S Jang

A Bouyssy

Nuclear elasticity applied to giant resonances of fast rotating nuclei

lsoscalar giant resonances in fast rotating nuclei are investigated within the framework of nu clear elasticity by solving the equation of motion of elastic nuclear medium in a rotating frame of reference. Both Coriolis and centrifugal forces are taken into account. The nuclear rotation re moves completely the azimuthal degeneracy of the giant resonance energies. Realistic large values of the angular velocity, which are still small as compared to the giant resonance frequencies, are briefly reviewed in relation to allowed high angular momenta. It is shown that for the A = 150 re gion, the Coriolis force is dominating for small values ( $ 0. 05) of the ratio of angular velocity to resonance frequency, whereas the centrifugal force plays a prominent part in the shift of the split resonance energies for larger values of the ratio. Typical examples of the resonance energies and their fragmentation due to both rotation and deformation are given.

I. INTRODUCTION

II. EQUATION OF MOTION OF ROTATING ELASTIC NUCLEI

As shown in I, the equation of motion of a uniform perfectly elastic nuclear medium in inertial system is given by

a 2 u (A. +2µ)V(V•u)-µVXVXu+F=p ot 2 , (2. 1)
where u is the displacement vector, F the external body force, and p the nuclear density. For the nuclear elasti city the Lame coefficients A and µ take the forms 1

[K 2 '1 2 l A= ----k} p 9 15 m • ' (2.2) 1 ,, 2 2 µ= s m• k 1P• (2.3)
where K is the nuclear compressibility, m • the effective nucleon mass, and k 1 the Fermi momentum. It is also noticed that the quantity A.+ fµ is the compression modulus of the elastic medium; that is, the ratio of iso tropic pressure to fractional rate of decrease of volume.

The Lame coefficient A is also related to the Landau pa rameter F 0 by A= l_ __K_k}pO+fF0).

(2. 4) 5 m* Generally, a dynamical equation of motion expressed in the system of rotating coordinates has the same form as the original equation of motion written in the inertial system, except for the body force term, which now be comes the effective body force. This was the case in the two-fluid model we formulated in a previous paper. 9

Therefore, to an observer, in the rotating system it ap pears as if the body is moving under the influence of this effective force, which is represented by F.tr=F-2p [ nx �; j -pnxmxrl , (2.5) where !l is the angular velocity. The velocity in the inertial system vi is related to the velocity in the rotating system v by vi = v + !l X r. The second term in the effective force (2.5) represents the Coriolis force and third term denotes the centrifugal force. If the angular velocity is time dependent, an extra term, -prXd!l/dt , is to be added to the effective force.

The fact that the displacement vector u plays the role of the usual radial vector r in the classical dynamics necessitates a careful formulation of the effective force in the approach of nuclear elasticity. Furthermore, the elastic medium under the influence of external potentials is not free from stress, even in the initial equilibrium state, and thus the actual form of the term correspond ing to the centrifugal force is slightly more complicated than the term shown in Eq. (2.5). For constant density, however, the effective force. can take the same form as before, except that the radial vector is now replaced by the displacement vector. Let the position vector of a typical point P 0 of the elastic body referred to the origin of a fixed rectangular cartesian system be s. At time t, the representative point P0 has moved to a point P with position vector r, and the position vector of the point P relative to P0 is given by the displacement vector u with u=r-s. In terms of general orthogonal curvilinear coordinates (q 1 ,q 2 ,q 3 ), the position vectors s and r can be written in the forms s= s(q 1 ,q 2 ,q3), r=r(q1 ,q 2 ,q3 ,t ), and it follows that u=u(q1 ,q 2 ,q 3 ,t ) . Therefore, for con stant density and in the absence of other potentials such as the Coulomb potential, the linearized equation of motion now takes the form au

(A + 2µ )V( V •u)-µ V XV X u-2p!l X at a 2 u -p!lX(!lXu)=p-2 (2.

6) at

Equations (2.5) and (2.6) show that the centrifugal force is or order 0 2 , whereas the Coriolis force is of order n. We now assume that the displacement vector can be separated into spatial and time parts as u(r ,t )=u(r)exp(iwt ). Applying the variational principle to Eq. (2.6) , as was done in I, we get, after rather tedious transformations, w 2 I p I u I 2 d7"=A I I V •u I 2 dT +2µ I l: I ei j I 2 d7" i, j + J pu*( 2iw!lXu)dT + J pu*!lX(!lXu)dT. (2.7), we shall now consider briefly the case of an elastic body whose density is a function of r and which, accord ingly, is assumed to be in a state of hydrostatic stress be fore being disturbed by external forces. In a slightly de formed elastic state, the stress is then the sum of the usual stress due to the small displacement u and the ini tial hydrostatic stress. The former is given by Eq. (2.2) of I and the latter is expressed as the negative of the original hydrostatic pressure at the initial representative points P0• Thus the initial stress is shown to be

-p(r+(-u))= -[p(r)-u•Vp] ,
where p is the hydrostatic pressure. Moreover, owing to disturbance due to the small displacement, the density in the disturbed state may be written as p'=p +Bp, where Bp is the change in density, which is calculated from the principle of mass conservation.

Assuming further that the body force can be associat ed with a potential \II , we can equate the body force F to the gradient of the potential, multiplied by the disturbed density. Since the nucleus under consideration is rotat ing, the potential \II may comprise the centrifugal term l/J,

(2.8)

as well as the Coulomb potential. It is to be remarked that the use of the potential \II instead of the explicit form of the centrifugal force modifies the expression (2.5) so that the centrifugal term is now included in F.

When we introduce the initial stress as well as the body-force term into the equation of motion written in terms of the stress-strain tensors, such as Eq. (3.1) of I, we see that the equation of motion now contains the terms

-V(p-u•Vp )+p V'l'+Bp V'I'.
(2.9)

When we neglect the contribution ansmg from the change in density and omit the initial stress, the expres sion (2.9) reduces then to the term which represents the usual centrifugal force. The pressure in (2.9) may also be associated with the potential \II by assuming that the pressure force in the hydrostatic equilibrium state, which is the negative of the pressure gradient, counterbalances the body force. Thus, for the rotating body with angular velocity 0, we have 1 0

Vp =p V\11 .

(2.10)

The pressure force will, in general, be very small, unless there is a very large pressure gradient. When we keep only the centrifugal term l/J in \II , we see, in view of the relation (2.10), that all terms of (2.9) are of order 0 2 • If

(A.+2µ)V(V•u0)-µVXVXu0= -(l)fpU0 , 2
there is no rotation, the potential \II reduces practically to the Coulomb term and the contribution from the ini tial stress to the usual nuclear elastic vibration will be inconsiderable, unless the relation (2.10) is reviewed so as to be still significant with regard to other specific external potentials. The form (2.9) of the centrifugal effect for the rotating elastic body contrasts with the corresponding term in the equation of motion (2.6) for constant density. We shall come back to the expression

(2.9) on the occasion of practical evaluation of various rotational effects.

III. PERTURBATION TREATMENT OF ROTATIONAL EFFECTS

Though an analytical solution of Eq. (2.6) cannot be expected, the equation of motion of this type can well be treated using the perturbation method, provided the magnitude of angular velocity is much smaller than the unperturbed initial frequency of oscillation without rota tion. From the quantum mechanical point of view the nuclear rotation makes sense only for deformed nuclei, whereas the rotation of a spherical body has meanings in the classical picture. In the following we first concen trate on the rotational effects by solving Eq. (2.6) without including the degree of freedom of deformation in the formulation and we subsequently take into ac count the nuclear surface deformation.

The displacement vector u as well as the frequency Ci) are now expanded in powers of 0 I (j)o as

u=uo+ [ � ] u1+ [ � r u 2 + . .. , (3.1) (j)= (j)o + [ �o ](j.) 1+ [ �o r(j.) 2 + ... ' (3.2)
where (j)o is the unperturbed initial frequency of oscilla tion without rotation. Upon introducing the expansion (3.1) and (3.2) into Eq. (2.6) and collecting the terms of the same power of 0/(1)0 up to second order, we get

(3.3) (j)o 2 (A.+2µ )V(V•u1 )-µVXVXu 1-2i n p0XUo= -p((l)oU1 +2{t)QW1Uo) ' (3.4) (j)fi (j)o(j) 1 [ {t)o ] 2 (A.+2µ )V(V •u 2 )-µV XV X U 2 -2i0p0 X Ut -2i0-p0 X Uo-O 0 X ( {). X Uo)
Multiplying Eq. (3.3) by u6 and integrating the result by parts, we obtain

wfi JP I Uo J 2 dr=A. J J V•u0 j 2dr 3 +2µ J � J e;j 0 > I 2 dr , (3.6)
i, j

= -p(wfiu 2 + 2(j)0w 1u1 +wru0+2wQW 2 u0) • (3.5) where eij are the strain tensors which were defined in I.

