Christophe Hohlweg
email: hohlweg.christophe@uqam.ca.canadian

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

Salvatore Stella
email: stella@math.haifa.ac.il.

Series and Algebraic Combinatorics (Hanover) Universal associahedra

est destinée au dépôt

Theorem. For any finite type initial exchange matrix B • , the g-vector fan F g (B • ) with respect to B • is the normal fan of a generalized associahedron Asso(B • ).

When we start from an acyclic initial exchange matrix, our construction precisely recovers the associahedra of [START_REF] Hohlweg | Permutahedra and generalized associahedra[END_REF]. These can all be obtained by deleting inequalities from the facet description of the permutahedron of the corresponding finite reflection group. The main difficulty to extend this approach to arbitrary initial exchange matrices lies in the fact that this property, intriguing as it might be, is essentially a coincidence. In fact, the hyperplane arrangement H supporting the g-vector fan is no longer the Coxeter arrangement of a finite reflection group. To overcome this situation, we develop an alternative approach based on a uniform understanding of the linear dependences among adjacent cones in the g-vector fan. In fact, not only we cover uniformly all finite type g-vector fans, but we actually treat them simultaneously with a universal object.

Theorem. For any given finite Dynkin type Γ, there exists a universal associahedron Asso un (Γ) such that, for any initial exchange matrix B • of type Γ, the generalized associahedron Asso(B • ) is a suitable projection of the universal associahedron Asso un (Γ). In particular, all g-vector fans of type Γ are sections of the normal fan of the universal associahedron Asso un (Γ).

See the long version [START_REF] Hohlweg | Polytopal realizations of finite type g-vector fans[END_REF] of this extended abstract for details and complete proofs.

Type A universal associahedra from triangulations

Before addressing the problem in full generality, we illustrate the results in type A using triangulations of polygons. We consider 2n + 6 points on the unit circle alternately colored black and white, and let Ω • (resp. Ω • ) denote the convex hull of the white (resp. black) points. Let T be a triangulation of Ω • (resp. of Ω • ), let δ ∈ T, and let γ be a diagonal of Ω • (resp. of Ω • ). When γ crosses δ, we define ε(δ ∈ T, γ) to be 1, -1, or 0 depending on whether γ crosses the quadrilateral formed by the two triangles of T incident to δ as a Z, as a Z, or in a corner. If γ and δ do not cross, then we set ε(δ ∈ T, γ) = 0. The following definition is illustrated in Figure 1. For two triangulations T

Definition 1.1. Fix an arbitrary reference triangulation T

• of Ω • . For a diagonal δ • of Ω • , we define the g-vector of δ • with respect to T • as g T • , δ • : = ε δ • ∈ T • , δ • δ • ∈T • ∈ R T • . For any diagonal δ • of Ω • and triangulation T • of Ω • with δ • ∈ T • , we define the c-vector of δ • ∈ T • with respect to T • as c T • , δ • ∈ T • : = -ε δ • ∈ T • , δ • δ • ∈T • . 7 • 6 • 2 • 3 • 1 • 4 • 8 • 5 • 1 • 5 • 2 • 4 • 8 • 6 • 3 • 7 • 2 • 4 • 2 • 8 • 4 • 6 • 4 • 7 • 4 • 8 •                                                             2 • 5 • 0 0 0 -1 -1 2 • 7 • 0 0 0 1 0 2 • 8 • 0 1 0 0 1 3 • 5 • -1 0 0 0 0 5 • 7 • 0 0 1 0 0 g(T • , T • ) 2 • 4 • 2 • 8 • 4 • 6 • 4 • 7 • 4 • 8 •                                                             2 • 5 • 0 1 0 0 -1 2 • 7 • 0 1 0 1 -1 2 • 8 • 0 1 0 0 0 3 • 5 • -1 0 0 0 0 5 • 7 • 0 0 1 0 0 c(T • , T • )
• of Ω • and T • of Ω • , the sets g(T • , T • ) : = {g(T • , δ • ) | δ • ∈ T • } and c(T • , T • ) : = {c(T • , δ • ∈ T • ) | δ • ∈ T • } are dual bases.
Moreover, the g-vectors sup- port a complete simplicial fan realizing the simplicial complex of collections of pairwise non-crossing diagonals of Ω • . See Section 3 for basic notions of polyhedral geometry.

Theorem 1.2. For any reference triangulation T

• of Ω • , the collection of cones F g (T • ) : = R ≥0 g(T • , T • ) T • triangulation of Ω •
, together with all their faces, forms a complete simplicial fan, called the g-vector fan of T • .

We prove that this fan is realized by the following associahedra. For a diagonal δ • of Ω • , we denote by h(δ • ) the number of diagonals of Ω • crossed by δ • . Theorem 1.3. For any reference triangulation T • of Ω • , the g-vector fan F g (T • ) is the normal fan of the T • -associahedron Asso(T • ) defined equivalently as (i) the convex hull of the points p T • , T

• : = ∑ δ • ∈T • h(δ • ) c(T • , δ • ∈ T • ) for all triangula- tions T • of Ω • , or (ii) the intersection of the hyperplanes H ≤ T • , δ • : = v ∈ R T • g(T • , δ • ) | v ≤ h(δ • ) for all diagonals δ • of Ω • .
When the triangulation T • has no internal triangle, the T • -associahedron was constructed by C. Hohlweg and C. Lange in [START_REF] Hohlweg | Realizations of the associahedron and cyclohedron[END_REF].

The geometry of these realizations depends on T • . However, the crucial observation is that h(δ • ) does not depend on T • .

Definition 1.4. Let ∆(Ω • ) be the set of all diagonals of Ω • . The universal T • -associahedron Asso un (Ω • ) ⊆ R ∆(Ω • ) is the convex hull of the points p un (T • ) : = ∑ δ • ∈T • h(δ • ) u(δ • ∈ T • ) for all triangulations T • of Ω • , where u(δ • ∈ T • ) : = -ε δ • ∈ T • , δ • δ • ∈∆(Ω • ) ∈ R ∆(Ω • ) .
This polytope has the following universal property. Theorem 1.5. For any reference triangulation T • of Ω • , the orthogonal projection of the universal associahedron Asso un (Ω • ) on the coordinate subspace R T • is the T • -associahedron Asso(T • ).

This extended abstract explains this result for arbitrary finite type cluster algebras.

Finite type cluster algebras

Cluster algebras. We work in the ambient field Q(x 1 , . . . , x n , p 1 , . . . , p m ) of rational expressions in n + m variables with coefficients in Q and denote by P m its abelian multiplicative subgroup generated by the elements

{p i } i∈[m] . Given p = ∏ i∈[m] p a i i ∈ P m we write {p} + : = ∏ i∈[m] p max(a i ,0) i and {p} -: = ∏ i∈[m] p
min(a i ,0) i so that p = {p} + {p} -1 -. A seed Σ is a triple (B, P, X) where • the exchange matrix B is an integer n × n skew-symmetrizable matrix, i.e., such that there exist a diagonal matrix D with -BD = (BD) T , • the coefficient tuple P is any subset of n elements of P m , • the cluster X is a set of cluster variables, n rational functions in the ambient field that are algebraically independent over Q(p 1 , . . . , p m ). To shorten our notation we think of rows and columns of B, as well as elements of P, as being labelled by the elements of X: we write B = (b xy ) x,y∈X and P = {p x } x∈X . Moreover we say that a cluster variable x (resp. a coefficient p) belongs to Σ = (B, P, X) to mean x ∈ X (resp. p ∈ P).

Given a seed Σ = (B, P, X) and a cluster variable x ∈ Σ, we can construct a new seed µ x (Σ) = Σ = (B , P , X ) by mutation in direction x, where:

• the new cluster X is obtained from X by replacing x with the cluster variable x defined by the following exchange relation:

xx = {p x } + ∏ y∈X, b xy >0 y b xy + {p x } -∏ y∈X, b xy <0 y -b xy
and leaving the remaining cluster variables unchanged so that X {x} = X {x }. Fix a seed Σ • = (B • , P • , X • ) and call it initial. Up to an automorphism of the ambient field we will assume that X • = {x 1 , . . . , x n } and drop X • from our notation. Finite type. We only deal with cluster algebras of finite type i.e., cluster algebras having only a finite number of cluster variables. Being of finite type is a property that depends only on the exchange matrix in the initial seed and not on the coefficient tuple.

The Cartan companion of an exchange matrix B is the symmetrizable matrix A(B) given by a xy = 2 if x = y and a xy = -|b xy | otherwise. In accordance with this statement, when talking about the (cluster) type of A(B • , P • ) or B • we will refer to the Cartan type of A(B). We reiterate that the Cartan type of A(B • ) need not be finite: being of finite type is a property of the mutation class.

For a finite type cluster algebra A(B • , P • ), we will consider the root system of A(B • ). To avoid any confusion later on let us state clearly the conventions we use here: for us simple roots {α x } x∈X • and fundamental weights {ω x } x∈X • are two basis of the same vector space V; the matrix relating them is the Cartan matrix A(B • ). Fundamental weights are the dual basis to simple coroots {α ∨

x } x∈X • , while simple roots are the dual basis to fundamental coweights {ω ∨

x } x∈X • ; coroots and coweights are two basis of the dual space V ∨ and they are related by the transpose of the Cartan matrix.

A finite type exchange matrix B • is said to be acyclic if A(B • ) is itself a Cartan matrix of finite type and cyclic otherwise. An acyclic finite type exchange matrix is said to be bipartite if each of its rows consists either of non-positive or non-negative entries.

