Abarna Lingeswaran

Alexia Royer

Coralie Metton

Céline Henry

Véronique Monnet

Vincent Juillard

Rozenn Gardan

The study of the pheromone exporter PptAB reveals crosstalk between different quorum sensing systems in Streptococcus thermophilus

In Gram+ bacteria, oligopeptides serve as the signaling molecules of quorum sensing (QS) mechanisms. They are secreted, matured and detected. In the case of SHP/Rgg mechanisms, peptides (SHP) need to be imported back to interact with their cognate regulators (Rgg) that modulate the expression of target genes involved in crucial functions such as virulence or competence.

Introduction

Lingeswaran, A. et al. Export of Rgg quorum sensing peptides is mediated by the PptAB ABC transporter in Streptococcus thermophilus strain LMD-9. Genes 11, doi:10.3390/genes11091096 (2020).

Perspectives

Results

1. Genes regulated by a SHP/Rgg mechanism in S. thermophilus strain LMD-9 are mainly located downstream of the rgg genes * veronique.monnet@inrae.fr

The SHP/Rgg QS mechanism in S. thermophilus

We have described such a SHP/Rgg mechanism in S. thermophilus. The SHPs are matured by the membrane protease Eep, exported by the ABC transporter PptAB and re-imported by the oligopeptide transporter Ami. Interestingly, this mechanism is widespread in streptococci and S. thermophilus has accumulated several copies of SHP/Rgg systems in its genome. The objective of this work was to characterize all targets of the 5 different systems present in S. thermophilus strain LMD-9.

• We will seek which SHPs regulates the 5 Rgg and will assess the affinity of the different pairs with the objective to have a global view of this regulation QS network in S. thermophilus.

• We will study the function of the targets of these systems • We hope that these results will help to better understand these SHP/Rgg mechanisms in streptococci with the objective of finely and precisely manipulating them without collateral effects.

• We have demonstrated cross-talk between the SHP/Rgg systems of S. thermophilus strain LMD-9, i.e. that the targets of a locus can be controlled by a distal SHP.

Conclusion

We compared the transcriptome of strain LMD-9 vs its isogenic ΔpptAB mutant* using a deep sequencing approach (RNAseq):

• the expression of genes located downstream of the 5 rgg genes was downregulated in the ΔpptAB mutant (in white on the figure; genes hatched in grey were unaffected),

• 2 shp out of 5 were strongly controlled (shp2 and shp3) and one was slightly controlled (shp5) by PptAB and thus by QS (in red). The 2 remaining shp transcripts were not detected by this approach (in red, hatched in white),

 Two shp genes (shp1 and shp4) do not seem to be expressed although the genes located dowstream their cognate rgg genes are controlled by QS * Both strains were inactivated for comR encoding the regulator of the competence as (i) the activity of ComR is positively controlled by QS via PptAB, (ii) the growth rate of a ΔcomR or ΔpptAB mutant is impaired compared to a WT strain. Thus, both strains (ΔcomR and ΔcomR ΔpptAB) had a strictly identitical growth rate. We only detected three SHPs out of the five (SHP2, SHP3 and SHP5; in grey in the table) by LC-MS/MS from culture supernatants in accordance with the results of the RNAseq approach.

Organizational schematic of the genes controlled by

Working hypothesis: as the SHP of two loci (1 and 4) were apparently not synthesized although QS target genes were expressed (RNAseq results), we hypothesized that other SHPs were involved in the regulation of the both loci. We chose the target of locus SHP4/Rgg4 (gene 7535) to study this phenomenon.

SHP3 is involved in the regulation of the locus shp4/rgg4 target (7535)

We studied the regulation of gene7535 using a transcriptional fusion with the reporter LuxAB (P7535-luxAB). We hypothesized that the SHP3/Rgg3 system could be involved in this regulation because the sequence of the promoters of gene RS10775 (the target of the SHP3/Rgg3 system) and 7535 are identical.