In obtaining Eq. The first and last terms on the right-hand side of Eq. (3.1 6) clearly represent the second order contribution from the Coriolis force, while the second term shows the centrifugal part. The expression for (J) j is formally the same as for (J) t> except that u1 is used in place of u0• Since E describes the difference between (J) i and (J)i. we can expect its contribution to ill 2 to be small. Actually, this is verified by computing explicitly the first order dis placement vector u1 by means of the usual perturbation method; that is, by expanding u1 in terms of u0 as Ut = l:va<">u&v>.

The angular integrals in Eq. (3.16) can easily be evalu ated and, finally, ill 2 can be expressed in terms of known radial integrals I n and I d of Eqs. (3.15). We have 1 [2( / 2 +l +m 2 -l)-(/ 2 +/-3m 2 ) I I n d ] } +E. ( 2/ -1 ) ( 21 + 3 ) (3.18) The result (3.18) is quite simple and its numerical computation is elementary, as in the case of the first order fre quency. We see immediately that the second order frequency does not vanish even for m =0 and the second order ro tational correction does also act to shift the eigenfrequency of m = 0 mode.

When the density is a function of rand when we keep only the term t/J in the potential \fl, the centrifugal contribu tion can be evaluated with the help of the expressions of (2.9) and (2.10). When we apply the principle of mass conser vation to the total density, the small deviation of density Bp may be equated to -V•(pu). Neglecting small quantities of the higher order, the second order frequency is now shown to be

ill 2 = 2 ! 0 {il l 1+ �� [ J u6•V•(puol•Vt/Jdr-J u6•V(pu0•Vt/J)dr ]/I d }+E.
(3. 19)

The angular integrals involved in this representation are rather cumbersome, but they are still analytically integrable. Thus, the second and third terms in the square brackets on the right-hand side become -(21 _1� 2 21 + 3 ) { 2[/(/ +O+m 2 -l] J il1 (r)U1 (r) r 2 dr+[3m 2 -l(l +O] J il1 (r) Vi(r)r2dr }/I d , (3.20) where il1 (r) =2p(r) [ 1u +0Vi(r)-2U1 (r)-r ; r U1 (r) ]-U1 (r)r ;7p(r) . In and I d , the ratio ill /ill o for a given value of multipo larity l and its projection m can be expressed as a func tion of !l/ill 0 , which is a measure of the angular velocity of the rotation.

Generally, rotational states with angular momentum I in nuclei can be characterized by their excitation ener gies EI> which usually lie low compared to the single particle excitations and which obey the I ([ + 1 ) rule. dependent nuclear elasticity constitutes an interesting new subject, the complexity of numerical calculations in the present problem may prevent one from gaining phys ical insight into the problem of rotating nuclear vibra tion. In the following we therefore turn our attention to the expression (3.16) together with (3.12).

IV. TYPICAL NUMERICAL RESULTS

When we substitute the results (3.14) and (3 .18) into the formula ( 3 .2) for the total frequency of vibration, we get the desired expression to second order for rotating nuclei. Thus, However, the well-known expression for rotational exci tation energies, Er=AI U+O+BUU+1)) 2 + • • • , converges poorly for I � 10 (R efs. 12 and 13). A much wider region of convergence can be realized if the angu lar velocity of nuclear rotation, !l, is used as an expan sion parameter. With the assumption of a rigid rotor, the nuclear angular velocity can be expressed 13• 14 as fi=fiVJ(I +1)/.7, which relates the angular velocity 0., the angular momentum I, and the nuclear moment of in ertia .7. It is thus seen that the angular velocity could attain, in principle very large values for higher angular momenta, provided the nuclear moment of inertia remains constant. However, the assumption of constant moment of inertia is not always reliable, as discussed in Ref. 13. The typical values of angular moment which can be accommodated in nuclei have already been estab lished in Ref. 14. In a previous paper 9 , we have also

given some examples of angular velocity characterizing various nuclei at different angular momenta. According to these results, reasonable values of angular velocity of rotation are rather small as compared to the giant reso nance frequencies. For sufficiently high spin states, e.g.,

I

V. SUMMARY AND CONCLUSION

The main purpose of the present work was to study the giant resonances of rotating nuclei, especially the iso scalar type of giant dipole and quadrupole resonances, by extending the concept of nuclear elasticity to the ro tating nuclei.

The following is a brief summary of the present work. The equation of motion of a perfectly elastic nuclear medium in the rotating frame of reference is derived. The effective forces, generated by the transformation of the equation of motion in the inertial system into that in the system of rotating coordinates, consists mainly of Coriolis and centrifugal forces. The case where the in tial pressure is related to the external potential is also discussed. We have shown how the equation of motion of rotating nuclei was solved with the help of the pertur bation method. The final result for the frequency of vi-bration of rotating nuclei is quite simple and its numeri cal evaluation becomes elementary. The angular velocity of nuclear rotation is briefly analyzed in relation to high angular momenta, which can be accomodated in nuclei.

It is illustrated that for the A = 150 region the Coriolis force dominates for small values ( :5' 0. 05) of the ratio of angular velocity to resonance frequency, but the centri fugal force plays a prominent part in the shift of the split resonance energies for larger values of the ratio. It is seen that the rotational effect removes completely the azimuthal degeneracy of the giant resonance energies.

  In a previous paper1 (hereafter referred to as I), we in vestigated the isoscalar giant resonances of deformed nu clei within the framework of the nuclear elasticity. In particular, we turned our attention to the fragmentation of the giant monopole and quadrupole resonances. It was thus shown how the nuclear surface deformation contributes to the fragmentation through the coupling between the monopole and quadrupole modes of oscilla tion. In the present paper we extend the concept of nuclear elasticity to rotating nuclei and deal with the isoscalar giant resonances of rotating elastic nuclear matter. The origin of the nuclear elasticity can be found in the fact that the equations of motion formulated in the dynam ics2• 3 of nuclear fluid in the framework of the time dependent Hartree-Fock theory can be approximated for some collective motions, such as the giant resonances, by the Lame equation which governs the classical theory of elasticity. Thus the isoscalar giant resonances of spheri cal nuclei2•3 and those of deformed nuclei1 have been ful ly studied recently in the spirit of the nuclear elastic vi bration. In a recent experiment, Newton et al.4 analyzed the high energy tail of y rays emitted following 4 0 Ar induced reactions and suggested that the nuclear giant dipole resonance may be excited in a nucleus which ro tates with very high angular momentum. Regarding the isovector giant dipole resonance of rotating nuclei, there have been several theoretical investigations in the past few years. For example, calculations of the position and splitting of the isovector giant dipole resonance were first performed on the basis of a simple schematic model of a rotating harmonic oscillator. 5• 6 More elaborate models7•8 were then applied to the same problem. In a recent paper,9 we also studied the isovector giant reso-nances of arbitrary multipolarity in fast rotating nuclei by solving the invicid two-fluid equation of relative motion written in a rotating frame of reference. Though the description we used in Ref. 9 is a rather semiclassical one, the resulting expressions display in a quite simple way general features of isovector giant resonances of fast rotating nuclei. In particular, we have shown how the splitting of the giant resonances arises from rotating de formed nuclei. For the rotating elastic nuclear matter, we have to transform first the equation of motion in I into one in a rotating frame of reference. Generally, the effective force in the new equation of motion in the rotating coor dinates contains the terms corresponding to Coriolis and centrifugal forces. The equation of motion of the rotat ing nuclear matter is, on the whole, more complicated than the corresponding equation of motion obtained in the two-fluid hydrodynamical model. 9 The reason is that the elastic medium under the influence of external potentials is not free from stress, even in the initial equi librium state, and thus the usual expression for the elas tic stress in the equation of motion in I has to be slightly modified by adding pressure-dependent terms. Never theless, if the density of the elastic nuclear matter is as sumed to be constant, the equation of motion can then preserve the same form of the rotational forces as in the case of the two-fluid model, except that the displacement vector now takes part in the force in place of the ordi nary radial vector. However, the presence of the rota tional forces in the equation of motion prevents us from obtaining an analytical solution of the equation, whatev er forms these forces may take. The derivation of the equation of motion in a rotating frame of reference is described in Sec. II. This is fol lowed by a method of solution of the equation. The per turbation method is shown to be of great utility for the treatment of the equation of motion that contains rota-tional forces. Section IV is devoted to discussions of particular examples of the resonance energies in terms of the angular velocity of rotating deformed nuclei. The meaning of nuclear fast rotations is briefly discussed in relation to high angular momenta which can be accomo dated in nuclei. A summary and final conclusions are presented in Sec. V.

