Yann Palu
email: yann.palu@u-picardie.fr.

Vincent Pilaud
email: vincent.pilaud@lix.polytechnique.fr.

Pierre-Guy Plamondon

Series and Algebraic Combinatorics (Hanover) Non-kissing complexes for gentle algebras

We introduce the non-kissing complex of any gentle bound quiver. This complex provides a powerful combinatorial model for support τ-tilting theory over gentle algebras, and it generalizes and unifies the previously considered situations of quivers defined from subsets of the grid or from dissections of a polygon (both generalizing the classical associahedron). In this extended abstract, we report on lattice theoretic and geometric properties of finite non-kissing complexes: we show that their flip graphs are Hasse diagrams of congruence-uniform lattices, and that they can be realized by convex polytopes.

Résumé.

Nous introduisons le complexe platonique d'un carquois aimable. Ce complexe offre un modèle combinatoire pour la théorie du τ-basculement à support des algèbres aimables et il généralise et unifie les cas particuliers définis à partir de sous-ensembles de la grille ou de dissections de polygones (contenant notamment le cas de l'associaèdre classique). Dans ce résumé étendu, nous présentons des propriétés combinatoires et géométriques des complexes platoniques finis: nous montrons que leurs graphes de flips sont les diagrammes de Hasse de treillis congruence-uniformes, et qu'ils peuvent être réalisés par des polytopes convexes.

Motivation: Non-kissing versus support τ-tilting

The non-kissing complex is a simplicial complex of pairwise non-kissing paths in a fixed shape of a grid. It was introduced by T. K. Petersen, P. Pylyavskyy and D. Speyer in [START_REF] Petersen | A non-crossing standard monomial theory[END_REF] for a staircase shape, studied by F. Santos, C. Stump and V. Welker [START_REF] Santos | Noncrossing sets and a Grassmann associahedron[END_REF] for rectangular shapes, and extended by T. McConville in [START_REF] Mcconville | Lattice structure of Grid-Tamari orders[END_REF] for arbitrary shapes. This complex is known to be a simplicial sphere, and it was actually realized as a polytope using successive edge stellations and suspensions in [START_REF] Mcconville | Lattice structure of Grid-Tamari orders[END_REF]Section 4]. Moreover, the dual graph of the non-kissing complex has a natural orientation which equips its facets with a lattice structure [START_REF] Mcconville | Lattice structure of Grid-Tamari orders[END_REF]Theorem 1.1,. Further lattice theoretic and geometric aspects of this complex were recently developed by A. Garver and T. McConville in [START_REF] Garver | Enumerative properties of Grid-Associahedra[END_REF].

The interest for non-kissing complexes is motivated by relevant instances arising from particular shapes. As already observed in [11, Section 10], when the shape is a ribbon, the non-kissing complex is an associahedron, and the non-kissing lattice is a type A Cambrian lattice of N. Reading [START_REF] Reading | Cambrian lattices[END_REF]. In particular, the straight ribbon corresponds to the Tamari lattice, an object at the heart of a deep research area [START_REF] Müller-Hoissen | Associahedra, Tamari Lattices and Related Structures[END_REF]. When the shape is a rectangle (or even a staircase), the non-kissing complex was studied in [START_REF] Petersen | A non-crossing standard monomial theory[END_REF][START_REF] Santos | Noncrossing sets and a Grassmann associahedron[END_REF] as the Grassmann associahedron, in connection to non-crossing subsets of [n].

Other instances of such complexes arise naturally from the representation theory of associative algebras. The notion of support τ-tilting module over an algebra was introduced by T. Adachi, O. Iyama and I. Reiten in [START_REF] Adachi | τ-tilting theory[END_REF], and has proved to be a successful generalization of tilting and cluster-tilting theory. Over a given algebra, indecomposable τ-rigid modules form a complex. For an account of the various algebraic interpretations of this complex, we refer the reader to [START_REF] Brüstle | Ordered exchange graphs[END_REF]. For example, in the case of the path algebra of a straight line quiver, the support τ-tilting complex is, again, an associahedron.

Our original motivation was to provide common interpretations to these different complexes. First, we realized any non-kissing complex as the support τ-tilting complex of a well-chosen associative algebra. The algebras that occur are certain gentle algebras, a special case of the well-studied string algebras of M. C. R. Butler and C. Ringel [START_REF] Butler | Auslander-Reiten sequences with few middle terms and applications to string algebras[END_REF]. Conversely, starting from any gentle bound quiver Q, we defined its blossoming quiver Q; and a non-kissing relation on the walks in Q; so that the following interpretation holds. Theorem 1.1. For any gentle bound quiver Q = (Q, I), the non-kissing complex of walks in the blossoming quiver Q; is isomorphic to the support τ-tilting complex of the gentle algebra kQ/I.

In short, to any walk in Q; corresponds a representation of Q, and this correspondence takes non-kissing walks to τ-compatible representations. This theorem provides a dictionary between the combinatorially-flavored non-kissing complex and the algebraically-flavoured support τ-tilting complex, thus opening a bridge to go back and forth between the two worlds. It allows us, for instance, to combinatorially define mutation of support τ-tilting modules. This seems worthwhile, as the mutation of support τ-tilting modules over an arbitrary algebra is generally difficult to carry out explicitly.

The precise statement and the proof of Theorem 1.1 can be found in the long version of this paper [START_REF] Palu | Non-kissing complexes and τ-tilting for gentle algebras[END_REF], as well as further representation-theoretic aspects of the project. In this extended abstract, we focus on combinatorial and geometric aspects. We first define in Section 2 the non-kissing complex of a gentle bound quiver and show that this complex is a pseudomanifold (meaning in particular that there is a well-defined notion of flips in non-kissing facets). In Section 3, we show that the graph of increasing flips is the Hasse diagram of a congruence-uniform lattice and describe its canonical join complex. Finally, Section 4 is devoted to the geometry of finite non-kissing complexes: we construct their g-vector fans and show that these fans are normal fans of convex polytopes. We refer to [START_REF] Palu | Non-kissing complexes and τ-tilting for gentle algebras[END_REF] for detailed proofs and further properties of non-kissing complexes.

Non-kissing complexes for gentle algebras 3 2 Non-kissing complexes of gentle bound quivers

Blossoming quivers and non-kissing walks

We fix a gentle bound quiver Q : = (Q, I), where Q : = (Q 0 , Q 1 , s, t) is a quiver with vertices Q 0 , arrows Q 1 , and source and target maps s, t : Q 1 → Q 0 , and I is a set of quadratic relations αβ = 0 with α, β ∈ Q 1 and t(α) = s(β) such that for any β ∈ Q 1 there is at most one α ∈ Q 1 such that t(α) = s(β) and αβ ∈ I (resp. αβ / ∈ I) and at most one γ ∈ Q 1 such that t(β) = s(γ) and βγ ∈ I (resp. βγ / ∈ I). See Figure 1. In all pictures, a relation αβ = 0 is indicated with an arc connecting the target of α to the source of β.

A string in Q is a word of the form

ρ = α ε 1 1 α ε 2 2 • • • α ε , where • α i ∈ Q 1 and ε i ∈ {-1, 1} for all i ∈ [ ],
• t(α

ε i i ) = s(α ε i+1 i+1 ) for all i ∈ [ -1], (by convention s(α -1 ) = t(α) and t(α -1 ) = s(α)),
• there is no αβ ∈ I such that αβ or β -1 α -1 appears as a factor of ρ, and • ρ is reduced, in the sense that no factor αα -1 or α -1 α appears in ρ, for α ∈ Q 1 . There is also an empty string ε v for each vertex v ∈ Q 0 . Strings are considered undirected, meaning that we implicitly identify ρ with ρ -1 . Let S( Q) be the set of strings of Q.

The blossoming quiver of Q = (Q, I) is the gentle bound quiver Q; = (Q ; , I ; ) obtained by adding arrows and relations at each vertex v ∈ Q 0 , so that v has precisely 2 incoming and 2 outgoing arrows and still fulfills the gentle conditions. See Figure 1.

A walk in Q is a maximal string in Q; , i.e. connecting two vertices of Q ; 0 Q 0 . A walk ω is bending if it has two opposite arrows and straight otherwise. For v ∈ Q 0 , the peak walk v peak (resp. the deep walk v deep ) is the walk with two outgoing (resp. incoming) arrows at vertex v and one incoming and one outgoing arrow at all its other vertices. A

substring of ω = ω ε 1 1 . . . ω ε is a factor σ = ω ε m+1 m+1 . . . ω ε n-1 n-1 with 1 ≤ m < n ≤ . We say that σ is a top (resp. bottom) substring of ω if ε m = -1 = -ε n (resp. ε m = 1 = -ε n ), mean-
ing that ω has two outgoing (resp. incoming) arrows at the endpoints of σ. Let Σ bot (ω) and Σ top (ω) be the sets of bottom and top substrings of ω respectively. Consider two walks ω, ω on Q. We say that ω kisses ω if Σ top (ω) ∩ Σ bot (ω ) = ∅, i.e. if there exists a common substring σ of ω and ω such that ω has two outgoing arrows incident to σ while ω has two incoming arrows incident to σ. See Figures 1 (right) & 3 (left). We authorize the case where σ is reduced to a vertex v, i.e. v is a peak of ω and a deep of ω . Note that ω can kiss ω several times, that ω and ω can mutually kiss, and that ω can kiss itself. The non-kissing complex of Q is the simplicial complex K nk ( Q) whose faces are the collections of walks which are not self-kissing and pairwise non-kissing. Note that no straight walk can kiss another walk by definition, so that they appear in all facets of K nk ( Q). We thus consider the reduced non-kissing complex C nk ( Q) to be the deletion of all straight walks from K nk ( Q). See Figure 2 (left).

Our definition of non-kissing complex is largely inspired from and specializes to simplicial complexes defined from subsets of the grid [START_REF] Petersen | A non-crossing standard monomial theory[END_REF][START_REF] Santos | Noncrossing sets and a Grassmann associahedron[END_REF][START_REF] Mcconville | Lattice structure of Grid-Tamari orders[END_REF][START_REF] Garver | Enumerative properties of Grid-Associahedra[END_REF] or from dissections of polygons [START_REF] Baryshnikov | On Stokes sets[END_REF][START_REF] Chapoton | Stokes posets and serpent nests[END_REF][START_REF] Garver | Oriented flip graphs of polygonal subdivisions and noncrossing tree partitions[END_REF][START_REF] Manneville | Geometric realizations of the accordion complex of a dissection[END_REF]. To the best of our knowledge, we actually provide the first connection between these two families, besides the observation that associahedra are instances of both. In fact, the example of Figure 2 is also a special case of both families.

Distinguished arrows and flips

We now show that the non-kissing complex C nk ( Q) is a pseudomanifold, i.e. that it is pure (all facets have the same dimension) and thin (there is a well-defined notion of flips).

A marked walk ω is a walk ω = α ε 1 1 • • • α ε with a marked arrow α ε i i . Consider two distinct non-kissing walks µ , ν marked at an arrow α ε ∈ Q ;

1 . Let σ denote their maximal common substring containing that occurrence of α. Since µ = ν , their common substring σ is strict, so that µ and ν split at least at one endpoint of σ. We define the countercurrent order at α by µ ≺ α ν when µ enters and/or exits σ in the direction of α, while ν enters and/or exits σ in the opposite direction. For a face F of K nk ( Q), we call distinguished walk of F at an arrow α the ≺ α -maximal walk dw(α, F), and we call distinguished arrows of a walk ω ∈ F the arrows da(ω, F) : = {α ∈ ω | dw(α, F) = ω}. The following statement is inspired from [11, Theorem 3.2] and illustrated in Figure 4 (left). (ii) The walk ω : = ρ στ is kissing ω but no other walk of F. Moreover, ω is the only other walk besides ω which is not kissing any other walk of F {ω}. We say that F {ω, ω } is obtained from F by flipping ω, and that the flip is supported by σ.

