Julien Henriet
email: julien.henriet@univ-fcomte.fr

Françoise Greffier
email: francoise.greffier@univ-fcomte.fr

AI-VT: An example of CBR that generates a variety of solutions to the same problem

Keywords:

AI-Virtual Trainer (AI-VT

) is an intelligent tutoring system based on case-based reasoning. AI-VT has been designed to generate personalised, varied, and consistent training sessions for learners. The AI-VT training sessions propose different exercises in regard to a capacity associated with sub-capacities. For example, in the field of training for algorithms, a capacity could be "Use a control structure alternative" and an associated sub-capacity could be "Write a boolean condition". AI-VT can elaborate a personalised list of exercises for each learner. One of the main requirements and challenges studied in this work is its ability to propose varied training sessions to the same learner for many weeks, which constitutes the challenge studied in our work. Indeed, if the same set of exercises is proposed time after time to learners, they will stop paying attention and lose motivation. Thus, even if the generation of training sessions is based on analogy and must integrate the repetition of some exercises, it also must introduce some diversity and AI-VT must deal with this diversity. In this paper, we have highlighted the fact that the retaining (or capitalisation) phase of CBR is of the utmost importance for diversity, and we have also highlighted that the equilibrium between repetition and variety depends on the abilities learned. This balance has an important impact on the retaining phase of AI-VT.

Introduction

We are interested in the issue of the personalisation of learning through training sessions. For us, a training session is a list of exercises suited to each learner. Motivation and repetition are key aspects in teaching. Nevertheless, repetition causes learners to be bored and to turn themselves off. Consequently, teachers must introduce originality and diversity and adapt the exercise level and nature to the learners' acquired skills. Furthermore, teachers must propose varied exercises and consistent sessions while providing training for the same skill over a given number of weeks. Thus, the elaboration of a cycle training session, suited to one particular learner, is a reasoning based on analogy in which it is necessary to introduce some kind of originality. Indeed, on the one hand, this elaboration is based on the past experiences of the trainer as well as the exercises previously proposed to the learner, and on the other hand, the exercises proposed to the learner must not be always the same. As a consequence, a case-based reasoning (CBR) system [START_REF] Kolodner | Case-based reasoning[END_REF], based on analogy reasoning, is a good answer to these kind of systems, but must be adapted in order to introduce diversity in the solutions to be proposed (the training sessions). In addition, this diversity varies from one domain to another. Indeed, the frequency with which an exercise must be proposed to learners in the field of sports is not the same as for learners in the field of algorithmics, for example. As a matter of fact, basic exercises will be proposed often by sports trainers since the body must practise a lot before integrating basic movements and attitudes. On the contrary, proposing an algorithmic exercise that the learner has already successfully completed twice will bore the learner.

Related works

This paper presents Artificial Intelligent -Virtual Trainer (AI-VT), a multiagent system (MAS) that uses CBR to provide consistent training sessions with widely differing progressions. CBR is widely employed in e-learning systems and intelligent tutoring systems (ITS) [START_REF] Graesser | Intelligent tutoring systems[END_REF]. J. L. Kolodner [START_REF] Kolodner | Case-based reasoninginspired approaches to education[END_REF] distinguished two types of CBR-inspired approaches to education: goal-based scenarios [START_REF] Schank | The design of goal-based scenarios[END_REF] where learners achieve missions in simulated worlds, and learning by design [START_REF] Kolodner | Case-based learning aids[END_REF], in which learners design and build working devices to obtain feedback. CBR is actually well-suited to the latter type of system [START_REF] Jamsandekar | Online learning -cbr approach[END_REF], as well as to other tools using artificial intelligence (AI) and distributed AI (DAI) systems, such as genetic algorithm (GA) [START_REF] Biswas | Learning by teaching: A new agent paradigm for educational software[END_REF], Artificial Neural Network (ANN) [START_REF] Gisolfi | Designing complex systems within distributed architectures: an intelligent tutoring systems perspective[END_REF] and MAS [START_REF] Tan | Personalized course generation and evolution based on genetic algorithms[END_REF]. A. Baylari and G. A. Montazer focused on the adaptation of tests to obtain a personalised estimation of a learner's level [START_REF] Gisolfi | Designing complex systems within distributed architectures: an intelligent tutoring systems perspective[END_REF]. They used an ANN in order to correlate learners' answers to the tests and the exercises proposed by teachers. The CBR-and GA-based e-learning system proposed by Huang et al. also provides lessons taking into account the curriculum and the incorrect-response patterns of a pre-test given to the learner [START_REF] Huang | Constructing a personalized elearning system based on genetic algorithm and case-based reasoning approach[END_REF]. O. P. Rishi et al. designed an ITS based on agents and a CBR system [START_REF] Rishi | Distributed case based reasoning for intelligent tutoring system: an agent based student modeling paradigm[END_REF] in which a Personal Agent is responsible for determining learner level. A Teaching Agent then determines the educative strategy with the help of CBR according to the description of the transmitted learner level. Finally, a Course Agent provides and revises the lessons and exercises corresponding to the strategy proposed by the system with the help of a tutor. All these tools provided by AI would therefore produce exactly the same exercises and lessons for training a single given skill, or would propose a large set of exercises as an answer to the diversity requirement and leave the teachers or the learners to choose the most adapted exercises themselves. In this particular domain, repetitive activities are a drawback, yet lesson planning is a process based on adaptation of past experiences. AI-VT tries to address the problem of balance between repetitiveness and the variety of the solutions proposed. Indeed, even if the exercises must be selected by analogy with previously proposed ones, the same exercise proposed too often to one learner may bore her/him. Moreover, the number of propositions varies according to the domain (algorithmic vs. sports, for example) and the level reached by the learner. The problem of variety in CBR-systems is close to the creativity one addressed in the literature [START_REF] Maher | Encouraging curiosity in case-based reasoning and recommender systems[END_REF][START_REF] Grace | Combining cbr and deep learning to generate surprising recipe designs[END_REF][START_REF] Grace | Surprise-triggered reformulation of design goals[END_REF][START_REF] Gero | Modeling creativity and knowledge-based creative design[END_REF][START_REF] Müller | Workflow streams: a means for compositional adaptation in process-oriented cbr[END_REF]. G. Muller and R. Bergmann proposed the introduction of novelty combining different solutions during the adaptation phase [START_REF] Müller | Workflow streams: a means for compositional adaptation in process-oriented cbr[END_REF]. In their approach, source case solutions are decomposed into elementary activities and elements and combined in original ways. This approach allows introduction of diversity and novelty in the solutions proposed by their system. Diversity is also addressed in applications dedicated to recommender systems [START_REF] Smyth | Similarity vs. diversity[END_REF][START_REF] Mcsherry | Diversity-conscious retrieval[END_REF][START_REF] Mcginty | On the role of diversity in conversational recommender systems[END_REF][START_REF] Bridge | Ways of computing diverse collaborative recommendations[END_REF]. These systems select products or services for customers in electronic commerce. In these approaches, the systems select and deliver a set of similar cases and their solutions. In this set, these systems also integrate cases which present some dissimilarities with the described problem part of the target case. The dissimilarities are computed according to different metrics, and the sets of cases are refined successively. In our problem, dissimilarity is not sufficient since the level acquired by the learner must be taken into account, as well as the ease with which each previously proposed exercise has been solved by the considered learner. More recent works go further into the introduction of unexpected results in order to surprise and retain attention. J. Gero and M.L. Maher present the basis of a new approach based on Deep Learning in order to introduce creativity [START_REF] Gero | Modeling creativity and knowledge-based creative design[END_REF]. K. Grace et al. went further with Deep Learning and proposed creative and unexpected concepts which were then adapted to a CBR-cycle process in order to generate original recipes [START_REF] Grace | Combining cbr and deep learning to generate surprising recipe designs[END_REF][START_REF] Grace | Surprise-triggered reformulation of design goals[END_REF]. This neural network is trained to introduce novelty (new ingredients) into a set of preferences by the end-user in order to give recipes with new ingredients [START_REF] Maher | Encouraging curiosity in case-based reasoning and recommender systems[END_REF]. Actually, in these approaches, creativity and originality are treated during the description and adaptation phases of the target case, whereas AI-VT addresses this particular aspect during the retaining and adaptation phases, giving much importance to these CBR-system phases.

Presentation of AI-VT

AI-VT is based on pedagogy which proposes repetition of exercises in order to attain levels of capacities. Once a lack of knowledge is detected by the teacher or the learner, she/he can decide to train for many weeks in order to reach this level. Then, when the user asks the AI-VT for a training session on a particular capacity with a specific duration, the system generates a session organised into sub-capacities and proposes exercises with regard to each sub-capacity. We also considered two different domains of application: practical (sports -Aïkido) and theoretical (computer science -algorithmics). As examples shown in Table 1, in the field of Aïkido, a capacity could be "Using a grip", and "Relaxing despite a grip" and "Pivoting around a grip" could be two associated sub-capacities. In the field of algorithmics, "Design an algorithm" is an example of capacity, and "Find inputs and outputs" and "Associate a type with a variable" are two sub-capacities which can be associated with it. In the first part of this section, we detail the session structure and the requirements of AI-VT. The distributed architecture and the data flows are presented in the second part. Finally, in the third part, we examine how a session is designed.

Lesson structure

In this sub-section, we describe the way a teacher elaborates a training session, the parameters and the way this generation is done, and the behaviour AI-VT should imitate.

We considered activities that guide each training session by reaching one capacity [START_REF] Tyler | Statistical methods for utilizing personal judgments to evaluate activities for teacher-training curricula[END_REF] divided into sub-capacities. These capacities and their order of appearance are decided at the beginning of each session. One specific skill can consequently be assigned to some consecutive sessions. The chosen capacity is then divided into elementary abilities (sub-capacities) that have to be mastered by the learner. We considered that in sports in particular, each skill may be shared by more than one capacity. In all the domains of application we considered (in sports training like Aïkido and theoretical disciplines like algorithmics), the mastery of each skill is a time-consuming process that is reached through the repetition of exercises [START_REF] Bernstein | The co-ordination and regulation of movements[END_REF]. Some will learn faster than others, and thus the teacher must adapt each session to the level of the learner.

