Alain Bourgeat
email: bourgeat@mcs.univ-lyon1.fr

Andrey L Piatnitski
email: andrey@sci.lebedev.ru

Homogenization of a diffusion convection equation, with random source terms periodically distributed

Keywords: Mathematics Subject Classification: 35K20, 35Q35, 35R60 Random operator; Homogenization;

Introduction

We are interested to study u(x, t) , the evolution in time of the concentration, which is transported by diffusion and convection from a "sources site" made 1 of a large number of similar "local sources". For this we consider a "local model" based on a general diffusion convection equation:

∂ t u ε -div(a(x)∇u ε ) + div(b(x)u ε ) = f ε ;
(1)

u ε t=0 = 0, ∂ ∂n a u ε • n(x) -b(x) • n(x)u ε + λu ε = 0. ( 2
)
where the sources density f ε comes from a set of "local sources" periodically repeated and lying on a same plan Σ; f ε (x, t) = j∈Z 2 f j (x, t). Assuming the release curve ( source emission vs. space and time),f j (, .), of each local source, being random, our aim is to give a mathematical model describing the global evolution of such a system .

In section 1, we define the geometry of the sources site and different randomness assumptions for the local sources. In this first section, we study a general "sources site" model assuming that the random field, x → f ε (x, t)(or space trajectory), is statistically homogeneous and ergodic. In section 2, adding different mixing properties to these very general assumptions, we estimate the rate of convergence expectation for (u ε -u 0 ). In section 3,under the same assumptions, we prove, the convergence in distribution , for the corrector, u 1 = (u ε -u 0 )/ε, at any fixed point (x, t) to a centered Gaussian random variable . Finally, assuming the time trajectory, t → f ε (x, t)is not random, we obtain the same convergence in distribution, for any finite joint distribution. This modelisation is used for describing contaminants transport and migration in aquifers, see for instance [START_REF] Bear | Dynamics of Fluids in Porous Media[END_REF], from a long-lived nuclear waste underground repository.In this case the above "local model" may be used for all the scaling up stages : from the set of containers to a repository unit, or from the set of repository units to a repository zone, or finally from the set of repository zones to the entire waste site. The general random behavior of the sources, as considered in sections 1 and 2, has only to be adapted to the different scaling up stages, by adding more precise assumptions on the randomness for each situation.

1 Definition of the problem 1.1 Description of the geometry Given a smooth bounded domain

Q ⊂ R 3 with diameter diam(Q) = R < ∞, such that Q + = {x ∈ Q : x 3 > 0} and Q -= {x ∈ Q : x 3 < 0} nonempty
Lipschitz domains; now we describe the geometry of the sources supports inside this domain. First denote ε a small positive number (measuring the typical length of a source support or the adimensionalised period of the source supports); assuming that, in the local variables, each source has its support K ε in a thin parallelepipedic set

K ε = [0, s 1 ] × [0, s 2 ] × ε γ-1 [-s 3 , s 3 ]
with 0 < s 1,2 < 1 and γ ≥ 1; then by repeating periodically a single source support, we define a "sources site" support B ε , and Bε a projection of the "sources site" on the middle plan Σ = {x ∈ Q : x 3 = 0} . First we assume that, all the source supports K ε of the entire "sources site" are situated inside a square Π ⊂ Σ, and not intersecting with the boundary ∂Π of Π; then:

B j ε = ε([0, s 1 ] × [0, s 2 ] + (j, 0)); Bj ε = B j ε Π = ε([0, s 1 ] × [0, s 2 ] + (j, 0)) Π, (3) Bε = j∈Z 2 Bj ε ; Bε ∂Π = Φ; (4) B j ε = Bj ε × ε γ [-s 3 , s 3 ]; B ε = j∈Z 2 B j ε . (5) S j ε = ε([0, 1] 2 + (j, 0)). (6)

Description of the randomness

Starting with (Ω, F, P) a standard probability space, for describing the randomness dependancy in space, assuming that the sources density f ε is statistically homogeneous with respect to the 2D variable x ′ = (x 1 , x 2 ), we define on this probability space a discrete random ergodic dynamical system

T z ; Z ∈ Z 2 .
Then by definition the random ergodic dynamical system T z , is a collection of measurable maps T z : Ω → Ω such that T z -preserves the measure P for all z ∈ Z 2 ; -has the group properties, T z+y = T z • T y for all z, y, T 0 = Id ; -is ergodic, i.e. the relation P(A)(1 -P(A)) = 0 holds for any invariant set A ∈ F.

