Issue 
EPJ Nuclear Sci. Technol.
Volume 1, 2015



Article Number  2  
Number of page(s)  9  
DOI  https://doi.org/10.1051/epjn/e2015500415  
Published online  27 November 2015 
https://doi.org/10.1051/epjn/e2015500415
Regular Article
Monte Carlo MSM correction factors for control rod worth estimates in subcritical and nearcritical fast neutron reactors
^{1} Laboratoire de Physique Corpusculaire de Caen, ENSICAEN/Université de Caen/CNRSIN2P3, 14050 Caen, France
^{2} SCK·CEN, Belgian Nuclear Research Centre, Boeretang 200, 2400 Mol, Belgium
^{3} Nuclear Fission Division, CIEMAT, Madrid, Spain
^{4} Laboratoire de Physique Subatomique et de Cosmologie, Université GrenobleAlpes, CNRS/IN2P3, 53, rue des Martyrs, 38026 Grenoble Cedex, France
^{5} Institut de Physique Nucléaire d’Orsay, CNRSIN2P3/Université Paris Sud, Orsay, France
^{6} Commissariat à l’Énergie Atomique et aux Énergies Alternatives, DEN, DER/SPEX, 13108 SaintPaullezDurance, France
^{*} email: lecouey@lpccaen.in2p3.fr
Received:
10
June
2015
Received in final form:
20
July
2015
Accepted:
10
August
2015
Published online:
27
November
2015
The GUINEVERE project was launched in 2006, within the 6th Euratom Framework Program IPEUROTRANS, in order to study the feasibility of transmutation in Accelerator Driven subcritical Systems (ADS). This zeropower facility hosted at the SCK·CEN site in Mol (Belgium) couples the fast subcritical lead reactor VENUSF with an external neutron source provided by interaction of deuterons delivered by the GENEPI3C accelerator and a tritiated target located at the reactor core center. In order to test online subcriticality monitoring techniques, the reactivity of all the VENUSF configurations used must be known beforehand to serve as benchmark values. That is why the Modified Source Multiplication Method (MSM) is under consideration to estimate the reactivity worth of the control rods when the reactor is largely subcritical as well as nearcritical. The MSM method appears to be a technique well adapted to measure control rod worth over a large range of subcriticality levels. The MSM factors which are required to account for spatial effects in the reactor can be successfully calculated using a Monte Carlo neutron transport code.
© J.L. Lecouey et al., published by EDP Sciences, 2015
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The GUINEVERE (Generator of Uninterrupted Intense NEutrons at the lead VEnus REactor) project [1] was launched in 2006, within the 6th Euratom Framework Program IPEUROTRANS [2], in order to study the feasibility of transmutation in Accelerator Driven subcritical Systems (ADS). This facility hosted at the SCK·CEN site in Mol (Belgium) is presently used in the followup FREYA project (7th European FP) [3]. It couples the fast subcritical leadmoderated reactor VENUSF with an external neutron source provided by the deuteron accelerator GENEPI3C via T(d,n)^{4}He fusion reactions occurring at the reactor core center (Fig. 1). It is partially dedicated to the investigation of techniques of online subcriticality monitoring.
Fig. 1 Overview of the GUINEVERE facility at SCK·CEN. 
The VENUSF reactor core is very modular and its reactivity can range from deep subcritical to critical by varying the number of fuel assemblies loaded in the core. It is also equipped with two boron carbide control rods which allow for a finer tuning of the reactivity. Fission chambers, spread throughout the reactor, allow recording count rates during either steadystate or timedependent measurements.
In order to test online subcriticality monitoring techniques, the reactivity of all the VENUSF configurations used must be known beforehand to serve as benchmark values. Thus, the reactivity worth of the control rods must be known as accurately as possible so that the reactivity of every new reactor configuration created by moving the control rods be estimated correctly.