AI-VT must (1) propose pertinent sub-capacities and exercises according to the capacity decided and the level already reached by the learner, (2) ensure that no exercise is proposed more than once during a given training session and that the sessions in the same training cycle are varied, and (3) build a consistent training session that begins with the simplest exercise and then continues with a list of exercises that relate sufficiently to the preceding and following ones. In a MAS, an agent is a physical or abstract entity having certain specific characteristics: perception of its environment (including itself and the other agents), the capacity to act (upon itself or the environment) and autonomy in its decisions and actions. In AI-VT, the choice of sub-capacities regarding a given capacity takes place via an autonomous process, as does the determination of exercises regarding a sub-capacity, or of any other exercises chosen and their priority levels. The initial choice of exercises regarding a sub-capacity must be an autonomous process: each agent's autonomy ensures a wise and free selection of the most suitable exercises. These processes can be undertaken simultaneously, coming after the determination of sub-capacities. In addition, each one must interact with the other processes and take their choices into account: the solution proposed by one agent influences the choices made by the others.

System architecture and communication model

As shown in Figure 1, the system is composed of four types of agents: the teacher, the learner, the capacity agent (CA) -which is responsible for choosing the sub-capacities regarding a capacity requested by the teacher -and the exercise agents. Each of these agents is responsible for proposing the exercises best suited to a given sub-capacity. CAs are directly connected to exercise agents, and they can exchange messages. The CA sends the set of sub-capacities it has chosen to one of the exercise agents. This first-contacted exercise agent (EA) endorses the role of coordinator between the CA and the other EAs. This EA assumes responsibility for the first-proposed sub-capacity, and then creates and sends the list to another EA which assumes the second sub-capacity, and so on. The EAs then communicate and share information in order to prepare the requested training session. Each EA proposes exercises concerning its assigned sub-capacity. Each EA takes into account the choices proposed by the other EAs: for example, one of the system's requirements is that each exercise is to be done only once during the entire training session. Thus, the choices of the EAs are shared. The referent version of the training session is transmitted from EA to EA until it fulfils all the requirements. Finally, the EA initially contacted by the CA sends the referent version of the training session back to the CA.

Determination of sub-capacities

The CA is responsible for choosing the set of sub-capacities and their duration. Once the training session has been chosen by the teacher, and after having analysed any additional learner needs, the CA follows the CBR approach to make these choices according to the sub-capacities already achieved and to the learners' degree of assimilation.

For the CA, a case is a set comprised of two parts: a problem and a solution.

Each problem part is composed of a capacity C, and the solution part of a set of (SC, D SC,C ) where SC is a sub-capacity and D SC,C the duration required to reach this SC regarding C. Thus, formally, a source case s is expressed as s = (C, {SC, D S SC,C }). The durations D SC,C are initialised by the teacher at the beginning of the season and updated by AI-VT after the training session, taking into account the evaluation of the learner's acquired level. Since learner levels of expertise rise, we can consider that durations decrease and thus call them 'remaining durations'. Indeed, if the teacher has considered that 60 minutes of training is usually necessary in order to master one sub-capacity, and after the learner has successfully trained herself/himself 20 minutes in this specific sub-capacity, we consider the average learner will then (after this training) need 60 -20 = 40 minutes during the next sessions for the considered sub-capacity.

First, all the sub-capacities associated with C are retrieved. The similarity between each source case s and the target case t is computed as follows:

SIM SC (t, s) = 1 if C t = C s 0 if C t = C s where C s (respectively C t ) is the capacity of s (respectively t).
In order to illustrate this phase, we can consider the cases stored in the case base reported in Table 2. In this example taken from an Aïkido training session, sub-capacities 'Breaking the partner's posture', 'Relaxing despite a grip', 'Making the partner lose balance' and 'Pivoting around a grip' have been associated with the capacity 'Using a grip' by the trainer in the initial process or in previous training sessions. Thus, if the capacity of t is C t ='Using a grip', the sub-capacities of Source Case 1 are reminded by analogy, since SIM SC (t, 1) = 1 and SIM SC (t, 2) = 0.

The adaptation phase consists of computing the duration of each sub-capacity. We assumed that these durations are somehow linked to the importance of practising each sub-capacity. We also assumed that these durations may depend at times on the given capacity. Indeed, one sub-capacity may be associated with two different capacities. And maybe, the considered sub-capacity may have to be mastered in order to master one capacity and only be known in order to master another capacity. As an example, "Give the types of simple variables" must be practised often in order to master the capacity "Design a simple algorithm", and only revised once or twice during the practice of the capacity "design object oriented algorithms". Actually, this is more frequently observed in sports training for which capacity progression is less linear than theoretical disciplines: in sports, capacities can be practised without a clear order ("using a grip" can be practised before of after "breaking a grip" in the season) whereas in theoretical fields there is usually a clearer order for mastering the capacities ("design simple algorithms" comes after "design object oriented algorithms").

Consequently, the adaptation module sorts the set of sub-capacities according to D SC,C (descending order). Then, the proposed durations are calculated according to the number of sub-capacities the teacher wants to work on and the duration of the entire session. Following the training session, the teacher and the learner evaluate the learner's acquired level of mastery (mark and duration spent on each exercise). Before the training session begins, each selected sub-capacity is transmitted to one EA that will have to associate the corresponding exercises. After the training session, each sub-capacity duration is modified in proportion to the evaluation from 0 to 10 of the learner's level for the proposed sub-capacity.

Finally, during the retaining (or capitalisation) phase, the system subtracts the durations from all the durations of the source cases for which the practised sub-capacity appear. During capitalisation, the effective time spent by the learner to resolve the sub-capacity exercises may differ from the resolution time (estimated time to be spent by the learner on the considered sub-capacity) initially allocated by the teacher. Since this difference between time really spent and time initially allocated gives information on the difficulties of the learner and the integration of the sub-capacity, it has been taken into account into the remaining time to spend on the sub-capacity.

The remaining duration of each worked sub-capacities is computed as follows:

D SC = D SC -M SC 10 × d alloc SC d real SC ,
where M SC is the mark (out of 10) obtained by the learner, d alloc SC is the predicted duration allowed to finish the exercises of the sub-capacity and d real SC is the real duration spent by the learner on the sub-capacity exercises.

Table 2 presents two source cases of the tests performed by one of the trainers (an Aïkido teacher) who evaluated AI-VT. The trainer chose three different sub-capacities per training session and decided that the total duration of the training session must be 60 minutes. Since the trainer chose the capacity The learner becomes capable of using a grip' for this training session, the CA recalled Case 1. The adaptation process then sorted the sub-capacities according to their durations and proposed the three first sub-capacities, allocating D SC = 60 3 = 20 minutes to 3 each sub-capacity. In this example, the times really spent were all equal to the times allocated. It was usual for sports training tests, but very unusual for algorithm trainings. Consequently, after capitalisation, the new durations were those reported in the last column of Table 2. Thus, the less assimilated sub-capacities ('Relaxing despite a grip' and 'Pivoting around a grip') became the most immediate ones. We also note that, as required for the system specification, when the same capcity ('Using a grip') was selected again, another set of sub-capacities (composed of the less assimilated ones, and others) were selected. Thus, as required, the proposed solutions changed even if the same capacity was requested again later.

Selection of varied exercises

This subsection presents how the exercises are chosen regarding the selected sub-capacities. At the allocation of its sub-capacity, each EA selects a set of exercises according to the CBR-cycle. For each EA, a source case is noted: σ = (SC, {(EX, AD σ EX , RD σ EX,SC )}), where AD σ EX is the estimated duration that must be allocated to the learner to resolve the exercise EX, and RD σ EX,SC , the estimated remaining duration to spend on this exercise EX before reaching the sub-capacitySC. Each source case σ contains the exercises possible regarding SC. Assuming Card(Sol σ ) is the number of exercises of the solution part of σ, the target case τ i (i.e. the part of the training session that will be proposed) taken into account by the EA EA i is noted:

τ i = (SC i , n∈{1..Card(Solσ)} {(EX n , AD τi
EXn , RD τi EXn,SCi )}).

Each EA i then retrieves the source case corresponding to SC i . The similarity between source case σ and target case τ i is computed as follows:

SIM EX (τ i , σ) = 1 if SC i = SC 0 if SC i = SC
The adaptation phase orders the exercises of the training session. Selected exercises for which the RD is the highest are proposed first. If two agents select the same exercise, the one with the highest RD EX,SCi prevails, and the one with the lowest must be changed. Then, exercises are ordered according to their complexity (ascending order). Finally, distances between consecutive exercises are computed, and permutations between consecutive exercises may occur in order to minimise these distances between one proposed exercise and the next one. This final adaptation step creates consistency for the training session.

During the revision phase, the teacher and learner evaluate the answers (give a mark between 0 and 10) proposed by the learner and give the real duration spent on each exercise.

As an example, Tables 3 and4 illustrate the adaptation of a training session dedicated to algorithms. Table 3 shows the sub-capacities retrieved by AI-VT. These sub-capacities are ordered by AI-VT according to their RD (descending order). In this Table 3, the first exercise (EX 6 ) deals with economy and has the highest complexity [START_REF] Rishi | Distributed case based reasoning for intelligent tutoring system: an agent based student modeling paradigm[END_REF], the second exercise (EX 7 ) deals with another context and has the same complexity. The third exercise (EX 2 ) deals with economy (like EX 6 ) and has a complexity of 18, and the last proposed exercise (EX 1 ) deals with geometry and is the simplest exercise (complexity is equal to 5). Thus, this first proposal begins with the most complex exercises and ends with the simplest one, and the context always changes. As it can be observed in Table 4, the adaptation process places the same exercises in a different order: the adapted training session will begin with the easiest exercise (EX 1 ) and the exercises that deal with economy (EX 2 and EX 6 ) are grouped. At the end of the CBR cycle, capitalisation will allow the system to prepare the next training session. Indeed, even if the same sub-capacities are required next, the system will have to propose a different set of exercises. Thus, the retaining phase of AI-VT is very important since it will give the history of the performed exercises. Furthermore, if one exercise has not been understood or successfully solved, or even solved but with great difficulty by the learner, the system must have the possibility to choose this exercise again. Otherwise, if one exercise has been successfully solved with no difficulty, AI-VT must not propose it again in the case of a theoretical knowledge acquisition. In addition, if an exercise has been done with much difficulty by an athlete, the duration of practice must not change. On the contrary, for a theoretical training, if the learner has spent a lot of time on an exercise and did not solve it, this exercise should be proposed once again to the learner with a higher resolution time.

