Thomas Gibaud

Doru Constantin
email: doru.constantin@u-psud.fr

Direct Liquid to Crystal Transition in a Quasi-Two-Dimensional Colloidal Membrane

Using synchrotron-based small angle x-ray scattering, we study rigid fd viruses assembled into isolated monolayers from mixtures with a non-absorbing polymer, which acts as osmotic

agent. As the polymer concentration increases we observe a direct liquid to crystal transition, without an intermediate hexatic phase, in contrast with many other similar systems, such as concentrated DNA phases or packings of surfactant micelles. We tentatively attribute this effect to the difference in stiffness. The liquid phase can be well described by a hard-disk fluid, while we model the crystalline one as a hexagonal harmonic lattice and we evaluate its elastic constants.

How does the ordering transition occur in two-dimensional systems of objects with isotropic interaction?

Over the last halfcentury, this deceivingly simple question received complex answers that reveal considerable hidden subtlety. In particular, according to the celebrated Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) scenario, 1 crystallization is preceded by the formation of an intermediate hexatic phase, characterized by two order parameters: a short-range positional one and a (sixfold) quasi-long-range orientational one.

Experimental measurements of the interaction between particles and of their degree of order have been performed on a wide variety of systems, such as adsorbed noble gases, [START_REF] Heiney | X-ray scattering study of the structure and freezing transition of monolayer xenon on graphite[END_REF][START_REF] Stephens | High-resolution xray-scattering study of the commensurateincommensurate transition of monolayer Kr on graphite[END_REF][START_REF] Dimon | Firstorder and continuous melting in a twodimensional system: Monolayer xenon on graphite[END_REF] monolayers of colloidal particles, [START_REF] Keim | Harmonic Lattice Behavior of Two-Dimensional Colloidal Crystals[END_REF][START_REF] Von Grünberg | Elastic behavior of a twodimensional crystal near melting[END_REF][START_REF] Brodin | Melting of 2D liquid crystal colloidal structure[END_REF][START_REF] Thorneywork | Two-Dimensional Melting of Colloidal Hard Spheres[END_REF] protein layers [START_REF] Koltover | Phase behavior and interactions of the membraneprotein bacteriorhodopsin[END_REF] or nano-objects inserted within lipid or surfactant bilayers (e.g. peptides [START_REF] Constantin | Membrane-mediated repulsion between gramicidin pores[END_REF] or inorganic nanoparticles. [START_REF] Constantin | Repulsion Between Inorganic Particles Inserted Within Surfactant Bilayers[END_REF][START_REF] Pansu | Insertion of Gold Nanoparticles in Fluid Mesophases: Size Filtering and Control of Interactions[END_REF] ) In many of these "flat" systems (with a transverse size comparable to the typical in-plane distance), the hexatic phase is indeed observed. [START_REF] Heiney | X-ray scattering study of the structure and freezing transition of monolayer xenon on graphite[END_REF][START_REF] Stephens | High-resolution xray-scattering study of the commensurateincommensurate transition of monolayer Kr on graphite[END_REF][START_REF] Dimon | Firstorder and continuous melting in a twodimensional system: Monolayer xenon on graphite[END_REF][START_REF] Keim | Harmonic Lattice Behavior of Two-Dimensional Colloidal Crystals[END_REF][START_REF] Von Grünberg | Elastic behavior of a twodimensional crystal near melting[END_REF][START_REF] Brodin | Melting of 2D liquid crystal colloidal structure[END_REF][START_REF] Thorneywork | Two-Dimensional Melting of Colloidal Hard Spheres[END_REF] A more complex case is that of packed elongated particles, where the out-of-plane degrees of freedom can also be important. The hexatic phase is present, for instance, in stacks of colloidal platelets, [START_REF] Petukhov | Observation of a hexatic columnar liquid crystal of polydisperse colloidal disks[END_REF] in systems of self-assembled micelles, [START_REF] Pal | Observation of the Chiral and Achiral Hexatic Phases of Selfassembled Micellar polymers[END_REF] but also in dense phases of doublestranded (ds) DNA, [START_REF] Podgornik | Bond orientational order, molecular motion, and free energy of high-density DNA mesophases[END_REF][START_REF] Strey | Refusing to Twist: Demonstration of a Line Hexatic Phase in DNA Liquid Crystals[END_REF][START_REF] Yasar | Continuity of states between the cholesteric → line hexatic transition and the condensation transition in DNA solutions[END_REF] which are similar to the very compact state of DNA encoutered, for instance, in bacteriophage capsids. [START_REF] Leforestier | The Bacteriophage Genome Undergoes a Succession of Intracapsid Phase Transitions upon DNA Ejection[END_REF] Here, we use fd viruses, quite similar to ds-DNA: they are monodisperse, chiral, charged and rigid rodlike objects that assemble under the influence of depleting agents into a wide variety of self-assembled structures, such as tactoids, [START_REF] Modlińska | Condensation and dissolution of nematic droplets in dispersions of colloidal rods with thermo-sensitive depletants[END_REF] colloidal membranes, [START_REF] Barry | Entropy driven self-assembly of nonamphiphilic colloidal membranes[END_REF][START_REF] Kang | Entropic forces stabilize diverse emergent structures in colloidal membranes[END_REF] twisted ribbons 22 and membrane rafts. [START_REF] Sharma | Hierarchical organization of chiral rafts in colloidal membranes[END_REF] We concentrate here on colloidal membranes, i.e. monolayers where the rods are parallel to the layer normal.

Despite its rich morphology, this system has only been studied at the macroscopic scale and in the fluid regime, via optical microscopy observation. Structural characterization, over length scales comparable to the distance between rods, is rendered difficult by the small size and reduced scattering power of the assemblies, although it has been carried out on bulk phases of related systems. [START_REF] Purdy | Measuring the nematic order of suspensions of colloidal fd virus by x-ray diffraction and optical birefringence[END_REF][START_REF] Grelet | Hexagonal Order in Crystalline and Columnar Phases of Hard Rods[END_REF] We perform small-angle X-ray scattering studies (SAXS) at a high-brilliance synchrotron beamline on suspensions of such objects. The number concentration n 2D of rods in the plane of the membrane is controlled by the bulk polymer concentration. As n 2D increases, we observe a first-order transition between an isotropic phase and a crystal phase. There is no hexatic phase, in contrast with many other twodimensional systems, such as dsDNA, and with the bulk phase of the semi-flexible wild-type fd viruses. [START_REF] Grelet | Hexagonal Order in Crystalline and Columnar Phases of Hard Rods[END_REF] As semiflexible rodlike particles we used the monodisperse filamentous bacteriophage fd -Y21M with a contour length of 880 nm, a diameter of 6.6nm and a persistence length of 9.9 ± 1.6µm. [START_REF] Barry | A model liquid crystalline system based on rodlike viruses with variable chirality and persistence length[END_REF] fd -Y21M viruses were synthesized using standard biological protocols [START_REF] Sambrook | Molecular cloning: a laboratory manual 3rd edition[END_REF] and dispersed at a concentration c V ranging from 0.5 to 10 mg/mL in 20 mM TRIS buffer at pH 8.0 and 100 mM NaCl.

In aqueous suspension, these virus particles exhibit purely repulsive interactions and behave as model system with respect to the Onsager theory. [START_REF] Barry | A model liquid crystalline system based on rodlike viruses with variable chirality and persistence length[END_REF] We mixed the viruses with a non absorbing polymer, Dextran (molecular weight 500 kDa, Sigma Aldrich) at a final virus concentration c V = 5 mg/mL and polymer concentrations in a range c D = 0 -58.5 mg/mL. Adding polymers to a dilute isotropic suspension of fd -Y21M induces attractive interactions between the rods via the depletion mechanism. [START_REF] Asakura | Interaction between particles suspended in solutions of macromolecules[END_REF] The membrane properties are mainly controled by the depletant concentration, [START_REF] Barry | Entropy driven self-assembly of nonamphiphilic colloidal membranes[END_REF]22 and their diameter ranges from 1 to 50 microns. [START_REF] Barry | Entropy driven self-assembly of nonamphiphilic colloidal membranes[END_REF] The line tension at the edge of the membranes is always too high to self-assemble monodispersed membranes with a well-defined diameter. 22 The thermal fluctuations of the membranes provide an entropic repulsion that keeps them isolated from each other along their normal. However, at high depletant concentrations, the induced attraction may be strong enough to overcome the entropic repulsion and the membranes tend to stack on top of each other 20 and eventually crystalize.

We prepared bulk samples of concentrated membranes in round glass capillaries, with a nominal outer diameter of 1 mm and 10 micronthick walls, purchased from WJM-Glas (Berlin, Germany). The capillaries were filled with the sample solution and sealed at the top to prevent evaporation. Some of them were mildly centrifuged for 5 minutes at 2000 g to concentrate the membranes at the bottom of the capillary and thus increase the strength of the scattered signal. The sediment occupies at least 1/5 of the capillary length, so the local concentration does not exceed c V = 25 mg/mL. We checked that there is no significant difference between the signal of the initial solutions and that of the centrifuged ones (see the Supplemental Material 29 for more details.) SAXS measurements were performed on the SWING beamline of the SOLEIL synchrotron (Saint-Aubin, France.) The sample-to-detector distance was 2 m and the wavelength λ = 1 Å, covering a scattering vector range 0.005 < q < 0.5 Å-1 . The measurements were performed at 22 • C. The capillaries were placed vertically. For the centrifuged ones, we made several measurements at different heights in the sedimented phase, at its upper interface and in the supernatant.

The intensity measured in the supernatant was used as a reference signal for that measured in the sediment.

The corrected intensity was then divided by the (orientationally averaged) form factor of the virus measured in isotropic solutions 29 to yield the in-plane averaged structure factor S(q). The structure factors are shown in Figure 1 for Dextran concentrations c D between 29.5 and 58.5 mg/mL. At lower c D we do not detect the formation of membranes (the solutions remain clear) and the scattering signal is weak and relatively flat. A two-dimensional system of isotropic particles can exhibit three phases: fluid, crystalline and hexatic. For the fluid phase we used the structure factor of hard disks, 31 which has a simple analytical expression and yields two parameters: the hard disk radius R HD and the in-plane number concentration n 2D (see Figure 2). From the low-q limit S(q → 0) of the fitted structure factor we extract the (twodimensional) osmotic pressure Π 2D .