Although the reactor asymptotic period measurement is a usual technique to determine the reactivity worth of control rods, it is limited to a small reactivity range (from ≈ –0.3 $ to +0.3 $) [4]. Consequently, it does not always allow measuring the total reactivity worth of the control rods. Furthermore, it is obviously inapplicable to control rod worth measurement in deep subcritical reactors.
This is the reason why the Modified Source Multiplication Method (MSM) [5] is under consideration to be used as an alternative method for estimating the reactivity worth of the VENUSF control rods when the reactor is largely subcritical as well as nearcritical. In this technique, the unknown reactivity is determined by comparing detector count rates driven by an external neutron source in the configuration of interest (in this paper it will be a new configuration obtained by moving the control rods) with those obtained with the same neutron source in another subcritical configuration whose reactivity is already known (reference configuration). However, to account for the flux shape differences between the two reactor configurations, some positiondependent correction factors (the socalled MSM factors) must be calculated using a neutron transport code.
In this paper, we first present the GUINEVERE facility and the various configurations of the VENUSF reactor studied. Then the principle of the MSM method is briefly exposed. The results of MSM factor calculations performed with the Monte Carlo neutron transport code MCNP are also shown. They were carried out in support to MSM experiments dedicated to the measurement of the VENUSF control rod worth when the reactor was either subcritical or near critical. In the former case, the GENEPI3C was used to generate the neutron external source. In the latter case, an AmBe neutron source was inserted in the reactor.
General trends in the MSM factor behavior which depend on the neutron source and detector locations, as well as on the reactor subcriticality level are outlined. Finally the calculated MSM correction factors are applied to the detector count rates measured during the MSM experiments. The consistency between the reactivity values given by the detectors is discussed.
2 The GUINEVERE facility
2.1 The VENUSF reactor
The VENUSF fast reactor is contained in a cylindrical vessel of approximately 80 cm in radius and 140 cm in height. A 12×12 grid surrounded by a square stainless steel casing can receive up to 144 elements of ≈8×8 cm^{2} in section which can be fuel assemblies, lead assemblies or specific elements for accommodating detectors or absorbent rods. The remaining room in the vessel is filled with semicircular lead plates, which act as a radial neutron reflector. In addition, the core is equipped with top and bottom 40 cmthick lead reflectors. Each fuel assembly (FA) contains a 5×5 pattern, filled with 9 fuel rodlets and 16 lead bars, surrounded by lead plates. The fuel is 30 wt.% enriched metallic uranium provided by CEA. Among the set of FAs, six are actually safety rods (SR) made of boron carbide and fuel followers with the absorbent part retracted from the core in normal operation. Two control rods (CR) made of natural boron carbide square cuboids can be positioned at various locations in the 12×12 grid. They can be moved vertically from 0 mm (fully inserted in the core) to 600 mm (fully retracted). Another absorbent rod, whose reactivity worth is very small, called PEAR (Pellet Absorber Rod) rod, is available for performing rod drop experiments.
Various configurations of the reactor in terms of reactivity can be studied thanks to the modular shape of the core. In this paper, since we are interested in measuring the reactivity worth of the set of two CRs, all the reactor configurations studied were obtained from either a nearcritical reactor configuration called CR0↓ or a subcritical configuration named SC1↓, by moving the two CRs together at various heights. Since the reactivities of the CR0↓ and SC1↓ configurations had been measured during previous experiments [6], they could serve as reference values for applying the MSM method.
The socalled CR0↓ configuration is represented in Figure 2. Ninetyseven FAs (in blue for the regular ones, in light blue for the SRs with fuel followers) are arranged in a way to create a pseudocylindrical core. The two boroncarbide CRs (in red) are located at the core periphery and retracted at approximately 515 mm in height. The CR0↓ configuration was created from a critical one by dropping the PEAR rod (in green). After analyzing the rod drop experiments using Inverse Point Kinetics, the reactivity of CR0↓ was found to be –136(2) pcm [6]. As shown in Figure 2, the reactor was equipped with 9 fission chambers (FCs) working in pulse mode. Three different types of FCs were used, either Photonis CFUL01 and CFUM21^{1}, or GE ReuterStokes (RS), whose specifications are listed in Table 1. In order to help localizing the various assemblies and detectors, an arbitrary coordinate system is used in the 12×12 grid: the upper left corner is labeled (–6,6) and the lower right one (6,–6), there is no (0,0) element. Outside the 12×12 grid, six cylindrical cavities bored in the outer reflector can receive experimental devices. They are labeled, from left to right: A1, B1, C1, A2, B2, and C2.