Principal coefficients and g-and c-vectors. Among all the cluster algebras having a fixed initial exchange matrix, a central role is played by those with principal coefficients. Definition 2.3 ([4, Definition 3.1]). A cluster algebra is said to have principal coefficients (at the initial seed) if its ambient field is Q(x 1 , . . . , x n , p 1 , . . . , p n ) and the initial coefficient tuple consists of the generators of P n i.e., P • = {p i } i∈ [n] . In this case we will write

A pr (B • ) for A(B • , {p i } i∈[n]
), and we reindex the generators

{p i } i∈[n] of P n by {p x } x∈X • .
Cluster algebras with principal coefficients are Z n -graded (in the basis {ω

x } x∈X • of V). The degree function deg(B • , •) on A pr (B • ) is obtained by setting deg(B • , x) : = ω x and deg(B • , p x ) : = ∑ y∈X • -b yx ω y for any x ∈ X • .
This assignment makes all exchange relations and all cluster variables in A pr (B • ) homogeneous [4] and it justifies the defini- tion of the following family of integer vectors associated to cluster variables.

Definition 2.4 ([4]

). The g-vector g(B • , x) of a cluster variable x ∈ A pr (B • ) is its degree. We denote by g(B • , Σ) : = {g(B • , x) | x ∈ Σ} the set of g-vectors of the cluster variable in the seed Σ of A pr (B • ).

The next definition gives another family of integer vectors, introduced implicitly in [START_REF] Fomin | Cluster algebras. IV. Coefficients[END_REF], that are relevant in the structure of A pr (B • ). Our next task in this section is to discuss a duality relation in between c-vectors and g-vectors. A first step is to recall the notion of the cluster complex of A(B • , P • ): it is the abstract simplicial complex whose vertices are the cluster variables of A(B • , P • ) and whose facets are its clusters. As it turns out, at least in the finite type cases, this complex is independent of the choice of coefficients, see [START_REF] Fomin | Cluster algebras. II. Finite type classification[END_REF]Theorem 1.13] and [START_REF] Fomin | Cluster algebras. IV. Coefficients[END_REF]Conjecture 4.3]. In particular this means that, up to isomorphism, there is only one cluster complex for each finite type: the one associated to A fr (B • ). We will use this remark later on to relate cluster variables of different cluster algebras of the same finite type. Note also that, again when A(B • , P • ) is of finite type, the cluster complex is a pseudomanifold [START_REF] Fomin | Cluster algebras. II. Finite type classification[END_REF].

For a skew-symmetrizable exchange matrix B • , the matrix B ∨

• : = -B T
• is still skewsymmetrizable. The cluster algebras A pr (B • ) and A pr (B ∨ • ) can be thought as dual to each other. Indeed their types are Langlands dual of each other. Moreover their cluster complexes are isomorphic: by performing the same sequence of mutations we can identify any cluster variable x of A pr (B • ) with a cluster variable x ∨ of A pr (B ∨ • ), and any seed Σ in A pr (B • ) with a seed Σ ∨ in A pr (B ∨ • ). More importantly the following crucial property holds. In view of the above results, and since A(B ∨ • ) = A(B • ) T , the c-vectors of a finite type cluster algebra A pr (B ∨ • ) can be understood as coroots for A(B • ) so that the g-vectors of A pr (B • ) become weights. This justify our choice to define g-vectors in the weight basis and c-vectors in the root basis.

Coefficient specialization and universal cluster algebra. We want to relate, within a given finite type, cluster algebras with different choices of coefficients. Pick a finite type exchange matrix B • and let A(B • , P • ) ⊂ Q(x 1 , . . . , x n , p 1 , . . . , p m ) and A(B • , P • ) ⊂ Q(x 1 , . . . , x n , p 1 , . . . , p ) be any two cluster algebras having B • as exchange matrix in their initial seed. As we said, cluster variables and seeds in these two algebras are in bijection because their cluster complexes are isomorphic. Let us write x ↔ x and Σ ↔ Σ for this bijection. We will say that A(B • , P • ) is obtained from A(B • , P • ) by a coefficient specialization if there exist a map of abelian groups η : P m → P such that, for any p x in some seed Σ of A(B • , P • ), we have η({p x } + ) = {p x } + and η({p x } -) = {p x } -and which extends in a unique way to a map of algebras that satisfy η(x) = x. Note that this is not the most general definition (see [4, Definition 12.1 and Proposition 12.2]) but it will suffice here. Armed with the notion of coefficient specialization we can now introduce the last kind of cluster algebra of finite type we will need. Definition 2.7 ([4, Definition 12.3 and Theorem 12.4]). Pick a finite type exchange matrix B • . The cluster algebra with universal coefficients A un (B • ) is the unique (up to canonical isomorphism) cluster algebra such that any other cluster algebra of the same type as B • can be obtained from it by a unique coefficient specialization.

Let us insist on the fact that, in view of the universal property it satisfies, A un (B • ) depends only on the type of B • and not on the exchange matrix B • itself. We keep B • in the notation only to fix an embedding into the ambient field.

Rather than proving the existence and explaining the details of how such a universal algebra is built, we will recall here one of its remarkable properties that follows directly from the g-vector recursion [8, Proposition 4.2 (v)] and that we will need later on. Denote by X (B • ) the set of all cluster variables in A un (B • ) and let {p[x]} x∈X (B • ) be the generators of

P |X (B • )| .
Theorem 2.8 ([9, Theorem 10.12]). The cluster algebra A un (B • ) can be realized over P |X (B • )| . The coefficient tuple P = {p x } x∈X at each seed Σ = (B, P, X) of A un (B • ) is given by the formula p x = ∏ y∈X (B • ) p[y] [g(B T ,y T );x T ] where [v; x] is the x-th coefficient of a vector v in the weight basis (ω x ) x∈X . Remark 2.9. In view of this result, it is straightforward to produce the coefficient specialization morphism to get any cluster algebra with principal coefficients of type B • from A un (B • ). Namely, for any seed Σ = (B , P , X ) of A un (B • ), we obtain A pr (B ) by evaluating to 1 all the coefficients p[y] corresponding to cluster variables y not in Σ .

Polyhedral geometry and fans

A polyhedral cone is a subset of the vector space V defined equivalently as the positive span of finitely many vectors or as the intersection of finitely many closed linear halfspaces. We write R ≥0 Λ for the positive span of a set Λ of vectors of V. The faces of a cone C are the intersections of C with its supporting hyperplanes. The 1-dimensional (resp. codimension 1) faces of C are called rays (resp. facets) of C. A cone is simplicial if it is generated by a set of independent vectors.

A polyhedral fan is a collection F of polyhedral cones of V such that

• if C ∈ F and F is a face of C, then F ∈ F ,
• the intersection of any two cones of F is a face of both. A fan is simplicial if all its cones are, and complete if the union of its cones covers the ambient space V. For a simplicial fan F with rays X , the collection {X ⊆ X | R ≥0 X ∈ F } of generating sets of the cones of F defines a pseudomanifold. The following statement characterizes which pseudomanifolds are complete simplicial fans. Proposition 3.1. Consider a pseudomanifold ∆ with vertex set X and a set of vectors {r(x)} x∈X of V. For X ∈ ∆, let r(X) : = r(x) x ∈ X . Then the collection of cones R ≥0 r(X) X ∈ ∆ forms a complete simplicial fan if and only if 1. there exists a facet X of ∆ such that r(X) is a basis of V and the open cones R >0 r(X) and R >0 r(X ) are disjoint for any facet X of ∆ distinct from X; 2. for any two adjacent facets X, X of ∆ with X {x} = X {x }, there is a linear dependence γ r(x) + γ r(x ) + ∑ y∈X∩X δ y r(y) = 0 on r(X ∪ X ) where the coefficients γ and γ have the same sign. (When these conditions hold, these coefficients do not vanish and the linear dependence is unique up to rescaling.)

A polytope is a subset P of V ∨ defined equivalently as the convex hull of finitely many points or as a bounded intersection of finitely many closed affine halfspaces. The faces of P are the intersections of P with its supporting hyperplanes. In particular, the dimension 0 (resp. dimension 1, resp. codimension 1) faces of P are called vertices (resp. edges, resp. facets) of P. The (outer) normal cone of a face F of P is the cone in V generated by the outer normal vectors of the facets of P containing F. The (outer) normal fan of P is the collection of the (outer) normal cones of all its faces. We say that a complete polyhedral fan in V is polytopal when it is the normal fan of a polytope in V ∨ . The following statement provides a characterization of polytopality of complete simplicial fans. Proposition 3.2. Consider a pseudomanifold ∆ with vertex set X and a set of vectors {r(x)} x∈X of V such that F : = R ≥0 r(X) X ∈ ∆ forms a complete simplicial fan. Assume that there exists a map h : X → R >0 such that for any two facets X, X of ∆ with X {x} = X {x }, we have γ h(x) + γ h(x ) + ∑ y δ y h(y) > 0, where γ r(x) + γ r(x ) + ∑ y δ y r(y) = 0 is the unique (up to scaling) linear dependence with γ, γ > 0 between the rays of r(X ∪ X ). Then, F is the normal fan of the polytope given by v ∈

V ∨ r(x) | v ≤ h(x) for all x ∈ X .   0 -1 1 1 0 -1 -1 1 0     0 -1 2 1 0 -2 -1 1 0   Figure 2:
The g-vector fan F g (B • ) for the type A 3 (left) and type C 3 (right) cyclic initial exchange matrices. To plot the figure, the 3-dimensional fans are intersected with the unit sphere and stereographically projected to the plane from the pole (-1, -1, -1).

