′)Φ(r,E′)〉E′〉E〉r,(15)β̃eff=∑m∑iβi(m)〈〈Φ+(r,E)χd,i(m)(E)〈ν(m)(E′)Σf(m)(r,E′)Φ(r,E′)〉E′〉E〉r∑m〈〈Φ+(r,E)χ(m)(E)〈ν(m)(E′)Σf(m)(r,E′)Φ(r,E′)〉E′〉E〉r,where m is the fissile isotope index, i the delayed neutron family index,χd,i(m) the delayed neutron spectrum for fissile isotope m and delayed neutron family i (basic data),χ(m) the total neutron spectrum for fissile isotope m,ν̅d,i(m) the average value of delayed neutrons emitted from fissile isotope m and delayed neutron family i, for a given incident neutron spectrum (from basic data),ν(m) the total neutrons emitted from fissile isotope m, and Σf(m) the macroscopic fission cross section for fissile isotope m.Furthermore,(16)βi(m)=ν̅d,i(m)ν̅ψ(m),
with ν̅ψ(m) being the average value of the number of total neutrons emitted from the fissile isotope m for a given incident neutron spectrum ψ (ψ has to be calculated by a cell or system neutronic calculation). In (14) and (15) the direct flux and the adjoint flux are solutions for the following eigenvalue equations, respectively:(17)Ω⋅∇φ(r,Ω,E)+Σt(r,E)φ(r,Ω,E)=1k∑mχ(m)(E)<ν(m)(E′)Σf(m)(r,E′)Φ(r,E′)>E′+<Σs(r,E′⟶E)Φ(r,E′)>E′(18)-Ω⋅∇φ+(r,Ω,E)+Σt(r,E)φ+(r,Ω,E)=1k∑mν(m)(E)Σf(m)(r,E)〈χ(m)(E′)Φ+(r,E′)〉E′+〈Σs(r,E⟶E′)Φ+(r,E′)〉E′.
Current Monte Carlo (MC) calculations approximate βeff by the following formula:(19)βeffMC=1-kpk,
where kp is the multiplication eigenvalue of the equation(20)Ω⋅∇φp(r,Ω,E)+Σt(r,E)φp(r,Ω,E)=1kp∑m[χ(m)(E)-∑iχd,i(m)(E)βi(m)(r)]×〈ν(m)(E′)Σf(m)(r,E′)Φp(r,E′)〉E′+〈Σs(r,E′⟶E)Φp(r,E′)〉E′.
Equation (20) is written in a general form, taking into account the spatial dependence of the delayed emission fraction β on position r. In this case βi(m)(r) is given by(21)βi(m)(r)=ν̅d,i(m)[〈ν(m)(E′)Σf(m)(r,E′)Φp(r,E′)〉E′/〈Σf(m)(r,E′)φp(r,E′)〉〈Σf(m)(r,E′)Φp(r,E′)〉E′].If we consider the system governed by (20) as a perturbation of the system governed by (17), we obtain as exact perturbation (11), that is,(22)1-kpk=∑m∑i〈〈Φ+(r,E)χd,i(m)(E)βi(m)(r)〈ν(m)(E′)Σf(m)(r,E′)Φp(r,E′)〉E′〉E〉r∑m〈〈Φ+(r,E)χ(m)(E)〈ν(m)(E′)Σf(m)(r,E′)Φp(r,E′)〉E′〉E〉r,where φ+ is the eigenfunction of (18) and φp the eigenfunction of (20). When comparing (22) with (15), it can be noticed that, besides the replacement of φ by φp, the differences lie in the presence of the terms βi(m)(r) (under integration) replacing the terms βi(m). If we assume in both (20) and (22) the following approximation:(23)βi(m)(r)≈βi(m)=ν̅d,i(m)ν̅ψ(m),
equation (20) takes the following form:(24)Ω⋅∇φp(r,Ω,E)+Σt(r,E)φp(r,Ω,E)=1k̃p∑m[χ(m)(E)-∑iχd,i(m)(E)βi(m)]×〈ν(m)(E′)Σf(m)(r,E′)Φp(r,E′)〉E′+〈Σs(r,E′⟶E)Φp(r,E′)〉E′
