G Subías

J García

J Blasco

J Herrero-Martín

M C Sánchez

Resonant x-ray scattering in 3d-transition-metal oxides: Anisotropy and charge orderings

The structural, magnetic and electronic properties of transition metal oxides reflect in atomic charge, spin and orbital degrees of freedom. Resonant x-ray scattering (RXS) allows us to perform an accurate investigation of all these electronic degrees. RXS combines high-Q resolution x-ray diffraction with the properties of the resonance providing information similar to that obtained by atomic spectroscopy (element selectivity and a large enhancement of scattering amplitude for this particular element and sensitivity to the symmetry of the electronic levels through the multipole electric transitions). Since electronic states are coupled to the local symmetry, RXS reveals the occurrence of symmetry breaking effects such as lattice distortions, onset of electronic orbital ordering or ordering of electronic charge distributions. We shall discuss the strength of RXS at the K absorption edge of 3d transition-metal oxides by describing various applications in the observation of local anisotropy and charge disproportionation. Examples of these resonant effects are (I) charge ordering transitions in manganites, Fe 3 O 4 and ferrites and (II) forbidden reflections and anisotropy in Mn 3+ perovskites, spinel ferrites and cobalt oxides. In all the studied cases, the electronic (charge and/or anisotropy) orderings are determined by the structural distortions.

Introduction

The discovery of new electronic properties, including high-T C superconductivity, colossal magnetoresistance and multiferroic modifications, in transition-metal oxides has fuelled a resurgence of interest in atomic charge, spin and orbital degrees of freedom in systems of highly correlated electrons. X-ray spectroscopy techniques are highly sensitive to these electron degrees of freedom. However, they probe short-distance correlations only. In contrast, x-ray diffraction reveal long-range ordered static correlations. Resonant x-ray scattering (RXS) combines absorption and diffraction as they have in common the x-ray atomic scattering factor (ASF), f, which is usually written as: f = f 0 + f '+ if " [START_REF] Materlik | Resonant Anomalous X-ray Scattering[END_REF]. It contains an energy independent part, f 0 , corresponding to the classical Thomson scattering and two energy-dependent terms, f ' and f ", also known as the anomalous ASF. RXS occurs when the x-ray energy is tuned near the absorption edge of an atom in the crystal. In this case, the anomalous ASF strongly depends on the photon energy, which manifests in marked variations of the scattered intensity. This dependence of the scattered intensity appears in any Bragg reflection when crossing the absorption edge of a constituent atom.

The study of the energy-dependent modulation of the diffraction intensity of intense Bragg peaks is the scope of the diffraction anomalous fine structure technique (DAFS) [START_REF] Stragier | [END_REF]3]. This technique allows the determination of the local structural information around the anomalous atom that is chemical and valence specific similar to that of Extended X-ray Absorption Fine Structure (EXAFS). The advantage of DAFS is that it is spatially and site-selective. Our interest here is focused on the study of either weak-allowed or forbidden reflections that appear in a phase transition. In the first case, the anomalous scattering contribution is comparable to the Thomson one and the peculiar characteristics of RXS can be studied. The intensity of weak superlattice reflections can be due to either a structural modulation, contributing to the structure factor as a Thomson term or because the anomalous ASF of atoms, now in different crystallographic sites, differ in some energy range. Normally, these differences are larger for photon energies close to an absorption edge, showing an enhancement (or strong decrease) of the scattered intensity just around the absorption threshold. In the case of symmetry forbidden reflections, only the resonant term is present. Since RXS involves virtual transitions of core electrons into empty states above the Fermi level, the excited electron is sensitive to any anisotropy around the anomalous atom so that the anomalous ASF has a tensorial character. Thus, a reflection can be observed on resonance if any of the components of the structure factor tensor is different from zero. In the case of weak superlattice reflections, the resonant atoms occupy different crystallographic sites and have different local structures. Thus, the anomalous ASF are different at energies close to the absorption edge and the scattered intensity shows a resonance reflecting the differences of the ASF between these non-equivalent atoms. RXS intensity is then observed coming from the difference between the diagonal terms of the ASF. In this case, there is a Thomson contribution and the analysis of the spectral shape must include it. On the other hand, resonances observed in forbidden reflections are related to atoms that occupy equivalent crystallographic sites. Symmetry elements with translation components (screw axes and glide planes) of the crystal space group transform an atom into another equivalent in the lattice, but with a differently oriented atomic surrounding. This makes that some of the offdiagonal terms change sign after these symmetry operations, giving rise to structure factor tensors that contain non-vanishing off-diagonal terms. Templeton & Templeton [4] first noticed such reflections that show a local polarization anisotropy of the x-ray susceptibility and they are now known as ATS reflections. We note that ATS reflections have strong polarization and azimuth dependences.

The vector potential of the electromagnetic field in the matter-radiation interaction term can be developed in a multipolar expansion in such a way that the symmetry of the excited levels can be chosen. Thus, we can speak on dipolar-dipolar transitions, dipolar-quadrupolar transitions, etc. Consequently, resonant methods are also sensitive to the symmetry of the electronic shells, which compose the intermediate states. For electric dipole-dipole (E1E1) transitions the wave vectors k i and k f of the incident and scattered x-rays do not enter the scattering amplitude. The strength of the x-ray resonances associated with electric quadrupolequadrupole (E2E2) and electric dipole-quadrupole transitions (E1E2) are less intense than for E1E1 process but must content the k i and k f dependence. A complete polarization and azimuthal analysis of the RXS experiment is needed to assign the multipolar origin of the RXS signal.

In this contribution, after a recall of the RXS amplitude, we will discuss several examples that correlate with the charge and orbital ordering concepts. We restrict to the metal K edges and mainly to the dipolar-dipolar channel, which describes most of the phenomenology. Two types of reflections are observed. The first one originates from the different anomalous scattering factors of non-equivalent atoms in the crystal. The occurrence of this type of resonance is correlated to the energy shift of the absorption edge (chemical shift) and it has been considered as a proof of charge ordering (CO). The second one arises from the anisotropy of the anomalous scattering factor of an atom at crystallographic equivalent sites. These ATS reflections have been assigned to orbital ordering (OO). However, the first type of reflections can also exhibit anisotropic behaviour. In all the studied cases, we conclude two main results: (i) the charge segregation is much smaller than one electron and consequently, it would be better described as a charge modulation and (ii) the anisotropy is present in the p-empty density of states and it seems not to be correlated with a real d-orbital ordering.

Resonant x-ray scattering tensor

In RXS, the global process of photon absorption, virtual photoelectron excitation and photon reemission, is coherent through the crystal, giving rise to the usual Bragg diffraction condition

) ( " ' 0 if f f e j F j j j j R Q i + + ∑ = ⋅ r r (1)
where R j r

is the position of the j-th scattering atom in the unit cell,

i f k k Q r r r - =
is the scattering vector ( and are the wave vectors of the incident and scattered beams) and is the Thomson scattering part of the atomic scattering factor. The resonant part, , is given by the expression [START_REF] Blume | Resonant Anomalous X-ray Scattering, Theory and Applications[END_REF] i k r

f k r j f 0 " ' if f + ∑ Γ - - - 〉 〉〈 〈 - = + n n g n g i n n f g g n e i E E O O E E m f i f 2 ) ( ) ( 1 * 3 2 " ' ω ψ ψ ψ ψ ω h h h (2)
In this expression, ω h is the photon energy, is the electron mass, ψ . The interaction of the electromagnetic radiation with matter is expressed by the operators and . By multipole expansion of these operators up to the electric quadrupole term, we have [START_REF] Hannon | [END_REF]:

i O ˆ* ˆf O ) 2 1 1 ( ˆ) ( ) ( ) ( r k i r O f i f i f i r r r r ⋅ - ⋅ = ε (3)
Here r r is the electron position measured from the absorbing atom and is the polarization of the incident(scattered) beam. Correspondingly, we get that there are three contributions to the resonant scattering factor: dipole-dipole (dd), dipole-quadrupole (dq) and quadrupole-quadrupole (qq).