Fig. 2 Schematic view of the CR0↓ configuration. The black star shows the position of the external neutron source (AmBe). Control rods (CR) are in red. 
Fission chambers used in configurations CR0↓ and SC1↓.
The socalled SC1↓ configuration is shown in Figure 3. It is derived from the CR0↓ configuration by removing the four central FAs. This removal also permits the insertion of the accelerator thimble inside the VENUSF core.
Fig. 3 Schematic view of the SC1↓ configuration. The black star shows the position of the external neutron source (GENEPI3C). Control rods (CR) are in red. 
Compared to CR0↓, some additional minor differences are itemized below:

the CRs are slightly more inserted inside the core (CR height is 479 mm instead of 515 mm);

the detector set is slightly different: the CFUL01673 detector is replaced by the CFUL01653 FC. The latter is replaced in the A1 location by the CFUL01658 FC which is identical to CFUL01659 and CFUL01653 FCs. The reactivity of SC1↓ was measured using the MSM method and was found to be –3824(96) pcm [6].
2.2 External neutron sources
The external neutron source used for performing the MSM experiments was different depending on whether the reference configuration was CR0↓ or SC1↓.
In the latter configuration, the external source was created at the center of the VENUSF core by deuterons interacting with a tritiated titanium target. The deuteron ions were accelerated up to an energy of 220 keV by the GENEPI3C particle accelerator [7] built by a collaboration of CNRSIN2P3 laboratories. The fusion reactions at core midplane generate a quasiisotropic field of ∼14MeV neutrons. The GENEPI3C can operate in pulsed mode, in continuous mode, and also in continuous mode with short beam interruptions. During the MSM experiments reported here, GENEPI3C delivered a continuous deuteron beam whose intensity ranged from ∼400 μA to ∼500 μA.
However, the intensity of the external neutron source created by the accelerator had to be monitored directly. Indeed, the tritium release and the beam tuning variations over time prevent the direct calculation of the neutron source intensity from that of the beam on target. This is the reason why the accelerator is equipped with two Si detectors which can detect either alpha particles from T(d,n)^{4}He reactions or protons from D(d,p)T reactions. The detection of α particles allows one to quantify the amount of 14MeV neutrons produced whereas the detection of protons allows estimating the parasitic production of 2.5MeV neutrons by D(d,n)^{3}He due to the implantation of deuterons in the target. During the MSM experiments reported here, the neutron source intensity varied from ∼1×10^{9} to ∼3×10^{9} 14MeV neutrons/s.
In the CR0↓ configuration, the external neutron source was an AmBe source inserted in the outer reflector slot A2 (denoted by a star in Fig. 2) which emitted only 2.2×10^{6} neutrons/s. Thus the AmBe neutron source intensity is lower than that induced by the GENEPI3C by three orders of magnitude. Furthermore, mainly because the AmBe source is offcentered, its importance is approximately eight times lower than that of GENEPI3C. In terms of detector count rates, these source dissimilarities are only (very) partially compensated by the difference in reactivity between the two CR0↓ and SC1↓ configurations. Therefore, since the two reference configurations are very dissimilar both in terms of reactivity and of source location, interesting differences in the results of the MSM experiments can be anticipated.
3 The MSM method
3.1 Principle
The MSM (Modified Source Multiplication) method is a technique for estimating the unknown reactivity of a subcritical configuration by comparing detector count rates driven by an external neutron source in this configuration with those obtained in another subcritical configuration whose reactivity is known.