 The expression of the gene 7535 does not seem to be controlled by SPH4 but is controlled by SHP3 and at least another SHP. The P7535-luxAB transcriptional fusion was introduced in different genetic background.

• Inactivation of SHP4 did not affect p7535 activity (red curve),

• Inactivation of SHP3 only partially affected p7535 activity (orange curve).

PptAB 2. Three SHPs are detected by mass spectrometry in strain LMD-9 SHP AA Sequence of the native SHPs * [M+H] + theorical [M+H] +

				experimental
	SHP1	MNKESFLAILLLIFESIIVIAVG	900.54010
	SHP2	MKKVIAIFLFIQTVVVIDIIIFPPFG	1018.56084	1018.56165
	SHP3	MKKQILLTLLLVVFEGIIVIVVG	898.56084	898.56189
	SHP4	MKKQKLLLLVVLVCEGIIVILVG	912.57649
	SHP5	MNKKALFSLLFVILEGIIVIGVG	856.51389	856.51599

* The AA sequence of the mature forms of the SHPs is underlined.


Guillaume Dupont

Sébastien Guenneau
email: sebastien.guenneau@fresnel.fr

Stefan Enoch

Guillaume Demesy

André Nicolet

Frédéric Zolla

André Diatta

Revolution analysis of three-dimensional arbitrary cloaks

Keywords:

 to cloaks with a surface of revolution. We derive the expression of the transformation matrix and show that one of its eigenvalues vanishes on the inner boundary of the cloaks, while the other two remain strictly positive and bounded. The validity of our approach is confirmed by finite edge-elements computations for a non-convex cloak of varying thickness.

Introduction

It was recently found that transformation optics open new avenues in electromagnetic cloaking, either through their heterogeneous anisotropic effective material parameters (transformation optics, [START_REF] Pendry | Controlling electromagnetic fields[END_REF][START_REF] Leonhardt | Optical conformal mapping[END_REF]) or through low index materials [START_REF] Alu | Achieving Transparency with Plasmonic and Metamaterial Coatings[END_REF] or negative refractive index materials [START_REF] Milton | On the cloaking effects associated with anomalous localized resonance[END_REF]. Interestingly, the invisibility is preserved in the case of an intense near field [START_REF] Zolla | Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect[END_REF], when the ray optics picture breaks down. The mathematics behind the scene have been known from researchers working in the area of inverse conductivity problems [START_REF] Greenleaf | On nonuniqueness for Calderons inverse problem[END_REF]. The first experimental realization of an invisibility cloak, chiefly achieved in the microwave regime [START_REF] Schurig | Metamaterial electromagnetic cloak at microwave frequencies[END_REF], suggests that cloaking will be limited to a very narrow range of frequencies. However, it will not be perfect since the cloak is necessarily dissipative and dispersive, and some of its tensor components are singular on the inner boundary. The latter drawback can be overcome by considering a generalized transform [START_REF] Zhang | Obtaining a nonsingular two-dimensional cloak of complex shape from a perfect three-dimensional cloak[END_REF][START_REF] Leonhardt | Broadband invisibility by non-euclidean cloaking[END_REF] in an upper dimensional space and then projecting the resulting metric on the physical space, and this leads to non-singular tensors of permittivity and permeability. Alternatively, one can design an approximate structured cloak via homogenization [START_REF] Cai | Optical Cloaking with metamaterials[END_REF][START_REF] Farhat | Achieving invisibility over a finite range of frequencies[END_REF][START_REF] Qiu | Spherical cloaking with homogeneous isotropic multilayered structures[END_REF], although the rapid growth of the field, fueled by a keen interest of the optics community, promises a large panel of new technological applications.

