Bertrand Pellenard

Pierre Alliez

Jean-Marie Morvan
email: morvan@math.univ-lyon1.fr

Isotropic 2D Quadrangle Meshing with Size and Orientation Control

We propose an approach for automatically generating isotropic 2D quadrangle meshes from arbitrary domains with a fine control over sizing and orientation of the elements. At the heart of our algorithm is an optimization procedure that, from a coarse initial tiling of the 2D domain, enforces each of the desirable mesh quality criteria (size, shape, orientation, degree, regularity) one at a time, in an order designed not to undo previous enhancements. Our experiments demonstrate how well our resulting quadrangle meshes conform to a wide range of input sizing and orientation fields.

Introduction

Quadrangle meshes are preferred over triangle meshes in a number of applications related to computer graphics, computer aided geometric design, computational engineering and reverse engineering. However, the automatic generation of isotropic quadrangle meshes for arbitrary 2D domains is still a scientific challenge due to the variety of requirements and quality criteria sought after.

The bare minimum requirement we impose that the input domain must be tiled with only convex quadrangles. We also wish to generate quadrangle meshes for which, locally, elements i) are well shaped in the sense of being close to squares, ii) are sized in accordance to an input sizing field, iii) are oriented in accordance to an input cross field (an orientation field modulo 90 degrees) and iv) have edges aligned to the domain boundary. A more global requirement practitioners often desire is that meshes should predominantly be composed of regular vertices, i.e., degree-4 vertices 4 inside the domain and (in general) degree-2 vertices on the domain boundary.

While these quality requirements are widely regarded as desirable, one key meshing difficulty is that they often conflict with one another: irregular vertices are often necessary for non-trivial sizing and cross fields, but they inevitably induce shape distortion and hence must be avoided whenever possible; a rapidly varying sizing field naturally induces both shape and orientation distortion; similarly, a rapidly varying cross field often results in both shape and size distortion. Another key challenge comes from the fact that some of the requirements or quality criteria, although locally defined, have global constraints-e.g., the number of edges on the domain boundary must be even, and the total index of irregular vertices must obey Gauss-Bonnet theorem [13].

Previous Work

The tension between local and global criteria may explain the variety of approaches proposed so far for the automatic generation of quadrangle meshes. Including methods devised to generate quadrangle surface tilings, the rich literature on this topic contains approaches which proceed by quadrangulation [START_REF] Bremner | Small strictly convex quadrilateral meshes of point sets[END_REF][START_REF] Bern | Quadrilateral meshing by circle packing[END_REF], square packing [START_REF] Shimada | Quadrilateral meshing with directionality control through the packing of square cells[END_REF], advancing front [START_REF] Owen | Q-morph: an indirect approach to advancing front quad meshing[END_REF], conversion [START_REF] Borouchaki | Adaptive triangular-quadrilateral mesh generation[END_REF], decimation [START_REF] Daniels | Localized quadrilateral coarsening[END_REF], Morse-Smale complexes [START_REF] Dong | Spectral surface quadrangulation[END_REF][START_REF] Zhang | A wave-based anisotropic quadrangulation method[END_REF][START_REF] Huang | Spectral quadrangulation with orientation and alignment control[END_REF], clustering [START_REF] Boier-Martin | Parameterization of triangle meshes over quadrilateral domains[END_REF][START_REF] Lévy | L p centroidal voronoi tessellation and its applications[END_REF], local and global operators [START_REF] Pietroni | Almost isometric mesh parameterization through abstract domains[END_REF][START_REF] Lai | An incremental approach to feature aligned quad dominant remeshing[END_REF], whisker weaving [START_REF] Wolfenbarger | A global minimization-based, automatic quadrilateral meshing algorithm[END_REF], medial axis [START_REF] Quadros | Laytracks: A new approach to automated quadrilateral mesh generation using medial axis transform[END_REF], streamlining [START_REF] Alliez | Anisotropic polygonal remeshing[END_REF][START_REF] Marinov | Direct anisotropic quad-dominant remeshing[END_REF] and parameterization [START_REF] Ray | Periodic global parameterization[END_REF][START_REF] Tong | Designing quadrangulations with discrete harmonic forms[END_REF][START_REF] Bommes | Mixed-integer quadrangulation[END_REF]. Among these approaches, some favor the conformance of the final mesh to an input cross field either by construction [START_REF] Marinov | Direct anisotropic quad-dominant remeshing[END_REF], or by solving for the smoothest cross field given a set of orientation constraints [START_REF] Bommes | Mixed-integer quadrangulation[END_REF]. Conformance to an input sizing field is either derived from the triangle mesh before conversion [START_REF] Borouchaki | Adaptive triangular-quadrilateral mesh generation[END_REF], or encoded in a density function before clustering [START_REF] Lévy | L p centroidal voronoi tessellation and its applications[END_REF]. Mesh regularity is controlled either explicitly by interactively placing a small number of irregular vertices before parameterization [START_REF] Tong | Designing quadrangulations with discrete harmonic forms[END_REF], or indirectly through a smooth cross field [START_REF] Bommes | Mixed-integer quadrangulation[END_REF]. Regularity may be improved a posteriori through, e.g., grid-preserving operators so as to generate simple base complexes [START_REF] Bommes | Global structure optimization of quadrilateral meshes[END_REF]. Strict local conformance to both sizing and cross fields is notoriously delicate for most approaches which involve a global variational formulation, and almost none of the fine-to-coarse approaches based on decimation [START_REF] Daniels | Localized quadrilateral coarsening[END_REF][START_REF] Pietroni | Almost isometric mesh parameterization through abstract domains[END_REF] leads to meshes that conform to both sizing and cross fields.

Contribution

In this paper we present a simple and practical isotropic quadrangle meshing algorithm for arbitrary 2D domains. We place a particular emphasis on having the resulting mesh conforming to both a sizing and a cross field. Our approach differs from previous work in that its methodology can be seen as antithetical to (e.g., Delaunay) refinement algorithms that locally refine a mesh one element at a time until all quality criteria are met. Rather than continuously fixing the element that most violate any of the requirements as in the Delaunay refinement strategy, we instead enhance the mesh by carefully addressing one requirement at a time, in a processing order designed not to undo previous enhancements. Table 1 describes how each step of the algorithm improves the various desirable quality criteria for flat 2D meshes.

Size Shape Orientation Degree Regularity

Initialization (2.2) • Relaxation (2.3) • Conforming relaxation (2.4) • • • • • Local parameterizations (2.5) • • • • Barycentric subdivision (2.6) • × • • Smoothing (2.7) • • • • .
Table 1. Each step of the algorithm improves different quality criteria. •: criterion is partially met; : criterion is met; ×: criterion may not be preserved; •: criterion is preserved.

Algorithm

The input of our algorithm is a closed domain, together with a sizing and a cross field. The sizing field is either user specified or automatically computed as a Lipschitz function from the local feature size estimate of the domain boundary [START_REF] Antani | Mesh sizing with additively weighted Voronoi diagrams[END_REF]. The cross field is either specified by the user or automatically computed as the smoothest field that is tangential to the domain boundary [13]. Note that the mixed-integer approach [START_REF] Bommes | Mixed-integer quadrangulation[END_REF] could also be used. The algorithm mainly proceeds by clustering and local parameterizations over a fine isotropic background triangle mesh M obtained by Delaunay refinement. We give pseudo-code of the algorithm below, while Figure 1 provides a visual depiction of its main steps.