and (22) becomes(25)β̃effMC=1-k̃pk=∑m∑iβi(m)×〈〈Φ+(r,E)χd,i(m)(E)〈ν(m)(E′)Σf(m)(r,E′)Φp(r,E′)〉E′〉E〉r∑m〈〈Φ+(r,E)χ(m)(E)〈ν(m)(E′)Σf(m)(r,E′)Φp(r,E′)〉E′〉E〉r.As shown in the previous section,the classicalβeffdefinition given in (15) is an “improved” PT first-order approximation of the relationship1-(kp/k)given in (25). Along the point of view of the perturbation approach, leading to a relationship among system integral properties like the eigenvalues and the delayed neutron effectiveness, (24) and (25) are intimately coupled, in the sense that the coherent prompt flux φp to be used in the βeff definition given by (25) is the eigenfunction of (24). We recall that (24) is an approximation of (20), owing to the assumption given in (23). If we want to remove this assumption, we have to make explicit the β(r) definition given in (21) and insert it into (20), thus obtaining(26)Ω⋅∇φp(r,Ω,E)+Σt(r,E)φp(r,Ω,E)=1kp∑m[∑iχd,i(m)(E)ν̅d,i(m)〈Σf(m)(r,E′)φp(r,E′)〉E′χ(m)(E)〈ν(m)(E′)Σf(m)(r,E′)Φp(r,E′)〉E′-∑iχd,i(m)(E)ν̅d,i(m)〈Σf(m)(r,E′)Φp(r,E′)〉E′]+〈Σs(r,E′⟶E)Φp(r,E′)〉E′.
Following the same perturbation approach, that is, perturbed system given by (26) and unperturbed system given by (17), we find(27)βeffMC=1-kpk=∑m∑iν̅d,i(m)〈〈Φ+(r,E)χd,i(m)(E)〈Σf(m)(r,E′)Φp(r,E′)〉E′〉E〉r∑m〈〈Φ+(r,E)χ(m)(E)〈ν(m)(E′)Σf(m)(r,E′)Φp(r,E′)〉E′〉E〉r.Analogously,the classicalβeffdefinition given in (14) is an “improved” PT first-order approximation of the relationship1-(kp/k)given in (27).Also in this case (26) and (27) are intimately coupled, in the sense that the coherent prompt flux φp to be used in the βeff definition given by (27) is the eigenfunction of (26). The previous elaboration can be summarized in the following way:(a)
as introduced in the previous section, the classicalβeff definitions, like those given in (14) and (15), which do not require any prompt flux calculation, are “improved” PT first-order approximations of the relationship 1-(kp/k) used by Monte Carlo calculations;(b)
the relationship1-(kp/k) provides a βeff estimate coherent with the assumptions made when performing the prompt flux calculation, that is, (24) provides the estimate given by (25), and (26) provides the estimate given by (27).


## 4. ERANOS Formalism and Calculation Set up

The followingβeff formulation is adopted in ERANOS [11] (given in the energy multigroup scheme, with g as energy group index):(28)βeff=∑m∑iν̅d,i(m)〈∑gΦg+(r)χ̅d,i,g∑g′Σf,g′(m)(r)Φg′(r)〉r〈∑gΦg+(r)χ̅g∑m∑g′νg′(m)Σf,g′(m)(r)Φg′(r)〉r,
where χ̅d,i,g is a unique delayed neutron spectrum (derived from basic data) for each fissile isotope m, delayed neutron family i, and(29)χ̅g=∑mχg(m)∑g′νg′(m)Σf,g′(m)ψg′∑m∑g′νg′(m)Σf,g′(m)ψg′.