The inhomogeneous transport equation associated with a subcritical configuration of a reactor driven by an external neutron source reads:(1)where P is the neutron production operator (by fission or (n,xn) reactions), A is the migration and loss operator and S is the external neutron source intensity. Φ is the neutron flux which is present inside the reactor when the external neutron source is inserted.
This transport equation can be made homogeneous by introducing the neutron multiplication coefficient k_{eff}:(2)
In that case, φ is the fundamental mode corresponding to the associated critical reactor. λ=1/k_{eff} is also an eigenvalue of the adjoint homogeneous equation:(3)
where and are the adjoint operators of A and P, respectively. is the adjoint flux, also called neutron importance function.
Multiplying the adjoint homogeneous equation (3) by Φ and integrating over space, angle and energy, one gets:(4)where denotes such an integration.
Then, multiplying the inhomogeneous equation (1) by and integrating over space, angle and energy leads to:(5)and combining equation (4) and equation (5), one gets:(6)
As in reference [5], we introduce the reaction rate in the detector , where Σ_{d} is the macroscopic reaction crosssection of the detector, and rewrite equation (6):(7)where is called the effective neutron source and the detector efficiency.
Now let us consider two subcritical configurations. Let configuration 0 be the subcritical configuration of known reactivity ρ_{0} and configuration 1 be that of unknown reactivity ρ_{1}. Assuming that the neutron external source and the detectors utilised are the same in both configurations, equation (7) can be used to find a relationship between ρ_{0}, ρ_{1}, and the detector count rates C_{0} and C_{1} in configurations 0 and 1:(8)where f_{MSM} is the MSM correction factor. One can also introduce the source importance which is defined as the ratio of the average importance of external source neutrons to the average importance of fissions in the reactor [8,9]:(9)
If one introduces the source multiplication coefficient k_{s} as [9]:(10)the source importance appears as the ratio of the neutron gain with the external neutron source to a hypothetical gain which would be obtained with a stabilized fission source in the same reactor:(11)
Then the MSM factor can be rewritten with the source importance of the two configurations:(12)where and are respectively the source importance and the source multiplication coefficient in configuration i. The parameter e_{i} is defined as:(13)
It represents the ratio of the reaction rate in the detector to the total rate of neutron produced in the reactor. Thus, formula (12) shows that the MSM factor accounts for the differences in neutron and source importance as well as in flux shapes between the two configurations considered.
However, if configurations 0 and 1 are very similar, such differences may vanish and formula (8) reduces to the Approximate Source Method (ASM) formula:(14)where the approximation k_{eff,0}/k_{eff,1} ≈ 1 is often made.
The MSM correction factors must be calculated using a transport code, either deterministic or stochastic. It is worth mentioning that the value of the MSM correction factor is expected to depend on the detector location. Indeed, any difference in the flux shape between the two configurations will result in positiondependent ratios in the f_{MSM} formula.
3.2 Calculation of MSM factors
Starting from equation (8), the MSM factor reads:(15)where the reactivity of configurations 0 and 1, ρ_{0} and ρ_{1}, as well as the detector count rates in configurations 0 and 1, C_{0} and C_{1}, can be calculated using a neutron transport code.
Although the use of deterministic codes is largely reported in literature, MSM factors can also be calculated using stochastic neutron transport codes. On one hand, the use of a Monte Carlo code advantageously allows one to transport neutrons in the reactor theoretically without any geometry simplification (to the extent that the reactor geometry be accurately known) and with pointwise energy dependent crosssections. On the other hand, Monte Carlo calculations are much more computertimeconsuming than deterministic ones and provide as results only statistical estimates of quantities of interest. In this paper, the Monte Carlo simulation code MCNP 5 [10] was employed, together with ZZ ALEPHLIBJEFF3.1.1, a continuous energy multitemperature library created at SCK·CEN and based on JEFF3.1.1 [11]. Once the geometry as well as the material composition of the various elements constituting the reactor have been described in an MCNP input file, the corresponding multiplication factor (hence the reactivity) can be estimated using a generationbased, iterative fission neutron source whose spatial distribution converges towards the fundamental mode of the reactor (the socalled “kcode” source). On the other hand, standard fixedsource calculations can provide estimates of reaction rates anywhere in the reactor. So, for the calculation of MSM factors, four Monte Carlo simulations must be run: two fixedsource simulations for calculating the source driven reaction rates C_{0} and C_{1} in the fission chambers for configurations 0 and 1, and two “kcode” simulations for estimating the reactivity of the same two configurations, ρ_{0} and ρ_{1}.