Preliminaries

The main steps of the algorithm (from initialization to local parameterizations through relaxations) act on the background mesh M. In the reminder of this paper, we will make heavy use of the following terms (see Figure 2):

• Tile. A tile is a union of triangles of M defining a subdomain homeomorphic to a disk. • Meta-vertex. A meta-vertex is a vertex of M which is either incident to 3 or more tiles inside the domain, or incident to 2 or more tiles on the domain boundary, or incident to a single tile when the vertex is tagged as a corner boundary vertex. The initialization clusters background mesh triangles (e) so that the tiling roughly meets the size and shape criteria; A relaxation (f) then improves the tiling for shape and orientation while preserving size; A conforming relaxation (g) improves the degree of the tiles and the regularity of the tiling; A series of local parameterizations (h) further improves the degrees and regularity; Barycentric subdivision (i) generates a pure quadrangle mesh; Smoothing (j) finally improves the shape of the quadrangles. We depict the conformance to the sizing field with a color ramp ranging from white to blue (resp. white to red), for elements smaller (resp. larger) than the specified sizing field (k). We depict the conformance to the cross field with a color ramp ranging from white to gray (l). Irregular vertices are outlined in red for degree excess and in blue for degree deficit. We show the distribution of angles, as well as distributions measuring conformance to sizing and orientation. 1000 quadrangles, total time: 40 s.

• Meta-edge. A meta-edge connects two meta-vertices through an edge path of M such that all edges along the path are incident to 2 tiles in the interior, or 1 tile on the boundary. A tile is thus surrounded by a cycle of meta-edges. • Side. A side is a chain of meta-edges around a tile. In particular, assuming a tile is surrounded by a cycle of at least 4 meta-edges, a subset of four of its meta-vertices can be chosen to represent quadrangle corners so that the tile has 4 sides. This assignment of sides will be useful in our algorithm since we target quadrangle elements.

Initialization

We construct the background mesh M as a 2D constrained Delaunay triangulation that fits a polyline approximating the domain boundary. We then refine M through Delaunay refinement [START_REF] Shewchuk | Delaunay refinement algorithms for triangular mesh generation[END_REF][START_REF] Rineau | A generic software design for delaunay refinement meshing[END_REF] until all triangles are both well shaped (isotropic) and sized in accordance to the input sizing field within a small fraction (by default 0.1).

The initialization step aims at generating a tiling that conforms to the sizing field, and that roughly meets the shape criterion (see Table 1). First, we proceed by generating one tile per triangle of M. We then recursively merge pairs of tiles using a priority queue of merging operations until the sizing field requirement is met (see Figure 1(e)). In order to favor isotropic tiles the merging operations are popped out of the queue in order of decreasing compactness, the latter being defined as the ratio between area and squared perimeter. We have experimented with another score which favors squares instead of discs but the simpler compactness score is sufficient. The idea of this step is similar to [START_REF] Garland | Hierarchical face clustering on polygonal surfaces[END_REF].

Relaxation

The first relaxation step aims at optimizing the initial tiling for the shape and orientation criteria while preserving the size (see Table 1). Optimization is carried out through a discrete clustering algorithm which operates over the background mesh so as to favor squares tiles which are both well sized and well oriented with respect to the sizing and cross fields ( [START_REF] Valette | Generic remeshing of 3d triangular meshes with metric-dependent discrete voronoi diagrams[END_REF]).

Similar in spirit to [START_REF] Lévy | L p centroidal voronoi tessellation and its applications[END_REF], we consider the L R ∞ metric related to a local Cartesian coordinate frame R = (u, v) specified by the cross field. We compute an approximate discrete Voronoi diagram over the background mesh instead of computing the exact continuous Voronoi diagram [START_REF] Boissonnat | Voronoi diagrams in higher dimensions under certain polyhedral distance functions[END_REF], so as to deal with a simple triangle labeling procedure. The L R ∞ distance between two points p, q ∈ R 2 is defined as:

d R ∞ (p, q) = max (|(p -q) • u|, |(p -q) • v|) .
Using the continuous formulation the minimized energy is defined as:

G {z i } N i=1 , {V i } N i=1 = N i=1 V i ρ(x) d R ∞ (x, z i ) dx,
where ρ is a density function defined on the domain: ρ(x) = s(x) -4 (s denotes the sizing function to be preserved); z i is a point generator and V i is the Voronoi cell of z i . Using a discrete formulation [START_REF] Valette | Approximated centroidal voronoi diagrams for uniform polygonal mesh coarsening[END_REF] now involving the background mesh M we consider a set of triangle generators {g i } N i=1 , and a tiling {T i } N i=1 . The energy to minimize is defined as:

H {g i } N i=1 , {T i } N i=1 = N i=1          t j ∈T i ρ(c(t j ))area(t j )d R ∞ (c(t j ), c(µ(T i )))          ,
where c(t) denotes the centroid of triangle t and µ denotes the triangle that contains the centroid of a tile. Energy H is iteratively minimized by alternating L R ∞ discrete Voronoi partitioning and relocation of the generators to their tile centroids. Algorithm 2 provides a pseudo-code for the relaxation step, where N is the number of tiles and K is the total number of iterations.

Algorithm 2: Relaxation

Input: Initial triangle generators {g 0 i } N i=1 and corresponding tiles

{T 0 i } N i=1 . begin while no convergence do Discrete partitioning {T k i } N i=1 partition associated to {g k i } N i=1
Relocate generators to centroids ∀i,

1 ≤ i ≤ n, g k+1 i = Γ(g k i )
Output: Optimized triangle generators {g K i } N i=1 and corresponding tiles

{T K i } N i=1 .
Discrete partitioning is achieved by flooding the domain one triangle at a time from their generators with a dynamic priority queue [START_REF] Cohen-Steiner | Variational shape approximation[END_REF]. Each tile is initialized to be its triangle generator, and a priority queue is filled with (up to three) incident triangles (candidates for flooding) per generator. In order to favor square tiles, the triangles are popped out of the queue and added to tiles in increasing order of L R ∞ distance to their respective triangle generator, where R is the local Cartesian coordinate frame specified by the cross field at the generator centroid.

Relocation is achieved through computing the (triangle) center of mass of each tile. In the continuous case the center of mass of a tile does not depend on the metric chosen to measure the object, and we observe a very similar behavior in our discrete setup. After the k-th iteration, for each tile T k i , we choose to find the triangle that minimizes the following energy:

f (t) = t j ∈T k i ρ(c(t j )) area(t j ) ||c(t j ) -c(t)|| 2 2 ,
where t ∈ T k i . We denote by Γ : g k i → g k+1 i the operation that computes the triangle centroid of the tile T k i associated to g k i . In such discrete algorithm convergence is reached when ∀i,

1 ≤ i ≤ N, Γ(g k i ) = g k i .
As convergence for the L R ∞ metric on arbitrary domains is not guaranteed, we also specify a maximum number of iterations (by default set to 50). Figure 3 depicts some iterations. As expected, the relaxation leads to a tiling of the domain with mostly square tiles which are well oriented and well sized (similar in spirit to a square packing approach [START_REF] Shimada | Quadrilateral meshing with directionality control through the packing of square cells[END_REF]), even if a varying sizing field inevitably implies shape distortion. We further observe in Figure 4 that although well shaped, the tiles are generally not conforming (see the many T-junctions), hence most of them would generate degree-6 polygons and the final mesh would contain many irregular vertices (generally of degree 3). We describe next a conforming relaxation procedure which aims at generating quasi 2-conforming configurations.