( 2 . 7 )

 27 This is the variational form of the equation of motion which consists of the main equation in the present inves tigation. Because of the presence of the additional force terms which represent nuclear rotation, we cannot ex pect to express the frequency w in the variational form as in Eq. (3.4) of I. We shall show, however, how the perturbation method enables us to acquire a quite simple expression for the frequency of vibration of rotating nu clei.Before going closely into the method of solution of Eq.

  (3.6), we have taken into account the boundary condition which states that the strain-stress tensor components vanish at the nuclear surface. Equa tion(3.6 ) is nothing but the variational expression we have already derived for the original Lame equation which governs the motion of a perfect elastic medium. The explicit form of the function u0 has been shown 1 to be u0=u0,n,+u08n8+u0tf>nt/>, where Uor = U1(r) Y 1 m (e,tf>) , aY1m Uoe= Vi(r)-ae-1 ay,m u Otf> = Vi ( r) sine -a;;;and n,, n8, and 11.,, denote, respectively, unit vectors in the directions of r, e, and </J. In Eq. (3.8), Yim {e,</J) are the spherical harmonics and the radial functions U1(r) and Vi (r) for constant density are expressed in terms of spherical Bessel functions ji(x) as (3.9) (3.10) where h2=p(J)'f,/(A.+2µ) and k2=p(J)'f,/µ. The constant multipliers A 1 and C1 are determined from the boundary conditions. To be more precise, the boundary condition we have previously stated leads to two expressions for the ratio C1/ A1• For example, L [ h(s) ) s=hR0 and 'fl=kR0, R0 being the radius of a spherical nucleus. The other expression for the same ra tio was shown in I. Having obtained the eigenvalues 5 and 'fl, we can proceed to the evaluation of various in tegrals in which the radial functions U1 and Vi take part. Equations (3.4) and (3.5) can now be solved with the help of the standard stationary perturbation technique. To this end, we multiply Eq. (3.4) by u6 and Eq. (3.3) by ur and then integrate the difference between two result ing equations. We then get . (J) o J pu6(0Xu0)dr (J)I =z-(3.12) n J P I u o I z drThe expression (3.12) represents the rotational effect aris ing from the Coriolis force alone. We now assume, without losing generality, that the angular velocity n is directed to the polar axis Oz of rectangular Cartesian axes Oxyz with the center of nucleus as the origin. In spherical polar coordinates, n then takes the form !l=(!l cos8)n, +( -!l sin8)n8+on.,, .

( 3 . 13 )( 3 1 0

 31331 Upon introducing the expression (3.13) into Eq. (3.1 2), we get, after performing angular integrations, I n (J)t =m(J) o I d ' where I n= J p[2U1(r) Vi(r)+ Vl(r)]r2dr , I d = J p[Ul(r)+IU+llVl(r)]r2dr see that, apart from two radial integrals, the effect of rotation due to the Coriolis force is simply proportional to the azimuthal component m multiplied by the initial unperturbed frequency. The present result is to be com pared with the corresponding formula obtained in the two-fluid model. 9 Since the eigenvalue equation obtained in the framework of the nuclear elasticity is much more complicated than the similar equation used in the hydro dynamical model,9 further simplication of the expression (3.14) as a function of eigenvalues cannot be expected even for constant density. The formula (3.14) is unfami liar to nuclear physicists but not to geophysicists. It is worth mentioning that the classical theory of elasticity has already been applied to the rotational problem of the Earth. The assumption of constant density allows us to integrate analytically two radial integrals (3.15a) and (3.15b) (see the Appendix). As we shall see, these are the only integrals to be computed through the present work. The result (3.14) shows that there is no rotational effect for m = 0 as far as the first order calculation is con cerned. The second order formulation removes, howev er, this restriction.In a way analogous to the derivation of the first order effect, we can derive the second order frequency of oscil lation, (J)z. Thus, we get, from Eq. (3.5),

(3. 21 )

 21 The expression(3.20) contrasts with the corresponding form(3 .18), obtained from the density-independent cen trifugal potential. In fact, the eigenvalues hR 0 and kR 0 that are involved in the integrals of(3.20) are not those evaluated from the eigenvalue equation formulated in I. Since both displacement vector and density in the origi nal Lame equation are now variables, the zero-order equation (3. 6) can be solved only numerically and thus the radial functions U1 ( r) and Vi ( r) are no longer given by the relations (3.9) and (3.10). Furthermore, according to the formulas (2.2) and (2.3), the Lame coefficients ').., and µ also change their values when the density varies (s ee Ref. 11).Though the study of the density-Upon introducing explicit results of the radial integrals

  = 78 for the 154Sm nucleus and I = 20 for the 24Mg nu cleus, the angular velocity 0. takes the value of 1.07 MeV for 154Sm and and 6. 59 MeV for 24Mg. These values imply that the ratio of the angular velocity to the isoscalar giant quadrupole resonance, O./w0, amounts to 0. 09 and 0.3, respectively. The large value of the ratio for the 24Mg nucleus is, of course, due to the rather high angular velocities of nuclear rotation which may occur in light nuclei.15Although the ratio 0./w0 in the present values covers a very wide region, it is to be understood that the most realistic large value of the ratio for the A= 150 region lies somewhere near 0.08. However, when we deal with light nuclei, such as 24Mg, a wider region of the ratio, up to 0.3 or more, should be taken into consideration. The ratio 0./wh, wh being the harmonic oscillator frequency, has also been used elsewhere, 6 •7•1 6 •17 instead of the present ratio 0. I w0, and some speculative values of the ratio have been tentatively interpolated.

Figure l displays theFIG. 1 . 2 FIG. 2 .

 122 Figure l displays the ratio w/w0 for the multipolari ties l = 1 and 2 for nuclei of A = 150 without deforma tion (l> =O). Obviously, the nuclear rotation does not make much sense for spherical nuclei, and thus Fig. 1 is drawn with the intention of showing how the rotation splits the resonance frequencies following values of the azimuthal component and the angular velocity. In this figure, a (a'), b (b'), and c (c') for l = l stand for m =0, l, and -1, respectively, and a (a'), b (b'), c (c'), d (d'), and e (e') for l =2 denote the components of m =0, 1, -1, 2, and -2, respectively. The thin curves represent the effects of the Coriolis force alone, whereas the broad curves show the effects of both the Coriolis and centrifugal forces. It is seen that for small values of the ratio of angular velocity to resonance frequency that is, 0. I w0 S 0. 05-the effect of the Coriolis force dominates, but the centrifugal force becomes prominent for larger values of the ratio. However, as was already discussed, the range of values of the ratio which is larger than 0.1 is a pure theoretical postulate, at least for nuclei of A= 150. In Fig. 1 the ordinate value of w I w0 = 1 cor responds to the unperturbed resonance energies 74.l A -1 1 3 MeV for l = 1 and 66.1 A -113 MeV for l =2.For deformed nuclei, w0 is not a single-valued quanti ty, but is already split into fragments due to the defor mation, as seen in I. A straightforward but hardly feasi-

  nance energies by rotation is very acute only for the fragments corresponding to p 1 and o 1 , while the other components change little and remain almost the same even for very large values of the ratio !l I ro 0 .

Figure 3

 3 Figure 3 shows the result of the same calculation for the isoscalar giant quadrupole resonance. The ordinate value 1 in this figure represents the giant quadrupole res onance of spherical nuclei, which is 66.1 A -113 MeV in the model of nuclear elasticity. The energies 60.6A -1 1 3 , 63.3A -113, 68. 8A -113, and 71.5A -113 MeV signify the components of the resonance energies without rotation for either prolate or oblate deformations. The bold curves are for prolate and dashed curves are for oblate. The letters a, b, c, d and e represent the different frag ments corresponding to m = 2, -2, 1, -1, and 0. The shift of the initial resonance energies by rotation is quite pronounced for the curves a and c, but not for the other fragments. It is interesting to note that there is essen tially no difference between the fragments b and d in ob late nuclei and that for very fast rotation-that is, !l/ro""'0.4, which is fairly far from realistic values-the number of fragments is again reduced to 3, which is the initial multiplet of the resonance energy without rota tion. The giant quadrupole resonance in rotating nuclei has also been discussed 17 by separating the quadrupole oscillation into toroidal, pulsation, and transversely skewed modes in the framework of distorted Fermi surface model. As we observe in the schematic represen tation of splitting of Fig. 1, the split energies of the iso scalar giant resonances in rotating nuclei are not gen erally symmetric with respect to the middle line corre sponding to the resonance energy of spherical nuclei without rotation, contrary to the case of isovector giant resonance splittings (Refs. 5, 8, and 9). It is interesting to ascertain that a recent calculation 18 based on the linear response theory reveals the same features.