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We describe the structure factor of the crystalline phase by the harmonic model of Dutta and Sinha 33 (denoted in the following as D & S), averaged over the azimuthal angle (in the plane of the membrane), since the recorded intensity is integrated over all orientations. There is no need to account for the experimental resolution, since this parameter would be correlated to the domain size (see the original paper for details.)

In the hexatic phase, many authors described the peaks by a Lorentzian function for simplicity: this choice yields an analytical functional form for the azimuthally averaged structure factor given by Heiney et al. 34 (H), which is close to a square-root Lorentzian function. [START_REF] Dimon | Firstorder and continuous melting in a twodimensional system: Monolayer xenon on graphite[END_REF] With respect to the original model, we convolute the peak function with a Gaussian to account for the experimental resolution. For completeness, we also included in the analysis the Pearson VII function (P) used by Grelet to model the signal of the hexatic phase in bulk systems. [START_REF] Grelet | Hexagonal Order in Crystalline and Columnar Phases of Hard Rods[END_REF] In the isotropic phase ( ), n 2D is a free parameter of the hard-disk model, while in the hexagonal one ( ) it is computed from the peak position. At coexistence (c D = 43.7 and 45.4 mg/mL), the lower concentration is linearly extrapolated from the isotropic values, while the higher is a free fit parameter. 29 For c D = 43.7 mg/mL and above, we fitted the curves in the range 0.043-0.077 Å-1 (around the first peak q 10 ) using the three models described above: D & S, H, and P respectively (see Figure 3). In all cases, we add to the models a linear slope and a constant contribution: α(q -q 10 ) + β, to account for imperfect background subtraction and for other sources of signal within the sample. Each model has six fit parameters, including α and β. In Table 1 we present the goodness-of-fit function χ 2 (divided by the number of data points N pts in the fitting range) for each curve and with each model. This parameter quantifies the agreement between data and model: we expect χ 2 /N pts ∼ 1 for a good fit, χ 2 /N pts > 1 signals an imperfect fit (the model is inaccurate or the data exhibits systematic uncertainties), while χ 2 /N pts 1 generally results from overestimat-ing the uncertainty of the data. To ascertain the importance of the linear term added to the models we also perform the fits where we set α = 0 (columns labeled "no slope".) The huge improvement in χ 2 /N pts justifies our use of this term. Clearly, the harmonic crystal model of Dutta and Sinha provides the best description of the four curves. However, while this agreement is excellent for the two higher concentrations, as shown by χ 2 /N pts < 1, it is clearly worse for c D = 43.7 and 45.4 mg/mL, where χ 2 /N pts = 1.75 and 4.59, respectively, because the model underestimates the data in the range 0.047-0.055 Å-1 . This also holds true for models H and P.

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Since none of the single-phase models fit the structure factor for c D = 43.7 and 45.4 [mg/mL], we tried to describe it as coexistence between the hexagonal phase found at higher concentrations and the hard-disk liquid phase found at lower concentrations. We therefore use the sum of the Dutta & Sinha model and the liquid model. The latter contribution has three parameters: the hard-disk radius R HD and the in-plane number density n 2D are obtained by extrapolation from the fits at lower concentrations, while the overall amplitude is free to vary. We therefore have a single additional coefficient with respect to the six in the D & S model. The best fit is shown as red line in Fig. 4 (and also in Fig. 1). For comparison, we plot as black line the best D & S fit.

The agreement is markedly better for the coexistence system: at c D = 43.7 mg/mL, adding one fit parameter decreases the χ 2 /N pts from 1.75 to 1.08, and over a much wider range. For comparison, trying to fit over the same range with only the D & S model yields χ 2 /N pts = 27.0, while the hard-disk liquid model yields χ 2 /N pts = 33.3 with fixed R HD and n 2D and χ 2 /N pts = 15.6 if these two parameters are allowed to vary (in which case they do not move more than 4% away from the initially fixed values.) Similar improvement in χ 2 /N pts (from 4.59 to 2.44) is obtained for c D = 45.4 mg/mL. We conclude that the best description of these two data points is given by the coexistence model.

Table 1: Goodness-of-fit function χ 2 (divided by the number of data points N pts in the fitting range) for the three models described in the text: Dutta and Sinha (DS), Heiney et The best values for the relevant fit parameters and their associated uncertainties (at the 68.3% confidence level) are shown in Table 2. L DS is the domain size in the Dutta and Sinha model (see the Supporting Information for the definitions.) The n 2D is common to the two models, being obtained from the peak position q 10 in the hexagonal phase as n 2D = √ 3 8π 2 q 2 10 . At coexistence, n 2D of the isotropic contribution is fixed as discussed in the text and only the value corresponding to the hexagonal phase is listed. The hard-disk radius R HD at coexistence is also fixed.

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!"!5! !"! ' !"! ! !"!'' !"!'! !"!&' *-./ 0# 1 B C
A question we cannot answer at this time is whether the coexistence is a genuine feature of the phase diagram (in which case it would imply that the transition is first-order) or an artifact due to sample heating by the x-ray beam and the system is in fact fully crystalline. We cannot rule out the latter possibility, although we limited the beam effect by reducing the exposure time for these two samples to the minimum achievable time (5 ms.) An important parameter of the crystalline phase is the exponent η(q, T ), which describes the power law divergence of the diffuse scattering around the Bragg peaks. It is given by the elastic moduli of the phase:

36 η(q, T ) = k B T q 2 4πµ 3µ + λ 2µ + λ = k B T q 2 8πµ (3 -σ) (1)
where µ (the shear modulus) and λ are the Lamé coefficients and σ is Poisson's ratio in two dimensions. [START_REF] Von Grünberg | Soft Matter[END_REF] In the following, we will denote by

η = η(q 10 , T ) = 4π 3 k B T µd 2 3µ + λ 2µ + λ (2)
the value of this parameter at room temperature and at the position of the first Bragg peak of the hexagonal lattice, q 10 = 4π √ 3d , where d 12 nm is the distance between first neighbors in the lattice. Close to q 10 , S(q) ∼ 1/ |q -q 10 | 2-η . /@3+A3B/: :35C/@+50:5 ''''''15C,<30,6 The values of η extracted from the D & S model are shown in Figure 5. In the hexagonal phase, they are 0.28 and 0.39, so we can estimate an average value η avg = 1/3, which happens to be the maximum value that η can reach in the KTHNY scenario. Plugging it into (2) and setting the final factor to 1 (under the assumption that λ µ, to be justified below) yields a shear modulus:

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µ 4π k B T d 2 = 0.09 k B T nm 2 = 3.6 10 -4 Pa m. (3)
From the osmotic pressure dependence. 29 we estimate the bulk modulus at µ + λ = 6.5 10 -3 Pa m, (4) about 18 times higher than the shear modulus. For context, this ratio is 3 in hexagonal lyotropic liquid crystals, [START_REF] Impéror-Clerc | An X-ray scattering study of flow-aligned samples of a lyotropic liquid-crystalline hexagonal phase[END_REF] 10 in magnetic colloids at the air-water interface [START_REF] Zahn | Elastic Properties of 2D Colloidal Crystals from Video Microscopy[END_REF] and about 12 in decanol monolayers. [START_REF] Zakri | Determination of the in-plane elastic tensor of crystalline decanol monolayers on water by x-ray diffraction[END_REF] The values of η estimated in the coexistence regime are unreliable, since the wide tails associated to high η are difficult to distinguish from the "bump" of the isotropic phase. The coexistence model thus yields η ∼ 0.1 -0.15, as shown in Figure 5. The pure D & S model, on the other hand, gives η ∼ 0.5 -0.6 (data not plotted.)

In the isotropic phase, the structure factor is well described by a hard core interaction, in agreement with recent macroscopic results. [START_REF] Grelet | Hard-Rod Behavior in Dense Mesophases of Semiflexible and Rigid Charged Viruses[END_REF] The osmotic pressure is much lower than expected from the concentration of Dextran, so the latter is probably not completely excluded from the membranes, as it happens in bulk sys-tems. [START_REF] Dogic | Isotropic-nematic phase transition in suspensions of filamentous virus and the neutral polymer Dextran[END_REF] Self-assembled membranes of stiff fd viruses represent an experimental quasi-twodimensional system of isotropic objects that does not follow the KTHNY scenario. In particular, they exhibit a direct transition between the crystalline and the fluid phases. We see no evidence for a hexatic phase, in contrast with literature results for bulk phases of a semi-flexible fd viruses [START_REF] Grelet | Hexagonal Order in Crystalline and Columnar Phases of Hard Rods[END_REF] and for dense DNA phases. [START_REF] Podgornik | Bond orientational order, molecular motion, and free energy of high-density DNA mesophases[END_REF][START_REF] Strey | Refusing to Twist: Demonstration of a Line Hexatic Phase in DNA Liquid Crystals[END_REF][START_REF] Yasar | Continuity of states between the cholesteric → line hexatic transition and the condensation transition in DNA solutions[END_REF] This result is puzzling, in view of the similarity between our fd viruses and the hexatic systems cited above. A notable difference is that the former is stiffer than dsDNA and also than the wild-type fd virus strains used in bulk studies. Our findings highlight the importance of flexibility in stabilizing the hexatic phase of elongated objects. This is a new result, since to our knowledge the role of flexibility has never been invoked in the literature in this context.

Figure 1 :

 1 Figure 1: Structure factors S(q) obtained by SAXS (open dots) for several Dextran concentrations, indicated alongside the curves, as well as the phase assignment. The fits with the appropriate models are shown as solid lines (see the text for details.) The curves are vertically shifted for clarity.