Consequently, AI-VT must capitalise cases of theoretical-domain training and cases of physical training differently.

In the case of physical training, only the RD is modified as follows: ∀SC, ∀EX, RD σ EX,SC = max(0, (RD σ EX,SC -M EX 10 × AD τ EX,SC )). In that case, the remaining duration of practise of the exercises are only decreased from the time spent over it during the training session.

And in the case of training on theoretical skills, AD and RD are modified as follows: ∀SC, ∀EX, AD σ EX,SC = max(0, (AD σ EX,SC -

M EX 10 × AD Real EX,SC AD τ EX,SC )), ∀SC, ∀EX, RD σ EX,SC = max(0, (RD σ EX,SC × (1 -M EX 10 × AD Real EX,SC AD τ EX,SC ))),
where M EX is the mark (out of 10) obtained by the learner for execise EX and AD Real EX,SC the real time spent over this exercise. For these types of learning, we considered that the time spent by the learner over an exercise (AD Real EX,SC ) can differ from the initial time allocated (AD τ EX,SC ). In order to illustrate the performance of AI-VT, Table 5 presents the different durations (RD, AD and real time spent) of the exercises proposed in the last training session and the durations of other exercises stored in the case base. We can see that the retaining phase modifies the priorities of the exercises stored in the case base. Indeed, the RDs of the successfully resolved exercises fall to 0: EX 1 and EX 2 will not be proposed next time. In addition, since EX 6 has been partially resolved (mark 5/10) with high diffulty (time spent 12 min. instead of 8 min. planned), its RD becomes inferior to other exercises in the case base: it could be proposed another time, but other exercises of the same sub-capacity will be selected first for the next training session. Finally, EX 7 has not been resolved at all (mark 0/10), and the learner has spent much time on it (15 min. instead of 8 min. planned). Consequently, its AD and RD stay the same, and

Results

AI-VT has been tested in two very different contexts: sports training (Aïkido, a traditional Japanese martial art) and algorithmics (computer science).

For the context of sports training, we asked seven Aïkido teachers to evaluate 10 consecutive training sessions for the same capacity. This corresponds to five weeks of training, with two 90-minute sessions per week. They evaluated the system through two aspects: the consistency of the proposed training sessions, and the diversity of the proposed exercises. It is important to note that all the trainers had different sessions and had initialised the system with different sub-capacities and exercise associations. Indeed, each trainer had his/her own way of teaching Aïkido and a set of favourite techniques.

For the second evaluation, we proposed to seven learners of computer science to use AI-VT for their training. These learners at our university (first year of studies) were having difficulties with algorithmics, and they were taking tutoring sessions. We proposed to them to resolve the exercises generated by AI-VT over four consecutive weeks for one 60-minute session per week. After each training session, we asked the learners to evaluate the session generated by the system through the same aspects: consistency and diversity of the proposed exercises.

Since there were two methods of capitalisation, we first compare and analyse the diversities obtained, and then we present the evaluations of AI-VT made by Aïkido trainers and university learners in algorithmics.

Table 6 presents the measures obtained for the diversity in both of the contexts of use. Since each Aïkido trainer asked for 10 training sessions and each university learner asked for four training sessions, AI-VT proposed more exercises in the context of Aïkido than in the context of algorithmics. Nevertheless, the difference is not so important: 28.71 exercises on average for Aïkido and 22.57 for algorithmic, whereas there are more than twice the sessions in Aïkido. This is due to the difference between both of the retaining processes: in the context of algorithmics, AI-VT ensures a greater turn-over of the exercises.

We observe that, in compliance with the requirements, an exercise is proposed more frequently in the context of Aïkido than in the context of algorith- 6 shows that each Aïkido exercise was generally proposed 1.54 times (Trainer #3) to 3.16 times (Trainer #1), and each exercise was globally proposed 2.26 times. In the context of algorithmics, each exercise was generally proposed 1.14 times (Student #3) to 2.0 times (Student #4), and each exercise was globally proposed 1.56 times. Figure 2 presents the evaluations of the Aikido trainers obtained by the training sessions generated by AI-VT. The mean marks obtained for each trainer are reported in this figure. The trainers were asked to give a mark from 0 to 10 for the consistency of the successive generated training sessions: 0 if the trainer felt that the exercises proposed in a session were not consistent at all with regard to the capacity and the sub-capacities trained, and 10 if the trainer was satisfied with the exercises proposed. The mean marks are reported in this figure. Six trainers considered the session consistencies between 7.2 and 7.8. Only the last trainer considered the mean consistency of the sessions was about 6.4. This was because AI-VT replaced many exercises with others deemed less important in the eyes of this trainer in the two last sessions.

The mean marks for the diversities of the training sessions are also reported in Figure 2. Six of the trainers gave mean marks between 7.4 and 8 for this aspect. There was only one mark of 6.8 for one trainer. This was due to the second session generated for this trainer, in which most of the exercises were the same as the ones proposed in the first session. This was due to the initialisation of the RD of the exercises and sub-capacities. Indeed, if these RD are too high for some sub-capacities and exercises, AI-VT will propose them until other exercises have a higher RD.

The mean marks obtained by AI-VT are reported in Figure 3. The mean mark obtained by AI-VT for consistency is 6.56. The consistency of the training session was not very good because the learners were disappointed by the repetition of exercises. Some of the learners also felt that the exercises were not adapted to their initial level (particularly in the first training session). Indeed, it would be appropriate to evaluate the levels of the learners before the first training sessions in order to propose exercises with appropriate levels of difficulty before the first time a capacity is worked on. This is the main reason why there are so many differences for AI-VT evaluations from one learner to another.

Discussion

The diversity of the AI-VT-generated solutions has been measured and evaluated by different kinds of users. In the case of Aïkido training, AI-VT generally mixed 28 to 29 exercises over 10 training sessions, and each exercise was proposed two to three times in all the sets of generated training sessions. Six out of the seven trainers who evaluated AI-VT were satisfied with the diversity and the consistency of the generated solutions (the mean mark obtained by AI-VT was 7.4/10). This proves the pertinence of the system for Aïkido training: in the particular field of sports training, the system can propose varied and consistent training sessions for many weeks even if the same capacity is requested several times consecutively. The consistency of the set of exercises proposed for each training session is guaranteed by the introduction of complexities and distances between the different exercises. Hence, these distances allow the system to propose exercises that were not initially chosen by the trainers and to sort the exercises in each session. The diversity introduced by the system may sometimes poorly influence the consistency of one session. Indeed, AI-VT may substitute an exercise with a less appropriate one in regard to the performed sub-capacity in order to satisfy the diversity requirement.

The performance of the system is less satisfactory for algorithmic training. The measures obtained show that 22 to 23 exercises were proposed to the learners during the four sessions it was tested for, and each exercise was proposed, on average, 1.56 times to each learner. This measure proves that the retaining phase of AI-VT allows the system to propose the same exercise less often than Aïkido. Nevertheless, as shown by the qualitative evaluation, we must develop further the diversity of the training sessions proposed. In that particular field, even if the learners who tested AI-VT were globally satisfied, they were very disappointed when the system proposed the same exercise twice or more. The diversity felt by the learners did not match the measured one. As a consequence, we will study some possible modifications for the formulas for the retaining phase so that exercises that were partially resolved or even resolved with high difficulty should not be proposed more than twice to the learners, and not proposed in consecutive training sessions. Nevertheless, in algorithmics, all the training sessions were consistent, and all the proposed exercises were appropriate. Other approaches like the one of B. Smyth and P. McClave [START_REF] Smyth | Similarity vs. diversity[END_REF] propose the introduction of dissimilarity measures in the retrieved phase. That kind of approach is focused on the extension of the scope covered by the set of retrieved cases, whereas ours deals with the variety of the successive sets of retrieved cases. In AI-VT, the dissimilarity is not sufficient; other parameters linked to the acquired level of the learner, the fact that an exercise has already been proposed, and the ease with which it has been solved are of the utmost importance and must be added to the metrics used to select the cases. AI-VT establishes a link between the adaptation and capitalisation of CBRsystems. Capitalisation is of the utmost importance in this system, which is required to give varied and creative solutions each time. Indeed, we have designed a way to use estimations of the levels acquired by users during the revision phase (based on the marks and the times really spent by learners on each exercise and sub-capacity) in order to enhance the accuracy of the adaptation process of CBR-systems. In addition, the introduction of remaining durations is of the utmost importance since it allows AI-VT to build varied solutions by analogy and thus to never propose the same session twice. In addition, the initialisation process of AI-VT is time-consuming for the teacher. Indeed, the teacher has to organise sessions and exercises into capacities and subcapacities and give the distances and the complexity of the exercises stored. In addition, the diversity of the exercises proposed in the training sessions depends on the number of stored exercises. For that reason, we will study the possibility of generating exercises and durations automatically and a way to help the trainer to initialise the system.