Description of the equations

Under the above assumptions, we are now considering the initial-boundary problem, with a random right hand side:

∂ t u ε -div(a(x)∇u ε ) + div(b(x)u ε ) = f ε , in Q × (0, ∞); (7)
u ε t=0 = 0, ∂ ∂n a u ε -b(x) • n(x)u ε + λu ε = 0 on ∂Q × (0, ∞), (8)
where a(x) (the diffusion tensor) is a uniformly and positive definite smooth matrix-function, b(x) ( the convection velocity) is a smooth vector field; and n a and n are the external co-normal and normal respectively.

In the following, we will use the notation:

x ′ = (x 1 , x 2 ) ∈ Σ; x = (x ′ , x 3 ), (9) and z = [x ′ /ε], x' = [x ′ ]; (10)
with [.] denoting the integer part.

Regarding the random source term , as usual, we will not show explicitly the dependency in ω, the outcomes, and write the random variables as follow:

φ ε (x, t) = φ(T z ω, t), Φ(x, t) = φ(T x' ω, t). (11)
and

f ε (x, t) = 1I Bε 1 ε γ φ(T z ω, t), f (x, t) = 1I ǫ -1 Bε 1 ε γ Φ(x, t). ( 12
)
We should notice first that the random function f ε (x, t), is independant of x 3 . In the next section, corresponding to the actual problem, we will assume x → f ε (x, t)(the space trajectory), to be statistically homogeneous and ergodic; but no special assumption will be done for the randomness of t → φ ε (x, t) . We will only assume the random function φ ε (., .)to be uniformly bounded, i.e. that there exists nonrandom constants Λ > 0 and C > 0 such that, for any fixed t ∈ (0, ∞) and any (x, ω) ∈ (Q × Ω),

|φ ε (x, t)| = |φ(T z ω, t)| ≤ Ce -Λt . (13)
A classical result says that for each ε > 0 and each ω ∈ Ω problem ( 7)-( 8) has a unique solution

u ε ∈ L 2 (0, T ; H 1 (Q)) ∩ C(0, T ; L 2 (Q)).
Our first aim is now to show that the limit problem takes the form

∂ t u 0 -div(a(x)∇u 0 ) + div(b(x)u 0 ) = F (t)δ Σ (x), in Q × (0, ∞); (14)
u 0 t=0 = 0, ∂ ∂n a u 0 -b • nu 0 + λu 0 = 0 on ∂Q × (0, ∞); (15)
where F (t) is the expectation ( constant in space, due to homogeneity and ergodicity of the Random field Φ in space):

F (t) = s 1 s 2 E{Φ(x, t)},
and δ Σ (x) is the surface Lebesgue measure with support Σ Π.

More precisely, we are going to prove that u ε converges to u 0 , as ε → 0, in L 2 (0, T ; H 1 (Q)) norm. To this end we introduce an auxiliary 2D source term with support on Σ Π

F ε (x, t) = 1I Bε φ(T z ω, t)δ Σ (x). ( 16
)
with Bε defined as as in (4), Lemma 1. Under assumption,(13) , for ay fixed t, the following bound holds:

f ε (•, t) -F ε (•, t) H -1 (Q) ≤ Cε γ/2 e -Λt
with C a nonrandom constant.

Proof. Letting

g ε (x, t) =        + x 3 -1 f ε (x ′ , y, t)dy, x 3 ≤ 0 - -1 x 3 f ε (x ′ , y, t)dy, x 3 > 0;
we have

∂ ∂x 3 g ε (x, t) = f ε (x, t) -F ε (x, t), and|g ε (x, t)| ≤ Ce -Λt .
It is also clear that supp(g ε ) ⊂ Σ. Therefore, for any

ϕ ∈ C ∞ 0 (Q), Q (f ε (x, t) -F ε (x, t))ϕ(x)dx = Q ∂ ∂x 3 g ε (x, t)ϕ(x)dx ≤ ≤ Q g ε (x, t) ∂ ∂x 3 ϕ(x)dx ≤ Σ (g ε (x, t)) 2 dx 1/2 Q |∇ϕ| 2 dx 1/2 ≤ ≤ Cε γ/2 ϕ H 1 0 (Q) e -Λt , which implies the statement.
Moreover, by Birkhoff's theorem the function (F ε (x, t) -F (t)δ Σ (x)) almost surely (a.s.) converges to 0 weakly in L 2 (Σ), as ε → 0, for all t > 0. As a consequence we obtain the following statement.