Conforming Relaxation

A closer look at Figure 4 reveals that the tiling is in general already conforming in one direction. We call this configuration 1-conforming. This is explained by the fact that a general tiling of the plane with equally sized square tiles is 1-conforming (see Figure 5, left). Our goal is to further relax the tiling so as to obtain quasi 2conforming configurations (5, middle).

An intuitive understanding of the conforming relaxation procedure can be gathered by looking at Figure 5(left) and realizing that we could shift the three square columns vertically as little as possible so as to tend toward a perfect 2-conforming configuration. In order to obtain the aforementioned shift, we propose to simply shift the centroid during the relocation of a relaxation iteration. The only remaining technicality now resides in the way to compute the shift. Although simple at first glance, recovering the local 1-conforming direction (vertical or horizontal) is already non trivial, and so is finding the shift magnitude and orientation (see Figure 6). In addition, both size and cross fields vary over the domain, requiring not just shifting the tiles but also sacrificing aspect ratio to reach conforming. Tiles which are already quadrangles or triangles (with 4 or 3 meta-vertices) are not shifted. While we assume that after relaxation all tiles are squares (geometry-wise), in general they are degree-6 tiles (see Figure 5(a)). We thus first need to choose which four of its metavertices form a square locally aligned to the cross field. We first compute the triangle centroid of the tile as during relaxation (see 2.3). Its cross (given by the input cross field) is now taken as local Cartesian coordinate frame R = (u, v). Denote by v 1 , . . . , v p (p > 4) the meta-vertices of the tile, ordered by circulating along the tile boundary.

To decide which of the meta-vertices are chosen as (ordered) corners c 1 , c 2 , c 3 , c 4 we maximize through dynamic programming the alignment of the sides (see 2.1) with the axes of R through maximization of the following energy:

E = max a⊂{u,v} {c 1 ,c 2 ,c 3 ,c 4 }⊂{v 1 ,...,v p } min |(c 2 -c 1 ) • a|, |(c 3 -c 2 ) • a 90 |, |(c 4 -c 3 ) • a|, |(c 1 -c 4 ) • a 90 | ,
where a 90 denotes vector a (which stands for u or v) rotated by 90 degrees.

In addition to assigning corners, the maximum of E for each side provides a local reference direction of the cross field (u or v). For each meta-edge, we then compute the length of its projection on its reference direction (u or v). Among all meta-edges, we then select only the ones with exactly one end meta-vertex coinciding to a corner meta-vertex, and pick the longest one, denoted by e. We then shift the centroid along a line parallel to a, in the opposite direction to the corner meta-vertex of e. The magnitude of the shift is chosen as a fraction (0.2 in our experiments) of the length of the projection of e (see inset). The number of iterations of conforming relaxations is a user-specified parameter (set to 20 for all examples shown). Figure 7 depicts how such conforming relaxation brings some initial 1-conforming configurations to quasi 2-conforming configurations. The latter can be fixed by a series of local parameterizations which we describe next.

Local parameterizations

The previous conforming relaxation step favored quasi-2-conforming configurations (Figure 5(b)) that we wish to turn into 2-conforming configurations (Figure 5(c)) so as to improve both degree and regularity criteria (see Table 1) while not negatively affecting the previous efforts made by previous steps.

Figure 7 reveals that the many quasi-2-conforming configurations exhibit similar topological arrangements of the tiles. We call butterfly a set of four tiles incident to a short meta-edge connecting two degree-3 meta-vertices (see Figure 8). Inspired by [START_REF] Pietroni | Almost isometric mesh parameterization through abstract domains[END_REF] we remove many of these butterfly configurations through local parameterizations on square domains, which merge 2 degree-3 meta-vertices into one degree-4 meta-vertex.

For each butterfly, we consider the union of its four tiles A, B, C, D as a subdomain Ω ABCD , and first perform a classification of its meta-vertices as depicted in Fig. 7. Conforming relaxation with uniform sizing (left) and non-uniform sizing (right). From top to bottom: initial, 1 iteration, 5 iterations and 10 iterations. In order to bring some initial 1-conforming configurations to quasi 2-conforming configurations, centroids of two adjacent tiles are locally shifted, which can induce size distortion. Fig. 8. Butterfly configuration. Left: a short meta-edge connects two degree-3 meta-vertices (outlined in black) which share four tiles. Middle left: among all meta-vertices of the four-tile subdomain we identify two inner meta-vertices (black), 4 interface meta-vertices separating the four tiles A, B, C, D pairwise (red), four corner meta-vertices (white) and the remaining meta-vertices (yellow). Middle right and right: the four tiles parameterized on a square. The parameterization preserves the topology of the interfaces on the boundary of the four tile subdomain. Some tiles may be smaller in parameter space so as to better fit the sizing field.

Figure 8. While the inner (black) and interface (red) meta-vertices can be easily classified from topological (adjacency) relationships between the tiles, distinguishing the four corners among the other meta-vertices (possibly many for rapidly varying sizing fields) requires incorporating a geometric criterion related to the local orientation of the tiles.

Contrary to the way we choose the corners for the conforming relaxation step, the orientation is, this time, not given by the cross field but instead depends on the relaxed four tiles (the rationale being, once again, to avoid undoing the previous enhancements). Denote by V AB , V BC , V CD , V DA the four interface meta-vertices. We estimate a reference Cartesian frame by fitting two lines through principal component analysis to the segment sets [START_REF]Principal component analysis in CGAL[END_REF]. Among these two lines the most reliable one (i.e., the line with minimum variance in the orthogonal direction) is chosen as reference direction. To select four corners among the meta-vertices we again resort to a dynamic programming approach similar to the one used in Section 2.4.

([V AB , V B ], [V CD , V D ]) and ([V DA , V D ], [V BC , V B ])
Our goal is to parameterize Ω ABCD on a square domain and to label its triangles (again with labels A,B,D,C) such that i) the chosen corner meta-vertices coincide with the corners of the square, ii) the interfaces at the boundary of Ω ABCD are preserved, iii) the two inner meta-vertices are merged into a degree-4 vertex and iv) Ω ABCD is partitioned with 4 tiles with a disk topology. Call φ the parameterization that maps Ω ABCD on the square domain. We first constrain the whole boundary of φ(Ω ABCD ) so as to respect the chosen corners and using an arc-length parameterization in-between these corners. We then parameterize Ω ABCD using the mean value coordinate approach [START_REF] Floater | Mean value coordinates[END_REF] and compute, in parameter space, the intersection point φ(v * ) between the two line segments (φ(V AB )φ(V CD )) and (φ(V BC )φ(V DA )). The nearest vertex (of degree at least 4) from φ(v * ) is then chosen as inner vertex, which means that the issuing vertex v * will be the center degree-4 meta-vertex. While simple at first glance once the inner and boundary vertices are decided upon, a naive triangle labeling step based on localization within quadrilaterals in parameter space can lead to improper topological partitioning. For this reason we trace four edge paths from φ(v * ) to φ(V AB ), φ(V BC ), φ(V CD ), φ(V DA ) in order of increasing length (shorter segments first) so as to determine proper interfaces between the triangle labels. These edge paths are constrained not to intersect except at the inner vertex and to connect no other boundary vertices than their target vertex V AB , V BC , V CD , V DA . Upon successful completion the triangles of Ω ABCD are labeled, and this ends the butterfly removal procedure.