Corollary 2.4. The reduced non-kissing complex C nk ( Q) is a pseudomanifold without boundary.

Non-kissing lattices

The flip of Proposition 2.3 and Figures 3 &4 exchanges two kissing walks ω, ω . The flip is increasing when their common substring is on top of ω and on the bottom of ω . This yields the increasing flip graph, where vertices are non-kissing facets and arcs are increasing flips. See Figure 2 (right). The main result of this section is the following statement. Congruence-uniform lattices will be properly defined in Section 3.4. To achieve Theorem 3.1, we use a technique developed by T. McConville for grid quivers [START_REF] Mcconville | Lattice structure of Grid-Tamari orders[END_REF]: we identify the non-kissing lattice with a quotient of a lattice of biclosed sets of strings.

Biclosed sets of strings

A closure operator on a finite set S is a map S → S cl on subsets of S such that ∅ cl = ∅, S ⊆ S cl , (S cl ) cl = S cl and S ⊆ T =⇒ S cl ⊆ T cl for any S, T ⊆ S. A subset S ⊆ S is closed if S cl = S, coclosed if S S is closed, and biclosed if it is both closed and coclosed. Let Bic(S ) be the inclusion poset of biclosed subsets of S. In [11, Theorem 5.5], T. Mc-Conville gave simple sufficient conditions for Bic(S ) to be a congruence uniform lattice. In [START_REF] Palu | Non-kissing complexes and τ-tilting for gentle algebras[END_REF]Theorem 3.21], we extended this criterion in the situation when the singletons of S are not biclosed so that we can apply it in our context of non-kissing complexes.

In a gentle bound quiver Q, we define the closure S cl of a set S of strings of Q as the set of all strings of the form

σ 1 α ε 1 1 σ 2 α ε 2 2 . . . α ε -1 -1 σ where σ i ∈ S, α i ∈ Q 1 and ε i ∈ {-1, 1}.
Let Bic( Q) be the inclusion poset on biclosed sets of strings of Q. For example, when Q is a path on n vertices with no relation, the strings are in bijection with pairs (i, j) ∈ ( n+1

2 ), the closure on strings translates to the concatenation (i, j) • (k, ) = δ j=k (i, ), biclosed sets of strings are in bijection with inversion sets of permutations of [n + 1], so that Bic( Q) is isomorphic to the weak order on S n+1 . Figure 5 (left) illustrates the poset Bic( Q) for another gentle bound quiver, with the empty set in the bottom and the set of all strings of Q on top. The criterion of [START_REF] Palu | Non-kissing complexes and τ-tilting for gentle algebras[END_REF]Theorem 3.21] yields the following result. Theorem 3.2. When Q has finitely many strings, the inclusion poset of biclosed sets Bic( Q) is a congruence-uniform lattice.

Lattice congruence

A lattice congruence of a lattice (L, ≤, ∧, ∨) is an equivalence relation ≡ on L compatible with meets and joins: x ≡ x and y ≡ y implies x ∧ y ≡ x ∧ y and x ∨ y ≡ x ∨ y . It defines a lattice quotient L/≡ on the congruence classes of ≡ where the order relation is given by X ≤ Y if and only if there exists x ∈ X and y ∈ Y such that x ≤ y and the meet X ∧ Y (resp. the join X ∨ Y) of two congruence classes X and Y is the congruence class of x ∧ y (resp. of x ∨ y) for arbitrary representatives x ∈ X and y ∈ Y. For a finite lattice L, an equivalence relation ≡ on L is a lattice congruence if and only if its congruence classes are intervals and the maps π ↓ and π ↑ , sending an element x ∈ L to the minimum and maximum of its congruence class respectively, are order preserving.

Following [START_REF] Mcconville | Lattice structure of Grid-Tamari orders[END_REF]Section 7], we associate to a biclosed set S ∈ Bic( Q) the sets

π ↓ (S) : = σ ∈ S( Q) Σ bot (σ) ⊆ S and π ↑ (S) : = σ ∈ S( Q) Σ top (σ) ∩ S = ∅ .
Here and throughout the paper, we denote by Σ bot (σ) the set of bottom substrings for a string σ = α ε 1 1 . . . α ε , i.e. the substrings α ε m m . . . α ε n n for 1 ≤ m ≤ n ≤ such that m = 1 or ε m-1 = -1, and n = or ε n+1 = 1 (and similarly for the set of top substrings Σ top (σ)). Proposition 3.3. For any S ∈ Bic( Q), the sets π ↓ (S) and π ↑ (S) are biclosed. Moreover,

(i) π ↓ (S) ⊆ S ⊆ π ↑ (S) for any element S ∈ Bic( Q), (ii) π ↓ • π ↓ = π ↓ • π ↑ = π ↓ and π ↑ • π ↑ = π ↑ • π ↓ = π ↑ , ( iii
) π ↓ and π ↑ are order preserving. Therefore, the fibers of π ↑ and π ↓ coincide and are the classes of a lattice congruence ≡ on Bic( Q).

For example, if Q is an oriented path with no relation, ≡ is a Cambrian congruence of the weak order [START_REF] Reading | Cambrian lattices[END_REF]. The congruence classes of ≡ appear as blue rectangles in Figure 5.

Non-kissing lattice

Coming back to our original problem, we now aim to show that the increasing flip graph on non-kissing facets is isomorphic to the Hasse diagram of the quotient of the lattice of biclosed set Bic( Q) of Section 3.1 by the lattice congruence of Section 3.2. The next two propositions provide explicit maps between biclosed sets of strings and non-kissing facets illustrated in Figure 6. It extends previous definitions of [START_REF] Mcconville | Lattice structure of Grid-Tamari orders[END_REF] for grid quivers.

Proposition 3.4. For S ∈ Bic( Q) and α ∈ Q ; 1 , let ω(α, S) : = α ε - -• • • α ε -1 -1 • α • α ε 1 1 • • • α ε r
r be the directed walk containing α defined by

• ε i = -1 if the string α ε 1 1 • • • α ε i-1
i-1 belongs to S, and

ε i = 1 otherwise, for all i ∈ [r], • ε -i = 1 if the string α ε -i+1 -i+1 • • • α ε -1
-1 belongs to S, and

ε i = -1 otherwise, for all i ∈ [ ]. Then the set ω(α, S) α ∈ Q ; 1 contains 2|Q 0 | -|Q 1 | straight
walks and |Q 0 | pairs of inverse directed bending walks, which are all pairwise non-kissing. We thus obtain a facet η(S) of K nk ( Q) by identifying these pairs of inverse directed bending walks.

Proposition 3.5. For any facet

F ∈ K nk ( Q), the set ζ(F) : = ω∈F Σ bot (ω) cl is biclosed.
When the quiver is an oriented path with no relation, the map η should be thought of as the map from permutations to triangulations defined in [START_REF] Reading | Cambrian lattices[END_REF]. Conversely, ζ maps a triangulation to the minimal permutation in its fiber under η. For the straight quiver, η plays the role of the binary search tree insertion while ζ selects the minimal linear extension of a binary tree. Using these maps, we show that the increasing flip graph on non-kissing facets is isomorphic to the Hasse diagram of the lattice quotient Bic( Q)/≡. Theorem 3.6. The maps η :

Bic( Q) → K nk ( Q) and ζ : K nk ( Q) → Bic( Q) satisfy: • η ζ(F) = F for any facet F ∈ K nk ( Q), • ζ η(S) = π ↓ (S) for any biclosed set S ∈ Bic( Q),
• for any facet F ∈ C nk ( Q) and σ ∈ ζ(F ), there exists an increasing flip F → F supported by σ if and only if ζ(F ) {σ} is biclosed. Therefore, the facets of K nk ( Q) are in bijection with the congruence classes of ≡ and the increasing flip graph is the Hasse diagram of the lattice quotient Bic( Q)/≡.

-→ η -→ ζ S η(S) ζ(η(S)) = π ↓ (S)

Canonical join complex

In a lattice (L, ≤, ∧, ∨), a join representation of x ∈ L is a subset J ⊆ L such that x = J. This representation is irredundant if x = J for a strict subset J J. The irredundant join representations of x ∈ L are ordered by containment of the lower ideals of their elements, i.e. J ≤ J if and only if for any y ∈ J there exists y ∈ J such that y ≤ y . When this order has a minimal element, it is called the canonical join representation of x. All elements of the canonical join representation x = J are then join-irreducible, i.e. cover a single element. Canonical meet representations and meet-irreducibles are defined dually.

A lattice L is congruence-uniform if its join-irreducible elements are in bijection with the join-irreducibles of its lattice of congruences, and similarly for meet-irreducibles. Congruence-uniform lattices behave nicely with join representations and congruence lattices. In particular, congruence-uniform lattices are semi-distributive, so that any element admits a canonical join representation. The collection of sets J which define canonical join representations in L is the canonical join complex of L.

To conclude our study of the non-kissing lattice L nk ( Q), we describe its canonical join complex. We say that a string σ ∈ S( Q) is distinguishable if there is a facet F ∈ K nk ( Q) and a walk ω ∈ F such that σ = ds(ω, F). These strings are characterized as follows.

Proposition 3.7. A string σ ∈ S( Q) is distinguishable if and only if Σ bot (σ) ∩ Σ top (σ) = {σ}.
One checks that Σ bot (σ) cl is biclosed so that we can define ji(σ) : = η Σ bot (σ) cl . Proposition 3.8. The map ji : σ → ji(σ) defines a bijection between the distinguishable strings of Q and the join-irreducible elements of the non-kissing lattice L nk ( Q).

Therefore, distinguishable strings are building blocks for canonical join representations in L nk ( Q). A descent of a facet F ∈ K nk ( Q) is a string σ which is the distinguished string of a ω of F and is a bottom substring of ω (so that the flip of ω in F is a descent in the non-kissing lattice). We denote by des(F) the set of descents of F. Proposition 3.9. The canonical join representation of F ∈ L nk ( Q) is given by F = σ∈des(F) ji(σ).

To conclude, we characterize which subsets of strings correspond to canonical join representations in the non-kissing lattice L nk ( Q). Following [START_REF] Garver | Enumerative properties of Grid-Associahedra[END_REF], we say that two strings are non-friendly if Σ top (σ) ∩ Σ bot (τ) = ∅ = Σ bot (σ) ∩ Σ top (τ). We call non-friendly complex the simplicial complex of sets of pairwise non-friendly distinguishable strings.

Theorem 3.10. The following assertions are equivalent for a set Σ of distinguishable strings of Q:

• Any two strings of Σ are non-friendly.

• {ji(σ) | σ ∈ Σ} is the canonical join-representation of a facet of K nk ( Q).

• Σ is the descent set of a non-kissing facet F ∈ K nk ( Q). In other words, the canonical join complex of L nk ( Q) is isomorphic to the non-friendly complex.

For example, when the quiver is a straight path with no relation, the non-friendly complex is isomorphic to the non-crossing partition complex.

