INTRODUCTION

Positron Emission Tomography (PET) is one of the most popular technique for the study of brain functional activity. Several studies as in Fox et al [START_REF] Fox | A stereotactic method of anatomical localization for positron emission tomography[END_REF], Neelin et al [START_REF] Neelin | Validation of an MRI/PET landmark registration method using 3D simulated PET images and point simulations[END_REF] show that PET is an in-vivo examination technique able to produce real images of cerebral activity, and is also neither destructive nor invasive. Unfortunately, PET images offer low resolution and signal-to-noise ratio. Moreover, they do not reflect the anatomy of patients. Accurate and reproducible analysis of PET images requires other informations, coming from atlases or other images such as Magnetic Resonance Image (MRI) of the same patient (Evans et al (3)). According to Rademacher et al [START_REF] Rademacher | Human Cerebral cortex : localization, parcellation and morphometry with Magnetic Resonance Imaging[END_REF], MRI offers accurate in-vivo localisation of anatomical landmarks. It has been used to estimate individual variations and left-right asymmetries of sulci and gyri. Hence, it is of great interest to superimpose functional PET data and anatomical MRI data. One approach to this problem is to use the stereotactic proportional grid of Talairach and Tournoux [START_REF] Talairach | Co planar stereotaxic atlas of the human brain[END_REF]. It defines spatial co-ordinates of brain sulci and gyri, and of cytoarchitectonic fields. Despite the application of scaling factors, the accuracy of standard brain atlas superimposition onto any given examination is about one to two centimetres at the cortical level, because of inter-individual variations described in Greitz et al [START_REF] Greitz | A computerized brain atlas : construction, anatomical content and some applications[END_REF]. Another approach, used by Jouandet et al [START_REF] Jouandet | Brainprints : computer generated two dimensional maps of the human ceerbral cortex in vivo[END_REF], determines cortical maps based on the topography of sulci and gyri. Our work is based on this method and our objective is to automatically recognise cortical sulci of any brain MRI examination, with respect to an anatomical atlas. Individual recognition of those structures is relevant to the general problems of image registration and search for a set of objects among a set of reference. It can be summarised by looking for the largest common structure in two different graphs, and can be performed using several methods as templates and springs described in Ballard and Braun [START_REF] Ballard | Computer vision[END_REF], maximal cliques in Miclet [START_REF] Miclet | Méthodes structurelles pour la reconnaissance des formes[END_REF], tree search in Vausselman (10) or relaxation techniques in Hummel and Zucker [START_REF] Hummel | On the Foundations of Relaxation Labeling Processes[END_REF]. The main problem is the large variation of dimensions and patterns of patients' brains and sulci. Classical features (length, orientation...) cannot be directly used to identify sulci and are very difficult to be determined. In this paper, we deal with representation and identification of sulci. A first step is to choose and to automatically extract anatomical knowledge from a database, in order to adapt it to any image where the recognition has to be performed. Then, we introduce a stochastic method using these features to recognise human cerebral sulci.

DATA AND FEATURE EXTRACTION

In a previous work, Desvignes et al [START_REF] Desvignes | Reconnaissance du sillon latéral sur image RMN 3D[END_REF] have established a heuristic search based on spatial location and relations among neighbouring sulci. Localisation, global and local orientations, local depth (depth of sulcus from cortical surface) and continuity are the main features of one sulcus. Connections with other sulci are also necessary relations for sulci identification. Data are issued from 3D MRI images (120x256x256 voxels, SPGR sequence, Signa 1.5T General Electrics Medical Systems unit). Brain is extracted and segmented into Cerebro Spinal Fluid (CSF), White Matter and Grey Matter as shown in Allain et al [START_REF] Allain | Entirely automatic 3D MRI brain analysis as a step in multi modal processing[END_REF]. Sulci are cerebral structures filled with CSF. 3D skeletonization and 3D curve thinning are applied to CSF. Sulci are then described by 3D digital curves on the cortical surface (Figure 1). Six sulci have been identified on a training set of nine brains. For each sulci, principal axis, centre of gravity, middle point, continuity (number and position of parts), localisation and connections with others sulci are computed. For each point, depth of the sulci is computed using a tangent vector method. Localisation, orientation and pattern are described by a search window. All the sulci of the training set are superimposed to obtain a cloud of points. 3D morphological operators are used to obtain a compact zone (dilation, hole filling) and to smooth its edge (conditional dilation). A 3D conditional thinning algorithm transforms this search window into a graph of segments, which is a close approximation of the iconic drawing used in medical atlases (Figure 2). Not all of these features are used during the matching process. At this stage, we introduce depths of sulci and search windows.

METHODS: RELAXATION PROCESS AND OPTIMISATION

Continuous relaxation [START_REF] Hummel | On the Foundations of Relaxation Labeling Processes[END_REF] is an iterative algorithm which correspond to a probabilistic approach of graph matching. We applied this process to match a graph of segments coming from a brain image, with another examination where sulci have already been identified. Each assignment i→ I between elements i and I from two sets is referred to a real coefficient p i t (I ) , that depends on all the coefficients p j t -1 (J ) computed at the preceding step. This process is based on the propagation, along the graphs, of constraints R ijk... (I,J,K...) that quantify the similarity between two structures of nodes (i,j,k...) and (I,J,K...).

Li [START_REF] Li | Matching: invariant to translations, rotations and scale changes[END_REF] explains that this technique offers the advantage of a global combinatorial solution, achieved through local propagation, which does not require any threshold to judge acceptance of individual matches .

Let n and N be the numbers of nodes from the two graphs to be matched, and r the number of nodes (i,j,k..) and (I,J,K..) considered in the constraints. The main characteristic of the relaxation process is its complexity in O(n r .N r ). Hence, the polygonal description into segments allows the use of binary constraints R ij (I,J) (equation IV), leading to a reduced CPU time. Unfortunately, continuous relaxation describes a homomorphism from one set onto the other, that suffers from the ambiguous matching and 'Nil class' problems: if each segment from the set of objects has only one assignment in the set of labels, one label can be assigned to several objects or can remain unassigned. During a relaxation process, a consistent labelling p , defined in Rosenfeld et al [START_REF] Rosenfeld | Scene labeling by relaxation operations[END_REF], is computed at each step t by maximising an average local consistency G t ( p ), called the gain of relaxation (equation I).

G t ( p) = R ij (IJ ).p i t (I ).p j t (J ) J =1 N ∑ j =1 n ∑ I =1 N ∑ i =1 n ∑ (I)
This maximum gain is obtained by updating the assignments p i t (I ) according to a gradient ascent method, where each p i t (I ) is moved towards the direction of this gradient (equations II and III).

p i t +1 (I ) = p i t (I ) 1 + q i t (I ) ⎡ ⎣ ⎤ ⎦ p i t (I ) 1 + q i t (I ) ⎡ ⎣ ⎤ ⎦ ( ) I ∑ (II) q i t (I ) = J =1 N ∑ j =1 n ∑ R ij (I, J ).p j t (J ) (III)
( denominator of equation II insures that

p i t I =1 N ∑ (I ) =1 )
Update leads, in the ideal case, to an individual labelling assignment (p i (I)=1 and p i (J)=0 ∀I###J), corresponding to a constant gain G t .

We now present a bi-relaxation method, which ensures a unique labelling, and is based on a relaxation algorithm where CPU time is reduced.

Optimisation

An experimental study (Figure 3) emphasises common properties from variations of the gain and variations of p i t (I ) . We can notice that from the time where ΔG t /G t process, starting again with a reduced number of segments. Individual labelling assignment is inevitably reached, through only a few iterations. Reduced CPU time then allows shape matching between sets of larger sizes in less time.

Bi-relaxation

Previous problems of ambiguous matching and 'Nil class' have been solved as in Legoupil et al [START_REF] Legoupil | Matching of curvilinear structures: application to the identification of cortical sulci on 3D Magnetic Resonance Brain Image[END_REF]. Two relaxation processes are applied, one from one set onto the other and conversely. The isomorphism defined allows assignments from each node of both two sets with at most one label, that is to say only one or none. The main principle of bi-relaxation is to apply a symmetrical processing of the two graphs, and to delete all the elements that do not respect the constraints. However, deletions require to achieve another birelaxation, in order to make sure that assignments are exactly the same as those obtained between the two initial sets (Figure 4). In the opposite case, the operation is iterated until no more suppressions are needed.

APPLICATION

3D MRI images of the head stem from clinical exams of sane patients. Each brain hemisphere is described by a graph where nodes are segments, and arcs are 3D geometrical relations between segments. To apply relaxation process to sulci matching, we first have to define similarity functions, that map geometrical difference between two pairs of segments (i,j) and (I,J) from two sulci. Then these functions have to be weighted and combined in the constraint function R ij (IJ), so as to define their relative influence. Finally, since the process is iterative, suitable initial values must be set for p i (I) and p I (i).

Similarity functions

Similarity functions required by the graph labelling process represent the main properties of sulci which are continuity and relative orientation of their segments. We selected the minimum distance between two segments, and a function representative of both the absolute angle defined by two segments i and j and the difference between the angles of two couples of segment (i,j) and (I,J). Due to our two original sets defined in the same volume, it must be noticed that one of our similarity functions has to be dependant on scale changes. Another imposed restriction is to detect rotations, in order to respect orientations of sulci, and to get rid of cross-assignments problems (Figure 5). As each sulcus is characterised by its continuity along segments, the first relation, which depends on scale changes and takes this feature into account, lies on the smallest distance between two segments. It is defined as

S 1 = | d min (i,j) -d min (I,J) |.
This relation being invariant to rotation, a second similarity is defined, based on the angles between segments. Because the mere difference between angles is not an invariant feature, two bisecting vectors are computed as follows:

The four segments i,j,I,J are considered as four vectors

OA → i , OA → j , OA
S 2 = | M 1 M 2 ⎯ → ⎯ | = (x 1 -x 2 ) 2 + (y 1 -y 2 ) 2 + (z 1 -z 2 ) 2
Using the notations θ1 = _(θ i +θ j ), θ2 = _(θ I +θ J ), ϕ1 = _(ϕ i +ϕ j ), ϕ2 = _(ϕ I +ϕ J ), k1= 2K cos[_(θ i +θ j )] and k2= 2K cos[_(θ I +θ J )], S 2 becomes:

S 2 = k 1 2 + k 2 2 -2k 1 k 2 sinθ 1 sinθ 2 1+ cos(ϕ 1 -ϕ 2 ) ( )+ cosθ 1 cosθ 2 ⎡ ⎣ ⎤ ⎦
This shows that S 2 depends on the difference between absolute angles between (i,j) and (I,J), and is also a factor of a difference between relative positions of the two pairs. Nevertheless, two couples of segments can have the same bisecting vector, provided that they are located on the surface of the same cone (Figure 6). So a null value of S 2 simply means that (i,j) and (I,J) define two cones of same direction and same opening angle. The next step is to combine S 1 and S 2 functions in order to control the relative influence of S 1 and S 2 .