A non trivial question to ask is whether one can design cloaks of non-spherical shapes. A parameterization of the cloak's boundaries was proposed by three of us to design cylindrical cloaks of an arbitrary cross-section [START_REF] Nicolet | Electromagnetic analysis of cylindrical cloaks of an arbitrary cross section[END_REF]. Its extension to the general three-dimensional case now requires to parameterize the inner and outer boundaries of the cloak as some surfaces with varying radii

ρ(θ , φ ) = a 0,0 + ∑ (m,n)∈N 2 \{(0,0)} {a m,n cos(mθ + nφ ) + b m,n sin(mθ + nφ )} , (1)
which is somewhat out of reach running the COMSOL multiphysics package on a 256 Gb RAM computer: Compared to a spherical cloak, the tetrahedral mesh needs be further refined due to the complexity of surfaces involved. However, to demonstrate the versatility of our Fourierbased approach, it is enough to analyse arbitrary cloaks built with surfaces of revolution i.e. generated by rotating a curve about an axis.

In the present paper, we discuss the design of axially invariant three-dimensional cloaks with an arbitrary cross-section described by two functions R 1 (φ ) and R 2 (φ ) giving an angle dependent distance from the origin. These functions correspond respectively to the interior and exterior boundary of the cloak. We shall only assume that these two boundaries can be represented by a differentiable function. Their finite Fourier expansions are thought in the form R j (φ ) = a j 0,0 + ∑ p n=1 a j 0,n cos(nφ ), j = 1, 2, where p can be a small integer, and a j 0,n = 0 for n > p for computational easiness. Note that our approach encompasses the case of toroidal cloaks [START_REF] You | Invisibility cloaks for toroids[END_REF] but cloaks with irregular boundaries such as cubes, fall beyond the scope of this study. Nevertheless, contrarily to previous three dimensional numerical studies [START_REF] You | Invisibility cloaks for toroids[END_REF][START_REF] Qiu | Spherical cloaking with homogeneous isotropic multilayered structures[END_REF], our scheme can be used in the design of cloaks with non-convex boundaries.

To illustrate our methodology, we compute the electromagnetic field diffracted by a cloak with rotational symmetry about the z-axis. We perform full-wave finite element simulations in the commercial package COMSOL, when the cloak is illuminated by an approximate plane wave (generated by a constant electric field on the upper surface of the computational domain).

Change of coordinates and pullbacks from optical to physical space

Material properties of the heterogeneous anisotropic cloak

The geometric transformation which maps the field within the full domain ρ ≤ R 2 (φ ) onto the annular domain R 1 (φ ) ≤ ρ ′ ≤ R 2 (φ ) can be expressed as:

       ρ ′ (ρ, φ ) = R 1 (φ ) + ρ R 2 (φ ) -R 1 (φ ) R 2 (φ ) , θ ′ = θ , 0 < θ ≤ 2π , φ ′ = φ , -π/2 < φ ≤ π/2 (2)
where 0 ≤ ρ ≤ R 2 (φ ). Note that the transformation maps the field for ρ > R 2 (φ ) onto itself through the identity transformation. This change of co-ordinates is characterized by the transformation of the differentials through the Jacobian:

J(ρ ′ , φ ′ ) = ∂ (ρ(ρ ′ , φ ′ ), θ , φ ) ∂ (ρ ′ , θ ′ , φ ′ ) . ( 3
)
This change of coordinates amounts to replacing a homogeneous isotropic medium with scalar permittivity and permeability ε and µ, by a metamaterial described by anisotropic heterogeneous matrices of permittivity and permeability given by [START_REF] Zolla | Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect[END_REF][START_REF] Nicolet | Transformation methods in computational electromagnetics[END_REF] ε ′ = εT -1 , and

µ ′ = µT -1 , (4)
where T = J T J/ det(J) is a representation of the metric tensor in the so called stretched radial coordinates. Note that there is no change in the impedance of the media since the permittivity and permeability undergo the same transformation.