Figure 2 :

 2 Figure 2: In-plane number density of rods within the membrane n 2D as a function of the Dextran concentration c D . The osmotic pressure P D imposed by the polymer[START_REF] Livney | Swelling of dextran gel and osmotic pressure of soluble dextran in the presence of salts[END_REF] is shown as top axis. In the isotropic phase ( ), n 2D is a free parameter of the hard-disk model, while in the hexagonal one ( ) it is computed from the peak position. At coexistence (c D = 43.7 and 45.4 mg/mL), the lower concentration is linearly extrapolated from the isotropic values, while the higher is a free fit parameter.29

Figure 3 :

 3 Figure 3: The top four curves in Fig. 1 (symbols) and fits with the three models (solid lines, different colors.)

Figure 4 :

 4 Figure 4: Structure factor at c D = 43.7 mg/mL (open dots) and fits with the Dutta & Sinha model (black solid line) and the coexistence model (red solid line). a -full range. b -detail of the peak region.

Figure 5 :

 5 Figure 5: Fluctuation parameter η in the D & S model as a function of the Dextran concentration c D .

  al. (H) and Pearson (P), with or without a linear slope term.

	c D [mg/mL]	D & S	H	P	χ 2 /N pts D & S (no slope) H (no slope) P (no slope)
	58.5	0.70	2.82 3.15	133.8	168.9	105.8
	50.2	0.93	3.62 2.04	72.3	68.8	57.7
	45.4	4.59	22.63 8.14	132.3	78.1	98.1
	43.7	1.75	4.92 2.30	18.1	13.0	12.8

Table 2 :

 2 Best values and associated uncertainties for the relevant fit parameters: Dutta and Sinha with linear slope term in the hexagonal phase, hard-disk liquid in the isotropic phase and their sum in the coexistence range.

	c D [mg/mL]	n 2D [10 -6 Å-1 ]	Value ± Uncertainty η L DS [10 3 Å]	R HD [ Å]
	58.5	80.12 ± 0.05 0.282 ± 0.005 2.9 ± 0.1
	50.2	76.04 ± 0.06	0.39 ± 0.01	3.4 ± 0.2
	45.4	71.40 ± 0.07	0.11 ± 0.01	2.9 ± 0.1	54.15
	43.7	72.40 ± 0.07	0.16 ± 0.01	2.8 ± 0.1	54.97
	40.5	63.37 ± 0.03			55.08 ± 0.01
	35.1	53.66 ± 0.06			54.15 ± 0.01
	29.5	46.80 ± 0.07			54.64 ± 0.01

Acknowledgement The SOLEIL synchrotron is acknowledged for the provision of beamtime (experiment 20150494) and Javier Perez for his support. This work was supported by the ANR under contract MEMINT (2012-BS04-0023).

Supporting Information Available

The following files are available free of charge.

• SI Constantin 2018.pdf: details on the fitting procedure, the evaluation of the form factor and of the osmotic pressure, the comparison of data from dilute and concentrated suspensions and on the implementation of the crystal model.


Pascal Chevalier

Jean Pierre Delmas

Mustapha Sadok

PERFORMANCE OF A THIRD-ORDER VOLTERRA MVDR BEAMFORMER IN THE PRESENCE OF NON-GAUSSIAN AND/OR NON-CIRCULAR INTERFERENCE

Keywords: Beamformer, non-circular, non-Gaussian, higher order, interferences, MVDR, Volterra, widely non linear, widely linear, third order, fourth order, sixth order

Linear beamformers are optimal, in a mean square (MS) sense, when the signal of interest (SOI) and observations are jointly Gaussian and circular. When the SOI and observations are zero-mean, jointly Gaussian and non-circular, optimal beamformers become widely linear (WL). They become non-linear with a structure depending on the unknown joint probability distribution of the SOI and observations when the latter are jointly non-Gaussian, assumption which is very common in radiocommunications. In this context, a third-order Volterra minimum variance distortionless response (MVDR) beamformer has been introduced recently for the reception of a SOI, whose waveform is unknown, but whose steering vector is known, corrupted by non-Gaussian and potentially non-circular interference, omnipresent in practical situations. However its statistical performance has not yet been analyzed. The aim of this paper is twofold. We first introduce an equivalent generalized sidelobe canceller (GSC) structure of this beamformer and then, we present an analytical performance analysis of the latter in the presence of one interference. This allows us to quantify the improvement of the performance with respect to the linear and WL MVDR beamformers.

INTRODUCTION

The conventional time invariant linear beamformers, such as the Capon beamformer [START_REF] Capon | Multidimensional maximum likelihood processing of a large aperture seismic array[END_REF], are only optimal for stationary Gaussian observations whose complex envelope is necessarily second-order (SO) circular. However in many applications such as in radiocommunications, signals are non-Gaussian and non-circular either at the SO and/or at a higher order (HO) [START_REF] Picinbono | On circularity[END_REF]. For this reason, several non-linear beamformers have been proposed in the literature. A widely linear [START_REF] Picinbono | Widely linear estimation with complex data[END_REF] MVDR beamformer has been introduced in [START_REF] Chevalier | Widely linear MVDR beamformer for the reception of an unknown signal corrupted by noncircular interferences[END_REF] to improve the performance of the Capon beamformer in SO noncircular contexts. A third-order Volterra GSC [START_REF] Griffiths | An alternative approach to linearly constrained adaptive beamforming[END_REF] structure has been proposed in [START_REF] Souloumiac | Improvement in non-Gaussian jammers rejection with a non linear spatial filter[END_REF] to improve the performance of the Capon beamformer in non-Gaussian contexts, without taking into account the potential non-circularity of interference. Then a family of third-order Volterra MVDR beamformers [START_REF] Chevalier | Third-order widely non-linear Volterra MVDR beamforming[END_REF] which takes into account both the potential non-Gaussian character and the potential SO, fourth-order (FO) and sixth-order (SIO) non-circularity of interference has been presented. But these beamformers have been implemented by an intricate GSC structure and their performance have been presented only by numerical illustrations.

In this context, the first purpose of this paper is to introduce an alternative GSC structure of these beamformers allowing much simpler implementations. The second purpose is to present an analytical performance analysis of these beamformers in the presence of interference. It has been shown in particular that, depending on the SO, FO and SIO interference statistics, some of these Volterra MVDR beamformers outperform the Capon beamformer for a circular non-Gaussian interference, while some others, outperform the WL MVDR beamformer for a non-circular non-Gaussian interference.

HYPOTHESES AND PROBLEM FORMULATION

Hypotheses

We consider an array of N narrowband sensors and we denote by x(t) the vector of the complex amplitudes of the signals at the output of these sensors. Each sensor is assumed to receive the contribution of an SOI corrupted by interference and a background noise. Under these assumptions, the observation vector x(t) can be written as follows

x(t) = s(t)s + n(t) ∈ C N , ( 1
)
where s(t) and s correspond to the complex envelope, assumed zero-mean, and the steering vector, assumed perfectly known, of the SOI respectively. The vector n(t) is the total noise vector, containing the background noise and the interference, and assumed to be zero-mean, potentially non-Gaussian and non-circular, and independent of s(t).

Problem formulation

The problem addressed in this paper is to estimate the unknown signal s(t) from the observation x(t). It is wellknown [START_REF] Picinbono | Widely linear estimation with complex data[END_REF] that the optimal estimate s(t) of s(t), in a MS sense, from the observation x(t), is the conditional expectation E[s(t)|x(t)]. Note, that for respectively circular or non-circular mutually Gaussian distributions of (s(t), x(t)), this conditional expectation becomes linear or widely linear [START_REF] Picinbono | Widely linear estimation with complex data[END_REF]. But for non-Gaussian distribution of (s(t), x(t)), the derivation of this conditional expectation needs this distribution which is unknown in practice.

To approximate this conditional expectation, a particular class of non-linear beamformers, the complex Volterra beamformers, has been introduced for the first time in [START_REF] Chevalier | Le filtrage de Volterra transverse réel et complexe en traitement du signal[END_REF][START_REF] Chevalier | Complex transversal Volterra filters optimal for detection and estimation[END_REF]. Then a particular third-order Volterra beamformer, whose output is defined by (2) has been proposed and briefly analyzed in [START_REF] Chevalier | Third-order widely non-linear Volterra MVDR beamforming[END_REF].

y(t) = w H 1,0 x(t) + w H 1,1 x * (t) +w H 3,0 [x(t)⊗x(t) ⊗ x(t)]+w H 3,1 [x(t)⊗x(t) ⊗ x * (t)] +w H 3,2 [x(t)⊗x * (t) ⊗ x * (t)]+w H 3,3 [x * (t)⊗x * (t) ⊗ x * (t)] def = w H x(t), (2)
where ⊗ denotes the Kronecker product. In (2), the thirdorder terms x 3,q (t) def = [x(t) ⊗(3-q) ⊗ x * (t) ⊗q ], q = 0, 1, 2, 3 are called cubic (C) terms and q are their indexes. The beamformers containing the linear term and the cubic terms q 1 , q 2 , .., q r are called L-C(q 1 , q 2 , .., q r ) and those containing the WL terms and the cubic terms q 1 , q 2 , .., q r are called WL-C(q 1 , q 2 , .., q r ). Note that only the L-C(1) beamformer have been considered in [START_REF] Souloumiac | Improvement in non-Gaussian jammers rejection with a non linear spatial filter[END_REF].