Constraints

The function R ij (IJ) denotes an estimation of the constraints that assignments p j (J) perform on an assignment p i (I) (equation III). To express it, we introduce a function ρ ij (I,J) ∈ [ 0,+∞ [ that measures the difference between the pairs (i,j) and (I,J). The function R(ρ) has to satisfy:

lim ρ → ∞ R(ρ) = 0 and lim ρ → 0 R(ρ) = 1 R(ρ) monotonically decreases in the range [0,1].
A typical choice for R(ρ) is e -ρ , which presents the maximal decrease and so allows the best discrimination of ρ ij (I,J) values ( similar values of ρ ij (I,J) yield different values of R(ρ) ). However, for a simple computation time problem, we prefer the following function:

R ij (IJ) = [ 1 + ρ ij (IJ) +[ρ ij (IJ)] 2 ] -1 (IV)
The easiest way to express ρ as a function of S 1 and S 2 is to use the linear combination: ρ = α S 1 + β S 2 , as in our previous work in 2D [START_REF] Legoupil | Matching of curvilinear structures: application to the identification of cortical sulci on 3D Magnetic Resonance Brain Image[END_REF], where a set of tests were performed, and a standard set of parameters (α,β) calculated. Further matching always used these standard parameters. Here we propose a more flexible approach, by automatically pre-computing parameters as functions of the two sets to match. First of all, ρ should have a mean value around 1, which is the sharpest zone of R(ρ) function.

Furthermore, S 1 and S 2 should have the same distribution in ρ expression. An example for gaussian variations would give:

ρ = 1 2 e - (S 1 -S1 ) 2 2σ S 1 2 + 1 2 e - (S 2 -S2 ) 2 2σ S 2 2
Unfortunately, experimental studies did not allow to attribute any analytical expression to the behaviour of these variables. So the best thing we can do is set that their average variations must be the same, by using the function:

ρ = S 1 2S1 + S 2 2S2

Initialisation

We have used two methods to initialise the bi-relaxation process:

The first one consists in assigning an equal probability of assignment to every segments:

### i,I, p i 0 (I ) = N -1 and p I 0 (i) = n -1 .
In the second one, we use prior knowledge from the database, such as mean depth and search window. The search window allows to diminish the number of segments involved in the process. Depth is used the following way:

### i,I, p i 0 (I ) = depth(I ) depth(I ) I ∑ and p I 0 (i) = depth(i) depth(i) i ∑ .

RESULTS AND PROSPECTS

The eighteen hemispheres of our training set came from 3D MRI examinations of sane patients, without any distinction of sex or age. Each sulcus is isolated from the hemisphere using an appropriate cloud of points.

Atlas is simulated with 3D MRI images where six main sulci are already identified (lateral, central, postcentral, precentral, superior frontal, superior temporal).

Tests have been performed on two kinds of sets. The first one was composed of isolated sulci, including 10 to 50 segments, matched with their corresponding sulci of the atlas. The second one was composed of whole hemispheres, including 200 to 500 segment, where one hemisphere is matched with another whole hemisphere.

We have found experimentally that using prior knowledge during the initialisation step was minimising probability variations, but also that gains were converging on their limits quite slowly. Therefore we choose to assign an equal probability of assignment to Figure 6: Construction of the bisecting vector every segments, which seems to lead faster to the same result. Results are estimated through attributing a mark from 0 to 4 to each matching. It takes into account the relative position of segments as well as their orientation, compared to a hand-made matching with the reference. The 216 matchings lead to a global mark of 83.3/100. Bad results are essentially due to a connectivity problem, inherent to the deletions done during the birelaxation process. Statistics on CPU time summarised in table 1 were performed on a Sun Sparc 10 workstation. Optimisation of the relaxation reduced CPU time up to a factor 20. The main interest of this method is its numeric and systematic representation of spatial relations among sulci, which is a solution to the large inter-individual variabilities of brain sulci. Our first objective is now to use more of this extracted knowledge, for a global validation of the relaxation process. It can be the perfect complement to our previous works [START_REF] Desvignes | Reconnaissance du sillon latéral sur image RMN 3D[END_REF]17), sometimes insufficient to differentiate neighbouring sulci such as central, precentral and post-central. Another improvement is also planned in the building of a real atlas.

Figure 1 :

 1 Figure 1: Lateral projection of a MRI image (left), polygonal description of sulci (right)

Figure 3 :Figure 2 :

 32 Figure 3: Relative variations of the gain G and of the probabilities p i versus number of iterations

→ I and OA → J 2

 →2 , normalised to K and placed at the origin. Bisecting vectors are given by is defined through expression:

Figure 4 :

 4 Figure 4: Bi-relaxation process Figure 5: Scale change (left) and cross-assignments (right) problems

TABLE 1 -

 1 Typical CPU times.

	Number of segments	Calculation time
	24x12	88''
	44x12	240''
	38x16	221''
	415x277	19 420''

Acknowledgements

This work takes within the framework of the "pôle Traitement et Analyse d'Images" ( Image Processing Center ).


Michel Desvignes

Nicolas Royackkers

Houssam Fawal

Marinette Revenu

Detection and identification of Sulci on 3D MRI

come

Introduction

Cross-subject analysis of structural and/or functional images often requires accurate localisation of cortical convolutions (sulci and gyri). This is particularly relevant to better understand the anatomical/functional relationships within the brain and to increase accuracy in neuro-surgical planning. Internal brain structures are easily identified on high resolution MRI, using the Talairach space, thanks to their low variability. Unfortunately, localisation and identification of sulci are very difficult tasks because of the high variability of cortical structures. In this work, we propose an approach for automatic localisation and identification of sulci on MR images.

Methods

Brain is automatically extracted and segmented into cerebro-spinal fluid (CSF), grey matter and white matter using morphological operators (thresolding, filtering, erosion, reconstruction, dilatation). 3D skeletonization and curve thinning are applied on CSF. Sulci are then described as 3D curves on the brain surface (fig. 1). For each sulcus, a set of features, related to localisation, orientation, shape, geometric measurements (e.g. length) and spatial relations between sulci are computed from a training set of images where sulci are identified [START_REF] Royackkers | Feature extraction for cortical sulci identification[END_REF]. To recognize sulci on a new brain image, the basic idea is to perform all these measurements in the coordinate space of the new brain, after registration of the training set on this brain. Linear registration is performed using the Talaraich space and all sulci of the training set are then projected onto the surface of the new brain [START_REF] Desvignes | Sulcal variability in a computerized brain atlas[END_REF]. We get all measurements, including mean and standard deviation of these features, in the space coordinates of this brain, allowing inter-subjects comparisons. Recognition is performed by searching a connected path on the surface of the brain with respect to the computed features, by minimising a cost function. Recognized sulci on the brain surface are propagated towards the center of the brain to obtain 3D surfaces, which are the medial parts of the convolutions (fig. 2).

Results and conclusion

Six main sulci on each hemisphere (lateral, central, pre-central, post-central, superior temporal, superior frontal sulci), have been identified on a training set of nine T1-weighted 3D MRI (SPGR sequence, 1.3 mm 3 ). The proposed method performs automatic detection and identification of these sulci on each hemisphere of thirteen T1-weighted MRI, with good results. We have developed a methodology for studying the morphology of the cerebral sulci and for building a sulci atlas. We get features for morphometry allowing inter-subjects comparisons. These features are used to identify sulci on 3D MRI, which are landmarks for inter-subjects or inter-modality (functional/anatomical) registration and mapping.

Figure 1 :

 1 Figure 1 : Identified sulci on the surface Figure 2 : 3D medial surface of sulci


Nicolas Royackkers
email: nicolas.royackkers@greyc.ismra.fr

Houssam Fawal

Michel Desvignes
email: michel.desvignes@greyc.ismra.fr

Marinette Revenu

J.-M Travère

Jean-Marcel Travere

Feature Extraction for Cortical Sulci Identification

HAL is

Introduction

Positron emission tomography (PET) is now widely used for the study of brain functional activity. PET is neither destructive nor invasive and is an in-vivo examination technique able to produce pictures of cerebral activity. Unfortunately, PET images do not reflect the anatomy of patients. In this way, when a particular activity area is identified in a PET examination, it cannot be localized in the corresponding brain. That is why a matching between PET and anatomical data from atlases or other images is necessary to make the most of PET information (multimodal data fusion) [START_REF] Revenu | Fusion individuelle de données cérébrales multimodales: informations issues d'images numériques et connaissance expertes[END_REF].

So as to allow the interpretation of a PET examination, a solution consists in the identification of the cortical sulci on a 3D magnetic resonance image (MRI) of the same patient. The resulting spatial network is sufficiently accurate to offer a good precision in the localization of functional areas. Cortical sulci identification is difficult because of the high interindividual variability of brain anatomy. In previous works [START_REF] Desvignes | Reconnaissance du sillon latéral sur des images RMN 3D[END_REF][START_REF] Revenu | Fusion individuelle de données cérébrales multimodales: informations issues d'images numériques et connaissance expertes[END_REF], we have described a method for the recognition of the superficial part of six major sulci from MRI. This identification is based on metric and topologic features adapted from paper atlases, which generally do not contain suitable information for automatic recognition.