Conclusion

We have designed a system based on case-based reasoning dedicated to the generation of varied training sessions for learners. AI-VT meets one of the most important requirements: its ability to generate varied training sessions. With this implementation of AI-VT, we highlighted the importance of the retaining phase of CBR for system diversity. Indeed, this retaining phase stores what the learners have used, i.e. the training sessions and training exercises stored. The process that stores these training sessions has an impact on whether an exercise should be proposed once again or not. In addition, we proved AI-VT's ability to adapt the diversity of the training-session exercises generated to the context of use. Indeed, the retaining phase of AI-VT is adapted to the context and type of learning it is used for. The results obtained for sports training are very different from the ones obtained for theoretical learning, like algorithmics. In the case of sports, learning can be based on repetition of the same exercises time after time. Indeed, even if an exercise is proposed at the beginning of each training session, it helps make certain actions automatic. On the contrary, being confronted with the same algorithmic exercise twice or more is disappointing for learners since they already have the resolution of the exercise stored somewhere on their computers. As a consequence, even if the process of generating a training session is based on analogy with past situations, an accurate balance between repetition and diversity is proposed by AI-VT, depending on the learned field.

Figure 1

 1 Figure1presents the architecture of AI-VT modelled as a multi-agent system (MAS). A MAS constitutes a paradigm designed to handle distributed systems.

Figure 1 :

 1 Figure 1: Overview of AI-VT Architecture

Figure 2 :

 2 Figure 2: Consistency and diversity marks obtained by the Aïkido training sessions

Figure 3 :

 3 Figure 3: Consistency and diversity marks obtained by the algorithmic training sessions

Table 1 :

 1 Examples of capacities and their associated sub-capaicties

	Domain	Sports -Aïkido	Algorithmic
	Capacity	Using a grip	Design an algorithm
	Associated	Break the partner's posture	Find inputs and outputs
	sub-capacities	Relax despite a grip	Give a formula for the calculs
		Make the partner loose balance Associate a type with a variable
		Pivote around a grip	Display a clear message

Table 2 :

 2 Example of modifications of durations after a training session.

	Source Capacity Sub-capacities	Initial	Teacher's	Stored
	case			duration	mark	duration
				(min.)	(points)	(min.)
	1	Using a	Breaking the partner's	90	7/10	90 -7 10 × 20 20 = 76
		grip	posture
			Relaxing despite a grip Making the partner	90 80	3/10 4/10	90 -3 10 × 20 20 = 84 80 -4 10 × 20 20 = 72
			lose balance
			Pivoting around a grip	80	-	80
	2	Breaking Breaking a single grip	90	-	90
		a grip	Relaxing despite a grip	70	3/10	70 -3 10 × 20 20 = 64

Table 3 :

 3 Example of exercises initially retrieved by AI-VT.

	Sub-capacities / Exercises	Dist. with Complex. RD
		next ex.
	-Sub-capacity SC3: Give a formula			20
	EX6: Retrieve the total price before taxe of product using	18	18	10
	knowing its price including taxe and taxe rate.
	Give the formula.
	EX7: Compute the fuel consumption of car	18	18	10
	knowing the distance and its mean speed. Give the formula.
	-Sub-capacity SC1: Find inputs & outputs			15
	EX2: Retrieve the total price before taxe of product using	18	18	10
	knowing its price including taxe and taxe rate.
	Give the inputs & outputs.
	EX1: Compute a rectangle perimeter. Give inputs & outputs.		5	5

Table 4 :

 4 Example of exercises finally proposed by AI-VT.

	Sub-capacities / Exercises	Dist. next Complex. RD
	-Sub-capacity SC1: Find inputs & outputs			15
	EX1: Compute rectangle perimeter. Give ...	18	5	5
	EX2: Retrieve the total price before taxe of product using ...	5	18	10
	-Sub-capacity SC3: Give a formula			20
	EX6: Retrieve the total price before taxe of product using ...	18	18	10
	EX7: Compute the fuel consumption of car knowing the ...		18	10

Table 5 :

 5 Example of capitalisation proposed by AI-VT.

	Exercises	Initial Initial Mark Real time	Capitalised	Capitalised
		AD	RD		spent	AD	RD
	Ex. of SC1:
	EX1 (proposed) EX2 (proposed)	5 5	5 10	10 10	5 5	5 -10 10 × 5 5 = 4 5 -10 10 × 5 5 = 4	5 × (1 -10 10 × 5 5 ) = 0 10 × (1 -10 10 × 5 5 ) = 0
	EX3 (case base)	5	5	-	-	5	5
	EX4 (case base)	5	5	-	-	5	5
	Ex. of SC3:
	EX6 (proposed) EX7 (proposed)	8 8	10 10	5 0	12 15	8 -5 10 × 12 8 = 7 8 -0 10 × 15 8 = 8 10 × (1 -0 10 × (1 -5 10 × 15 8 ) = 3 10 × 15 8 ) = 10
	EX8 (case base)	8	10	-	-	8	10

Table 6 :

 6 Measures of the diversities obtained

	User	NB of EX Frequency	User	NB of EX Frequency
	Aïkido			Algorithmic
	Trainer #1	19	3.16	Student #1	18	1.39
	#2	37	1.62	#2	25	1.72
	#3	39	1.54	#3	21	1.14
	#4	34	1.76	#4	17	2.00
	#5	22	2.73	#5	21	1.86
	#6	24	2.72	#6	25	1.48
	#7	26	2.31	#7	31	1.32

it will most probably be proposed next time with EX 8 .

Acknowledgement

The authors whish to thank the FR-EDUC (Research Federation for ED-UCation) of University of Franche-Comté for their financial help, and Mary Moritz (WordStyle Traductions) for her help with the English language.


G Noguere

D Bernard

P Blaise

O Bouland

L Leal

P Leconte

O Litaize

Y Peneliau

B Roque

A Santamarina

J-F Vidal

Improved MOX fuel calculations with the Evaluated Nuclear Data Library JEFF-3.2

An overestimation of the k eff values for MOX fuels was identified with Monte-Carlo (TRIPOLI4) and deterministic (APOLLO2) calculations based on the Evaluated Nuclear Data Library JEFF. The overestimation becomes sizeable with the Pu-ageing, reaching a reactivity change of ∆ρ ≃ +700 pcm for integral measurements carried out with MOX fuel containing a large amount of Americium. This bias was observed on various critical configurations performed in the zero power reactor EOLE of the CEA Cadarache, France. The present work focuses on the improvements achieved with the new 239 Pu and 241 Am Evaluated Nuclear Data Files available in the latest version of the JEFF library (JEFF-3.2). The Resolved Resonance Range of the plutonium evaluation

Introduction

Numerous integral experiments on MOX fuel configurations were carried out in the zero power reactor EOLE of the CEA Cadarache, France. The interpretation of the experimental results is routinely performed with the deterministic code APOLLO2 [1] and Monte-Carlo code TRIPOLI4 [START_REF] Both | Tripoli4 -A three dimensional polykinetic particle transport Monte Carlo code[END_REF] developed at the CEA Saclay, France. The data application library used by both codes for neutron transport calculations is based on the nuclear data available in Preprint submitted to Nucl. Sci. Eng.

the Joint Evaluated Fission and Fusion library (JEFF). The integral results provide valuable nuclear data trends for the continuous improvement of JEFF under the non-regression principle. Results given in this paper are illustrated with k eff values obtained for the MH1.2, FUBILA, MISTRAL-2, MISTRAL-3 and MISTRAL-4 programs where the residual reactivity are measured through critical fuel loading or/and critical soluble boron concentration.

The nuclear data of interest for MOX fuel calculations were significantly improved since the release of the JEF-2.2 library in 1992. The present work focuses only on the description of the resolved resonance range of the neutron cross sections for which an incremental method was adopted for modifying and correcting resonance parameters of the plutonium ( 239 Pu, 240 Pu, 241 Pu) and americium ( 241 Am) isotopes. The method is described in Ref. [START_REF] Santamarina | The JEFF-3.1.1 Nuclear Data Library[END_REF]. Despite the evaluation works performed on these isotopes for the JEFF-3.1.1 library, we still observe systematic overestimation of the experimental k eff by results of calculations that increases with the Pu-ageing, which defines the 241 Am build up from 241 Pu beta decay with time. For the reference configuration of the MISTRAL-4 experiments, the bias on the calculation-to-experimental ratios reaches +735(8) pcm. Such biases were recently reduced with the latest version of the JEFF library, namely JEFF-3.2. The improvements are discussed in the present document.

The second section addresses the context of the present study. A review of the evaluation activities performed on the Plutonium and Americium isotopes is presented in the third section. The experimental programs selected for the "benchmarking" of the new Evaluated Nuclear Data Files are briefly described in section 4. Results expressed in terms of reactivity effects are discussed in section 5.

Context

The resolved resonance range of the Plutonium and Americium isotopes of interest for MOX fuel calculations was elaborated in three steps. Explanations are reminded in Ref. [START_REF] Santamarina | The JEFF-3.1.1 Nuclear Data Library[END_REF]. New evaluated nuclear data files for 240 Pu and 241 Pu were proposed for the release of JEFF-3.0, while corrections for 241 Am and 239 Pu were included in JEFF-3.1 and JEFF-3.1.1 respectively. Figure 1 compares the evaluated and experimental neutron cross sections for the first resonances of each isotope. 2.1 From JEF-2.2 to JEFF-3.0

Inconsistent descriptions of the low energy capture and fission cross sections of several actinides compiled in the JEF-2.2 library were identified with an Integral Data Assimilation procedure (called RDN) on a large variety of integral data. The results exhibited the need of new resonance analysis of 240 Pu and 241 Pu. The evaluation work was performed with the SAMMY code [START_REF] Larson | Updated User's Guide for SAMMY[END_REF] in collaboration with ORNL.

For 241 Pu, the SAMMY analysis focuses on the first s-wave resonance at 0.26 eV [6]. The new resonance parameters evaluated with the Reich-Moore approximation of the R-Matrix theory allowed a correct prediction of the 242 Pu and 243 Am build-up as a result of an increasing of the capture resonance integral (+6.5%) and of the partial radiation width Γ γ from 32.6 meV (in JEF-2.2) to 34.6 meV (in JEFF-3.1.1). Comparisons with data retrieved from the experimental data base EXFOR are shown in Fig. 1.

