buildings against outdoor noise. This trend of a widespread use of porous materials highlights the need of tools allowing for the prediction of their response to several kinds of excitations. From a scienti®c point of view, a better understanding of the vibro-acoustic behavior of these materials is of great interest. Practically, industrials and design departments use porous media in multi-layered materials in their wish to develop high acoustical performance products. Therefore, there is a need for evaluating their eciency.

Dierent models have been proposed to depict a porous material, considering the ¯uid phase displacement. If the solid phase of a porous medium is assumed motionless, the wave propagates only through the ¯uid phase. This approach is valid for an acoustical excitation above a decoupling frequency [START_REF] Zwikker | Sound absorbing materials[END_REF] beyond which the two phases of a porous material are then considered to be uncoupled. The assumption of a motionless frame has allowed quite accurate descriptions of the response of porous materials such as soils. Sabatier et al. [START_REF] Sabatier | The interaction of airborne sound with the porous ground: The theoretical for mulation[END_REF] found that semi-in®nite sandy soils excited by a plane wave could be correctly modeled using an equivalent ¯uid model. In that particular case, frame motion appeared to have no in¯uence on the computed surface impedance. On the other hand, this hypothesis is not valid anymore when considering some hard backed ®nite foams for example. Some results by Allard [START_REF] Allard | Propagation of sound in porous media[END_REF] for a plane wave excitation show that an equivalent ¯uid model is unable to depict phenomena such as quarter-wave resonance. Tooms [START_REF] Tooms | Propagation form a point source above a porous and elastic foam layer[END_REF] also showed the greater accuracy of elastic-porous models for the prediction of response of low-density foams to a point source excitation. Hence, it seems that even for the case of acoustic excitations, taking into account the elastic frame motion is necessary. Besides, equivalent ¯uid models are obviously inappropriate in the case a mechanical force excitation, since the solid phase is directly stimulated.

Problems involving plane wave excitations are easier to solve than those considering a volume source or a mechanical excitation for example. In that way, this kind of excitation has been mainly considered. Studies on two-dimensional con®gurations have been made in the case of impinging plane-waves using a complete description of porous media. Sabatier et al. [START_REF] Sabatier | The interaction of airborne sound with the porous ground: The theoretical for mulation[END_REF] proposed a description of a porous material based on a modi®ed form of the Biot±Stoll [START_REF] Stoll | Theoretical aspects of sound transmission in sediments[END_REF] equations. Considering a ®nite thickness porous material backed by a rigid wall, they applied the boundary conditions between the air and the porous medium in order to get the amplitude coecients of the propagating waves in the porous medium. Allard [START_REF] Allard | Propagation of sound in porous media[END_REF] used a formalism allowing replacement of a material layer by a transfer matrix linking the stresses and velocities at each side of the given layer, so that a surface impedance could be computed. The utility of this latter method lies in the characterization of a porous layer by one matrix. Hence, velocities and stresses can be linked at each side of a multi-layered material by merging layer matrices into a global matrix. Other works based on a ®nite element model have been achieved for multi-layers of ®nite lateral size. Bolton and Kang [START_REF] Bolton | Finite element modeling of isotropic elastic porous materials coupled with acoustical ®nite elements[END_REF] were interested in solving the coupled elasto-poroacoustic two-dimensional problem of a porous material excited by an impinging plane wave under normal incidence. Also, Panneton and Atalla [START_REF] Panneton | An ecient ®nite element scheme for solving the three dimensional por oelasticity problem in acoustics[END_REF] used a formulation involving the solid and ¯uid phase displacement (known as the {u,U} formulation) to describe the porous material in the case of a coupled elasto-poro-acoustic three dimensional problem. They applied their theory to the study of the transmission loss of a double plate with a porous material in between. Recently, these authors [START_REF] Atalla | Validation of examples for the mixed pressure displacement formulation for poroelastic media[END_REF] have developed a {u,P} formulation for the porous material using the solid phase displacement and the air pressure in the pores.

Apart from the plane wave excitation, the acoustic ®eld due to a point source over an impedance plane has been investigated. In the case of a locally reacting material [START_REF] Nobile | Acoustic propagation over an impedance plane[END_REF][START_REF] Habault | Ground eect analysis: surface wave and layer potential representations[END_REF] or an equivalent ¯uid [START_REF] Habault | Sound propagation over ground: analytical approximations and experi mental results[END_REF], analytical solutions or approximations by series or integrals have been obtained for three dimensional cases. The problem induced by a volume source radiating over a porous layer has also been solved when considering the elasticity of the frame in the porous medium [START_REF] Attenborough | Solid particle motion induced by a point source above a poroelastic half space[END_REF][START_REF] Tooms | Sound propagation in a refracting ¯uid above a layered ¯uid saturated porous elastic material[END_REF]. The axisymmetric displacement potentials are expressed as inverse Hankel transforms. Each of these Hankel transforms stand for plane-wave displacement potentials in the porous medium and are determined by boundary conditions at the interface. Attenborough [START_REF] Attenborough | Solid particle motion induced by a point source above a poroelastic half space[END_REF] proposed a heuristic approach to compute the solution expressed in an integral form. This approach gives reasonable accuracy though not completely rigorous. Fast Fourier techniques have also been developed to compute these axisymmetric displacement potentials. These methods, known as Fast Fourier Program methods (FFP programs) initially introduced by DiNapoli and Davenport [START_REF] Dinapoli | Theoretical and numerical Green's function solution in a plane layered medium[END_REF], have been successfully applied to problems of sound propagation in a system composed of a horizontally strati®ed ¯uid overlying a horizontally strati®ed elastic and porous solid [START_REF] Tooms | Sound propagation in a refracting ¯uid above a layered ¯uid saturated porous elastic material[END_REF].

Two dimensional models have also been developed for line source excitations. For the propagation above a ¯at interface, analytic solutions can be found by considering a porous material depicted as locally reacting [START_REF] Wilde | Ground eects in environmental sound propagation[END_REF]. This kind of problem has also been studied for geophysical applications in water saturated strati®ed soils [START_REF] Boutin | Green functions and associated sources in in®nite and strati®ed poroelastic media[END_REF] using a {u,P} formulation for the description of the porous material.

In this paper, the two dimensional problem of an acoustical line source radiating over a planar multi-layers in®nite in two of its dimensions will be investigated. The line source above a ¯at interface model is often relevant in building acoustics (e.g., ¯uid conveying ducts radiating noise near a multi-layered wall) and environmental acoustics (trac noise in presence of a ground). The method used is the discrete wave number method, referred as Bouchon's method. Initially, this method has been designed in seismic sciences for the computing of the response of an horizontal set of layers to seismic excitations in the case of complicated shaped grounds [START_REF] Boutin | Dynamique des milieux poreux sature s de formables. fonctions de Green Perme ameÁ tre dynamique[END_REF]. In the present work, the method is adapted to an acoustic problem in the case of a planar interface. This approach looks like FFP in its formulation, but is rather dierent in its physical content. Moreover, a Fourier transform is used instead of a Hankel transform. Finally, Bouchon's method can be extended to more complicated pro®le for the interface between air and the poroelastic material.