Theorem 5.2. For any finite type exchange matrix B • and exchange submodular function h, the g-vector fan F g (B • ) is the normal fan of the B • -associahedron Asso h (B • ) defined equivalently as (i) the convex hull of the points

p h B • , Σ : = ∑ x∈Σ h(x) c B ∨ • , x ∨ ∈ Σ ∨ for all seeds Σ of A pr (B • ), or (ii) the intersection of the halfspaces H h ≤ (B • , x) : = v ∈ V ∨ g B • ,
x v ≤ h(x) for all cluster variables x of A pr (B • ).

Our next step is to discuss the existence of exchange submodular functions for any finite type cluster algebra with principal coefficients. The important observation here is that the definition of exchange submodular function does not involve in any way the coefficients of A pr (B • ) so that it suffices to construct one in the coefficient free cases. Indeed, if h is exchange submodular for A fr (B • ), and η is the coefficient specialization morphism given by η :

A pr (B • ) -→ A fr (B • ) p i -→ 1
one gets the desired map by setting h(x) : = h(η(x)) for any cluster variable x of A pr (B • ).

Recall that, up to an obvious automorphism of the ambient field, there exists a unique cluster algebra without coefficients for each given finite type [START_REF] Fomin | Cluster algebras. IV. Coefficients[END_REF]. We can therefore, without loss of generality, assume that B • is bipartite and directly deduce our result from [START_REF] Stella | Polyhedral models for generalized associahedra via Coxeter elements[END_REF]Proposition 8.3] obtained as an easy consequence of [1, Lemma 2.4] which we recast here in our current setting.

When B • is acyclic, the Weyl group of A(B • ) is finite and has a longest element w

• . A point λ ∨ : = ∑ x∈X • λ ∨ x ω ∨
x in the interior of the fundamental Weyl chamber of A(B ∨ • ) (that is to say λ ∨ x > 0 for all x ∈ X • ) is fairly balanced if w • (λ ∨ ) = -λ ∨ . Proposition 5.3. Let A fr (B • ) be any finite type cluster algebra without coefficients and assume that B • is bipartite. To each fairly balanced point λ ∨ corresponds an exchange submodular function h λ ∨ on A fr (B • ).

A particular example of fairly balanced point is the point ρ

∨ : = ∑ x∈X • ω ∨
x . Note that ρ ∨ is both the sum of the fundamental coweights and the half sum of all positive coroots of the root system of finite type A(B • ). In particular h ρ ∨ is the half compatibility sum of x, i.e., the half sum of the compatibility degrees h ρ ∨ (x) : = 1 2 ∑ y =x ( y x ) over all cluster variables distinct from x. The point ρ ∨ is particularly relevant in representation theory and its role in this context has already been observed in [START_REF] Chapoton | Polytopal Realizations of Generalized Associahedra[END_REF]Remark 1.6]. We call balanced B • -associahedron and denote by Asso(B • ) the B • -associahedron Asso h ρ ∨ (B • ) for the exchange submodular function h ρ ∨ . Remark 5.4. When B • is acyclic, the B • -associahedron Asso(B • ) was already constructed in [START_REF] Hohlweg | Permutahedra and generalized associahedra[END_REF]. It is then obtained by deleting inequalities from the facet description of the permutahedron of the Coxeter group of type A(B • ). This requires the fact that the Coxeter arrangement refines the Cambrian fan [START_REF] Reading | Cambrian fans[END_REF]. Similar properties still hold for any type A initial seed: on the one hand, the g-vector fan is refined by the arrangement of hyperplanes normal to the c-vectors of A pr (B • ); on the other hand, the B • -associahedron is obtained by deleting inequalities from the facet description of the zonotope obtained as the Minkowski sum of all c-vectors of A pr (B • ). Although the fan refinement still holds for all the other finite types, the B • -associahedron in general cannot be obtained by deleting inequalities in the facet description of a zonotope.

We sum up by restating our main result. Corollary 5.5. For any finite type exchange matrix B • , the g-vector fan F g (B • ) is polytopal.

Universal associahedron

For each initial exchange matrix B • of a given type, we constructed in Section 5 a generalized associahedron Asso h (B • ) by lifting the g-vector fan using an exchange submodular function h on the cluster variables of A pr (B • ). As already observed though, the func- tion h is independent of the coefficients of A pr (B • ), so that all g-vector fans can be lifted with the same function h. This motivates the definition of a universal associahedron.

Consider the finite type cluster algebra A un (B • ) with universal coefficients, and let X (B • ) denote its set of cluster variables. Consider a |X (B • )|-dimensional vector space U with basis {β x } x∈X (B • ) and its dual space U ∨ with basis {β ∨

x ∨ } x ∨ ∈X (B ∨ • ) . As before, the cluster variables of A un (B • ) and A un (B ∨ • ) are related by x ↔ x ∨ . For X ⊆ X (B • ), we denote by H X the coordinate subspace of U spanned by {β x } x∈X .

Given a seed Σ in A un (B • ), the u-vector of a cluster variable x ∈ Σ is the vector

u(B • , x ∈ Σ) : = ∑ y∈X (B • ) u yx β y of exponents of p x = ∏ y∈X (B • ) (p[y]
) u yx . Remark 2.9 then reformulates in terms of u-and c-vectors as follows. Choose a seed Σ = (B , P , X ) in A un (B • ) that you want to make initial. Then, for any cluster variable x in a seed Σ, the c-vector c(B , x ∈ Σ) is the orthogonal projection of the u-vector u(B • , x ∈ Σ) on the coordinate subspace H X . (Here and elsewhere we identify H X with V and H X ∨ with V ∨ in the obvious way.) We are now ready to define the universal associahedron. Note that Asso h un (B • ) does not depend on B • but only on its cluster type. We keep B • in the notation to fix the indexing. Our interest in Asso h un (B • ) comes from the following. Theorem 6.2. Fix a finite type exchange matrix B • and an exchange submodular function h. For any seed (B , P , X ) of A un (B • ), the orthogonal projection of the universal associahedron Asso h un (B • ) on the coordinate subspace H X ∨ of U ∨ spanned by {β ∨ x ∨ } x ∨ ∈X ∨ is the Bassociahedron Asso h (B ). Remark 6.3. Consider the normal fan F of the universal B • -associahedron Asso h un (B • ). Then for any seed Σ = (B , P , X ) in A un (B • ), the section of F by the coordinate subspace H X of U spanned by {β x } x∈X is the g-vector fan F g (B ). We therefore call universal g-vector fan the normal fan F g un (B • ) of the universal B • -associahedron Asso un (B • ). As an immediate consequence of Theorem 6.2, we obtain that the vertices of the universal associahedron Asso h un (B • ) are precisely the points p h un (B • , Σ) for all seeds Σ of A un (B • ), and that the mutation graph of the cluster algebra A un (B • ) is a subgraph of the graph of Asso h un (B • ). However, this inclusion is strict in general. We conclude with three observations on the universal B • Although a priori defined in U ∨ , Asso h un (B • ) seems to be of codimension 1. • In general Asso h un (B • ) is neither simple nor simplicial. Table 1 presents some statistics for the number of vertices per facet and facets per vertex in type A n for n ∈ [START_REF] Fomin | Cluster algebras. IV. Coefficients[END_REF].

• The face lattice (and thus the f -vector) of Asso h un (B • ) seems independent of h.

Figure 1 :

 1 Figure 1: Two triangulations T • and T • with the corresponding g-and c-vectors.

  • the row (resp. column) of B indexed by x is the negative of the row (resp. column) of B indexed by x, while all other entries satisfy b yz = b yz + 1 2 |b yx |b xz + b yx |b xz | , • the elements of the new coefficient tuple P are obtained by a related mutation rule. As mutations are involutions, they define an equivalence relation on the set of all seeds.

Definition 2 . 1 (

 21 [START_REF] Fomin | Cluster algebras. IV. Coefficients[END_REF] Definition 2.11]). The (geometric type) cluster algebra A(B • , P • ) is the ZP m -subring of the ambient field generated by all the cluster variables in all the seeds mutationally equivalent to the initial seed Σ • .The simplest possible choice of coefficient tuple in the initial seed, namely m = 0 and P • = {1} i∈[n] , gives rise to the cluster algebra without coefficients A fr (B • ).

Theorem 2 . 2 ([ 3 ,

 223 Theorem 1.4]). The cluster algebra A(B • , P • ) is of finite type if and only if there exists an exchange matrix B obtained by a sequence of mutations from B • such that its Cartan companion is a Cartan matrix of finite type. Moreover the type of A(B) is determined by B • .

Definition 2 . 5 .

 25 Given a seed Σ in A pr (B • ), the c-vector of a cluster variable x ∈ Σ is the vector c(B • , x ∈ Σ) : = ∑ y∈X • c yx α y of exponents of p x = ∏ y∈X • (p y ) c yx . We denote by c(B • , Σ) : = {c(B • , x ∈ Σ) | x ∈ Σ} the set of c-vectors of a seed Σ.

Theorem 2 . 6 (

 26 [START_REF] Nakanishi | On tropical dualities in cluster algebras[END_REF] Theorem 1.2]). For any seed Σ of A pr (B • ), let Σ ∨ be its dual in A pr (B ∨ • ).Then the set of g-vectors g(B • , Σ) and the set of c-vectors c(B ∨ • , Σ ∨ ) form dual bases, that is g(B • , x) c(B ∨ • , y ∨ ∈ Σ ∨ ) = δ x=y for any two cluster variables x, y ∈ Σ.

Figure 3 :

 3 Figure 3: The associahedra Asso(B • ) for the type A 3 (left) and type C 3 (right) cyclic initial exchange matrices whose g-vector fans are depicted in Figure 2.

Definition 6 . 1 .

 61 For any finite type exchange matrix B • and any exchange submodular function h, the universal B • -associahedron Asso h un (B • ) ⊆ U ∨ is the convex hull of the points p h un B • , Σ : = ∑ x∈Σ h(x) u B ∨ • , x ∨ ∈ Σ ∨ for all seeds Σ of A un (B • ).