4 FIG. 3 .

 43 FIG. 3. Fragmentation of the isoscalar giant quadrupole resonance of rotating prolate and oblate nuclei of A = 150 with 1>=0.2. The unit for both wand !l is as for Fig. 1.

ACKNOWLEDGMENTS

We are grateful to Dr. P. Ring for fruitful discussions. Institut de Physique Nucleaire is an associated laborato ry of CNRS.

APPENDIX

The radial integrals In and Id of Eqs. (3.15) for constant density can be evaluated analytically using the explicit forms of the functions U 1 and Vi of Eqs. (3.9) and (3.10). Thus,


Paul Legentil

Fabrice Leroux
email: fabrice.leroux@uca.fr

Sandrine Therias

Damien Boyer

François Reveret

Geneviève Chadeyron
email: genevieve.chadeyron@sigma-clermont.fr

Reliability study under thermal and photonic stresses of sulforhodamine B (SRB) confined in layered double hydroxide (LDH)

When sulforhodamine B (SRB) is entrapped in a tightly packed hybrid material composed of dodecylsulfate anions interleaved in layered double hydroxide, the organic dye is considered as a possible red-emitting phosphor for white light-emitting diodes (WLEDs). To confirm such promising potential, a reliability study is here undertaken and the materials and their associated silicone films are subjected to different thermal and photonic stresses. Optical properties, photoluminescence quantum yields and emission spectra are recorded after photo-aging studies. Interestingly, the composite silicone film is found to be completely stable under blue LED excitation, while in the absence of the hybrid LDH the emission of SRB decays rapidly, thus underlining the protective role of the LDH hybrid cargo. At this stage, these results confirm the potential offered by a system consisting of such an emitter film and a blue LED. This system also opens up new possibilities for interesting organic dyes that are sensitive to photonic and/or thermal stresses.

Introduction

In lighting applications, white light-emitting diodes (WLED) are now emerging very strongly and constitute a reference in solid-state lighting sources because of their excellent properties, such as high light efficiency, energy-saving properties, long lifetime and absence of toxic mercury or other heavy metals. A commercial WLED consists of a blue chip, emitting between 450 and 480 nm, combined with a yellow-emitting phosphor [START_REF] Nair | A review on the advancements in phosphor-converted light emitting diodes (pc-LEDs): Phosphor synthesis, device fabrication and characterization[END_REF], YAG:Ce 3+ (yttrium aluminium garnet doped with Ce 3+ cations), and a red-emitting phosphor is commonly added to improve the colorimetric characteristics of light [START_REF] Xia | Recent developments in the new inorganic solid-state LED phosphors[END_REF]. A recent study concerning a well-known organic dye, sulforhodamine B (SRB), underlined its interest as a red phosphor and its potential under commercial blue LED excitation when highly dispersed in a host structure, layered double hydroxide (LDH) [START_REF] Legentil | Sulforhodamine B-LDH composite as a rare-earth-free red-emitting phosphor for LED lighting[END_REF]. Indeed, the inorganic LDH vessel intercalated with surfactant molecules (dodecylsulfate (DS)) is found to display a suitable arrangement for SRB to be well ensconced, avoiding intermolecular interactions deleterious for the red emission as well as providing an affinity with silicone in the preparation of homogeneous films. The properties of an optimized system composed of alternating a blue chip, a YAG: Ce 3+ /silicone composite film and a LDH-DS-SRB/silicone composite film are satisfactorily met, with a correlated color temperature (T(K)) associated with a color rendering index (CRI) suitable for a commercial WLED application.

LEDs are known to have an operating temperature of up to 100 or 120°C, and hence phosphors must have the ability to sustain such harsh conditions without any degradation Several studies have been undertaken on inorganic phosphors. Da Lago et al. [START_REF] Dal Lago | Phosphors for LED-based light sources: Thermal properties and reliability issues[END_REF] published one of the first reliability studies concerning a commercial YAG: Ce phosphor under thermal stresses (between 85 and 145 °C) for remote-phosphor technology. Thus, it was demonstrated that the heat generated can significantly limit the lifetime as well as the performance of the phosphor. We can also mention the works of Shao et al [START_REF] Shao | Temperature-dependent photoluminescence properties of (Ba,Sr)2SiO4:Eu2+ phosphors for white LEDs applications[END_REF], which have shown the modification of the color coordinates and the spectral shift of BaSiO 4 : Eu 2+ emission (a greenemitting phosphor under blue LED excitation) with respect to temperature. In order to provide better thermal stability, they suggested substituting the Ba sites by Sr atoms in this phosphor, and in this way they succeeded in lowering thermal quenching. Since then, other works have reported the limited stability of phosphors formulated with rare earth elements under thermal stresses [START_REF] Yazdan Mehr | Accelerated life time testing and optical degradation of remote phosphor plates[END_REF][START_REF] Zhou | Thermal influence of phosphor to GaNbased white LEDs[END_REF].

However, the reliability of organic phosphors has scarcely been reported so far. Indeed, the use of organic dyes for WLED applications is appealing in terms of their luminescence properties but the stability studies are only superficial [START_REF] Boonsin | Development of rare-earth-free phosphors for eco-energy lighting based LEDs[END_REF][START_REF] Kajjam | Triphenylamine based yellowish-orange light emitting organic dyes (donor-π-acceptor) for hybrid WLEDs and OLEDs: synthesis, characterization and theoretical study[END_REF]. As mentioned before, to further address a practical application in commercial lighting devices, a major specification required of the fluorophore and its associated system is to present long-term stability with respect to various types of stress in real operating conditions. Several published works mentioned the use of fluorophore with a potential application for replacing traditional phosphors with rare earth elements in commercial WLED, such as fluorescein, pyranine, rhodamine or triphenylamine derivatives for instance [START_REF] Das | Fluorescein isothiocyanate and rhodamine B dye encapsulated mesoporous SiO2 for applications of blue LED excited white LED[END_REF][START_REF] Legentil | Revisiting fluorescein and layered double hydroxide using a synergistic approach: A complete optical study[END_REF][START_REF] Nyalosaso | Towards rare-earth-free white light-emitting diode devices based on the combination of dicyanomethylene and pyranine as organic dyes supported on zinc single-layered hydroxide[END_REF][START_REF] Zhang | A triphenylamine derivative as an efficient organic light color-conversion material for white LEDs[END_REF]. However, the authors do not usually pay attention to the reliability of the dyes in stressful conditions.

Traditionally, LDH materials find applications as precursors in catalysis and as scavengers in environmental science, but are also finding renewed interest in the domain of energy [START_REF] Wu | A review on fabricating heterostructures from layered double hydroxides for enhanced photocatalytic activities[END_REF][START_REF] Yin | CoNi-layered double hydroxide array on graphene-based fiber as a new electrode material for microsupercapacitor[END_REF] as well as in biology as vessels for drug delivery [START_REF] Conterosito | High-Throughput Preparation of New Photoactive Nanocomposites[END_REF][START_REF] Lonkar | Nanohybrids of phenolic antioxidant intercalated into MgAl-layered double hydroxide clay[END_REF][START_REF] Yan | Sulforhodamine B Intercalated Layered Double Hydroxide Thin Film with Polarized Photoluminescence[END_REF]. Their general chemical formula is as follows

[M(II) 1-x M(III) x (OH) 2 ] x+ [A n-] x/n •mH 2 O
where M(II) is a divalent cation such as Zn 2+ , Mg 2+ , Fe 2+ , etc… and M(III) a trivalent cation such as Al 3+ , Fe 3+ , etc. and A n-is an interleaved anion. The anisotropic twodimensional structure is built from edge-sharing octahedral. Compared to brucite Mg(OH) 2 , the partial substitution of divalent cations by trivalent cations implies for LDH sheets a net positive charge. To ensure electroneutrality, anions of interest, here the dye molecules together with the surfactant molecules, are located in the interlayer space of the LDH with the presence of water molecules.