A more general approach has then been developed, where a set of features is automatically extracted from a database containing previously recognized instances of sulci. It allows to build a model of the cortical topography, describing sulci and their topologic relations. An individual correction method, based on Talairach's grid, is also proposed. It aims at making the data coming from various patients more homogeneous. The models of sulci we get should increase the efficiency of the identification process.

Anatomical and functional data fusion: different approaches

Computerized atlases

Computerized brain atlases are easy-to-use tools in the frame of the fusion of anatomical data with different modalities. They are built with information from paper atlases [START_REF] Rademacher | Human cerebral cortex: localization, parcellation and morphometry with magnetic resonance imaging[END_REF], from post-mortem examinations of frozen brains [START_REF] Gee | Elastically deforming 3D atlas to match anatomical brain images[END_REF], from the two latter sources [START_REF] Greitz | A computerized brain atlas: construction, anatomical content, and some applications[END_REF], or from MR images [START_REF] Evans | MRI-PET correlation in three dimensions using a volume-of-interest (VOI) atlas[END_REF]. Typically, computerized atlases are composed of axial sections in which the major elements of brain anatomy can be found. In each section, anatomical elements are represented by a set of closed contours limiting "regions of interest" (ROI). 3D approaches have been proposed too; they use "volumes of interest" (VOI) [START_REF] Evans | MRI-PET correlation in three dimensions using a volume-of-interest (VOI) atlas[END_REF][START_REF] Rademacher | Human cerebral cortex: localization, parcellation and morphometry with magnetic resonance imaging[END_REF].

All computerized atlases represent a single model of human cerebral anatomy. The underlying assumption is that all normal and adult brains, at least at a certain level of representation, have the same structure [START_REF] Bajcsy | Multiresolution elastic matching[END_REF]. It can be verified for the general organization of the brain. However, the large intersubject local variability of cerebral anatomy must be underlined (e.g. shape, depth and localization of the sulci). Global size and pattern can also be very different from one brain to another. Therefore, the superimposition of a computerized atlas to an image needs the use of correction mechanisms.

Proportional atlases

Proportional atlases contain the representation of a generic anatomy in particular coordinate systems in which the anatomical structures are localized proportionally to brain dimensions. The superimposition of such an atlas to any examination is performed by applying scaling corrections. Among the numerous available methods, the proportional grid system of Talairach [START_REF] Talairach | Co-planar stereotaxic atlas of the human brain[END_REF] produces the most stable results [START_REF] Joliot | Traitement du signal de résonance magnétique nucléaire in vivo[END_REF][START_REF] Steinmetz | Cerebral cortical localization: application and validation of the proportional grid system in MR imaging[END_REF]. Yet, it cannot fully correct anatomical differences between subjects, since the localization variation interval represents more than 10% of the length of an average brain [START_REF] Steinmetz | Cerebral cortical localization: application and validation of the proportional grid system in MR imaging[END_REF][START_REF] Talairach | Co-planar stereotaxic atlas of the human brain[END_REF].

Deforming atlases

Deforming atlases, as well as proportional atlases, for the result of an individual matching between anatomical data (deformed atlas) and functional data (PET image), require the use of an intermediate image (usually MRI). This picture allows the identification of the anatomical structures of the studied patient, in order to guide the matching of the atlas. Global matching of the atlas onto the image is the first step in these methods. It consists in applying rigid transformations to the atlas so as to correct major differences between the two representations. Performing local deformations on the atlas is the second stage of the matching. Local rigid transformations on each region of interest can then be carried out [START_REF] Evans | MRI-PET correlation in three dimensions using a volume-of-interest (VOI) atlas[END_REF]. But usually an elastic matching of the atlas is performed [START_REF] Bajcsy | Multiresolution elastic matching[END_REF][START_REF] Gee | Elastically deforming 3D atlas to match anatomical brain images[END_REF]. This technique consists in representing the atlas as a kind of rubber ball which is submitted to external forces, and in matching its surface with the surface of the brain contained in the image [START_REF] Bajcsy | Multiresolution elastic matching[END_REF]. Matching can also be carried out with the contours of some anatomical elements or with the whole grey matter/white matter interface [START_REF] Gee | Elastically deforming 3D atlas to match anatomical brain images[END_REF].

The elastic matching methods seem quite elegant, but they do not manage to reconcile simplicity with efficiency. Indeed, when the matching process only uses the surface of the brain, the atlas surface is successfully mapped on the image brain surface, but it is not the same for anatomical structures. So the method finally fails in taking into account intersubject local variability of cerebral anatomy. On the contrary, when matched elements are the anatomical landmarks themselves, the mapped atlas represents the studied brain anatomy in a satisfying way. But these elements must be interactively (i.e. manually) pointed at in the image, which represents a tedious task [START_REF] Allain | Imagerie par résonance magnétique du cerveau: analyse automatique tridimensionnelle et segmentation; application au traitement de données en tomographie par émissions de positons[END_REF].

An identification method for six major sulci on 3D MR images

In collaboration with the CYCERON PET research center of Caen, we have developed a method for the automatic identification of the superficial part of six major sulci on 3D MR images. It consists in a direct recognition of anatomical elements for each subject. A full description of the identification procedure can be found in [START_REF] Desvignes | Reconnaissance du sillon latéral sur des images RMN 3D[END_REF].

FIG. 1. MR images (sagittal projections): top of a head and isolated brain

The brain contained in the pre-processed image is composed of three constituents: white matter, grey matter and cerebro-spinal fluid (CSF). The CSF allows to detect the sulci it fills, thanks to its particular grey level values. So the volume composed of all the sulci is isolated with a multi-level thresolding operation. A 3D skeletonization is then applied to this volume. The Tsao algorithm is used [START_REF] Tsao | A parallel thinning algorithm for 3D pictures[END_REF], after having added an artificial layer on the surface of the brain, in order to obtain the skeleton at the surface level. The resulting structure is thin in one direction, centered inside the initial volume and respectful of the original object topography. Nevertheless, the superficial layer blocks the skeletonization in a perpendicular direction to the surface. A curve thinning is applied on the surface. Superficial part of the sulci then appears as 3D digital curves (Fig. 2).

Sulci identification is performed mainly on superficial part of the sulci. All the same, we compute a depth for each superficial sulci point to take into account partially the structure present inside the brain [START_REF] Desvignes | Calcul de la profondeur en un point des sillons du cortex sur des images RMN tridimensionnelles[END_REF].

FIG. 2. The three steps of sulci detection: thresolding, skeletonization and thinning

Identification of a sulcus

Our method identifies the superficial part of each one of the recognized sulci separately (at the present time: lateral, precentral, central, postcentral, superior frontal and superior temporal sulci in the two hemispheres).

A rough localization of the sought sulcus in a small area makes up the first step of the identification process. Thus, a search area is defined. It is usually composed of the union of several parallelepipeds defined in Talairach's grid. The linear structure that constitutes the superficial line of the whole set of sulci is then partitioned in "topological segments" defined as sets of voxels joining two terminal points or nodes of the structure. In order to prepare and to facilitate the sulcus identification, three kinds of processes can be applied: the deletion of segments whose features are not compatible with the sought curve ones, the fusion of connected segments after such suppressions, and the division of long and very curved segments (to keep the average direction meaningful).

The identification by itself is initialized with the choice of a first segment. As a first step, long enough segments whose directions in axial and sagittal projections are compatible with expected values are selected. The best candidate, chosen for its maximum depth (lateral sulcus) or length (other sulci) is then recorded. Next, a sulcus following is performed from each one of the end of the segment, in order to find the missing parts of the sulcus. If existing, connected segments are examined and the one whose direction is closest to the expected slope is selected. On the contrary, the near enough segment which verifies best this criterion is chosen. Once selected, a segment is merged with the first one. The sulcus following continues until no segment can be found to extend the sulcus in the search area (Fig. 3).

FIG. 3. The six identified sulci on the surface of the left and right hemispheres

The need for a more general approach

Some major weaknesses of the developed identification process can be stressed. At first, there is no synthetic and explicit model of the sulci features. Next, most of the knowledge is determined or adapted from paper atlases. A heavy experimental work is then needed on each sulcus in order to obtain good results. Furthermore, as well as the information used, the identification processes developed are different for each recognized sulcus. Thus, they cannot be re-used to extend the identification to other sulci. At last, the treatment is static: a single solution is always produced, and no indication is given to the user about its validity (it is the best result according to the sulcus model, but can it be considered as satisfactory? And do equivalent solutions exist?).

To tackle these issues, a new approach has been developed.

Cortical topography description & automatic knowledge extraction 4.1. Purposes

From a database composed of some 3D MR images where sulci have already been recognized, the aim is to extract complete, homogeneous and ready-to-process statistical characteristics in order to establish a quantitative model of the cortical topography.

The defined processes and the extracted models should allow to validate and improve the choice of the information used for sulci recognition. The study of anatomical interindividual and interhemisphere variations (asymmetries) should be made easier. It should also enable the production and registration of data that are independent to algorithms, as well as the development of stages for the validation and comparison of potential solutions. Finally, the generalization of identification processes and their application to new sulci should be facilitated by the use of a unique sulci model.

A model of the cortical topography

Model features

Single sulcus features: each sulcus identified in the database is characterized by its principal direction, its gravity center, its length, some statistics on its global depth, its spatial and geodesic straightness, and its continuity.

Geometrical relations between sulci: for each couple of sulci appearing in the same database image, the connection points (if they exist), the relative localization of gravity centers, and the angle between the axes are computed.

Extraction and validation of the data

All the processes involved in the automatic extraction of the above-mentioned features have been implemented. At the present time, they are using already identified sulci contained in a database composed of 9 MR images.