Lemma 2. Due to the assumption of ergodicity for the discrete random dynamical system T z ,and under assumption (13), we have the convergence, a.s., in L 2 norm:

lim ε→0 F ε -F δ Σ L 2 (0,T ;H -1 (Q)) = 0. ( 17
)
Proof. For an arbitrary ϕ ∈ L 2 (0, T ;

H 1 0 (Q)) the relation holds T 0 Q (F ε (x, t) -F (t)δ Σ (x))ϕ(x, t) dxdt = = T 0 Σ (F ε (x ′ , 0, t) -F (t))ϕ(x ′ , 0, t) dx ′ dt ≤ ≤ T 0 ϕ(•, t) H 1/2 (Σ) (F ε (•, t) -F (t) H -1/2 (Σ) dt ≤ ≤ C ϕ L 2 (0,T ;H 1 0 (Q)) T 0 (F ε (•, t) -F (t) 2 H -1/2 (Σ) dt 1/2 (18)
Due to the compactness of embedding of L 2 (Σ) into H -1/2 (Σ), the a.s. weak convergence of ((

F ε (x ′ , 0, t) -F (t)) to 0 in L 2 (Σ) implies that a.s. for all t ∈ [0, T ] lim ε→0 (F ε (•, t) -F (t)) H -1/2 (Σ) = 0. Since (F ε (•, t) -F (t)) L 2 (Σ) ≤ Ce -Λt
, by the Lebesgue theorem we get a.s.

lim ε→0 T 0 (F ε (•, t) -F (t)) 2 H -1/2 (Σ) dt = 0
for any T > 0. The desired result now follows from (18).

Remark 1. Later on, we will also use the estimate

F ε (•, t) -F (t)δ Σ H -1 (Q) ≤ Ce -Λt , ∀(ω, t) ∈ Ω × (0, →); (19)
which is an easy consequence of (13).

We now proceed with the convergence result.

Theorem 1. Under the same assumptions as in Lemma2

lim ε→0 u ε -u 0 L 2 (0,∞;H 1 (Q)) = 0 a.s.,
with u 0 solution of ( 14),(15).

Proof. Subtracting ( 14) from [START_REF] Bourgeat | A homogenized model of an underground waste repository including a disturbed zone[END_REF] we conclude that the difference (u ε -u 0 ) satisfies the equation

∂ t (u ε -u 0 ) -div(a(x)∇(u ε -u 0 )) -b(x)∇(u ε -u 0 ) = f ε -F (t)δ Σ (x), in Q × (0, ∞);(20) (u ε -u 0 ) t=0 = 0, ∂ ∂n a (u ε -u 0 ) + (λ -b • n)(u ε -u 0 ) = 0 on ∂Q × (0, ∞).
The standard energy estimate for this problem reads

u ε -u 0 L 2 (0,T ;H 1 (Q)) ≤ C(T ) f ε -F δ Σ L 2 (0,T ;H -1 (Q)) ; (21)
it then follows from Lemma 1 and 2 that lim

ε→0 u ε -u 0 L 2 (0,T ;H 1 (Q)) = 0 a.s. ( 22
)
for any T > 0. The proof of the convergence on the infinite time interval relies on the dissipative properties of the studied problem. Consider a problem

∂ t w -div(a(x)∇w) -b(x)∇w = 0, in Q × (0, ∞); (23)
w t=0 = w 0 , ∂ ∂n a w + (λ -b • n)w = 0 on ∂Q × (0, ∞). Lemma 3. A solution of problem (23) satisfies the estimate w(•, t) C(Q) ≤ Ce -κt w 0 L 2 (Q) , t ≥ 1,
with constants κ > 0 and C > 0.