Gentle associahedra

In this section, we provide polyhedral realizations for finite non-kissing complexes, using tools inspired from the finite type cluster algebras of S. Fomin and A. Zelevinsky [START_REF] Fomin | Cluster algebras. II. Finite type classification[END_REF].

g-vectors and c-vectors

Let m V : = ∑ i∈[m] e v i ∈ R Q 0 be the multiplicity vector of a multiset V = {v 1 , . . . , v m } of Q 0 . For a string σ ∈ S( Q), let m σ : = m V(σ) where V(σ) is the multiset of vertices of σ.
For a walk ω on Q, we denote by peaks(ω) (resp. by deeps(ω)) the (multi)set of peaks (resp. deeps) of ω. The g-vector of ω is the vector g(ω

) : = m peaks(ω) -m deeps(ω) ∈ R Q 0 .
For a set Ω of walks, g(Ω) : = {g(ω) | ω ∈ Ω}. Note that g(ω) = 0 for a straight walk ω.

Consider a bending walk ω in a facet F ∈ C nk ( Q). By Proposition 2.1, ω has two distinguished arrows da(ω, F) around its distinguished string ds(ω, F). The c-vector of ω ∈ F is the vector c(ω ∈ F

) : = ε(ω, F) m ds(ω,F) ∈ R Q 0 , where ε(ω, F) : = 1 if ds(ω, F) ∈ Σ top (ω) and ε(ω, F) : = -1 if ds(ω, F) ∈ Σ bot (ω). Denote by c(F) : = {c(ω ∈ F) | ω ∈ F}.
F • • • • • •                 1 0 0 -1 0 0 0 2 0 0 0 0 0 -1 3 1 0 0 1 0 0 4 0 0 0 -1 0 0 5 0 0 0 1 1 1 6 0 1 0 0 0 0 g(F) • • • • • •                
1 0 0 -1 0 0 0 2 0 0 0 0 1 -1 3 1 0 0 0 0 0 4 1 0 0 -1 1 0 5 0 0 0 0 1 0 6 0 1 0 0 0 0 c(F)

g-vector fans and gentle associahedra

We now use g-and c-vectors to construct polyhedral realizations of finite non-kissing complexes. Since the g-vectors of the walks ω, ω , µ, ν involved in the flip of Figure 3 satisfy the linear dependence g(ω) + g(ω ) = g(µ) + g(ν), we get the following statement.

Theorem 4.2. For a gentle bound quiver Q with finite non-kissing complex C nk ( Q), the collection of cones F g ( Q) : = R ≥0 g(F) F non-kissing face of C nk ( Q) forms a complete simplicial fan, that we call the g-vector fan of Q.

This fan is illustrated in Figure 8 (left). Note that it was constructed in [START_REF] Manneville | Geometric realizations of the accordion complex of a dissection[END_REF] for dissection quivers and in [START_REF] Garver | Enumerative properties of Grid-Associahedra[END_REF] for grid quivers. Both constructions extend the type A Cambrian fans of N. Reading and D. Speyer [START_REF] Reading | Cambrian fans[END_REF] obtained for path quivers with no relations. We now aim at constructing a polytope whose normal fan is the g-vector fan of Q. For two walks ω, ω on Q, denote by κ(ω, ω ) the number of distinct kisses of ω to ω . The kissing number of ω and ω is kn(ω, ω ) : = κ(ω, ω ) + κ(ω , ω). When C nk ( Q) is finite, we can define the kissing number of a walk ω on Q as kn(ω) : = ∑ ω kn(ω, ω ). Theorem 4.3. For a gentle bound quiver Q with finite non-kissing complex C nk ( Q), the g-vector fan F g ( Q) is the normal fan of the Q-associahedron Asso( Q) defined equivalently as:

(i) the convex hull of the points p(F) : = ∑ ω∈F kn(ω) c(ω ∈ F) for all facets F ∈ C nk ( Q), or (ii) the intersection of the halfspaces H ≤ (ω) : = x ∈ R Q 0 g(ω) | x ≤ kn(ω) for all walks ω on Q.

For path quivers with no relation, we recover the associahedra of C. Hohlweg and C. Lange [START_REF] Hohlweg | Realizations of the associahedron and cyclohedron[END_REF]. The latter are obtained by deleting inequalities in the facet description of the classical permutahedron. This property is lost for arbitrary gentle quivers: on the one hand, the Coxeter arrangement supporting the g-vector fan F g ( Q) is not necessarily of finite type; on the other hand, the Q-associahedron is not always obtained by deleting inequalities in the facet description of the Minkowski sum of all c-vectors. See [START_REF] Palu | Non-kissing complexes and τ-tilting for gentle algebras[END_REF] for a detailed discussion. The Q-associahedron was constructed in [START_REF] Manneville | Geometric realizations of the accordion complex of a dissection[END_REF] in the special case of dissection quivers. An example of Asso( Q) is shown in Figure 8. For grid quivers, our construction proves the polytopality conjecture for the g-vector fan stated in [START_REF] Garver | Enumerative properties of Grid-Associahedra[END_REF].

This realization of the non-kissing complex has the following relevant property regarding the non-kissing lattice studied in Section 3. Proposition 4.4. When oriented in the linear direction (-1, . . . , -1) ∈ R Q 0 , the graph of the Q-associahedron is (isomorphic to) the increasing flip graph.

Figure 1 :

 1 Figure 1:A gentle bound quiver Q (left), its blossoming quiver Q; (middle), and some walks in Q; (right). The dotted red and orange walks are non-kissing, but both are kissing the plain blue walk. See Figure4for examples of non-kissing facets.

Figure 2 :

 2 Figure 2: A reduced non-kissing complex (left) and its flip graph (right).

Figure 3 :

 3 Figure 3: Two kissing walks (left) and a flip (right).

Proposition 2 . 1 .

 21 Each bending (resp. straight) walk of a facet F ∈ K nk ( Q) contains precisely 2 (resp. 1) distinguished arrows pointing in opposite directions (resp. in the direction of the walk).

Corollary 2 . 2 .Proposition 2 . 3 .

 2223 The reduced non-kissing complex C nk ( Q) is pure of dimension |Q 0 |.Define the distinguished string of a bending walk ω in a facet F ∈ K nk ( Q) as the substring ds(ω, F) of ω located between the two distinguished arrows of ω. Consider a facet F ∈ K nk ( Q) and a bending walk ω ∈ F. Write ω = ρστ where σ : = ds(ω, F). Let {α, β} : = da(ω, F), and α and β be the other two arrows of Q ; 1 incident to the endpoints of σ and such that α α ∈ I or αα ∈ I, and β β ∈ I or ββ ∈ I. Let µ : = dw(α , F {ω}) and ν : = dw(β , F {ω}). See Figures 3 (right) & 4. Then (i) The string σ splits the walk µ into µ = ρ στ and the walk ν into ν = ρστ .

Figure 4 :

 4 Figure 4: Flipping the red walk ω to the orange walk ω . The walks µ, ν involved in the flip are the blue and green walks. Distinguished arrows are marked with triple arrows.

Theorem 3 . 1 .

 31 If C nk ( Q) is finite, the increasing flip graph is the Hasse diagram of a congruenceuniform lattice, that we call non-kissing lattice and denote by L nk ( Q).

Figure 5 :

 5 Figure 5: The inclusion lattice of biclosed sets Bic( Q) with congruence classes of ≡ in blue (left), and the corresponding lattice of increasing flips on facets of K nk ( Q) (right).

Figure 6 :

 6 Figure 6: The maps η (left) and ζ (right) between non-kissing facets and biclosed sets.

Proposition 4 . 1 .

 41 For any non-kissing facet F ∈ C nk ( Q), the set of g-vectors g(F) and the set of c-vectors c(F) form dual bases.

Figure 7 :

 7 Figure 7: The g-and c-matrices of a facet F form dual bases.

Figure 8 :

 8 Figure 8: The g-vector fan F g ( Q) (left) and the gentle associahedra (right).

Yann Palu, Vincent Pilaud, and Pierre-Guy Plamondon

Supported by the French ANR grant SC3A (15 CE40 0004 01). § pierre-guy.plamondon-psud.fr. Supported by the French ANR grant SC3A (15 CE40 0004 01).


Kevin Gillet

Guido Roma

Jean-Paul Crocombette

Dominique Gosset

The influence of irradiation induced vacancies on the mobility of helium in boron carbide

Keywords: Density functional theory, Boron carbide, Helium, Vacancies, Kinetics, Nudged elastic bands

come

The influence of irradiation induced vacancies on the mobility of helium in boron carbide

Introduction

Boron carbide, due to the very high neutron absorption cross-section of the boron isotope 10 B, and fairly smooth dependence of it on neutron energy, has been extensively employed as neutron absorber in the nuclear industry for several decades, including for fast breeder reactors [START_REF] Rouault | Handbook of Nuclear Engineering[END_REF][START_REF] Gosset | Structural Materials for Generation IV Nuclear Reactors[END_REF].

It constitutes the core of the control rods, placed among the fuel rods in the reactor, which are meant to regulate the rate of fuel fission or to shutdown the reactor. It is crucial to maintain the integrity of control rods for security and operational management, but due to the degradation of irradiated boron carbide, a brittle material, their lifetime, which is around 10 to 15 years in thermal neutron reactors, drops to less than 5 years in fast neutron reactors [START_REF] Gosset | Structural Materials for Generation IV Nuclear Reactors[END_REF]. Being at the core of the nuclear power plant, the replacement of control rods is expensive and has to be carried out when the reactor is shut down. Therefore, optimizing boron carbide behavior under irradiation is one of the key features of next generation nuclear power plants viability.

Boron carbide is also increasingly widespread among other important technological applications, in particular related to its hardness -one of the highest known-, low density and good chemical stability, like for example armor plating [START_REF] Hayun | [END_REF]4], processing tools, composites fibers and more. As a semiconductor, it could perhaps be envisaged for functional applications; however the wide stoichiometry range and still not fully understood atomic disorder has, up to now, prevented such applications.

In spite of the technological importance of the material both self-and impurity diffusion, which are important in processing, microstructure control, and irradiation behavior, are very scarcely understood in this material.

In particular, the interplay of helium and the point defects created by neutron irradiation, and their mobility, is crucial in order to understand and control the formation of helium bubbles, which ultimately control internal strain [5] and the lifetime of control rods [START_REF] Gosset | Structural Materials for Generation IV Nuclear Reactors[END_REF].

Boron carbide, somewhat improperly often referred to as B 4 C, occurs in variable stoichiometry. In order to explain these features various defective structures were investigated, in particular chain vacancies [6,7], antisites [8,9] and, very recently, boron interstitials [10]. The goal was, however, mainly phase stability and not the properties of isolated defects, nor their role in mass transport kinetics. In spite of a number of experimental works devoted to helium bubbles [11,12,13,14,15,16,17,18,19,20,21] and the related question of helium kinetics [20,[START_REF] Clayton | The release of helium from slightly irradiated boron carbide and boron carbide-silicon carbide plates[END_REF][START_REF] Svetukhin | [END_REF][START_REF] Beyer | A physically based model for helium release from irradiated boron carbide[END_REF], not much is known on self-diffusion.

In a previous theoretical work we showed how helium interstitial 2D diffusion could bias the shape of helium bubbles [START_REF] Schneider | [END_REF]. A subsequent theoretical study [26] has confirmed our results for He and investigated the mobility of lithium, the other product of the nuclear reaction triggered by 10 B neutron absorption. Although some results on neutral vacancies where presented in these two works, no attempt was made to assess the influence of such defects on the mobility of He.

In order to gain a deeper understanding of He bubble formation under irradiation conditions it is necessary to assess the role of intrinsic defects created by irradiation and, in particular, the role of vacancies, which constitute possible bubble nucleation sprouts.

In this work we focus on the influence of intrinsic vacancies created under irradiation on the diffusion of helium, which is produced by nuclear reactions in boron carbide submitted to a neutron flux. Such a picture can also be simulated by helium implantation [20], possibly coupled with ion irradiation. We investigate the possible mechanisms at the atomic scale by means of first principles calculations in the framework of Density Functional Theory (DFT).