The quantity χ̅g is a unique fission spectrum, used by ERANOS for spatial calculations, for all the fissile isotopes. The weighting flux ψg is obtained by the fuel cell cross section calculation. In (28) the direct flux and the adjoint flux are solutions for the following eigenvalue equations, respectively:(30)Ω⋅∇φg(r,Ω)+Σt,g(r)φg(r,Ω)=1kχ̅g∑m∑g′νg′(m)Σf,g′(m)(r)Φg′(r)+∑g′Σs,g′→g(r)Φg′(r);(31)-Ω⋅∇φg+(r,Ω)+Σt,g(r)φg+(r,Ω)=1k∑mνg(m)Σf,g(m)(r)∑g′χ̅g′Φg′+(r)+∑g′Σs,g→g′(r)Φg′+(r)..It can be seen that, besides the approximation on the delayed neutron spectrum and on the total fission spectrum, (28) is the energy-multigroup version of (14). Considering as perturbed state with respect to the reference state in (30) the solution for the following equation:(32)Ω⋅∇φp,g(r,Ω)+Σt,g(r)φp,g(r,Ω)=1kp∑m∑g′[χ̅gνg′(m)-∑iχ̅d,i,gν̅d,i(m)]Σf,g′(m)(r)Φp,g′(r)+∑g′Σs,g′→g(r)Φp,g′(r),
we obtain as exact perturbation(33)βeffMC=1-kpk=∑m∑iν̅d,i(m)×〈∑gΦg+(r)χ̅d,i,g∑g′Σf,g′(m)(r)Φp,g′(r)〉r〈∑gΦg+(r)χ̅g∑m∑g′νg′(m)Σf,g′(m)(r)Φp,g′(r)〉r.On the basis of the theory outlined in the previous section, (28) is the “improved” PT first-order approximation of (33). Unfortunately, ERANOS cannot calculate the solution for (32) and consequently (33) cannot be applied because, as mentioned before, it uses a unique fission spectrum for all the fissile isotopes (and in (32) both χ̅g and the various χ̅d,i,g appear). In addition, to use (for our scopes) the βeff modules built-in in ERANOS we need a further assumption, that is, we have to evaluate(34)ν̅(m)=∑g′νg′(m)Σf,g′(m)ψg′∑g′Σf,g′(m)ψg′,
with ψg as weighting flux obtained by the cell cross-section calculation. Thus, our new reference state is given by the following equation:(35)Ω⋅∇φg(r,Ω)+Σt,g(r)φg(r,Ω)=1kχ̅g∑mν̅(m)∑g′Σf,g′(m)(r)Φg′(r)+∑g′Σs,g′→g(r)Φg′(r),
with the associated adjoint(36)-Ω⋅∇φg+(r,Ω)+Σt,g(r)φg+(r,Ω)=1k∑mν̅(m)Σf,g(m)(r)∑g′χ̅g′Φg′+(r)+∑g′Σs,g→g′(r)Φg′+(r),
where (of course, because of (34), (37) provides the same result as (29)):(37)χ̅g=∑mχg(m)ν̅(m)∑g′Σf,g′(m)ψg′∑mν̅(m)∑g′Σf,g′(m)ψg′.