When possible, results have been compared to sulci features given in anatomical books. The extracted values were then proved to be very close to those measured by neuroanatomists. For instance, the difference between the average brain volume computed and the measure obtained in [START_REF] Filipek | Magnetic resonance imaging-based brain morphometry: development and application to normal subjects[END_REF] with a similar protocol is less than 2%. The average difference between the average sulci depths computed in our model and the values given by [START_REF] Ono | Atlas of the cerebral sulci[END_REF] is about 7%, though the atlas used post-mortem examinations.

Nevertheless, these results cannot be used directly for an individual identification.

Correction mechanisms

The need for an individual data correction

All the computed features are influenced by the shape and the size of the different brains: localization, direction, depth and other sulci characteristics depend on brain morphology (Fig. 4). Now, with only 9 MR images, some very important variations can be noticed about brain dimensions. For instance, a difference of 13% on length-to-width ratio, 32% on width and 75% on volume can be found. That is why computing features in different images to use them for sulci identification in another image is a nonsense: atlases show general statistics that are not adapted to each individual examination where the recognition has to be performed.

So as to solve this problem, we propose an original approach based on Talairach's proportional grid system, in order to adapt extracted features to any brain image.

FIG. 4. Influence of the shape and the size of an object on its superficial curve features

Adapting mechanisms

Our approach consists in converting the coordinates of database sulci points towards the coordinate system of the picture where the recognition has to take place, by means of Talairach's grid [START_REF] Talairach | Co-planar stereotaxic atlas of the human brain[END_REF]. Next, all the previously defined sulci features are computed on this image. Thus, all data are homogeneous because they are obtained in the same anatomical coordinate system. Moreover, they take into account the studied brain characteristics. Actually, our approach can be considered as the opposite of Talairach's one: the grid is not used to localize cortical structures in a standard brain, but the anatomical information contained in a database are adapted to the brain under study.

Statistical models of sulci

Sulci and types of sulci

There are as many computed individual models of a sulcus as available images. models are used to build a unique model for each type (or class) of a sulcus (e.g. the lateral sulcus from left hemisphere model). It can then be used for identification tasks. To build the statistical model of a sulcus class, the variation interval, the average and the standard deviation of each individually extracted feature are determined.

The case of depth is particular: our depth is not corrected by the use of Talairach's grid. Depth correction is made proportionally to the ratio of brain volumes. The statistics defined for other features are then computed on depth information.

Other knowledge

Some data used during the identification are not obtained from the previously defined models. They are the search area and the expected local directions and depths of a sulcus. This intrinsically statistical knowledge is automatically determined from the surface of the brain contained in the picture where the identification is performed.

Search area: By means of Talairach's grid system, the database instances of a sulcus type are transferred in the analyzed image (proportional correction) and projected onto the surface of the brain it contains (non-linear correction). The resulting area shows the localization of the same sulcus in a learning set of images. It is then extended by an automatic multi-dilation process in order to reach a compact zone whose edges look as if they were manually drawn (Fig. 5).

FIG. 5. Building of a search area (lateral sulcus from the left hemisphere)

Local depth: Each point of the search area is associated to a local average depth. During the identification of a sulcus, the expected local depth can then be easily computed for each voxel on the brain surface with some of its closest neighbours belonging to the area.

Local direction: As well as local depths, local directions are very useful to get a precise representation of the sulci, especially the most curved ones. Two different approaches have been developed so as to build a shape model of a set of sulci. One is based on a morphologic treatment of the search area, the other on a parametric description of the set of sulci used to build this area. They both build a curve onto the surface of the brain under examination. Computing the local direction associated to a point of the search zone then amounts to take into account the closest part of the curve.

Results and conclusion

In order to perform the fusion of functional data from PET images and anatomical information from 3D MR images, a fully automatic method for the identification of the superficial part of six major sulci has been developed. Identified sulci have been compared to manual identifications done by two independent medical specialists teams. On the learning set of 9 brain images, 106 of 108 sulci have been correctly identified. On a test set of 4 other brain examinations, 45 on 48 sulci seem to be correctly recognized (this last result is not yet validated), which would give a success rate greater than 90%. Our method seems to be efficient and robust, but it cannot be easily applied to other sulci.

Automatically extracting homogeneous knowledge is a first step towards the generalization of the process. The sulci identified in a database are first projected in the analyzed image, by means of Talairach's grid system. A set of features is then extracted so as to represent these sulci and their geometrical relations. A statistical model of each type of sulcus is finally built. It is ready-to-process and respectful of the studied brain morphology. Search area and local information are determined in the same way.

Our first objective is now to use all the extracted data. Once a unique sulci model is used, common identification processes must be developed. The identification mechanisms should not change much as far as their results are good. However, several curves could be recorded first in each search area in order to select the best one later. As a matter of fact, an individual validation stage must be developed so as to find the best solution among a few sets of segments, using comparison with the extracted model of the sulcus under examination. Finally, a collective validation step must also be developed. It will choose the best set of sulci, compared with the whole cortical topography model described as a graph. The issue is then to determine the closest graph to the model.

Other improvements are also planned, such as the extension of the sulci model in order to take into account ramified structures, or the building of a more declarative representation of the processes so as to improve the identification flexibility.

3. 1 .

 1 Method description 3.1.1. Cortical topography detection 3D MR images are obtained from adult volunteers without any cerebral pathology. They are composed of 120 sections of 256x256 isotropic voxels. An automatic processing localizes the midsagittal plane and the anterior and posterior commissures (AC & PC) of Talairach's grid. Next, the brain is isolated from the rest of the head [1] (Fig. 1).

Dist a n c e O

 e ri e nt a tio n Le n g th Re l a tiv e ori e nt a tio n

Acknowledgements

The authors wish to thank J. Yelnik from INSERM U106 (La Salpétrière Hospital, Paris) and J.M. Constans from the Hospital of Caen for their contribution to this work.


Régis Clouard

Marinette Revenu

Abderrahim Elmoataz

A Software Workbench for Knowledge Acquisition and Integration in Image processing

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Andrés Serna

Beatriz Marcotegui
email: beatriz.marcotegui@mines-paristech.fr

Franc ¸ois Goulette

Jean-Emmanuel Deschaud
email: jean-emmanuel.deschaud@mines-paristech.fr

Paris-rue-Madame database: a 3D mobile laser scanner dataset for benchmarking urban detection, segmentation and classification methods

Keywords: 3D database, mobile laser scanner, urban analysis, segmentation, classification, point-wise evaluation

This paper describes a publicly available 3D database from the rue Madame, a street in the 6 th Parisian district. Data have been acquired by the Mobile Laser Scanning (MLS) system L3D2 and correspond to a 160 m long street section. Annotation has been carried out in a manually assisted way. An initial annotation is obtained using an automatic segmentation algorithm. Then, a manual refinement is done and a label is assigned to each segmented object. Finally, a class is also manually assigned to each object. Available classes include facades, ground, cars, motorcycles, pedestrians, traffic signs, among others. The result is a list of (X, Y, Z, reflectance, label, class) points. Our aim is to offer, to the scientific community, a 3D manually labeled dataset for detection, segmentation and classification benchmarking. With respect to other databases available in the state of the art, this dataset has been exhaustively annotated in order to include all available objects and to allow point-wise comparison.

INTRODUCTION

Nowadays, LiDAR technology ("light detection and ranging") has been prospering in the remote sensing community thanks to developments such as: Aerial Laser Scanning (ALS), useful for large scale buildings survey, roads and forests; Terrestrial Laser Scanning (TLS), for more detailed but slower urban surveys in outdoor and indoor environments; Mobile Laser Scanning (MLS), less precise than TLS but much more productive since the sensors are mounted on a vehicle; and more recently, "stop and go" systems, easy transportable TLS systems making a trade off between precision and productivity.

Thanks to all these technologies, the amount of available 3D geographical data and processing techniques has bloomed in recent years. Many semiautomatic and automatic methods aiming at analyzing 3D urban point clouds can be found in the literature. It is an active research area. However, there is not a general consensus about the best detection, segmentation and classification methods. This choice is application dependent. One of the main drawbacks is the lack of publicly available databases allowing benchmarking.

In the literature, most available urban data consist in close-range images, aerial images, satellite images but a few laser datasets (ISPRS, 2013;[START_REF] Ign | IGN -Geospatial and Terrestrial Imagery[END_REF].

Moreover, manual annotations and algorithm outputs are rarely found in available 3D repositories [START_REF] Nüchter | Robotic 3D Scan Repository[END_REF]CoE LaSR, 2013).

Some available data include Oakland dataset [START_REF] Munoz | Contextual Classification with Functional Max-Margin Markov Networks[END_REF], which contains 1.6 million points collected around Carnegie Mellon University campus in Oakland, Pittsburgh, USA. Data are provided in ASCII format: (X, Y, Z, label, confidence) one point per line, vrml files and label counts. The training/validation and testing data contains 5 labels (scatter misc, default wires, utility poles, load bearing and facades). Ohio database [START_REF] Golovinskiy | Shape-based recognition of 3D point clouds in urban environments[END_REF] is a combination of ALS and TLS data in Ottawa city (Ohio, USA). It contains 26 tiles 100 × 100 meters each with several objects such as buildings, trees, cars and lampposts. However, ground truth annotations only consists in a 2D labeled point in the center of each object. In that sense, segmentations results cannot be evaluated point by point. Enschede database [START_REF] Zhou | Mapping curbstones in airborne and mobile laser scanning data[END_REF] contains residential streets approximatively 1 km long in the Enschede city (The Netherlands). Ground truth annotation consists in 2D geo-referenced lines marking curbstones. A well-defined evaluation method is available using buffers around each 2D line. The drawback of this dataset is that no other objects are annotated. Paris-rue-Soufflot database [START_REF] Hernández | Filtering of artifacts and pavement segmentation from mobile Li-DAR data[END_REF] contains MLS data from a street 500 m long in the 5 th Parisian district. Six classes have been annotated: facades, ground, cars, lampposts, pedestrians and others.

In this paper, we present a 3D MLS database for benchmarking detection, segmentation and classification methods. Each point in the 3D point cloud has been segmented and classified, allowing pointwise evaluations. Additionally, our annotation includes all available objects in the urban scene. Data have been acquired and processed in the framework of TerraMobilita project (http://cmm.ensmp.fr/ TerraMobilita/ ).