) ( f i ε r
Using the Cartesian reference coordinate system defined in figure 1 for the photon polarization and wave vector, we can develop the scalar product of equation (3) and the resonant scattering amplitude of equation ( 2) can be written in the form:

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∑ ∑ ∑ + - - ∑ Γ - - - - = + β α αβγ αβγδ αβγδ δ γ β α βαγ γ αβγ γ β α αβ β α ε ε ε ε ε ε ω ω , * * * * 3 2 " ' 4 1 ) ( 2 2 ) ( ) ( 1 Q k k I k I k i D i E E E E m f i f i f i f f i i f i f n n g n g n e h h h (4)
where α, β, γ, δ are indexes that vary independently over the three Cartesian directions x, y, z, and the transition matrix elements , and associated to dd, dq and qq contributions are characterized by the following Cartesian tensors of second, third and fourth rank, respectively: It is important to determine the symmetry properties of the D, I and Q tensors [7] that depend on two factors: (i) the transformation properties of a space group itself and (ii) the local symmetry of the resonant atoms position. Referring to non-magnetic samples, the structure factor tensor is a second rank symmetric tensor (parity-even) with six independent components in the dd approximation. This number reduces upon taking into account the local site symmetry of the atom. The qq term is also symmetrical under the inversion symmetry, whereas the dq contribution is only antisymmetrical. As a result, if an atom sits in an inversion centre, the I tensor must be zero and the dipole-quadrupole transition is only allowed for atoms breaking the inversion symmetry.

αβ D αβγ I αβγδ Q 〉 〉〈 ∑ 〈 = 〉 〉〈 ∑ 〈 = 〉 〉〈 ∑ 〈 = g n n n g g n n n g g n n n g r r r r Q r r r I r r D ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ δ γ β α αβγδ γ β α αβγ β α αβ (5)
In the following we shall describe some significant RXS experiments following the two types of resonant reflections, weak-allowed and forbidden. In the first case, we shall relate them to the charge modulation, intimately joined to the transition metal-ligand bond distances and in the second case we shall relate them to the anisotropy of the geometrical local structure around the resonant atoms. Since the charge is a time-reversal invariant quantity, we shall deal either with pure electric dd and qq transitions that are even under parity or with parity-odd electric dq transitions. Time-reversal odd events that are related to the magnetic properties of the system will be not considered.

Resonant effects due to different crystallographic sites (charge) ordering

Understanding the charge state in the high and low temperature phases of mixed valence transition-metal oxides is of fundamental interest in the context of metal-insulator transitions that are assumed to be driven by CO. The sensitivity of RXS to the CO relies on the fact the energy values of the absorption edge for the two different valence states of the transition-metal atom are slightly different (known as chemical shift). If long-range order of these valence states exists, superlattice reflections due to the contrast between the atomic scattering factors of the two valence states will exhibit a resonance enhancement. The point is that CO is intimately correlated with the associated crystal distortions coming from the structural transitions that accompanied the metal-insulator ones. Thus, a question arises whether the electronic CO is the cause or the effect of the lattice distortions.

The concept of CO in solids was first applied by E. J. W. Verwey to the metal to insulator transition that occurs in magnetite (Fe 3 O 4 ) at T v ∼ 120 K, now known as the Verwey transition [8]. Above T v , Fe 3 O 4 has the inverse spinel cubic AB 2 O 4 structure, where A and B are the tetrahedral and octahedral Fe sites, respectively. Verwey originally proposed that the hopping of valence electrons on the octahedral B-site sublattice is responsible for metallic conductivity. In the insulating phase, spatial localization of the valence electrons on these B-sites gives rise to an ordered pattern of Fe 3+ and Fe 2+ ions in successive [001] planes (cubic notation). The B-site sublattice, shown in the inset of figure 2, can be regarded as a diamond lattice of tetrahedra of nearest-neighbour Fe atoms sharing alternate corners. This simple model was implying the observation of (0,k,l) reflections with k+l=4n+2 in the low temperature phase. These reflections are forbidden above T v because of the diamond glide plane of the spinel structure. The first RXS experiments in magnetite showed that (002) and (006) reflections belonging to this type of cubic forbidden reflections originates from the local structural anisotropy of the Fe atoms at the Bsites that have a trigonal point symmetry ( 3 m) [9][10][11]. These works discarded the Verwey's CO model but they did not guarantee the lack of CO with other periodicities of the cubic unit cell.

In order to investigate other possible CO periodicities, we need to start from the low temperature crystallographic structure. The symmetry lowering Cc m Fd → 3 generates 8 and 16 non-equivalent Fe sites at tetrahedral and octahedral positions, respectively; each one can have its own local atomic charge. However, the exact structure is not yet perfectly known [12][13][14]. A good approach to the real structure consists of a cell with lattice parameters

c P / 2 c c c a a a 2 2 / 2 / × × ≈
, a c being the cubic cell parameter [12,14]. Complexity is greatly reduced because there are only 6 non-equivalent Fe atoms, two in tetrahedral sites (A 1 and A 2 ) and four in octahedral sites (B 1 , B 2 , B 3 and B 4 ). It can be noticed that atomic displacements in this low temperature structure result in two main types of superlattice reflections, which are indexed in the cubic notation: (I) (h,k,l) reflections such that h+k=even resulting form the loss of the translation that give rise to charge modulations with wave vector

fcc c q ) 1 , 0 , 0 ( = r
and (II) half-integer (h,k,l+1/2) reflections arising from the doubling of the cell along the c axis that corresponds to charge modulation with c q ) 2 / 1 , 0 , 0 ( = r . We examine now the limit for the possible charge segregation over the octahedral atoms along the c axis given by the sensitivity of the RXS technique in a highly stoichiometric single crystal of Fe 3 O 4 (T V =123.5 K) [15].

Figure 2 (left panel) shows the energy dependence of the intensities for some characteristic Bragg and forbidden reflections at the Fe K-edge and at 60 K compared to the fluorescence spectrum. Experimental data have been corrected for absorption. (a) Energy scans of the permitted (0,0,1) and (1,1,0) reflections show three resonant peaks, a first one at the Fe K threshold (7118 eV) and the other two around 7124-7129 eV, which corresponds to the white line in the fluorescence spectrum. The observed strong resonant effect is a consequence of electronic and structural differences among the Fe atoms at B1 and B2 octahedral sites. This can be parameterised in terms of valence as a charge segregation δ=0.23e. This result confirms the lack of ionic CO in terms of Fe 2+ and Fe 3+ ions, in agreement with previous RXS [16][17][18] and synchrotron powder diffraction [14] studies.