To avoid distorting the shape and orientation of the final mesh, we only remove butterflies whose inner meta-edge length is smaller than a fraction of the local sizing field (0.5 in our experiments). We use a dynamic priority queue to gracefully deal with butterflies in order of increasing inner meta-edge length. Figure 9 depicts how local parameterizations improve degree and regularity criteria for both uniform and non-uniform sizing field cases.

Barycentric Subdivision

After conforming relaxation there is no guarantee that all tiles are quadrangles. Experimentally we obtain an order of 75% quadrangles for uniform sizing and of 60% quadrangles for non-uniform sizing. We thus resort to barycentric subdivision (see Figure 1(i)) in order to meet the degree criterion (Table 1). The edge lengths of the quadrangles are reduced by a factor of two compared to the mesh before barycentric subdivision. This factor is taken into account during all previous steps of the algorithm.

Smoothing

Near T-junctions the previous barycentric subdivision step generates highly distorted quadrangles. We thus resort to a few iterations of Laplacian smoothing (see Figures 1(j) and 10) so as to enhance the shape criterion.

Results

Our algorithm is implemented in C++ using the CGAL library [10]. All examples were computed on a 2.40GHz PC with a single thread. Each of our figures highlights irregular vertices in blue for degree deficit, and in red for degree excess. To depict conformance to the input cross field we use a color ramp ranging from white to gray, where white means perfect conformance and gray means 45 degree distortion.

To depict conformance to the input sizing field we use a color ramp ranging from white to blue (resp. red) for tiles smaller (resp. larger) than targeted. We also depict distributions of angles, conformance to sizing, and conformance to cross field. We ran a large number of examples of various size and complexity in order to assess our results and compare them to state-of-the-art methods. Figure 11 depicts how a final quadrangle mesh conforms to both size and orientation on a geographic map. The input sizing field is set to be the largest 1-Lipschitz function constrained to match the local feature size estimate of the input domain boundary [START_REF] Antani | Mesh sizing with additively weighted Voronoi diagrams[END_REF]. The cross field is set to be the smoothest cross field constrained to be tangential to the input domain boundary. The distribution of angles is shown, with over 80% of angles within the interval [75 -105]. The algorithm takes 350 seconds, with two third of the time spent on local parameterizations. The peak memory usage is 200 MBytes. Figure 12 depicts an example with a constant cross field combined with a rapidly varying sizing field. Irregular vertices inevitably appear between dense and sparse mesh areas, and the orientation is partially distorted. Figure 13 shows a trivial domain example with a constant cross field and compares uniform vs. non-uniform sizing. Figure 14 depicts examples of uniform sizing and varying cross fields set to be smooth and tangential to the input domain boundary.

Figure 15 compares our results with [START_REF] Bommes | Mixed-integer quadrangulation[END_REF] using a uniform sizing as this method is not primarily aimed at handling rapidly varying sizing fields. Our approach better preserves orientations near the domain boundary at the price of a larger number of irregular vertices.

We do not provide direct control over the final number of vertices of the final mesh as it depends on both the input sizing field and the number of degree-4 tiles after local parameterizations (experimentally near 70%). The efficiency of the algorithm can be improved by accelerating the relaxation step [START_REF] Valette | Approximated centroidal voronoi diagrams for uniform polygonal mesh coarsening[END_REF] and by numerical optimizations during the local parameterizations. . The cross field is set to be tangential to the input domain boundary. The meshes produced by [START_REF] Bommes | Mixed-integer quadrangulation[END_REF] are more regular at the price of increased shearing distortion.

Conclusion

We proposed a principled approach for the automatic generation of quadrangle meshes from arbitrary 2D domains. The main methodology consists of enhancing a rough initial tiling by carefully addressing one meshing requirement at a time (size, shape, orientation, degree, regularity) in an order designed not to undo previous enhancements. While other similar approaches (such as [START_REF] Shimada | Quadrilateral meshing with directionality control through the packing of square cells[END_REF]) only show examples on simple domains and smooth cross fields, our experiments confirm that the output quadrangles meshes conform both to the input sizing and cross fields, even on complex domains and for rapidly changing fields. Size and orientation conformance comes at the price of a larger number of irregular vertices. For applications requiring coarse base complexes, we could potentially improve our approach by allowing the user to trade conformance to input fields for increased regularity. We also wish to improve the conforming step by resorting to a different strategy when it fails locally, and the parameterization step by it more general for an arbitrary set of tiles clustered around a target irregular vertex.

Our approach is primarily based on relaxation and local parameterizations. Because the main steps of the algorithm only deal with labeling the triangles of a background triangle mesh, our implementation is simple and reliable, and is less prone to numerical robustness issues. Resorting to local parameterizations on trivial convex domains such as [START_REF] Pietroni | Almost isometric mesh parameterization through abstract domains[END_REF] provides us with scalability and robustness. For these reasons our approach could be extended to reliable quadrangle surface remeshing (including anisotropic quadrangle surface remeshing). Most of the main steps also seem to generalize to hexahedron domain tiling, except for the barycentric subdivision step. As future work we plan to alleviate this problem so as to extend the main principles behind our approach for hexahedron domain tiling.

Fig. 1 .

 1 Fig. 1. Overview. The algorithm takes as input a closed domain (a), a sizing field (b) and a cross field (c). It then operates on a triangle background mesh refined according to the sizing field (d).The initialization clusters background mesh triangles (e) so that the tiling roughly meets the size and shape criteria; A relaxation (f) then improves the tiling for shape and orientation while preserving size; A conforming relaxation (g) improves the degree of the tiles and the regularity of the tiling; A series of local parameterizations (h) further improves the degrees and regularity; Barycentric subdivision (i) generates a pure quadrangle mesh; Smoothing (j) finally improves the shape of the quadrangles. We depict the conformance to the sizing field with a color ramp ranging from white to blue (resp. white to red), for elements smaller (resp. larger) than the specified sizing field (k). We depict the conformance to the cross field with a color ramp ranging from white to gray (l). Irregular vertices are outlined in red for degree excess and in blue for degree deficit. We show the distribution of angles, as well as distributions measuring conformance to sizing and orientation. 1000 quadrangles, total time: 40 s.

Fig. 2 .

 2 Fig. 2. Terminology defined in Section 2.1.

Fig. 3 .

 3 Fig. 3. Discrete L R ∞ iterations with a non-uniform sizing field (the cross field is shown).

Fig. 4 .Fig. 5 .

 45 Fig. 4. After relaxation, for uniform (top) and non-uniform (bottom) sizing field. Tiles are in general conforming to one direction of the local cross field. The black lines depict some local conforming directions.

Fig. 6 .