THIRD-ORDER VOLTERRA MVDR BEAMFORMER

To impose no distortion on s(t) at the output y(t), the spatial filters of the first order terms of (2) must verify the constraints:

w H 1,0 s = 1 and w H 1,1 s * = 0. To obtain the constraints on the cubic terms, we must decompose the random variable n(t) on a fixed orthogonal basis

(s, u 1 , ...u N -1 ) of C N : n(t) = n 0 (t)s + ∑ N -1 i=1 n i (t)u i .
To cancel the SOI contributions in the cubic term w H 3,q x 3,q (t), it is equivalent to cancel all the terms of w H 3,q x 3,q (t) excluding the terms containing the u i 's only. For example, for q = 1, we must impose the

1 + 3(N -1) + 3(N -1) 2 = N 3 -(N -1) 3 constraints: w H 3,1 (s ⊗ s ⊗ s * ) = 0 w H 3,1 (u i ⊗ s ⊗ s * ) = 0, w H 3,1 (s ⊗ u i ⊗ s * ) = 0, w H 3,1 (s ⊗ s ⊗ u * i ) = 0, 1 ≤ i ≤ N -1 w H 3,1 (u i ⊗ u j ⊗ s * ) = 0, w H 3,1 (u i ⊗ s ⊗ u * j ) = 0, w H 3,1 (s ⊗ u i ⊗ u * j ) = 0, 1 ≤ i, j ≤ N -1, (3)
or equivalently C H 1 w 3,1 = 0 N 3 -(N -1) 3 . For any other index q = 0, 2, 3, the constraints can be written as C H q w 3,q = 0 N 3 -(N -1) 3 and the global set of constraints takes the form:

C H w = f , ( 4
)
where

C = Diag(s, s * , C 0 , C 1 , C 2 , C 3 ) and f is the vector (1, 0 T 1+4[N 3 -(N -1) 3 ] ) T .
Thus the best SO estimate s(t) exploiting the noise statistics only, corresponds to the output of the third-order Volterra beamformer w MVDR which minimizes the time-averaged output power, w H R x w under the previous constraint.

w MVDR = arg{ min C H w=f w H R x w}, (5)
where

R x def = < E[ x(t) x H (t)
] > is the time-averaged correlation matrix of x(t). As the extended observation x(t) has redundant components for N > 1, R x is singular and consequently the solutions of the constrained optimization problem ( 5) are difficult to derive (see e.g., [10, sec.19.3c]). To solve this difficulty, an intricate equivalent GSC structure, which transforms a constrained least MS problem to an unconstrained least MS one, has been proposed in [START_REF] Chevalier | Third-order widely non-linear Volterra MVDR beamforming[END_REF].

EQUIVALENT THIRD-ORDER VOLTERRA GSC STRUCTURE

We present here a simpler equivalent GSC structure. It is based on the existence of a full column rank matrix B:

B def = Diag(B 1,0 , B * 1,0 , B 3,0 , B 3,1 , B 3,2 , B 3,3 ), (6)
with

B 1,0 def = [u 1 , ..., u N -1 ] and B 3,q = [B ⊗(3-q) 1,0 ⊗B * 1,0 ⊗q ],
q = 0, .., 3, whose columns span span(C) ⊥ , which implies

B H C = O [2(N -1)+4(N -1) 3 ]×[2+4(N 3 -(N -1) 3 )] .
Consequently, any filter w that may be decomposed into two components:

w = w f -v where w f def = [w T f , 0 T N +4N 3 ] T , such that w f is an N × 1 filter satisfying w H f s = 1, satisfies the constraint (4) if and only if v = w f -B w a ,
where w a is an unconstrained [2(N -1) + 4(N -1) 3 ] × 1 vector. Consequently, the constrained minimization problem ( 5) is equivalent to the unconstrained one:

min wa < E|( w f -B w a ) H x(t)| 2 > . ( 7
)
Using z(t

) def = [z T (t), z H (t), [z(t) ⊗ z(t) ⊗ z(t)] T , [z(t) ⊗ z(t)⊗z * (t)] T , [z(t)⊗z * (t)⊗z * (t)] T , [z * (t)⊗z * (t)⊗z * (t)] T ,
where z(t) def = B H 1,0 x(t), it is straightforward to prove that ( 7) is equivalent to the unconstrained minimization:

min wa < E|w H f x(t) -w H a z(t)| 2 > . ( 8
)
To solve easily this minimization (8), we have to withdraw the redundancies in the 4 terms z 3,q (t)

def = [z(t) ⊗(3-q) ⊗ z * (t) ⊗q ] of z(t).
This can be obtained by using selection matrices K q such that z ′ 3,q (t) = K q z 3,q (t). Consequently z(t) can be replaced by z ′ (t) = K z(t) with K = Diag(I N -1 , I N -1 , K 0 , K 1 , K 2 , K 3 ) and the minimization ( 8) is equivalent to the minimization:

min w ′ a < E|(w H f x(t) -w ′ H a z′ (t)| 2 >, ( 9
)
whose solution is

w ′ a,opt def = R -1 z′ R z′ ,x w f = [KB H R xBK H ] -1 KB H R x w f = [KB H R ñBK H ] -1 KB H R ñ w f , ( 10
)
where

R z′ def = < E[z ′ (t)z ′ H (t)] >, R z′ ,x def = < E[z ′ (t)x H (t)] >.
The output y(t) of the GSC structure is then given by:

y(t) = w H f x(t) -w ′ H a,opt z′ (t) = s(t) + w H f n(t) -w ′ H a,opt KB H x(t) = s(t) + ( w f -BK H w ′ a,opt ) H n(t). (11) w H f G LC(0) = 1 + α 2 β 6 ϵ 4 j |κ j,nc,2 -3γ j | 2 (1+ϵ j )A 0 -α 2 β 6 ϵ 4 j |κ j,nc,2 -3γ j | 2 , (21) G LC(1) = 1 + α 2 β 6 ϵ 4 j (κ j,c -2) 2 (1 + ϵ j )A 1 -α 2 β 6 ϵ 4 j (κ j,c -2) 2 , (22) G LC(2) = 1 + α 2 β 6 ϵ 4 j |(κ j,nc,2 -γ j ) + 2γ j /(β 2 ϵ j )| 2 (1+ϵ j )A 2 -α 2 β 6 ϵ 4 j |(κ j,nc,2 -γ j )+2γ j /(β 2 ϵ j )| 2 , ( 23
)
G LC(3) = 1 + α 2 β 6 ϵ 4 j |κ 2 j,nc,1 | (1 + ϵ j )A 3 -α 2 β 6 ϵ 4 j |κ 2 j,nc,1 | . ( 24
)
Expressions ( 14) to ( 17) and ( 21) to (24) of the SINR and gain G of the considered L-C(q) MVDR beamformers depend on the statistical properties of the interference and more precisely on the coefficients γ j , κ j,c , κ j,nc,1 , κ j,nc,2 and χ j,c . Consequently many insights can be deduced from these expressions. In the following, we concentrate on the behavior of these beamformers for a strong interference (i.e., ϵ j ≫ 1): For χ j,c = κ 2 j,c (which means that |j(t)| takes at most two values corresponding to zero and a non-zero constant value) and β ̸ = 0, (15) and ( 22) become:

SINR LC(1) ≈ ϵ s ( 1- α 2 (5κ j,c -4) κ j,c (κ j,c +1)-α 2 (κ j,c -2) 2 ) (25) G LC(1) ≈ 1 + α 2 (κ j,c -2) 2 κ j,c (κ j,c + 1) -α 2 (κ j,c -2) 2 , (26)
In particular for CPM, FSK and non-filtered PSK interference:

SINR LC(1) ≈ ϵ s [1 -α 2 /(2 -α 2 )] and G LC(1) ≈ 1+α 2 /(2-α 2 ). For impulsive interference, such that |j(t)| is
Bernoulli distributed and noting p def = P (|j(t)| ̸ = 0), we can verify that χ j,c = κ 2 j,c = 1/p 2 and we obtain as p decreases to zero, i.e., for very impulsive interference, SINR LC(1) ≈ ϵ s and G LC(1) ≈ 1 + α 2 /(1 -α 2 ). In this case, the SINR is the one obtained in the absence of interference.

For χ j,c = |κ j,nc,2 | 2 (which means that j 2 (t) takes at most two values corresponding to zero and a non-zero constant value, and consequently that j(t) is necessarily rectilinear) and β ̸ = 0, (14), ( 16), ( 21) and (23) become:

SINR LC(0) ≈ ϵ s ( 1- 9α 2 (κ j,c -|γ 2 j |) |κ j,nc,2 | 2 +9κ j,c -6Re(γ j κ * j,nc,2 )-α 2 |κ j,nc,2 -3γ j | 2 ) ( 27
)
SINR LC(2) ≈ ϵ s ( 1- α 2 (9κ j,c -|γ 2 j | -4Re(γ j κ * j,nc,2 )) |κ j,nc,2 | 2 +9κ j,c -6Re(γ j κ * j,nc,2 )-α 2 |κ j,nc,2 -γ j | 2 ) ( 28
)
G LC(0) ≈ 1+ α 2 |κ j,nc,2 -3γ j | 2 |κ j,nc,2 | 2 +9κ j,c -6Re(γ j κ * j,nc,2 )-α 2 |κ j,nc,2 -3γ j | 2 (29) G LC(2) ≈ 1 + α 2 |κ j,nc,2 -γ j | 2 |κ j,nc,2 | 2 +9κ j,c -6Re(γ j κ * j,nc,2 )-α 2 |κ j,nc,2 -γ j | 2 . ( 30
)
In particular for a non-filtered BPSK interference:

SINR LC(0) ≈ ϵ s , G LC(0) ≈ 1 + α 2 1 -α 2 (31) SINR LC(2) ≈ ϵ s (1 -α 2 ), G LC(2) ≈ 1. (32)
In this case, (32) shows that the L-C(2) MVDR beamformer does not improve the Capon beamformer, whereas (31) shows that the L-C(0) MVDR beamformer outperforms the Capon beamformer by completely canceling the interference whatever α . For χ j,c = |κ j,nc,1 | 2 (which means that j 4 (t) takes at most two values corresponding to zero and a non-zero constant value) and β ̸ = 0, (17) and (24) become:

SINR LC(3) ≈ ϵ s ( 1- 9α 2 κ j,c 9κ j,c +(1-α 2 )|κ j,nc,1 | 2 ) ( 33
)
G LC(3) ≈ 1 + α 2 |κ j,nc,1 | 2 9κ j,c + (1 -α 2 )|κ j,nc,1 | 2 . (34)
In particular, for non-filtered BPSK and non-filtered QPSK interference, we obtain:

SINR LC(3) ≈ ϵ s ( 1 - 9α 2 10 -α 2 ) ( 35
)
G LC(3) ≈ 1 + α 2 10 -α 2 , ( 36
)
which proves that the L-C(3) MVDR beamformer improves only slightly the Capon beamformer and remains less powerful than the WL beamformer for a rectilinear interference.

5.3.