Paper organization is as follows. Section 2 reminds some basic definitions. Section 3 describes Paris-rue-Madame database. Section 4 explains the MLS system and acquisition details. Section 5 briefly presents our manually assisted annotation protocol. Finally, Section 7 concludes the work.

BACKGROUND

A typical 3D urban analysis method includes 5 main steps: i) data filtering/down-sampling in order to reduce outliers and redundant data; ii) Digital Terrain Model (DTM) generation; iii) object detection in order to define object hypotheses and regions of interest; iv) object segmentation in order to extract each individual object; and v) object classification in order to assign a semantic category to each object. In the scientific community several definitions can be found for detection, segmentation and classification steps. For clarity, let us define these concepts in the way we intend they should be understood with this dataset:

Detection: An object is considered detected if it is
included in the list of object hypotheses, i.e. it has not been suppressed by any filtering/downsampling method and it has not been included as part of the DTM. Note that an object hypothesis may contain several connected objects or even contain only a part of an object. In the detection step, we are only interested in keeping all possible objects. This is important because in most works reported in the literature, non-detected objects cannot be recovered in subsequent algorithm steps.

Segmentation: An object is considered segmented if it is correctly isolated as a single object, i.e. connected objects are correctly separated (there is no sub-segmentation) and each individual object is entirely inside of only one connected component (there is no over-segmentation). This is important because many algorithms based on clustering and connected components can wrongly gather objects touching each other, e.g. motorcycles parked next to the facade, pedestrians walking together, cars closely parked to others, among others. Classification: In the classification step, a category is assigned to each segmented object. Each class represents an urban semantic entity. Depending on the application, several classes can be defined: facade, ground, curbstone, pedestrian, car, lamppost, etc.

DATA DESCRIPTION

Paris-rue-Madame dataset contains 3D MLS data from rue Madame, a street in the 6 th Parisian district (France). Figure 1 shows an orthophoto from the test zone, approximatively a 160 m long street section between rue Mézières and rue Vaugirard. The acquisition was made on February 8, 2013 at 13:30 Universal Time (UT). The dataset contains two PLY files with 10 million points each. Each file contains a list of (X, Y, Z, reflectance, label, class) points, where XYZ correspond to geo-referenced (E,N,U) coordinates in Lambert 93 and altitude IGN1969 (grid RAF09) reference system, reflectance is the laser intensity, label is the object identifier obtained after segmentation and class determines the object category. An offset has been subtracted from XY coordinates with the aim of increasing data precision: X 0 = 650976 m and Y 0 = 6861466 m, respectively. The available files are "GT Madame1 2.ply" and "GT Madame1 3.ply", both of them coded as binary big endian version 1. This database contains 642 objects categorized in 26 classes, as shown in Table 1. It is noteworthy that several objects inside buildings have been acquired through windows and open doors, these objects have been annotated as facades. Other several "special" classes have been added because they are too different to be mixed with others. For instance, fast pedestrians and pedestrians+something have different geometrical features than a simple pedestrian. The idea of this annotation is including as much classes as possible, then each user may gather or exclude classes depending on the application.

It is noteworthy that this database is different from others available in the state of the art since the entire 3D point cloud has been segmented and classified, i.e. each point contains a label and a class. Thus, pointwise evaluation of detection, segmentation and classification methods is possible.

ACQUISITION

The acquisition has been carried out by the MLS system L3D2 from the robotics laboratory CAOR-MINES ParisTech [START_REF] Goulette | An integrated on-board laser range sensing system for On-the-Way City and Road Modelling[END_REF]. This system is equipped with a Velodyne HDL32, as shown in Figure 3. In this system, several lasers are mounted on upper and lower blocks of 32 lasers each and the entire unit spins, giving much denser point clouds than classic Riegl sensors [START_REF] Velodyne | The HDL-32E Velodyne LiDAR sensor. Velodyne Lidar[END_REF]. Annotation has been carried out in a manually assisted way. An initial segmentation is obtained using an automatic method based on elevation images [START_REF] Serna | Detection, segmentation and classification of 3D urban objects using mathematical morphology and supervised learning[END_REF]. The work-flow is shown in Figure 4. For further details and complete analyses in each step, the reader is also encouraged to review the following three works [START_REF] Hernández | Filtering of artifacts and pavement segmentation from mobile Li-DAR data[END_REF]) (Serna and Marcotegui, 2013a) (Serna and Marcotegui, 2013c). After this automatic method, a manual refinement is carried out in order to correct possible errors in detection and segmentation steps. Then, a label is assigned to each segmented object. Finally, a class is also manually assigned in order to categorize each segmented object.

During manual refinement, three typical segmentation errors are both found and corrected: i) bad segmentation of some connected objects, e.g. motorcycles parked close to each other are not correctly separated, as shown in Figure 5; ii) over-segmentation of some objects due to artifacts or noise, i.e. some cars are over-segmented on their roof, as shown in Figure 6; iii) sub-segmentation of some objects, i.e. some objects touching the facades are not well separated, as shown in Figure 7.

DOWNLOAD & LICENSE

CONCLUSIONS

We have presented a 3D MLS database manually annotated from rue Madame, a street in the 6 th Parisian district. Each 3D point has been labeled and classified, resulting in a list of (X, Y, Z, reflectance, label, class) points.

The database has been acquired by the L3D2 vehicle, a MLS system from the Robotics laboratory (CAOR) at MINES ParisTech. A distinctive feature of this system is that it uses a Velodyne sensor aligned in a similar way to a Riegl sensor but providing much denser 3D point clouds.

Annotation has been carried out in a manual assisted way by the Center for mathematical morphology (CMM) at MINES ParisTech. First, an automatic segmentation method is applied. Then, manual refinement and classification are done. This approach is faster than a completely manual approach and it provides accurate results. This dataset is different from others available in the state of the art since each point has been segmented and classified, allowing pointwise benchmarking.

In future works, other datasets acquired in the framework of TerraMobilita project will be annotated and made available to the scientific community.

Figure 1 :

 1 Figure 1: Rue Madame, Paris (France). Orthophoto from IGN-Google Maps.

Figure 2

 2 Figure 2 presents one of the 3D point clouds of this database colored by the point height (Z coordinate), the reflectance, the object label and the object class.

Figure 2 :

 2 Figure2: "GT Madame1 2.ply" file: 3D point cloud colored by its available fields. For the object label, each color represents a different object (only for visualization purposes, some colors have been repeated). For object class visualization: facades (blue), ground (gray), cars (yellow), motorcycles (olive), traffic signs (goldenrod), pedestrians (pink).

Figure 3 :

 3 Figure 3: MLS system L3D2 from CAOR-MINES Paris-Tech

Figure 4 :

 4 Figure 4: Work-flow of our automatic segmentation methodology.

Figure 5 :

 5 Figure 5: Manual refinement of connected motorcycles.

Figure 6 :

 6 Figure 6: Manual refinement of over-segmented cars.

Figure 7 :

 7 Figure 7: Manual refinement of sub-segmented objects touching the facade.

Table 1 :

 1 Available classes and number of objects in the Paris-Rue Madame dataset.

	Number of objects

ACKNOWLEDGEMENTS

The work reported in this paper has been performed as part of Cap Digital Business Cluster TerraMobilita Project.


V Scagnoli

S W Huang

M Garganourakis

R A De Souza

U Staub

V Simonet

P Lejay

R Ballou

K Marty

E Ressouche

V A Chernyshov

Dyadkin

V Dmitriev

S V Maleyev

E V Moskvin

D Menzel

A Rosch

R Neubauer

Georgii

T Kimura

T Goto

H Shintani

K Ishizaka

T Arima

R D Johnson

L C Chapon

D D Khalyavin

P Manuel

P G Radaelli

Phys C Martin

Rev

Lett

P Bordet

M Loire

C Darie

J Kljun

P Bonville

O Isnard

B Za

H D Zhou

L L Lumata

P L Kuhns

A P Reyes

S Petit

P Le- 667 Jay

J Ollivier

M Enderle

C Stock

A Schneidewind

Y Su

D F Radaelli

A Mcmorrow

N Bombardi

Lee

A Zorko

M Pregelj

A Potočnik

J Van Tol

A Ozarowski

R Bal

T Brückel

M Lippert

T Köhler

J R Schneider

Dzyaloshinsky-Moriya driven helical-butterfly structure in Ba 3 NbFe 3 Si 2 O 14

We have used soft x-ray magnetic diffraction at the Fe 3+ L2,3 edges to examine to what extent the Dzyaloshinsky-Moriya interaction in Ba3NbFe3Si2O14 influences its low temperature magnetic structure. A modulated component of the moments along the c-axis is present, adding to the previously proposed helical magnetic configuration of co-planar moments in the a, b-plane. This leads to a "helical-butterfly" structure and suggests that both the multi-axial in-plane and the uniform out-of-plane Dzyaloshinsky-Moriya vectors are relevant. A non zero orbital magnetic signal is also observed at the oxygen K edge, which reflects the surprisingly strong hybridization between iron 3d and oxygen 2p states, given the nominal spherical symmetry of the Fe 3+ half filled shell.

I. INTRODUCTION

The term chirality was first utilized in science by Lord Kelvin. His original definition has evolved with time and we now speak about a chiral system if such a system exists in two distinct (enantiomeric) states that are interconverted by space inversion, but not by time reversal combined with any proper spatial rotation. 1 Chirality permeates natural sciences from biochemistry to solid state physics. The fact that living organisms use only the left enantiomers of amino acids is still not well understood. Chirality is also found in magnets. 2,3 An example is the left-or right-handedness associated with the helical order of magnetic moments. In principle, the two states are degenerate, resulting in an equipopulation of chiral domains. However, competing interactions or external effects such as strain, can unbalance this ratio, favoring one particular state. In particular, in non centrosymmetric crystals, characterized by the absence of parity symmetry, a single domain might be selected. Despite having 65 non centrosymmetric (including 22 chiral) space groups allowing chiral crystal structures, out of 230, only few single handed magnetic compounds were reported. [4][5][6][7] Interest in such systems is two-fold. First, they can exhibit interesting physical properties such as magnetic Skyrmion lattices 8 or helimagnons. 7 The second is related to the discovery of magnetically induced multiferroics 9 where researchers struggle to find materials with a stronger electrical polarization. 10 The latter is directly affected by the imbalance between chiral domains, which possess opposite electric polarizations.