Theory

The cases studied in this paper consider an in®nitely extended planar multi-layered material made up of acoustic, elastic, and poroelastic media. The equations governing the wave propagation in the media involved are presented in the following, assuming an exp(jÁ3Át) time dependence. These equations, valid for a general three dimensional case, are used in the next section for a two-dimensional con®guration.

The propagation equations

In the ¯uid

The propagation of a wave in a ¯uid is governed by Helmholtz equation

r 2 p M k 2 f Áp M s M À M 0 Y I
where p(M) denotes the pressure at point M(x,z), k f 3aC f is the wave number in the ¯uid, C f the sound speed in the ¯uid, s(M À M 0 ) a line source located in M 0 (x 0 ,z 0 ).

In the porous material

Porous materials are described by Biot±Allard equations [START_REF] Allard | Propagation of sound in porous media[END_REF], and are rewritten here using the frame displacement Ũs and pressure p inside the pores as variables. When there is no source in the porous material, the equations are as follows:

For the solid phase

À 3 2 Á &11 À &2 12 &22 Á Ũ ÈÁ &12 &22 Á3 2 Árp ! P Q À N r r Á Ũ Nr 2 Ũ QÁ È &22 Á3 2 rrÁrpY P For the ¯uid phase ÀÈÁrp RÁÈ &12 Á3 2 rrÁrp Q À RÁ &12 &22 rrÁ Ũ Y Q
where N, P, Q, R denote Biot coecients which are characteristics of the porous material. These coecients depend on the bulk modulus of the ¯uid K f , the bulk modulus of the elastic solid K s , the bulk modulus of the frame K b , and shear modulus " (Lame coecient). Thermal dissipative eects are taken into account through a correcting function G(B 2 3) introduced in K b , where B 2 is the Prandtl number. &11 and &22 are the eective densities representing the densities of each phase modi®ed so as to take into account dissipative viscous and inertial eects. &12 is an eective density accounting for the coupling between both phases. The correcting factor G(3) is introduced to include the viscous eects, G being the same function as previously.

The expressions of the eective densities are:

&11 1 À È & & À jÁ'ÁÈ 2 Á G 3 3 Y R &22 ÈÁ& f & À jÁ'ÁÈ 2 Á G 3 3 Y S &12 À& jÁ'ÁÈ 2 Á G 3 3 Y T with & ÀÈÁ& f Á I À 1 U
representing the inertial coupled eects. & f and & S are the ¯uid and in the above equations, the solid density, ' is the ¯ow resistivity È is the porosity, I is the tortuosity of the porous material.

The system formed by Eqs. ( 2) and ( 3) is solved by expressing the displacement ®elds as a sum of a compressional and a rotational wave. Using this expansion, two kinds of compressional waves (P 1 , P 2 ) and one kind of shear wave are shown to propagate simultaneously in the porous material. For the restricting case of a motionless frame, the porous material is reduced to its ¯uid phase and is then governed by an Helmholtz-like equation obtained from Eq. (3). In that case, only one compressional wave P 2 can propagate.

In the elastic material

Using Ũ as the displacement ®eld of the elastic material, the displacement equation of motion is given by

l "rrÁ Ũ "Ár 2 Ũ &Á3 2 Á Ũ 0Y V
where ! and " are the Lame coecients, & is the density of the material. In this case, the expansion in terms of propagation modes highlight the possibility for one kind of compressional wave and one kind of shear wave to propagate.

General response at a receiver

In the following, the two dimensional problem of a multi-layered material excited by a harmonic line source located in the air is considered. As stated in the introduction, the approach is based on a one-dimensional spatial Fourier transform. The response ~i 3 MY M 0 in medium i at receiver M due to an harmonic acoustical line source located in M 0 and its one-dimensional Fourier transform ~i k x Y z are linked by the following equation:

~i 3 MY M 0 I I exp ÀiÁk x Á x M À x M 0 À Á À Á Á ~i k x Y z M À z M 0 À Á Á dk x X W
In the following, f denotes the Fourier transform of f along x. For a porous

medium ~kx Y z M Ũs P & ' , for an elastic medium ~kx Y z M Ũs & '
and for a ¯uid ~kx Y z M P .

The quantity exp ÀiÁk 9) can be seen as the superposition of all plane wave components with wave-number =3/C i (C i velocity of the considered kind of wave in the medium) and horizontal wave number k x .

x Á x M À x M 0 Á ~kx Y z M À z M 0 in Eq. (
One needs now to evaluate ~kx Y z M À z M 0 in each medium i.

Response in air for an acoustical line source excitation

In free ®eld conditions the Green's function for the pressure due to a line source in air associated with Helmholtz equation is [START_REF] Boutin | Dynamique des milieux poreux sature s de formables. fonctions de Green Perme ameÁ tre dynamique[END_REF]

H 2 0 f ÁÀM 0 j 4Ái I I 1 4ÁiÁ%Ák z exp ÀiÁk x Á x M À x M 0 À Á À Á Áexp ÀiÁk z Á M Àz M 0 j À Á Á dk x Y IH
where H 2 0 is the zero order Hankel's function of the second kind, k z denotes the vertical wave number in air with

k 2 z f 2 Àk 2 x X Both propagating (k x < 3/C f ) and evanescent waves (k x > 3/C f ) are considered.
In presence of a material, the incident plane wave contribution is re¯ected and refracted. The response in air can be deduced from Eqs. ( 9) and ( 10) and written as follows.

`

k 1 Y M 1 4ÁiÁ%Ák z Á exp ÀiÁk z Á M Àz M 0 j À Á R k x Áexp ÀiÁk z Á M z M 0 j À Á Â Ã Y II
where R(k x ) is the plane wave re¯ection coecient on the air-multilayered material interface which only depends of the horizontal wave number k x . R(k x ) has the usual sense for propagating waves, and this notion is extended to evanescent waves [START_REF] Brouard | Measurements and prediction of the re¯ection coecient of porous layers at obli que incidence and for inhomogeneous waves[END_REF].

Response in a multi-layered material for a line source

To simplify, the case of a single material of thickness h and backed by a rigid wall is considered (Fig. 1). The principle of the method presented in this section can be extended to a large number of materials. The calculation will be detailed for the case of one porous material. The main aspects of the method can be reused for an elastic material or a ¯uid.

In a porous material, the ®elds can be expanded in propagation modes (compressional and shear waves), to give

~porous 3 k x Y z M À Á Ũ r 0 4 1 r 0 5 1 r 0 4 2 r 0 5 2 r 24 25 P 1 À È r Ũ Èr Ũf V b b b b X W b b a b b Y IP
with 1 À K aK and À È aK Èa K X One shall note that the ¯uid phase displacement Ũf can be expressed as Ũf " 1 Ár 0 1 " 2 Ár 0 2 " 3 Ár 2Y where " i (i=1,3) denote complex amplitude ratio of the ¯uid displacement over the solid displacement associated with each kind of wave. Expressions for " i are given in Ref. [START_REF] Boutin | Dynamique des milieux poreux sature s de formables. fonctions de Green Perme ameÁ tre dynamique[END_REF]. Each scalar potential ( 0 4 i for ascending wave and 0 5 i for descending waves) or vectorial potential ( 24 for ascending waves and 25 for descending waves) involved in the solid displacement expression may be expressed using the transmission coecients in the porous medium.