As a first step towards the calculation of MSM factors, MCNP input files had to be built for the configurations CR0↓ and SC1↓ as well as their variants created by moving the CRs. In order to save computing times (a factor of ∼4.5 was gained), it was decided to use a simplified reactor geometry. Indeed, the MSM method bears interest only if the calculation of MSM factors turns out to be rather insensitive to the details and errors on the reactor geometry, as well as to uncertainties on material compositions and on nuclear data: since MSM experiments are carried out to estimate the unknown reactivity of a reactor configuration, one can imagine that the reactor itself could be not very well known either. Fortunately, this robustness of MSM factor calculations has already been observed for previous MSM experiments at the VENUSF reactor and can be understood by recalling that MSM factors are double ratios of quantities: one can expect that any reasonable difference between the calculated reactivity values and the real ones will be at least partially compensated by corresponding differences between the calculated reaction rates and the measured ones [6].
Since the control and safety rods are nearly homogeneous, the principal source of geometrical simplification was the homogenization of the fuel assemblies. Additionally, some details of the bottom reactor reflector geometry were not considered. Also, the GENEPI3C accelerator was not modelled. Instead, a 14MeV point source was placed in vacuum at the core center. For the AmBe source, the average source energy of 5 MeV was used. Finally, the FCs were not modelled at all. Instead, use was made of the nextevent estimator MCNP tally F5 (point detector) to estimate the fission rates of the FC deposits, at the center of each detector location.
One MCNP input file was created for each CR height selected for the MSM experiments (from 0 to 600 mm by step of 60 mm around the reference CR position of the SC1↓ configuration and by step of 50 mm around the one of CR0↓). Then, prior to calculating the four terms of formula (15), the reactivity scale of the MCNP models of VENUSF configurations had to be adjusted so that the calculated reactivities of CR0↓ and SC1↓ be approximately equal to their measured values of –136 pcm and –3824 pcm, respectively. This allowed an overall consistency between experimental results concerning the configurations used as references and the subsequent calculations. This was achieved by multiplying the average number of neutrons per fission ν used inside the MCNP code by a factor of 1.001071. This slightly modified value of ν was then used for calculating the reactivity of all the other configuration variants obtained by changing the CR heights.
3.3 Results of MSM factor calculations
Figure 4 shows the evolution of MSM factors as a function of the new height of the CRs after moving them away from their position associated with the reference configuration SC1↓ (479 mm). Error bars were calculated using the quadratic sum of the uncertainties on the four terms of formula (15). The relative uncertainty is basically dominated by that on the fission reaction rates for which a precision of less than 1% could be achieved in a reasonable computing time.
Fig. 4 MSM factors as a function of FC position and CR height using SC1↓ as reference. 
The first observation to be made is that, as expected, regardless of the detector position, there is roughly no MSM correction to consider when the CRs do not move much around 479 mm. However, as the amplitude of the CR motion and thus the dissimilarity between the neutron flux shapes increases, the MSM factors tend to deviate more and more from 1. Regarding the evolution of the MSM factors, the detectors seem to fall into three or four different groups. In the first group (positions (–2,–6), (–6,6) and (6,–6)), the MSM factors do not deviate much from 1, even for the largest CR motion. This can be explained by the fact that those three detectors are rather far from the source and far from the CRs and therefore rather protected from the modifications of the flux shape caused by the CR motion. It is less and less the case as we move from the first group to the second one (detectors in (6,–2), (–6,–6), (6,6)) and then to the third one (detectors in A1, C2 and C1).