Therefore, materials showing a single chiral domain are promising candidates to host a significant macroscopic electrical polarization, which makes them an ideal model system to study. Ba 3 NbFe 3 Si 2 O 14 gathered attention in this respect, exhibiting fully chiral magnetism 5 and magnetoelectric coupling phenomena. [11][12][13] Ba 3 NbFe 3 Si 2 O 14 crystallizes in a trigonal P321 space tion vector (0, 0, τ ) with τ close to 1/7 (see Fig. 1a). An 66 extremely appealing discovery was that the single crys-

III. RESONANT X-RAY SCATTERING

The x-ray cross section for magnetic scattering is normally very small, though at synchrotron photon sources such weak signals are routinely measurable. [22][23][24][25] However, when working close to an atomic absorption edge the magnetic scattering signals are significantly enhanced and are element sensitive. Resonant x-ray diffraction occurs when a photon excites a core electron to empty states, and is subsequently re-emitted when the electron and the core hole recombine. [26][27][28] This process introduces anisotropic contributions to the x-ray susceptibility tensor, [29][30][31] the amplitude of which increases dramatically as the photon energy is tuned to an atomic absorption edge. In the presence of long-range magnetic order, or a spatially anisotropic electronic distribution, the interference of the anomalous scattering amplitudes may lead to Bragg peaks at positions forbidden by the crystallographic space group. An example of such a resonant enhancement of the diffracted intensity as a function of energy occurring in the vicinity of the Fe L 3 edge in Ba 3 NbFe 3 Si 2 O 14 is given in Fig. 3. X rays thus prove to be a valid alternative or complementary tool to neutron diffraction for the study of magnetic structures. [32][33][34][35][36] Its superior resolution in reciprocal space can be advantageous, simplifying for instance the precise determination Rotating the sample about the diffraction wave vector might result in a smooth change of the diffracted intensity which helps to reconstruct the magnetic moment motif.

F E1 ǫ ′ ,ǫ = (ǫ ′ • ǫ)F (0) -i(ǫ ′ × ǫ) • ẑn F (1) + (ǫ ′ • ẑn )(ǫ • ẑn )F (2) , (1)
It is worth emphasizing that Eq. ( 1) is an approximation for the resonant magnetic scattering cross section which, strictly speaking, is only valid for a cylindrical symmetrical environment of the resonant ion. When this approximation does not hold the diffracted intensities must be described as exemplified in Ref. 2, 38-42.

IV. RESULTS

Once the sample is cooled below the Néel temperature T N , superstructure peaks (0, 0, nτ ) of order n up to three arise from magnetic ordering and magnetically induced lattice distortions (Fig. 2). The observation of such reflections is remarkable as, given the magnetic motif suggested by neutron diffraction, they should be absent. They are of resonant nature and they disappear when the energy of the incident x rays is detuned from the iron L edges (Fig. 3). Non-resonant magnetic intensity could be zero or too small to be visible. Resonant xray diffraction is sensitive to the spin, orbital and charge degrees of freedom. 28,[43][44][45] In order to assert their origin and refine the magnetic structure, we collected their energy, azimuthal and temperature dependence. shows the energy dependence of the three superstructural The dashed line is the expected mean-field theory dependence. The (0, 0, 2τ ) satellite is 7 times more intense than the (0, 0, τ ). The same ratio holds between the (0, 0, τ ) and the (0, 0, 3τ ) satellites. Data was measured with π incident photon energy of 710 eV.

to the oxygen K edge, which corresponds to an electric dipolar transition from the 1s to the 2p level. Upon cooling below T N a signal is observed at this energy.

Figure 7 shows its resonant nature. Observation of a resonant signal on an anion is not unusual. [54][55][56] A resonant signal can arise, given a non zero overlap between the initial and the final state, whereas a difference exists in the up/down spin dipolar overlap integrals. The difference can be induced by polarization of the orbitals. 57 Such an asymmetry can arise also in case of a difference in the lifetime of the up/down spin channels. Recently Beale et al. 55 observed a resonant signal at the oxygen K edge in TbMn 2 O 5 , which they interpreted as a signature of an antiferromagnetically ordered spin polarization on the oxygen site. Such an observation is quite remarkable and we share their opinion that the study of oxygen spin polarization may lead to new insight in the understanding of the magnetoelectric coupling mechanism. As a matter of fact, an antiferromagnetic order at the oxygen site is consistent with neutron diffraction experiments that have already suggested a spin polarization of the oxygen by finding a value of 4 µ B instead of the expected 5 µ B for the spherical Fe 3+ half filled ion magnetic moment. 5,11 In our case the signal at the oxygen K edge is 90 times weaker than the corresponding one observed at the iron L 3 edge. Note that at the K edge the signal originates solely from the orbital magnetic moment component, given the absence of spin-orbit splitting of the hole in the core state. [START_REF] Lovesey | Ro-731 driguez[END_REF]58,59 No intensity was observed at the (0, 0, 2τ ) and (0, 0, 3τ ) satellites at the oxygen K edge. open square] obtained in the vicinity is also shown. Full (red) circle results from a fit of the integrated intensity of a reciprocal lattice scan along the c* reciprocal lattice direction. Open (red) circle are a result of an energy scan with fix momentum transfer. The (black) continuous line is a Gaussian fit of the oxygen resonance with a FWHM = 1.4±0.1 eV.

V. SYMMETRY CONSIDERATIONS 324

Insights into the results can be obtained from group 325 representation analysis, [START_REF] Ballou | Neutron Scattering from 788 Magnetic Materials[END_REF] provided that a single irre-326 ducible representation is selected at the magnetic order- I summarizes such results for the cyclic group 3.

357

The choice of a propagation vector amounts to select

Characters

Basis Vectors

1 3 + 3 - Γ1 1 1 1 3 ν=1 vν (θ, φ + (ν -1) 2π 3 ) Γ2 1 e i 2π 3 e i 4π 3 3 ν=1 e -i(ν-1) 2π 3 vν (θ, φ + (ν -1) 2π 3 ) Γ3 1 e i 4π 3 e i 2π 3 3 ν=1 e -i(ν-1) 4π 3 vν (θ, φ + (ν -1) 2π
3 ) TABLE I. Irreducible representations of the cyclic point group 3, little co-group of the propagation vectors τ± = (0, 0, ±τ ) in the space group P321, and associated invariant basis vectors. vν (θ, φ) symbolizes a vector associated to a Bravais lattice Lν at an angle θ from the c-axis and the projection of which in the perpendicular plane is at an angle φ from the a-axis.

v ν (π/2, φ ν )e -iξν + σǫ H v ν (π/2, φ ν + π/2)e -i(ξν -π/2) = 395 m a,b (x ν + iσǫ H ŷν )e -iξν
I σ ′ σ = I π ′ π = 0 , (2)
I π ′ σ = I σ ′ π ∝ sin 2 θ B .
where θ B is the Bragg angle. Noteworthy is the absence 474 of any azimuthal dependence. We therefore expect no

I R -I L =Im{F * σ ′ π F σ ′ σ + F * π ′ π F π ′ σ }.
the scattering vectors (0, 0, ±τ ) would be obtained only 502

I σ ′ σ = 0 , (3)
I π ′ σ = I σ ′ π = k 1 cos 2 θ B , I π ′ π = k 2 sin 2 (2 θ B ) ,
where the constants k i depend on the amplitude of the

F σ ′ σ = - 1 √ 6 Ψ 2 0 , (4)
F π ′ σ = F σ ′ π = 0 , F π ′ π ∝ 1 √ 6 (1 + cos 2 θ B ) Ψ 2 0 , F σ /F π = -1/(1 + cos 2 θ B ) .
A derivation of such relations is presented in the Appendix. Such a model suggests no azimuthal dependence in all the diffraction channels and a ratio I σ /I π = 0.6 in relative agreement with the azimuthal dependence shown in Fig. 5 with a χ 2 = 6.1. Agreement is improved (χ 2 = 2.2) by letting the ratio value vary as a free parameter, with the experimental value of 0.54±0.02, still reasonably close to the one derived by Eq.(4). However, such a ratio, as exemplified in Fig. 3, is not constant as a function of energy. These deviations might arise from a small symmetry break resulting in a loss of the 3-fold axis which would cause extra terms to appear in the structure factor. The latter has also been suggested recently by terahertz spectroscopy. 65 Experimental uncertainties are however too big to extract more quantitative conclusions on the presence of such contributions.