The paper is organised as follows : in section 2 we discuss methodological aspects of our calculations. The results section 3 is splitted in two subsections: the first, 3.1, devoted to vacancy formation and migration, the second, 3.2 to substitutional helium and its migration mechanisms involving vacancies. We conclude with a discussion on the possible contribution of those mechanisms to He diffusion (section 4).

Computational Details

The results presented in this paper rest upon first-principles calculations as formulated in the Density Functional Theory (DFT) framework. Defect stability and defect mobility are respectively quantified in term of formation energy and migration energy obtained through total energy calculations of several configurations.

Boron carbide rhombohedral phase exists over a large range of stoichiometry: the literature reports boron carbides from 8.6 to 18.8 at.%C [27,28] even if precise boundaries are still under debates [29]. These stoichiometry changes in boron carbide are thought to be accomodated thanks to local disorder in the structure [30]. Thus model structures of boron carbide are necessarily approximations. In this study we chose to represent the boron carbide bulk crystal by the carbon-rich crystallographic structure B 11 C p CBC (C p ≡ carbon in polar position) which is considered as the most stable theoretical structure over the whole homogeneity range at 0 K [31,32,33]. This structure is described by a 15-atom elementary cell of rhombohedral symmetry with an icosahedron B 11 C at each vertex and a chain C-B-C on the main diagonal ( 111 direction). The 12 icosahedron sites are divided in two distinct types resulting from their chemical environment : the 6 equatorial sites which are bonded to the chains (B5, B6, B7, B8) and the 6 polar sites which are directly bonded to other icosahedra (C3, B2, B3, B4) ; the chain sites correspond to the carbon extremities (C1, C2) called chain end and the boron chain center B1 (Fig. 1).

Boron carbide is experimentally observed of rhombohedral symmetry [34], and carbon atoms on the icosahedra are supposed to be found randomly distributed between the six crystallographically equivalent polar sites. Thus the use of periodic boundary conditions leads to an unphysical periodic arrangment of the carbons on the icosahedra which breaks the rhombohedral symmetry into monoclinic symmetry. The resulting distorsion being relatively weak, the cell symmetry was constrained to rhombohedral symmetry by using the averaged monoclinic cell parameters of the equilibrium structure. The different configurations were then relaxed at the equilibrium volume to conserve the rhombohedral symmetry.

Total energies were computed with the open-source code Quantum Espresso [35] using a GGA-based vdW-DF functional [36] which combines an optimized Becke86 exchange component with non-local van der Waals correlation part, as used in a previous work on He in boron carbide [START_REF] Schneider | [END_REF]. Defect calculations were carried out in a 2×2×2 supercell to approach the dilute limit of the defect. The Brillouin zone of the supercell was sampled with a 2×2×2 Monkhorst-Pack k-points mesh and the wave function kinetic energy cutoff was set to 80 Ry. In order to overcome the DFT band gap problem [37], calculations were also performed with the HSE06 hybrid functional [38,39] to get a better description of charge transition levels of defects. Charged defect calculations with the supercell method induce spurious interactions between the periodic images of the defect; thus long-range electrostatic interactions between charged defects were corrected with the Madelung monopole term of the Makov & Paine approach [40,41]. The latter is expressed, for a lattice of charges q separated by a typical length L, as q 2 α 2L where α stands for the Madelung constant of the lattice and stands for the dielectric constant of the material. Using the static dielectric constant calculated for the B 11 C p CBC structure 0 ≈ 8, the monopole correction was evaluated at 0.27q 2 eV. This first order correction was found sufficient to obtain converged energies within 0.05 eV of the asymptotic value with our 2×2×2 supercells, as shown in Fig. 2.

Then the formation energy of a given defect in charge state q is computed as :

E q f (µ e ) = E def ect tot -E perf ect tot + i ∆n i µ * i + q × (E V BM + µ e ) + E M adelung corr
where ∆n i and µ * i are associated respectively with the lack or excess of the atomic species i and its chemical potential in the thermodynamic conditions of the material. In the following sections, formation energies are expressed in the carbon-rich thermodynamical condition. Carbon chemical potential was set to the atomic energy of graphite and helium chemical potential was set to the energy of an isolated He atom. The zero of the electron chemical potential was set to the valence band top of the perfect structure.

Eventually migration energies were calculated thanks to the Nudged Elastic Band method [42] as implemented in the PWneb module of Quantum Espresso distribution. Minimum energy migration paths were interpolated based on at least 5 discrete images, each relaxed below atomic forces of 0.05 eV/ Å.

Results

In order to consider the interaction of vacancies with helium impurities and the possibility of vacancy-driven diffusion mechanisms, several key migrations are investigated. First we focus on the stability and migration of an isolated vacancy, a prerequisite to any mechanism involving vacancies. Then the association and dissociation of He impurities with vacancies, i.e., the formation and annihilation of substitutional impurities, are examined.

Stability and mobility of an isolated vacancy

For the sake of completeness, vacancy stability was studied on the 11 nonequivalent sites of the B 11 C p CBC structure. Table 1 reports the formation energies of neutral vacancies and their charge transition levels.

Our charged vacancies calculations confirm that chain vacancies are more favorable than icosahedral vacancies [START_REF] Schneider | [END_REF]26], especially the boron chain center vacancy which is the most stable vacancy in the structure by far.

Because of its predominance, the boron chain center (B1) vacancy stability was investigated using semilocal and hybrid functionals. The latter yielded slightly higher formation energies than GGA calculations but the qualitative picture is unchanged. In Fig. 3 the formation energies are represented with two different conventions:

In panel a) the GGA and HSE formation energies are compared with the usual x-axis convention, aligning at µ e = 0 both valence band maxima, whereas in Fig. 3b GGA and HSE band structures are referred to a common absolute level [43] and only HSE valence band maximum is set to µ e = 0. The experimental band gap at 2.09 eV [44] is also displayed (as "Exp. CBM") to account for the discrepancies coming from the idealized B 11 C p CBC structure of boron carbide. This double representation highlights the fact that the shift of the charge transition levels is mostly explained by the valence band shift between GGA and HSE calculations, as previously shown for a variety of defects in materials [43].

Based on our results, the chain center vacancy is expected to be positively charged or neutral in strongly or moderate p-type materials, and negatively charged in intrinsic and n-type conditions. The boron chain center vacancy adopts different geometries depending on its charge state. The distance between the two carbon atoms of the chain, evaluated at 2.90 Å for the neutral defect with HSE functional, increases to 3.14 Å when charged +1 and drops to 1.90 Å when charged -1. The negatively charged geometry change is similar to the formation of the C-C bond described by Raucoules et al. [7] in high pressure LDA calculations. This is confirmed by the analysis of the charge density along the neutral and charged defective chains, as shown in Figure 4, with the significative increase of the electron density between the two carbon atoms in the negative charge state.

Icosahedral vacancies have formation energies ranging from 4.42 to 5.82 eV in the neutral state and they become positively charged near the valence band edge and negatively charged above µ e ≈ 1.0 eV (see Table 1). Due to their very high formation energy, icosahedral vacancies are metastable configurations towards the chain center vacancy. Even though icosahedral vacancies have similar formation energies, polar and equatorial vacancies cannot be considered as equally stable. Indeed the relaxation starting from some of the equatorial configurations converged to the boron chain center configuration, showing that equatorial vacancies may be simply unstable. Probing the energy landscape around icosahedral vacancies confirmed this behavior showing that, for equatorial vacancies, the barrier to convert to a chain vacancy is either zero or extremely small and that the barrier between polar vacancies and the chain center vacancy is high enough to guarantee their stability ; thus in the following polar vacancies are considered stable and equatorial vacancies are considered unstable.

In order to limit the complexity of the study, carbon vacancies were not considered in the migration processes. Their migration would imply the creation of antisite defects whose study goes beyond the scope of this work. Moreover, equatorial vacancies being unstable, carbon chain vacancies would not have obvious migration paths. Therefore vacancy diffusion mechanisms are restricted to migrations involving polar boron vacancies and the chain center vacancy.

The B1 vacancy being the most stable configuration, elementary jumps between two chain centers were examined at first. The least energetic direct jump between two chain centers has been determined with an energy barrier of 4.73 eV in the neutral state, which makes it very unlikely to be the main diffusion mechanism for vacancies. To find other migration paths for vacancies, polar vacancies are considered as intermediate configurations.

The straightforward chain-polar mechanism combines a jump from the chain center to an icosahedra pole, possibly an exchange of the polar vacancy with one of its neighbour and eventually the landing of the vacancy on an other chain center (Fig. 5). These steps have respectively been evaluated with energy barriers of 3.72 eV, 0.80 eV and 0.85 eV. The limiting step is indeed the chain center to polar vacancy jump due to the large formation energy difference. Even if this chain-polar mechanism has a very high activation energy, it is still more favorable than the direct chain-chain migration path. Theoretically polar vacancies could also jump directly to other icosahedra with a calculated energy barrier of 1.99 eV (in the neutral charge state) and provide long-range diffusion through a polar-polar mechanism with lower activation energy. The impact of this mechanism should be assessed as a function of vacancy supersaturation; however, at equilibrium, the contribution of this kind of mechanism is thought to be rather limited because the formation energy of polar vacancies is very high. The main diffusion mechanism of vacancies is thus the chain-polar mechanism, with a migration energy of 3.72 eV and an activation energy (formation+migration) of 5.3 eV for the neutral vacancy and that might be a few tenths of an eV lower for charged species, depending on the Fermi level position. This high activation energy for vacancy diffusion is not surprising if we compare it to other semiconducting carbides like silicon carbide. [45] A recent work by You and coworkers [26], devoted to neutral defects, reached similar conclusions concerning migration energies. However, slightly higher energy barriers for the equatorial to chain center jump are reported, leading to consider the equatorial site as a stable position for migrations.

In particular, the authors suggest that the migration from the polar to the chain center site is composed of two elementary jumps: one of 1.00 eV between the polar and the equatorial site and another jump of 0.18 eV between the equatorial and the chain center site. The migration from polar to chain center that we propose instead is a direct jump between the polar and the chain center site with an energy barrier around 0.80 eV. These discrepancies, which might arise from slightly different approximations (in particular the exchange-correlation functional) do not change the global picture of vacancy diffusion. Our study, in addition, clearly highlights the importance of the treatment of charged defects in boron carbide as illustrated with the stability of the chain center vacancy.

Helium impurities and vacancies

The introduction of vacancies in the model allows us to consider diffusion mechanisms other than the direct interstitial mechanism previously studied by Schneider et al. [START_REF] Schneider | [END_REF]. Indeed with the presence of intrinsic vacancies and interstitial impurities comes the possibility of substitutional impurity formation. Table 1 reports the formation energies and the charge transition levels of substitutional helium atoms and of vacancies.

Being chemically inert, the neutral substitutional helium adopts quite the same site preference as the neutral vacancy: the chain center helium is the most stable substitutional defect, the polar icosahedral helium is very expensive in terms of energy, and the equatorial helium is to be considered unstable. However, in the case of the substitutional impurity, the carbon chain sites are also unstable in all charge states.

Compared to the isolated vacancy, the charge transition levels of the chain center helium are shifted towards the middle of the band gap (Fig. 6a,6b) while the polar sites charge transition levels are only marginally affected by the presence of helium. This behavior is presumably the consequence of the geometric nature of the chain center vacancy: the helium impurity is more favorable in an open chain and its presence prevents the formation of a C-C bond.