Proof of Lemma. Without loss of generality we may assume that w 0 ≥ 0. Then, by the maximum principle, the solution w is positive for any time. Moreover, by the Harnack inequality and standard parabolic estimates,

max x∈Q,1≤t≤2 w(x, t) ≤ C min x∈Q,1≤t≤2 w(x, t) ≤ w 0 L 2 (Q)
Integrating by parts the equation ( 23) over the set {(x, t) :

x ∈ Q, 1 ≤ t ≤ 2} we obtain Q w(x, 2)dt - Q w(x, 1)dt = -λ 2 1 dt ∂Q w(x, t)dσ,
where dσ is the surface volume element. It follows from the last two inequalities that

Q w(x, 2)dt ≤ ν Q w(x, 1)dt with ν < 1.
Iterating this procedure we conclude get

Q w(x, N )dt ≤ ν N Q w(x, 1)dt.
Letting -κ = ln ν, the last estimate reads

Q w(x, N )dt ≤ e -κN Q w(x, 1)dt.
¿From this estimate the statement of the Lemma easily follows by the Harnack inequality.

To complete the proof of Theorem denote by G(x, y, t) the Green function of problem (23). By Lemma 3 and Harnack inequality we have

G(x, y, t) ≤ Ce -κt , t ≥ 2,
and, by the parabolic estimates,

G(•, y, t) H 1 (Q) ≤ Ce -κt , t ≥ 2.
Combining this bound with (19) and the estimates of Lemma 1 and making use of the integral representation of a solution to parabolic problem (20) gives

∞ T u ε (•, t) -u 0 (•, t) 2 H 1 (Q) dt ≤ Ce -min(Λ,κ)t .
Together with the convergence on finite intervals obtained in (22)this result implies the desired statement.

Estimates for the rate of convergence

From the applications point of view the convergence result alone is not of great interest, if it is not accompanied by any rate of convergence estimates. In this section, under natural additional assumptions on the behaviour of the source term correlation function or mixing coefficients, see for instance [START_REF] Pozhidaev | On the error of averaging of symmetric elliptic systems[END_REF] and [START_REF] Yurinski | Averaging of Symmetric diffusion in random medium[END_REF] , we provide a number of bounds for the convergence rate.

Let us start by recalling the definition of the correlation function of a Random Field (x

′ , t) ∈ R 2 × R → f (x ′ , t) R(t, s, x ′ , y ′ ) ≡ E ( f (x ′ , t) -E f (x ′ , t))( f (y ′ , s) -E f (y ′ , s)) ; (24)
which is ≡ R(t, s, x ′ -y ′ ) if the Random Field f , is statistically homogeneous, in the variable x ′ .

Mixing assumptions

Here, and in the following, we will take :

f (x ′ , t) ≡ Φ(x, t) = φ(T x' ω, t); (25)
as defined in (9) and (11). Moreover, we will use ,in the following, additional assumptions on the random field t → f (x ′ , t); i.e. :

• The upper bound for the correlation of f (x ′ , t) and f (y ′ , s) is only depending on the difference between x ′ and y ′ and does not depend on any particular choice of t or s; and

• the closer the two times s, t are, the bigger is the dependancy (with global exponential decay); namely, for ∀t, s

∈ [0, → [ and ∀x ′ , y ′ ∈ Q, ∃ R(y ′ ) ≥ 0; |R(t, s, x ′ , y ′ )| ≤ e -Λ min(s,t) R(x ′ -y ′ ). ( 26
)
In this last assumption,((26)) assumes that there is a locally, uniform in t and in s, estimate for the correlation of f (•, t) and f (•, s). We should also notice that the bound (26) is consistent with the previous assumption(13).

Concerning R(x ′ , y ′ ) we will now assume that at least one of the following conditions holds true:

R0. R(y ′ ) = 0 if |y ′ | > R 0 .
for some R 0 > 0.

R1.