The nuclear system 240 Pu+n is characterized by a broad resonance at 1.06 eV with a cross-section peak that is as high as 20 kbarns. Small changes on the parameters that describe this resonance may have a significant impact on MOX calculations because it represents 95% of the 240 Pu capture rate and almost 30% of the total capture rate in MOX fuel. The resonance parameters compiled in JEFF-3.1.1 were established from two sets of transmission data measured at the ORELA facility with a thin and thick samples at nitrogen temperature (T = 77 K) [START_REF] Harvey | Solid State Effects on Thermal Neutron Cross Sections and on Low Energy Resonances[END_REF]. Detailed explanations on the resonance analysis performed up to 5.7 keV are given in Ref. [START_REF] Bouland | [END_REF]. The main issue was the assessment of the effective temperature (T eff = 109.6 K) to account for the Doppler broadening.

Performances of the evaluated nuclear data files for 240 Pu and 241 Pu were investigated with experimental results obtained from irradiated fuel samples. Calculations performed with the DARWIN package are presented in Ref. [START_REF] San Felice | Experimental validation of the DARWIN-2.3 package for fuel cycle applications[END_REF]. The prediction of the 240 Pu and 242 Pu build-up are summarized in Fig. 2.

2.2

From JEFF-3.0 to JEFF-3.1

The impact of the first four s-wave resonances of 241 Am at 0.3 eV, 0.6 eV, 1.3 eV and 1.9 eV on average integral trend was investigated prior the release of JEFF-3.1. The main motivations for revising the lower part of the 241 Am neutron cross sections were based on integral trends that suggested an increase of the thermal capture cross section and of the epithermal resonance integral [START_REF] Bernard | [END_REF]. The evaluation work was performed with the SAMMY code by using the total cross sections measured by Belanova [11], Slaughter [START_REF] Slaughter | Oak Ridge National Laboratory Report[END_REF] and Derrien [START_REF] Derrien | The total cross section and the fission cross section of 2 41Am in the resonance region, resonance parameters[END_REF] with the transmission technique. Figure 3 shows the differences between the 241 Am(n,γ) cross section of the JEFF-3.0 and JEFF-3.1.1 (=JEFF-3.1) libraries. This evaluation work led to TRIPOLI4 calculations in better agreement with reactivity worth measurements carried out at the MINERVE facility of the CEA Cadarache in the frame of the OSMOSE program [START_REF] Leconte | OSMOSE programme: Validation of Actinide Nuclear Data for LWR applications[END_REF]. Results are summarized in Table 1. However, calculations based on JEFF-3.1.1 were not able to fully explain differences observed on older integral results carried out in the MELUSINE reactor of the CEA Grenoble (experimental programs ICARE-S, ICARE-R and SHERWOOD) [START_REF] Bernard | [END_REF], as well as the systematic overestimation of the critical k eff of 100% MOX cores of the EOLE facility of CEA Cadarache. The origin of these long standing problems is clarified in section 3.2.

2.3

From JEFF-3.1 to JEFF-3.1.1

Evaluation work on 239 Pu was focused on the prompt neutron multiplicity and on the α ratio (Fig. 4). The α ratio of 239 Pu, meaning the ratio of the neutroninduced capture cross-section to the neutron-induced fission cross-section, was investigated in the thermal energy range via Reactivity Temperature Coefficient (RTC) analysis performed with the APOLLO2 code on the MISTRAL-2 and MISTRAL-3 experiments carried out in the EOLE facility [16,[START_REF] Erradi | Analysis of the MISTRAL experiment on LWR Reactivity Temperature Coefficient[END_REF]. The interpretation of the experimental results demonstrated that part of the negative bias reported in Table 2 for the low-temperature range (below 80 • C) was linked to the thermal spectrum shift effect, which dependents on the thermal shapes of the capture and fission cross-sections of 239 Pu.

Improved RTC calculations were achieved on the temperature range [10 • C-80 • C] by adding a bound state with a negative energy (E=-0.02 eV) close to the binding energy. The parameters were optimized to modify the energy dependence of the α ratio below the thermal energy of 25.3 meV. Such a modification led to an increase of the thermal capture cross section of 2.1 barns and a decrease of the thermal fission cross section of 0.6 barns.

The 239 Pu evaluation was also improved with a new modelisation of the prompt neutron multiplicity up to 20 eV. The performances were demonstrated via ICSBEP benchmarks. As shown by the extensive work of Van Der Mark [START_REF] Van Der Marck | [END_REF] performed over 368 Plutonium Solutions in Thermal spectrum (PST), the average value of the C/E -1 results is +203 pcm for JEFF-3.1.1, compared to +462 pcm for ENDF/B-VII.1 and +633 pcm for JENDL-4.0.

Plutonium and Americium Evaluated Nuclear Data Files

New investigations were conducted on the resolved resonance range of the 239 Pu and 241 Am isotopes with the aim to explain the systematic overestimation of the MOX reactivity at zero power and room temperature. The differences between the calculations and the experimental results range from few tens to several hundred of pcm on many MOX fuel configurations carried out in the EOLE facility.

The evaluation activities performed on each isotope is described in sections 3.1 and 3.2. The resulting thermal quantities and values for the resonance integrals are reported in Table 3.

New description of the 239 Pu resolved resonance range

The update of the 239 Pu resolved resonance range was a necessity because of the obsolete data format chosen to store the numerous parameters. In the eighties and early nineties Derrien and others performed an evaluation in a collaborative work including CEA and ORNL. At that time, due to computer limitations for data storage and processing decision was made to split the resonance region in three parts, namely, [10 -5 eV -1 keV], [1 keV -2 keV], and [2 keV -2.5 keV]. Some additional issues with the evaluation arise from the use of three distinct sets of resonance parameters. The cross-sections calculated at the energy boundary of two consecutive disjoint resonance parameter sets could be different leading to discontinuity. An artificial background cross section was also needed to correct the fission cross section in the second energy range [1 keV -2 keV]. Another concern relates to data uncertainty assessment using resonance parameter covariance. For data uncertainty analysis the use of a single resonance parameter set covering the entire energy region would be preferable since the disjoint set of resonance parameters do not permit the determination of a fully uncertainty correlation in the entire energy region.

A collaborative effort between the Oak Ridge National Laboratory (ORNL) and the CEA of Cadarache was initiated under the auspice of the US Department of Energy and the Nuclear Energy Agency (NEA) to address issues pertinent to the performance of the 239 Pu cross-sections in benchmark calculations. A new resolved resonance analysis was performed in the frame of the WPEC subgroup SG-34 of the Nuclear Energy Agency. Results and discussions are reported in the final report of the working group [START_REF] Saint | Co-ordinated Evaluation of Plutonium-239 in the Resonance Region[END_REF]. The main effort related to the determination of resonance parameters up to 2.5 keV was performed with the SAMMY code at ORNL [START_REF]Pu resonance evaluation for thermal benchmark system calculations[END_REF]. The lower energy range of the neutron cross sections was finalized in collaboration with the CEA Cadarache Fig. 5. Comparison of the (C-E) results obtained with JEFF-3.1.1 and JEFF-3.2 for a selected set of thermal solutions of plutonium (PST) [START_REF] Peneliau | JEFF-3.2T3 benchmarking -Criticality experiments to full reactor calculations[END_REF][START_REF] Cabellos | Processing and benchmarking of the JEFF32T4 (JEFF-3.2) library[END_REF].

in order to account for constraints on the thermal shape of the α ratio (see section 2.3) in conjunction with the fluctuations of the prompt neutron multiplicity ν p (E) established for JEFF-3.1.1. The fitting of the 239 Pu nuclear data was continuously monitored during the evaluation procedure with a selected set of ICSBEP benchmarks. This step was crucial to conserve the good performances of the JEFF library on PST benchmarks (Fig. 5).

Thermal quantities and values for the resonance integral are reported in Table 3. An overall good agreement is obtained with the values compiled in Ref. [START_REF] Mughabghab | Atlas of Neutron Resonances[END_REF]. The main issue is related to the difference between the total neutron multiplicity ν t = 2.8745 and the value of 2.8836 ± 0.0047 recommended by the "standard" expert group of the AIEA. Although the difference (-0.0091) remains within the reasonable limit of two standard deviation, a deeper investigation could reveals compensation effects with the Prompt Fission Neutron Spectrum (PFNS). Few lessons can be drawn from results reported in Ref. [START_REF] Peneliau | Pu Prompt Fission Neutron Spectra Impact on a Set of Criticality and Experimental Reactors Benchmarks[END_REF]. The major conclusion is that calculations of PST benchmarks with softer PFNS of different origins lead to a systematic increase of the k eff values ranging from 50 pcm to 800 pcm. These trends are not compatible with an increase of the neutron multiplicity in the thermal energy range. Complementary benchmarking works are under progress in the frame of the WPEC subgroup SG-40 ( CIELO pilot project) with the aim to clarify these oposite results.

Improved description of the 241 Am capture cross section

A new resonance analysis of the nuclear system 241 Am+n was performed using recent transmission and capture data measured at the GELINA facility of the Institute for Reference Materials and Measurements (IRMM, Belgium). The resonance parameters reported by Lampoudis [START_REF] Lampoudis | [END_REF] suggest to increase by 20% the capture resonance integral. A similar conclusion is obtained with the parameters compiled in JEFF-3.2 [26]. The thermal quantities and resonance integral are compared in Table 3.

Several integral experiments have been designed to measure the neutron capture cross-sections of heavy nuclides involved in fission reactor design calculations. For the nuclear system 241 Am+n, we focused our benchmarking activities on the experimental programs carried out in the MELUSINE facility of the CEA Grenoble and in the MINERVE and EOLE facilities of the CEA Cadarache.