(b) Resonant intensity is only observed in the σ-π' channel for the (0,0,7/2) reflection, indicating that this is a forbidden reflection in the low temperature phase. The energy scan shows a three-peak structured resonance nearly at the same energies as the (0,0,1) and (1,1,0) reflections. In this case, the electronic anisotropy comes from interference among equivalent crystallographic sites. Six different sites are present for the Fe atoms so up to six different terms could contribute to the resonant signal [19]. The superlattice (4,4,3/2) corresponds to the same periodicity along c as the forbidden (0,0,l/2) reflections. However, it displays hardly any resonant effect opposite to what is expected to occur at those very weak reflections. Therefore, we can conclude that no charge segregation exists with (0,0,1/2) periodicity and the ATS (0,0,l/2) reflections have its origin in the loss of octahedral and tetrahedral symmetry originated by the structural phase transition.

In order to establish the correlation between the lattice distortion and the charge segregation and anisotropy orderings, we have measured the temperature dependence of the intensity of the following superlattice reflections: (4,0,1/2) off-resonance (E=7.1 keV) and (0,0,1) and (0,0,7/2) on resonance (E=7.125 keV), which is reported in figure 2 (right panel). The resonant and nonresonant signals simultaneously disappear at T v (±0.5 K) and the intensity of all these reflections is zero at temperatures above 125 K. This result shows that RXS in Fe 3 O 4 comes from the ordering of local distortions at the structural transition, which leads to an ordered formal charge segregation and electronic anisotropy at the Fe atoms [15].

We will comment now on the half-doped manganites such as Nd 0.5 Sr 0.5 MnO 3 and Bi 0.5 Sr 0.5 MnO 3 . It was proposed the ordering of an alternating pattern of Mn 3+ and Mn 4+ ions was predicted leading to the onset of superlattice reflections doubling the b axis of the orthorhombic Pbnm (Ibmm) cell [20]. The observed modulation wave vectors are (0,k,0) and (0,k/2,0) with k odd for the ordering of charge and orbital degrees of freedom, respectively. The occurrence of orbital order (OO) is also predicted independently from the CO because Mn 3+ are Jahn-Teller distorted ions. We have measured the energy dependence of the intensity at the Mn K edge at Q=(0,3,0) and Q=(0,5/2,0) for both, Nd:Sr [21] and Bi:Sr [22] single crystals. Figure 3 summarizes the results obtained. Non-resonant intensity can be observed away from the K-edge of Mn and a broad resonance is found at the absorption edge at (0,3,0) reflections in the non-rotated (σ-σ') channel. On the other hand, a strong Gaussian-shaped resonance is observed at energies close to the K-edge of Mn at (0,5/2,0) reflections only in the rotated (σ-π') channel, which identifies these half-integer reflections as structurally forbidden ones. The variation with azimuth of these resonances shows a characteristic oscillation with π periodicity. We note that in this case, the two kinds of behaviour mixed since the marked different anisotropy of the two types of atoms. The resonant scattering at either (0,k,0) or (0,k/2,0) reflections does not depend on the wave vector and it arises from E1 (1s→ np) transition. The temperature dependence of the resonant intensities shows that the two types of reflections disappear at the metal-insulator phase transition.

k

The complete analysis of the energy line-shape and the azimuth and polarization dependence of the resonant intensities is carried out using a semi-empirical structural model [23]. The checkerboard arrangement of two types of crystal Mn sites gives an excellent agreement with the experimental data. These two sites differ in their local geometric structure: one site is anisotropic (tetragonal distorted oxygen octahedron) and the other site is isotropic (nearly undistorted oxygen octahedron). Estimates of the valence modulation between the two Mn atoms are slightly variable depending on the manganite but all fall below the ideal charge segregation of ±0.5e. As shown in [21] and [22], ±0.08 for Nd 0.5 Sr 0.5 MnO 3 and ±0.07 for Bi 0.5 Sr 0.5 MnO 3 , respectively.

CO for x different from 0.5 is in general not well defined. We have also explored further the Bi 1-x Sr x MnO 3 system toward the Mn3+-rich side, that is for x=0.37 [24]. Superlattice reflections of (0,k,0) and (0,k/2,0) types with k odd were observed at the Mn K-edge. This is to be compared with the observation of the K-edge resonances in Bi 0.5 Sr 0.5 MnO 3 . Figure 4 shows an example of the resonant enhancement as observed at (0,3,0) and (0,7/2,0) reflections in Bi 0.63 Sr 0.37 MnO 3 . The energy dependence of the intensity for both, weak-allowed (0,k,0) and forbidden (0,k/2,0) reflections is identical to that observed in the half-doped bismuth manganite (figure 3). The dependencies of the resonant intensities with azimuthal angle reveals identical twofold symmetry too. We note that the σ-σ' intensity of the (0,3,0) peak approaches the non-resonant intensity at the minimum opposite to the constant evolution expected for a pure CO reflection. These results indicate that the checkerboard ordering of two types of Mn atoms in terms of the local structure in the ab plane in a ratio 1:1 is strongly stable and extends to doping concentrations x<0.5. Intermediate valence states lower than +3.5 are deduced for Bi 0.63 Sr 0.37 MnO 3 , where the charge disproportion is found to be ±0.07.

The last example is provided by La 0.33 Sr 0.67 FeO 3 , which shows a charge modulation that is commensurate with the carrier concentration (n=1-x) and corresponds to a wave vector q=(2π/a p ) (1/3,1/3,1/3), being a p the primitive cubic lattice parameter. Ordered layers of Fe 3+ and Fe 5+ ions in a sequence of …Fe 3+ Fe 3+ Fe 5+ … along the cubic [111] direction were originally proposed to explain the metal-paramagnetic to insulator-antiferromagnetic transition at 200 K [25]. We have investigated this three-fold CO using the RXS technique [26]. Superlattice (h/3,h/3,h/3) reflections in the pseudocubic cell notation were observed in the non-rotated σ-σ' polarization channel for h=2, 4 and 5. These reflections are forbidden in both cubic (Pm-3m) and rombohedral (R-3c) symmetries. However, they exhibit a resonant behavior at energies close to the Fe K edge on top of a non-resonant signal, as shown in figure 5. Concerning the local charges of the two types of Fe cations, the chemical shift between the expanded and compressed Fe atoms was found to be 0.7±0.1 eV. This is to be compared to the energy shift of 1.26 eV between Fe 3+ and Fe 4+ [27]. Assuming a linear relationship between energy shift of the K absorption edge and local charge, we determined that the amount of charge segregation is 0.6e. Since the resonant intensity is constant as a function of the azimuth angle over a range of 180 º, as shown in the inset of figure 5, the Fe atoms have not a local anisotropy in this case. This result indicates the presence of a pure charge density wave ordering in the mixed-valent perovskite La 0.33 Sr 0.67 FeO 3 .

Resonant effects due to anisotropy (orbital) ordering

The direct observation of OO is a difficult task because it is accompanied by other effects such as lattice distortions and charge segregations. The study of the RXS and the anisotropic character of the x-ray scattering were developed in crystallography more than 20 years ago [4]. In particular, Dmitrienko [28,29] developed a general theoretical treatment of the "forbidden" ATS reflections, deriving new extinction rules valid near the absorption edge. In the microscopic point of view, these ATS reflections are due to the presence of the absorbing atom in an anisotropic chemical environment, which brings about the orientation of unoccupied electronic levels.