For P waves

i 1Y 2 X 0 i A i k x 4i%k i z exp Àik i z M À Áz M 0 j À Á X IQ
For an S wave 2

A 3 k x 4i%k z exp Àik z M À Áz M 0 j À Á Y IR
where =1 for an ascending wave and À1 for a descending wave. A i are the transmission coecients in the porous medium for each kind of wave (these coecients being dierent whether the wave is ascending or descending). k i z and k z are the vertical wave numbers in the porous medium depending on the kind of wave (the wave numbers are opposite for an ascending and a descending wave). The expressions of these wave numbers are given in Ref. [START_REF] Boutin | Dynamique des milieux poreux sature s de formables. fonctions de Green Perme ameÁ tre dynamique[END_REF].

For the case of an elastic material, `i k x Y z M f U S g is expanded in propagation modes involving compressional and shear waves potentials, in the same way as previously. The expressions of these potentials are similar to the ones given in Eqs. ( 13) and ( 14), but the vertical wave numbers k z and k z are now those in the elastic medium.

In the case of a ¯uid, only compressional waves can propagate. The potential 0 associated with pressure P in the ¯uid has the same expression as in Eq. ( 13), considering the vertical wave number k z in the ¯uid. Once A i have been determined, ~33 M can be reconstructed using Eq. ( 9).

Calculation of the transmission and re¯ection coecients

As said previously, the waves propagating in the dierent media are expanded in terms of propagation modes. The solid phase displacement and pressure inside the pores together with the acoustical pressure are expressed in terms of potentials whose expression involve re¯ection and transmission coecients at each interface of the multi-layered material. These coecients are determined by applying boundary conditions between materials. These conditions are given for various con®gurations by Deresiewieckz and Sfalak [START_REF] Deresiewieckz | On uniqueness in dynamic poroelasticity[END_REF]. For an interface between two porous media, these are: (a) the solid phase normal displacement continuity, (b) the solid phase tangential displacement continuity, (c) the continuity of the normal relative displacement expressed as È ŨS À Ũf , (d) the saturating ¯uid pressure continuity, (e) the solid phase normal strain continuity, (f) the solid phase tangential strain continuity. For a air-porous material interface, boundary conditions above are replaced by (a H ) total normal displacement continuity, (d), (e), (f H ) zero solid phase tangential constraint. For a porous-elastic material interface, these conditions are (a), (b), (c HH ) zero relative displacement in the porous material at the interface, (e HH ) normal total strain continuity, (f HH ) tangential total strain continuity. For boundary conditions in other con®gurations, the reader may refer to Ref. [START_REF] Deresiewieckz | On uniqueness in dynamic poroelasticity[END_REF].

Numerical implementation

The calculation of exp ÀiÁk

x Á x M À x M 0 Á ~kx Y z M À z M 0
for the case of an in®nitely thick porous material is now presented. The same process is used for a ®nite thickness or a multi-layered material.

Apart from the incident wave, another compressional wave is considered in air accounting for the re¯ected ®eld. Two compressional and one shear waves, all descending, propagate in the porous medium. Fig. 2 summarizes the dierent waves involved. The calculation of exp ÀiÁk x Á x M À x M 0 Á ~kx Y z M À z M 0 can be seen as the determination of the total response in air of the material (incident + re¯ected ®eld) to an impinging plane wave excitation. For the sake of simplicity, a unit amplitude for the incident plane wave potential is assumed for this ®rst step of the calculation. The potentials associated with each kind of wave have the following expression:

For the incident wave (in air)

0 exp ÀiÁk x Áx Áexp ÀiÁk ir z Áz À Á Y IS
For the re¯ected wave (in air)

0 A 0 Áexp ÀiÁk x Áx Áexp iÁk ir z Áz À Á Y IT
where A 0 is the re¯ection coecient on the interface.

For the transmitted waves (in the porous material) P 1 wave where, A 1 , A 2 , A 3 are the transmission coecients associated with each kind of wave, and 1 Y 2 Y are the vertical wave numbers for P 1 , P 2 , S waves, respectively, in the porous material.

0 I A 1 Áexp ÀiÁk x Áx Áexp ÀiÁ I Áz À Á Y IU P 2 wave 0 P A 2 Áexp ÀiÁk x Áx Áexp ÀiÁ P Áz À Á Y IV S wave 2 A 3 Áexp ÀiÁk x Áx Áexp ÀiÁ Áz Y IW
The boundary conditions between the porous material and air (conditions (a H ), (d), (e), (f H )) are now expressed using the above potentials. These conditions yield:

2Á"Á 2 I l a Á ÈÁ " 1 À 1 Á 2 I h i ÁA 1 h 2Á"Á 2 P l a Á ÈÁ " 2 À 1 Á 2 P i ÁA 2 2Á"Ák x Á ÁA 3 & ir Á3 2 Á 1 R Y PH 2Á"Ák x Á I Áe 1 2Á"Ák x Á P ÁA 2 "Á k 2 x À 2 À Á ÁA 3 0Y PI 1a Á À È ÈÁ" 1 Á 2 I 1a Á À È ÈÁ" 2 Á 2 P & ir Á3 2 Á 1 R Y PP 1 À È ÈÁ" 1 Á I 1 À È ÈÁ" 2 Á P 1 À È ÈÁ" 3 Á k x ir 1 À R Y PQ
, are de®ned in Section 2.2.

The system formed by Eqs. ( 20)±( 23) can by written in matrix form

MC Â A n MLY PR

where [A n ] is the vector of the re¯ection and transmission coecients in the porous material. More generally, [A n ] contains the re¯ection and transmission coecients in the multi-layered material.

[MC] expresses the boundary conditions between re¯ected and transmitted waves.

[ML] is the incident wave vector. Each row of [MC] and [ML] corresponds to boundary conditions (20)± [START_REF] Campillo | Synthetic SH seismograms in a laterally varying medium by the discrete wavenumber method[END_REF]. Column i of [MC] corresponds to a kind of wave considered in the multi-layered material and associated to the coecient A i . System (24) is solved using the Gauss pivot method. In that way, A i are obtained and the scalar or vectorial response ~kx Y z M is computed for any value of k x and z, by multiplying coecients A i by the Green's function amplitude coecient (see Eq. ( 11)).