Post Irradiated Experiments carried out in the MELUSINE reactor from 1986 to 1988, namely ICARE-S, ICARE-R and SHERWOD, were recently reanalyzed with the APOLLO code [START_REF] Bernard | [END_REF]. The programs ICARE-S and ICARE-R were dedicated to investigate the neutronic properties of Light Water Reactors with high conversion ratio. They are characterized by a neutron spectrum covering the epithermal energy range. The core configuration of the SHERWOOD program was representative to standard PWR conditions. The combination of these three experiments allows to cover a wide energy range from thermal to few tens of eV. Each experiment consisted in irradiating one or two experimental pins composed of isotopically enriched samples of actinides. Integral trends on the capture cross section can be deduced from the isotopic analyses before and after the irradiation period.

Additional information on the 241 Am capture cross section was also deduced from the OSMOSE program carried out in the MINERVE facility with the oscillation technique. Four experimental results were obtained with two 241 Am samples oscillated in UOX (R1UO2) and MOX (R1MOX) core configurations [START_REF] Leconte | OSMOSE programme: Validation of Actinide Nuclear Data for LWR applications[END_REF]. The measured reactivity worth is nearly proportional to the effective capture cross section of 241 Am.

Complementary results were obtained from the measurements of the Pu ageing which is obtained from the following difference: within the upper limit of the experimental uncertainties.

∆(δt) = k eff (t 0 + δt) -k eff (t 0 )
For the fission cross section, we use an integral trend based on fission rates measured with fission chambers placed in the center of the ERASME-R configuration. The latter program was performed in the EOLE facility to study fundamental lattice parameters associated with the use of MOX fuel in undermoderated spectrum.

The integral trends collected during the benchmarking of the 241 Am resonance parameters are listed in Table 4. A better agreement between the calculations and the experimental results is achieved with the latest JEFF-3.2 library. One of the remaining ambiguous results concerns the new trend observed for the OSMOSE experiment (≃ +7%). It is not consistent with the calculation of the 241 Pu ageing. Two experiments are under investigation for solving this problem. The first one will use the oscillating technique in the MINERVE facility with new 241 Am samples previously used at the IRMM to measure the 241 Am(n,2n) cross section [27]. The second one will consists to perform transmission measurements at the IRMM with the MINERVE samples.

Integral benchmarks

Several experimental programs were conducted by CEA to investigate the use of MOX fuel in commercial PWRs or innovative concepts for various moderation ratios. In the EOLE facility, integral experiments with MOX fuel started in 1984 with the ERASME program devoted to High-Conversion LWRs (under-moderated MOX lattices) [START_REF] Santamarina | Undermoderated PWR neutronic qualification through the ERASME experiment[END_REF]. These HCLWR studies were followed in 1989 by the EPICURE 30% MOX recycling program in standard PWR [START_REF] Mondot | EPICURE : an experimental program devoted to the validation of the calculational schemes for plutonium recycling in PWRs[END_REF][START_REF] Fougeras | [END_REF].

The MISTRAL programs (MISTRAL-2, MISTRAL-3 and MISTRAL-4), devoted to 100% MOX recycling in advanced PWR, were undertaken between 1997 to 2000 [START_REF] Cathalau | MISTRAL: an experimental program in the EOLE facility devoted to 100% MOX core physics[END_REF][START_REF] Yamamoto | Core Physics experiment of 100% MOX core MISTRAL[END_REF][START_REF] Cathalau | First validation of neutronic lattice parameters of over moderated 100% MOX fuelled PWR cores on the basis of the MISTRAL experiment[END_REF][START_REF] Fougeras | MISTRAL-4 : an experimental mockup in the EOLE facility devoted to high moderation 100% MOX core Physics[END_REF][START_REF] Hibi | Analysis of MISTRAL and EPICURE experiments with SRAC and MVP code systems[END_REF][START_REF] Tatsumi | Analysis of High Moderation PWR MOX Core MISTRAL-4 with SRAC and MVP[END_REF][START_REF] Yamamoto | Analysis of MISTRAL Experiments with JENDL-3.2[END_REF][START_REF] Cathalau | Full MOX recycling in ALWR : Lessons Drawn through the MISTRAL Program[END_REF][START_REF] Litaize | Analysis of the MISTRAL experiment with APOLLO2 qualification of neutronic parameters of UOX and MOX cores[END_REF][START_REF] Blaise | Monte Carlo modeling of MISTRAL-4 100% MOX 17 × 17 ALWR mock-up cores using TRIPOLI-4.6[END_REF]. Finally, the BASALA (2000)(2001)(2002)(2003) and FUBILA (2005)(2006) programs were designed to study MOX core configurations for BWR Physics [START_REF] Yamamoto | BASALA: Advanced BWR MOX core physics experiments[END_REF][START_REF] Yamamoto | FUBILA: core physics experimental program aimed at validation of core analysis method for high burn up BWR full MOX cores[END_REF][START_REF] Yamamoto | Experimental results and analysis of core physics experiments, FUBILA, for high burn-up BWR full MOX cores[END_REF][START_REF] Blaise | [END_REF]. These programs and their main achievements are described elsewhere [START_REF] Blaise | 12 years of Franco-Japanese instrumental program in EOLE for the validation of 100% MOX recycling in ALWRS[END_REF].

We have focused our attention on the MH1.2 configuration of the EPICURE program, on the MISTRAL-2-3-4 configurations and on the FUBILA program.

Results obtained with the Monte-Carlo code TRIPOLI and the JEFF-3.1.1 library are expressed in term of ∆ρ = C -E, where E and C stand for the experimental and calculated reactivities, respectively. Owing to the collaboration with the Japanese organisms (NUPEC and JNES), complementary calculations were performed with the Monte-Carlo code MVP [START_REF] Nagaya | MVP/GMVP II: general purpose Monte Carlo codes fro neutron and photon transport calculations based on continuous energy and multigroup methods[END_REF] with the Japanese Library JENDL. Results are summarized in Table 5.

PWR mixed-oxide fuel experiments MH1.2 in the EPICURE program

The EPICURE program (1989-1994) was jointly agreed upon 1988 by the CEA and its industrial partners. This program consisted in 5 configurations (UH1.2, UH1.4, UH1.4-abs, MH1.2, UM-17 × 17, UM-Zone). They provide a complete and high-quality experimental database for evaluating the uncertainties in reactor physics calculations of plutonium-recycling PWRs.

The present work focuses on the mixed uranium-plutonium core MH1.2. The central zone was composed of MOX fuel pins (7% enrichment). The reactivity is correctly predicted with the JEFF-3.1.1 library: ∆ρ(MH1.2) = +51 ± 260 pcm.

The experimental uncertainty of ±260 pcm (1σ) involves the measurement uncertainty of the core residual reactivity and the technological uncertainties (fuel density, Pu composition, overclad diameter, lattice pitch).

During the evaluation work, particular attention had to be paid to this result in order to maintain the good agreement with the experimental value.

MISTRAL program for high-moderation 100% MOX core physics

The The MISTRAL-2 configuration was a highly moderated full-MOX core consisting of about 1600 MOX (7% enrichment) fuel pins in a lattice pitch of 1.32 cm. The criticality was obtained by adjusting the number of fuel pins.

The core configuration of MISTRAL-3 was devoted to the physical study of a 100% MOX lattice with higher moderation than MISTRAL-2. This configuration consisted of about 1350 MOX 7% fuel pins in the lattice pitch of 1.39 cm. In that case, the criticality was obtained by adjusting the soluble boron concentration. The fourth configuration, MISTRAL-4, has been studied for the third generation of LWRs in order to alow 100% MOX recycling.

Several parameters were measured and investigated (critical mass, boron concentration, buckling measurement using reaction rate distribution measurements, pin-by-pin fission maps, boron worth, spectrum indices measurement, modified conversion factor, isothermal temperature coefficients, reactivity worth of several absorbers, void coefficient). In this work, we limit ourselves to k eff calculations. Results obtained with the JEFF-3.

MOX core configurations in the FUBILA program for BWR Physics

The FUBILA program was a first-of-a-kind full MOX core physics experiment with actual BWR fuel pins undertaken by CEA and JNES (Japanese organism). Eight different core configurations (2005)(2006) were designed to obtain core physics data of high burn up for 9 ×9 and 10 ×10 BWR MOX assemblies. The reference configuration of the FUBILA program is of great interest because the experimental zone was composed of fresh mixed oxide fuel while the driven zone is characterized by a large content of 241 Am. As a consequence, we expect to have ∆ρ results of similar magnitude between the reference configuration of FUBILA and the MH1.2 configuration (section 4.1). TRIPOLI calculations based on the JEFF-3.1.1 library confirm this assumption: ∆ρ(FUBILA Ref.) = +119 ± 250 pcm.

5 Results and discussions

Reactivity effect breakdown calculations

This section aims at presenting the origins of the discrepancies observed on ∆ρ (section 4) based on reactivity breakdown calculations performed in Ref. [START_REF] Santos | Optimisation de lapproche de representativite et de transposition pour la conception neutronique de programmes experimentaux dans les maquettes critiques[END_REF]. Table 6 shows the evolution of the plutonium isotopic vector with the Pu ageing for standard MOX fuel. The production of americium in MOX fuel pins is illustrated over a large time scale that corresponds to the beginning [START_REF] Gastaldo | High-order discrete ordinate transport in non-confirming 2D cartesian meshes[END_REF][START_REF] Tellier | [END_REF] by using the cross-sections covariance matrices COMAC [START_REF] Saint | Estimation of multi-group cross section covariances[END_REF], excepted for 241 Am. For this isotope, the uncertainty was increased to account for the underestimation of the capture resonance integral by at least 20% (see Table 3). Table 7 shows the decomposition of the nuclear data uncertainty components per isotope for standard EPICURE MOX fuel pins (4.3% enrichment). The calculations were performed with the exact perturbation formalism implemented in the PARIS code [START_REF] Gastaldo | High-order discrete ordinate transport in non-confirming 2D cartesian meshes[END_REF][START_REF] Tellier | [END_REF] by using the cross-sections covariance matrices CO-MAC [START_REF] Saint | Estimation of multi-group cross section covariances[END_REF]. They were developed within the nuclear data activities of the CEA Cadarache. For 241 Am, the reported uncertainties take into account the results reported in Table 3. The comparison of the capture resonance integral:

Isotopes

I γ (JEFF-311) = 1526.4 barns, I γ (JEFF-32) = 1826.1 barns, suggests to apply a relative uncertainty of about 20% over the whole resolved resonance range.