R 2 R(y ′ )dy ′ < ∞. R2. R(y ′ ) ≤ C(1 + |y ′ |) -ν , ν > 0.
To obtain other estimates for the rate of convergence u ε → ε→0 u 0 , including higher moments estimates for the discrepancy |u ε -u 0 |, we may assume that the functions f (x ′ , t) ≡ Φ(x, t) = φ(T x' ω, t) possess one of the following mixing property:

M1. for each t ≥ 0 the strong spatial mixing coefficient α t 0 (s) of f (t, •) decays fast enough so that

α t 0 (s) < Ce -Λt 0 (1 + s) -ν 1 , ν 1 > 0; ( 27
)
where the strong spatial mixing coefficient, α t 0 (s), is defined as follows:

α t 0 (s) = sup G 1 , G 2 sup E 1 ∈F G 1 E 2 ∈F G 2 |P(E 1 ∩ E 2 ) -P(E 1 )P(E 2 )| with F G 1 = σ{ f (y ′ 1 , t 1 ) : y ′ 1 ∈ G 1 , t 1 ≥ t 0 }, F G 2 = σ{ f (y ′ 2 , t 2 ) : y ′ 2 ∈ G 2 , t 2 ≥ t 0 };and the first supremum taken over all sets G 1 , G 2 ⊂ R 2 such that dist(G 1 , G 2 ) ≥ s.
M2. for each t ≥ 0 the maximum spatial correlation coefficient β t 0 (s) of f (•, t) decays fast enough so that

β t 0 (s) < Ce -Λt 0 (1 + s) -ν 1 , ν 1 > 0 ( 28
)
with the maximum spatial correlation coefficient is defined by:

β t 0 (s) = sup G 1 , G 2 sup ξ, η |E(ξη)|;
where the second supremum is taken over all random variables ξ and η which are respectively F G 1 -and F G 2 -measurable and satisfy the conditions

Eξ = Eη = 0, ξ L ∞ (Ω) = η L ∞ (Ω) = 1; and the first supremum is taken over all sets G 1 , G 2 ⊂ R 2 such that dist(G 1 , G 2 ) ≥ s.
Remark 2. Notice that the condition R0 is fulfilled if the strong mixing coefficient α t 0 (s) is equal to 0 for s ≥ R 0 , and also that the condition M1 implies the condition R2 with ν = ν 1 /3.

Replacing the random source density f ε (x, t), distributed in a small neighbourhood of the plane Σ, by a random source density F ε (x, t) concentrated on the plane Σ, as defined in (16), we consider the auxiliary problem:

∂ t ûε -div(a(x)∇û ε ) + div(b(x)û ε ) = F ε , in Q × (0, ∞); (29)
ûε t=0 = 0, ∂ ∂n a ûε -b(x) • n(x)û ε + λû ε = 0 on ∂Q × (0, ∞); (30)
and obtain the next convergence estimates.

Lemma 4. Under assumptions on the domain geometry in sec. 1.1 and assumptions on the coefficients of equations (( 7)-((8) in sec. 1.3,with assumption(13), the bounds hold

u ε -ûε L ∞ (Q×(0,T )) ≤ Cε γ | ln ε|, u ε -ûε L ∞ (0,T ;L p (Q)) ≤ C(p)ε γ , 1 ≤ p < ∞.( 31
)
Proof. Denote by G(t -s, x, y) the Green function of problem ( 7)-(8). Using Aronson's estimates for fundamental solutions on finite time interval [START_REF] Aronson | Bounds for the fundamental solution of a parabolic equation[END_REF], the function G(t, x, y) admits the following bounds

G(t, x, y) ≤ C t 3/2 exp(-c |x -y| 2 t )
with strictly positive C and c. Moreover,

|∇ y G(t, x, y)| ≤ C t 3/2 |x -y| t exp(-c |x -y| 2 t ) (32)
Clearly, the difference (f ε -F ε ) can be represented as follows

f ε (x, t) -F ε (x, t) = ∂ ∂x 3 ϑ x 3 ε γ 1I Bε (x ′ )f (T z ω, t), with ϑ(r) =    r + 1, -1 ≤ r < 0, r -1, 0 ≤ r ≤ 1, 0, otherwise.
By Green formula and estimates (13), (32), we have