 6 Fig. 6. Shifting centroids. We depict examples of shifts with increasing ambiguity. A tile which is already quadrangle is not shifted (left). A tile with one side split into two meta-edges (middle left), the chosen quadrangle corners are depicted in black. A tile with two parallel sides split (middle). A tile with three sides split (middle right). Another ambiguous case (right).

Fig. 9 .

 9 Fig. 9. Local parameterizations. Uniform sizing field (top left and top right) and non-uniform sizing field (bottom left and bottom right) before (left) and after (right) local parameterizations. The cross field is shown.

Fig. 10 .

 10 Fig. 10. Smoothing. Mesh before (a) and after smoothing (b). We compare the distribution of angles before (blue) and after (red) smoothing.

Fig. 11 .Fig. 12 . 2 , 0 < x < 1 Fig. 13 .

 111220113 Fig. 11. France. The final mesh (a) contains 77% regular vertices (b). It conforms to the sizing field (c) as well as to the cross field (d). The close-up depicts an area where size and orientation vary rapidly. 4500 quadrangles, total time: 300 s.

Fig. 14 .

 14 Fig. 14. Non-uniform cross fields. Sizing fields are uniform. Total time: 45 and 450 s.

Fig. 15 .

 15 Fig.[START_REF] Floater | Mean value coordinates[END_REF]. Comparison with Bommes et al.[START_REF] Bommes | Mixed-integer quadrangulation[END_REF]. The cross field is set to be tangential to the input domain boundary. The meshes produced by[START_REF] Bommes | Mixed-integer quadrangulation[END_REF] are more regular at the price of increased shearing distortion.

A degree-k vertex has k adjacent inside cells.

Acknowledgments

The authors are grateful to Mathieu Desbrun for discussion and advice, and to David Bommes for experiments. This work was funded by the European Research Council (ERC Starting Grant 'Robust Geometry Processing', Grant agreement 257474), and by a French ANR Grant (GIGA ANR-09-BLAN-0331-01).


Isabelle Llorens

Eric Lahera

William Delnet

Olivier Proux

Aurélien Braillard

Jean-Louis Hazemann

Alain Prat

Denis Testemale

Quentin Dermigny

Frederic Gelebart

Marc Morand

Abhay Shukla

Nathalie Bardou

Olivier Ulrich

Stéphan Arnaud

Jean-François Berar

Nathalie Boudet

Bernard Caillot

Perrine Chaurand

Jérôme Rose

Emmanuel Doelsch

Philippe Martin

Pier Lorenzo Solari

High energy resolution five-crystal spectrometer for high quality fluorescence and absorption measurements on an X-ray Absorption Spectroscopy beamline

Fluorescence detection is classically achieved with a solid state detector (SSD) on X-ray Absorption Spectroscopy (XAS) beamlines. This kind of detection however presents some limitations related to the limited energy resolution and saturation. Crystal analyzer spectrometers (CAS) based on a Johann-type geometry have been developed to overcome these limitations. We have tested and installed such a system on the BM30B/CRG-FAME XAS beamline at the ESRF dedicated to the structural investigation of very dilute systems in environmental, material and biological sciences. The spectrometer has been designed to be a mobile device for easy integration in multi-purpose hard X-ray synchrotron beamlines or even with a laboratory X-ray source. The CAS allows to collect X-ray photons from a large solid angle with five spherically bent crystals. It will cover a large energy range allowing to probe fluorescence lines characteristic of all the elements from Ca (Z = 20) to U (Z = 92). It provides an energy resolution of 1-2 eV. XAS spectroscopy is the main application of this device even if other spectroscopic techniques (RIXS, XES, XRS...) can be also achieved with it. The performances of the CAS are illustrated by two experiments that are difficult or impossible to perform with SSD and the complementarity of the CAS vs SSD detectors is discussed.

I. INTRODUCTION

The BM30B/CRG-FAME 1 beamline at the European Synchrotron Radiation Facility (ESRF, Grenoble, France) is dedicated to X-ray absorption spectroscopy (XAS) applied to a wide variety of research fields: condensed matter physics, materials science, biophysics, chemistry and mainly geochemical topics, with special emphasis on highly diluted samples.

As such, the optics of the beamline has been designed to optimize both the incident photon flux on the sample and the optical stability to reduce non-statistical noise. A 30-element used for an optimal acquisition of fluorescence signal. As we show in this article, a way to improve the fluorescence detection significantly in the case of complex or highly diluted samples (see § III) is to use a crystal analyzer spectrometer. With this aim, a focusing Johann type spectrometer has been built and recently commissioned on the beamline so as to improve both the sensitivity in terms of sample concentration and the signal quality.

In general, it is considered that such a spectrometer can be used in the following fields:

(i) X-ray absorption fine structure (XAFS) spectroscopy in samples with multi elemental composition matrices, (ii) lifetime-broadening-removed XAFS spectroscopy 2 , (iii) siteselective XAFS spectroscopy 3 , (iv) X-ray emission spectroscopy (XES) 4 , (v) resonant inelastic X-ray scattering (RIXS) 5 , (vi) X-ray Raman scattering (XRS) 6,7 . A complete overview of possible applications can be found in reviews by de Groot 8 , Kotani and Shin 9 , Schülke [START_REF] Schülke | Series of Synchrotron Radiation 7[END_REF] and Rueff and Shukla [START_REF] Rueff | [END_REF] . In our case, the spectrometer is dedicated for an improvement of fluorescence detection to improve signal to noise in XAFS measurements and mostly used to distinguish a weak fluorescence emission signal from a multitude of undesirable intense peaks.

A description of the prototype of the present spectrometer which used solitary crystal analyser has already been published 12 as well as results of experiments performed with it 13 .

The main limitation of this prototype is the small solid angle of detection corresponding to one crystal (0.03 sr) and thus the limited counting statistics of the measurement. To overcome this, a new spectrometer including five bent crystals has been designed and installed on the beamline. With this design, the integrated solid angle of detection is increased to 0.15 sr. In this paper, we report the complete original design of this spectrometer and results from first experiments.

II. SOURCE

The spectrometer (CAS) is installed on the BM30B/CRG-FAME beamline at the ESRF (schematic description in FIG 1). The source is a 0.8 T bending magnet (critical energy E c =20 keV giving a maximum of the photons flux around 17 keV). The maximal horizontal divergence integrated by the optical components is 2 mrad. The main optical elements are two parabolic Rh-coated mirrors and a liquid-nitrogen-cooled Si (220) pseudo-channelcut doublecrystal monochromator 14 . The spot size (300 x 100 µm 2 , HxV FWHM) is kept constant on the sample with (i) the dynamical sagittal focusing of the second crystal of the monochromator 15 , in the horizontal plane, and (ii) the dynamic adjustment of the height of the experimental bench during an energy scan which compensates the vertical deviation of the beam.

The energy resolution is close to the intrinsic value of the monochromator crystal 14 , i.e. keV (for 1.5 mrad horizontal divergence). Finally, for a given energy, the absolute noise on the intensity of the incident beam ranges between 0.02 and 0.05% for a 1s integrating time.

III. LIMITATIONS OF A SOLID STATE DETECTOR

Energy dispersive solid-state detectors (SSD) are classically used on most of X-ray Absorption Spectroscopy beamlines to record spectra in fluorescence detection mode.