Performance of WL-C(q) and L-C(q 1 , q 2 ) MVDR beamformers

Closed-form expressions of G WL-C(q1) /G WL and G L-C(q1,q2) are too intricate to derive. However, using symbolic math toolboxes and the results of Section 5.2, it is possible to prove in particular, that for a strong BPSK interference:

G WL-C(0) ≈G WL-C(1) ≈ G WL-C(3) ≈ G L-C(0,1) ≈ G L-C(0) ≈ 1+ α 2 1-α 2 > G WL ≈ 1+ α 2 2-α 2 > G L-C(3) ≈ 1, (37)
whereas for a strong QPSK interference:

G L-C(1,3) ≈ 1 + α 2 1 -α 2 > G L-C(1) ≈ 1 + α 2 2 -α 2 > G L-C(3) > G WL = G L-C(0) = G L-C(2) = 1. (38)
This result shows in particular that in this latter case, SINR L-C(1,3) ≈ ϵ s , which proves the quasi-optimality (among the beamformers which use the total noise statistics only) of the L-C(1,3) MVDR beamformer for a strong QPSK interference. Finally, let us note that in all cases, the WL-C(0,1,2,3) MVDR beamformer reaches at least the performance of the best WL-C(q 1 ) and L-C(q 1 , q 2 ) MVDR beamformer and is thus quasi-optimal not only for strong BPSK and QPSK interference but also for very impulsive interference, circular or not.

Performance illustrations

We consider throughout this section a two-element array with omnidirectional sensors and we assume that the SOI has a signal to noise ratio (SNR) π s /η 2 , equal to 10dB. This SOI is assumed to be corrupted by a single interference whose interference to noise ratio (INR) π j /η 2 , is equal to 30dB. Under these assumptions, Fig. 2 displays, for a non-filtered BPSK interference, the variations of SINR B at the output of different MVDR beamformers. Note that this figure confirms the results (37), i.e., the equivalent performance of the L-C(0) and WL-C(0) MVDR beamformers and the better performance of the L-C(0) MVDR beamformer with respect to the WL MVDR beamformer, itself better than the L-C(3) MVDR beamformer, the latter being equivalent to the Capon beamformer. Moreover, Fig. 2 shows the very weak information brought by the WL-C(0,1), WL-C(0,1,3) and WL-C(0,1,2,3) MVDR beamformers with respect to the L-C(0) or L-C(1,3) MVDR beamformers which are quasi-optimal.

CONCLUSION

An alternate equivalent GSC structure of a previously introduced third-order Volterra MVDR beamformer has been proposed. It permits both simpler implementations of this beamformer and analytical SINR computations at its output, which has been done in the paper. This analytical performance analysis allows us to specify how this beamformer outperforms the Capon and the WL beamformers for respectively, circular or not circular non-Gaussian interference, depending on the SO, FO and SIO interference statistics.

Fig. 2 SINR

 2 Fig.2 SINR B as a function of α, non-filtered BPSK interference.

Fig. 3 Fig. 3 SINR

 33 Fig.3displays the same variations as Fig.2, but for a QPSK interference. Again, this figure confirms the results (38), i.e., the better performance of the L-C(1,3) MVDR beamformer with respect to the L-C(1) MVDR beamformer, itself better than the L-C(3) MVDR beamformer, itself better than the Capon beamformer.

Fig. 4

 4 Fig.4displays the variations of G B at the output of the different MVDR beamformers as a function of α, for an impulsive circular interference, such that |j(t)| follows a Bernoulli distribution with P (|j(t)| ̸ = 0) = 0.001 associated with κ j,c = 1000. This figure confirms that in this case, G LC(1) ≈ 1 + α 2 /(1 -α 2 ) is quasi-optimal for very high value of κ j,c . As a consequence, the beamformers L-C(1,3) and L-C(0,1,2,3) bring no further gains with respect to L-C(1).

Fig. 4 G

 4 Fig.4 G B as a function of α, circular Bernoulli impulsive interference.


Philippe Guillaume
email: philippe@ifas.org.za

Myriam Houssay-Holzschuch

Myriam Houssay

Territorial strategies of South African informal dwellers Submitted to Urban Forum (Special issue on "Informal settlements in South Africa: case studies of contemporary processes

Introduction

Dramatic land invasion in Kempton Park, Johannesburg, in July 2001 has once again attracted international attention to the plight of land-hungry Black South Africans. The squatters held up placards in front of the removers and addressed Minister of Safety and Security Steve Tshwete in no uncertain terms: "Steve: have you got a nice big bedroom? All we want is a piece of ground the size of your kitchen. How many cars sleep in your garage? We don't want land that belong to others, there is enough land in this country for everyone. Please tell us Steve, how long must we wait?" (Mail and Guardian, 13/7/2001).

Political parties used this opportunity for their own particular political agendas: the Pan-Africanist Congress supported the invasion; the Democratic Alliance gave the squatters opportunistic material assistance. Moreover, the Kempton Park incident shed light on the situation of squatters in South African cities. As is the case elsewhere in developing countries, the rapid growth of the metropolis has created a huge housing problem, partly solved by illegal occupancy -under harsh living conditions -of urban land. In South Africa, the apartheid regime and its legacies worsened the problem by: 1) Enforcing a stringent control on African mobility to the cities and thus creating 'illegal' migrations;

2) Restricting land ownership, especially in urban areas;

3) Restricting African residency in urban areas to certain, delimited neighbourhoods, the townships, where accommodation was only to be found in low quality houses provided by the authorities. Reluctant to spend too much money on housing for the African population, the authorities never delivered enough accommodation for the African urban population. In some cases, e.g. in Cape Town, building of accommodation for Africans even stopped for decades, generating overcrowding in the existing townships as well as squatting on the Cape Flats. [START_REF] Fast | An Overview of African Settlement in the Cape Metropolitan Area to[END_REF] 4) As a logical consequence of apartheid policies limiting the presence of Africans in cities, the official reaction towards squatters was brutal eviction. No alternative accommodation was offered and removals took place even to remote homelands. [START_REF] Cole | Crossroads: the Politics of Reform and Repression, 1976-1986[END_REF][START_REF] Silk | A Shanty Town in South Africa: The Story of Modderdam[END_REF].

The last years of the apartheid regime saw a rapid growth of informal settlements in and around big cities from the second half of the 1980s (e.g. [START_REF] Crankshaw | The roots of homelessness: causes of squatting in the Vlakfontein settlement South of Johannesburg[END_REF]. At first, it was seen as a consequence of the 1986 repeal of influx control legislation: informal dwellers were then understood as new migrants, directly arriving from the countryside, especially from the homelands. Life in the settlements was also described as semi-traditional and semi-rural, thus providing an interstitial space where migrants could adapt to urban ways.

Detailed field studies proved that this interpretation was not correct [START_REF] Guillaume | Le logement en camps de squatters dans la région de Johannesbourg. Etude comparée de trois sites[END_REF][START_REF] Sapire | Informal settlement monitoring project: survey of 3071 black households in the PWV Region[END_REF] and offered an alternative, more complex, view of the situation: many squatters were in fact long-term residents of the agglomeration and had decided to resort to squatting as they felt it offered a 'suitable' alternative to their former residence and way of life. It was therefore a 'choice', obviously in a very limited set of options -mainly due to lack of financial resources. We will show that this choice took place in a holistic strategy to get access to urban and personal opportunities. Squatting was seen as a mean to attain personal independence and to gain access to private space, as opposed to the overcrowding of the surrounding townships. As these townships are usually located far from the city centre, squatting allows people to settle in places with better location: building a shack in centrally located areas gives easier access to urban resources (whether jobs or less legal sources of income). Whereas the informal dwellers may be poor and marginalized by the urban society, they are nevertheless able to make choices, to develop specific strategies and have a significant impact on the shape and life of their city. Since these choices are made by individuals, they are diverse and the squatters, as well as the informal settlements, should not be seen as an homogeneous group. Different people happen to settle in different places, for various reasons, thus creating different landscapes. Our findings match here with a strong current in social sciences: the view from 'below' rehabilitates the role of the poor in the construction of the city, the creation of its landscapes and its economy by fostering active strategies, whether through their choice of residence or of occupation.

Understanding the territorial strategies of the informal dwellers is an important challenge: according to the 1999 October Household Survey, 59% of African-headed households in urban areas were living in formal dwellings while 25% were living in shacks, 10% in backyard rooms and 4% in other accommodation such as a tent or caravan. The last 1% was living in traditional dwellings. Almost 40% of African-headed households in South African cities were housed in unsuitable dwellings (October Household Survey, 1999). Meeting the housing needs of these citizens is one of the most urgent tasks of the democratically elected government. What is more: as revealed in the Kempton Park placard, housing (or lack thereof) and the tenure of land are a key criteria in understanding new fractures in the postapartheid South African society.

Before we go into further analysis of the territorial strategies of informal dwellers, we must make some methodological remarks:

1. The data used in this paper comes from the existing literature (see reference list) and from our own field studies: in the Johannesburg area, in the Thembalihle, Kliptown (Soweto), Newtown and Power Station squatter camps (city centre), during the years 1995 to 1997; in the Cape Town area, in the settlements of Lower Crossroads, Khayelitsha and Mfuleni during the years 1996 and 1997. We have conducted in-depth interviews with randomly chosen residents and we have collected life histories.

2. The large scope of our data allows us to offer a generalised analysis of the territorial strategies of the informal dwellers. Comparing different settlements, in different cities is however a difficult task, and requires careful analysis. We are very much aware of the methodological difficulties of such a broad analysis. We must acknowledge that local contexts and policies have constructed strong differences. The official prohibition of squatting was enforced even more ruthlessly in Cape Town, for example, and this prohibition lasted and well into the 1980s. Building of accommodation for Africans was interrupted for many years, enormously increasing the housing backlog: people had no access to legal accommodation since vacant housing was non-existent; they could not resort to squatting without being evicted.

3. The internationally-used concept of 'informal' settlements covers many different realities.

While the exploration of these differences is one of our subjects here, let us state that a clear distinction needs to be made between informal tenure and informal housing.

Informal tenure includes non-western, traditional occupancy of the land, as well as squatting (i.e. occupation of a piece of land without its owner's consent). Informal housing can be found on squatted land but also in backyard shacks in townships or in siteand-service schemes. Harsh living conditions linked with informal housing (vulnerability to wind, rain and fire, problematic access to water, electricity and sewer, etc.) are not only the squatters' lot.