After some elementary algebra, we find that

T -1 =      c 2 13 +ρ(ρ ′ ,φ ) 2 c 11 ρ ′2 0 - c 13 ρ ′ 0 c 11 0 - c 13 ρ ′ 0 c 11      , (5)
where

c 11 (φ ′ ) = R 2 (φ ′ ) R 2 (φ ′ ) -R 1 (φ ′ ) , (6)
and

c 13 (φ ′ ) = R 2 (φ ′ ) ρ ′ -R 2 (φ ′ ) (R 2 (φ ′ ) -R 1 (φ ′ )) 2 dR 1 (φ ′ ) dφ ′ + R 1 (φ ′ ) R 1 (φ ′ ) -ρ ′ (R 2 (φ ′ ) -R 1 (φ ′ )) 2 dR 2 (φ ′ ) dφ ′ , (7)
for R 1 (φ ′ ) ≤ ρ ′ ≤ R 2 (φ ′ ). Elsewhere, T -1 reduces to the identity matrix (c 11 = 1, c 13 = 0 and ρ = ρ ′ for ρ ′ > R 2 (φ ′ )).

Singularity analysis of the transformation matrix

To exemplify the symmetric nature of the coefficients of T -1 , and due to the role played by its first entry in the following singularity analysis, we show the variation of (T -1 ) 11 in Fig. 1. We note that this coefficient varies between zero and three. The fine mesh on the inner boundary of the cloak is also apparent from Fig. 1, as are the non-convex and symmetric features of the cloak.

In the case of a spherical cloak, c 13 vanishes and T -1 reduces to Diag( ρ 2 c 11 ρ ′2 , c 11 , c 11 ). We note that the first eigenvalue vanishes on the inner boundary and the other two remain constant, which is consistent with the singularity analysis led in [START_REF] Kohn | Cloaking via change of variables in electric impedance tomography[END_REF]. This is unlike the circular cylindrical case whereby one eigenvalue goes to zero while the other one goes to infinity on the inner boundary [START_REF] Zolla | Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect[END_REF].

Fig. 1. 3D plot of (T -1 ) 11 , as given by Eq. 5, Eq. 6 and Eq. 7 within the cloak with boundaries given by Eq. 10. The symmetries are noted.

However, in our case, the cloak is of an arbitrary shape, and it is therefore illuminating to look at the behaviour of its permittivity and permeability tensors's eigenvalues. The eigenvalues of ( 5) are found to be

λ j = c 2 13 +ρ 2 +c 2 11 ρ ′2 2c 11 ρ ′2 + (-1) j 2 c 2 13 +ρ 2 +c 2 11 ρ ′2 c 11 ρ ′2 2 -4 ρ 2
ρ ′2 , j = 1, 2 , and λ 3 = c 11 .

We can therefore see that λ j , j = 1, 2 are spatially varying functions of ρ ′ (ρ, φ ), such that λ 1 = 0 when ρ = 0 i.e. at the inner boundary, while λ 2 > 0. Importantly, λ 3 is a strictly positive constant for a given angle φ ′ (i.e. a function independent upon ρ). Last, we checked that when there is no longer a symmetry of revolution about one axis, all three eigenvalues are also spatially varying with θ ′ , but λ 3 remains independent upon ρ. This reflects the fact that the geodesics for light within the arbitrary cloak follow more and more complex trajectories when we perturb the geometry away from the spherical cloak.

Finite elements computations

We would now like to further investigate the electromagnetic response of the cloak to an incident plane wave from above, see Fig. 2. For this, we choose the electric field E as the unknown:

∇ × µ ′ -1 ∇ × E -k 2 ε ′ E = 0 (9)
where k = ω √ µ 0 ε 0 = ω/c is the wavenumber, c being the speed of light in vacuum, and ε ′ and µ ′ are defined by Eqs. [START_REF] Milton | On the cloaking effects associated with anomalous localized resonance[END_REF]. Also, E = E i + E d , where E i is the incident field and E d is the diffracted field which satisfies the usual outgoing wave conditions (to ensure existence and uniqueness of the solution). The weak formulation associated with Eq. 9 is discretised using second order finite edge elements (or Whitney forms) which behave nicely under geometric transforms (pull-back properties) [START_REF] Nicolet | Transformation methods in computational electromagnetics[END_REF]. For the sake of illustration, cf. Fig. 2, let us consider a cloak with inner and outer boundaries expressed as We report the computations for the magnitude of the total electric field in Fig. 2 (3D plot), Fig. 3 (2D plot in the xz-plane) and Fig. 4 (2D plot in the yz-plane) for a plane wave incident from above at wavenumber k = 2π/0.3 (units are in inverse of a length, say µm -1 for nearly visible light (UV)).