The experimental application of RXS to the study of OO started in 1998, when Murakami and co-workers [30] 3+ sites by detecting the (2n+1/4,2n+1/4,0) forbidden reflections. The angular dependence around the scattering vector of the intensity (azimuthal dependence) confirms that they result from the anisotropic character of the anomalous part of the atomic scattering factor. However, it was later demonstrated that the Jahn-Teller distortion of the oxygen octahedra surrounding the Mn atoms is sufficient to reproduce the experimental results at the K edge without invoking any contribution of the 3d-OO [31].After this first observation, RXS is widely used to study the orbital degree of freedom in several manganites [32][33][34][35][36] and other transition-metal oxides [37][38][39][40]. In particular, the following experiments provide key information for the mechanism of RXS and OO.

(1) The parent compound of colossal magnetoresistive manganites, LaMnO 3 , was expected to show an alternating ordering of and orbitals in the ab plane below 780 K due to the greatly distorted MnO ) 3 ( 2

2 r x - ) 3 ( 2 2
r y -6 octahedron and the A-type antiferromagnetic order [START_REF] Goodenough | Magnetism and Chemical Bond[END_REF]. The observation of resonance in the σ-π' scattered intensity of the (3,0,0) forbidden reflection at the Mn K-edge was initially interpreted as a direct probe of this OO [32]. Moreover, the increase of the resonant signal with decreasing temperature as T N was approached from above was interpreted as a direct correlation of the magnetic order with OO.

Recently, we have revisited the RXS at the Mn K-edge of LaMnO 3 , both experimentally and theoretically [36]. We have observed two independent forbidden reflections, (0,3,0) [or, equivalently (3,0,0)] and (0,0,3), which are related to two different nonzero off-diagonal elements of the second-rank atomic scattering factor tensor. The different energy dependence of the RXS spectra for the two types of forbidden reflections has been explained within the multiple scattering theory in terms of long-range ordered structural distortions around Mn atoms using a cluster that includes up to 63 atoms beyond the first oxygen neighbors [START_REF] Benfatto | [END_REF], as demonstrated in figure 6(a). Since no change in either the intensity or its azimuthal dependence was observed when crossing T N at 140 K (see figure 6 (2) Another example of the excitement of forbidden reflections on a cubic crystal structure that deserves discussion are the (0,0,l) (l=4n+2) reflections in Fe 3 O 4 . Once the Verwey model of CO was discarded, these reflections offer the possibility to study RXS excited by a dq-transition which is only allowed on the tetrahedral site whereas the octahedral site allows, on the contrary, only for dd and qq transitions. Two distinct resonant lines are observed in the RXS spectra of (0,0,2) and (0,0,6) forbidden reflections in Fe 3 O 4 at the pre-edge and at the main edge energies of the Fe K-edge [10]. The comparison of the energy and azimuthal dependence of these reflections at the Fe and Co K edges in These results confirm that the pre-peak resonance originates from dq transitions at the tetrahedral Fe atoms and the main-edge resonance is due to the anisotropy of the trigonal ( ) point symmetry of octahedral B sites of the spinel structure. The azimuthal dependence corroborates the dd character of the main-edge resonance, which does not depend on either the type or the formal valency of the transition-metal atom that occupies the octahedral site. We note here that we have observed anisotropic RXS, mainly of a structural origin, in a system where the local atom environment is nearly isotropic since there is not a significant distortion of the ligand configuration [43].

(3) Recently, the correlation of OO and magnetic ordering has been studied on layered cobalt oxides, RBaCo 2 O 5.5 with R=rare-earth [39,40]. Resonances have been observed at the Co K edge for (0,k,0) reflections with k odd in TbBaCo 2 O 5.5 whereas (h,0,0) reflections with h odd were not detected either on or off resonance in any phase transition [39]. The cusp of the resonant scattering is either up -(0,3,0) and (0,7,0)-or down -(0,1,0) and (0,5,0)-depending on the k value. This behavior arises from displacements of the Tb and Ba ions from the ideal tetragonal positions while the occurrence of RXS and its azimuthal dependence comes form the presence of two different environments for Co atoms (octhedron and pyramid), ordered along the b axis. Therefore, resonant (0,k,0) reflections with k odd corresponds to ATS reflections and no further contribution due to OO has been deduced from this experiment [40]. On the other hand, superlattice (1,0,l) with l even reflections were reported occurring at the metal-insulator transition in GdBaCo 2 O 5.5-x [40]. RXS was observed at these reflections, which is sensitive to the difference in anisotropy of adjacent Co pyramids and/or Co octahedral and it has been associated to the presence of OO.

Conclusions

RXS has demonstrated its potential to solve old and new problems on strong correlated transition-metal oxides. The results on the low temperature insulating phase of magnetite has reopened the discussion on the origin of the Verwey transition. The classical CO model has been overcome and many recent theoretical papers gives a more realistic interpretation without invoking to ionic ordering. In general, all these RXS results discard the description of mixedvalence transition-metal oxides as a bimodal distribution of integer valence states and the socalled CO phase implies the segregation in different crsytallographic sites whose electronic differences are mainly determined by the structural distortions. These structural distortions induces a different charge density on different crystallographic sites and in some cases, an electronic anisotropy by lowering the local symmetry. Finally, RXS has been also fundamental to determine the type of ordering in many cases, such as Bi 0.67 Sr 0.33 MnO 3 and La 0.33 Sr 0.67 FeO 3 .

Γ

  are the energy and inverse lifetime of the intermediate excited states n

Figure 1 .

 1 Figure 1. Scheme of a general RXS experiment (vertical scattering plane)

Figure 2 .

 2 Figure 2. The left panel shows some of the experimental RXS spectra around the Fe K-edge in Fe 3 O 4 , corrected for absorption. The right panel shows the temperature dependence of the integrated intensities, on and off-resonance, normalized to the low-temperature value.

Figure 3 .

 3 Figure 3. The left panel shows the normalized intensity for the non-rotated polarization channel at the (0,3,0) peak associated to CO whereas the right panel shows the rotated polarization channel at the (0,5/2,0) peak associated to OO in Nd 0.5 Sr 0.5 MnO 3 and Bi 0.5 Sr 0.5 MnO 3 .

Figure 4 .

 4 Figure 4. RXS results in Bi 0.63 Sr 0.37 MnO 3 at the Mn K-edge. (a) Polarization analysis of the (0,3,0) reflection as a function of photon energy. (b) energy dependence of the forbidden (0,7/2,0) reflection for the σ-π' channel. The insets show the respective azimuthal angle dependence on resonance.

Figure 5 .

 5 Figure 5. Energy scans of the σ-σ' intensity for the (h/3,h/3,h/3) reflections measured at 10 K and corrected for absorption. The inset shows no dependence of the resonant intensity (7129.5 eV) on the azimuth angle for (4/3,4/3,4/3) reflection.

  investigated the CO and OO in La 0.5 Sr 1.5 MnO 4 at the Mn K-edge. It was concluded the successful observation of OO of the electrons

Figure 6 .