Squatting and the city: a socio-spatial approach

Landscapes of informality, landscapes of poverty

A first glance at informal settlements will present the homogeneity of the landscapes it produces: the lack of financial resources leads the residents to use recycled material for their shacks. Asbestos, wood, plastic and stones make up for shelter.

Widespread poverty among the residents can be clearly seen in the private space: the shacks are generally poorly equipped and sparsely furnished [START_REF] Houssay-Holzschuch | Le Cap, ville sud-africaine. Ville blanche, vies noires[END_REF]. Their internal organisation is designed along rationalised lines to optimise the use of the little available space. The few rooms can generally boast only a few mattresses, clothes hanging on walls covered with recycled wrapping paper: Campbell soups, Cadbury's or Liebig, emblems of a consumer society, ornate the walls of the very poor. In a corner, pots and pans indicate the kitchen. Cooking takes place either on a paraffin stove or, if the informal settlement has been provided with electricity as part of a regularisation process, one electric plate. An old couch, the pride of the family, makes room for parents and visitors.

Inside the shack, a semi-public, semi-private space is thus defined: it is a space for social relations, for reception and representation. This 'social space' is used to enhance the resident's prestige, to offer people an ideal representation of him/herself and to publicly display his/her values. It has at first a strongly Western flavour: the home ideal of many informal dwellers is the common representation of a modern, Western, middle-class interior.

The 'riches' or material possessions of the dweller are displayed: TV or music player if any, bibelots on the coffee-table, crockery in a glassed dresser. Even the poorest among informal dwellers tend to reconstitute this environment, e.g. with second-hand furniture. It is kept immaculate and receives many people. A high frequency of visits is seen as socially and culturally desirable and is made possible by the high unemployment rate. The walls around the couch are further decorated by photographs showing formally dressed, graduating children: education is here both internally recognised and socially displayed as an important source of prestige and the only way to social upliftment. This inside organisation and decoration of the shack can be found in the vast majority of informal settlements in South African cities. It is also similar to the ones existing in formal African townships [START_REF] Houssay-Holzschuch | Le Cap, ville sud-africaine. Ville blanche, vies noires[END_REF]. This homogeneity of private space organisation is mirrored by similarities in the public space across various settlements. As their 'informal' status would suggest, they are ill-equipped: water, electricity, sewer, roads are lacking more often than not. Public amenities such as schools or clinics are absent. Residents try to tackle their own needs either by providing informal services (such as commerce, crèches, etc.) or by organising and lobbying for in situ upgrading. They have often attained some success and this creates a first differentiation between settlements: some have been 'regularised' (their existence and their occupation of a specific piece of land has been juridically recognised) and have been delivered with communal taps and toilets; others are still battling or are simply too recent.

These arrangements of both internal, private space and public space in informal settlements give us some leads to understand the socio-spatial processes of place-making and settlementmaking for informal dwellers:

1. Most evidently, poverty characterises the squatters' space.

2. Relatively homogeneous design appears, by contrast to the variation in external landscapes shown by a careful analysis (see next paragraph). Not surprisingly, this design tries to answer simultaneously the need for a shelter and the need for a home: for instance, thermal insulation and decoration are both provided by the recycled wrapping papers on the walls.

3. Social values, relations and networks pervades this internal space and transform it into place. Similarly, we will show that settlement-making (i.e. localisation, produced landscape, economic and land profile...) also follows both different networks of sociability and the squatter's expectations and values in terms of family life, quality of life, etc.

The poor are here building their own city, trying to put their individual and collective mark on an area to locally transform a space imposed by poverty into an appropriated place reflecting their own values.

Different stories, different spaces

While this is the general picture that can be found throughout South African cities, informal settlements each have their specificity, linked to their particular history. The seven specific cases that follow (four of them in the Johannesburg area, the other three in Cape Town) give an insight into different dynamics.

-Thembalihle and Mfuleni

The informal settlement of Thembalihle (Good Hope in Zulu) has been built on an open space in Lenasia, an Indian township located 30 km southwest of Johannesburg. The first residents of Thembalihle arrived in 1986, settled on land belonging to the then PWV Region and the squatter camp grew to 6,000 residents in 1990 and 28,000 in 1996 (Guillaume, 2001:41). The first residents organised themselves into a civic association with the proclaimed objective of organising and controlling the space of the settlement. Each newcomer was assigned a 250 m 2 plot on which he could build his/her shack. Today, the 5,800 shacks of the settlement do not have access to water or to electricity, even though the Lenasia power lines pass directly above the settlement. Since the 1995 local elections, the new Thembalihle's mayor has been trying to improve the living conditions in the settlement: public toilets, garbage containers, and caravans offering police or health services have been placed at the settlement's entry. The general landscape in Thembalihle is made of large, green plots where shacks or houses are build (20% of the shacks are consolidated, or are in the process of being consolidated into permanent structures). The grid-iron plan and large streets planned by the civic association add to the calm and order of the place. This landscape resembles many popular neighbourhoods in French-speaking Africa, even if the economic profile of Thembalihle is much more strictly urban: cities of various sizes and status, such as Abidjan [START_REF] Antoine | Abidjan " côté cours[END_REF], Lomé (Gervais-Lambony, 1994), Kinshasa [START_REF] Pain | Kinshasa, la ville et la cité[END_REF], Toliara (Houssay-Holzschuch, 1994), etc. have been described by scholars as retaining a "rural" character, especially in their peripheral landscape. There, relatively large plots house the extended family; they are often planted with trees and allow some poultry or even cattle-raising.

Traditional networks are still at work. Nevertheless, all authors make clear that this rural character or ruralité is not at all incompatible with urban life, roots and values. Thus, whereas Thembalihle may well be an illegal settlement, it is nevertheless an organised and controlled space where residents lead a peaceful life by South African standards. The squatter camp in Mfuleni, an African township in the periphery of Cape Town, offers the same characteristics in terms of landscape: located at the very periphery of the City of Cape Town area, it is also a peaceful neighbourhood. There is only little competition for space, which allows residents to occupy large plots. The sandy soil of the Cape Flats has been planted with plants and greenery, flourishing with reasonable success. Some services and an important housing project were encouraging signs for the squatters' future in 1997.

-Lower Crossroads Peripheral informal settlements do not always conform to the model exemplified by the Thembalihle case. The Lower Crossroads settlement of the Cape Flats is far from the city centre and is situated in a low-density area. Nevertheless, the settlement itself is very densely populated. As from 1986, its residents chose to live here as a refuge against the violence in Crossroads [START_REF] Cole | Crossroads: the Politics of Reform and Repression, 1976-1986[END_REF]. They were mainly supporters of the controversial leader Nxobongwana and followed his political (mis)fortune. Established in a context of urgency and violence, the settlement has not been planned and its sandy soil does not allow easy gardening or waste and garbage disposal. The basic facilities have taken a long time to arrive, even if they now compare to those of Thembalihle.

-Kliptown and Nyanga : Kliptown, one of the oldest neighbourhoods of Soweto, offers a landscape in striking contrast to that of Thembalihle. The informal settlement has occupied the interstitial, vacant spaces of an historic suburb; it was a racially mixed neighbourhood at the beginning of the 20 th century and was later proclaimed African. Many buildings and the commercial centre are relics of Kliptown's past. The density of services and shops, reaching a level seldom found in African townships, resemble East African urban areas: the bustling streets of Nairobi [START_REF] Vennetier | Les villes d'Afrique tropicale[END_REF] or Kampala [START_REF] Calas | Kampala, la ville et la violence[END_REF] offer a similar atmosphere and landscape. Indeed, the bungalow-style old houses, the arcades under which Indian-owned shops are located, the mixing of the formal and the informal (in terms of commerce, services or housing) strengthen this impression. In the early 1980s, Kliptown's residential component was 'informalised': lacking space, many residents or tenants of formal houses built shacks on the same plot or nearby. No general plan is apparent, as the shacks were built one after another to suit the Kliptownians' needs. Many shacks partly occupy public space, such as the street. A nearby vacant space has also been occupied in the second half of the 1980s. The landscape created by the 3,000 or so informal dwellers in Kliptown is characterised by a feeling of urban complexity if not lack of readable spatial organisation similar to the one encountered in the Nyanga township of Cape Town: squatters settled on any kind of vacant space, in between formal houses and hostels. The bustling informal market on the main road through Nyanga adds to the "urban" atmosphere (see further for specific developments on the concept of "urbanity").

-Newtown and Power Station

The Johannesburg city centre has also been a locality for squatters. Its informal settlements such as Newtown or Power Station are yet another type. Though they are much smaller (between 100 and 200 residents in Newtown in 1996, some 20 shacks in Power Station), they function as 'blot on the landscape', as defined by [START_REF] Dixon | Ideology, Geography, Racial Exclusion: The Squatter Camp as 'Blot on the Landscape[END_REF]: they are surrounded by the modernist architecture of high-rise and prestigious buildings such as the de Beers building. The Johannesburg Stock Exchange, the Market Theatre, museums or the French cultural and scientific centre IFAS (Institut Français d'Afrique du Sud) are also in the immediate proximity. Particularly dilapidated shacks were densely build in 1995 on a vacant space next to the Newtown taxi station, before being forcibly relocated in December 1996 to the peripheral site of Orange Farm (some 50 km south of Johannesburg). The inner city squatters were among the poorest of the poor, barely scratching together a living in the big city by any possible means, legal or not. Whereas they were in a very central location, they did not have access to any services: there were no toilets, no electricity and a standpipe at the taxi station was the only source of water. Newtown was a hopeless space, a last refuge for desperate people. Power Station, a disused industrial building nearby, also accommodated some fifty squatters: the walls and roof, though in bad shape, were sheltering a few makeshift shacks, hidden from the street. Their makeshift construction, enabled by the protection of the power station, show that the line between squatting and homelessness can be a very tenuous one.

Squatting and poverty

As observed by some specialists of popular housing in Third World cities, "informal housing is the most visible dimension of poverty" (Gilbert and Gugler, 1992:114). [START_REF] Sapire | Informal settlement monitoring project: survey of 3071 black households in the PWV Region[END_REF] have shown that people living in squatter camps are, by far, in worse economic conditions than formal townships residents.