Table 1 :

 1 • -associahedron Asso h un (B • ) obtained by computer experiment: n ambient dim. dim. # vertices # facets # vertices / facet # facets / vertex Some statistics for the universal associahedron of type A n for n ∈[START_REF] Fomin | Cluster algebras. IV. Coefficients[END_REF].

	1	2	1	2	2	1	1
	2	5	4	5	5	4	4
	3	9	8	14	60	9 ≤ • ≤ 10	30 ≤ • ≤ 42
	4	14	13	42	8960	14 ≤ • ≤ 28	3463 ≤ • ≤ 4244

The g-vector fan

We first recast a well known fact concerning the cones spanned by the g-vectors of a any finite type cluster algebra with principal coefficients. Theorem 4.1. For any finite type exchange matrix B • , the collection of cones

, together with all their faces, forms a complete simplicial fan, called the g-vector fan of B • .

There are several ways to deduce Theorem 4.1 from the literature. One option is to use Proposition 3.1, whose second condition is implied by the following description of the linear dependence between the g-vectors of two adjacent clusters, which is crucial later. Lemma 4.2. For any finite type exchange matrix B • and any adjacent seeds (B, P, X), (B , P , X ) in A pr (B • ) with X {x} = X {x }, the g-vectors of X ∪ X satisfy precisely one of the following two linear dependences

Note that which of the two possible linear dependences is satisfied by the g-vectors of X ∪ X depends on the initial exchange matrix B • . Remark 4.3. When the exchange matrix B • is acyclic, the g-vector fan is the Cambrian fan constructed by N. Reading and D. Speyer [START_REF] Reading | Cambrian fans[END_REF]. Cyclic examples are shown in Figure 2.

Polytopality

In this section, we show that the g-vector fan F g (B • ) is polytopal for any finite type exchange matrix B • . As mentioned in the introduction, this result was previously known for acyclic finite type exchange matrices [START_REF] Hohlweg | Permutahedra and generalized associahedra[END_REF]. We first consider some convenient functions which will be used later in Theorem 5.2 to lift the g-vector fan. The existence of such functions will be discussed in Proposition 5.3. The following statement is our central result.


Samy Mzabi

Daniel Berghezan

Stéphane Roux

François Hild

Costantino Creton

A critical local energy release rate criterion for fatigue fracture of elastomers

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Pierre Farger

Blaise Haidon

Pascal Roussel

Bénédicte Arab-Chapelet

Murielle Rivenet

Single-Crystal to Single-Crystal Transformation in the Thorium-TEDGA-Oxalate-Nitrate System: Description and Comparison of the Main Structural Features

Keywords: thorium, oxalate, tetraethyldiglycolamide, mixed ligands, nitrate, crystal structure, spectroscopic data, single-crystal to single-crystal phase transitions, oxalic acid geometry

), the actinide atom is surrounded by two oxalate ligands and two TEDGA, leading to a 10-fold coordination. The dimensionality of the networks changes from linear chains (1D) (1) to isolated entities (0D) (2) or dimeric units (0D) (4). Compound (3) is formed by the assembly of 12-fold coordinated monomeric entities (0D) in which the thorium cation is surrounded by four TEDGA. This compound is the first example of such a coordination number without nitrate anion included in the coordination sphere of Th.

Introduction

In the nuclear industry, the long term radiotoxicity of the spent fuel is mainly due to plutonium and minor actinides. [START_REF] Ansari | Chemistry of Diglycolamides: Promising Extractants for Actinide Partitioning[END_REF] So, in order to reduce the radiotoxicity of ultimate wastes, researches for new generations of nuclear plants focus on the recycling of plutonium and minor actinides with the aim, on one hand, to fabricate new Pu and U-based mixed oxides nuclear fuel pellets (MOX), on the other hand, to synthesize Minor Actinides-based transmutation targets. In the current industrial PUREX process, uranium(VI) and plutonium(IV) are recovered from a nitric solution containing the fission products and minor actinides by liquid-liquid phase extraction thanks to the use of tributylphosphate (TBP) in dodecane. [START_REF] Warf | Extraction of Cerium(IV) Nitrate by Butyl Phosphate[END_REF][START_REF] Shoun | Actinide Extractants: Development, Comparison, and Future[END_REF] The extracting molecules have to meet various specifications such as (i) good extraction of actinides from a highly concentrated nitric solution to the organic diluent (ii) stability in aliphatic diluents ie resistance to radiolytic and hydrolytic degradation (iii) ease of synthesis (iv) low environmental impact. The use of tributylphosphate as an extracting agent may induce the presence of residual phosphate groups in the wastes which explain that new extracting molecules containing only C, H, O and N elements, fully incinerable, has been the subject of intense researches these last years. [START_REF] Sun | Ionic Liquids-Based Extraction: A Promising Strategy for the Advanced Nuclear Fuel Cycle[END_REF][START_REF] Drew | Lanthanide(III) Complexes of a Highly Efficient Actinide(III) Extracting Agent -2,6-Bis(5,6-Dipropyl-1,2,4-Triazin-3-Yl)Pyridine[END_REF] The amide family containing a carbonyl group and a nitrogen atom in alpha-position has shown promising results for the complexation of actinides and liquid-liquid extraction. [START_REF] Nair | Extraction of Uranium(VI) and Plutonium(IV) with Dihexylbutyramide and Dihexylisobutyramide from Nitric Acid Medium[END_REF] Within this family of compounds, the diglycolamide molecules (DGA) of general formula (R 2 N(CO)(CH 2 )O(CH 2 )(CO)NR 2 ) are promising An(IV) complexing agents due to their large number of electronegative sites. The solubility of DGA molecules and their complexing properties strongly depends on their substituent groups, R. By example, tetraoctyldiglycolamide (TODGA) is deemed to be a very good extractant of transuranic elements in dodecane [START_REF] Ansari | Chemistry of Diglycolamides: Promising Extractants for Actinide Partitioning[END_REF][START_REF] Brown | Plutonium Loading of Prospective Grouped Actinide Extraction (GANEX) Solvent Systems Based on Diglycolamide Extractants[END_REF][START_REF] Sasaki | Extraction Behaviors of Eu, Th, U and Am with Diamides+thenoyltrifluoroacetone[END_REF] while tetraethyldiglycolamide (TEDGA) shows good abilities for precipitating actinides and other metals in acidic conditions. [START_REF] Sasaki | Complexation and Back Extraction of Various Metals by Water-Soluble Diglycolamide[END_REF] Despite the possible use of DGA as extracting and/or complexing agents of f-elements, few structural studies have been conducted in the DGA-lanthanides or -actinides systems. The crystal structures of lanthanides complexes coordinated by DGA ligands, L, have been reported for lanthanum [La(pic) 3 (L 2 ) with L containing RR' = Et and Ph; 10 (LaL 3 )(La(NO 3 ) 6 ) with L substituted by R = i-Pr or by i-Bu], for cerium and ytterbium [(CeL)(Ce(NO 3 ) 6 )(CH 3 OH)(C 4 H 10 O), (YbL)(NO 3 ) 3 where L is a tris-DGA ligand and (YbL 3 )(NO 3 ) 4 with L holding R = i-Pr] [START_REF] Matloka | Highly Efficient Binding of Trivalent F-Elements from Acidic Media with a C3-Symmetric Tripodal Ligand Containing Diglycolamide Arms[END_REF] and for the heavy lanthanides series (Ln = Tb, Dy, Ho, Er, Tm, Yb, and Lu) [(LnL 3 )(NO 3 ) 3 .4H 2 O with L substituted by R = Et]. [START_REF] Okumura | Crystal Structures of Lanthanoid(III) (Ln(III), Ln = Tb, Dy, Ho, Er, Tm, Yb, and Lu) Nitrate Complexes with N , N , N ′, N ′-Tetraethyldiglycolamide[END_REF] Main of the crystal structures of the actinides-DGA complexes were reported for An-yl salts with a DGA:actinyle ratio of 1:1 [(UO 2 L)X 2 with L carrying R = i-Pr or R = i-Bu and X = Cl -or NO 3 -] 13 or 1:2 [(UO 2 L 2 )(X) 2 [START_REF] Kannan | Lanthanum(III) and Uranyl(VI) Diglycolamide Complexes: Synthetic Precursors and Structural Studies Involving Nitrate Complexation[END_REF][START_REF] Tian | Quest for Environmentally Benign Ligands for Actinide Separations: Thermodynamic, Spectroscopic, and Structural Characterization of UVI Complexes with Oxa-Diamide and Related Ligands[END_REF] , (PuO 2 L 2 )(X) 2 15 and (NpO 2 L 2 )(X) [START_REF] Tian | Optical Absorption and Structure of a Highly Symmetrical Neptunium(V) Diamide Complex[END_REF] with R =Me and X = ClO 4 -]. But, to the best of our knowledge, the crystal structures reported in the An(IV)-DGA system have remained limited to (Pu IV L 3 )(NO 3 ) 3 .MCN and (Pu IV L 3 )(NO 3 ) 3 .ETOH with R = Me attached to L. [START_REF] Reilly | Plutonium(IV) Complexation by Diglycolamide Ligands-coordination Chemistry Insight into TODGA-Based Actinide Separations[END_REF] This paper reports the influence of tetraethyldiglycolamide (TEDGA) on the crystallization process in the An(IV)-oxalate system, where Th(IV) was used as an analog of Pu(IV). Under evaporation, the presence of TEDGA leads to the crystal growth of four new compounds hereafter named (1), ( 2), ( 3) and ( 4) which crystal structures were determined by singlecrystal X-Ray Diffraction. This paper aims to describe and compare the structural arrangement of these compounds and to report the spectroscopic study of compound (3), obtained as a pure phase.