|u ε (x, t) -ûε (x, t)| = t 0 Q G(t -s, x, y) ∂ ∂y 3 ϑ y 3 ε γ 1I Bε (y ′ )f (T y'/ε ω, s) dyds = t 0 Q ∂ ∂y 3 G(t -s, x, y)ϑ y 3 ε γ 1I Bε (y ′ )f (T y'/ε ω, s) dyds ≤ ≤ t 0 Q C (t -s) 2 |x -y| √ t -s exp -c |x -y| 2 t -s 1I {|y 3 |≤ε γ } exp(-Λs) dsdy.
Integrating first in time and then in space, after straightforward computation, one gets

t 0 1 (t -s) 2 |x -y| √ t -s exp -c |x -y| 2 t -s exp(-Λs) ds ≤ t 0 1 s 2 |x -y| √ s exp -c |x -y| 2 s ds ≤ 1 |x -y| 2 ∞ 0 1 s 5/2 exp - c s ds = c 2 |x -y| 2 .
For |x 3 | ≤ 2ε γ this gives (31)

|u ε (x, t) -ûε (x, t)| ≤ C 2ε γ 0 dy 3 |y ′ |≤R 1 |y| 2 dy ′ ≤ C(Q, γ)ε γ | ln(ε)|, (33)
where, here and later on, R = diam(Q).

For |x 3 | ≥ 2ε γ we have

|u ε (x, t) -ûε (x, t)| ≤ Cε γ |y ′ |≤R 1 x 2 3 + |y ′ | 2 dy ′ ≤ C(Q)ε γ | ln(|x 3 |)|. (34)
Finally, (33)-(34) yield

|u ε (x, t) -ûε (x, t)| ≤ C(Q, γ)ε γ | ln(max{|x 3 |, 2ε γ })|; (35)
and in particular,

u ε -ûε L ∞ (0,∞;L p (Q)) ≤ C(p)ε γ (36)
for all p ∈ [1, ∞); which is (31) and completes the proof.

Remark 3. The proof of Lemma 4has nothing to do with the randomness of f ε . We only used the Green function properties of the initial problem, the structure of the support of f ε , and the fact that this function is bounded.

Next we denote

F ε (t) = E{Φ(x, t)}1I Bε (x ′ )δ Σ (x) (37)
and consider a deterministic auxiliary problem withF ε as non random source term:

∂ t ǔε -div(a(x)∇ǔ ε ) + div(b(x)ǔ ε ) = F ε , in Q × (0, ∞); (38)
ǔε t=0 = 0, ∂ ∂n a ǔε -b(x) • n(x)ǔ ε + λǔ ε = 0 on ∂Q × (0, ∞). ( 39
)
We will now use the following statement derived from [START_REF] Bourgeat | A homogenized model of an underground waste repository including a disturbed zone[END_REF] and [8].

Proposition 1. Let be ǔε solution of ( 38)-( 39) and u 0 solution of (( 7)-((8);

then ǔε -u 0 L ∞ (0,T ;L 2 (Q)) ≤ Cε. ( 40
)
In view of Lemma 4 and proposition 1, in order to estimate the discrepancy u ε -u 0 L ∞ (0,T ;L 2 (Q)) , it suffices to obtain an upper bound for the expression ûε -ǔε

L ∞ (0,T ;L 2 (Q))
. This is the main and most technical part of this section.

Proposition 2. Let ûε solution of ( 29)-( 30) and ǔε solution of ( 38)-( 39), with condition R0 fulfilled; then the following estimate holds

E ûε -ǔε 2 L 2 (0,T ;L 2 (Q)) ≤ Cε 2 . ( 41
)
Proof. The difference U ε ≡ (ǔ ε -ûε ) solves the problem

∂ t U ε -div(a(x)∇U ε ) + div(b(x)U ε ) = F ε -F ε , in Q × (0, ∞); (42)
U ε t=0 = 0, ∂ ∂n a U ε -b(x) • n(x)U ε + λU ε = 0 on ∂Q × (0, ∞). ( 43
)
Our aim is to estimate the expression

E U 2 (t, •) 2 L 2 (Q) .
To this end we first obtain a point-wise in x bound. Using the notation F ε 0 = F ε -F ε , by the Green formula we have