To evaluate the effect of vacancies on the direct interstitial mechanism, the interactions between interstitial helium impurities and vacancies were studied. To remind the work of Schneider et al. [START_REF] Schneider | [END_REF], the two dominant types of helium interstitials sites are called "Type 2" and "Type 3" (or P2 and P3, see Fig. 7). The Type 2 position corresponds to 6 quasi-equivalent sites Our calculations show that the Type 2 interstitial configurations are unstable in presence of a positive and neutral B1 vacancy located nearby in the same {111} slice (the region containing a CBC chain between two {111} planes). In these charge states, the substitutional impurity is energetically more favorable and the migration barrier from Type 2 interstitial is less than 0.1 eV. However for the negative chain center vacancy, the substitutional helium is less favorable and the migration barrier reaches 0.98 eV (Fig. 8c). Again these results are directly related to the geometric behavior of the B1 vacancy, which in the negative charge state forms a C-C bond. Polar icosa-hedral vacancies are also strong attractors for interstitial helium atoms : adjacent Type 2 and 3 sites are no longer stable in presence of polar vacancies in any charge state. If we except the negative B1 substitutional atom, substitutional helium defects are favorable by more than 1.00 eV compared to the interstitial helium defects in presence of vacancies.

As stated previously, in presence of substitutional impurities we can consider other diffusion mechanisms, in particular vacancy mechanisms and Frank-Turnbull dissociative ones [46,[START_REF] Bocquet | Diffusion in Metals and Alloys[END_REF][START_REF] Bracht | [END_REF].

Let us consider first classical vacancy mechanisms [START_REF] Bocquet | Diffusion in Metals and Alloys[END_REF][START_REF] Bracht | [END_REF]; they involve the exchange between a He substitutional impurity and a nearby vacancy, their association/dissociation as a bound complex, and the migration of the two members close or far from each others. A helium/vacancy exchange between two chain centers or a chain center and a polar site provided migration paths with energy barriers included between 0.60 and 1.60 eV. However, according to some test calculations we performed, the migration of vacancies in proximity of a substitutional helium atom involves barriers that are even higher than for the chain-polar mechanism described earlier in this paper for the isolated vacancy, whose activation energy is 3.72 eV. Thus, in spite of the relatively low exchange barriers, the very low mobility of vacancies compared to the mobility of interstitial helium atoms precludes the straightfoward identification of convenient vacancy-driven mechanisms.

Let us now discuss the Frank-Turnbull dissociative mechanism [46,[START_REF] Bocquet | Diffusion in Metals and Alloys[END_REF][START_REF] Bracht | [END_REF]; for this the main defect reactions to be considered are the association of a He interstitial with a vacancy to form a substitutional impurity and its dissociation. Given the high formation energies of the polar helium atoms we focus here only on the chain center vacancy: as a nearby Type 2 interstitial site is found to be unstable, the migration describing this mechanism involves the nearest Type 3 interstitial (Fig. 8a and8c). For neutral and positive charge states, the migration energy from the interstitial to the substitutional configuration depends only on the Type 3 -Type 2 interstitial jump, whose energy barrier varies between 0.10 and 0.30 eV [START_REF] Schneider | [END_REF]. For the negative charge state the migration energy required to form the substitutional impurity corresponds to the migration barrier between the Type 2 interstitial and the B1 site, i.e., 0.98 eV. As suggested above, this barrier is related to the energy necessary to break the C-C bond formed on the B1 site.

The dissociation of the B1 substitutional helium into an interstitial-vacancy pair requires to overcome an energy barrier of 1.41, 1.05 and 0.76 eV respectively for charge +1, 0 and -1. From this dissociative mechanism on the chain center, two scenarios arise, depending of the charge state. The direct interstitial mechanism has a migration energy of ≈ 1.20 eV, which corresponds to the migration between two Type 2 interstitial helium atoms around the C-B-C chain (Fig. 9a). [START_REF] Schneider | [END_REF] Therefore in p-type conditions, when the vacancies are mostly neutral or positive, they clearly act as traps: in order to fully dissociate from the vacancy, a substitutional B1 helium has to cross the combined energy of the dissociation and Type 2 -Type 2 migration; an upper bound for the activation energy of this mechanism corresponds to the sum of the two steps, giving ≈ 2.3 eV for the neutral state and ≈ 2.6 eV for the positive state (Fig. 9b). On the other hand, in intrinsic or n-type conditions, the chain center vacancies are negative and even reduces locally the migration barrier. Indeed with the negative B1 vacancy, Type 2 interstitial positions are still stable and the Type 2 -Type 2 jump around the chain is still possible. This jump around the C-B-C chain has an energy barrier of ≈ 1.20 eV as reminded before, but in the presence of the negative vacancy on the chain center, this energy barrier drops to 0.89 eV (Fig. 8c). This means that the direct jump between two Type 2 interstitial positions is more likely than the formation of a B1 substitutional which has an energy barrier of 0.98 eV in negative conditions for the B1 vacancies. Thus the direct interstitial mechanism would essentially be unaffected by negative vacancies, except from subtle correlation effects due to the alternation of slightly different barriers.

There is a further possibility that we have not yet considered : it is the trapping of an interstitial He by a vacancy without the formation of a substitutional He. Our calculations show that this can happen when an interstitial helium travels in a {111} slice directly above or below a chain center vacancy (Fig. 8b). In particular, as shown in Fig. 8d, this trapping is stronger for a negative vacancy and only moderate for a neutral and positive one. This seems to be related both to a geometric and electrostatic effect, because in the negative charge state the defective chain is contracted forming a C-C bond, leaving thus more space for the Type 3 interstitial. But also for the neutral vacancy, when the He passes through the Type 3 position, the defective chain shrinks to a distance of 1.76 Å, suggesting a charge transfer from the chain end to the chain center, and an attractive electrostatic interaction between the electron depleted He region and the electron augmented chain center; this is coherent with the previously observed correlation between charge depletion and He insertion [START_REF] Schneider | [END_REF].

The trapping tendency is now reversed with respect to the dissociative mechanism, where the strongest trap was the positive vacancy. Combining the two mechanisms -the dissociative and the interstitial trappingbasically we can observe trapping from vacancies of any charge state, with capture energies between 2.1 and 2.6 eV, and a slightly stronger effect in p-type conditions.

Discussion

The results presented in the previous section help us setting up a picture of the role of vacancies on He diffusion in boron carbide. To fix ideas let us recall that interstitial He diffusion, without vacancies, is expected to occur in 2D along {111} slices with an activation energy of 1.2 eV and in 3D with a higher activation energy of 2.2 eV. [START_REF] Schneider | [END_REF] Most stable vacancies, i.e., chain center ones, can migrate, at best with an activation energy estimated at 3.7 eV through a chain-polar mechanism. In case of high vacancy supersaturation, i.e., when the fraction of icosahedral (polar) vacancies represents a non-negligible concentration, a direct intericosahedral mechanism with 2 eV activation energy has been found.

We consider two possible vacancy driven helium diffusion mechanisms [START_REF] Bracht | [END_REF]: the first, a usual vacancy mechanism, where He atoms, always in substitutional positions, exchange with vacancies; the second, a dissociative mechanism, where He is alternatively in substitutional and interstitial positions.

In the first case, although the exchange itself has low activation energy (0.6 eV at best), the migration energy of the isolated vacancy is very high (3.7 eV) ; the migration of vacancies around He are not lower than that, and clearly high enough to exclude this mechanism to be the dominant one. However, when the concentration of vacancy is very high the mechanism involving only polar vacancy could be activated, with a migration energy of 2 eV; this is still higher than 2D interstitial diffusion, but slightly lower than 3D interstitial diffusion [START_REF] Schneider | [END_REF].

For the second case, the dissociative mechanism, we have to distinguish two further scenarios, depending on the Fermi level position. If the material is p-type, B1 vacancies are expected to be neutral or positive; in this case, the vacancy will trap helium in substitutional positions and the energy landscape will look like the one in Fig. 9b, where the activation energy, apart from subtle correlation effects between the He-V dissociation and He interstitial migration, is 2.3-2.6 eV. If, conversely, the material is n-type, vacancies are expected to be in their negative state; in this case the dissociative mechanism is not any more relevant, because He interstitials would easily turn around vacancies in {111} slices with an even slightly lower barrier (0.89 eV, see Fig. 9b and dashed line in Fig. 8c). However, vacancies in nearby {111} slices would trap He in interstitial form (Fig. 9c) and eventually produce a similar effect as the dissociative mechanism in p-type conditions, with an overall activation energy that, according to the concentration of vacancies, can reach 2.1 eV.

The results reported in this paper add a piece of information in order to interpret experiments where the mobility of He in irradiated materials is an important parameter. In a recent experimental work of one of us [49] B 4 C samples were implanted with He atoms, then annealed and He profiles were extracted through Nuclear Reaction Analysis (NRA). An activation energy of E a = 2.03 ± 0.18 eV was obtained following a classic Arrhenius analysis for the intra-granular diffusion of He atoms in boron carbide.

Considering only interstitial mechanisms for helium diffusion, the threedimensional diffusion path with an activation energy of 2.2 eV is in better agreement with the experimental value than the two-dimensional diffusion path with its activation energy of 1.2 eV. However, the measured profiles are supposed to integrate the 2D and 3D diffusion and, as such, to measure the lowest (2D) activation energy. In view of our results on the influence of vacancies we are keen to interpret the measured activation energy as a consequence of the presence of vacancies in the material, whose effect is to slow down the 2D diffusion to an activation energy of ≈2 eV. Vacancies are certainly present following He implantation in the measured sample. The profile of the damage peak almost coincides with the one of as-implanted He. This means that upon annealing, He can leave the damaged region because its mobility is in any case larger than that of vacancies. However, the vacancy concentration in the queues of the damage peak might be sufficient to influence He diffusion. Moreover, it has been suggested that antisites and vacancies and various forms of defective chains are difficult to avoid in boron carbide [50,51], which strengthens the hypothesis that vacancies may influence He diffusion even outside of the damage peak.

We should nevertheless mention two further hypotheses excluding the role of vacancies : the first is that self-interstitial defects might diffuse faster than helium outside of the damaged region and provide other mechanisms to reduce He mobility. The second is that small (immobile) He bubbles (not visible with TEM) are formed during irradiation, and that the detrapping energy of He atoms from those bubbles might also affect the measured activation energy.

Conclusion

In summary, we presented here a first principle study of the stability and kinetics of vacancies and substitutional He impurities in various charge states in carbon-rich boron carbide (B 4 C). We showed that the most stable vacancy, the chain center boron one, is negatively charged in intrinsic and n-type conditions, and neutral or positive in p-type conditions. The charge state of the chain center vacancy clearly affects its topology: the positive and neutral vacancies correspond to wide open chains whereas the negative vacancy is better described by a shrinked chain with a C-C bond. We also show that, independently of the charge state, the mobility of vacancies is very low (E mig >3 eV) at equilibrium, with a possible enhancement (E mig 2 eV) in case of supersaturation of polar icosahedral boron vacancies.

It is found that substitutional He impurities are more favorable than interstitial helium in presence of vacancies in p-type conditions, when chain center vacancies are neutral or positively charged. The influence of vacancies on helium mobility is to slow down the 2D He diffusion along {111} planes in two possible ways : in intrinsic and n-type conditions, when vacancies are negatively charged, 2D diffusion still occurs via an interstitial mechanism, with trapping of interstitial helium by vacancies located in a nearby {111} slice. In p-type conditions, diffusion occurs via a dissociative mechanism, where He is occasionally trapped in vacancies as a substitutional impurity. In both cases the activation energy for detrapping is slightly higher 2 eV, which is compatible with very recents experimental results [49].