Then, setting(38)χ̅p,g=∑m[χ̅gν̅(m)-∑iχ̅d,i,gν̅d,i(m)]∑g′Σf,g′(m)ψg′∑m[ν̅(m)-∑iν̅d,i(m)]∑g′Σf,g′(m)ψg′,
Equation (32) becomes(39)Ω⋅∇φp,g(r,Ω)+Σt,g(r)φp,g(r,Ω)=1kpχ̅p,g∑m[ν̅(m)-∑iν̅d,i(m)]∑g′Σf,g′(m)(r)Φp,g′(r)+∑g′Σs,g′→g(r)Φp,g′(r).On the basis of (16), (39) is equivalent to:(40)Ω⋅∇φp,g(r,Ω)+Σt,g(r)φp,g(r,Ω)=1kpχ̅p,g∑m[1-∑iβi(m)]ν̅(m)∑g′Σf,g′(m)(r)Φp,g′(r)+∑g′Σs,g′→g(r)Φp,g′(r)
Considering now (39) as the perturbed state with respect to the reference state in (35), taking into account (38), we obtain(41)βeffMC=1-kpk=∑m∑iν̅d,i(m)×〈∑gΦg+(r)χ̅d,i,g∑g′Σf,g′(m)(r)Φp,g′(r)〉r〈∑gΦg+(r)χ̅g∑mν̅(m)∑g′Σf,g′(m)(r)Φp,g′(r)〉r.
The corresponding “improved” PT first-order approximation of (41) is(42)βeff=∑m∑iν̅d,i(m)〈∑gΦg+(r)χ̅d,i,g∑g′Σf,g′(m)(r)Φg′(r)〉r〈∑gΦg+(r)χ̅g∑mν̅(m)∑g′Σf,g′(m)(r)Φg′(r)〉r.
and Equation (39), both sides of (41) and (42) can be calculated by ERANOS.


## 5. ERANOS Results

The GUINEVERE system, shown in Figure1, is selected to perform the numerical analysis. The fuel fissile zone is made of Uranium 30% (weight) enriched in U235. The fuel matrix has the following volumetric fractions: Uranium 17%, Stainless Steel 16%, Lead 60% and Air 7%. The zone Air + SS Sheet has the following volumetric fractions: Stainless Steel 93% and Air 7%. The RZ system is a simplified schematization of the GUINEVERE start-up (at critical) configuration (after start-up the central zone is replaced by the beam tube of the deuteron accelerator delivering, in continuous or pulsed mode, 14 MeV neutrons by deuterium-tritium reactions).Figure 1
Simplified RZ model of the GUINEVERE start-up (at critical) configuration. Dimensions, not in scale, are given in cm.ERANOS transport calculations are performed with 49 energy groups (see the appendix), P0 transport approximation, and angular quadrature S4 (module BISTRO [8]). The neutron data library JEFF 3.1 [9] is used for the calculations. In Table 1 the delayed neutron data ν̅d,i(m) [12] used for the analysis are shown. Delayed neutron spectra χ̅d,i,g are the ones built-in in ERANOS and taken from [12].Table 1
Delayed neutron dataν̅d,i(m) (from [12]).
Family12345678SumU2350.000550.002450.001620.003260.005090.001520.001420.000390.01630U2380.000390.004840.001740.006370.013670.009210.005950.004330.04650The followingν̅(m) values were obtained by (28): ν̅(235)=2.49 and ν̅(238)=2.75. This high ν̅(238) value is due to the fact that about 61% of U(238) total fissions occur in the energy range 2 ÷ 8 MeV, where v(238) values span from 2.67 to 3.48 (only about 5% of U(238) total fissions occur above 2 MeV). In Figure 2 a comparison between the total fission spectrum χ̅g obtained by (37) and the prompt fission spectrum χ̅p,g obtained by (38) is shown. In Table 2 a synthesis of the results is reported.Table 2
Synthesis of the results.
CaseEquationkkpβeff (pcm)1−kp/k (pcm)(a)(28) for βeff723(b)(31) for φ1.01955(c)(31) for φ+1.01955(d)(35) for φ1.02149(e)(36) for φ+1.02149(f)(39) for φp1.01402(g)(41) for 1−kp/k732(h)(41) for βeff (using φp)732(i)(42) for βeff (using φ)731Table 3
The 49 energy group scheme.