Computation of ~i 3 xY z from ~kx Y z M

To compute the ®elds in the dierent media, an inverse wave number transform must be performed on ~i k x Y z (see Eq. ( 9)). However, this inverse wave number transform cannot be performed analytically and a numerical method has to be applied. The discrete wave number method, referred to as Bouchon's method [START_REF] Bouchon | Calculation of complete seismograms for an explosive source in a layered medium[END_REF], and used in seismology, is proposed. In this method, the ®eld radiated by a unique source is approached by the ®eld created by an in®nite set of periodic sources S 3 . These latter sources are periodic along one axis, parallel to the interface, and distant from each other of distance L (see Fig. 3). By construction, each source of this set is a superposition of plane waves. The angular spacing between the propagation direction of two consecutive plane waves is linked to L by Eq. (28). L tending to in®nity corresponds to a unique source. The response to a periodic set of sources is also periodic with the same period L. The approximation of the ®eld ~i 3 M radiated by a unique source is obtained by adding the contributions of all the periodic sources ~i 3 O j Y M À Á , where O j is the position of the periodic source j. Its expression is given by

~i 3 j periodized M I j I ~i 3 O j Y M À Á X PS
By using Eq. ( 9) for each term of the sum in Eq. ( 25), the response at M(x 1 ,x 3 ) can ®nally be expressed as [START_REF] Bouchon | Calculation of complete seismograms for an explosive source in a layered medium[END_REF] ~i

3 M % ~i 3 j periodized M nI n I exp ÀiÁk n Áx Á ~kn Y z ÁÁkY PT with k n nÁÁk PU and Ák 2% L X PV
The horizontal wave number k x has only discrete values (Eq. ( 27)), and the spectral spacing Ák is linked to L (Eq. ( 28)). Practically, Eq. ( 26) becomes a ®nite sum, with a summation index varying from ÀN max to N max . Both values of N max and L will be important for the convergence of the approximation of Eq. ( 26). First, the periodic length L must be large enough to insure that the discretization of the wave number spectrum leads to a suitable approximation of ~3 k x Y x 3 . Also N max must be large enough so as to take into account at least all signi®cant contributions, i.e. homogenous and evanescent wave contribution. Suitable values for N max will be discussed later.

The expression obtained for the approximation of the ®elds using a set of periodic sources is equivalent to the computation of the response in the material by performing the calculation in one point of a set of receivers by an inverse discrete Fourier transform (IDFT) instead of a calculation of this response by direct integration for that receiver as expressed in Eq. ( 9). In that way, the use of an IFFT algorithm can be considered in order to ``optimize'' Bouchon's method, and the ®elds are computed on a grid of receivers parallel to the air±porous interface. For each receiver M j with coordinates z ®xed and x j jÁÁx (j=ÀN max ,N max ), expression (26) is then approximated by

~3 M j À Á % nN mx n N mx exp ÀiÁk n Áx M j À Á Á ~kn Y z ÁÁkX PW
This expression can either be processed with for each receiver M j or with an IFFT algorithm using the array

~kj Y z À Á ÁÁk & ' j
(j=ÀN max ,N max ). However if an IFFT algorithm is used, the positions of the receivers and their spatial spacing will be determined once Ák and N max are given. We have x 1j =jÂÁx(j=ÀN max ,N max ) with Áx=2%/k max , and x max =%/(Ák) according to the Shannon criterion. Ák and N max are chosen so as to insure the convergence of the approximation (29). The algorithm presented above allows for the computation of any ®elds at any point of a multi-layered material. The characteristics of the material such as surface impedance or the transmission loss for either a plane wave or a line source excitation can also be calculated.

Validation and numerical results

Basically, the developed algorithm requires solving a series of plane wave problems associated with the discrete wave number k j . Firstly, the validation of the calculation of ~3 k x Y x 3 is considered. Then, some validation results regarding the computation of pressure levels due to an acoustical line source radiating over a porous material are presented together with some original results.

Calculation of ~i k x Y z

The ®rst example considers a plate-mineral wool-plate system, in®nite in two of its dimensions and surrounded by air on each side (see Fig. 4). The characteristics of the materials are given in Table 1. The excitation is an impinging plane wave under normal incidence. Fig. 5 compares the transmission loss for this con®guration obtained with the developed algorithm and those obtained by an analytical formula of Lesueur [START_REF] Lesueur | Rayonnement acoustique des structures[END_REF]. The latter approach originally considered a ¯uid between the two plates. In the present example, the characteristics of the ¯uid are those of the porous material depicted by the equivalent ¯uid Biot±Allard [START_REF] Allard | Propagation of sound in porous media[END_REF] model using characteristic Table 2 Characteristics of the dierent layers used in Fig. 5 I & 1 (kg/m 3 ) viscous and thermal length. The same model has been used in the proposed algorithm for the porous material. The results on Fig. 5 show a very good agreement between the proposed model and the analytical solution.

' (N Â s/m 4 ) È N (N/cm 2 ) # Ã (m) Ã H (m) Thickness (cm)
The second example considers a porous material made up of four dierent layers (see characteristics in Table 2) of ®nite thickness bonded onto a rigid wall. The poroelastic structure is excited by an impinging plane wave under normal incidence. The real and imaginary part of the surface impedance of the structure are computed using the developed approach. The results are compared to those given by Panneton and Atalla [START_REF] Panneton | An ecient ®nite element scheme for solving the three dimensional por oelasticity problem in acoustics[END_REF] for a ®nite element model, on Fig. 6. An excellent agreement is obtained between the two models. Moreover, both of them emphasize the importance of taking into account the frame movement. Actually they highlight resonance phenomenon around 500 Hz that would not have been apparent with an equivalent ¯uid model.

In the following, a line source in air radiating over a porous material is considered. The noise levels at any point will be predicted using the ®elds approximation presented in Section 2.3.

Study of the dierent parameters in¯uential on the convergence

As was seen in Section 2.3, Eq. ( 29) provides an approximation of the ®elds whose computation can be optimized by the use of an IFFT algorithm. However, criteria are to be found for the convergence of this approximation.

Criterion on k max

The con®guration represented on Fig. 7 is considered with the in®nite thickness foam depicted in Table 1. The levels are calculated for a receiver at the interface, right under the source at distance d=0.5 m, using Bouchon's method with L=500 m for the considered frequency range. First of all, a suitable value for k max has to be found to insure the convergence of the sum approaching the ®eld. Fig. 8 represents Fig. 6. Case of a ®nite thick multilayered material excited by a plane wave with normal incidence. The surface impedance is predicted by FEM and the present approach with Biot Allard description of the porous media.

the results as a function of frequency with a value for k max varying between 0.95 Â 3/C air and 1.5 Â 3/C air . As one could have expected, it appears that all contributions for k x <3/C air must be taken into account. Actually, these values for k x correspond to propagating plane waves, so that they cannot be neglected. The value k max =1.3 Â 3/C air seems to insure convergence of the computed ®eld. Studies made on other materials give similar values for k max for the considered source-receiver con®guration.

The second parameter that in¯uences convergence of the sum is the position of the receiver M(x,z) with respect to the source M 0 (0,z 0 ). It appears that when the sourcereceiver distance is large, i.e. Àz 0 j ) 1, the required value for k max gets close to 3/ C air . Also the x coordinate of the receiver (the source is assumed to be at the x-axis origin) has little in¯uence on the required k max . Actually, considering the previous remarks, the criterion k max =2 Â 3/C air is so applied for all con®gurations and materials to insure convergence.