Acknowledgements

This work was granted access to the HPC resources of TGCC and CINES under the allocation 2017A0010906018 made by GENCI and under the allocation by CEA-DEN. We acknowledge partial financial support by the french research program NEEDS Matériaux.

Data Statement

The raw/processed data required to reproduce these findings cannot be shared at this time for legal reasons and because the data also forms part of 21 an ongoing study.

Figure 1 :

 1 Figure 1: The B 11 C p CBC structure, atoms are divided in 4 groups : polar (C3, B2, B3, B4), equatorial (B5, B6, B7, B8), chain end (C1, C2) and chain center (B1).

Figure 2 :

 2 Figure 2: Convergence of the B1 vacancy formation energy versus supercell volume (number of atoms on the top x-axis) for neutral and negative charge states with and without the monopole correction. The two points for the 404-atom supercell stands for Γ and 2×2×2 BZ sampling.

Figure 3 :

 3 Figure 3: Formation energy of the chain center vacancy as a function of the Fermi level. Two different representations are displayed: a) GGA and HSE valence band maxima (VBM) are aligned at µ e = 0, b) HSE and GGA charge transition levels are referred to a common absolute level[43] and only HSE VBM is aligned at µ e = 0.

Figure 4 :

 4 Figure 4: A comparison of the charge density along the line connecting the two carbon atoms C 1 and C 2 adjacent to the chain center boron vacancy in neutral and negative charge state. The black points above the curve indicate the positions of the carbon atoms in both configurations. The vertical dashed lines show the separation between icosahedra and chain regions. For the Q=+1 charge state (not shown) the charge density is almost identical to the neutral case.

Figure 5 :

 5 Figure 5: Migration energy profiles for the chain-to-polar conversion mechanism for vacancies in positive, neutral and negative charge state.

Figure 6 :

 6 Figure 6: Formation energy of the chain center helium impurity as a function of the Fermi level. Two different representations are displayed: a) GGA and HSE VBM are aligned at µ e = 0, b) HSE and GGA charge transition levels are referred to a common absolute level[43] and only HSE VBM is aligned at µ e = 0.

Figure 7 :

 7 Figure 7: Helium interstitial positions in the rhombohedral elementary cell. Polar atoms are not represented for convenience. Type 2 and Type 3 helium sites are respectively coloured in green and blue.

Figure 8 :

 8 Figure 8: The two trapping mechanisms (some icosahedral atoms are not shown for readability): a) trapping with formation of chain-center helium substitutional; b) trapping on the Type 3 interstitial helium site. Migration barriers associated with these mechanisms are : c) Formation of chain center helium from Type 3 interstitial in three charge state (dashed line corresponds to the direct interstitial jump between two Type 2 positions); d) Type 2 -Type 3 migration with trapping from a vacancy located in an adjacent {111} slice (stronger in negative charge state).

Figure 9 :

 9 Figure 9: Migration energy profile for a neutral interstitial helium atom along a {111} slice in boron carbide far from vacancies (a), encountering a chain center vacancy in the slice (b) or approaching a chain center vacancy in a neighboring {111} slice (c). Three charge states for the vacancy are shown.

1

  He

	chain center	B1	1.55	-	0.36 3.59 0.41 1.51
	chain end	C1 C2	2.78 2.92	--	0.82 0.80	× ×	× ×	× ×
		C3	4.59 0.20 1.47 5.85 0.24 1.50
	polar	B2 B3	4.42 0.58 1.11 5.77 0.59 1.33 5.82 -1.00 7.38 0.15 0.76
		B4	5.36 0.52 0.96 6.71 0.57 1.16
		B5	5.19 0.93 1.07	×	×	×
	equatorial	B6 B7	× ×	× ×	× ×	× ×	× ×	× ×
		B8	4.49 0.28 0.90	×	×	×

Table 1 :

 1 Formation energies and charge transition levels for vacancies and substitutional helium in boron carbide with the GGA functional; the × symbols indicate unstable configurations and the -symbols indicate charge transition levels lying outsite the band gap. Energy units are eV.


J-F Eschbach

G Martin

Grassi

Axel Rizzo

Claire Vaglio-Gaudard

Gilles Noguere

Romain Eschbach

Julie-Fiona Martin

Gabriele Grassi

Assessment of the 153 Eu and 154 Eu neutron capture cross sections from the Integral Data Assimilation of used nuclear fuel experiments

Keywords: Nuclear data, DARWIN2.3, CONRAD, Integral data assimilation, 153 Eu, 154 Eu, capture cross section

Assessment of the

153 Eu and 154 Eu neutron capture cross sections from the Integral Data Assimilation of spent nuclear fuel experiments

INTRODUCTION

The europium isotopes are fission products of interest for nuclear applications. As shown in Fig. 1, 153 Eu and 154 Eu have a high capture cross-section in the thermal and epi-thermal energy ranges. The main international evaluated nuclear data libraries (JEFF, ENDF/B, JENDL) recommend nearly similar capture cross sections for 153 Eu and 154 Eu. As indicated in Table 1, the discrepancies between the capture resonance integrals lie below 4%. However, new time-of-flight experiments [START_REF] Leinweber | Europium resonance parameters from neutron capture and transmission measurements in the energy range 0.01-200 eV[END_REF] carried out at the LINAC facility of the Rensselaer Polytechnic Institute (RPI) suggest increasing the 153 Eu capture resonance integral of JEFF-3.1.1 [START_REF] Santamarina | The JEFF-3.1.1 Nuclear Data library[END_REF] by at least 11%. For 154 Eu, analyses of Post-Irradiation Examinations (PIEs) in light and heavy water reactors reported in Ref. [START_REF] Skutnik | Proposed re-evaluation of the 154 Eu thermal (n,γ) capture cross-section based on spent fuel nuclear benchmarking studies[END_REF] also suggest increasing the 154 Eu capture cross section recommended in the ENDF/B-VII.1 library.

The aim of the present work is to confirm the integral trends reported on the 153 Eu(n,) and 154 Eu(n,) reactions by using a large set of used nuclear fuel experiments performed in typical Pressurized Water Reactors (PWRs) energy spectra. The selected integral trends are those calculated with the DARWIN2.3 package [START_REF] San-Felice | Experimental validation of the DARWIN2.3 package for fuel cycle applications[END_REF]. DARWIN2.3 is the French reference calculation package for fuel cycle studies that computes physical quantities of interest for the fuel cycle, at any irradiation and cooling time. The originality of our study is to combine the Integral Data Assimilation procedure and the marginalization technique, implemented in the nuclear data code CONRAD [START_REF] De Saint | Status of CONRAD, a nuclear reaction analysis tool[END_REF][START_REF] Archier | CONRAD evaluation code: Development Status and Perspectives[END_REF], for determining integral trends on the 153 Eu and 154 Eu capture cross sections by taking into account most sources of uncertainties: the uncertainties considered for the study are the ones associated with nuclear data, the ones associated with the interpretation of the PIEs with DARWIN2.3, and uncertainties that account for methodological approximations introduced by DARWIN2.3 deterministic solvers. The DARWIN2.3 package will be presented in section 2, as well as the experimental data employed for the assimilation and the method used to compute an experimental correlation matrix. The fitting and marginalization techniques used in this work will be presented in section 3. The results will be presented and discussed in section 4.

FUEL INVENTORY CALCULATION WITH DARWIN2.3

This section gives a short presentation of both the DARWIN2.3 package and the used nuclear fuel experiments employed in the present study. [START_REF] San-Felice | Experimental validation of the DARWIN2.3 package for fuel cycle applications[END_REF] is the French reference calculation tool used for fuel cycle studies. It solves the Boltzmann and Bateman equations to compute fuel cycle parameters, at any irradiation and cooling time. The quantities of interest for the fuel cycle are fuel inventory, masses, residual decay heat, neutron emissions, α, ß, γ sources and spectra, and radiotoxicity. A flow chart of the DARWIN2.3 package for PWR calculations is presented in Fig. 2. For PWRs, DARWIN2.3 includes the APOLLO2 deterministic transport code [7], which provides neutron data to the PEPIN2 depletion solver [START_REF] Tsilanizara | DARWIN: an Evolution Code System for a Large Range of Applications[END_REF]. These data are self-shielded cross-sections libraries and neutron fluxes.

Presentation of the DARWIN2.3 package

DARWIN2.3
The nuclear data used in the DARWIN calculations come from the JEFF-3.1.1 library. For the multigroup nuclear data, a spatial self-shielding calculation is carried out with the SHEM group structure (281 groups) [START_REF] Hfaiedh | Determination of the Optimized SHEM mesh for neutron transport calculation[END_REF]. The neutron flux is computed with a Probability Collision method-based calculation. Self-shielded cross-sections and neutron flux are re-calculated as a function of the burnup in order to take into account the impact of the fission products and actinides production.

Experimental database

The material balance and decay heat calculations with the DARWIN2.3 package have been experimentally validated for PWR [START_REF] San-Felice | Experimental validation of the DARWIN2.3 package for fuel cycle applications[END_REF]. For the material balance, this experimental validation relies on the analysis of dedicated PIEs, mostly consisting in the chemical characterization of in-pile irradiated fuel samples. In this work, the experimental results come from portions of fuel rods extracted from four power reactors, namely Gravelines, Bugey, Dampierre (operated by EdF) and Gösgen (operated by Kernkraftwerk Gösgen-Däniken). The calculated-to-experimental ratios of interest for the 153 Eu and 154 Eu build-up are reported in Table 2. The associated uncertainties account for the experimental uncertainties of the isotopic ratio measurements as well as systematic uncertainties related to the fuel temperature (±50°C), to the moderator temperature (±2°C) and to the burnup scaling procedure (±2%) [START_REF] San-Felice | Experimental validation of the DARWIN2.3 package for fuel cycle applications[END_REF].

Numerical biases

DARWIN2.3 is a package relying on deterministic calculation schemes and solvers.

Methodological approximations, such as self-shielding calculation, spatial discretization and scattering anisotropy, may introduce numerical biases on the computed physical quantities. In this work, specific studies were carried out to provide an order of magnitude on the numerical biases by comparison to a new Monte-Carlo procedure for calculating the evolution of used fuel as a function of burnup as well as numerical biases induced by the new treatment of the up-scattering resonance effect. Results obtained from these two studies are discussed hereafter.

Comparative pin-cell depletion calculations were carried out between the deterministic code APOLLO2 and the Monte-Carlo code TRIPOLI-4 [START_REF] Brun | TRIPOLI-4®, CEA, EDF, and AREVA reference Monte Carlo code[END_REF]. They use the same filiation chain optimized for reactivity calculations. Recent developments enables the coupling between TRIPOLI-4 and the MENDEL [START_REF] Lahaye | First Verification and Validation steps of MENDEL release 1.0 cycle code system[END_REF] deterministic depletion solver for performing depletion calculations [START_REF] Damian | ORPHEE research reactor: 3D core depletion calculation using Monte-Carlo code TRIPOLI-4®[END_REF][START_REF] Blancher | On corereflector interface effects in the ASTRID sodium cooled fast reactor[END_REF][START_REF] Archier | Validation of a multi-purpose depletion chain for burnup calculations through TRIPOLI-4 calculations and IFP perturbation method[END_REF][START_REF] Chambon | Validation of HORUS3D/N against TRIPOLI-4®D for code depletion calculation of the Jules Horowitz reactor[END_REF]. The independent replicas method is used, meaning that random seeds are used in order to perform the same depletion calculation times. The concentration value and its associated uncertainty are derived from the two first moments of the distribution of the concentration values. The comparative study between APOLLO2 and TRIPOLI-4 was carried out with various fuels, each of them being representative (i.e. having the same initial composition) of the PIEs presented in Table 2. Table 3 shows the differences between APOLLO2 and TRIPOLI-4 pin-cell calculations for 153 Eu and 154 Eu concentration as a function of burnup. One can observe that the resulting differences lie below 1% and remain nearly constant with the burnup. The quoted uncertainties are low. They were obtained with 64 independent replicas. Such low uncertainties should be handled with care: it has been shown that in some specific cases, the independent replicas method could propagate numerical biases along the successive transport / depletion steps [START_REF] Dumonteil | Biases and statistical errors in Monte Carlo burnup calculations: an unbiased stochastic scheme to solve Boltzmann/Bateman coupled equations[END_REF]. These possible numerical biases are not taken into account by the statistical uncertainty reported in Table 3.