Experimental section

Synthesis Caution! Thorium nitrate (Th(NO 3 ) 4 .4H 2 O) is a radioactive and chemically toxic reactant, so precautions with suitable care and protection for handling such substances have been followed. Reactants (NO 3 ) 4 .4H 2 O was purchased from Prolabo (99%), oxalic acid from Alfa Aesar (98%) and TEDGA from Pharmasynthèse SA (>99%).

Synthesis

of [Th(TEDGA) 2 (C 2 O 4 )][NO 3 ] 2 [H 2 C 2 O 4 ].6H 2 O (1), [Th(TEDGA) 2 (C 2 O 4 ) 2 ][H 2 C 2 O 4 ] 2 .2H 2 O (2), [Th(TEDGA) 4 [NO 3 ] 4 .4H 2 O (3) and [Th 2 (C 2 O 4 ) 3 (TEDGA) 4 ][NO 3 ][HC 2 O 4 ][H 2 C 2 O 4 ] 4 .7H 2 O (4)
Thorium nitrate (0.2 mmol) and TEDGA (1.4 mmol) were added in 10 mL of deionized water. The solution was heated at 50°C for 2h then a solution of oxalic acid (1.4 mmol in 10 mL of deionized water) was added within 2h. The colorless solution obtained after adding all reactants was left under the hood for slow evaporation. After ten days, a viscous solution containing crystals of (1) was obtained. The crystals of (1) had to be retained into their solution otherwise a rapid degradation was observed. The solution was sealed to prevent further evaporation which leads to the massive crystallization of oxalic acid. The compound (1) was initially retained into the viscous solution but after five weeks it was transformed into a mixture of (2) and (3). Four months later the mixture has turned into a new mixture, containing crystals of (1) and (4). It has to be noted that the single-crystal to single-crystal transformation is reproducible except for the formation of (2) which has occurred just at once and couldn't be reproduced. Several trials were conducted in order to obtain the compound (2) from (1). In particular, the effect of the temperature on the transformation was studied. Despite a careful monitoring on the solutions at different temperatures, the compound (2) was not obtained. These transformations are accompanied by a morphological change of the crystals (Figure 1). Indeed, the compound (1) displays a rod-like shape (Figure 1a) while intermediate compounds ( 2) and (3) have a platelet shape (Figure 1b), the smallest single crystals being typical of compound (2) and the biggest corresponding to (3). In the final mixture, the rod-like singlecrystals of (1) appear again whereas compound (4) displays a block type morphology (Figure 1c). 3) was obtained as a pure phase by adding thorium nitrate (0.2 mmol), TEDGA (0.8 mmol) and oxalic acid (50 mg) to 10 mL of deionized water, at room temperature. Solid oxalic acid was used in place of oxalic acid solution to favor slow diffusion and then achieve a better crystal growth. The colorless solution obtained after adding all the reactants was left under the hood. After ten days, a precipitate matching the X-Ray powder diffraction pattern of compound (3) was obtained (Figure S2) with a yield of 25%. This pure powder was afterward used to perform further chemical analysis.

Single crystal diffraction

Except for (2), the collections of the diffraction data were performed at 100K, on a Bruker DUO-APEX2 CCD area detector diffractometer using Mo-Kα radiation (λ= 0.71073 Å) with an Incoatec microsource. The data reduction was accomplished using SAINT V7.53a. [START_REF]SAINT Plus Version 7.53a[END_REF] The substantial redundancy in data allowed a semi empirical absorption correction (SADABS V2.10) to be applied, on the basis of multiple measurements of equivalent reflections. Due to the low stability of (1), data were collected with 5s frame exposure while data for (2), (3) and (4) were recorded with 20s frame exposure. The structures were solved by direct methods, developed by successive difference Fourier syntheses, and refined by full-matrix least-squares on all F 2 data using OLEX2 and SHELXL program software. [START_REF] Dolomanov | It OLEX2: A Complete Structure Solution, Refinement and Analysis Program[END_REF][START_REF] Sheldrick | A Short History of It SHELX[END_REF] Hydrogen atoms of the TEDGA were included in calculated positions and allowed to ride on their parent atoms. Hydrogen atoms of water molecules and oxalic acid parts were found on Fourier-difference maps. The low quality of the data for the compound (2) has led to consider several C-O distance constrains for free oxalic part.

Selected crystallographic data are reported Table 1. The CIF files of compounds 1-4 are provided as supplementary materials under the CCDC numbers xxx for (1) etc.

Powder diffraction

X-ray powder diffraction was performed on Bruker D8 Advance diffractometer (LynxEye detector) in a Bragg-Brentano - mode using Cu-K radiation. Each powder pattern was recorded within an angular range of 5-70° in 2, with steps of 0.02° and counting time of 0.5s per step.

Infrared spectroscopy

Infrared spectroscopy was performed with a PerkinElmer Spectrum Two spectrometer equipped with a single reflection attenuated total reflectance (ATR), between 4000-400cm -1 with a 1 cm -1 resolution.

Thermal analysis

Thorium based compounds, thermal analysis was performed on a thermoanalyser 92 SETARAM TGA up to 900°C under air condition with a heating rate of 5°C.min -1 .

Results and discussion

Structural description

Crystal structure of (1) 2). Due to symmetry consideration (4-fold axe) the oxalate ligands are twisted of 90° on both sides of the SBU.

Compound (1) [Th(TEDGA) 2 (C 2 O 4 )][NO 3 ] 2 [H 2 C 2 O 4 ].6H 2 O
The SBU share their bis-bidentate oxalate ligands,  2 - 2 : 2 , to form cationic chains, [Th(TEDGA) 2 (C 2 O 4 )] 2+ ∞ , running along the c axis (Figures 2c and2d). The distance Th-Th through the oxalate ligands is equal to 6.490 Å and is similar to the one encountered in the literature for most of the ten-fold coordinated Thorium(IV) compounds connected by a bisbidentate  2 -oxalate. [START_REF] Clavier | X-Ray Diffraction and μ-Raman Investigation of the Monoclinic-Orthorhombic Phase Transition in Th1-xUx(C2O4)2•2H2O Solid Solutions[END_REF][START_REF] Zhang | Spectroscopic Studies and Crystal Structures of Double Thorium(IV) Oxalates with Sodium Ions[END_REF] In (1), the [Th(TEDGA) 2 (C 2 O 4 )] 2+ ∞ chains are linear with successive Th-Th-Th atoms forming an angle equal to 180°, close to the one found in ∞ chains are corrugated with a An-An-An angle close to 158°. [START_REF] Blanchard | Solid State Chemistry of Ten-Fold Coordinate Thorium(IV) Complexes with Oxalates in Presence of Ammonium and Hydrazinium Ions[END_REF][24][START_REF] Andreev | Polymeric Structure of Oxalato-Bridged Complexes of Tetravalent Actinides Th, U, Np and Pu[END_REF][START_REF] Akhtar | The Crystal Structure of Tetrapotassium Tetraoxalatothorium(IV) Tetrahydrate, K4Th(C2O4)4.4H2O[END_REF] Four nitrates, found along the (110) and (-110) directions, are found in the environment of one out of two [Th(TEDGA) 2 (C 2 O 4 ) 2 ] SBU. These nitrate anions are involved in a H-bonding network along with the oxalic acid molecules, found on the 2 fold axes parallel to c axis, and the water molecules. Only the water molecules bond the [Th(TEDGA)(C 2 O 4 )] 2+ ∞ chains, which could explain the low stability of compound (1) in air (Figure 2d). Crystal structure of (2)

Compound (2), [Th(TEDGA) 2 (C 2 O 4 ) 2 ][H 2 C 2 O 4 ] 2 .2H 2 O
, crystallizes in the monoclinic space group P2 1 /n (n°14) with the unit cell parameters: a = 12.246(2) Å, b = 32.253(5) Å, c = 12.256(2) Å and  = 106.741(1)°. The ten oxygen atoms surrounding Th 4+ form a bicapped square antiprism polyhedron (Figure 3a). The 10-fold coordination involves six oxygen atoms coming from two tridentate TEDGA ligands and four oxygen atoms provided by two bidentate oxalate ligands, leading to structural building units (SBU) with the same formula as observed in (1), [Th(TEDGA) 2 (C 2 O 4 ) 2 ] (Figure 3b). The Th-O distances range between 2). The two TEDGA ligands show, both, a planar geometry. In TEDGA 1, the ethyl groups are in a syn-anti conformation. In TEDGA 2 the two conformations, syn-anti and syn-syn, are found since the C3 atoms half occupy two crystallographic positions leading to two orientations of the C3-C4 ethyl groups (Figure 3b). The oxalate ligands are in a bis-monodentate coordination mode leading to a 0D compound. The neutral SBU are assembled by hydrogen bonds which involve two oxalic acid uncoordinated to Th 4+ and two water molecules (Figure 3c). The hydrogen bonds network between

H 2 C 2 O 4 , H 2 O and [Th(TEDGA) 2 (C 2 O 4 ) 2 ] forms [Th(TEDGA) 2 (C 2 O 4 ) 2 ][H 2 C 2 O 4 ] 2 .2H 2 O
layers laying in the aoc plane (Figure 3c). The layers are stacked in b direction thanks to Van der Walls interaction between the TEDGA molecules (Figure 3d). The absence of strong bonding interaction between the [Th(TEDGA

) 2 (C 2 O 4 ) 2 ][H 2 C 2 O 4 ] 2 .2H 2 O
layers most probably explain the lack of stability of compound ( 2) and the difficulties in reproducing its synthesis. It has also to be noted that it is the only compound of the series with no nitrate included in the crystal structure. Th 4+ is coordinated by four tridentate TEDGA leading to an uncommon coordination number of 12 (Figure 4a) instead of the 8 to 10-fold oxygen coordination usually reported. [START_REF] Tutson | Thorium Coordination: A Comprehensive Review Based on Coordination Number[END_REF] The Th-O distances, between 2.500(1) Å and 2.681(1) Å (table 2), are much shorter than the Th-O distances calculated for a 12-fold coordination (2.87 Å). [START_REF] Gong | A Tetrapositive Metal Ion in the Gas Phase: Thorium(IV) Coordinated by Neutral Tridentate Ligands[END_REF] Indeed, the Th-O distances are equivalent to those calculated for Th(DGA) 3 [START_REF] Gong | A Tetrapositive Metal Ion in the Gas Phase: Thorium(IV) Coordinated by Neutral Tridentate Ligands[END_REF] which shows the strong interaction existing between the Th 4+ cations and TEDGA molecules in compound (3).