Criterion on N max

The second criterion of convergence lies in the number N max of terms, or equivalently the wave number spacing Ák (=k max /N max ) chosen for either a term by term calculation or the IFFT algorithm. When considering the physical approach of Bouchon's method, Ák is linked to the periodization length L. Obviously, the number of sampling values for the approximation is the important parameter. Consequently, values of L for convergence are frequency dependent.

Moreover, required values for N max also depend on the distance between the source and the receiver and the kind of material. This latter dependency is highlighted by a study on two materials described in this section. A foam with characteristics given in Table 1, and a soil with a characteristics given in Table 3, are considered, both having with in®nite thickness. Suitable values of N max vary from 300 to 1100 for the foam and from 300 to 1700 for the soil when the distance between the source and receiver varies from 0 to 500 m.

Validation of ~i 3 M

An acoustic line source radiating over an in®nite foam is examined. The characteristics of the foam are given in Table 1. Fig. 9 displays the sound levels are Fig. 9. The use of Bouchon's method for the prediction of pressure levels due to a line source radiating over a semi in®nite thick foam, modeled as an equivalent ¯uid. The receiver is straight below the source, at the air porous interface. The results are compared to those given by Chandler Wilde's analytical formula. computed for a receiver on the interface, right under the source at distance d=0.5 m using Bouchon's method and Chandler±Wilde's analytical formula [START_REF] Wilde | Ground eects in environmental sound propagation[END_REF]. For the ®rst method, k max =23/(C air ) and a value of L=500 m have been chosen. Since Chand-ler±Wilde's approach requires to consider the material as locally reacting, Delany± Bazley model was utilized to characterize the foam for the analytical comparison. In the developed approach, the Biot±Allard model mentioned earlier was chosen. The two models used give close descriptions of the porous materials in this case except perhaps at low frequencies. The results are represented on Fig. 9. Agreement between the two approaches is excellent.

Eciency of the optimization of Bouchon's method by IFFT algorithm

In the present case, pressure levels due to a line acoustic source are calculated on a grid of receptors at the air±porous interface. The same material as in the previous example is considered. The results computed at 1000 Hz are represented on Fig. 10, as a function of the distance from the source. There is a good agreement between IFFT and the point by point calculation for the considered distances. However, the results given by IFFT are erroneous when getting close to x max . This phenomenon is known as aliasing. This is due to the fact that IFFT algorithm requires that the ®eld vanishes outside out of the range interval considered, which is not the case here. From there, the ®eld out of this range interval will be wrapped around in the neighboring windows. This means that the ®eld calculated in the receptors will be the sum of the signals in all range windows having the same width as the studied range interval. As far as the horizontal wave number spectrum is concerned, this phenomenon can be accounted for by an undersampling of the spectrum in the neighborhood of a pole of the integrand in Eq. ( 9). Improvements can be reached by adaptative integration scheme, or complex integration paths of Eq. ( 9). These techniques are described elsewhere [START_REF] Jensen | Computational ocean acoustics[END_REF].

To conclude, an IFFT algorithm can be used rather than a point by point computation to solve in a more ecient way the considered problem with a plane interface, as far as the computational time is concerned. However, Bouchon's method is of great interest since it give a physical background to what could be seen as a simple integration scheme.

Numerical results for an acoustical line source

Finally a con®guration of a ®nite thickness foam on a rigid support and excited by a line source is considered. The same emitter±receiver con®guration as for the previous example is chosen. The characteristics of the foam are given in Table 4. The levels at the receiver are given by Bouchon's method. A periodization length L=500 m has been taken to insure convergence in the considered frequency range. The results are represented on Fig. 11. In this case, the in¯uence of the frame motion is studied. These results show resonance peaks at about 900 Hz which highlight the importance of taking into account the frame motion. Outside of these resonances, the levels are very similar for the frequency range studied, with or without accounting for skeleton motion.

Conclusion

This paper investigated the prediction of the ®elds induced by an acoustical line source exciting an in®nite lateral multi-layered material with an in®nite planar interface. The ®rst original aspect of this work is to take into account the propagation phenomena in the whole material. In particular, a complete description of porous materials based on Biot±Allard equations was employed. The second aspect is related to the method used to solve the problem, namely Bouchon's method. In this method, the ®elds created by the original source are approached by those induced by a set of in®nite sources radiating in discrete directions. These sources are periodic along the axis parallel to the interface, and the interspacing is related to the angular spacing of the discrete directions in which they emit. An IFFT algorithm can also be used to compute the levels on a grid of receptors from the values of the Fourier transform of the ®elds in a ®nite numbers of points. The Fourier transform of the transmitted and re¯ected ®elds computed by the implemented algorithm has been validated. Several types of materials were considered: sandwich plates with porous material in-between, ®nite thickness porous multi-layered material bonded on a rigid wall. The results obtained are in complete agreement with analytical formulas or other numerical method found in the literature and testify to the reliability of the algorithm. Moreover, the second case studied also highlighted the limits of the motionless frame hypothesis, which had already been underlined in literature.

For a line source excitation, Bouchon's method was used. The results obtained are again in excellent agreement with those given by an analytical solution for the case of a porous material considered as locally reacting. Besides, the importance of the frame movement was demonstrated once again in the case of a ®nite foam on a rigid backing. Criteria have been established to insure the convergence of the ®elds approximations. First, the reconstruction of the ®elds requires accounting for all signi®cant contributions in the sums approaching the ®elds expression. In that way, the criterion k max >(2 Â 3/C max ) was met. Also, a large periodization length L, linked to Ák, had to be taken. For a given frequency, suitable values for N max highly depend on the source-receiver distance and the material. The use of an IFFT allowed for an optimization of the designed algorithm in terms of computational time. But then, the ®elds are processed on grid of receivers and a minimal width x max is imposed when a maximum Ák at ®xed k max is chosen, since x max =%/(Ák).

The quality of the results obtained enables one to think about Bouchon's method for an irregularly shaped air±material interface. In particular, future works could consist in ®nding the response of a linear source radiating over a road coupled with a soil of any pro®le. This particular problem ®nds applications in outdoor sound propagation, where one intends to predict sound levels near a roadway. Bouchon's method has already been used for similar con®gurations in seismology [START_REF] Campillo | Synthetic SH seismograms in a laterally varying medium by the discrete wavenumber method[END_REF]. The approach consists in replacing the pro®le by a set of periodic sources with unknown amplitudes distributed on the interface, both in air and in the material. The whole con®guration obtained is then periodized in one direction, the same way as was done in this paper. In that way, a periodic pro®le which has to be known on the whole periodization length L is considered. This ``double periodization'' leads to an approximation of the ®elds by a double sum. Bouchon's method could also be employed for a mechanical line force acting on a multi-layered porous material [START_REF] Boutin | Dynamique des milieux poreux sature s de formables. fonctions de Green Perme ameÁ tre dynamique[END_REF]. In this case, the source is considered to lie just slightly below the interface. So rather than an acoustical source one has to deal with a poro-elastic or elastic source according to the kind of material excited. The ®elds in the excited medium are then determined as was done in this paper. The only changes concern the incident ®elds; the amplitudes of each propagation mode are related to Green's coecients in the excited medium.