 6 Figure 6. (a) Experimental (circles) RXS of the (0,3,0) and (0,0,3) forbidden reflections in LaMnO 3 . The inset shows the MXAN calculations done using the 63-atoms cluster (upper panel) and the 6-atoms cluster (lower panel). (b) Temperature dependence of the integrated intensity of the (0,3,0) and (0,0,3) reflections on resonance, normalized for comparison. The inset shows the respective azimuthal angle dependence on resonance.

Figure 7 .

 7 Figure 7. Energy scans of the (0,0,2) (closed symbols) and (0,0,6) (open symbols) forbidden reflections at the Fe K edge in the spinel ferrites. The inset compares the Co K-edge RXS of the (0,0,2) and (0,0,6) reflections in CoFe 2 O 4 .

Acknowledgement

The authors thank financial support from the Spanish MICINN (FIS08-03951 project) and DGA (Camrads). We also acknowledge ESRF for granting beam time and technical support.


Naouel Ben
email: naouel.benali@ensi.rnu.tn

Joanna Moulierac
email: joanna.moulierac@irisa.fr

Abdelfattah Belghith
email: abdelfattah.belghith@ensi.rnu.tn

Miklós Molnár
email: miklos.molnar@irisa.fr

mQMA: multi-constrained QoS Multicast Aggregation

Traditional IP Multicast has been proposed in order to manage group communications over the Internet in a bandwidth efficient manner. Although this proposition has been well studied, there are still some problems for its deployment. In this paper, we propose a new algorithm mQMA that deals with two important problems of traditional IP multicast, i.e., multicast forwarding state scalability and multi-constrained QoS routing. The algorithm mQMA builds few trees and maintains few forwarding states for the groups thanks to the technique of multicast tree aggregation, which allows several groups to share the same delivery tree. Moreover, the algorithm mQMA builds trees satisfying multiple QoS constraints. We show, trough extensive simulations, that mQMA leverages the same QoS performances as Mamcra which is the main multi-constrained multicast routing algorithm. Moreover, mQMA reduces dramatically the number of trees to be maintained.

I. INTRODUCTION

Recently, the Internet has shown a tremendous growth. Emergent multimedia applications like audio/video conferencing, video on demand, IP-telephony usually have other requirements than traditional Internet services such as e-mail and file transfer services. IP multicast techniques have been proposed to support group communications over the Internet with the aim of reducing the network resource consumption.

Although it has been studied for a long time, IP multicast is not well deployed over the Internet. The multicast forwarding state scalability problem and the lack of QoS support can be considered among the reasons for IP multicast not to be widely deployed.

Multicast tree aggregation [START_REF] Cui | Aggregated Multicast -A Comparative Study[END_REF], [START_REF] Guitton | Scalable Tree Aggregation for Multicast[END_REF] is a recent approach that deals with the problem of multicast forwarding state scalability. In this approach, multiple groups share the same delivery tree within a domain whereas in traditional IP multicast, a separate tree is built and maintained for each group. With the multicast tree aggregation, fewer trees are maintained which reduces both the number of forwarding states in routers and the overhead induced by multicast control messages. Some information has to be added in the border routers of the domain in order to multiplex the data for the groups onto an aggregated tree. Several algorithms have been proposed to perform tree aggregation: AM [START_REF] Cui | Aggregated Multicast -A Comparative Study[END_REF] (Aggregated Multicast), STA [START_REF] Guitton | Scalable Tree Aggregation for Multicast[END_REF] (Scalable Tree Aggregation) are the first proposed, BEAM [START_REF] Cui | BEAM: A Distributed Aggregated Multicast Protocol Using Bi-directional Trees[END_REF] deals with distributed multicast tree aggregation, AMBTS [START_REF] Liu | AMBTS: A Scheme of Aggregated Multicast Based on Tree Splitting[END_REF] and TALD [START_REF] Moulierac | Multicast Tree Aggregation in Large Domains[END_REF] deal specifically with tree aggregation in large domains.

However, these algorithms aggregate multicast groups without considering any QoS requirements. Aggregation is based on trees computed by IP multicast routing protocols which use the shortest path tree algorithm optimized on one single metric, typically the hop count. Two solutions have been proposed in the literature to deal with QoS multicast aggregation: AQoSM [START_REF] Cui | AQoSM: Scalable QoS multicast provisioning in Diff-Serv networks[END_REF] and Q-STA [START_REF] Moulierac | QoS Scalable Tree Aggregation[END_REF]. The goal of these two algorithms is to aggregate groups to trees while respecting bandwidth requirements of groups. However, multicast applications today need to satisfy more than one or two QoS criteria that's why multiple QoS multicast aggregation is needed. Tree shared by multicast members must fulfill flow requirements such as delay, bandwidth, delay variation (also known as jitter), packet loss that can be tolerated and/or number of hops. To achieve QoS multicast aggregation under several QoS constraints, the forwarding structure must be computed by multi-constrained multicast routing algorithms.

In this paper, we propose a new algorithm called mQMA (multi-constrained QoS Multicast Aggregation) which performs multicast aggregation taking into account multiple QoS requirements. To the best of our knowledge, it is the first proposed protocol that deals with these two important problems of multicast forwarding state scalability and of multi-constrained multicast routing. Our protocol mQMA reduces the number of trees to be maintained and yet it builds trees satisfying several additive constraints.

The rest of the paper is organized as follows. Section II sets up the needed requirements by presenting the previous work that dealt with multicast tree aggregation and multi-constrained multicast routing. Section III describes our proposed algorithm mQMA. Section IV shows how mQMA algorithm improves the multi-constrained routing structure. Section V presents and analyzes the simulation results. Finally, we present some concluding remarks in section VI.

II. RELATED WORK

This section describes first the work related to multicast aggregation and then that related to multi-constrained multicast QoS routing.

A. Multicast Tree Aggregation

Multicast tree aggregation is proposed as a solution to the multicast forwarding state scalability problem. It is a multicast scheme which forces multicast groups to share the same delivery tree within a domain. In aggregated multicast, arriving groups must be matched to aggregated trees. The group-tree matching can be performed in different ways : the match is perfect when the aggregated tree covers exactly all the members of the arriving group and has a cost (in terms of bandwidth consumption) not larger than that of the native tree (computed by common multicast routing). When bandwidth waste is allowed, the leaky aggregation may associate a group to a higher cost tree. In this case, a bandwidth loss threshold must be specified, and if the bandwidth consumption of a tree exceeds the specified threshold, then it is discarded and not considered as an aggregated tree for the group. The authors in [START_REF] Moulierac | On the number of MPLS LSP using Multicast Tree Aggregation[END_REF] studied the number of aggregated trees to be configured depending on the network topology and on the multicast routing algorithm. When aggregation is centralized, a tree manager stores the information concerning the group memberships and the trees maintained in the domain. The multicast Tree Set (MT S) contains all the multicast trees configured into the domain. So, whenever a border router receives a IGMP join or leave message for a group, it contacts the tree manager which is in charge of finding an adequate tree for the given group. In this phase, if the tree manager finds a tree respecting the bandwidth threshold, then the aggregation is performed. If not, the tree manager configures a brand new tree for the group. This tree becomes in turn a candidate for further aggregations. In all the cases, the tree manager informs the border router of the matching group-tree. Then, the border router will be able to route the packets for multicast groups using aggregated trees.