These results confirm the increasing (decreasing) contribution of 241 Am ( 241 Pu). The contributions of the other uranium, plutonium and light isotopes remain unchanged with the Pu ageing. The global uncertainty calculated with the covariance data of COMAC is dominated by the 239 Pu and 241 Am contributions.

For 241 Am, the difference ∆ > 700 pcm calculated between t 0 and t 0 +20 years is compatible with the ∆ρ results obtained for the reference configurations of the MISTRAL programs (597 pcm < ∆ρ < 735 pcm). The trend of the k eff overestimation from the MH1.2 experiment to the MISTRAL-2/MISTRAL-4 experiments can be partially explained by the increase of the moderation ratio V H 2 O /V M OX from 1.25 to 2.2. However, the large k eff shift confirms the underestimation of the 241 Am(n,γ) reaction in the JEFF-3.1.1 library. The improvements achieved with JEFF-3.2 are detailed in sections 5.2 (k eff calculations) and 5.3 (uncertainty propagation calculations).

Benchmaking of the JEFF-3.2 library

Interpretations of the integral experiments carried out in the EOLE facility are routinely performed with the deterministic and Monte-Carlo codes APOLLO2 and TRIPOLI4 [START_REF] Santamarina | Validation of the new code package APOLLO2.8 for accurate PWR calculations[END_REF]. We focus our work on material buckling B 2 m and critical k eff measurements.

Results obtained with the JEFF-3.1.1 and JEFF-3.2 libraries are shown in Fig. 6 as a function of the Pu ageing. The reactivity of Pu-fueled lattices calculated with the APOLLO2 code within the fundamental mode hypothesis is mainly checked through material buckling values deduced from radial and axial flux measurements. Three results are reported for the MH1.2, MISTRAL-2 and MISTRAL-3 programs.

The MVP code provide results consistent with those calculated with the TRIPOLI4 code. The observed trend confirms the increasing discrepancies with the Pu ageing between the calculations and the experiments. The worst result reaches a maximum close to ∆ρ ≃ +800 pcm for the reference configuration of the MISTRAL-4 program carried out in 1999. This trend is explained with the perturbation calculations reported in section 5.1.

The latest version of the JEFF library (JEFF-3.2) improves significantly the reactivity calculations of MOX configurations over a wide range of moderation ratios. The mean value ∆ρ calculated over the five reference configurations of the FUBILA, MH1.2 and MISTRAL-2-3-4 programs becomes: ∆ρ = +50 pcm, with a standard deviation of:

σ ∆ρ = 180 pcm.
This satisfactory trend is mainly achieved due to the new 241 Am evaluation performed in collaboration with the IRMM (section 3.2). The capture resonance integral was increased by 20% (Table 3). The impact of the 239 Pu evaluation (section 3.1) and of the whole JEFF-3.2 library can be deduced from the reference configuration of FUBILA and from the MH1.2 configuration of EPICURE because their experimental zone was composed of MOX fuel pins characterized by a low americium content (< 2%). The ∆ρ values of -148(6) pcm and -122(40) pcm remain within the limit of the benchmark uncertainties (±250 pcm). These results satisfy the non-regression principle established for improving the JEFF library.

Propagation and reduction of the nuclear data uncertainties

This section presents the propagation of the resonance parameter uncertainties on the calculated k eff . Owing to the results of the nuclear data uncertainty calculations presented in Table 7, we have focused our attention on the 239 Pu and 241 Am contributions. A Monte-Carlo propagation technique was applied on the MISTRAL benchmarks calculated with the APOLLO2 code. Detailed explanations are given in Ref. [START_REF] Noguere | [END_REF]. Similar technique was used to propagate the 237 Np nuclear data uncertainties on oscillation measurements performed in the MINERVE facility [54].

The Resonance Parameter Covariance Matrices (RPCM) for 239 Pu and 241 Am were generated by using the marginalization procedure implemented in the CONRAD code. The marginalization technique consists of distinguishing the contributions of the "statistical" and "systematic" uncertainties during the evaluation of the resonance parameters. The "statistical" uncertainties are mainly related to uncertainties that do not produce long range correlations.

The "systematic" uncertainties represent the uncertainties of experimental corrections such as the sample composition and the normalization used in agreement with the result provided by the deterministic approach (895 pcm in Table 7). Such a result is dominated by the uncertainty of the 239 Pu(n,γ) reaction. The latter ranges from 4% to 7% up to 2.5 keV (Figure 8).

A reduction of the uncertainties affecting the accuracy of the 239 Pu cross sections can be achieved through appropriate rejection criteria. Such a procedure could become a powerful Integral Data Assimilation technique when a large number of experimental results is included in the analysis. In the present work, we aim only to establish a raw estimate of the target accuracy for the 239 Pu cross section uncertainties in order to get an overall 239 Pu contribution of about 400 pcm. Final results corresponding to this criteria are reported in Table 9. The capture cross section uncertainty is reduced by 2% up to 4 eV without changing the uncertainty on the fission cross section. The Monte-Carlo procedure also provide useful feedbacks and guidelines on the thermal quantities, resonance integrals and equivalent K1 for future evaluation works. The resulting relative uncertainties and correlation matrix for the 239 Pu(n,γ) and 239 Pu(n,f) reactions are shown in Fig. 10.

Conclusions

The evaluation activities performed on the resonance range of the plutonium ( 239 Pu, 240 Pu, 241 Pu) and americium ( 241 Am) isotopes within the JEFF project are summarized in the present work. Despite the good performances of JEFF-3.1.1 on PST benchmarks, integral experiments carried out in the EOLE facility of CEA Cadarache have pointed out sizeable C-E discrepancies on k eff with the Pu ageing. The observed trend was successfully corrected by using the latest version of the JEFF library (JEFF-3.2). The agreement between the calculations and the experimental values (reference configurations of MH1.2, MISTRAL and FUBILA programs) is close to 50 pcm on average with a standard deviation of 180 pcm. This improvement was achieved via the increase of the 241 Am capture cross section (+20%) based on the experimental work carried out at the IRMM.

In the case of the MISTRAL programs, uncertainty calculations based on JEFF-3.1.1 covariances have shown the non-negligible contributions (≃ 900 pcm) of the 241 Am and 239 Pu capture cross section uncertainties. The 241 Am contribution to the k eff uncertainty is significantly reduced (≃ 150 pcm) by using the JEFF-3.2 evaluation together with the corresponding Resonance Parameter Covariance Matrix. By contrast, the 239 Pu contribution is not improved with JEFF-3.2 because the 239 Pu evaluation was elaborated with capture data whose normalization uncertainty ranges from 4% to 7%.

A preliminary study performed to provide target accuracies for the 239 Pu(n,γ) reaction suggests to decrease the relative uncertainty down to 2% up to 4 eV, leading to a thermal capture cross section of 270.1±6.8 barns and an equivalent K1 of 1161.5±12.0 barns. These trends could provide useful feedbacks and guidelines for future evaluation works.

Fig. 1 .

 1 Fig.1.239 Pu, 240 Pu, 241 Pu and 241 Am neutron cross sections reconstructed with the resonance parameters of the JEFF libraries. The experimental data are taken from the EXFOR data base[START_REF] Henriksson | The art of collecting experimental data internationally: EXFOR, CINDA and the NRDC network[END_REF].

Fig. 2 .

 2 Fig. 2. MOX fuel inventory calculations performed with the DARWIN package by using the JEFF-3.1.1 library [9].

Fig. 3 .

 3 Fig. 3. Comparison of the 241 Am broad-group average capture cross section calculated with the JEFF-3.0 and JEFF-3.1.1 library.

Fig. 4 .

 4 Fig. 4. α ratio (left hand plot) and prompt neutron multiplicity (right hand plot) of 239 Pu reconstructed with the JEFF-3.1 and JEFF-3.1.1 libraries. The theoretical curves are compared with EXFOR data.

1 . 1

 11 library show an increasing discrepancy with the Pu-ageing between the experimental and the calculated reactivity: ∆ρ(MISTRAL-2 Ref.) = +597 ± 220 pcm. ∆ρ(MISTRAL-3 Ref.) = +640 ± 210 pcm. ∆ρ(MISTRAL-4 Ref.) = +735 ± 250 pcm.

Fig. 6 .

 6 Fig. 6. Integral trends obtained with the JEFF-3.1.1, JENDL-3.2 and JEFF-3.2 libraries for material buckling B 2 m (APOLLO2 calculations) and critical k eff measurements (TRIPOLI and MVP calculations).

σFig. 7 .

 7 Fig.7. Relative uncertainties and correlation matrices for the neutron induced reactions on 241 Am up to 150 eV from JEFF-3.2[26].

σFig. 8 .

 8 Fig.8. Relative uncertainties and correlation matrices for the neutron induced reactions on 239 Pu up to 2.5 keV from JEFF-3.2[START_REF] Saint | Co-ordinated Evaluation of Plutonium-239 in the Resonance Region[END_REF].

Fig. 9 .

 9 Fig. 9. Distributions of the k eff values obtained by Monte-Carlo for the MISTRAl-2 and MISTRAL-3 experiments by using the 241 Am and 239 Pu Resonance Parameter Covariance Matrix of JEFF-3.2.

Fig. 10 .

 10 Fig. 10. Final relative uncertainty and correlation matrix for the 239 Pu capture and fission cross sections (up to 2.5 keV) obtained from our Monte-Carlo uncertainty reduction procedure.