An additional study was performed to investigate the resonant up-scattering effect [START_REF] Ouisloumen | A model for neutron scattering off heavy isotopes that accounts for thermal agitation effects[END_REF] on fuel inventory calculations. This physical effect is accounted for in APOLLO2 through a dedicated nuclear data library. Table 4 reports the differences between two APOLLO2 pin-cell calculations performed with and without the resonant up-scattering effect. It can be observed that the differences increase with the burnup. For the 154 Eu build up, the differences between the two APOLLO2 calculations become close to 1% above 60 GWd/t.

These two studies made it possible to quantify numerical biases related respectively to the use of deterministic solvers and to the up-scattering effect. In the following section, these numerical biases are considered as additional sources of systematic uncertainties.

Correlation matrix between calculated-to-experimental ratios

The calculated-to-experimental ratios reported in Table 2 share the same sources of systematic uncertainties (section 2.2) and numerical biases (section 2.3). We have used the AGS method [START_REF] Bastian | AGS, a computer code for uncertainty propagation in time-of-flight cross-section data[END_REF] to compute the correlation matrix M E between the C/E-1 values. The originality of the AGS method is to combine statistical and systematic uncertainties as follows:

, ( 3
)
in which is a diagonal matrix containing variances of the uncorrelated uncertainties, and is a rectangular matrix containing the systematic uncertainties for each of the sources of uncertainty. No correlation between the experimental isotopic ratios was provided alongside the results of the chemical analyses. Therefore, the -matrix only contains the experimental uncertainties coming from the Inductively Coupled Plasma Mass Spectrometry (ICP-MS) technique:

, ( 4
)
in which represents the uncertainty associated with the experiment. For the S-matrix, we have considered four sources of systematic uncertainties: ,

.

Parameters p 1 and p 2 are related to the (1) fuel and (2) moderator temperatures. Their respective contributions  1,i and  2,i were deduced from direct perturbations in the DARWIN calculations.

Parameters p 3 and p 4 are related to both numerical biases investigated in section 2.3. Their corresponding contributions  3,i and  4,i are deduced from values reported in Tables 3 and4.

Fig. 4 shows the final correlation matrix between the calculated-to-experimental ratios obtained with Eq. ( 3). As expected, non-negligible correlations exist between the various concentration ratios confirming the need of using M E in the Integral Data Assimilation procedure.

INTEGRAL DATA ASSIMILATION PROCEDURE

The Integral Data Assimilation procedure, as implemented in the CONRAD code [START_REF] De Saint | Status of CONRAD, a nuclear reaction analysis tool[END_REF][START_REF] Archier | CONRAD evaluation code: Development Status and Perspectives[END_REF], consists in adjusting neutron cross-sections on a given set of calculated-to-experimental ratios and in propagating statistical and systematic uncertainties via a two-step calculation sequence. The first step is a standard least-square fitting procedure, and the second step is a marginalization procedure that aims at accounting for nuisance parameters uncertainties.

Model parameters

In this work, free parameters are the 153 Eu and 154 Eu capture cross sections. They will be adjusted with the CONRAD code in order to improve the calculated-to-experimental ratios reported in Table 2. To avoid the resolution of underestimated systems, capture cross sections were averaged over the neutron flux . Throughout the paper, 1-group cross-sections are defined as follows:

(8)

and the 2-group representation is given by: (9) [START_REF] Brun | TRIPOLI-4®, CEA, EDF, and AREVA reference Monte Carlo code[END_REF] where E c =0.5 eV represents the cadmium cut-off energy which is used to calculate the capture resonance integral:

As a consequence, the 2-group description of the neutron cross sections will provide useful information on the capture resonance integral. The method is discussed in section 4.2.

The final uncertainties on the 1-group or 2-group cross sections depend on the complex formation chain of the europium isotopes. The latter is illustrated in Fig. 3. The CYRUS tool [START_REF] Vallet | Deterministic approach of the decay heat uncertainty due to JEFF-3.1.1 nuclear data uncertainties with the CYRUS tool and the DARWIN2.3 depletion code[END_REF] was used to provide an exhaustive list of nuclear data involved in the 153 Eu and 154 Eu build-up. The 1-group cross-sections, fission yields, and decay constants of interest for this work are listed in Table 6. Parameter represents the thermal cumulative fission yield of nuclide X for actinide and ) is the decay period of . All these parameters are nuisance parameters whose uncertainties are taken either from the covariance database COMAC-V2.0 [START_REF] Archier | COMAC -Nuclear data covariance matrices library for reactor applications[END_REF][START_REF] Terranova | Covariance Evaluation for Nuclear Data of Interest to the Reactivity Loss Estimation of the Jules Horowitz Reactor[END_REF], developed at the CEA Cadarache, or from the evaluated nuclear data library JEFF-3.1.1.

These uncertainties are marginalized after the fitting procedure.

Fitting procedure

The fitting algorithm implemented in the CONRAD code [START_REF] Privas | Generation of 238 U covariance matrices by using the integral data assimilation technique of the CONRAD code[END_REF] relies on the Bayes' theorem: [START_REF] Damian | ORPHEE research reactor: 3D core depletion calculation using Monte-Carlo code TRIPOLI-4®[END_REF] where contains the free model parameters, represents the experimental values, denotes the prior information, is the prior probability density of , is the likelihood probability density, corresponding to the calculated-to-experimental ratios, and represents the posterior probability density of . Vector can contain the 1-group cross sections defined by Eq. ( 8):

.

Alternatively, we can use the 2-group representation to get integral trends on the 153 Eu capture cross section above the cadmium cut-off energy (Eq. ( 10)):

. ( 14
)
Few assumptions are needed in order to find the two first moments of . According to the principle of maximum entropy, a multivariate joint normal distribution can be chosen for the prior probability density. If the Laplace approximation is made, the posterior density probability is also assumed to be a normal distribution. Then, the posterior parameters and covariance matrix correspond to the minimum of the following cost function:

, ( 15
)
in which contains the calculated values and is the experimental correlation matrix (section 2.4). A Gauss-Newton iterative scheme [START_REF] Privas | Generation of 238 U covariance matrices by using the integral data assimilation technique of the CONRAD code[END_REF] is normally used to solve Eq. ( 15), ensuring a good convergence of the minimizing process in spite of the non-linearity of this equation when adjusting resonance parameters of a cross section for instance. In this study, the problem may be considered linear given the number of fitted parameters and the number of experiments. Hence, a single iteration is used in this work, and it gives a global trend on the capture cross section of the europium isotopes that can prove useful for future experimental and evaluation works.

Marginalization procedure

The marginalization technique [START_REF] Habert | Retroactive generation of covariance matrix of nuclear model parameters using marginalization techniques[END_REF] was implemented in CONRAD, to account for the uncertainties on nuisance parameters (non-fitted model parameters ) listed in Table 6. It consists in building a "full" covariance matrix between fitted and nuisance parameters from their and respective covariance matrices [START_REF] Noguere | Zero variance penalty model for the generation of covariance matrices in integral data assimilation problems[END_REF]: ,

The -matrix represents the posterior covariance matrix provided by the fitting algorithm. The and partial derivative matrices contain the derivatives of the calculated values with respect to the fitted and non-fitted parameters:

(19) (20)
To avoid Peelle's pertinent puzzle [START_REF] Smith | Probability, statistics, and data uncertainties in nuclear science and technology[END_REF], namely the occurrence of abnormal values of quantities fitted on experimental data with both statistical and large systematic uncertainties, it was also chosen to marginalize the uncertainty associated to the burnup scaling procedure (±2%). This strategy was applied in Ref. [START_REF] Saint | Evaluation of neutron-induced cross sections and their related covariances with physical constraints[END_REF] to treat the first resonance of 239 Pu.

RESULTS

The Integral Data Assimilation procedure presented in section 3 was used to extract integral trends on the 1-group capture cross sections of 153 Eu and 154 Eu (Eq. ( 13)), and also on the capture resonance integral of 153 Eu (Eq. ( 14)).

Integral trends on 1-group capture cross sections

Prior uncertainties on the 1-group capture cross sections of 153 Eu and 154 Eu were calculated from the COMAC-V2.0 covariance database. These prior relative uncertainties are then introduced in the fitting procedure, with values of ±5% for and ±11% for . Two series of calculations were performed: one with the fission yield uncertainties coming from COMAC-V2.0 and the other with those from the JEFF-3.1.1 library.

Table 6 shows the final integral trends deduced from the calculated-to-experimental ratios of Table 2. The marginalization procedure provides more realistic uncertainties than the fitting procedure only. One can also observe a slight reduction of the uncertainty of about 1.5% when using the COMAC-V2.0 uncertainties for thermal cumulative fission yields. This is mainly explained by the strong reduction of the uncertainty for the , i.e. 241 Pu fission yield on 151 Sm. The results suggest an increase of both capture cross-sections, as compared to the JEFF-3.1.1 evaluation. The proposed modifications of +4.9% and +7.1% are within the range of the prior uncertainties. Fig. 5 and Fig. 6 show that the posterior C/E-1 values are closer to zero and the associated uncertainties are reduced, especially for the 154 Eu/ 238 U isotopic ratios.

This trend is explained by the strong correlation coefficient of 0.98 between the two fitted parameters.

Table 7 reports some integral trends dedicated to the experimental validation of the 153 Eu evaluation available in the JEFF-3.1.1 library. They come from the analysis of oscillation and activation measurements carried out in critical mock-up with PWR-like neutron spectra [START_REF] Saint | Evaluation of neutron-induced cross sections and their related covariances with physical constraints[END_REF][START_REF] Dean | Validation of important fission product evaluations through CERES integral benchmarks[END_REF][START_REF] Gruel | Interpretation of Fission Products Oscillations in the MINERVE reactor, from Thermal to Epithermal Spectra[END_REF][START_REF] Bernard | Validation of JEFF-3.1.1 Thermal and Epithermal neutron-induced capture cross-sections through MELUSINE experiment analysis[END_REF].

The negative sign indicate that the 153 Eu capture cross section in JEFF-3.1.1 is underestimated.

Our results confirm these trends.

Regarding 154 Eu capture cross-section, a recent work based on ENDF/B-VII.1 also indicates an overestimation of the 154 Eu concentration in Light and Heavy Water Reactors [START_REF] Skutnik | Proposed re-evaluation of the 154 Eu thermal (n,γ) capture cross-section based on spent fuel nuclear benchmarking studies[END_REF]. As indicated in Table 1, the evaluations in JEFF-3.1.1 and ENDF/B-VII.1 are nearly similar. Therefore, conclusions from Ref. [START_REF] Skutnik | Proposed re-evaluation of the 154 Eu thermal (n,γ) capture cross-section based on spent fuel nuclear benchmarking studies[END_REF] are consistent with the calculated-to-experimental ratios of Table 2, and then confirm the sizeable underestimation of the capture cross-section in both libraries.