R 1 (φ ) = 0.2 + 0.02 cos(4φ ) , R 2 (φ ) = 0.4 + 0.02 cos(8φ ) . (10)
Around 4.10 5 elements were used in this computation, which corresponds to about 2.7 10 6 degrees of freedom. While the convergence of the numerical scheme has been checked by considering different types of meshes, the large size of the system means we were not able to further refine the mesh for the computational resources at hand.

Conclusion

In conclusion, we have proposed a design of an arbitrarily shaped cloak using a Fourier approach. We only assumed that the cloak displays a symmetry of revolution about one axis in order to reduce the computational complexity of the problem (extension to completely arbitrarily shaped cloaks is a straightforward matter). Cloaking has been confirmed numerically for an incident plane wave in resonance with the concealed region and an analysis of the cloak's singularity has been carried out.

A. Diatta and S. Guenneau acknowledge funding from EPSRC grant EP/F027125/1.

Fig. 2 .

 2 Fig. 2. 3D plot of the magnitude E 2 1 + E 2 2 + E 2 3 of the total electric field for a plane wave of wavenumber k = 2π/0.3 incident from above on an non-convex invisibility cloak.

Fig. 3 .

 3 Fig. 3. 2D plot of E 2 1 + E 2 2 + E 2 3 generated by a slice of Fig. 2 in the xz-plane for y = 0.

Fig. 4 .

 4 Fig. 4. 2D plot of E 2 1 + E 2 2 + E 2 3 generated by a slice of Fig. 2 in the yz-plane for x = 0.


A Aydi

A Simon

D Michau

R Von Der Mühll

N Abdelmoula

H Khemakhem

Dielectric study of ferroelectric or relaxor Behavior in Ba 1-x Na x (Ti 0,8 Sn 0,2 ) 1-x NbO 3 ceramics

Keywords: Ceramic, Permittivity, Ferroelectric, Relaxor, Piezoelectricity

The ferroelectric and piezoelectric properties of Ba 1-x Na x (Ti 0,8 Sn 0,2 ) 1-x-Nb x O 3 (BTSnNx0.2), the new lead-free ceramic, are investigated. At room temperature, X-ray diffraction (XRD) study suggests that the compositions have a single phase with cubic symmetry. The dielectric study reveals that the material has a ferroelectric or a relaxor behavior. The polarization state is checked by hysteresis and piezoelectric measurements.

Introduction

A large number of perovskite materials exhibit a ferroelectric or relaxor behavior [1,2,3,4]. These materials can be very useful for different solid-state devices [5,6,7]. The best known relaxor is Pb(Mg 1/3 Nb 2/3 )O 3 , a lead-based ceramic with a perovskite structure. Its derived compounds are also the best leadbased relaxors. In accordance with environmental concerns, we have to avoid PbO-related problems; thus, researchers have moved toward lead-free relaxors [8,9,10,11].

The relaxor effect was then related to the substitution within octahedral site in lead-free perovskites which are derived from BaTiO 3 [12,13,14]. This behavior can be related to either a cationic or anionic disorder in the A or B sites of ABO 3 structure [15,16].

The aim of the present work is to study the dielectric response temperature and frequency of BTSnNx0.2 ceramics to experimentally point out the ferroelectric behavior of these materials using dielectric, hysteresis and piezoelectric measurements.