B. QoS Multicast Routing

The problem of QoS routing, even in the unicast case, is known to be NP-complete and has been extensively studied by the research community [START_REF] Van Mieghem | Quality of Future Internet Services[END_REF]. Before presenting the mechanisms used to construct multi-constrained QoS multicast delivery structure, we first specify some hypothesis used to solve this problem and the notation used throughout this paper. Then we present, rather briefly, the main existing approaches to achieve multicast QoS routing.

a) Hypothesis: QoS routing approaches assume that the network-state information is temporarily static and has been distributed (known) throughout the network. This networkstate information is accurately maintained at each node and we assume that it can be collected by any appropriate traffic engineering mechanism. The QoS metrics are categorized into additive (e.g., delay, jitter,..) and min (resp. max) metrics (e.g. bandwidth). The constraints on min (resp. max) QoS measures can easily be treated by pruning all links (and disconnected nodes) which do not satisfy the requested min (resp. max) QoS constraints. In contrast, constraints on additive QoS measures cause more difficulties. We consider, hereafter and without loss of generality, that all QoS measures are additive.

A network topology is modeled as an undirected and valuated graph G = (V, E), where V is the set nodes and E the set of links. Each link is characterized by an m-dimensional link weight vector of m non-negative QoS weights (w j i , for i = 1, 2, ..., m and j is a link in E). The m QoS constraints (limits) L i for i = 1, 2, ..., m are represented by the constraint vector. A multicast group g is composed of a source s and a set of members D = {d 1 , d 2 , ..., d k } where k is the number of multicast members.

b) Multi-constrained multicast routing proposals: For the multicast case, a number of QoS routing algorithms based on single, dual and multiple metrics have been proposed. Single and dual metric QoS multicast routing algorithms have been proposed for cost, delay and the combinations: cost-delay and delay-jitter [START_REF] Chen | An overview of quality-of-service routing for the next generation high-speed networks: Problems and solutions[END_REF]. Few works have dealt with multiple QoS metrics for multicast routing. One way to tackle the multicast QoS routing problem is to compute a set of unicast paths from the source to the multicast members using a unicast QoS routing algorithm. Then, the obtained sub-graph can be reduced to optimize network utilization without violating the constraints. The aim of this reduction phase is to minimize the number of duplicated packets in links and to have a structure that approximates a tree structure as it is not necessarily the case. This strategy is adopted by the Multicast Adaptive Multiple Constraints Routing Algorithm (Mamcra) proposed in [START_REF] Kuipers | Constrained-based multicast routing algorithm[END_REF] as the multicast extension of Samcra, a unicast QoS routing algorithm [START_REF] Van Mieghem | Hop-by-hop quality of service routing[END_REF] and the Taboo based QoS Multicast Routing algorithm (Taboo-QMR) [START_REF] Ali | Résolution du probleme de routage multicast multicritere[END_REF].

III. THE MQMA ALGORITHM

In this section, we describe our proposed mQMA algorithm to achieve aggregated multi-constrained multicast routing.

A. Algorithm mechanisms

The mQMA algorithm achieves multicast tree aggregation in a centralized way by considering a centralized tree manager. The mQMA algorithm proceeds in three main steps as described below. When a new multicast group g arrives, the tree manager is contacted by the border router and after executing the first two steps, namely the computing path step and the tree decomposition, the tree manager tackles the aggregation step. In the following, we detail each of the three steps of the mQMA algorithm.

Step (1): Computing paths. The aim of this step is to compute multi-constrained paths from the source node to each destination node of the group. To perform this step, any unicast QoS routing algorithms can be used. In our simulation, we used the Samcra algorithm [START_REF] Van Mieghem | Hop-by-hop quality of service routing[END_REF]. This first step is mandatory to determine one multi-constrained Feasible Paths Set (F P S) for a given group. The quality of the paths in F P S depends on the unicast routing algorithm used. If Samcra is used, feasible paths from source node to each destination node are found if they exist. The Samcra algorithm chooses the shortest path among all paths from the source to the destination according to the objective function used by Samcra (this function denoted by "Samcra length" is defined in section V). A destination (i.e., a group member) for which no path is found is then excluded, from the group as QoS requirements cannot be fulfilled. In this case, the F P S contains no path for this destination.

Step (2): Tree decomposition. The F P S obtained in the first step isn't necessarily a tree, it may contain cycles. The aim of this step is to decompose the F P S into several trees. The obtained trees will form the Feasible Trees Set (F T S). If the F P S contains no cycle, the F T S contains only one tree which corresponds to the graft of all the paths in the F P S and the algorithm goes directly to step 3. This tree decomposition works in a greedy manner by attempting to graft iteratively the paths for the members onto a tree of the F T S. If during an iteration, the graft of a path implies a cycle in all trees of the F T S, then this path is added to the F T S as a new tree. To proceed greedily, the F P S must be arranged according to a certain order criterion. In this paper and in the simulations, paths of the F P S were arranged according to the number of members. So, the path containing the most number of members is considered as the first tree in the F T S and then it is used to eventually graft the other paths of the F P S. The effect of adopting different order criteria can be explored. Another issue that can be investigated is the choice of the tree within the F T S to be used to graft paths when the path can be grafted on more than one tree. Actually, our algorithm selects the first tree in the F T S to which a new path can be aggregated. All these issues are beyond the scope of the present work.

Step (3): Aggregation.

After executing the first two steps, the tree manager attempts to achieve a global aggregation of the whole multicast group. It consists on finding from the MT S a tree MT i that covers all the members covered by the F P S without violating the constraints and with a cost (in terms of the number of links) at most the cost of the F P S. If an MT i exists, it is used by the multicast group G and all data for G are multiplexed onto that tree. If, however, several such MT i can be used to aggregate G, the algorithm chooses the one of minimum cost. If such an MT i doesn't exist and if the F T S contains only one tree, then the aggregation of G isn't possible and the tree in F T S must be added to the multicast tree set MT S. If the F T S contains more than one tree and if the global aggregation isn't possible, mQMA proceeds to a partial aggregation. Indeed, each tree t i (1 ≤ i ≤ |F T S|) of the F T S covers a sub-group G i of the initial multicast group G. The mQMA algorithm attempts to aggregate all the obtained sub-groups and if the aggregation is not possible, the corresponding tree is added to the multicast tree set (MT S) until all the members of G are satisfied. In this case, a multicast group is associated to several trees and the tree manager is in charge to add information that allows routers to multiplex data onto these trees.

B. mQMA on an example

Let us consider the topology presented in Figure 1 composed of 7 nodes and 7 links where s is the source node and nodes d 1 and d 2 are the members of the multicast group. We suppose that two additive metrics are considered, the values of which are portrayed aside to the different links. We consider the constraint limits to be [START_REF] Cui | AQoSM: Scalable QoS multicast provisioning in Diff-Serv networks[END_REF][START_REF] Cui | AQoSM: Scalable QoS multicast provisioning in Diff-Serv networks[END_REF]. Execution of the first step of mQMA provides the set FPS of shortest paths to d 1 and d 2 .