Importantly, organic dyes are known to be sensitive to different stresses (thermal, photonic…) [START_REF] Lamouche | Optical properties of dye molecules as a function of the surrounding dielectric medium[END_REF][START_REF] Sultana | Role of ammonium ion on the aggregation and adsorption properties of sodium dodecylsulfate[END_REF]. To overcome such issues, an elegant solution consists in immobilizing the SRB molecules in an inorganic host lattice like layered double hydroxide (LDH) as previously reported [START_REF] Legentil | Sulforhodamine B-LDH composite as a rare-earth-free red-emitting phosphor for LED lighting[END_REF]. Pioneer research has underlined the benefit of confining photoactive species to a restricted space such as a two-dimensional host in terms of 1) their strong adsorption through host-guest interaction to avoid possible migration and aggregation within a polymer, and 2) their dispersion with the presence of co-adsorbates to avoid as far as possible guestguest interactions [START_REF] Ogawa | Photofunctions of Intercalation Compounds[END_REF]. However, a reliability study of such a hybrid organic dye/LDH host matrix for LED applications must be conducted before considering any practical application; this aspect has never been published to the best of our knowledge.

In the following, a reliability study of the SRB dye in the presence of thermal and photonic stresses in different conditions when placed in a LED set-up is presented in order to confirm the promising features of the synthetized hybrid LDH-SRB. To address this issue, comparative studies were performed, first on the hybrid LDH/SRB powder and then on a composite film by dispersing the powder in silicone polymer. In the first case only temperature effects were studied. Then, both thermal and photonic stresses were concomitantly applied and recorded for silicone films. Behaviour in accelerated photo-aging was also observed. Photoluminescence properties such as emission spectra and photoluminescence quantum yields were scrutinized during all the different tests to better understand the mechanisms involved in the loss of optical properties.

Experimental section

Materials

Sulforhodamine B sodium salt C 27 H 30 N 2 NaO 7 S 2 , sodium dodecylsulfate CH 3 (CH 2 ) 11 OSO 3 Na and terephthalic acid C 8 H 6 O 4 were purchased from Sigma-Aldrich, and YAG:Ce 3+ from Phosphortech. Zn(NO 3 ) 2 .6H 2 O (purity 99.9+%), Al(NO 3 ) 3 .9H 2 O (purity 99.9+%) and NaOH (97%) were obtained from Sigma Aldrich. The two-component silicone elastomer, Bluesil RTV 141 part A and part B, was supplied by Elkem.

Synthesis procedure for LDH hybrid materials using the coprecipitation method

The sulforhodamine B (SRB)-dodecylsulfate (DS) LDH phase, Zn 2 Al-DS-SRB, called LDH-DS-SRB, was prepared using the coprecipitation method. The synthesis of [Zn 2 Al 1 (OH) 6 ] + [DS -] 1-x [SRB -] x •mH 2 O was performed using 50 mL of sulforhodamine B and dodecylsulfate aqueous solution (with de-ionized water). 50 mL of an aqueous solution of Zn 2+ (3.2 mmol) and Al +3 (1.6 mmol) was added dropwise over a period of 3 hours under magnetic stirring. To synthesize the [Zn 2 Al 1 (OH) 6 ] + [DS - ] 0.9995 [SRB -] 0.0005 •mH 2 O sample, 0.8x10 -3 mmol and 1.5992 mmol were used for SRB salt and sodium dodecylsulfate, respectively. This formulation was determined in a previous article as that leading to the best optical performance under commercial blue LED and UV excitation [START_REF] Legentil | Sulforhodamine B-LDH composite as a rare-earth-free red-emitting phosphor for LED lighting[END_REF].

The pH was maintained at 8.5 by adding 0.25 M NaOH during the synthesis process. Coprecipitation was performed under nitrogen at 20 °C. The mixture was centrifuged at 5,000 rpm for 5 minutes and the slurry on the bottom of the flask was then washed several times with deionized water until a clear and transparent supernatant was obtained. A paste was recovered and dried overnight at room temperature to obtain the LDH-DS-SRB powder.

The post-synthesis hydrothermal treatment was performed by dispersing the paste, obtained before the drying step, in 20 ml of de-ionized water in a sealed container at 110 °C under autogenous pressure during 48 hours. The slurry was then centrifuged (5 minutes at 5,000 rpm) and the paste was dried overnight at room temperature to obtain the LDH-DS-SRB-TH powder.

Similarly, the terephthalate LDH phase, Zn 2 Al-TP-SRB, was prepared by following the method previously described for LDH-DS-SRB by replacing DS by terephthalic acid (TP). However, TP being dianionic, the input of TP was half the molar concentration compared to DS.

Elaboration of Silicone/HDL-DS-SRB composite films

The LDH-DS-SRB powder was used to produce the silicone/hybrid composite material. A loading rate of 40 wt. % was chosen (beyond this loading rate, it was difficult to obtain a homogeneous distribution of the powder in the polymer matrix, as shown in figure S1). The two-component silicone elastomer (silicone Bluesil-RTV 141 A&B) was composed of a viscous liquid, called part A, cured by a polyaddition reaction with a catalyser, part B.

The silicone/hybrid composite material was manufactured by mixing the LDH powder with part A of the silicone elastomer using a mechanical mixer ("Thinky Mixer") for 10 min at 1200 rpm. Then the obtained mixture was processed through a three-roll Exakt80E (spacing of 30 m between the first two rolls and 50 m between the last two rolls) to achieve a better dispersion of the LDH platelets by shearing and to obtain a homogeneous hybrid component. Part B was added at 10phr and further homogenised using the mechanical mixer for 10 min at 1200 rpm.

The silicone/hybrid composite film (Si-LDH) was prepared by casting onto a Teflon surface using an Elcometer 4340 automatic film applicator. The knife blade height was set at 200 µm and the casting speed was 30 mm/s. This two-component silicone film was cured at 80 °C for 2 hours.

A reference film composed of SRB powder dispersed in the silicone, called Si-SRB, was also performed by dispersing 100 μL of an SRB solution in ethylene glycol (10 g.L -1 ) in 4.45 g of part A of the silicone elastomer using the "Thinky Mixer" mechanical mixer for 10 min at 1200 rpm. Then 0.45 g of part B was added and homogenised with the mechanical mixer for 10 min at 1200 rpm. The film was prepared by casting onto a Teflon surface in the same way as for Si-LDH mentioned above.

Film thickness was measured using an Elcometer 456 coating thickness gauge.

Characterisation

X-ray diffraction

LDH powders were characterised by X-Ray Diffraction; the XRD patterns were recorded with a Philips X-Pert Pro diffractometer operating with Cu-Kα radiation (λ = 1.5418 Å). The data were collected in a 2θ range between 5° and 70° with a step size of 1°/min.

Thermogravimetric analysis

Thermogravimetric (TG) analyses were performed on a Setaram TGA 92 instrument with a linear heating rate of 5 °C.min -1 under air.

UV-Visible absorption

The UV-visible absorption spectra of the samples were recorded in a wavelength range of 200 to 800 nm with a UV-vis spectrophotometer (SP-3000 Plus) equipped with an integrating sphere and UV-Probe software.

Luminescence

Quantum yield efficiencies and emission spectra were measured using a C9920-02G PL-QY integrating sphere measurement system from Hamamatsu Photonics. The setup consisted of a 150 W monochromatized Xe lamp, an integrating sphere (Spectralon coating, ⌀= 3.3 in.) and a highsensitivity CCD camera.

Emission stability of films under LED irradiation

A photo-aging study was performed in a SEPAP 12/24 unit. This system was designed to achieve accelerated artificial weathering conditions related to natural ageing. The composite films were placed on a rotating carousel positioned in the centre, and four polychromatic mercury lamps (400 W) with wavelengths higher than 300 nm (90 W.m -2 ) were installed at the corners of the chamber. The temperature was set at 60 °C.

Reliability studies were carried out using a home-made setup consisting of a power-controlled blue LED emitting at 460 nm as the excitation source and a HR4000 high resolution spectrometer (Ocean Optics) as the PL analyser. The samples were positioned on a heating element whose temperature was adjusted to 80°C. The emission spectra of the composite films were acquired every 20 min for 24 hours. Their area was integrated to obtain the total emission intensity. LED power was measured using a Scientech Model Mentor MA 10 with a MC2501 calorimetric head unit (25.4 mm aperture). The measurement was performed by centring the head unit over the LED source and measuring the power of the LED light emitted through the aperture. The blue LED power was 183 mW. The LED power density can be expressed in W/m 2 and was calculated by LED power (in watt) per unit surface of the sample (0.25 cm 2 ). The power density of the blue LED was 9300 W/m 2 ; for our experiments, a filter was used in order to reduce this value to 2480 W/m 2 .