The dihedral angles between the planes including the amide functions of each TEDGA are close to 140° which indicates a strong constrain of the TEDGA ligands. Moreover, the ethyl groups are in a syn-anti conformation, most probably in order to reduce the steric hindrance (Figure 4b). Four nitrate anions surround the isolated entities [Th(TEDGA)] 4+ (Figure 4c). They contribute to stabilize the overall structure by electrostatic interaction involving [Th(TEDGA)] 4+ and hydrogen bonds involving water molecules and forming (NO 3 ) 4 (H 2 O) 4 clusters (figure 4d). 2)°. The cationic environment is similar for the two independent thorium which are surrounded by ten oxygen atoms, forming a bicapped square antiprism (Figure 5a). As observed in compounds ( 1) and ( 2), six oxygen atoms come from the TEDGA ligands and four come from two oxalate ligands (Figure 5b). The terminal oxalates, Ox1a and Ox1b, part form the hydrogen-bond network with water molecules and oxalic acids (Figure 5d). In order to respect the global neutral charge of the network, among the different oxalate acids, one must be partially deprotonated. According to the crystal structure determination (H positioning and Wickoff positions), the planar oxalate Ox3 is this peculiar one. Finally, seven water molecules are located in the interstices, two of them (O30 and O32) are involved in the hydrogen bonding with the terminal oxalate of the dimeric units (Ox1a and Ox1b), and the other five are involved in the hydrogen bonding with the uncoordinated oxalic parts, the nitrate anions and the bridging oxalate, Ox2a and Ox2b, thereby ensuring the stability of the network.

Thorium coordination

Although the connectivity of the networks differs between (1), ( 2) and ( 4), the first coordination sphere of the thorium cation remains equivalent in these three compounds. The actinide is located at the center of a bicapped square antiprism (CN: 10) whose summits of the square faces are made of two carbonyl oxygen atoms from one TEDGA and two oxygen atoms coming from two different oxalate groups. The two ether oxygen atoms of the TEDGA ligands occupy the capping positions. Unlike in ( 1), ( 2) and ( 4), the actinide in compound (3) presents a 12-fold coordination forming an icosahedron whose summits are composed by the oxygen atoms of TEDGA parts, without oxalate's bonding. An overview of the literature indicates that 12-coordinated thorium atoms are found in thorium nitrate compounds [Th(NO 3 ) 6 ][L] with L = bypyridinium, [START_REF] Rammo | Characterization and Crystal Structure of 4,4′-Bipyridinium Nitrate Trinitratodioxo-Uranium(VI) and 4,4′-Dipyridinium Diaqua Hexanitrato-Thorate(IV)[END_REF][30][START_REF] Abram | Bis2-[1-(Thiosemicarbazono)Ethyl]Pyridinium Hexakis(Nitrato-O,O')Thorate(IV) Tetramethanol Solvate[END_REF] oxadiazole, 32 phosphonium, [START_REF] Dulong | Nitrite Complexes of Uranium and Thorium[END_REF] oxonium, [START_REF] Wang | Structure of the Extraction Complex Bis[(Dicyclohexano-18-Crown-6)Oxonium] Hexanitratothorate(IV) Isomer It A[END_REF] or [Th(NO 3 ) 3-5 ][M] with M = phosphate, [START_REF] Charushnikova | Crystal Structure and Spectral Properties of Complexes of Tetravalent Th, U, Np and Pu Nitrates with Trimethyl Phosphate and Dimethyl Phosphate[END_REF] sulfoxide 36 and phosphine [START_REF] Morse | Th(IV) Complexes with Cis-Ethylenebis(Diphenylphosphine Oxide): X-Ray Structures and {NMR} Solution Studies[END_REF] but remain rarely encountered (only 26 results in the conquest database). To the best of our knowledge, it is the first compound presenting such coordination number without any nitrate anion belonging to the coordination sphere. The presence of TEDGA in the coordination sphere of each of the compounds shows the great complexing behavior and stability of Th(TEDGA) x complexes despite the presence of oxalic acid which is known to form insoluble and stable thorium oxalate Th(C 2 O 4 ) 2 .nH 2 O. [START_REF] Tyrpekl | Alterations of Thorium Oxalate Morphology by Changing Elementary Precipitation Conditions[END_REF][START_REF] Abraham | Actinide Oxalates, Solid State Structures and Applications[END_REF] The Th-O ether mean distances (2.648 to 2.719Å) are almost equivalent through the series despite the 4-fold TEDGA coordination of Th(IV) in (3) and the higher ionic radius expected for a twelve-fold coordinated Th(IV) compared to a ten-fold one (1.21Å vs 1.13Å). These Th-O ether bond values are higher than the mean Th-O oxalate and Th-O carbonyl distances (Table 2), while the Th-O oxalate distances, between 2.423(7) and 2.509(2)Å, are similar to those encountered in the literature for  2 - 2 : 2 and  2 - 2 bridging oxalate ligands. [START_REF] Zhang | Spectroscopic Studies and Crystal Structures of Double Thorium(IV) Oxalates with Sodium Ions[END_REF][START_REF] Thuéry | Solid State Structure of Thorium(IV) Complexes with Common Aminopolycarboxylate Ligands[END_REF] The Th-O carbonyl bond values are close for compounds (1), ( 2) and ( 4). Their values, between 2.421(6) and 2.532(1)Å, lie within the same range than previously described for carbonyl derivative based compounds [START_REF] Benetollo | Oxydiacetato Complexes of Thorium(IV), the Crystal Structures of Tetra-Aquo Bis(Oxydiacetato)Thorium(IV) Hexahydrate and Di(Sodium Nitrate) Disodium Tris(Oxydiacetato)Thorium(IV)[END_REF]42 however the Th-O carbonyl bond seems affected by the ionic radius variation of Th(IV) since compound (3) presents a larger Th-O carbonyl mean distance (2.516Å) than compounds (1), ( 2) and (4) (2.427, 2.461 and 2.463Å, respectively). These values remain consistent with the Th-O carbonyl distances determined by the calculation. [START_REF] Gong | A Tetrapositive Metal Ion in the Gas Phase: Thorium(IV) Coordinated by Neutral Tridentate Ligands[END_REF] So far, only two compounds based on An(IV) have been described in the literature: (Pu IV (TMDGA) 3 )(NO 3 ) 3 .MCN (TMDGA = tetramethyldiglycolamide ; MCN = acetonitrile) and (Pu IV (TMDGA) 3 )(NO 3 ) 3 .ETOH. [START_REF] Reilly | Plutonium(IV) Complexation by Diglycolamide Ligands-coordination Chemistry Insight into TODGA-Based Actinide Separations[END_REF] In these examples, the Pu(IV) cations are surrounded by nine oxygen atoms coming from the three TMDGA which form a twisted tricapped trigonal prism. The ether oxygen atoms occupy the capping position while the oxygen atoms belonging to the carbonyl function form the summits of the trigonal faces. The Pu-O distances show the same trend than observed in the present study in that the Pu-O ether distances are longer than the Pu-O carbonyl distances (0.18Å difference). The Pu-O distance are, on average, ca. 0.15-0.19Å shorter than the Th-O distances which may account for the larger ionic radius of Th(IV) than Pu(IV) together with the higher coordination numbers of Th(IV) in the present compounds than the one of Pu(IV) in the previously reported Pu IV (TMDGA) 3 )(NO 3 ) 3 solvates. The presence of an additional TEDGA around the thorium cation in (3) may also be related to the larger ionic radius of Th(IV) than Pu(IV) (0.09Å difference for a same coordination number), thereby allowing a higher coordination number. The DFT calculations performed by Gong et al [START_REF] Gong | A Tetrapositive Metal Ion in the Gas Phase: Thorium(IV) Coordinated by Neutral Tridentate Ligands[END_REF] on the Thorium-tetramethyldiglycolamide (TMDGA) system showed that the structure of Th(TMDGA) 4 is less stable than the one of Th(TMDGA) 3 due to larger Th-O distances. One may note that the experimental Th-O distances, more specifically the Th-O ether distances experimentally determined in the present study, are shorter than the calculated distances. In (1) the angle formed by two oxalate parts coordinating Th is of 90° while it is lower in ( 2) and ( 4) with a mean value of about 71° and 80°, respectively (Table S6). The opposite TEDGA ligands coordinating Th in compounds (1), ( 2), ( 3) and ( 4) are turned by an angle varying from 31° to 69° (Table S7). No relationships could be established between the torsion angle of the TEDGA molecules and the dihedral angle formed by their mean planes (Table S8).