In conclusion, in the case of the SC1↓ configuration, it seems possible to estimate the CR worth using a simple ASM approach without any calculated correction factor as long as the detectors are carefully selected.
Figures 5 and 6 deal in the same way with the MSM factors for the CRs moving when the reactor is almost at critical, that is when the reference configuration used for the MSM calculation is CR0↓ (CR height at 515 mm). The color code for detector positions is identical to that of Figure 4. First of all, it is worth mentioning that the statistical error bars are significantly larger than those shown in Figure 4. Indeed, since CR0↓ is almost critical, the reactivity values are rather close to zero and the relative uncertainties on the calculated reactivity tend to be much larger than those calculated for the configurations based on SC1↓. Furthermore, some MCNP fixedsource calculations needed for estimating detector fission rates can become very computertimeconsuming as the CR height increases, and hence as the multiplication factor k_{eff} becomes very close to 1. To quantify this evolution, one can make use of the Figure of Merit (FOM) [10] which is defined as:
Fig. 5 MSM factors as a function of FC position and CR height using CR0↓ as reference. 
Fig. 6 MSM factors associated with detector CFUL01673 in C2 as a function of CR height using CR0↓ as reference. 
As in the case where the reference configuration is SC1↓, the MSM factors tend to deviate more and more from 1 as the amplitude of the CR motion increases, as expected by the associated larger perturbation of the reference flux. However, compared to Figure 4, two main differences are visible in Figures 5 and 6.
The most striking one concerns the FC located in C2 (Fig. 6). The behavior of the associated MSM factor is so different from the others that it had to be shown in a separate figure. This is due to the extremely short distance from the AmBe source (13.5 cm) combined to the high energy threshold of ^{238}U (∼1 MeV) which represents 99.965% of the CFUL01673 deposit mass. Thus, on the one hand, CFUL01673 is proportionally much more sensitive than the other FCs to the fast neutrons originating directly from the AmBe source and, on the other hand, it is much less sensitive to the regular neutron multiplication in the core.
When looking at Figure 5, it also appears that the clear division of detectors in groups proposed for Figure 4 does not hold any longer. Although some detectors very close to one CR exhibit strong correction factors, such as RS10075, it is not the case for the FC located in (6,–2). On the other hand, the detector located in (–6,6) is very far away from the CRs and from the AmBe source and still, its MSM correction factor is far from remaining close to 1 when the CRs are moved.
To understand why these MSM factors exhibit a much more complex behavior than in the case of those calculated for SC1↓ and its variants, it is worth using formula (12). The latter relates the MSM factors to the ratio of the source importance in the configuration of interest to that in the reference configuration. The source importance can be easily calculated by combining the results of MCNP kcode and fixedsource calculations.
As already mentioned herein above, the AmBe source importance is much smaller (∼0.3) than the GENEPI3C one (∼2.5), mainly because of the difference in the source location. Hence, it is more fruitful to compare the source importance variations (compared to the value of φ* taken arbitrarily at 0 mm) as a function of CR height for the variants of the configurations based on CR0↓ and for those based on SC1↓. Results (in %) are shown in Figure 7.
Fig. 7 Source importance variations as a function of the CR height. 
The difference in behavior of the source importance between the two sets of reactor configurations is striking. On one hand, in the case of SC1↓derived configurations (in red), the variation of source importance as the CR height changes, if any, is very small. On the other hand, when considering CR0↓derived configurations (in black), the source importance appears to increase significantly as the CRs are raised. This is obviously due to the very short distance between the AmBe source and one of the two CRs, which makes the source multiplication very sensitive to the motion of the neutron absorbent (while the external neutron source was at the core center in the case of the variants based on SC1↓). Hence, whereas the MSM factor evolution seems to be mainly explained by the flux shape modification occurring around the CRs when they are moved from the 479 mm position for SC1↓, the MSM factor variation with the CR motion around CR0↓ seems to be more complex. In this case, the source importance and the flux shape are both modified.