Here FPS ={F P 1 =(s-a-d 0 -e-d 1 ),F P 2 =(s-b-d 0 -e-d 2 )}. During the second step of mQMA, the graph structure obtained as the union of paths in the F P S is decomposed into trees. The tree

F T S={F T 1 ={(s-b-d 0 -e-d 1 )},F T 2 ={(s-a-d 0 -e-d 2 )}}.
The last step of mQMA, aims to find trees in the MT S, the multicast tree set, configured into the domain covering members and fulfilling the constraints. The global aggregation attempts to find one tree T to aggregate the entire group. If, it is not possible, mQMA attempts to aggregate the trees of the F T S tree per tree. Moreover, when paths are computed in the first step using a unicast QoS routing algorithm such as Samcra, only optimal paths are considered (optimal according to the multi-criteria length function defined by the algorithm). That's why during the aggregating step and according to the already accepted and configured groups, feasible but not optimal tree, covering the new coming group, may exist in the MT S and may be used to achieve global aggregation.

Let us consider another case where the constraint limits for a new request are fixed to (7, 7) and let us suppose that the MT S contains already the tree MT 1 ={s-d 0 -e-d 1 -d 2 }. This tree is feasible for the new request so it can be used to aggregate the group (d 1 ,d 2 ). If global aggregation is not possible, partial aggregation is executed in order to find in the MT S, trees on which F T 1 and/or F T 2 can be aggregated. If the MT S does not contain any tree that can be used to aggregate one of F T S trees, then the considered tree is added to the MT S. So, to summarize this example, four scenarios are possible: [START_REF] Cui | Aggregated Multicast -A Comparative Study[END_REF] The MT S contains one tree covering the member of the group and fulfilling the constraint limits; (2) if not, the MT S contains two trees covering, each one, one of the F T S trees (F T 1 and F T 2 ); (3) if not, the MT S contains one tree that covers one of the two F T S trees (F T 1 or F T 2 ), so the second uncovered tree of the F T S must be added to the MT S and must be configured into the domain; (4) if the two trees of the F T S can not be matched to any of the MT S trees, F T 1 and F T 2 are both added to the MT S and must be configured into the domain.

IV. MQMA: A PROPOSAL TO ENHANCE MULTI-CONSTRAINED ROUTING STRUCTURE

mQMA algorithm performs multicast aggregation taking into account multiple members requirements. In this section, we detail the cycles problem and how mQMA contributes in the reduction of forwarding structure.

A. Problem statement

Multicast routing aims to forward multicast data efficiently by sending single packets through the shared links and duplicating them if it is necessary through a tree structure. When multi-constrained multicast routing is considered, this philosophy may be no longer respected. Indeed, multiconstrained multicast routing computes a routing structure that satisfies multiple additive QoS metrics and this structure is not necessarily a tree. Not to affect the multicast philosophy, multi-constrained proposal takes into account the reduction of the computed routing structure in order to eliminate as many redundancies as possible without violating the members requirements.

Cycles problem on an example: Figure 1 illustrates the cycle problem. If the constraint limits are [START_REF] Chen | An overview of quality-of-service routing for the next generation high-speed networks: Problems and solutions[END_REF][START_REF] Chen | An overview of quality-of-service routing for the next generation high-speed networks: Problems and solutions[END_REF], optimal paths (F P 1 and F P 2 ) for d 1 and d 2 form the cycle (s-a-d 0 -bs). When rerouting data for d 2 through F P 1 or data for d 1 through F P 2 , the cycle (s-a-d 0 -b-s) is removed leading to a better use of network resources. When the constraint limits are fixed to [START_REF] Cui | AQoSM: Scalable QoS multicast provisioning in Diff-Serv networks[END_REF][START_REF] Cui | AQoSM: Scalable QoS multicast provisioning in Diff-Serv networks[END_REF]. The path (s-b-d 0 -e-d 1 ) does not respect the constraints for d 1 and the path (s-a-d 0 -e-d 2 ) does not respect the constraints for d 2 . Consequently, the structure used for the group of source s and of members d 1 and d 2 contains the cycle (s-a-d 0 -b-s) which can not be removed. In this case, d 0 receives packets twice, once from a and once from b and the link (d 0 -e) transfers duplicated packets. A specific routing has to be in place as traditional IP routing cannot support that case.

B. mQMA contribution to solve cycles problem

Besides the reduction of routing entries, the algorithm mQMA can be considered as an alternative to solve the cycles problem. In addition to the aggregation benefit, the routing structure used by mQMA to deliver multi-constrained multicast data has been implicitly reduced.

mQMA removes cycles during the two last steps. (1) During the tree decomposition, some cycles can be removed. Indeed, the F T S grows in a greedy manner with the add of paths. When a path is added, it may cover some members not already considered. In that case, and if the path is feasible for these members, they won't be considered at all. If the paths for these members were forming some cycles, then with tree decomposition, these cycles are removed. (2) During the phase of aggregation, one global tree that is feasible and that covers the members can exist even if the structure F T S contains more than one tree. In that case, the cycles are removed from the group quite easily as global aggregation is made.

To apprehend how mQMA eliminates cycles, let's consider Figure 1 where s is the source node and nodes d 0 , d 1 and d 2 the multicast members.

If the constraints are fixed to [START_REF] Moulierac | QoS Scalable Tree Aggregation[END_REF][START_REF] Moulierac | QoS Scalable Tree Aggregation[END_REF], and if Samcra is used as the unicast QoS routing, the first step of mQMA algorithm returns the set F P S={F P 1 =(s-a-d 0 -e-d 1 ),F P 0 =(sd 0 ),F P 2 =(s-b-d 0 -e-d 2 )}. The paths in the F P S do not form a tree, it contains 3 cycles which are (s,a,d 0 ,s), (s,b,d 0 ,s) and (s,a,b,d 0 ,s).

Cycles reduction during the tree decomposition step: The decomposition step consider as a first path, the path containing the most member nodes. So, the first computed tree is F T 1 ={sa-d 0 -e-d 1 }. The path F P 2 =(sad 0ed 2 ) can not be grafted into the F T 1 due to the cycle (s,a,b,d 0 ), so it must be added into the F T S as F T 2 and finally the path F P 0 =(s-d 0 ) used to forward data to d 0 is already in F T 1 , so this path will not be added to the F T S as a third tree. Thanks to this step, the cycles (s,a,d 0 ), (s,b,d 0 ) are removed and then the F T S contains only 2 trees. The third step in mQMA attempts to aggregate these two trees to exiting trees of MT S.

Cycles reduction during the aggregation step : If the MT S, contains the tree {s-d 0 -e-d 1 -d 2 } and as the constraint limits are fixed to [START_REF] Moulierac | QoS Scalable Tree Aggregation[END_REF][START_REF] Moulierac | QoS Scalable Tree Aggregation[END_REF], this tree can be used to aggregate the multicast group. So, the routing structure used to forward data is now a tree and does no longer contain cycles. If the constraints are fixed to [START_REF] Cui | AQoSM: Scalable QoS multicast provisioning in Diff-Serv networks[END_REF][START_REF] Cui | AQoSM: Scalable QoS multicast provisioning in Diff-Serv networks[END_REF], global aggregation is not possible. In fact, the path (s-d 0 -e-d 1 ) and the path (s-d 0 -e-d 2 ) are not within constraints and so the tree {s-a-d 0 -e-d 1 -d 2 } can not be used to aggregate the whole group.