Results and discussion

Hydrothermal post-treatment

Structural effect

X-ray diffractogram patterns of both LDH-DS-SRB and LDH-DS-SRB-HT powders before and after hydrothermal treatment respectively are plotted in Fig. 1. The two patterns closely overlap, and the associated diffraction peaks are located at similar angles in 2θ. The observed interlamellar space is 2.58 nm for both powders. However, thinner diffraction peaks are observed for LDH-DS-SRB-HT than for LDH-DS-SRB powder, in particular the peaks with l Miller peak indices, underlining the fact that the hydrothermal treatment induced more ordered platelet stacking. The average size of crystallites can be calculated using the Scherrer equation L = kλ/(B.cosθ) where B is the FWHM (Full Width at Half Maximum) of diffraction peak (003), λ is the Cu K α radiation wavelength in nm, θ is the diffraction angle of (003) in radian et k is the shape factor (k = 0,9) [START_REF] Cullity | Elements of X-ray Diffraction[END_REF]. The sizes calculated for LDH-DS-SRB and LDH-DS-SRB-HT are 27.2 nm and 42.3 nm respectively, these values corresponding roughly to 11 and 16 structurally coherent stacked layers, thus confirming a better crystallisation in the stacking direction for LDH-DS-SRB-HT.

As shown in a previous paper [START_REF] Legentil | Sulforhodamine B-LDH composite as a rare-earth-free red-emitting phosphor for LED lighting[END_REF] and illustrated on the scheme of the Fig. 1(right), the sulfonate groups are in the vicinity of the hydroxyl-covered LDH layers, thus leaving the benzene and xanthene rings close to the surfactant alkyl chains. The relative amount of SRB per DS is very small, SRB is highly diluted in the LDH-DS environment.

Optical properties

Internal photoluminescence quantum yields (PL QY int ) were recorded from the samples before and after hydrothermal treatment to know whether a change in the ordered stacking may affect the optical response of the interleaved dye molecules (Table 1). Rather counter-intuitively, LDH-DS-SRB-HT powder exhibits lower PL QY int than LDH-DS-SRB. Even if the hydrothermal treatment leaves intact the confinement of the SRB molecules inside the host structure (similar basal spacing is observed), a significant decrease in the luminescence properties is observed for the three excitation wavelengths of interest (570 nm is the excitation wavelength leading to the best PL QY ab and int, [START_REF] Legentil | Sulforhodamine B-LDH composite as a rare-earth-free red-emitting phosphor for LED lighting[END_REF] (365 nm and 480 nm wavelengths correspond to UV and commercial blue LED, respectively).

Table 1. PL QY int of LDH-DS-SRB and LDH-DS-SRB-HT powder for three excitation wavelengths (570 nm, 480 nm and 365 nm). Three wavelengths are of interest: the 365 nm and 480 nm wavelengths correspond to UV and commercial blue LED respectively, while 570 nm is the excitation wavelength leading to the best PL QY ab .

Excitation wavelength (nm)(((nm)nm) LDH-DS-SRB LDH-DS-SRB-TH

PL QY int (%) 570 nm 73.1 ± 3.5 52.6 ± 2.5 480 nm 68.9 ± 3.5 42.8 ± 2.1 365 nm 73.1 ± 3.7 39.0 ± 2.0

We can note that the PL QY ab recorded for these samples before and after hydrothermal treatment (which corresponds to the ratio between the number of photons emitted by the phosphor and the number of photons emitted by the excitation source) leads to the same conclusion (Table S2).

To better understand the modification undergone by the dye molecule, the emission spectra under blue excitation (480 nm) were recorded for both LDH-DS-SRB and LDH-DS-SRB-HT powders (Fig. 2). The emission spectra being similar for the other two excitation wavelengths, they are not presented here. The maximum intensity of the emission band is shifted towards higher wavelengths (from 608 to 614 nm) after the hydrothermal treatment, thus indicating a modification of the environment for the SRB molecules. In detail, the intensity ratio between the main emission peak and the associated shoulder is also modified, with the values of I shoulder1 / I peak1 = 0.58 and I shoulder2 / I peak2 = 0.67 before and after the hydrothermal treatment, respectively. Since the main peak is attributed to SRB monomer molecules and the shoulder peak to SRB aggregates (dimer and trimer) [START_REF] Ray | Adsorption of Sulforhodamine Dyes in Cationic Langmuir-Blodgett Films: Spectroscopic and Structural Studies[END_REF][START_REF] Yan | Sulforhodamine B Intercalated Layered Double Hydroxide Thin Film with Polarized Photoluminescence[END_REF], the hydrothermal treatment leads to a greater aggregation of the SRB molecules into the LDH-DS hybrid material. From these observations, the LDH-DS-SRB sample without hydrothermal treatment is selected in the following.

Effect of thermal stress

Powder LDS-DS-SRB

Thermogravimetric analyses (TGA)

Thermogravimetric analysis shows that the SRB molecule is stable up to about 220 °C (Fig. 3a). Below 100 °C, the mass loss can be attributed to the departure of the weakly-adsorbed water molecules (weight loss n°1 of 6,8% around 80-100 °C). After 220 °C, decomposition of the SRB 250 molecules (weight loss n°2, 3 and 4) occurs until the final mass loss observed at 500 °C. 251 For the hybrid LDH-DS-SRB material, several mass losses are observed. After the water 257 molecules that are weakly bound to the surface of the HDL platelets have been removed (weight loss 258 1), deshydroxylation of the LDH layers occurs from 180 up to 300 °C, concomitantly with the 259 decomposition of the interleaved organic molecules (weight losses 2 and 3). This is consistent with 260 DS decomposition, well documented in the literature [START_REF] Degoutin | Organosilica Mesoporous Materials with Double Functionality: Amino Groups and β-Cyclodextrin Synthesis and Properties[END_REF][START_REF] El-Kharrag | Synthesis and Characterization of Mesoporous Sodium Dodecyl Sulfate-Coated Magnetite Nanoparticles[END_REF]. 261

The associated mass loss starts at 200 °C and continues up to 400 °C. The mass loss observed at 500 262 °C is attributed to the final decomposition of the organic residue, while the mass loss at around 740 263 °C is assigned to the decomposition of ZnSO 4 formed during the thermal treatment with the collapse 264 of the LDH layers reacting with the functionalized head of the surfactant (as well as possibly SRB). 265

Thermal stress in oven (120 °C) 266

The thermal stability study was completed in static conditions by placing the LDH-DS-SRB in an 267 oven at 120 °C for 24 hours. X-ray diffraction patterns were recorded after 5 and 24 hours (Fig. 4). 268 After both exposure times at 120 °C, the LDH-DS-SRB powder exhibits a similar XRD pattern. As the peak at 61 ° in 2θ assigned to the diffraction peak (110) merged with (113) is still observed, this is indicative that the LDH layer structure is maintained. However, the harmonic peaks (00l), visible at angles lower than 25 ° (2θ) attributed to the strongly pronounced stacking for LDH-DS-SRB disappear after the thermal treatment at 120 °C. Only a few peaks of relatively low intensity are observed; the first peak located at 2 = 7.5 ° corresponds to a basal spacing of 11.8 nm. Unexpectedly at this low temperature, this strong modification can be explained by the decomposition of the interleaved organic DS molecules. Even if the decomposition of DS occurs at slightly higher temperatures under dynamic mode (TGA), the surfactant is found to be thermally unstable in static conditions. A basal spacing of 11 to 12 nm for the degraded hybrid phase seems to indicate the presence of interleaved sulfate anions [START_REF] Sotiles | Converting Mn/Al layered double hydroxide anion exchangers into cation exchangers by topotactic reactions using alkali metal sulfate solutions[END_REF] in association with a significant loss of coherent stacking, which can be further interpreted by a process of exfoliation.

Variations in internal photoluminescence quantum yields (PL QY int ) as a function of the excitation wavelengths are shown for the samples treated 5 and 24 h at 120°C and for pristine LDH-DS-SRB powder in Fig. 5. The PL QY int slightly decreases after the thermal treatment (for both exposure times) while the SRB molecule is not expected to decompose in this temperature range (Fig. 3 a). The partial loss of the stacked structure reduces the optical properties, most probably by changing the local environment of the SRB molecules. Indeed, initially stabilised by the LDH host structure as well as well dispersed and ensconced within the space supplied by the interleaved DS, the modification may cause the SRB molecules to move closer to each other. This could explain the decrease in the photoluminescence properties due to the non-radiative de-excitations.