Different commercial detectors are available; mainly high-purity germanium (HPGe, cryogenically cooled, optimized for hard X-rays) and silicon drift detectors 16 (SDD, Peltier cooled, optimized for soft to hard X-rays). These solid state detectors are usually easy to use and they allow to probe preferentially diluted elements. The input count rate can be limited by the high photon flux delivered by third generation synchrotron beamlines, but the maximum counting rate can be increased in some cases using an appropriate dead-time correction 16 .

Nevertheless, several limitations for optimal XAS acquisition still occur.

A. Saturation

The use of a solid state detector can find limitations when a high counting rate due to the scattered beam and/or a fluorescent signal from the matrix containing the diluted specimen does not allow detection the signal of interest. For example, in the particular case of Co adsorbed on iron oxide nanoparticles, the absorption spectrum at the Co K-edge (K 1 = 6.930 keV) is difficult to measure with an energy-dispersive detector because the Fe K fluorescence lines (7.058 keV) produce a large signal which saturates the detector.

B. Energy resolution

The second limitation comes from the SSD energy resolution which is fundamentally limited to about 120 eV (FWHM) at 6 keV (Fano statistics) 17 and can be experimentally approached only for low counting rates (i.e. large shaping time). A typical energy resolution for a XAS experiment ranges around 150-300 eV, the choice in this range depending on the compromise between an optimal counting rate and a reasonable energy resolution. This energy resolution can be improved by replacing a "conventional" SSD with a superconducting tunnel junction cryogenically cooled detector with an energy resolution ~10-20 eV 18 .

However, the complexity of these detectors, mainly due to the required very low temperature of the sensor area (around 100-500 mK), actually limits their use to particle physics and astrophysics.

This limited energy resolution induces a partial overlap of the measured signal with the low-energy tail of the scattered beams (elastic or Thompson scattering, inelastic or Compton scattering...). An illustration (FIG. 2) is given by the study of bromide aqueous solutions at 8ppm (0.1 mM), 40ppm (0.5 mM) and 80ppm (1 mM). The sample-holder is a glassy carbon cell located inside a high pressure vessel equipped with 1.5 mm Be windows for incident, transmitted and fluorescence beams 19 . The Br K peaks intensity is of course related to the amount of Br. Consequently, the fluorescence signal for low Br concentrations becomes significantly small with respect to the tail of the Compton peak. It is generally considered that the signal should be at least 3 times the background, and then the Br concentration detection limit is about 23ppm (0.3 mM). This value is consistent with lowest concentrations previously mentioned for XAS measurement on BM30B/CRG-FAME at ESRF 1 and on BioCAT at APS 20 .

C. Spatial resolution

Solid state detectors do not have any spatial detection resolution. Thus, any fluorescence, elastic and inelastic scattering signal from a sample holder, or more generally from the experimental setup, cannot be filtered. One solution is to install fluorescence soller slits between the sample and the detector but this does not give significant improvement 21 .

IV. SPECTROMETER

A. Mechanics

The spectrometer has been entirely designed by the staff of the BM30B/CRG-FAME beamline and built at the Néel Institute (CNRS, Grenoble).

During the design, emphasis was given to user-friendly operation and on high adaptability of the sample environment. Different views of the spectrometer are shown in FIG 4: a top-view of the final drawing, a detailed view of the mechanical assembly of crystal and a 3D view with the 5 Rowland circles that intersect at the sample and detector points. The main parameters of the spectrometer and characteristics are summarized in Table 1. As already mentioned 12 , all the different mechanical motions are achieved using standard linear (y and z) and rotation ( and ) motorized stages. The role of the rotation stages is to align vertically the bent crystal, i.e. place the crystal in a normal position to the y axis. These adjustments -verticality and normality -of the crystals are achieved during the preliminary alignment procedure; these motions are not used during energy scans. Each crystal is therefore always in vertical position. For this reason, the mechanical angular ranges of the rotation stages are limited to ±2°. The technological solutions for the hinges are weak link systems as they allow a precise positioning, without any mechanical clearance. Following the same idea, the individual height of the 5 crystals can be finely adjusted (z motion, FIG.

3) in order to compensate for small variations of the crystals characteristics, such as the radii of curvature. The Bragg energy selection is then achieved using only the linear motions along the perpendicular beam axis (y) and vertical axis (z). For this purpose, large high-load linear motions are used for the long-range linear movements. The five y-translation motions are fixed on an aluminum alloy (Fortal) plate with a 27.5° offset angle between two adjacent translations. The detector is placed on 4 motorized motions (1 rotation and 3 translations) to position it at the desired angle, just above the sample and at the focal point. With such mechanical and technical characteristics, the attainable Bragg angle ranges from 45 to 86°. The lower limit of 45° was arbitrary fixed during the conception phase. For such low Bragg angles, the energy resolution of the spectrometer is poor due to the Johann geometry error (see for example ref 27). The higher limit is a consequence of the Rowland circle geometry used: for a Bragg angle of 90°, the ideal detector position is on the sample.

Thus, due to the spatial clearance necessary to install large sample environment set-ups, we set the higher limit to 86°.

Such a geometry (each crystal is maintained vertically, the detector is above the sample), allows to increase the number of crystals symmetrically around the axis sample-detector. The different Rowland circles intersect then exactly at the same points, the sample and the detector, without any additional angular adjustment. This geometry is also used for example in a 5-crystal 22 , 14-crystal 23,24 and 40-crystal 24

B. Optics

Different geometries exist depending on the application: Cu foil absorber coupled to a point focusing spatial filter 26 for XRS, Bent crystal Laue analyzer in a logarithmic spiral shape 27,28,29,30 (Zhong, Kropf, Adams, Kalaja,) for XAFS. We choose to work with spherical bent crystals in the Johann geometry 31 .

Spherically bent crystals (0.5 m bending radius) with a diameter of 0.1 m are used as dispersive optical elements in the spectrometer. The crystals are assembled as already described by Collart and co-workers 31 . High resolution X-ray analyzers are obtained using anodic bonding technique, which is a method of permanently joining glass to silicon without the use of adhesives. This method is a common process used in microelectronics device

C. Detection

In the Rowland circle geometry, the detector is located above the sample, along the z det axis (FIG. 4). The total path length from the sample to the diffracting bent crystal and then to the detector equals 1 m; therefore operation under helium atmosphere is compulsory in order to minimize the absorption, especially in the low-energy region (from 4 to 8 keV).

Currently, two detectors are available. The first one is a 5 mm thick NaI(Tl)

scintillator from FBM Oxford with a large active surface (7 cm 2 ), an energy resolution of 50% and a low maximum counting rate. The second detector is a silicon drift detector (SDD) from SII Nanotechnology (VORTEX-90EX  ) with a small active area (50 mm 2 ) and a high dynamic range, it is also more compact and offers a useful energy resolution (2-4%).

The use of a 2D hybrid pixel detector such as XPAD3S 32 (soon available on the beamline), Pilatus 33 , Medipix2 34 is also possible. The advantage of using 2D detectors is the possibility to have a single threshold adjustable per pixel and so to suppress the background counts (as with the SDD). Moreover such a detector can be used to isolate the appropriate signal induced by the X-ray beam -sample interaction, so as to discriminate the signals from the sample and from its container. Finally it also allows to focus the diffracted photons on different areas on this detector, to monitor each crystal separately.