Nitrate positioning

In all the compounds reported herein, the nitrate anions are found in the interstices, as for instance in (Ln(TEDGA) 3 )(NO 3 ) 3 .4H 2 O with Ln = Tb, Dy, Ho, Er, Tm, Yb, and Lu 12 , (Yb(tris-DGA))(NO 3 ) 3 , (Yb(i-PrDGA) 3 )(NO 3 ) 4 , [START_REF] Matloka | Highly Efficient Binding of Trivalent F-Elements from Acidic Media with a C3-Symmetric Tripodal Ligand Containing Diglycolamide Arms[END_REF] (Pu IV (TMDGA) 3 )(NO 3 ) 3 .MCN and (Pu IV (TMDGA) 3 )(NO 3 ) 3 .ETOH. [START_REF] Reilly | Plutonium(IV) Complexation by Diglycolamide Ligands-coordination Chemistry Insight into TODGA-Based Actinide Separations[END_REF] One may note that no solid compounds containing simultaneously nitrate and DGA ligands in the same coordination of the f-element was reported up to date and that the only mixed ligands compounds including DGA (L) are found only with the lightest lanthanides, La and Ce. These compounds are heteroleptic which means that Ln(NO 3 ) 6 polyhedra coexist in the crystal structure besides the Ln(DGA) n entities leading to (LaL 3 )(La(NO 3 ) 6 ) and (CeL)(Ce(NO 3 ) 6 )(CH 3 OH)(C 4 H 10 O) formula. In the present study, the nitrate anions remain close to the thorium cations, but do not integrate the coordination sphere. They occupy either the free spaces between the two arms of TEDGA ligands in compounds ( 1) and (3) (Figures 2b and4c) or the free spaces between the TEDGA ligands in compound (4) (Figure 5c). According to these structural results and to the crystal structure previously reported, one can assume that the presence of DGA ligands in the coordination sphere of the f-cations preclude the integration of nitrate in the same environment.

Geometry of the free di-and mono-protonated oxalic acids in (1), (2) and (4)

The oxalate ligands coordinating Th(IV) in compounds ( 1), ( 2) and ( 4) are nearly planar (Table S4). In these oxalate ligands, the C-O distances are slightly different in accordance with the bridging mode of the oxalate groups. In the µ2- 2 : 2 oxalates, all the C-O distances are equivalent to 2.24-2.26Å while in the µ2- 2 terminal oxalates the C-O distances with O atoms in the coordination sphere of Th(IV) tend to be slightly longer (1.25-1.28Å) than the C-O distances with free oxygen atoms (1.22-1.26Å). The oxalic acid molecules can be planar [START_REF] Jeffrey | The Structure of the Oxalate Ion[END_REF] , as the oxalate groups, but their carboxylate plans can also display a twisted angle up to 90°. [START_REF] Jeffrey | The Structure of the Oxalate Ion[END_REF][START_REF] Dewar | Structure of the Oxalate Ion[END_REF] In the present compounds the free di-and mono-protonated oxalic acids exhibit a twisted angle varying from 7 to 88° which shows that the oxalic acid can adopt different geometries (Figure 6, Table S9). The nomenclature introduced by Nieminen et al. uses the lower case letters c (cis) and t (trans) to describe the relative orientation of the O-H and C-C bonds and the upper letters C (Cis) or T (Trans) to define the relative orientations of the C=O bond. [START_REF] Nieminen | Matrix-Isolation and Ab Initio Studies of Oxalic Acid[END_REF] In compound (1) the oxalic acid has a conformation tTt and exhibit a torsion angle, , equal to 19°. In compound (2) the two oxalic acids with the same conformations tTt, Ox2 and Ox3, shows torsion angles of about 7-8°. In compound (4) different conformations of oxalic acid coexist: the tCt conformation with torsion angles equal to 7, 19 and 88° for the oxalates ox4a, ox4b and ox5a, respectively, and the tTt conformation for the oxalate ox5b ( equal to 39°) (Figure 6). On the basis of these results, it is impossible to establish a relation between the torsion angle and the oxalic acid conformation. However, a careful examination of the C-O distances in compounds ( 2) and ( 4

Comparison of the spectroscopic data for TEDGA, (1), and (3)

We were able to obtain the compound (1) and (3) as pure powders, the last by modifying the temperature of synthesis. The X-ray diffraction pattern and thermal decomposition of compounds ( 1) and ( 3) are shown in Figures S2 andS5, while IR spectra are presented in Figure 7. Between 4000 cm -1 and 2000 cm -1 , the spectra show the three bands characteristic of the  C-H stretching of the methylene and methyl groups. One can also notice the presence of a large band between 3600 cm -1 and 3400 cm -1 due to the presence of water molecules in the structures. In the region between 2000 cm -1 and 400 cm -1 , the peak at 1639 cm -1 corresponding to the carbonyl stretching of the TEDGA is shifted at 1612 cm -1 and 1607 cm -1 in (1) and (3) spectra, respectively, because of the coordination of the TEDGA to the actinide. [START_REF] Okumura | Crystal Structures of Lanthanoid(III) (Ln(III), Ln = Tb, Dy, Ho, Er, Tm, Yb, and Lu) Nitrate Complexes with N , N , N ′, N ′-Tetraethyldiglycolamide[END_REF][START_REF] Narita | Study on the Extraction of Trivalent Lanthanide Ions with N,N′-Dimethyl-N,N′-Diphenyl-Malonamide and Diglycolamide[END_REF] The C-O-C of the TEDGA exhibits two vibrational bands at 1125 cm -1 ( as ) and 1039 cm -1 ( s ) for the free ligands. [START_REF] Kuncheria | Thorium(IV) Nitrate Complexes with Some Schiff Bases of 4-Aminoantipyrine and Certain Carbonyl Compounds[END_REF] Those bands are slightly shifted due to the coordination of the ether oxygen atom. The bands at 826, 1282 and 1385 cm -1 correspond to the vibrational bands of uncoordinated nitrate anions. [START_REF] Oliver | The Structure of Aqueous Thorium Nitrate Solutions by Laser Raman Spectroscopy[END_REF][START_REF] Kuncheria | Thorium(IV) Nitrate Complexes with Some Schiff Bases of 4-Aminoantipyrine and Certain Carbonyl Compounds[END_REF][START_REF] Goebbert | Infrared Spectroscopy of the Microhydrated Nitrate Ions NO3-(H2O)1-6[END_REF] The compound (1) spectrum displays one additionnal band at 1730 cm -1 due to the presence of oxalic acid molecules in the crystal structure. For this compound, the band at 1607 cm -1 might be attributed to the combination of the carbonyl stretching arising from the TEDGA part or to the asymmetric stretching of the oxalate part. [START_REF] Andreev | Polymeric Structure of Oxalato-Bridged Complexes of Tetravalent Actinides Th, U, Np and Pu[END_REF][START_REF] Tamain | Coordination Modes of Americium in the Am2(C2O4)3(H2O)6•4H2O Oxalate: Synthesis, Crystal Structure, Spectroscopic Characterizations and Comparison in the M2(C2O4)3(H2O)6•nH2O (M = Ln, An) Series[END_REF][START_REF] Duvieubourg-Garela | Adaptable Coordination of U(IV) in the 2D-(4,4) Uranium Oxalate Network: From 8 to 10 Coordinations in the Uranium (IV) Oxalate Hydrates[END_REF] The symmetric stretching bands are observed in 1300-1350 cm -1 range. [START_REF] Andreev | Polymeric Structure of Oxalato-Bridged Complexes of Tetravalent Actinides Th, U, Np and Pu[END_REF][START_REF] Tamain | Coordination Modes of Americium in the Am2(C2O4)3(H2O)6•4H2O Oxalate: Synthesis, Crystal Structure, Spectroscopic Characterizations and Comparison in the M2(C2O4)3(H2O)6•nH2O (M = Ln, An) Series[END_REF]

Conclusions

Even if the use of diglycolamide molecules for the nuclear waste treatment has been studied for long time, the ability of these organic compounds for complexing the lanthanides or the renewable elements, plutonium or uranium, is still a hot topic and the knowledge of the different possible phases able to be formed during a process, a very crucial point. The precomplexation of the water soluble TEDGA and thorium(IV), followed by an addition of oxalic acid, has allowed to isolate and characterize four new compounds containing Th(IV) and TEDGA, by successive single-crystal to single-crystal phase transformations. Three of them are homoleptic mixed ligands compounds containing two TEDGA and two oxalate in the coordination sphere of Th(IV) while the fourth compound exhibits four TEDGA in the cation environment leading then to an uncommon coordination number of 12. Although the

TEDGA (3) (1)
mixed oxalate-TEDGA ligands compounds present the same coordination sphere, the dimensionality of the networks varies from linear chains to isolated entities or dimeric units. The TEDGA molecule shows then a great ability to complex the actinide cation (thorium in the present case) despite the presence of oxalic acid. The influence that TEDGA might have on the structural and microstructural properties of the precipitate during the next step of the industrial process, i.e. the oxalic precipitation, is under study by carrying out oxalic precipitation and crystal growth of Th(C 2 O 4 ) 2 .6H 2 O and Pu(C 2 O 4 ) 2 .6H 2 O in presence of TEDGA.

Figure 1 :

 1 Figure 1: Micrographs of the single-crystals of compound (1) (a); the mixture of (2) and (3) (b) and the mixture of rods like crystals (1) and blocks (4) (c).