Emission spectra of the LDH-DS-SRB powder before and after thermal treatment are compared (Fig. 6). In both cases, a wide and asymmetric emission band is observed. The maximum intensity is strongly shifted (by about 15 nm) after treatment at 120 °C. Moreover, emission intensity is found to decrease and the contribution associated with the SRB aggregates increases. At this temperature the loss of the lamellar structure, as explained before, leads to significant interactions between the SRB molecules, which are close to each other and are prone to aggregation by π-stacking (parallel stacking) due to conjugated xanthene rings or by hydrogen bonding (head-to-tail). Xanthene dyes such as SRB or fluorescein are well known to interact mutually, resulting in luminescence quenching [START_REF] De | Spectroscopic studies with fluorescein dye-Protonation, aggregation and interaction with nanoparticles[END_REF]. The LDH-DS-SRB powders treated at 120 °C for 5 and 24 hours are shown in Fig. 7 under daylight and UV excitation (365 nm). The heat-treated powder exposed for 24 hours appears slightly darker under daylight (Fig. 7 c) than the initial powder or the sample treated for 5 hours (Fig. 7 a) and b), respectively). Under UV radiation (Fig. 7 d), e) and f), the three powders are photoluminescent with no real distinction in terms of emission intensity. Replacing dodecylsulfate (DS) by terephthalate anions (TP)

With the aim of providing more stability, a study was performed by replacing the monofunctional surfactant DS by terephthalate anions (TP). Pioneering work showed the pillaring effect of such a di-anion form [START_REF] Drezdzon | Synthesis of isopolymetalate-pillared hydrotalcite via organic-anion-pillared precursors[END_REF], which should be of interest here in stabilizing the host structure. However, the amount of interleaved TP was half the quantity of DS, while the amount of SRB was not modified. The LDH TP SRB sample was treated at 120 °C in an oven for 24 hours. The XRD patterns before and after this thermal treatment are presented in Fig. 8 a). Before thermal treatment, the powder is well crystallized, leading to a basal spacing of 1.41 nm (diffraction peak (003) at 6.3 ° in 2θ), in agreement with the literature [START_REF] Drezdzon | Synthesis of isopolymetalate-pillared hydrotalcite via organic-anion-pillared precursors[END_REF]. As before in the case of DS, the presence of SRB molecules cannot be detected, due to the very small amount used. After treatment, the pattern is strongly modified but the diffraction peak (003) at 6.3 ° in 2θ is still observed, meaning that the lamellar structure, even if largely degraded, is still somehow present. As expected, the 2D host structure is more robust under thermal stress thanks to a pillaring effect supplied by the TP molecules. Indeed, bi-tethered TP molecules allow an increase in the cohesion strength between layers. PL QY int was recorded before and after the thermal treatment for excitation wavelengths between 300 and 600 nm (Fig. 8 b). Before thermal treatment, the optical performances of the sample LDH-TP-SRB are generally weaker than for LDH-DS-SRB (68.1 % and 73.8 % at 570 nm respectively, for example). Moreover, photoluminescence is also thermally more affected, and the PL QY int of LDH-TP-SRB decreases significantly after the treatment at 120°C; this is even more pronounced below 450 nm. At first glance, the structural cohesion supplied by TP molecules does not help to stabilize the photoluminescence properties more than in the case of the DS spacer. This is tentatively interpreted by the fact that the interlayer molecular packing is halved when replacing DS by TP, leaving the SRB molecules more space and more potential to diffuse freely, possibly causing their intermolecular interaction and thus optical quenching. For this reason, the focus is on LDH DS SRB in the following.

HTK-XRD analysis: thermal stability up to 220 °C

To unravel more precisely the structural changes occurring in the temperature domain between 25 and 220 °C, the LDH-DS-SRB powder pattern was recorded with a progressive temperature rise. The series of stacked XRD patterns recorded for each 20 °C step is shown in Fig. 9. Cooling, at 15 °C/min until reaching 25 °C, is followed by a 1-hour stabilisation period; then a final acquisition for another hour is recorded. The three intense peaks at 40°, 46° and 65 ° in 2θ are due to the platinum holder.

Between 25 and 60 °C, the XRD patterns are superimposed, then the intensity of the diffraction peaks (00l) decreases with the temperature above 80°C, underlining that the LDH matrix is gradually losing its crystallinity in the stacking direction. Above 160°C, the diffraction peaks (00l) observed below 30 ° in 2θ disappear, thus showing the absence of any stacking. However, the LDH sheets remain intact, with the presence of diffraction peaks at 35 and 60° in 2θ. This study confirms that the layer structure of the LDH hybrid material is lost with the decomposition of the surfactant molecules, while the LDH sheets remain after treatment at 220°C. Photographs of LDH-DS-SRB powder under daylight and UV excitation at 365 nm before and after HTK-XRD are presented in Fig. 10. In daylight, the powder treated at 220 °C (b) is still pink, but slightly less colourful than the initial powder (a). Under UV radiation (365 nm), photoluminescence is also still observed (d) but the emission is much weaker than initially (c). Moreover, the colour emitted by the powder excited under UV radiation is slightly different, as shown in Fig. 11. Indeed, the thermal treatment has an effect on the trichromatic coordinates, with a pronounced shift in the XY CIE diagram. The effect of the thermal treatment on the UV-vis absorption spectra of LDH-DS-SRB is shown in Fig. 12 a. The main absorption band of the SRB molecule, centred at 570 nm, is still observed after the thermal treatment at 220°C. However, the overall absorption is found after such heat treatment to increase significantly in the 400-600 nm range. This could be explained by the formation of absorbent residual product from the decomposition of the DS molecules from 120 to 150°C (Fig. 3 &4). This is confirmed by the UV-vis absorption spectra of the LDH-DS powder (Fig. 12 b). The LDH-DS powder initially presents a white colour, whereas after heat treatment at 220°C it is black, which is consistent with the formation of carbonaceous products. The PL QY int of the LDH-DS-SRB powders before and after thermal treatment at 220°C are reported in Table 2 for λ exc = 480 and 365 nm. The drop in optical performance is clearly shown by these PL QY int values. λ exc = 480 nm 68.9 ± 3.5 15.0 ± 0.8 λ exc = 365 nm 73.1 ± 3.7 4.3 ± 0.2

Silicone/LDH-DS-SRB composite film (Si-LDS) behaviour under thermal stress

The thermal stability of the Si-LDS composite film with the optimal formulation was further studied at 120 °C for 24 h.

X-ray diffraction

XRD patterns of the Si-LDS composite film before and after the thermal treatment are presented in Fig. 13 and were compared to the pattern of the LDH-DS-SRB reference powder. First, the LDH matrix relative diffraction peaks are observed on the composite film pattern. After the thermal treatment the harmonic peaks (00l) disappear, indicating that the dispersion of the LDH-DS-SRB powder in silicone does not improve its thermal stability in terms of structural 2D integrity.

Optical properties

The PL QY int of the composite film were recorded as a function of the excitation wavelengths before and after the thermal treatment in an oven at 120 °C for 24 hours, and are shown for each sample in Fig. 14 a. After the thermal treatment, the PL QY int decreases significantly in the studied spectral range of 300-600 nm.

The emission spectrum of the composite film is shifted after the thermal treatment towards higher wavelengths (Fig. 14 b). This is the result of the aggregation of SRB molecules, as already observed with the LDH-DS-SRB reference powder (Fig. 6). The UV-Vis emission spectra also support these findings, with a significant broadening and an increase in absorption over the 550-350 nm range for the sample heat-treated at 120°C. Finally, for this thermal treatment, the PL QYint of the Si-LDS composite film and of the LDH-DS-SRB powder (Fig. 5), exhibit the same change in their spectral features. However, the PL QY int values recorded for the film are lower than for powder. This is in part explained by the dilution effect provided by the silicone matrix.

A film composed exclusively of silicone was also produced; it underwent the same 24-hour heat treatment at 120°C as the Si-LDS film. Recording the UV-Vis and infrared spectra led to the conclusion that the silicone does not undergo any modification after this treatment. The decrease in optical performance after the heat treatment at 120°C is thus explained, as for the powder, by an aggregation phenomenon generated by temperature.