Experimental process

The polycrystalline ceramic samples of BTSnNx0.2 were prepared via solid-state reactions at 1030°C for 15 hours under oxygen atmosphere, using BaCO 3 , Na 2 CO 3 , TiO 2 , SnO 2 and Nb 2 O 5 powders in stoichiometric amounts.

Weighed ingredients were pressed under 100 MPa into disks of 8 mm in diameter and about 1 mm in thickness. Calcination was done at 1030 °C for 15 hours, followed by a two-hour sintering period at 1250°C for 0.60 ≤ x ≤ 0,90. XRD patterns were recorded on a Philips diffractometer using the CuKα radiation, the angular range 10 ≤ 2θ≤ 110° (with each step = 0.02°) and 10 seconds counting.

Dielectric measurements were performed on ceramic disks after deposition of gold electrodes on the circular faces by cathodic sputtering. The real and imaginary parts of relative permittivity, which are respectively ' r ε and " r ε , were determined under helium atmosphere versus both temperature (80 -600K) and frequency (10 2 -2.10 6 Hz) using a Wayne -Kerr 6425 frequency analyzer.

A conventional Sawyer-Tower circuit was used to measure the polarisation hysteresis (P-E) loop at 50 Hz. The electromechanical coupling factor k p and the piezoelectric coefficient d 31 were determined by the resonance method according to IEEE standard using an impedance analyser (HP 4194 A).

A Scanning Electronic Microscope (SEM), JEOL 6360A, was used to characterize the microstructure.

Results and discussion

The SEM micrograph of BTSnNx0.2 ceramic is shown on Fig. 1. All samples were sintered at 1250°C for 2 hours. We obtain a well dense ceramic with an average grain size of about 2µ m. XRD measurements of BTSnNx0.2 (x = 0.80) powder are reported in Fig. 2. The results reveal a single perovskite phase with cubic structure. The cell parameters were calculated by least-square refinement. Good agreement between the observed and the calculated interplaner spacing suggests that this composition has a cubic structure at room temperature. Final refinement with the cubic space group P m 3 m gives the estimated value of unit cell parameter a = 4,010 Å. The powder diffraction data are given in Table 1. For x ≥ 0,85, The samples reveal ferroelectric behavior compositions. Three anomalies are observed when the temperature increases gradually for BTSnNx0.2 (x = 0.90). These anomalies are related to the trigonalorthorhombic, orthorhombic-tetragonal and tetragonalcubic phase transitions at T C . The dielectric measurement shows that ' r ε rmax = 6000 for ceramic solution BTSnNx0.2 (x = 0.90) but for the BSNN0.9 ceramic, ' r ε rmax = 1947 [4]. Therefore, the addition of titanium (Ti) shows an increase of the maximum value of permittivity.

The variation of the temperature of dielectric anomalies as a function of composition x is shown in Fig. 5. It can be observed that the value of T m or T c increases when x increases. The relaxor character becomes more important when the composition parameter x decreases. This behavior is then similar to the others reported in the literature [4]. In order to demonstrate the ferroelectric character of the material, we attempt to measure the hysteresis or piezoelectric effect of polarized ceramics.

Ambient temperature (T = 300K) which is far below T C = 324; 388; 392K (for x = 0.825; 0,85; 0,875 respectively), ferroelectric (P-E (polarisation-field)) loops are observed in Fig7a; 7b and 7c. Spontaneous polarizations are Ps = 1, 92; 1.54 and 1,37 µ C/ cm 2 and coercive fields Ec = 3,31; 4.53 and 5,1 kV/cm for compositions x = 0.825; 0,85 and 0,875, respectively. These figures show that the material has good ferroelectric nature. The shape of P-E loops varies with x. It can be seen that for low applied fields, the P-E loops is far from saturation. Sharp and well saturation squared P-E loops could be observed for increasing E showing full domain switching. The piezoelectric measurements were performed at room temperature on the ceramic with composition BTSnNx0.2 (x = 0.85). The admittance and susceptance of this ceramic disk were measured versus frequency around the main transverse resonance mode. The variation of the admittance (G) and the susceptance (B) versis frequency is shown on Fig. 8

Conclusions

BTSnNx0.2 ceramics were prepared by a solid-state technique. The XRD study shows that the compound exists in a single phase at room temperature. Dielectric measurements reveal two kinds of behavior depending on the type of composition: classical ferroelectric or relaxor.