R1

  crystallizes in the tetragonal noncentrosymmetric space group P4 2 2 1 2 (n°94) with a = b = 18.7140(12) Å and c = 12.9212(9) Å. The thorium (IV) ions are surrounded by ten oxygen atoms which form a bicapped square antiprism (Figure2a). Six oxygen atoms are provided by the coordination of two tridentate TEDGA ligands and four oxygen atoms come from two bidentate oxalate ligands leading to structural building units (SBU) of formula [Th(TEDGA) 2 (C 2 O 4 ) 2 ]. The TEDGA ligands are planar and possess two ethyl groups in a syn-syn conformation and two ethyl groups in a synanti conformation (Figure2b). The Th-O distances range between 2.421(6) and 2.648(6) Å. The longest Th-O distances are representative of the Th-O ether bonds while the shortest lengths are attributed to the Th-O carbonyl ones. The Th-O oxalate distances have intermediate values close to the Th-O carbonyl (Table

  (NH 4 ) 4 Th(C 2 O 4 ) 4 •4H 2 O, 23 while in compounds M 4 [An(C 2 O 4 ) 4 ].nH 2 O (An = Th, M=K, N 2 H 5 , n=4 ; M=C(NH 3 ) 2 , n=2 ; M=Na, n=5.5 -An = U, 22 M=K, n=4 ; M=C(NH 3 ) 2 , n=2 and An=Np, Pu, M= C(NH 3 ) 2 , n=2) the [An(C 2 O 4 ) 4 ] 4+

Figure 2 :

 2 Figure 2: Crystal structure of [Th(TEDGA) 2 (C 2 O 4 )][NO 3 ] 2 [H 2 C 2 O 4 ].6H 2 O (1). The ten-fold coordination sphere around Th forms a bicapped square antiprism (a) which summits are occupied by oxygen atoms coming from two bidentate oxalates and two tridentate TEDGA molecules. Two ethyl-groups of the TEDGA ligands have a syn-syn conformation while the other two have a syn-anti conformation (b). One out of two structural building units [Th(TEDGA) 2 (C 2 O 4 ) 2 ] is surrounded by four nitrate groups (b). The [Th(TEDGA) 2 (C 2 O 4 ) 2 ] share their oxalate ligands and form [Th(TEDGA) 2 (C 2 O 4 )] 2+ ∞ chains (c) running along the c axis (d). The chains are maintained in a three-dimensional arrangement by means of a H-bond network (blue dotted lines) involving the nitrate anions, oxalic acid and water molecules (d). Legend: Th 4+ are in green, C in grey, O in red and N in blue. The black arrows show the orientation of the ethyl groups over the plane defined by N-C-C atoms: full line: up, dotted line: down. The TEDGA molecules were removed for clarity in the H-network representation.

  (6) and 2.721(6) Å, the longest Th-O distances being representative of the Th-O ether bonds (2.7Å) and the shortest of the Th-O carbonyl and Th-O oxalate ones (2.4Å) (Table

Figure 3 :

 3 Figure 3: Crystal structure of [Th(TEDGA) 2 (C 2 O 4 ) 2 ][H 2 C 2 O 4 ] 2 .2H 2 O (2). The ten-fold coordination sphere around Th forms a bicapped square antiprism (a) which summits are occupied by oxygen atoms coming from two bidentate oxalates and two tridentate TEDGA molecules. Two ethyl-groups of the TEDGA ligands have a syn-syn conformation while the other two have either a syn-anti or a syn-syn conformation (b). The [Th(TEDGA) 2 (C 2 O 4 ) 2 ] SBU are maintained in a two-dimensional arrangement by means of a H-bond network (blue dotted lines) involving two oxalic acid and two water molecules (c). Legend: Th 4+ are in green, C in grey, O in red and N in blue. The black arrows show the orientation of the ethyl groups over the plane defined by N-C-C atoms: full line: up, dotted line: down. The TEDGA molecules were removed for clarity in the H-network representation.

Figure 4 :

 4 Figure 4: Crystal structure of [Th(TEDGA) 4 ][NO 3 ] 4 .4H 2 O (3). Each of oxygen atoms of the twelve-fold coordinated Th (a) belongs to a tridentate TEDGA molecules. The ethyl-groups of the four TEDGA ligands have

Figure 5 :

 5 Figure 5: Crystal structure of [Th 2 (C 2 O 4 ) 3 (TEDGA) 4 ][NO 3 ][HC 2 O 4 ][H 2 C 2 O 4 ] 4 .7H 2 O (4). The ten-fold coordination sphere around Th forms a bicapped square antiprism (a) which summits are occupied by four oxygen atoms coming from two bidentate oxalates and by 6 oxygen atoms originating from two tridentate TEDGA molecules. One ethyl-group of the TEDGA ligands has a syn-syn conformation while the other three have a syn-anti conformation (b). One out of two dimeric units [Th 2 (Ox) 3 (TEDGA) 4 ] 2+ is surrounded by two nitrate which are included in the second coordination sphere (c). The dimeric units are maintained in a threedimensional arrangement by means of a H-bond network (blue dotted lines) between the nitrate anions, oxalic acid and water molecules (d). Legend: Th 4+ are in green, C in grey, O in red and N in blue. The black arrows show the orientation of the ethyl groups over the plane defined by N-C-C atoms: full line: up, dotted line: down. The TEDGA molecules were removed for clarity in the H-network representation.

  ), which crystal structure was solved at 100K, reveal undoubtedly that the C-O distances around 1.2Å can be attributed to the C=O bonds while the C-O distances around 1.3Å are related to the C-OH bonds. The 1.3Å distance characteristic of C-OH is also found in the monoprotonated oxalate ox3 of compound (4).

Figure 6 :

 6 Figure 6: Oxalic acids in the crystal structure of [Th 2 (C 2 O 4 ) 3 (TEDGA) 4 ][NO 3 ][HC 2 O 4 ][H 2 C 2 O 4 ] 4 .7H 2 O (4).

Figure 7 :

 7 Figure 7: Infrared spectra of TEDGA (in black), [Th(TEDGA) 2 (C 2 O 4 )][NO 3 ] 2 [H 2 C 2 O 4 ].6H 2 O (1) (in blue) and [Th(TEDGA) 4 [NO 3 ] 4 .4H 2 O (3) (in red).

Table 1 :

 1 Crystallographic data for compounds1-4.

	Compound	1	2	3	4
	Chemical formula C28 H60 N6	C32 H48 N4	C48 H104 N12	C64 H117 N9
		O26 Th	O24 Th	O28 Th	O54 Th2
	M (g.mol -1 )	1128.86	1104.78	1529.47	2340.74
	Temperature (K)	100	293	100	100
	Symmetry	tetragonal	monoclinic	monoclinic	triclinic
	Space group	P 4 2 2 1 2	P 2 1 /n	C 2/c	P -1
	a (Å)	18.7140(12)	12.246(2)	26.5966(13)	15.6611(7)
	b (Å)	18.7140(12)	32.253(5)	15.4489(7)	17.9082(9)
	c (Å)	12.9212(9)	12.256(2)	18.5582(9)	18.1814(7)
		90	90	90	89.896(2)
		90	106.741(12)	116.528(1)	65.549(2)
		90	90	90	87.623(2)
	V Å 3	4525.2(7)	4635.6(13)	6822.5(6)	4637.2(4)
	Z	4	4	4	2
	Density calc.	1.657	1.583	1.489	1.676
	(g.cm -3 )
	mm  	3.387	3.301	2.273	3.310
	 range	1.539 -26.393 1.846 -22.002 1.527 -30.511	1.449 -26.351
	Limiting indices	-23 ≤ h ≤ 18	-12 ≤ h ≤ 12	-37 ≤ h ≤ 37	-19 ≤ h ≤ 19
		-16 ≤ k ≤ 22	-33 ≤ k ≤ 33	-22 ≤ k ≤ 22	-22 ≤ k ≤ 22
		-12 ≤ l ≤ 16	-12 ≤ l ≤ 12	-26 ≤ l ≤ 26	-22 ≤ l ≤ 22
	Collected	21750	39816	51745	61346
	reflections
	Unique reflections 4652	5668	10367	18660
	R(int)	0.0739	0.0879	0.0287	0.0338
	Parameters	289	575	416	1206
	Goodness of fit	1.026	1.015	1.070	1.020
	Final R indices [I
	> 2(I)]

Table 2 :

 2 Minimal, maximal and mean Th-O distances in (1), (2), (3) and (4).

			Th-O (min -max)					Th-O (mean)
		1	2	3	4		1	2	3	4	ref
	Th-	2.648 (6) 2.717(6)	2.666(1)	2.622(2)	Th-	2.648	2.719	2.674	2.657	2.697 41
	O ether		-	-	-	O ether					2.623 43,44
	(Å)		2.721(6)	2.681(1)	2.684(2)	(Å)
	Th-	2.421(6)	2.450(7)	2.500(1)	2.434(2)	Th-	2.427	2.461	2.516	2.463	2.458 25
	O carbonyl	-	-	-	-	O carbonyl					2.411 41
	(Å)	2.432(6)	2.475(6)	2.532(1)	2.481(2)	(Å)
	Th-	2.487(5)	2.423(7)		2.416(2)	Th-	2.490	2.447		2.461	2.491 40
	O oxalate	-	-		-	O oxalate					2.485 25
	(Å)	2.492(5)	2.468(6)		2.509(2)	(Å)

Acknowledgements

The authors would like to thanks the ANR-ASTUTE (projet ANR-15-CE08-0011-01) for the financial support and to Chevreul Institute (FR 2638), Ministère de l'Enseignement Supérieur et de la Recherche, Région Nord -Pas de Calais and FEDER for supporting and funding partially this work.

Supporting information

Comparison between simulated and experimental powder pattern for (1), ( 3) and the mixtures (2) + ( 3) and ( 1) + ( 4) can be found on figure S1-S4 in supporting information. The figure S5 contains the thermal analysis of (3). The crystallographic data of (1), ( 2), ( 3) and ( 4) are available in CIF format.