Composite film: durability under photonic and thermal stresses

Accelerated photo-aging in a SEPAP 12/24 unit

The photo-aging of the Si-LDS composite film was studied using a SEPAP 12/24 unit. In order to have a reference, a second Si-SRB composite film was produced by dispersing 0.05 %wt. pure SRB in the silicone matrix. Thus, the amounts of SRB in both films are comparable. Fig. 15 plots PL QY int as a function of irradiation time for three excitation wavelengths of interest (365, 480 and 570 nm) for Si-LDS composite film. Only the variation in the PL QY int for excitation at 570 nm is provided for the S-SRB film because its luminescence is initially very low or even nonexistent for excitations at 365 and 470 nm. At 570 nm, after only 3 hours, the Si-SRB film has entirely lost its luminescence, whereas the luminescence of the Si-LDS film remains identical to that observed initially, showing the benefit of the inorganic vessel for stabilizing the SRB optical properties. Quantitatively, after 35 hours the PL QY int of the Si-LDS film is still above 30 %. For excitation at 365 and 480 nm, the kinetic of luminescence extinction is close to that at 570 nm. In order to have a better understanding of these observations, UV-vis spectra were recorded after photo-aging. The asymmetric band relative to SRB molecules is observed for both composites before the irradiation process starts. Maximum absorbance is the same for both composites, which confirms the similar quantity of SRB molecules in both composites. However, the maximum is shifted by 20 nm between the two, and can safely be explained by a higher amount of SRB aggregates in Si-SRB than in Si-LDS. During the irradiation process, the main absorbance band decreases for both composites (Fig. 16 a). The SRB molecules are irreversibly damaged by irradiation in the SEPAP 12/24 unit. However, the degradation kinetic is considerably faster for the Si-SRB film (Fig. 16 b) than for Si-LDS. Indeed, absorbance is almost extinguished after only 4 hours for the Si-SRB film, whereas a value still subsists after 34 hours for Si-LDS. This observation matches the decrease in the PL QY int described above (Fig. 15).

In conclusion, the LDH host matrix makes it possible to elaborate a homogeneous composite luminescent film, but also to significantly increase SRB stability under stresses (thermal and photonic). The conditions in the SEPAP 12/24 unit were very drastic ( > 300 nm and T = 60 °C) and are not well representative of the photo-aging of composites associated with commercial blue LEDs. Nevertheless, these experiments provide information on the stability of the Si-LDS film under UV LED excitation. In the following section, the behaviour of these composites is studied in direct association with a commercial blue LED.

Aging under blue LED

In order to assess the stability of the Si-LDS composite film in operating conditions close to those encountered in LED devices, the Si-LDS film and the Si-SRB composite film were irradiated by a powerful blue LED (460 nm, power: 2480 W/m 2 ) for 1400 minutes at 80°C in a black chamber. A spectrophotometer recorded the emission spectra every 20 minutes for 24 hours.

The variation in the integrated areas of the emission spectra as a function of irradiation time is illustrated in Fig. 17.

The Si-LDS composite film, using the LDH hosting the SRB molecules, is not affected at all by the stresses, and photoluminescence intensity is stable. On the contrary, the intensity of Si-SRB decreases significantly with irradiation time. A loss of 32 % was observed at the end of the first 24 hours. The beneficial effect of the LDH matrix is thus clearly demonstrated in such real conditions of use, and confirms that this Si-LDS film is fully compatible with a blue LED device. Photographs of both composite films, Si-SRB and Si-SDS, are presented in Fig. 18 before and after the 24 h irradiation process. The colour fading of Si-SRB reveals the degradation of the SRB molecules. The aspect of Si-LDS is not modified by photo-aging; the role of LDH is well highlighted here again.

Conclusions

This study emphasises the ability of a LDH host structure to protect the SRB luminescent dye against various stresses. The LDH interior, filled with spacer, impedes SRB intermolecular interaction. This effect is found to be more pronounced for tightly-packed interlayer space obtained from a mono-functional co-intercalate such as DS than from TP. In the latter case, SRB molecules are freer to diffuse and thus to quench the optical properties. It is pointed out that the hybrid LDH-DS-SRB and the associated composite film exhibit a suitable orange -red emission under blue LED excitation.

The progressive loss in the optical properties of the LDH-DS-SRB hybrid powder after a treatment at 120 °C is explained by a breakdown of the LDH stacking structure, as confirmed by XRD analysis in temperature. This structural disruptive effect modifies the environment of the SRB molecules, most probably provoking their partial aggregations and thus leading to non-radiative energy transfers. Satisfactorily, a significant photoluminescence quantum yield is still observed after the thermal treatment.

Similarly, the LDH-DS-SRB/silicone composite film exhibits luminescence after thermal exposure, and this is also the case after being exposed to severe photo-aging conditions. Interestingly, the emission of the composite film when irradiated by a commercial blue LED at 80 °C to reproduce "real conditions of use" is completely stable after 24 hours. These results highlight the stability provided by the LDH host structure for dyes, and the potential of LDH-DS-SRB for commercial lighting applications based on a blue LED.

Figure 1

 1 Figure 1. X-ray patterns of LDH-DS-SRB and LDH-DS-SRB-HT powders and on the right an idealized scheme of the LDH-DS-SRB.

Figure 2 .

 2 Figure 2. Emission spectra of LDH-DS-SRB and LDH-DS-SRB-HT for λ exc = 480 nm

Figure 3 .

 3 Figure 3. Thermogravimetric analysis and heat flow of a) SRB (2 °C/min, 25 °C-800 °C, under air) and 255 b) LDH-DS-SRB powder (2 °C/min, 25 °C-1000 °C, under air). 256

Figure 4 .

 4 Figure 4. X-ray diffraction patterns of LDH-DS-SRB powder before after thermal treatment (5 and 24 hours)

Figure 5 .

 5 Figure 5. Variation in PL QY int as a function of the excitation wavelength for LDH-DS-SRB powders, before and after thermal treatment at 120 °C.

Figure 6 .

 6 Figure 6. Emission spectra of the LDH-DS-SRB powder before and after thermal treatment at 120 °C for 5 h and 24 h (λ exc = 480 nm).

Figure 7 .

 7 Figure 7. Initial LDH-DS-SRB powder and powders treated at 120 °C for 5 and 24 hours under daylight (a), b) and c), respectively) and under UV radiation (365 nm) (d), e) and f), respectively).

Figure 8

 8 Figure 8. a) XRD patterns of LDH-TP-SRB powder and b) PL QY int before and after thermal treatment at 120 °C for 24h.

Figure 9 .

 9 Figure 9. XRD patterns of LDH-DS-SRB powder as a function of temperature (HTK-XRD): temperature elevation of 5 °C/min followed by 15 minutes of temperature stabilisation then 1 hour of acquisition.Cooling, at 15 °C/min until reaching 25 °C, is followed by a 1-hour stabilisation period; then a final acquisition for another hour is recorded. The three intense peaks at 40°, 46° and 65 ° in 2θ are due to the platinum holder.

Figure 10 .

 10 Figure 10. LDH-DS-SRB powder in daylight before (a) and after (b) HTK-XRD analysis (220 °C then cooled to 25 °C) and under UV radiation at 365 nm before (c) and after (d) this analysis.

Figure 11 .

 11 Figure 11. XY CIE 1931 diagram representing the trichromatic coordinates of the LDH-DS-SRB powder before and after HTK analysis.

Figure 12 .

 12 Figure 12. UV-visible spectra of a) LDH-DS-SRB and b) LDH-DS powders before and after HTK-XRD analysis.

Table 2 .

 2 PL QY int for λ exc = 480 nm and 365 nm of LDH-DS-SRB powder before and after the HTK-XRD analysis (after cooling to 25 °C)

Figure 13 .

 13 Figure13. XRD patterns of the Si-LDS composite film before and after thermal treatment as well as the pattern of the LDH-DS-SRB powder

Figure 14 .

 14 Figure 14. a) Variation in PL QY int as a function of the excitation wavelength for the Si-LDS composite film before and after thermal treatment. b) Emission spectra ( exc = 480 nm), and UV-visible spectra (inset) of the composite film before and after thermal treatment.

Figure 15 .

 15 Figure 15. Variation in PL QY int as a function of irradiation time in the SEPAP 12/24 unit for both Si-SRB and Si-LDS composite films for several excitation wavelengths (365, 480 and 570 nm).

Figure 16 .

 16 Figure 16. Variation in UV-visible spectra (a) and maximum absorbance (b) for Si-SRB and Si-LDS composites as a function of irradiation time.

Figure 17 .

 17 Figure 17. Variation in normalized photoluminescence intensity with respect to irradiation time for the Si-LDS and Si-SRB composite films heated to 80 °C and irradiated by a commercial blue LED (460 nm, 2480 W.m -2 ).

Figure 18 .

 18 Figure18. Photographs of Si-SRB and Si-LDS composite films before (a) and c), respectively) and after (b) and d), respectively) aging under photonic (blue LED) and thermal (80°C) stresses.

CONFLICTS OF INTEREST

There is no conflict to declare.

FORMATTING OF FUNDING SOURCES

This work was supported by CPER DEFI MMASYF through its 2016 «MetaProfile» project. Thus, the authors would like to thank the European Union in the framework of the European Regional Development Fund (ERDF), and the Région Auvergne Rhône-Alpes, which co-funded this project.