D. Theoretical and experimental resolutions

The global theoretical resolution of the spectrometer includes both the incident beam characteristics (the beam vertical size, h, -E beam vertical size -) and those of the crystal (the intrinsic resolution of the chosen reflexion at the emission energy E -E reflexion -, the Johann's approximation -E Johann -):

∆ = ∆ = . √ ∆ = ² ²
Where  is the Bragg angle (°), E the fluorescence emission energy (eV), f hkl the structure factor of the reflexion, R the crystal curvature radius (m) and h the vertical beam size (mm).

The contributions being independent, the total energy resolution E CAS of the spectrometer is the convolution of all these individual contributions. By approximating all these contributions as Gaussian functions, E CAS (can be easily calculated using the following formula:

        2 2 2 2 size vertical beam Johann reflexion CAS E E E E       
The experimental resolution is determined by recording the quasi-elastic peak of the incident beam from the sample. The FWHM of the peak, fitted by a Gaussian function, gives the total (incident beam and spectrometer contribution) experimental resolution value. in the crystal structure which ultimately broaden the bandwidth of the reflection 35,36 . A solution to overcome this effect is to use diced analyzer crystals, which are built by fixing a large number, typically 10 4 , of small flat single crystals (dices) on a spherical substrate, thus providing a polygonal approximation to the Rowland circle geometry 37 . These crystals allow a better resolution: 10-300 meV for diced Si(nnn) crystals with n=3 to 7 to be compared with 500 -2000 meV for bent Si(nnn)/Si(nn0) crystals with n = 3 to 8 for the same curvature radius 38 .

The main effect of the Johann geometrical aberrations is on the energy resolution result. This is a consequence of a vertical spreading of the diffracted spot on the detector due to the incidence angle  of the X-ray arriving at the center of the bend crystal, higher than an X-ray arriving in another area of it (FIG. 3). To decrease this contribution and so to increase the energy resolution, it is possible to limit the detection area, i.e. to limit the crystal collecting area to its center.

V. EXPERIMENTAL VALIDATION

We have seen in § III several limitations of solid state detectors (saturation, energy and spatial resolution). This paragraph presents a solution to overcome some of these limitations.

The first measurements were dedicated to XAFS spectroscopy applications: (i) the matrix fluorescence removal (see § III.A) with the study of Co adsorbed on iron oxide nanoparticles and (ii) the probe of a weak fluorescence at high energy (see § III.B) in a crystalline matrix with the example of diluted Sr in UO 2 simulated irradiated nuclear fuel.

A. Beyond the saturation: Co adsorbed on iron oxide nanoparticles

During the last decade, interest in nanotechnology and manufactured nanoparticles has grown and a wide range of applications for emerging nanomaterials has been proposed. For instance differences in reactivity might be exploited to improve surface-based reactions as it could be used for arsenic removal processes. Oxyhydroxide iron particles smaller than 20 nm exhibit an increase of adsorption capacity and those of 11 nm in diameter adsorbs 3 times more As per nm 2 than 20 nm particles 39,40,41 .

In our experiment, we focus on the mechanisms of Co adsorption at the surface of nanomaghemite (-Fe 2 O 3 ). The difficulty, here, is to probe an element (Co, atomic no. Z) adsorbed on another (Fe, atomic no. Z-1) which dominates the fluorescence signal. Co Kedge total fluorescence and High Energy Resolution Fluorescence Detected (HERFD)-XANES spectra are presented in FIG. 5. The integrated counts after edge are 4.8x10 5 using SSD and 3.6x10 4 using CAS corresponding to count-rates of 8.10 4 s -1 and 300 s -1 respectively. These values do not reflect the data quality. One way to quantify the detector efficiency for a given kind of sample is then to calculate the effective number of counts (N eff ) 12 defined as background signal signal N N N  2 . For this particular system, N eff is ~150 c/s for SSD and 250 c/s for CAS. In the last case, the total count-rate is dominated by useful events (250 out of 300 s ) are effective counts. It is thus reasonable to multiply the number of acquisitions to increase the data quality. Moreover, the background on CAS spectrum is very low, constant and due to photons scattered (not diffracted) by the crystals. Inversely, the background on SSD spectrum is high and increases with energy. Independently from statistical considerations, the spectrum shape associated with very small absorption edge (~0.002) makes the normalization of SSD data very complicated for this particular system. HERFD data have been analyzed and preliminary results show that with this high surface coverage, Co atoms are adsorbed on ferrihydrite surface.

B. Energy resolution: Sr in simulated irradiated UO 2 nuclear fuel.

One of the main goals of research in nuclear energy is to improve the economic and safety performance of nuclear fuels. One solution is to extend its life time in reactor. But in this case, the behavior of fission products becomes the limiting factor and more specifically their segregation/precipitation. Thus, increasing fuel burn up must be accompanied by an effort to improve our understanding of the nature and behavior of the material as fission products accumulate. One course of action is to collect experimental data relative to irradiated fuel. Due to the very high radioactivity of the samples, this data can only be obtained through post-irradiation examination of irradiated fuels in dedicated facilities. To overcome the former difficulty, the use of simulated high burn up UO 2 fuel, termed SIMFUEL, is a good alternative 42 . This experiment has a double interest. The first one is to probe a diluted element (Sr at 1400 ppm) in a crystalline UO 2 sample doped with 11 elements (Ba, Ce, La, Mo, Sr, Y, Zr, Rh, Pd, Ru, Nd) simulating the chemical composition of irradiated nuclear fuel. The second more technical interest is to see if we can limit the impact on the XAS spectra of the Bragg peaks originating from the well crystallized UO matrix. For the experiment detailed here, we focused our interest on Sr.

The data are collected both in HERFD and total fluorescence modes at Sr K-edge. In total fluorescence detection using SSD, the XANES/EXAFS spectra cannot be exploited due to Bragg peaks in the pre-edge and at the end of the EXAFS region (FIG. 6). Several orientations of the sample relative to the incident beam are tested but provided no improvement. However, different solutions exist to reduce the spurious signal given by Bragg peaks using rotating 43 or vibrating 44 sample holder. With such systems, the Bragg peak energy position changes with the sample angular orientation. By integrating the absorption signal on different angular positions, Bragg peaks are averaged on a given energy range. They are not deleted but their effects are effectively reduced. Indeed, the photons diffracted (so elastically scattered) by sample crystallites are not detected by the CAS since their energy is different from the selected fluorescence line energy (in this case Sr K 1 ). This enables us to probe a relatively diluted element (Sr) within the UO 2 crystalline matrix.

Based on the comparison and a combination fitting, the HERFD spectra analysis demonstrates that Sr is distributed between two chemical forms: SrO type (40%) and a perovskite SrZrO 3 type (60%).

VI. DISCUSSION: SSD vs. CAS

As already mentioned, the main application of this spectrometer on BM30B/CRG-FAME beamline is the matrix fluorescence removal. In this case, it is interesting to compare fluorescence and HERFD modes, and thus quantify in which case using a CAS is more appropriate than a SSD. We chose here to develop this idea in the particular case of Co included in a Fe-rich matrix 45 . In this particular experiment ( § V.A), the interest of using the CAS is shown in FIG. 7 We used the formalism developed by Bunker 46 estimate the number of effective counts. For the CAS, we reasonably assume that the signal is only due to the Co fluorescence contribution:

∆ Δ

On the other hand, for the SSD the contribution of the background has to be considered:

. .