Average monthly household income

Matchbox 900

Backyard shack 749

Table 1: Average monthly household income (in ZAR) in 1990, in relation to the housing structure. (Source: Sapire & Schlemmer, 1990, p. 73).

Our 1996-1997 fieldwork in Gauteng has produced similar results. The average monthly household income amounted to R 744 in Thembalihle and to R 481 in Kliptown, but in both cases, the per capita income was only R 120. In 1995, some 40% of South African Black urban families were living below the poverty line, estimated at R 840 per month for an urban family of two adults and three kids (Republic of South Africa, 1994). [START_REF] Crankshaw | The roots of homelessness: causes of squatting in the Vlakfontein settlement South of Johannesburg[END_REF] also showed that poor qualification or employment characterised the Vlakfontein squatter population, compared to the situation in the townships: 51% of professionally active people were unqualified as opposed to 'only' 27% in surroundings townships. What is more, 68% of the Vlakfontein residents were working in the informal sector.

Unemployment is also widespread among informal dwellers as families dramatically experience the loss of a job: unable to pay for their former matchbox house or flat rent, they move to a more affordable form of shelter, usually in informal settlements. This was the case for the vast majority of the Johannesburg CBD squatters. In addition, people in Newtown or Power Station squatter camps had no family to turn to in difficult times. For them, squatting was the only option.

As elsewhere in the Third World, poor education, low qualification and skills, piece jobs and low wages are some obvious explanations for the presence of informal squatters camps in South African cities. In most cases, squatters appear to be living "without any visible means of subsistence" (Koch, 1983:151).

However, other, very specific explanations for squatting can be drawn from the South African 2. The lack of available accommodation for urban Africans generated a very high population density in the "matchbox houses", as the formal houses built under the apartheid regime are often designated by their residents. Many legal tenants generated an income from renting rooms or backyard shacks at relatively high rates: in 1997, an illegal tenant in a backyard shack could pay between R 40 and R 100 per month to the legal resident (who paid less in formal rent for the whole matchbox house). Arbitrary decisions from the landlord are common and concern the amount and increase of the rent as well as curfew hours at night, maximum number of people allowed to visit, or the restricted use of water, electricity or toilets. Moving to an informal settlement can then appear as a suitable alternative to a landlord's whims: the rent would be reduced to zero, there might be no loss of comfort and the former tenant would gain independence. Even when the relations between tenant and landlord are good, the unwritten agreement between them is always at risk, as observed in our fieldwork: a family spent fourteen years subletting rooms in a Sowetan matchbox house, and feeling part of the household. When the legal tenant passed away, this family had no legal right to further occupy the dwelling. The family had to move to an informal settlement.

3. In many cases, industrial or mine workers are housed by their employers. During the various economic crises that have affected the mining sector in the last decades, many workers that lost both their job and their accommodation found refuge in surrounding squatter camps.

4. We also came across a large number of people, aged fifty to sixty, that had been squatters in urban areas for their whole life. Because of apartheid legislation, such as the 1951 Prevention of Illegal Squatting Act, they had repeatedly experienced forceful removal, and their shacks had been destroyed many times. Lacking any alternative, they could only reproduce one residential strategy: build another shack somewhere else.

If poverty is indeed one of the first causes for squatting, it is not the only one. When asked what led them to their informal home, all the residents interviewed in the Johannesburg CBD squatter camps, without exception, said that they were in desperate need of a place to stay. In Kliptown, only 41% gave the same answer, while this figure dropped to 19% in Thembalihle.

Thus, reasons to resort to squatting differ. Are they part of a more holistic personal, collective or territorial strategy?

Movement into squatter settlements

Squatters are not newcomers to the urban environment. In Kliptown and Thembalihle, we found that most squatters entered the Johannesburg area between 1971 and 1980. Other studies, such as [START_REF] Sapire | Informal settlement monitoring project: survey of 3071 black households in the PWV Region[END_REF], confirm the squatters' relatively early arrival.

We also found that the 1986 abolition of Influx Control legislation did not necessarily cause rural-urban migration into informal settlements. Once in the Johannesburg area, the urban poor move within the metropolis: they change places in order to access different urban opportunities. If we add together the number of people previously housed in matchbox houses and in backyard shacks, 76% of Thembalihle's people and 60% of Kliptown's people had been living in formal townships before resorting to squatting. Overcrowding, cost of living and lack of personal freedom were strong reasons for this move.

Johannesburg area

Squatter networks

The landscape, localisation and socio-economic characteristics of the various informal settlements we have studied differ significantly. We suggest that different social networks and motivations have presided to their birth and development. Squatters, according to their means and needs, develop an adapted territorial and social strategy. Three different kinds of solidarity are at work: kinship, group, and project solidarity.

Kinship solidarity

In the Kliptown squatter camp, the settlement process has largely been organised along family lines. One third of the squatters who arrived in Kliptown to find a space of their own joined a member of their family. This type of network has long been described as an important access to city facilities, both in Third World countries and in the West during the Industrial Revolution (e.g. [START_REF] Bairoch | De Jéricho à Mexico, Villes et économie dans l'histoire[END_REF]. Let us add that here, that access to kinship relations in a context of an acute housing shortage is particularly important and shows itself in the residents interviews: some came to occupy a shack freed by a family member who found better accommodation elsewhere. Others built their shacks on family land, next to a formal house inhabited by an aunt or cousin. The historic character of the Kliptown neighbourhood, with many long-time residents, enables this type of solidarity. Many newcomers used their family's facilities or social connections in the area.

Other migrations to squatter settlements functioned in the same way, but used religious networks instead of family relations: people settled with the help of members of the same church. For example, members of the Bantu Congregational Church of Zion in South Africa moved inside Khayelitsha, Cape Town, to be closer to their place of worship in Site B. Their new localisation enabled them to enjoy the day-to-day presence and solidarity of other church members: small monetary loans, some help for looking after the children, or visits to bedridden members were common (fieldwork, 1997). Religious solidarity can also act by providing information: Eida, member of a Zionist church, left her shack in Dhlamini to settle in Kliptown, following the advice of another church member.

Group solidarity

'Groups' as we define them are less formal categories of people, momentarily sharing a common interest or living through the same difficulties. For instance, homeless people and street children of the Johannesburg CBD share the same and extremely difficult survival conditions.

Group solidarity was an important strategy in the formation of the Newtown squatter settlement. Before their removal by Police forces in 1995, a group of squatters had occupied a piece of land near Braamfontein Railway Station. After their shacks were destroyed, they sought new accommodation. The majority stayed together as a 'group' and settled nearby, creating the Newtown Squatter Camp. The collective intra-urban migration allowed each individual to find a minimal shelter. The number (a few dozens) of the former Braamfontein

Station squatters was also a deterrent against new eviction and police intervention, easier against individuals. In consequence, the existence of the group had some strategic reasons.

Furthermore, the group had an economic raison d'être: most of its members were engaged in semi-legal or illegal activities such as trade in taxi spares or dagga (wild hemp or marijuana).

Newtown Squatter Camp housed individuals that all shared the same socio-economic profile:

they were all destitute and marginalized in and by the urban society. They had no housing, no formal economic activity, and no family. After the December 1996 removal of the camp and the resident's relocation in Orange Farm, the solidarity between them faltered and the group ceased to exist: the social and economic networks that underpinned the group no longer existed. On an individual basis, most of the former Newtown squatters returned to the city, a strategy to reduce transport costs. The city centre also offered them better prospects for involvement in small or illegal businesses. In 1997 we re-encountered some of the former Newtown squatters in the Power Station squatter camp, where they tried to organise themselves into a new "residents committee" and to further consolidate their situation in the centre of the city. However, the Power Station building was cleared and barricaded in mid-

2000.

Conclusive remarks: urbanity, citadinité and citizenship and the South African urban crisis

The squatter crisis has continued unabated in the last years and is a key political issue for the South African authorities, at national, provincial and local level. To better understand the real stakes of this crisis, we will use two concepts recently developed by French-speaking geographers: "urbanity" (urbanité) and citadinité, linking them to the third notion of "citizenship" (citoyenneté).

"Urbanity" was defined by Jacques Lévy as the specific trait of cities, what makes a city out of a settlement, or "putting together a maximum of social objects within a minimal distance"

("mise en coprésence du maximum d'objets sociaux dans une conjonction de distances minimales", Lévy, 1994:286). Following Max Weber and the Chicago School, Lévy considers that urban space enables and facilitates meeting between people of different backgrounds and origins. Using, as for instance Weber did, the European city as a reference, he shows that it is both a heuristic and normative model. Heuristic, because it is a model where compacity, social heterogeneity, use of public transports, etc. have historically developed. Asian or Arab cities, while having led to fewer literature and theorisation, offer a similar pattern. As such, this model of a European city, with a high level of social and racial mixing (of urbanity), is a tool to understand other cities with a comparative method.

Nevertheless, Lévy underlines also that the European city has long been a normative model by planners and urban thinkers: an urban settlement with distinctly non-European characteristics was denied the quality of the city. His highly sophisticated conception of urbanity and the evaluation thereof [START_REF] Lévy | L'espace légitime. Sur la dimension politique de la fonction géographique[END_REF]2000) uses the heuristic model of the European city as a starting point to build a methodology to compare cities around the world: they have varied level of efficiency in providing meeting places or opportunities for their own inhabitants and it can be evaluated in terms of different levels of urbanity. For instance, highly segregated cities, and South African cities among them, are considered as having a low level of urbanity.

The 1997 Urban Development Framework (Republic of South Africa, 1997) implicitly recognised this lack of urbanity when it planned a more integrated, new South African, city.

Unfortunately, if the squatter's lot has been improved by housing and service delivery in recent housing schemes such as Devland near Soweto or Delft South in Cape Town [START_REF] Guillaume | Le logement en camps de squatters dans la région de Johannesbourg. Etude comparée de trois sites[END_REF]Oldfield, 1999), they are still located in the periphery of the city: for financial reasons, the newly-developed land are far from the city centres, therefore far from jobs and other opportunities. The squatters' spatial integration into a new, racially and socially integrated city is thus compromised.