VII. CONCLUSION

We have studied the magnetic structure of the in- calculations. 64 The orbital magnetic signal observed at the oxygen K edge reflects the strong hybridization be-tween iron 3d and oxygen 2p states. Finally, the energy

57 S

 57 54 group (a = b = 8.539, c = 5.241, γ = 120 • ). It displays 55 an antiferromagnetic order below T N =27 K. The mag-56 netic moments are localized on the Fe 3+ ions (L ≃ 0, = 5/2). These occupy the Wyckoff position (3f) 58 (0.2496, 0, 0.5) with .2. site symmetry, forming trian-59 gular units in the a,b-planes. Elastic neutron scatter-60 ing studies 5 suggest that the same triangular configura-61 tion of co-planar moments at 120 • from each other is 62 stabilized within each triangle of an a,b-plane and that 63 this arrangement is helically modulated from a,b-plane 64 to a,b-plane along the c-axis according to the propaga-65

67

  tals are grown enantiopure and that the low tempera-68 ture magnetic structure is single domain, with a single 69 chirality of the triangular magnetic arrangement on the 70 triangles and a single chirality of the helical modulation 71 of the magnetic moments, which was dubbed helicity. 5 72 It was suggested that the Dzyaloshinsky-Moriya 14,15 ex-73 change interaction might be responsible for selecting the 74 ground state configuration 5 and for the opening of a small 75 gap in the magnetic excitation spectrum. 16 Another in-76 elastic neutron scattering study proposed the latter to 77 arise from single ion anisotropy, 17 but recent spin reso-78 nance experiments support the first scenario indicating 79 furthermore that not only the uniform component along 80 the c-axis of the Dzyaloshinsky-Moriya vector but also 81 its multiaxial component within the a,b-plane might be 82 sizeable. 18 The latter could generate an additional com-83 ponent to the magnetic structure not necessarily detected 84 by neutron scattering. To find evidence for such a mag-85 netic motif we have used resonant x-ray diffraction at the 86 Fe L edges. Our results show clear deviations from the 87 magnetic structure previously proposed, confirming the 88 existence of such a component. Powders of Ba 3 NbFe 3 Si 2 O 14 were synthesized by solid state reaction from stoichiometric amounts of Nb 2 O 3 , Fe 2 O 3 , SiO 2 oxides and BaCO 3 barium carbonate, at 1150 • C in air within an alumina crucible. The reagents were carefully mixed and pressed at 1GPa to form compact cylinders before annealing. The phase purity was checked by x-ray powder diffraction. Single crystals were grown from the as-prepared polycrystalline cylinders by the floating-zone method in an image furnace. 19 The single crystal used in the present investigation was extracted from the same batch as the one used in Ref. 5 and has the same structural chirality ǫ T , to be precise ǫ T = -1. After polishing the surface perpendicular to the [001] direction it was annealed to improve the surface quality. We have performed resonant x-ray diffraction experiments at the Fe L 2,3 edge. These energies correspond to a wavelength of approximately 17 Å and are associated to an electric dipole resonance from the iron 2p to 3d levels. Experiments were performed with the RE-SOXS chamber 20 at the X11MA beamline 21 of the Swiss Light Source. The twin Apple undulators provide linear, horizontal π and vertical σ, and circularly, right R and left L, polarized x rays with a polarization rate close to 100%. The polarization of the diffracted beam was not analyzed. The sample was attached to the cold finger of an He flow cryostat with a base temperature of 10 K. Azimuthal scans were achieved by rotation of the single crystal, with an accuracy of approximately ±5 • .

37 146

 37 FIG. 1. (Color online) a) Perspective view of the magnetic structure as suggested by neutron diffraction experiments. Fe ions are in black and different colors are used for the moments on the three Bravais lattices. b) (left to right) shows different types of magnetic ordering: a simple spiral, a ferromagnetic (conical) spiral, a complex spiral (or butterfly) and a static longitudinal wave. c) Pictorial view of the magnetic structure suggested by the present study. The dark colored moments describe the same pattern as in a). The light colored moments represent the magnetic structure as the sinusoidal modulation along the c-axis is superimposed to the basal helical order previously reported. Cones visualize the rotation of the magnetic moments about the c-axis and visualize the change in the modulation amplitude. Shaded areas are parallel to (00ℓ) planes. Note that the tilting out of the a, b-plane is exaggerated for clarity.

  FIG. 2. (Color online) Scan along the [001] direction in reciprocal space at an incident photon energy of 709.8 eV corresponding to the Fe L3 edge. r.l.u. denotes reciprocal-lattice units. Dashed (black) line represents data collected at 8 K while the continuous (red) line represents data collected above TN at 32 K. The peak visible in the vicinity of 0.5 r.l.u. corresponds to higher harmonic contamination from the (001) reflection.

  FIG. 3. (Color online)Intensity versus energy of the three satellite reflections in the vicinity of the Fe L3 edge. Spectra collected with incident π [(blue) square] and σ polarizations [(red) filled circle] at 10 K. Spectra are scaled [(0, 0, τ ) and (0, 0, 3τ ) were multiplied by 2.5 and 80 respectively] and shifted for clarity and lines are guides to the eye. The reflectivity contribution has been evaluated and subtracted by performing the same scan above TN . The black continuous line represents the sample absorption spectra collected in fluorescent mode.

197=0

  FIG. 4. (Color online) Azimuthal angle dependence of theIσ/Iπ (red) and IL/IR (black) ratio for the (0, 0, τ ) magnetic reflection. The (black) line represents the predictions of the model described in the text (χ 2 = 4.0 for comparison with both dataset, χ 2 = 1.5 for the ratio IL/IR alone). Measurements were performed in the vicinity of the Fe L3 edge (E=709.8 eV). The azimuthal angle equals zero when the [100] direction is in the scattering plane.

284A

  FIG.6. (Color online) Normalized integrated intensity vs temperature of the three satellite reflections. The solid lines show the best fit to power-law behavior Inτ ∝ (TN -T ) 2βn . The dashed line is the expected mean-field theory dependence. The (0, 0, 2τ ) satellite is 7 times more intense than the (0, 0, τ ). The same ratio holds between the (0, 0, τ ) and the (0, 0, 3τ ) satellites. Data was measured with π incident photon energy of 710 eV.

  FIG. 7. (Color online) Intensity [(red) circle] vs energy of the (0, 0, τ ) satellite reflection at the oxygen K edge collected at 10 K with π incident x rays. The fluorescence spectra [(blue) open square] obtained in the vicinity is also shown. Full (red) circle results from a fit of the integrated intensity of a reciprocal lattice scan along the c* reciprocal lattice direction. Open (red) circle are a result of an energy scan with fix momentum transfer. The (black) continuous line is a Gaussian fit of the oxygen resonance with a FWHM = 1.4±0.1 eV.

327ing.

  The analysis is simplified by the fact that the space 328 group P321 associated with the paramagnetic phase is 329 symmorphic. It is, to be precise, a semi-direct product 330 of the abelian translation group associated with a hexag-331 onal lattice and the dihedral point group 32, which con-332 sists of the identity 1, the anti-clockwise rotation 3 + and 333 the clockwise rotation 3 -= (3 + ) 2 about the ternary c-334 axis and the dyads (π-rotations) about the three binary 335 axes at 120 • to each other within the a,b-plane. A vector 336 along the reciprocal c ⋆ -axis is reversed under the dyads 337 and is invariant otherwise. It follows that the star of the 338 magnetic propagation vector consists of the two vectors 339 τ ± = (0, 0, ±τ ) each being associated with the little space 340 group P3, which is a semi-direct product of the transla-341 tion group of the paramagnetic phase and the abelian 342 cyclic point group 3. An abelian group G of n G elements 343 has n G conjugacy classes (each being reduced to a single-344 ton owing to the commutativity), which implies that it 345 has n G irreducible representations Γ i (i = 1, ... , n G ). It 346 follows that these are necessarily all of dimension d i = 1, 347 to comply with the identity nG i=1 d 2 i = n G . Each Γ i co-348 incides then with its character χ i . The value of χ i on any 349 group element g is an n G -th root e i2πp/nG (p = 1, ... , n G ) 350 of 1, because the order of g always divide n G . The charac-351 ter table is then built by making use of the orthogonality 352 theorems. The basis vector of the invariant subspace of 353 each Γ i is also easily deduced by applying the projection 354 operator P i = di nG g∈G χ i (g) ⋆ g on trial vectors. Table

358 an irreducible representation of the translation group 359 and determines a dephasing of moments within each 360

 359360 Bravais lattice. Information on the phase relations be-361 tween moments of distinct Bravais lattices can be ex-362 tracted only from the irreducible representations of the 363 little co-group. Three Bravais lattices L ν (ν = 1, 2, 364 3) are associated with the positions (0.2496, 0, 0.5), 365 (0, 0.2496, 0.5) and (-0.2496, -0.2496, 0.5) of the Fe 3+ 366 ions on the 3f site. Under the symmetry operation 3 + 367 a moment of L 1 (resp. L 2 , L 3 ) is rotated by an an-368 gle of 120 • about the c-axis and is transported into L 2 369 (resp. L 3 , L 1 ) whereas under the symmetry 3 -it is ro-370 tated by an angle of 240 • about the c-axis and is trans-371 ported into L 3 (resp. L 1 , L 2 ). This defines a repre-372 sentation Γ of the cyclic group 3 of dimension 9 whose 373 character χ takes the values χ(1) = 9, χ(3 + ) = 0 and 374 χ(3 -) = 0 on the group elements. Γ reduces into irre-375 ducible components as : Γ = 3Γ 1 ⊕3Γ 2 ⊕3Γ 3 . A magnetic 376 structure can be most generally regarded as composed 377 of several sine-wave amplitude modulations of moments: 378 1 2 ( v ν (θ ν , φ ν )e -iξν e -i τ±• rνn + c.c.), where r νn = r ν + R n 379 defines the position of the moment of L ν in the n-th unit 380 cell, ξ ν stands for an initial phase and c.c. means to 381 take the complex conjugate. The reduction of Γ then 382 suggests that, whatever the selected irreducible repre-383 sentation Γ i , three independent directions of the mo-384 ments are allowed by symmetry and can be combined, 385 for instance along two orthogonal unit vectors in the a,b-386 plane, xν = (π/2, φ ν ) at an angle φ ν from the a-axis and 387 ŷν = (π/2, φ ν +π/2) at an angle φ ν +π/2 from the a-axis, 388 and along the unit vector ẑν = (0, 0) of the c-axis, with 389 possibly vectors v ν (θ ν , φ ν ) of different lengths. 390 VI. DISCUSSION 391 It was shown, 5 from collected neutron diffraction inten-392 sities, that a helicoidal modulation is stabilized within 393 each L ν , associated with a combination of the form 394