Another possibility to express when there is an advantage to use the CAS vs. the SSD is to estimate when the noise is smaller with one system or with the other: and . . .

Δ ∆

Calculations are performed with  CAS and  SSD equal 0.3 and 0.0013, respectively and considering that the conditions are identical, i.e. sample, , the SSD at 25 cm from the sample with the element size of 5x5mm² (solid angle: 4.10 -4 all paths under vacuum, the number of emitted photons and the integration time are the same (FIG. 8).

FIG. 8 shows that it is often preferable to use a CAS instead of a 13-element SSD.

Detection systems available on BM30B/FAME beamline, i.e. CAS and 30-element SSD, are also compared with µ Fe (E incident ) = 305.6 cm 2 .g -1 and µ Co (E incident ) = 324.8 cm 2 .g -1 at 8 keV,  Fe = 0.340 and  Co = 0.373 47 , leads to a ( ) limit equal to 0.015 if using 5 crystals. The spectrometer have been also duplicated and successfully tested on the MARS beamline dedicated to the characterization of radioactive samples at SOLEIL synchrotron 48 .

FWHM 0 .

 0 40 eV at Co and 0.83 eV at Sr K-edges respectively with Si (220) crystal monochromator. The flux measured is about 5x10 11 photons/s/200mA between 7.5 and 13.5

FIGURE 1 .

 1 FIGURE 1. Schematic view of CRG BM30B/CRG-FAME beamline at the European Synchrotron Radiation Facility in Grenoble (France).

FIGURE 2 .

 2 FIGURE 2. Influence of scattered beams on X-ray fluorescence emission spectra collected using the CANBERRA 30-elements SSD for bromide aqueous solutions at different concentrations.

FIGURE 3 .

 3 FIGURE 3. Visualisation of the intersection between the incident beam and the detection cone and resolution effect -vertical spreading of the focal spot on the detector.

  fabrication. The analyzer consists of a 225 µm thick silicon wafer spherically bent between convex and concave polished PYREX glass substrates of 0.5 m curvature radius. A dedicated experimental set-up has been developed by the 'Cellule Project' of the Institut de Minéralogie et Physique des Milieux Condensés in order to clamp silicon wafer and glass substrates at high force (1500 N) and high temperature (~ 350°C). A high DC potential (up to >1.7 kV) is then applied between silicon and glass creating an electrical field which drifts the ions in the glass. The depletion of sodium ions near the surface and the creation of surface charge generate a large electrostatic force and bring the silicon and glass into intimate contact, eventually creating chemical bonds. Such analyzers have been successfully produced and have improved the energy resolution 31 . Currently, spherically bent Si crystals with (111) and (110) orientations are available on the beamline. They have been characterized by measuring the energy resolution on the spectrometer (see § III.D). Two others orientations (331) and (311) will be available in the future. These sets of different crystals will allow to cover a broad energy range from 4 to 19.7 keV. Such an energy range is sufficient to probe the K, K, L and L fluorescence lines of all the elements from Ca (Z = 20) to U (Z = 92).

FIGURE 4 .

 4 FIGURE 4. View of the spectrometer: detail of crystal assemblage (left -top position), view of the spectrometer on the beamline with the Vortex EX-90 as detector (left -down position) and top-view of a spectrometer drawing (right).

FIGURE 5 .

 5 FIGURE 5. Co K-edge XANES of Co adsorbed on ferrihydrite collected in total fluorescence yield using a solid state detector (SSD) and in high energy resolution fluorescence detected using the crystal analyzer spectrometer (CAS). Integrated counts after edge are ~5.10 5 using SSD and ~4.10 4 using CAS for counting times of 6s and 120s respectively.

FIGURE 6 .

 6 FIGURE 6. Normalized Sr K-edge XANES spectra collected in total fluorescence (SSD) and HERDF mode (CAS) on a crystalline UO 2 sample doped with 1400 ppm Sr. The arrows signal parasite effects due to Bragg peaks from UO matrix.

  photons: no contribution from the scattered photons (from the sample or from the crystals) and from the fluorescence lines tails of the main constituents of the sample. The signal received by the detector of the CAS system centered around the Kα1 fluorescence energy of Co (energy width of the emission line: Δ) after optimization of the spectrometer () can be then expressed as:∆ ΔWhere is the spectrometer cross-section.Conversely, the signal measured with a SSD with a typical energy resolution around 250 eV includes the contribution of 1) the entire K 1 and K 2 fluorescence lines of Co and 2)the K fluorescence lines of Fe as a background:

FIGURE 7 .

 7 FIGURE 7. Fluorescence spectra measured using the high resolution crystal analyzer spectrometer (top) and the 30-element solid state detector (bottom) on Co adsorbed on nano-Fe 2 O 3 .

FIGURE 8 .

 8 FIGURE 8. Comparison of the CAS with 13-element and 30-element SSD to determine which is the more appropriate in the case of Co adsorbed on iron sample.

TABLE 1 .

 1 Technical parameters of the spectrometer

	Bragg angle range (°)

  spectrometers installed on the 6-2 beamline at SSRL. The spectrometer uses in total 21 small stepper motors to control all the motions: 4 stepper motors for each crystal bloc, one for the main vertical translation and 3 for the detector motions. Many commercial electronics for such devices exist but our choice was to develop our own controller, with less features and less versatile than commercial devices, but clearly cheaper. We chose to use WAGO modules, an 750-841 Ethernet 100 Mbit/s TCP/IP programmable fieldbus controller, and stepper motor modules for current up to 1.5 A. The

commands necessary to control the motors are sent via ESRF standard control program SPEC

25

to each rack controller. The main advantages of our system are its small size, a connection via Ethernet, a standalone operation feature to test the motor either with a web client or console based program, and the low cost. The spectrometer can be relatively easily removed from the beamline, and reinstalled and controlled by a laptop anywhere.

table 2 .TABLE 2 .

 22 Theoretical and experimental resolutions obtained for the first experiments are gathered in Experimental and theoretical resolutions for HERDF-XAS experiments.

	Emission line	Crystal	Bragg angle	Theoretical	Best experimental
			(°)	resolution (eV)	resolution (eV)
	Fe K 1	Si(333)	67.9	1.9	2.3
	(6.404 keV)
	Co K 1	Si(440)	68.7	1.9	3.0
	(6.930 keV)
	Fe K 	Si(440)	66.1	1.9	2.0
	(7.058 keV)
	Cu K 1	Si(444)	79.3	0.7	1.4
	(8.048 keV)
	Sr K 1	Si(880)	65.7	5.0	12.9
	(14.165 keV)
	As it can be seen, experimental values are in general worse than the theoretical expected
	values. However, for low hkl values, experimental resolutions are close to theoretical values
	while for high hkl value, a better resolution can be expected. This probably due to the
	microstrains developed during the crystal bending stage. Bending causes elastic deformations

ACKNOWLEDGEMENTS

This project is financially supported by the ANR NANOSURF (coordinator: C. Chaneac, Laboratoire de Chimie de la Matière Condensée de Paris, France), CEREGE laboratory (Aix en Provence, France) and the INSU CNRS institute.