In short, the behavior of the MSM correction factors as a function of the detector position appears to be much more complex when using the AmBe source instead of the external neutron source provided by means of the GENEPI3C accelerator. This difference is likely to originate from the offcentered position, close to one CR, of the AmBe source.
4 Application to MSM experiments
It is beyond the scope of this paper to apply the MSM factors calculations presented herein to the numerous measurements performed at the VENUSF reactor. Instead we present hereinafter a few results which illustrate the most the performances of the MSM method with Monte Carlo simulations.
4.1 MSM experiments with the SC1↓ configuration
As shown in Figure 4, the farther from that of the reference configuration (479 mm) the CR height is, the larger the MSM correction factor is. This is the reason why the results corresponding to the CR heights settled at 0 mm have been selected and are shown in Table 2.
ASM reactivity, MSM factor and MSM reactivity when CRs are lowered to 0 mm (from SC1↓) for each of the nine FCs used.
First the ASM reactivity of the VENUSF reactor when the CRs are positioned at 0 mm was calculated from each detector count rate as follows:(16)where is equal to –3824 ± 96 pcm. Since the neutron external source was created by means of the GENEPI3C accelerator, a specific normalization had to be applied to the detector count rates, and , measured respectively in the reference configuration (SC1↓, CRs at 479 mm) and in the other selected configuration (SC1↓, CRs at 0 mm). In order to take into account the fluctuations of the neutron production in the tritiated target, the total numbers of counts in the detectors were normalized per alpha particle detected in the dedicated Si detector, instead of per second.
As can be seen in Table 2, the ASM reactivity varies by about 550 pcm, depending on the detector considered. Some spatial effects are definitely at work. Now, applying the MSM factors calculated with MCNP simulations which are shown in Figure 4 and also listed in Table 2, the MSM reactivity can be built for each detector as follows:(17)
After multiplication by the MSM factors, the dispersion of the detector reactivity values is successfully reduced to ∼120 pcm.
To finish with the SC1↓related experiments, we make use of one interesting result of the MSM factor calculations. In Figure 4, one can see that the MSM factors for three FCs (positions (–2,–6), (–6,6) and (6,–6)) are very close to 1. This suggests that the simple ASM method applied to these three detectors might give the same results as the MSM method does. The comparison of the reactivity results of the ASM method applied to the three detectors with the MSM ones using the full set of FCs, as a function of CR height, is presented in Figure 8. For each CR position and each method, the reactivity was calculated as the weighted mean of the values given by the two different sets of selected detectors (3 and 9 FCS for the ASM and MSM methods, respectively):(18)
Fig. 8 Reactivity of VENUSF as a function of CR height when CRs are moved together from 0 to 600 mm, around the SC1↓ configuration. 
As can be seen in Figure 8, the agreement between the two methods is excellent over the whole range of CR heights. Not only it validates the use of MSM factor calculations to choose the right detector subset to use an ASM approach but it also shows that the measurements made with all the other detectors can be well corrected. It is also remarkable that, once the kcode MCNP reactivity values are properly scaled by adjusting the calculated reactivity of SC1↓ to the measured one, they (in blue) also are in very good agreement with the experimental results over the whole range of CR height. In addition, it is worthwhile to mention that some dynamical reactivity measurements carried out in pulsed neutron source experiments [12] or in experiments with programmed interruptions of a continuous beam [13], with the CR heights at 0, 240, 479 and 600 mm, gave results consistent with those presented here.
4.2 MSM experiments with the CR0↓ configuration
Since, as in the case of the SC1↓ experiments, the MSM corrections to the ASM reactivity values are expected to be the strongest at 0 mm for the CR0↓ experiments, the results obtained for this height are gathered in Table 3. The spread observed among the ASM reactivity values is much more dramatic (∼850 pcm) because of the presence of the CFUL01673 FC with ^{238}U as main deposit (Sect. 3.3). However, once the MSM factors have been applied, the dispersion of the results drops down to ∼80 pcm. Even though the CFUL01673 final reactivity value seems to be slightly overestimated (in absolute value), one has to keep in mind that the MSM correction is a tourdeforce for this detector considering the amplitude of the correction to be applied.