Fig. 1 .

 1 Fig. 1. Case of an acoustical line source radiating over a ®nite thick porous material.

Fig. 2 .

 2 Fig. 2. Re¯exion and transmission in a porous material for a plane wave excitation. The Biot Allard description of the porous material accounts for the propagation of txo compressional waves and a shear wave in the porous medium.

Fig. 3 .

 3 Fig. 3. Bouchon's method theoretical principal. The initial source is replaced by a set of in®nite sources radiating in discrete directions.

Fig. 5 .

 5 Fig.5. Prediction of the transmission loss for the sandwich plate using Lesueur's analytical formula and the present approach. The porous medium is modeled as an equivalent ¯uid in both cases.

Fig. 4 .

 4 Fig. 4. Con®guration of a sandwich plate excited by an impinging plane wave.

Fig. 8 .

 8 Fig. 8. Suitable frequency dependant values for k max to insure the convergence of Bouchon's method approximation of the ®elds. A ®xed k max /(3/C air ) ratio can be determined for the frequency range considered.

Fig. 7 .

 7 Fig. 7. The dierent parameters in¯uential on the convergence of Bouchon's method approximation. Source receiver con®guration.

Fig. 10 .

 10 Fig. 10. Computation of pressure levels on a grid of receivers at 1000 Hz. Results are represented for a point by point and an optimization by the use of an IFFT algorithm.

Fig. 11 .

 11 Fig. 11. Case of a ®nite thick foam on a rigid backing. In¯uence of the solid phase motion on the pre diction of pressure levels for a line source excitation. Quarter wave resonances for a complete description of the porous medium is highlighted.

Table 3

 3 Characteristics of the soil

	I	È	& S (kg/m 3 )	' (N Â s/m 4 )	Shape factor	Thickness (m)
	1.93	0.269	2650	366 000	1.38	I


Tb, Dy, Ho Er C Opagiste

R M Galéra

Ferromagnetism in novel compounds of the R 3 Pt 23 Si 11 series with heavy rare earth: Gd

Keywords:

Ferromagnetism in novel compounds of the R3Pt23Si11

series with heavy rare earth: Gd, Tb, Dy, Ho and Er

Christine Opagiste, Rose-Marie

Introduction

The ternary Ce-Pt-Si phase diagram has been the focus of great interest due to the discovery of the non-centrosymetric heavy fermion superconductor CePt 3 Si [1]. In the course of the study of this Ce-Pt-Si ternary system, Gribanov et al. have reported for the first time the existence of a defined cubic compound with the composition Ce 3 Pt 23 Si 11 that can be found in equilibrium in the cerium deficient CePt 3 Si compound [2].

We recently started the investigation of the new series R 3 Pt 23 Si 11 with R a light rare earth element (Ce, Pr and Nd) [3,4,5,6]. Our studies on a very high quality single crystal of Ce 3 Pt 23 Si 11 have revealed that a ferromagnetic order is stabilised at T c = 0.44 K. In the ferromagnetic phase, a weak but significant magnetic anisotropy exists with an easy magnetization axis along the threefold axes of the cube. In the paramagnetic phase, the magnetic susceptibility follows a Curie-Weiss law, fully consistent with a normal trivalent Ce 3+ ion. However, the observation by neutron spectroscopy of two inelastic lines of magnetic origin remains an intriguing point that is still not understood [3,4]. In the same series we have synthesized the novel compounds with Pr and Nd. Our studies on polycrystalline samples show that at low temperatures Pr 3 Pt 23 Si 11 becomes a Van Vleck type paramagnetic due to a non magnetic crystalline electric field ground state and that the magnetization of Nd 3 Pt 23 Si 11 has a metamagnetic behaviour [5,6].

To our knowledge on the side of the heavy rare earth elements, except Yb 3 Pt 23 Si 11 [7], no other compound has been synthesized and studied up to now. Quite recently we have successfully synthesized five new compounds in the R 3 Pt 23 Si 11 series with Gd, Tb, Dy, Ho and Er. In this paper, we report investigations of the crystallographic structure, magnetic behaviour and heat capacity properties of these compounds in the temperature range 2-300 K.

Samples synthesis and crystallographic studies

High quality polycrystalline samples of R 3 Pt 23 Si 11 have been prepared in an induction furnace. Stoichiometric proportions of the different constituents; R= Gd, Tb, Dy, Ho and Er (99.9%, Johnson Matthey), Pt (99.95%, Alfa Aesar) and Si (99.9999%, Alfa Aesar), were melted in a water cooled copper crucible under a highly-purified argon atmosphere. The samples were melted several times to improve the homogeneity. Mass losses during this step were less than 0.6%. No further heat treatment was applied.

The crystal structure and the sample quality were checked by conventional X-ray powder diffraction (Cu-Kα radiation on a Philips PW1730 diffractometer), whilst structural refinements were performed using the FullProf program [START_REF] Rodriguez-Carvajal | Satellite Meeting on Powder Diffraction of the XV Congress of the IUCr[END_REF]. The results of the refinements are fully consistent with the fcc crystal structure (Fm 3m space group) already reported for Ce 3 Pt 23 Si 11 [9]. Figure 1 1. The atomic coordinates in the elementary cell are summarized in tables 2 to 6 for each compound. The variation of the lattice parameter with the atomic number of the rare earth elements is reported in figure 2. The decrease of the lattice parameter with the atomic number is consistent with the effect of the lanthanide contraction.

Magnetic Properties

Magnetization measurements were performed on polycrystalline samples using different magnetometers based on the extraction method in the temperature range 2-300 K. We used a commercial Quantum Design MPMS magnetometer for measurements in fields up to 5 T and a second experimental set-up, equipped with resistive detection coils in fields up to 8 T. The thermal variation of the magnetization under an applied field of 100 Oe has been measured between 2 and 300 K for Dy-Ho-and Er 3 Pt 23 Si 11 . For each compound, below a certain temperature the thermal variation presents a steep increase and remains constant down to 2 K. This is illustrated, for In the paramagnetic phase, isothermal magnetization curves M(H) have been measured up to 300K. The inverse of the magnetic susceptibility is deduced from the Arrott plots [10]: M 2 = f (H/M). For all compounds 1 χ varies

Table 7: The Curie temperatures T C deduced from the magnetic and heat capacity measurements are reported in column 2 and 3 respectively. C th is the Curie constant of the trivalent rare earth ion in emu.K per mol of R ion (emu.K/mol R). n ex is the value of the exchange constante used in the calculation of the Curie-Weiss law : 1 χ = T C thn ex . Columns 6 and 7 give the saturation magnetic moment and the spontaneous magnetization at 2 K respectively. 4 represent the Curie-Weiss law : 1 χ = T C th -n ex , with C th the theroretical Curie constant of the trivalent rare earth ion and n ex the exchange constant (see table 7). Theoretically n ex = θp C th , with θ p is the paramagnetic Curie temperature.