Table 1

 1 Average integral trends C/E -1 obtained on the 241 Am(n,γ) reactions from the interpretation of the pile-oscillation experiment OSMOSE[START_REF] Leconte | OSMOSE programme: Validation of Actinide Nuclear Data for LWR applications[END_REF] carried out at the MINERVE facility (CEA Cadarache, France). The calculations were performed with the TRIPOLI code by using the JEFF-3.1.1 and ENDF/B-VII.0 libraries.

	MINERVE	Exp.	Evaluated Nuclear Data Libraries
	configurations	unc.	ENDF/B-VII.0	JEFF-3.1.1
	R1UO2 (UOX lattice) ±2.8%	-14.5%	-3.8%
	R1MOX (MOX lattice) ±2.4%	-12.6%	-4.2%

Table 3 .

 3 Thermal quantities, resonance integrals and equivalent K1 calculated at 293.6 K with the NJOY code. Results are compared with the data compiled in the Atlas of Neutron Resonance of Mughaghab[START_REF] Mughabghab | Atlas of Neutron Resonances[END_REF].

	Isotope	ν t	σ f	σ γ	I f	I γ	K1
	239 Pu Atlas	2.879±0.006 748.1±2.0 269.3±2.9 303±10	180±20
	JEFF-311	2.875	746.9	272.7	303.6	181.5	1155.4
	JEFF-32	2.875	747.2	270.1	308.8	180.1	1161.5
	241 Am Atlas		3.2±0.1	587±12	14.4±1.0 1425±100
	JEFF-311	3.239	3.2	647.0	17.3	1526.4
	JEFF-32	3.102	3.2	747.8	15.2	1826.1

Table 4

 4 Integral trends for 241 Am deduced from experimental programs carried out in the MELUSINE, MINERVE and EOLE facilities[START_REF] Leconte | OSMOSE programme: Validation of Actinide Nuclear Data for LWR applications[END_REF][START_REF] Bernard | [END_REF].

	Experimental Experimental	Physical	Exp.	JEFF libraries
	facilities	programs	quantities	unc.	JEFF-311 JEFF-32
	MELUSINE	ICARE-R	(n,γ)	±4.8%	-21.4%	-9.0%
		ICARE-S	(n,γ)	±2.7%	-15.2%	-4.0%
		SHERWOOD	(n,γ)	±2.1%	-16.5%	-3.0%
	MINERVE	OSMOSE (R1U02)	(n,γ)	±2.8%	-3.8%	+7.7%
		OSMOSE (R1MOX)	(n,γ)	±2.4%	-4.2%	+7.3%
		OSMOSE (R1U02)	241 Pu ageing ±3.1%	-10.6%	-1.8%
	EOLE	FUBILA	Pu ageing	±4.0%	-3.1%	+5.3%
		MISTRAL3	Pu ageing	±3.0%	-4.0%	+4.4%
		ERASME-R	(n,f)	±7.2%	+9.5%	+0.7%

Table 5 (

 5 C-E) results in pcm obtained with the TRIPOLI and MVP codes.

	EOLE	TRIPOLI4	MVP
	Configurations	with JEFF-3.1.1 with JENDL-3.2
	MH1.2	+51(6)	[22]	+220(20) [42]
	FUBILA Ref	+119(8)	[45]	+270(20) [44]
	MISTRAL-2 Ref +597(8)	[40]	+640(20) [42]
	MISTRAL-3 Ref +640(8)	[40]	+700(20) [42]
	MISTRAL-4 Ref +735(8)	[40]	+880(20) [42]

  MISTRAL program (MOX investigation of Systems which are TechnicallyRelevant of Advanced Light water reactors) was carried out in the EOLE reactor in the frame of a broad collaboration between NUPEC (Japanese government-sponsored corporation), CEA and their associated industrial partners. It was designed to measure the main neutronic parameters of interest for Advanced Light Water Reactors with high moderation 100% MOX fuel

cores. The configurations of interest for this work are

MISTRAL-2 (1997MISTRAL-2 ( - 1998))

,

MISTRAL-3 (1998MISTRAL-3 ( -1999) )

and

MISTRAL-4 (1999MISTRAL-4 ( -2000))

. MISTRAL-2 and MISTRAL-3 are regular cores, whereas MISTRAL-4 consists of a mock-up of 17 × 17 overmoderated assemblies.

  The experimental region was composed of four full MOX BWR assemblies placed in the center of the EOLE core tank. The BWR type MOX rods were specifically fabricated for this program in the MOX fabrication plant MELOX located in CEA Marcoule. They are composed of MOX pellets of Pu enrichments of 3.0, 5.0, 8.5 and 11.5%. In January 2005, the Pu composition was 238 Pu(2%), 239 Pu (56%), 240 Pu (25%), 241 Pu (9%), 242 Pu (9%) and 241 Am (1%). The experimental region was surrounded by a driver region composed of EPICURE MOX rods (7.0% enrichment) with the following Pu composi-

tion: 238 Pu(1%), 239 Pu (58%), 240 Pu (25%), 241 Pu (5%), 242 Pu (5%) and 241 Am (6%) (January 2005).

Table 6

 6 Evolution of the Pu isotopic vector with the Pu ageing for standard MOX fuel corresponding to recycled UO2 fuel for a burnup of 33 GWd/t, t 0 being the begining of irradiation.

	Isotopes	t 0	t 0 + 20 years
	238 Pu	2.1%	1.8%
	239 Pu	55.8%	56.2%
	240 Pu	25.2%	25.43%
	241 Pu	8.8%	3.1%
	242 Pu	6.6%	6.6%
	241 Am	1.5%	7.0%

Table 7

 7 Isotopic uncertainty components as a function of the Pu ageing for standard EPI-CURE MOX fuel pins (4.3% enrichment). The calculations were performed with the exact perturbation formalism implemented in the PARIS code

  Twenty years after the fabrication of the EPICURE MOX fuel pins, the 241 Am content is nearly equivalent to the initial content of 241 Pu.

		t 0	t 0 + 20 years	∆
	235 U	16 pcm	18 pcm	+2 pcm
	238 U	260 pcm	263 pcm	+3 pcm
	239 Pu	802 pcm	895 pcm	+93 pcm
	240 Pu	70 pcm	73 pcm	+3 pcm
	241 Pu	125 pcm	50 pcm	-75 pcm
	242 Pu	132 pcm	132 pcm	-
	1 H	41 pcm	44 pcm	+3 pcm
	16 O	44 pcm	46 pcm	+2 pcm
	27 Al	23 pcm	23 pcm	-
	of the EPICURE program (section 4.1) to the end of the FUBILA program
	(section 4.3).

241 Am > 200 pcm > 900 pcm > +700 pcm

Table 8

 8 Uncertainties on k eff calculated by Monte-Carlo by using the 241 Am and 239 Pu covariance data available in the JEFF-3.2 library. These results take only into account the uncertainties on the resonance parameters.

	Isotopes MISTRAL-2 MISTRAL-3	∆
	241 Am	±131 pcm	±143 pcm	+12 pcm
	239 Pu	±954 pcm	±978 pcm	+24 pcm
	time-of-flight measurements. Relative uncertainties on total, capture, fission
	and elastic cross sections calculated over a broad energy mesh (15 groups) are
	shown in Figs. 7 and 8.
	Impacts of the nuclear data uncertainties on the calculated lattice reactivity
	were investigated on the MISTRAL-2 and MISTRAL-3 configurations. The
	Monte-Carlo calculations were performed based on the Cholesky decomposi-
	tion of the Resonance Parameter Covariance Matrices assuming the resonance
	parameters to be Gaussian distributed. Results obtained from 1000 APOLLO2
	calculations are displayed on Fig. 9. The standard deviation of each distribu-
	tion is reported in

Table 8 .

 8 For 241 Am, the contribution of the resonance

Table 9

 9 Comparison between the relative uncertainties on the 239 Pu capture cross section, fission cross section and thermal quantities calculated with JEFF-3.2 and obtained in this work (target accuracy). The magnitude of the 241 Am contribution is significantly improved compared to the contribution of about 900 pcm reported in Table7. By contrast, the 239 Pu resonance parameter uncertainties available in JEFF-3.2 still lead to a non-negligible contribution slightly greater than 900 pcm, in

	Energy range	(n,γ)	Relative uncertainty
	(eV)	JEFF-3.2	JEFF-3.2 Target accuracy
	< 0.1	297.3 barns	4.3%	2.5%
	0.1 -0.54	693.9 barns	4.4%	2.5%
	0.54 -4.0	4.7 barns	3.7%	2.0%
	4.0 -22.6	52.2 barns	6.9%	6.9%
	22.6 -454.0	16.1 barns	6.8%	6.8%
	454.0 -2500.0	4.2 barns	5.6%	5.6%
	Energy range	(n,f)	Relative uncertainty
	(eV)	JEFF-3.2	JEFF-3.2 Target accuracy
	< 0.1	762.6 barns	0.9%	0.9%
	0.1 -0.54	1057.9 barns	1.9%	1.7%
	0.54 -4.0	24.1 barns	1.2%	1.2%
	4.0 -22.6	77.9 barns	3.1%	3.0%
	22.6 -454.0	19.1 barns	3.5%	3.5%
	454.0 -2500.0	5.0 barns	3.4%	3.3%
	Thermal	Value	Relative uncertainty
	quantities	JEFF-3.2	JEFF-3.2 Target accuracy
	σ f	747.2 barns	0.9%	0.9%
	σ γ	270.1 barns	4.4%	2.5%
	I f	308.8 barns	2.3%	2.3%
	I γ	180.1 barns	5.7%	5.7%
	K1	1161.5 barns	1.7%	1.0%
	parameter uncertainties reaches 131 pcm for MISTRAL-2 and 143 pcm for
	MISTRAL-3.

Acknowledgements

This work was supported by the french industrial partners (EDF and AREVA) through the SINET project of the Nuclear Energy Division of CEA.

was re-evaluated at ORNL (Oak Ridge National Laboratory) with the SAMMY