Upper energy (eV)Group #Upper energy (eV)Group #Upper energy (eV)Group #Upper energy (eV)Group #1.96E + 0712.470E + 06141.11E + 05271.49E + 02401.73E + 0722.23E + 06156.74E + 04289.17E + 01411.49E + 0732.02E + 06164.09E + 04296.79E + 01421.38E + 0741.65E + 06172.48E + 04304.02E + 01431.16E + 0751.35E + 06181.50E + 04312.26E + 01441.00E + 0761.22E + 06199.12E + 03321.37E + 01458.19E + 0671.11E + 06205.53E + 03338.32E + 00466.70E + 0681.00 + 06213.35E + 03344.000E + 00476.07E + 0699.07E + 05222.03E + 03355.40E − 01485.49E + 06108.21E + 05231.23E + 03361.00E − 01494.49E + 06114.98E + 05247.49E + 02373.68E + 06123.02E + 05254.54E + 02383.04E + 06131.83E + 05263.04E + 0239Figure 2
Total fission spectrumχ̅g (37) and prompt fission spectrum χ̅p,g (38).Case (a) is the classicalβeff formulation adopted in ERANOS, with φ and φ+ from cases (b) and (c), respectively. Cases (d) and (e) differ from (b) and (c) for the presence of ν̅(m) in the fission production terms, and this approximation provides a difference of +186 pcm with respect to the standard treatment of the fission production terms. The prompt case (f), again with the presence of ν̅(m) in the fission production terms, provides a difference of −721 pcm (~βeff) with respect to cases (d) and (e).Case (g), representing the approach used by Monte Carlo calculations, provides exactly the same value as in case (h), that is, the last term in (41). This numerical result confirms the theoretical analysis described above, linking the Monte Carlo formulation to the PT approach. Finally, case (i) shows that the replacement of the prompt flux in case (h) by the total flux does not affect appreciably the βeff value (at least for the fast system considered).Actually, the differences betweenφ and φp, from cases (d) and (f) in Table 2, respectively, are not remarkable, as it can be seen from Figure 3 (where φ and φp are undistinguishable). φ and φp in Figure 3 are spatial average values over the core volume, same area under φ and φp curves.Figure 3
Total and prompt fluxes, together with spectral differencesΔ=(φp,g-φg)/φg. (φp,g from case (f) and φg from case (d) in Table 2, spectral differences on right axis).


## 6. Conclusions

When evaluatingβeff by the relationship 1-(kp/k), the quality of the obtained results depends on the quality of the description of the delayed neutron emissions assumed in the kp prompt calculation. It could seem obvious, but if we look at the standard route used in ERANOS (and in general in deterministic codes) to calculate βeff, by a formulation which does not require any prompt flux calculation, we realize that great detail is given to the characteristics of the delayed neutron emission through the βeff formulation, and without the need to transpose such a detail into a full system prompt neutron calculation. For example, in the ERANOS case, it is not possible to set up a prompt calculation having the same quality of the delayed neutron emission description available in the βeff formulation. And, in fact, in the present case we are obliged to lower the level of the information about the delayed neutron emission in the βeff definition in order to run the corresponding prompt calculation.Following the perturbation theory approach, a rigorous relationship may be established between theβeff evaluation by the formula 1-(kp/k), used in Monte Carlo codes, and the corresponding βeff calculation by the classical formulation involving direct and adjoint fluxes, and it has been shown how the classical formulation can be considered an “improved” PT first-order formulation of the formula 1-(kp/k). By means of a transport computational analysis using a 49 energy group structure and carried out in a coherent and consistent way, that is, using the same deterministic code ERANOS and neutron data library JEFF 3.1 for the βeff evaluation in different ways, the theoretical analysis is numerically confirmed.Both theoretical and numerical results confirm the effectiveness of theβeff evaluation by the relationship 1-(kp/k), at least in cases where spectral differences between total and prompt fluxes are negligible with respect to the value of the functionals entering the classical βeff formulation. For other material configurations more investigations are needed to obtain a detailed quantification of the effects involved.



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*Source: 584256-2011-08-22.xml*