The concept of citadinité, on the other hand, does not describe the characteristics and efficiency of the urban space: it is people-centred and refers to someone's capacity of being an active element in the urban society and environment, in which he/she feels at home.

Similar notions have been developed by the Chicago school or Henri [START_REF] Lefebvre | Le Droit à la ville[END_REF], under the notion and/or translation of urbanism, understood, as in Wirth's expression, as a way of life. Elisabeth Dorier-Apprill adds that "the term of citadinité applies to the city dwellers, their spatial practices and their representations of urban space, their appropriation of urban territories and their cultural values rooted in urban space" ("Le terme de citadinité s'appliquerait plutôt aux habitants des villes, à leurs pratiques et representations des espaces urbains, leurs formes d'appropriation de ces espaces, leurs ancrages culturels dans la ville", -Apprill, 2001:81). The squatters we met during our fieldwork have been rooted for a long time in the urban environment. Their values and practices are typically urban and they definitely feel at home in the city. Their citadinité is thus both a fact and a claim: that they should have a right to the city, Lefebvre's "droit à la ville" (1968), to its equipments, jobs, and opportunities (Gervais-Lambony, 2001:96).

Dorier

Lastly, "citadinité" is etymologically close to "citoyenneté" (citizenship), the political, juridical, conscious and active belonging to one's place (e.g. to one's country). The closeness of these two terms has long been recognised as not only linguistic: both "city" (with a high level of urbanity) and "nation" are imagined communities of diverse individuals, sharing space, resources and even values (e.g. [START_REF] Ghorra-Gobin | Penser la ville de demain : qu'est-ce qui institue la ville ?[END_REF]. Again, the political nature of space comes to light. Well known by the apartheid regime, it was recognised by the RDP when it stated that cities, rebuild in a more compact, central and integrated way, have an important role to play in the birth of the new, democratic nation. The planners reflecting on the squatter crisis and trying to find solutions should be reminded of this fact.

  violence characterised the 1980-1990 decade. The conflict between hostel and township residents created no-go areas, e.g. in the East Rand areas of Thokoza, Vosloorus or Katlehong. Township residents living close to the hostels, and therefore on the front line of conflict, may have seen their homes destroyed. These families resorted to squatting.

Table 2 :

 2 Respective dates of arrival in the Johannesburg area and in the Kliptown squatter camp (%). Source: own fieldwork, 1996.

	Kliptown camp

Table 3 :

 3 Previous type of housing of Thembalihle and Kliptown squatters (%). Source: ownfieldwork, 1995-1996.

Similar projects

In Thembalihle, another form of network was prevalent and explained the socio-spatial formation of the settlement. The residents were neither kin, nor linked by group solidarity as described above. They accessed the area with a monetary transaction within a specific housing market. They were also sharing a same residential project.

Whereas squatters are commonly seen as people who build their house themselves, this was not the norm in Thembalihle: 57% of shacks were not built by their residents, but bought. A housing market is thriving in this settlement. Starting at R 400 for the smallest shack, a tworoomed shack is worth around R 2500. The maximum price of R 7500 corresponds to a big shack transformed into a permanent building. The shacks in secondary ownership were bought at an average price of R 2700 (Fieldwork, 1995(Fieldwork, -1996)). This represents a considerable amount of money for squatters, often more than one year's rent for a large sublet room in a formal matchbox house. To buy a second-hand shack is then a substantial project, which requires financial savings and sacrifices. From a background of deprivation in terms of freedom, mobility, land ownership and choice to ask a builder for a "small house" or to buy one, even in an informal settlement with no legal titles, is proof of one's independence, autonomy, respectability and security. For the people of Thembalihle, their seemingly pathetic shack is the first realisation of an ambitious project in their life, in the city and in society: they are able to live independently and peacefully with their family in their own private space.

Succeeding in such a project represents a victory over the past, over their own poverty that led them only to overpopulated townships or to squatter camps under constant threat of eviction.


Nans Roques

Philippe Gerbier
email: gerbier@univ-montp2.fr

Inhar Imaz

Philippe Guionneau

Jean-Pascal Sutter

Bis(hexafluoroacetylacetonato)manganese(II) complex of 1,1-dimethyl-2,5-bis-(3-N-tert-butyl-N-phenylaminoxyl) -3,4-diphenylsilolacyclopentadiene

published or not. The documents may come

Bis(hexafluoroacetylacetonato)manganese(II) complex of 1,1-dimethyl-2,5-bis-(3-N-tert-butyl-N-phenylaminoxyl)

-3,4-diphenylsilolacyclopentadiene.

Comment

The design and construction of molecule-based magnetic materials possessing a well defined structural dimensionality along with interesting magnetic properties has been a challenging proposition over the past twenty years (Miller et al., 2000;Veciana et al., 2000). In this field of research, one of the most successful approaches towards molecule-based magnetic materials involves the coordination of stable organic radicals to high-spin metal ions, such as manganese in the form Mn(hfac) 2 (hfac = hexafluoroacteylacetonate). Following this approach, numerous examples of molecule-based materials possessing a large variety of dimensionality and architecture (Mathevet et al., 2001) have been reported on account of the large number of combinations available through variations of the nitroxide radical nature and the organic spacers to which the radical unit(s) is (are) bound.

We have already reported the synthesis, structure and detailed magnetic properties of the first example of silacyclopentadiene (or silole) substituted by two phenyl-tert-butyl nitroxide radicals (I) (Roques et al., 2003). Our goal was to take advantage of the photo-excited triplet state that was evidenced in the silole spacer to generate a reversible light-induced modification of the magnetic behaviour of this molecule (Roques et al., 2004).

With the idea of further pursuing these investigations, we report herein the synthesis and structure of the first manganese bis-adduct of (I): complex (II). It is worth noting that II is only the second example of a tert-butyl nitroxide based mononuc-lear manganese complex (Rajca et al., 2001). This discrete unit is a promising model for the exploration of photo-induced magnetic states in low dimensional metal-organic magnetic materials.

Complex (II) is obtained as dark red diamond-like blocks by reacting silole 1 with Mn(hfac) 2 (see experimental section).

Structure examination reveals a linear bis-adduct structure, I-Mn(Hfac) 2 -I (Fig. 1). The Mn 2+ ion coincides with an inversion centre and is bound to two chelating hfac ligands and two monodentate nitroxides in a distorted octahedral configuration, as previously reported in the literature (Iwamura et al., 1998;Inoue et al., 2000) for extended one-and two-dimensional structures. The distortions from regular octahedral geometry about Mn are rather small, with no bond angle deviating by more than 6° from its ideal value. The largest distortions involve the pincer angles of ca 84° for the chelating hfac ligands.

The Mn-O bond distances lie in the range of 2.119 (2)-2.176 (2) Å. These are typical values for Mn 2+ complexes of hfac and nitroxide ligands [START_REF] Shibata | [END_REF][START_REF] Dickman | [END_REF]. Coordination of the nitroxide ligands to the metal centre affords strong changes in its structure. As usually observed in tetraarysiloles, the molecule displays a propeller-like arrangement of the four benzene rings (Yamagushi et al., 2000). However, the C2 symmetry axis that passes through the central silicon atom in the structure of (I) is lost (Roques et al., 2003). The dihedral angles between the five membered silole ring and the phenyl rings bearing the nitroxide radicals are 45.92 (9) and 35.61 (9)°, the latter being considerably smaller than the value of 49° encountered in the parent molecule. Another difference is observed for the dihedral angles made by the nitroxide groups and the phenyl rings to which they are bonded. While in the parent silole the dihedral angles are similar to that found here for the free nitroxide, 16.1 (4)°, the complexed nitroxides in II are only twisted by angles of 7.3 (4)°. These structural modifications are ascribed to the presence of the bulky hfac ligands in the coordination sphere of the metal center.

To conclude, compound II is the first bis-adduct of 1,1-dimethyl-2,5-bis-(3-N-tert-butyl-N-phenylaminoxyl) -3,4-diphenylsilolacyclopentadiene with bis(hexafluoroacetylacetonato)manganese(II) to have been synthesized and structurally characterized. Experiments are currently underway in our laboratory to use this bis-adduct as a building block to construct magnetic chains. Our efforts are also ç directed at studies of the photo-excited magnetic states of this metal-organic radical species.

Experimental

A sample of Mn(hfac) 2 •H 2 O (12 mmol) was suspended in 30 ml of n-heptane and the mixture was refluxed for 24 h to remove water by azeotropic distillation. To the cooled solution was added 12 mmol of bis-nitroxide silole (I) in 10 ml of freshly distilled dichloromethane. After stirring for 10 min, the resulting deep brown solution was concentrated to ca 5 ml under reduced pressure. The dark microcrystalline powder formed during the concentration step was removed by filtration, and the resulting filtrate let to evaporate to dryness to yield dark red diamond-like crystals of (II) after four days.

Refinement

All H atoms were positioned geometrically (C-H = 0.96 Å for the H atoms of methyl groups and C-H = 0.93 Å for the other H atoms) and allowed to ride on their parent atoms [U iso (H) = 1.2Ueq(C)]. Non-H atoms were refined anisotropically.

Computing details

Data collection: KappaCCD Server Software [START_REF] Nonius | KappaCCD Server Software[END_REF]; cell refinement: DENZO-SMN (Otwinowski & Minor, 1997); data reduction: DENZO-SMN; program(s) used to solve structure: SHELXS97 (Sheldrick, 1990); program(s) used to refine Selected geometric parameters (Å, °)

Figures

  FiguresFigure 1. The structure of complex (II), showing the atom-numbering scheme and displacement ellipsoids at the 30% probability level. For clarity, only the manganese atom and the atoms belonging to the silole ligand have been labelled.

  FiguresFigure 1. The structure of complex (II), showing the atom-numbering scheme and displacement ellipsoids at the 30% probability level. For clarity, only the manganese atom and the atoms belonging to the silole ligand have been labelled.

Acknowledgements

This work was supported by the French CNRS and by the European Commission through the Network of Excellence MAG-MANet (NMP3/CT/2005/515767).