According to the dielectric properties, the addition of Titanium (Ti) shows the increase of the maximum value of permittivity.

The piezoelectric coefficient d 31= 69,5pC/N and the Spontaneous polarizations Ps = 1, 92 µ C/ cm 2 make these lead-free ceramics suitable for diverse applications (dielectrics for capacitors, actuators...).

  -final diameter / initial diameter) and the compactness (experimental density / theoretical density) were systematically determined. Their average values were in the range of 11-16 % and 91-95%, respectively depending on the composition.

Fig. 1 :

 1 Fig.1 : SEM picture of the surface of BTSnNx0.2 ceramic ( x = 0.85).

Fig. 2 .

 2 Fig. 2. XRD pattern of a BTSnNx0.2 (x = 0.8) powder.

Fig. 3a ;

 3a Fig. 3a; 3b; 3c; 3d; 3e and 3f show temperature dependence of the real permittivity ' r ε in the frequency range 0.1 -200 kHz for BTSnNx0.2 ( x = 0,6; 0,7; 0,8; 0,825; 0,85 and 0,90). For x < 0,85, the samples are of relaxor behavior compositions. There is only one broad peak in temperature evolution of ' r ε at T m . A frequency dispersion occurs while T < T m with a shift of T m when the frequency varies. The temperature T m of maximum ' r ε ( ' r ε rmax ) shifted to higher values on frequency increase. T m is near 300 K for x = 0,8.For x ≥ 0,85, The samples reveal ferroelectric behavior compositions. Three anomalies are observed when the temperature increases gradually for BTSnNx0.2 (x = 0.90). These anomalies are related to the trigonalorthorhombic, orthorhombic-tetragonal and tetragonalcubic phase transitions at T C .

Fig. 4a ;

 4a Fig. 4a; 4b; 4c and 4d show the thermal evolution of the imaginary part of the permittivity for some ceramics. The dielectric losses are about 2 %.

Fig 4 .

 4 Fig 4. Thermal variation of permittivity ε" r for BTSnNx0.2 (x = 0,6; 0,8; 0,85; and 0,9).

Fig. 5 .

 5 Fig. 5. Variation of T C or T m vs composition forBTSnNx0.2 ceramics.

Fig.6a and

  Fig.6a and 6b depict the temperature dependence of ' / 1 r ε at 1 kHz for the compositions x = 0,8 and x = 0,85, respectively. For x = 0.85, T c = T 0 , where T 0 and C is the

Fig 6 .

 6 Fig 6. Thermal variation of 1/ε' r (1kHz) for BTSnNx0.2 (x = 0.8 and 0.85).

Fig. 7 .

 7 Fig. 7. P-E hysteresis loops of BTSnNx0.2 (x =0,825; 0,85 and 0,875)

  and Fig 9.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Variation of the susceptance (B) as function of frequency at T = 300 K for BTSnNx0.2 ceramic with x = 0.85.

Fig. 10 .

 10 Fig. 10. Piezoelectric admittance circle at the main radial resonance for BTSnNx0.2 (x = 0.85 ), at 300 K.

Table 1 .

 1 X-ray powder diffraction data for BTSnNx0.2 (x = 0.8).

	Bragg	h k l
	22.46	0 0 1
	31.90	1 0 1
	40.02	1 1 1
	45.72	0 0 2
	51.52	1 0 2
	56.82	1 1 2
	66.68	2 0 2
	71.30	0 0 3
	76.02	1 0 3
	80.22	1 1 3
	84.52	2 2 2
	93.22	2 1 3
	98.82	2 0 3
	101.92	0 0 4
	106.42	1 0 4
	111.02	1 1 4