T mag C (K) T HC C (K) C th n ex g J J (µ B ) M s (µ B /
In the ferromagnetic phase the magnetization curves have been measured down to 2 K, the base temperature of the experimental setups. The magnetization curves for Gd 3 Pt 23 Si 11 , Tb 3 Pt 23 Si 11 and Dy 3 Pt 23 Si 11 are reported in figure 5 together with the thermal variation of the spontaneous moment. The value of the spontaneous moment is deduced from the Arrott's plots. At 2 K the spontaneous moment is found systematically smaller than the expected value M S (0 K) = g J J for all the compounds except for Gd 3 Pt 23 Si 11 (see table 7). In this last compound the expected 7 µ B /Gd for a S=7/2 ion are found (see figure 5). From the values observed at 2 K for the spontaneous magnetization of all the other compounds it is obvious that the saturation of the magnetic moment is not reached at zero K. This reduction of the magnetic moment is ascribed to the crystalline electric field (CEF) effects that in crystals partly lift the degeneracy of the fundamental multiplet. Note that, contrary to the light rare earth compounds [4,5], the magnetic susceptibility shows no deviation to the Curie-Weiss law at low temperatures in the heavy rare earth side of the series (see figure 4). This deviation is also a CEF effect associated with the population of the excited levels as the temperature increases. It gets rapidly reduced as soon as the excited CEF levels become closer in energy. This is likely the case for the heavy rare earth compounds owing to the high degeneracy of the J=L+S multiplet of heavy rare earth ions. The evolution of the Curie temperature with the rare earth ion in the series is compared with that of the de Gennes factor in figure 6. A very good consistency is observed between both evolutions in the whole series.

Thermodynamic properties: heat capacity, entropy

In the temperature range 1.9 -300K, the heat capacity was measured using the relaxation method with a commercial Quantum Design PPMS. The measurements were focused at low temperature in order to determine Figure 6: (Color online) Comparison of the evolution with the rare earth ion between the Curie temperature (dots) and the de Gennes factor (diamonds) [11] in the R 3 Pt 23 Si 11 series. Note that the Pr compound behaves as a Van Vleck paramagnet at low temperature [5].

precisely the temperature of the ferromagnetic transition. For all the compounds a clear lambda anomaly is observed at T C as shown for instance in figure 7 for the specific heat of Er 3 Pt 23 Si 11 and Dy 3 Pt 23 Si 11 . The Curie temperatures deduced from the inflexion point of the lambda anomaly are reported in table 7. The values are consistent with those determined from the magnetic measurements.

A tentative determination of the magnetic entropy from the experimental specific heat has been performed for all the compounds. This is done with the aim to determine the degeneracy of the magnetic ground state. As stated above it is expected that the CEF interactions partly lift the 2J + 1 degeneracy of the fundamental multiplet. The Curie temperature of some compounds is relatively high. Thus, it is necessary to subtract properly the phonon contribution to the total specific heat in order to get the magnetic contribution C mag . For these compounds the phonon contribution is determined from the specific heat curve measured in the non magnetic La 3 Pt 23 Si 11 compound [4] after normalizing the temperature by the factor F = 3 θ D (R 3 P t 23 Si 11 ) θ D (La 3 P t 23 Si 11 ) , where θ D is the Debye temperature of the compound [12]. For instance, in figure 7 the total specific heat of Dy 3 Pt 23 Si 11 is reported with the phonon contribution deduced from the La 3 Pt 23 Si 11 specific heat.

Figure 8 gathers the thermal variation of the estimated magnetic entropy of all the studied compounds. At the Curie temperature, the entropy for Gd 3 Pt 23 Si 11 reaches only 92.6% of 3 × R ln 8, the expected value for a S=7/2 ion. The magnetic entropy is obtained by integrating Cmag T from 0 K to T. As the lowest temperature reached in the present measurements is only 1.9 K, it is likely that the magnetic entropy is underestimated. Nonetheless for the other compounds the entropy at T C reaches values far lower than the 3 × R ln(2J + 1) upper limits. According to the magnetic measurements, this reduction is likely due the CEF effects.

Conclusion

In the present study we show that compounds of the R 3 Pt 23 Si 11 series exist with Gd, Tb, Dy, Ho and Er. This proves the high stability of this ternary series. The magnetic studies show that, all compounds follow a Curie-Weiss law and that a ferromagnetic order is stabilized at low temperatures. As expected in a scenario of normal RKKY-type exchange interactions, the highest value of T C is found in the Gd compound. It is also observed that in the series the evolution of the values of T C or T N (case of Nd 3 Pt 23 Si 11 ) with the atomic number of the rare earth ion is in a fair agreement with the de Gennes law. Though no deviation to the Curie-Weiss law is observed at low temperature on the magnetic susceptibility, the reduced values of the spontaneous magnetization together with a magnetic entropy at T C smaller than 3 × R ln(2J + 1), are consistent with the lifting of the multiplet degeneracy by the crystalline electric field effects in the Tb, Dy, Ho and Er compounds.

Figure 1 :

 1 Figure 1: (Color online) X-ray diffraction patern of Gd 3 Pt 23 Si 11 with the results of the FullProf refinements.

Figure 2 :Figure 3 :

 23 Figure 2: (Color online) Comparison between the variation of the lattice parameter and the ionic radius of the trivalent rare earth ion in the R 3 Pt 23 Si 11 series. The dashed line is a guide for the eyes.

11 Figure 4 :

 114 Figure 4: (Color online) Thermal variation of the inverse of the susceptibility of the five studied compounds.

Figure 5 :

 5 Figure 5: (Color online) Upper part magnetization curves measured on Gd 3 Pt 23 Si 11 , Tb 3 Pt 23 Si 11 and Dy 3 Pt 23 Si 11 respectively at different temperatures below and above T C . Lower part thermal variation of the spontaneous magnetization for Gd 3 Pt 23 Si 11 , Tb 3 Pt 23 Si 11 and Dy 3 Pt 23 Si 11 respectively.

Figure 7 :

 7 Figure 7: (Color online) a) Thermal variation of the specific heat in Er 3 Pt 23 Si 11 in the low temperature range. The Curie temperature T C is deduced from the inflexion point of the lambda anomaly. b) Thermal variation of the total specific heat in Dy 3 Pt 23 Si 11 (open dots) and of the phonon contribution deduced from the La 3 Pt 23 Si 11 measurements (full squares)(see text).