V. SIMULATIONS

This section presents the results of our conducted simulations to compare our proposed algorithm mQMA to the Mamcra algorithm. In the first step of mQMA, we have to use a unicast QoS routing algorithm. In our experimental study, we adopt Samcra as the unicast QoS routing algorithm to compute the set of feasible paths F P S. The optimality is obtained according to the Samcra length function.

A. Parameters of the simulations

The simulations were ran on the Abilene network [START_REF]Abilene network[END_REF] which contains 11 routers and 14 links. In this network, 1 000 groups were generated randomly with size between 2 and 11 members. The groups were source-oriented and the source for these groups was chosen among four routers which are known as specific delivering sources. The links l on the network were given values w l i (for the ith metric) randomly from 1 which was 5 during the simulations. We considered 2 additive metrics and we generated the constrained limits L i (for the ith metric) randomly for all the groups between 5 and 500. For mQMA, the bandwidth threshold t b varies from 0% for perfect aggregation to 40% for leaky aggregation. We repeat each scenario 1000 times. The simulation tool used to achieve this simulation is a graph simulator developed at IRISA and can be found at [START_REF]Graph Simulator[END_REF].

B. Results of the simulations

This subsection presents the results of the simulations considering the number of trees, the network resource usage, the mean Samcra length per member and the number of cycles removed.

1) Number of trees: Figure 2 shows the reduction of the number of trees for mQMA. This performance is due to the multicast tree aggregation. We recall that Mamcra routing structure is not always a tree.

While Mamcra needs to maintain, as many structures as groups in the domain, (i.e. 1 000), mQMA reduces the number of trees to only 200. As mQMA builds few trees, few forwarding states are maintained and the control messages to maintain the trees are reduced.

2) Network resource usage: Figure 3 shows the network resource usage in terms of relative utilization. This metric is denoted by

U R = C(TmQMA)-C(TMamcra) C(TMamcra) = C(TmQMA)
C(TMamcra) -1 where C(T algo ) represents the cost of the routing structures of the algorithm algo used to forward data for all the generated groups. In the simulations, the cost is measured in terms of number of links used for each trees set. In other words, C(T algo ) = g∈G |t(g)|, where G represents the set of all the groups generated, t(g) represents the structure used for g (which can be a set of trees), and |t(g)| represents the number of links of t(g). In mQMA, if the same tree is used to aggregate 2 groups, it is counted twice.

The relative utilization depends on the tolerated bandwidth threshold. When the bandwidth threshold exceeds 0%, mQMA can find an already existing tree that covers the members, that is feasible and that has a lower cost. This occurs when mQMA lists all the trees and chooses among the ones that are feasible, the tree that has the lowest number of links. The results show that even when leaky aggregation is adopted and for lower threshold (≈ 17%), the aggregated trees used by mQMA to forward multicast traffic have lower cost than the Mamcra structure used when no aggregation is performed. So, even when leaky aggregation, mQMA may involve an effective gain in resource usage without violating the multiconstraint requirements of multicast groups. When the threshold increases, the network resource usage is not so high, the 3) Mean Samcra length per member: Figure 4 shows the mean Samcra length for the members, which is the main metric used for multi-constrained routing algorithm. This metric reflects the quality of the communications for the members of the group, expressed in function of the different QoS metrics. The Samcra length from s to d is denoted by : Samcra length(s,d) = max i<# metrics ej ∈path(s,d)

w ej i L i ,
where w ej i is the value of the metric number i for the edge e j (e j ∈ P ), and L i is the requirement of the group for the metric i. The algorithm Samcra computes the optimal structure in minimizing the value of this metric. The results show that mQMA behaves in the same way as Samcra with sligthly higher results for this metric. Then, we can say that the two algorithms are comparable and that mQMA gives good performance in terms of Samcra length as its results are close to the optimal.

4) Number of cycles removed:

To achieve multi-constrained multicast routing, the routing structure is not always tree as it is detailed in section IV. The graph structure computed by the unicast QoS routing algorithm may contain cycles. Figure 5 illustrates the number of cases with cycles contained on the structure computed by the unicast algorithm to reach all the multicast members (namely Samcra). It represents the number of cases in which the set F P S of feasible paths contains cycles. Figure 5 mQMA eliminates cycles to the number of cases in which Mamcra succeed to do it. We recall, that cycles reduction obtained by mQMA include the one achieved by the tree decomposition and the one achieved by the aggregation. In some cases, neither Mamcra nor mQMA removes the cycles found in F P S. In most of these cases, the cycles can not be removed and the structure F P S is the only one possible that respects the constraints fixed for the group. The results show that the two algorithms are comparable in terms of cycle reduction as they behave in the same way. The table I shows the number of cycles removed by Mamcra, mQMA and tree decomposition when generating 30000 test cases.

In 74% of the generated cases, the structure F P S found by Samcra contains at least one cycle. This confirms the necessity of finding an algorithm that removes the cycles. At the end of the simulations, mQMA has removed slightly more trees than Mamcra, that means that in 95.59% of cases, Mamcra has removed all the cycles from the F P S while mQMA has removed all the cycles in 96.84% of cases. From these 96.84% cases obtained by mQMA, 68.72% are implied by the tree decomposition step. The other remaining cases are implied by global aggregation.

VI. CONCLUSION

We presented in this paper a new algorithm mQMA that deals with the two main problems of multicast deployment: multicast forwarding state scalability and multi-constrained QoS routing. Our protocol is based on the techniques of both multicast aggregation and multi-constrained routing. To the best of our knowledge, mQMA is the first protocol that deals with these two main problems. With mQMA, in the worst case, a group can be associated to several trees and the data for the group are multiplexed onto these trees.

Conducted simulations showed that our algorithm achieves indeed comparable performances in terms of Samcra length as Samcra itself which provides the optimal routes for this metric. Moreover, mQMA spares network resources by using much less than Mamcra. Finally, mQMA gains all the benefits of the multicast tree aggregation by building very few trees (at maximum 200 trees for 1 000 groups for Abilene network) and maintaining consequently few forwarding states for these groups.

Fig. 1 .

 1 Fig. 1. mQMA : Network topology used to illustrate different mQMA mechanisms

Fig. 2 .

 2 Fig. 2. Reduction of the total number of trees

Fig. 3 .Fig. 4 .

 34 Fig. 3. Relative resource utilization : mQMA vs Mamcra

Fig. 5 .

 5 Fig. 5. Number of cycles removed

  compares the number of cases in which

		Structures with cy-	Without cycles -	Without cycles -	Without
		cles	Mamcra	mQMA	cycles	-	Tree
					Decomposition
	Maximum Value metric = 5	22402	21559	21754	15757
	Total number of cycles re-		95.59% of cycles	96.84% of cycles	68.72% of cycles
	moved (in %)		removed	removed	removed

TABLE I MQMA

 I REMOVES MORE CYCLES FROM THE STRUCTURES THAN MAMCRA DOES

		3800						Number of structures with cycles
											Mamcra
		3600									mQMA
		3400
		3200
	Number of cycles	2600 2800 3000
		2400
		2200
		2000
		1800
		10	15	20	25	30	35	40	45	50	55	60	65
						Mean value of constraints