Integral trend on the capture resonance integral

In a second step, we tried to discriminate the contribution of the thermal energy range and of the resonance range of the 153 Eu capture cross-section using the 2-group representation defined by Eqs ( 9) and ( 10). Below 0.5 eV, the 1/v behaviour of the capture cross-section is mainly conditioned by the value at the thermal energy. Therefore, strongly depends on the thermal value  0 . Similarly, a direct perturbation of could be seen as a perturbation of the capture resonance integral I 0 . As shown in Fig. 1, such an approach cannot be applied to 154 Eu, because the first resonance is too close to the thermal cross section.

resonance integral of JEFF-3.1.1 from I 0 =1409 barns to I 0 1502 ± 96 barns (using the fission yields uncertainties of JEFF-3.1.1), or to I 0 1502 ± 68 barns (using the fission yields uncertainties of COMAC-V2.0). This tendency is in good agreement with the I 0 = 1560 barns value deduced from the time-of-flight data measured at the Rensselaer Polytechnic Institute [START_REF] Leinweber | Europium resonance parameters from neutron capture and transmission measurements in the energy range 0.01-200 eV[END_REF].

Such an agreement confirms our choice to marginalize the uncertainties of and to apply the calculated-to-experimental biases on the 153 Eu capture resonance integral of JEFF-3.1.1.

CONCLUSION

The results obtained in the present work confirm the relevancy of using the fitting algorithm of the CONRAD code in association with the marginalization of the nuisance parameter uncertainties for determining reliable integral trends from PIE analyses. Previous studies with the DARWIN2. Experimental correlation matrix generated with the AGS method between the 14 calculated-to-experimental ratios reported in Table 2.

Figure 1 .

 1 Figure 1. 153 Eu and 154 Eu radiative capture cross-sections recommended in the evaluated nuclear data library JEFF-3.1.1 compared to a PWR neutron spectrum (in arbitrary unit).

Figure 3 .

 3 Figure 3. Build up of the 153 Eu and154 Eu isotopes in a PWR-like spectrum for a 4.5 wt.% 235 U UOX fuel and a 6.7wt.% Pu amount MOX fuelboth of them irradiated at 40 GWd/tgiven by the CYRUS tool[START_REF] Vallet | Deterministic approach of the decay heat uncertainty due to JEFF-3.1.1 nuclear data uncertainties with the CYRUS tool and the DARWIN2.3 depletion code[END_REF].

Figure 4 .

 4 Figure 4. Experimental correlation matrix generated with the AGS method between the 14 calculated-to-experimental ratios reported in Table2.

Figure 5 .

 5 Figure 5. Calculated-to-experimental values related to the 153 Eu/ 238 U isotopic ratios before adjustment (top plots) and after adjustment (bottom plots) of the 153 Eu and 154 Eu 1-group capture cross sections.

Figure 6 .

 6 Figure 6. Calculated-to-experimental values related to the 154 Eu/ 238 U isotopic ratios before adjustment (top plots) and after adjustment (bottom plots) of the 153 Eu and 154 Eu 1-group capture cross sections.

  3 package of several153 Eu/ 238 U and 154 Eu/ 238 U isotopic ratios as a function of the burnup indicated a possible underestimation of the 153 Eu and 154 Eu capture cross sections This result is consistent with the capture resonance integral I 0 =1560 barns deduced from measurements carried out at the RPI facility. As a consequence, the resonance parameters extracted from the RPI data can be used in the future release of the JEFF library.

	barns to 1502±68 barns.
	recommended in the JEFF-3.1.1 library. The use of the CONRAD algorithms allowed to suggest
	an increase of (4.9 ± 3.2)% and (7.1 ± 6.0)% of the 153 Eu(n,) and 154 Eu(n,) reaction cross
	sections, respectively. For 153 Eu, integral trends of similar amplitude, ranging from 6% to 11.6%,
	were reported in the literature.
	Characteristics of the 153 Eu capture cross section allow distinguishing the contribution of the
	thermal and epithermal energy ranges. Since the 153 Eu thermal cross-section recommended in
	JEFF-3.1.1 ( 0 =312 barns) is in good agreement with the experimental values reported in the
	literature, we assumed that the calculated-to-experimental biases are mainly due to the
	underestimation of the 153 Eu capture resonance integral in JEFF-3.1.1. The CONRAD
	calculations suggested to increase by +(6.6 ± 4.8)% the capture resonance integral from 1409

Table 1 .

 1 Thermal capture cross section σ 0 and capture resonance integral I 0 of the 153 Eu and 154 Eu isotopes given in the main international evaluated nuclear data libraries.

	Evaluated nuclear	153 Eu		154 Eu
	data library	σ 0 (barns)	I 0 (barns)	σ 0 (barns)	I 0 (barns)
	JEFF-3.1.1	313	1409	1356	1326
	ENDF/B-VII.0	312	1409	1353	1299
	ENDF/B-VII.1	358	1420	1353	1299
	JENDL-4.0	313	1412	1353	1354

Table 2 .

 2 Calculated-to-experimental ratios related to the 153 Eu/ 238 U and 154 Eu/ 238 U isotopic ratios provided by DARWIN2.3[START_REF] San-Felice | Experimental validation of the DARWIN2.3 package for fuel cycle applications[END_REF], i.e. before the adjustment performed in this work. The origins of the relative uncertainties σ are explained in section 2.2.

	Reactor name and fuel	Burnup	153 Eu / 238 U		154 Eu / 238 U
	characteristics	(GWd/t)	C/E-1 (%)	σ (%)	C/E-1 (%)	σ (%)
	Gravelines -4.5%	40	2.0	2.6	1.4	3.5
	235 U-enriched UOX fuel	50	2.3	2.3	3.9	2.8
	(17×17 rods)	60	4.1	1.9	7.5	2.3
	Bugey -3.1% 235 U-enriched UOX fuel (17×17 rods)	20 25	2.2 2.9	2.9 3.3	0.6 6.3	4.0 3.8
	Gösgen -4.3%
	235 U-enriched UOX fuel	70	4.8	1.6	9.0	1.8
	(15×15 rods)
	Dampierre -6.7% Pu
	amount in the central zone	58	2.3	1.2	4.5	2.6
	(17×17 rods)

Table 3 .

 3 Differences obtained on the 153 Eu/ 238 U and 154 Eu/ 238 U isotopic ratios between APOLLO2 (AP2) and TRIPOLI-4 (T4) pin-cell calculations as a function of burnup. The relative uncertainties σ T4 represent the statistical uncertainties of the Monte-Carlo simulations.

			153 Eu / 238 U		154 Eu / 238 U
	Fuel characteristics used	Burnup
	in the pin-cell calculations	(GWd/t)	C(AP2)/C(T4)-1	σ T4	C(AP2)/C(T4)-1	σ T4
			(%)	(%)	(%)	(%)
	4.5 wt.% 235 U UOX fuel (Gravelines)	40 50 60	0.09 0.10 0.11	0.02 0.02 0.01	0.45 0.47 0.42	0.02 0.02 0.02
	3.1% wt.% 235 U UOX fuel	20	-0.04	0.02	0.29	0.02
	(Bugey)	25	0.01	0.02	0.37	0.02
	4.3% wt.% 235 U UOX fuel (Gösgen)	70	0.11	0.01	0.42	0.02
	6.7 wt. % Pu amount MOX (Dampierre)	58	0.06	0.02	-0.21	0.02

Table 4 .

 4 Differences obtained on the 153 Eu/ 238 U and 154 Eu/ 238 U isotopic ratios between two APOLLO2 pin-cell calculations with (AP2-UPS) and without (AP2) up-scattering effects.

	Fuel characteristics used	Burnup	153 Eu / 238 U	154 Eu / 238 U
	in the pin-cell calculations	(GWd/t)	C(AP2-UPS)/C(AP2)-1	C(AP2-UPS)/C(AP2)-1
			(%)	(%)
	4.5 wt.% 235 U UOX fuel (Gravelines)	40 50 60	0.19 0.16 0.14	0.64 0.72 0.81
	3.1% wt.% 235 U UOX fuel	20	0.24	0.54
	(Bugey)	25	0.22	0.58
	4.3% wt.% 235 U UOX fuel (Gösgen)	70	0.14	0.81
	6.7 wt. % Pu amount MOX (Dampierre)	58	-0.07	0.31

Table 6 .

 6 Trends on the 153 Eu and 154 Eu 1-group capture cross-sections obtained from the Integral Data Assimilation of the C/E-1 values reported in Table2. The positive signs indicate an increase of the capture cross section as compared to the JEFF-3.1.1 evaluations.

	Type of fitting procedure	153 Eu(n,γ)	154 Eu(n,γ)	Correlation
	a) Fit only	+ (4.9 ± 0.3 )% + ( 7.1 ± 0.6 )%	0.694
	b) Fit + Marginalization (JEFF-3.1.1 Fission Yields)	+ (4.9 ± 4.7 )% + ( 7.1 ± 7.5 )%	0.978
	c) Fit + Marginalization (COMAC-V2.0 Fission Yields)	+ ( 4.9 ± 3.2 )% + (7.1 ± 6.0 )%	0.984

ACKNOWLEDGMENTS

We would like to thank all the colleagues from CEA who helped us during this work, especially David Bernard, Coralie Carmouze, Pierre Leconte, Pascal Archier, and Vanessa Vallet.

A few experimental values for the 153 Eu thermal capture cross section [START_REF] Leconte | Thermal neutron activation experiments on Ag[END_REF][START_REF]Pile oscillator measurements of resonance absorption integrals[END_REF][START_REF] Mowon | The neutron capture cross-section of 151 Eu and 153 Eu in the energy range 0.1 to 100 keV[END_REF][START_REF] Arboccò | Experimental determination of k o , Q o factors, effective resonance energies and neutron cross-sections for 37 isotopes of interest in NAA[END_REF][START_REF] Basunia | Determination of the 151 Eu(n,γ) 152m1,g Eu and 153 Eu(n,γ) 154 Eu reaction cross-sections at thermal neutron energy[END_REF] are reported in Table 8. They are all in reasonable agreement within the uncertainties range, confirming the JEFF-3.1.1 value of  0 =312 barns. Therefore, we decided to adjust and to marginalize the uncertainty of together with the uncertainties of the nuisance parameters listed in Table 5.

For , we use the relative uncertainty that corresponds to the standard deviation of the experimental values depicted in Table 8, which is approximately 5%. For , its prior relative uncertainty was deduced from COMAC-V2.0, with a value close to 7%.

If the fission yield uncertainties of JEFF-3.1.1 are introduced in the marginalization procedure of the nuisance parameters (Table 8), the two-step CONRAD calculation provides the following trends for 153 Eu and 154 Eu:

( 153 Eu) = + (6.6 ± 6.8) %, ( 154 Eu) = + (6.7 ± 7.4) %.

When the fission yields uncertainties of COMAC-V2.0 are used, we obtain smaller uncertainties:

( 153 Eu) = + (6.6 ± 4.8) %, ( 154 Eu) = + (6.7 ± 5.9) %.

Both results still suggest an increase of JEFF-3.1.1 capture cross sections. The integral trend for the 154 Eu(n,) reaction remains consistent with the previous trends (+7.1%) reported in Table 6, while the trend for the 153 Eu(n,) reaction between 0.5 eV to 20 MeV is slightly higher than the one given in Table 5. Nuisance parameters and relative uncertainties introduced in the marginalization procedure.

Model parameters Relative uncertainties

Origin of the uncertainties