Figure 8 :

 8 Figure 8: (Color online) Thermal evolution of the estimated magnetic entropy in Gd-, Tb-, Dy-, Ho-, Er 3 Pt 23 Si 11 .

Table 1 :

 1 Crystallographic details and results of the FullProf refinements realized assuming a cubic structure of the Ce 3 Pt 23 Si 11 type: space group Fm 3m (n o 225) and Z=8. Column 4 gives the number of measured reflections. For all compounds the same 28 parameters have been refined. * presence of ≈ 5% of the phase Pt 2 Si. Pt 23 Si 11 *

	Compound		a( Å)	D cal (g/cm 3 ) Reflexions χ 2
	Gd 3 Pt 23 Si 11	16.8249(7)	14.693	296	5.88
	Tb 3 Pt 23 Si 11		16.8181(8)	14.725	286	5.82
	Dy 3 Pt 23 Si 11	*	16.814(1)	14.767	299	7.41
	Ho 3 Pt 23 Si 11		16.8015(5)	14.820	272	7.92
	Er 3 16.7979(5)	14.849	283	5.92

Table 2 :

 2 Atomic coordinates and isotropic displacement parameters for Gd 3 Pt 23 Si 11 .

	Atom Wyckoff site	x/a	y/b	z/c	U iso ( Å2 ) Occupation
	Gd	24d	0	1/4	1/4	1.2
	Pt 1	32f	0.0808(1) 0.0808(1) 0.0808(1)	1.0
	Pt 2	32f	0.3088(1) 0.3088(1) 0.3088(1)	0.8
	Pt 3	24e	0.3745(2)	0	0	0.5
	Pt 4	96k	0.0848(1) 0.0848(1) 0.2525(1)	0.9
	Si 1	24e	0.188(1)	0	0	1.0
	Si 2	32f	0.1723(6) 0.1723(6) 0.1723(6)	1.0
	Si 3	32f	0.3889(8) 0.3889(8) 0.3889(8)	1.5

Table 3 :

 3 Atomic coordinates and isotropic displacement parameters for Tb 3 Pt 23 Si 11 .

	Atom Wyckoff site	x/a	y/b	z/c	U iso ( Å2 ) Occupation
	Tb	24d	0	1/4	1/4	1.0
	Pt 1	32f	0.0826(2) 0.0826(2) 0.0826(2)	1.2
	Pt 2	32f	0.3093(1) 0.3093(1) 0.3093(1)	0.4
	Pt 3	24e	0.3738(3)	0	0	0.4
	Pt 4	96k	0.0847(1) 0.0847(1) 0.2524(1)	0.7
	Si 1	24e	0.184(2)	0	0	1.5
	Si 2	32f	0.1759(8) 0.1759(8) 0.1759(8)	1.5
	Si 3	32f	0.3918(8) 0.3918(8) 0.3918(8)	1.5

Table 4 :

 4 Atomic coordinates and isotropic displacement parameters for Dy 3 Pt 23 Si 11 .

	Atom Wyckoff site	x/a	y/b	z/c	U iso ( Å2 ) Occupation
	Dy	24d	0	1/4	1/4	1.0
	Pt 1	32f	0.0828(2) 0.0828(2) 0.0828(2)	1.5
	Pt 2	32f	0.3089(2) 0.3089(2) 0.3089(2)	0.8
	Pt 3	24e	0.3753(4)	0	0	1.1
	Pt 4	96k	0.0850(1) 0.0850(1) 0.2527(1)	0.8
	Si 1	24e	0.194(2)	0	0	1.5
	Si 2	32f	0.170(1) 0.170(1) 0.170(1)	1.5
	Si 3	32f	0.386(1) 0.386(1) 0.386(1)	1.5

Table 5 :

 5 Atomic coordinates and isotropic displacement parameters for Ho 3 Pt 23 Si 11 .

	Atom Wyckoff site	x/a	y/b	z/c	U iso ( Å2 ) Occupation
	Ho	24d	0	1/4	1/4	1.3
	Pt 1	32f	0.0829(2) 0.0829(2) 0.0829(2)	0.7
	Pt 2	32f	0.3093(1) 0.3093(1) 0.3093(1)	0.2
	Pt 3	24e	0.3770(3)	0	0	0.4
	Pt 4	96k	0.0848(1) 0.0848(1) 0.2524(1)	0.8
	Si 1	24e	0.182(2)	0	0	1.5
	Si 2	32f	0.1711(8) 0.1711(8) 0.1711(8)	1.5
	Si 3	32f	0.388(1) 0.388(1) 0.388(1)	1.5

Table 6 :

 6 Atomic coordinates and isotropic displacement parameters for Er 3 Pt 23 Si 11 .

	Atom Wyckoff site	x/a	y/b	z/c	U iso ( Å2 ) Occupation
	Er	24d	0	1/4	1/4	0.9
	Pt 1	32f	0.0829(2) 0.0829(2) 0.0829(2)	1.2
	Pt 2	32f	0.3081(1) 0.3081(1) 0.3081(1)	0.8
	Pt 3	24e	0.3747(3)	0	0	0.8
	Pt 4	96k	0.0849(1) 0.0849(1) 0.2525(1)	0.9
	Si 1	24e	0.190(2)	0	0	1.6
	Si 2	32f	0.1653(8) 0.1653(8) 0.1653(8)	1.6
	Si 3	32f	0.3897(8) 0.3897(8) 0.3897(8)	1.6

Acknowledgments

We are grateful to R. Haettel, A. Hadj-Azzem, J. Balay, Y. Deschanels, E. Eyreaud, D. Dufeu and P. Lachkar from the Institut Néel, for their technical assistance.


Claude Boutin

Study of thermosensitive heterogeneous media via space-time homogenization

The second-order terms then result in divy[A(Vx01 + VyO2)] + divx[A(Vx8 0 + vyol)] = pC[OTt7 ° +cOtO 2] -S Successive spatial and temporal averaging then lead to the macroscopic description (A13) previously found.

Case iv-c: for this last case el/ft --O(6t), denotes et = e and el = e 2. The renormalised equationreads div

[A grad0] = e3pCO -e4S (A16)
while the conductive term, on account of the double-scale space derivative, is now written div(AV8) = [divv + e2div.~,][Uv + e2X~x]{80 q-e01 q-e282 q-...}

The zero-and first-order terms lead to/7 o = tT°(x, T, t) and 01 = 0l(x, T, t), while the second-order terms give divy[A(Vy02 + Vxg°)] = 0, hence the classic solution ~72 = ((y)Vx8 ° + O2(x, T, t). The third-order terms give divy[A(Vy~73 + V T01)] = pCOt~ O. Taking the space average, this leads to the conclusion that 8 = 8°(x,T).

Finally, the fourth-order terms result in

divy[A(V~0 4 + V.,.O2)] + div.~.[A(Vyt? 2 + Vz0°)] = [OT O° +OtO 1] --S
The space-time average again leads to the completely macroscopic description (A13).

The following table summarises the space-time descriptions according to the scale factors.

Table AH .

 AH Space-time descriptions according to the scale factors.

		t
		= O(1)	<<1
	= O(1)	Case i	Case ii
		(y, t)	(y, T)
	<< 1	Case iii	Case iv
		(x, t)	(x, T)