ASM reactivity, MSM factor and MSM reactivity when CRs are lowered to 0 mm (from CR0↓) for each of the nine FCs used.
5 Conclusions
The socalled Modified Source Multiplication Method (MSM) technique consists in determining the unknown reactivity of a reactor configuration by comparing detector count rates driven by an external neutron source in the configuration of interest with those obtained in another subcritical configuration whose reactivity is already known (reference configuration). This method can be used as an alternative method to the asymptotic period measurement for determining control rod worth.
This paper focused on the use of the Monte Carlo neutron transport code MCNP to calculate positiondependent MSM correction factors needed to account for the flux shape differences between the reference reactor configuration and the configuration whose reactivity is to be measured.
A comparison was made between the MSM factors obtained for a set of nine detectors spread in the leadmoderated fast neutron reactor VENUSF, for a largely subcritical configuration and a nearcritical one. It was found that the MSM factors exhibited some common trends but also that the behavior of these factors was much more difficult to explain simply in the nearcritical case because of the specific location of the external neutron source.
However, in both cases, the MSM factors calculated with the MCNP Monte Carlo code were successfully applied to ASM reactivity values obtained experimentally: the reactivity spread among the detectors was strongly reduced by the MSM correction.
In conclusion, the MSM method seems to be a technique well adapted to measure control rod worth over a large range of subcriticality levels. The required MSM factors can be easily calculated using a Monte Carlo neutron transport code, although the computing time can become very large when the studied reactor configurations are very close to criticality. Consequently, appropriate variance reduction techniques remain to be investigated.
Acknowledgments
This work was partially supported by the 6th and 7th Framework Programs of the European Commission (EURATOM) through the EUROTRANSIP contract # FI6WCT2005516520 and FREYA contract # 269665, and the French PACEN and NEEDS programs of CNRS. The authors want to thank the VENUS reactor and GENEPI3C accelerator technical teams for their help and support during experiments.
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Cite this article as: JeanLuc Lecouey, Anatoly Kochetkov, Antonin Krása, Peter Baeten, Vicente Bécares, Annick Billebaud, Sébastien Chabod, Thibault Chevret, Xavier Doligez, FrançoisRené Lecolley, Grégory Lehaut, Nathalie Marie, Frédéric Mellier, Wim Uyttenhove, David Villamarin, Guido Vittiglio, Jan Wagemans, Monte Carlo MSM correction factors for control rod worth estimates in subcritical and nearcritical fast neutron reactors, EPJ Nuclear Sci. Technol. 1, 2 (2015)
All Tables
ASM reactivity, MSM factor and MSM reactivity when CRs are lowered to 0 mm (from SC1↓) for each of the nine FCs used.
ASM reactivity, MSM factor and MSM reactivity when CRs are lowered to 0 mm (from CR0↓) for each of the nine FCs used.
All Figures
Fig. 1 Overview of the GUINEVERE facility at SCK·CEN. 

In the text 
Fig. 2 Schematic view of the CR0↓ configuration. The black star shows the position of the external neutron source (AmBe). Control rods (CR) are in red. 

In the text 
Fig. 3 Schematic view of the SC1↓ configuration. The black star shows the position of the external neutron source (GENEPI3C). Control rods (CR) are in red. 

In the text 
Fig. 4 MSM factors as a function of FC position and CR height using SC1↓ as reference. 

In the text 
Fig. 5 MSM factors as a function of FC position and CR height using CR0↓ as reference. 

In the text 
Fig. 6 MSM factors associated with detector CFUL01673 in C2 as a function of CR height using CR0↓ as reference. 

In the text 
Fig. 7 Source importance variations as a function of the CR height. 

In the text 
Fig. 8 Reactivity of VENUSF as a function of CR height when CRs are moved together from 0 to 600 mm, around the SC1↓ configuration. 

In the text 
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