  with σ = +1 for τ + and σ = 396 -1 for τ -. It is implicitly assumed that the vectors 397 v ν (π/2, φ ν ) = m a,b xν and v ν (π/2, φ ν + π/2) = m a,b ŷν 398 have the same length m a,b , which leads to a circular 399 helix. An elliptic helix would have been obtained oth-400 erwise, which a priori cannot be excluded. ǫ H = ±1 401 defines the magnetic helicity, that is to say the sense of 402 the rotation of the moments in the helix as one moves 403 along the propagation vector: m( r νn ) × m( r νn + c) = 404 ǫ H m 2 a,b sin(2πτ )( c/| c |) whatever the chosen description 405 between τ + and τ -. If we impose φ ν=2,3 -φ 1 according 406 toTable I then we must have ξ 1 = ξ 2 = ξ 3 , which can 407 be set to 0, together with φ 1 , without loss of general-408 ity. Table I illustrates that a triangular configuration of 409 the moments on each triangle is associated with Γ 1 with 410 a magnetic triangular chirality +1, that is to say with 411 an anti-clockwise sense of the rotation of the moments 412 as one moves anti-clockwise on a triangle. A triangu-413 lar configuration of the moments on each triangle with 414 the opposite magnetic triangular chirality -1, that is to 415 say with a clockwise sense of the rotation of the mo-416 ments as one moves anti-clockwise on a triangle, emerges 417 from Γ 2 (resp. Γ 3 ) when ǫ H = +1 (resp. ǫ H = -1), 418 in which case Γ 3 (resp. Γ 2 ) describes a ferro-collinear 419 configuration of the moments on each triangle. Inten-420 sity asymmetry of the pairs K ± τ of magnetic satel-421 lites about reciprocal nodes K indicated that for, a left-422 handed structural chirality ǫ T = -1, if ǫ H = -1 then 423 Γ 1 is selected and if ǫ H = +1 then Γ 2 is selected. This424 interdependence of the dephasing of moments within and 425 between the Bravais lattices L ν was explained as arising 426 from the twist in the exchange paths connecting the mo-427 ments of consecutive a,b-planes, which depends on the 428 structural chirality ǫ T and imposes the magnetic trian-429 gular chirality ǫ T ǫ H . X-ray anomalous scattering con-430 firmed that the structural chirality of the investigated 431 crystal is ǫ T = -1. Neutron spherical polarimetry finally 432 demonstrated that only the magnetic helicity ǫ H = -1, 433 and therefore only the (ǫ H , ǫ T ǫ H ) = (-1, +1) magnetic 434 helicity-triangular chirality pair, is selected, which was 435 ascribed to the uniform Dzyaloshinsky-Moriya interac-436 tions with the Dzyaloshinsky-Moriya vectors all along the 437 c-axis. This model 5 was later confirmed by polarized neu-438 tron inelastic scattering with polarization analysis, which 439 allowed probing both the symmetric and antisymmetric 440 nature of the dynamical correlations associated with the 441 magnon excitations emerging from the magnetic order. 16 442 A crucial point of the reported model of the circular 443 helices with moments within the a,b-plane is that the 444 dephasing of the moments associated with the triangu-445 lar configuration of moments on each triangle leads to 446 zero magnetic structure factors at the scattering vec-447 tors (0, 0, ±τ ). One however may recall that the neu-448 trons detect only the components of the moments per-449 pendicular to the scattering vectors. An additional 450 sine-wave amplitude modulated component along the c-451 axis of the moments is therefore not to be excluded, 452in which case we would rather have the combination453 v ν (π/2, φ ν )e -iξν + σǫ H v ν (π/2, φ ν + π/2)e -i(ξν -π/2) + v ν (0, 0)e -iξ ′ ν = m a,b (x ν +iσǫ H ŷν )e -iξν +m c ẑν e -iξ ′ ν . The length m c of the vector v ν (0, 0) should however be small 456 enough so that the neutron intensities to which it should 457 give rise at the other scattering vectors, (h, k, ℓ ± τ ) with 458 h = 0 or k = 0, are drowned beneath the statistical un-459 certainties of the neutron intensities associated with the 460 main helical modulation component. Table I actually il-461 lustrates that this c-component of the moments would 462 lead to a zero magnetic structure factor for the scatter-463 ing vectors (0, 0, ±τ ), and therefore would no longer be reducible representation is either Γ 2 or Γ 3 . A non-zero 466 magnetic structure factor vectorially oriented along the 467 c-axis is computed only in the case of the irreducible rep-468 resentation Γ 1 : F Γ1 m = (0, 0, f z ). The magnetic intensity 469 I ǫ ′ ǫ = F ǫ ′ ǫ F * ǫ ′ ǫ ( * stands for complex conjugation) in the 470 different diffraction channels 61 (ǫ = σ, π and ǫ ′ = σ ′ , π ′ ), 471 associated with this amplitude modulated c-component, 472 can be calculated with the help of Eq. (1) leading to 473

475

  modulation of the intensity as we rotate the sample about 476 the scattering wave vector. Moreover, we expectI σ = 477 (I σ ′ σ + I π ′ σ ) = I π = (I σ ′ π + I π ′ π ) and I R = I L . The478 latter equality can be derived from Eq. (A1) in Ref. 62 479 which states

486

  if either the Γ 2 irreducible representation or the Γ 3 irre-487 ducible representation were to be stabilized as the main 488 helical modulation component of the magnetic structure, 489 but this is ruled out from the neutron diffraction data. 490 A mixing of the irreducible representation Γ 1 with the 491 irreducible representation Γ 2 (or Γ 3 ) finally is a priori not 492 to be excluded, though this would imply that the mag-493 netic transition is necessarily first order. Nevertheless, 494 the additional magnetic component should be extremely 495 tiny to escape standard powder neutron detection, since 496 it should lie in the a,b-plane to produce a non zero mag-497 netic structure factor. In the case of the ferro-collinear 498 configuration in the a,b-plane, associated with irreducible 499 representation Γ 2 for ǫ H = -1, which gives rise to a mag-500 netic structure factor of the form F Γ2 m = (f x , f y , 0), one 501 calculates with the help of Eq. (1) the intensities:

503

  component of the moments associated with the irre-504 ducible representation Γ 2 and their orientation in the 505 a,b-plane with respect to the moments associated with 506 the main irreducible representation Γ 1 . Even in this 507 case there is no azimuthal angle dependence, but we find 508 I σ < I π and I R = I L . Including both Γ 1 and Γ 3 con-509 tributions will lead to an azimuthal angle dependence in 510 the rotated channels and again I σ < I π and I R = I L . 511 We are now in the position to compare the x-ray ex-512 perimental data with the prediction from representation 513 theory. Fig. 4 shows that the ratio I R over I L is con-514 stant as a function of the azimuthal angle and equals 515 one. Also the ratio I σ over I π is roughly constant within 516 the error bars and is very close to one. It is thus clear 517 that no mixing of irreducible representations is detected 518 and that the magnetic structure abides by only the irre-519 ducible representation Γ 1 but involves components of the 520 moments along the three orthogonal direction in space. 521 As a whole it consists of moments in a triangular arrange-522 ment on each triangle in the a,b-plane helically modu-523 lated along the c-axis and exhibiting small up and down 524 oscillations along the c-axis in phase with each other and 525 with the same period as the helical modulation, as de-526 picted for a single helix in Fig. 1b and for the three 527 lattices in Fig. 1c. Such a motif is reminiscent of the 528 beatings of butterfly wings (although these wings here 529 are three in number and not four), that lead us to dub 530 it as "helical-butterfly". The existence of the butterfly 531 component is consistent with the Dzyaloshinsky-Moriya 532 interactions. Owing to the presence of the three 2-fold 533 axes at 120 • of each other in the a,b-plane, each being 534 perpendicular to one of the three sides of every trian-535 gle of moments, the Dzyaloshinsky-Moriya vector asso-536 ciated with each pair of moments must by symmetry lie 537 within the plane containing the link connecting the two 538 moments. 63 The Dzyaloshinsky-Moriya vector field may 539 therefore have a uniform component along the c-axis and 540 a multi-axial component along the side of each triangle. 541 It is this last component that gives rise to the butterfly 542 component. It has been suggested that its contribution 543 might be significant 64 if not dominating. 18 544 Let us now analyze the azimuthal-angle dependence of 545 the (0, 0, 2τ ) reflection. According to the Γ 1 magnetic 546 structure factor F Γ1 m = (0, 0, f z ) and the formalism to cal-547 culate magnetic diffraction intensity in Ref. 28 we should 548 observe intensity only in the unrotated π ′ π scattering 549 channel which is at odds with the data shown in Fig. 5. 550 To reconcile the observations with theoretical prediction 551 we must adopt a more sophisticated model which does 552 not rely on the fact that the resonant ion environment is 553 cylindrically symmetrical. We need a tensorial structure 554 factor Ψ K Q where the positive integer K is the rank of the 555 tensor, and the projection Q can take the (2K +1) integer 556 values which satisfy -K ≤ Q ≤ K. For a dipole tran-557 sition, tensors up to rank 2 contribute (K ≤ 2). K = 0 558 reflects charge contribution, K = 1 time-odd dipole, and 559 K = 2 time-even quadrupole. For our superstructural re-560 flection we are interested in the quadrupolar contribution and given the presence of the 3-fold axis parallel to the c-axis we have Ψ K Q (0, 0, 2τ ) = (-1) 2τ T K Q [1 + 2 cos(Qα)] which is non-zero only for Q = 0. T K Q is an atomic tensor that describes the contribution of each atom to the structure factor. Making use of the formula in appendix C of Ref. 41 we obtain the following results for the structure factor in the different polarization channels:

triguing compound Ba 3

 3 NbFe 3 Si 2 O 14 with resonant x-ray diffraction at the Fe L edges and O K edge. These experiments give new insight into the details of the magnetic structure recently determined by neutron diffraction. Our experiments have found an extra sinusoidal modulation of the Fe magnetic moments along the crystallographic c-axis, concomitant with the helical order in the a,b-plane, generating an helical-butterfly magnetic structure. Such sinusoidal modulation arises from the Dzyaloshinsky-Moriya interaction as suggested by symmetry consideration and recent linear spin-wave theory

Limiting ourselves to the quadrupolar contribution 617 (K=2) and taking advantage of the structure factor 618