E U 2 (x, t) = E t 0 Q G(t -s, x, y)F ε 0 (s, y)dyds 2 ≤ ≤ E t 0 Q t 0 Q G(t -s, x, y)G(t -r, x, z)F ε 0 (s, y)F ε 0 (r, z) dydsdzdr ≤ C t 0 Q ′ t 0 Q ′ G(t -s, x, (y ′ , 0))G(t -r, x, (z ′ , 0)) R |z ′ -y ′ | ε dy ′ dsdz ′ dr = C t 0 Q ′ t 0 Q ′ G(s, x, (y ′ , 0))G(r, x, (z ′ , 0)) R |z ′ -y ′ | ε dy ′ dsdz ′ dr ≤ C 1 t 0 Q ′ t 0 Q ′ 1 s 3/2 r 3/2 exp -c x 2 3 + |x ′ -y ′ | 2 s × exp -c x 2 3 + |x ′ -z ′ | 2 r R |z ′ -y ′ | ε dy ′ dsdz ′ dr;
here we have denoted Q ′ = Q ∩ {y 3 = 0}. We first integrate in s. This gives

t 0 1 s 3/2 exp -c x 2 3 + |x ′ -y ′ | 2 s ds = 1 (x 2 3 + |x ′ -y ′ | 2 ) 1/2 t/(x 2 3 +|x ′ -y ′ | 2 ) 0 1 s 3/2 exp - c s ds ≤ C 2 (x 2 3 + |x ′ -y ′ | 2 ) 1/2
with C 2 = ∞ 0 t -3/2 exp(-c/s)ds. Substituting this estimate in the previous inequality, one gets

E U 2 (x, t) ≤ C Q ′ Q ′ R |z ′ -y ′ | ε dy ′ (x 2 3 + |x ′ -y ′ | 2 ) 1/2 dz ′ (x 2 3 + |x ′ -z ′ | 2 ) 1/2 (44)
Without loss of generality we assume that 0

∈ Q. If we denote Q 0 = {y ′ ∈ R 2 , |y ′ | < 2diam(Q)} and perform the change the variables ỹ′ = y ′ -x ′ , z′ = z ′ -x ′ , then E U 2 (x, t) ≤ C Q 0 Q 0 1 (x 2 3 + |ỹ ′ | 2 ) 1/2 1 (x 2 3 + |z ′ | 2 ) 1/2 R |z ′ -ỹ′ | ε dỹ ′ dz ′ (45) For brevity denote Q 2,ε 0 = {(ỹ ′ , z′ ) ∈ Q 0 × Q 0 : |ỹ ′ -z′ | ≤ R 0 ε}. Due to the assumption R0, we have E U 2 (x, t) ≤ C Q 2,ε 0 1 (x 2 3 + |ỹ ′ | 2 ) 1/2 1 (x 2 3 + |z ′ | 2 ) 1/2 dỹ ′ dz ′ .
In order to achieve an upper bound for this integral we divide the integration area into two parts, namely

Q 2,ε 1 = Q 2,ε 0 ∩ {|ỹ ′ | < 2R 0 ε, |z ′ | < 2R 0 ε} and Q 2,ε 2 = Q 2,ε 0 \ Q 2,ε 1 .
The integral over Q 2,ε 1 can be estimated as follows

Q 2,ε 1 dỹ ′ (x 2 3 + |ỹ ′ | 2 ) 1/2 dz ′ (x 2 3 + |z ′ | 2 ) 1/2 ≤ {|y ′ |<2R 0 ε} dỹ ′ (x 2 3 + |ỹ ′ | 2 ) 1/2 2 = = 2R 0 ε 0 rdr (x 2 3 + r 2 ) 1/2 2 = 4R 2 0 ε 2 0 ds 2(x 2 3 + s) 1/2 2 ≤ C(R 0 )ε 2 ; ( 46
)
the explicit formula for the last integral has also been used here. For any 47) and (46), we arrive at the desired point-wise upper bound:

(x ′ , y ′ ) ∈ Q 2,ε 2 it holds 1 (x 2 3 + |y ′ | 2 ) 1/2 dy ′ (x 2 3 + |z ′ | 2 ) 1/2 ≤ C(R 0 ) 1 (x 2 3 + |y ′ | 2 ) Hence, Q 2,ε 2 dỹ ′ (x 2 3 + |ỹ ′ | 2 ) 1/2 dz ′ (x 2 3 + |z ′ | 2 ) 1/2 ≤ Q 2,ε 2 dỹ ′ dz ′ (x 2 3 + |ỹ ′ | 2 ) ≤ ≤ C(R 0 )ε 2 |ỹ ′ |≥2R 0 ε dỹ ′ (x 2 3 + |ỹ ′ | 2 ) = C(R 0 )ε 2 2diam(Q) 2R 0 ε rdr (x 2 3 + r 2 ) ≤ (47) ≤ C(R 0 , Q)ε 2 ln(x 2 3 + ε 2 ). Combining (
E U 2 (x, t) ≤ C(R 0 , Q)ε 2 ln(x 2 3 + ε 2 ). (48)
Now, the estimate (41) is straightforward, we should just integrate (48) over Q.

The above statements allow us to estimate the rate of convergence in Theorem 1.

Theorem 2. Let the assumptions of Theorem 1 be fulfilled, and assume in addition that the condition R0 holds true. Then

E u ε -u 0 2 L 2 (0,T ;L 2 (Q)) ≤ C(R 0 , Q)(ε 2 + ε 2γ ),
where u ε is a solution of the original problem ( 7)-( 8), and u 0 is a solution of homogenized problem ( 14)-(15).

Proof. This assertion is a straightforward consequence of Lemma 4 and Propositions 1 and 2.

Taking into account the dissipative properties of the boundary conditions (8) and the bounds (13), (26), and following the line of the proof of Proposition 2, one can obtain the estimate Theorem 3. Under the assumptions of Theorem 2 the inequality

E u ε -u 0 2 L 2 ((t,∞);L 2 (Q)) ≤ C(R 0 , Q) exp(-κt)(ε 2 + ε 2γ ), κ > 0. (49)
holds with κ > 0 which only depends on Λ, the operator in ( 7)-(8) and the domain Q.

Our next goal is to relax the mixing assumptions on the source function f (x, ω). We want to show that the statement of the last theorem remains valid if the correlation function of f or its strong mixing coefficient satisfy certain polynomial decay conditions. Theorem 4. Suppose that either the condition R2 is fulfilled with ν > 2 or the strong spatial mixing coefficient α t 0 (s) satisfies the upper bound

α t 0 (s) ≤ C(1 + s) -ν 1
with ν 1 > 6. Then the inequality holds

E u ε -u 0 2 L 2 (0,T ;L 2 (Q)) ≤ C(ν, Q)(ε 2 + ε 2γ ),
Proof. It suffices to show that the estimate (48) holds. To this end we consider the auxiliary problem (42)-( 43) with in the R.H.S.F ε -F ε and notice that the upper bounds (44) and (45) are valid with assumptions of theorem 4. It follows from (45) and the standing assumptions that

E U 2 (x, t) ≤ C Q 0 Q 0 1 (x 2 3 + |ỹ| 2 ) 1/2 1 (x 2 3 + |z| 2 ) 1/2 1 (1 + ε -1 |z -ỹ|) ν dỹdz = = Cε 2 ε -1 Q 0 ε -1 Q 0 1 (X 2 3 + |ỹ| 2 ) 1/2 1 (X 2 3 + |z| 2 ) 1/2 1 (1 + |z -ỹ|) ν dỹdz
with ν > 2 and X 3 = x 3 /ε; for the notation simplicity we write ỹ and z instead of ỹ′ and z′ . We divide the domain ε

-1 Q 0 × ε -1 Q 0 into three parts, namely ε -1 Q 0 × ε -1 Q 0 = Q 1 ∪ Q 2 ∪ Q 3 = (ỹ, z) : |ỹ| ≤ 1 2 |z| ∪ (ỹ, z) : 1 2 |z| ≤ |ỹ| ≤ 2|z| ∪ (ỹ, z) : |ỹ| ≥ 2|z| ,
and estimate the contribution of each subdomain separately. In

Q 1 we have |z ′ -ỹ′ | ≥ 1 2 |z ′ |, thus ε -1 Q 0 Cε 2 dz (X 2 3 + |z| 2 ) 1/2 {2|ỹ|≤|z|} 1 (X 2 3 + |ỹ| 2 ) 1/2 1 (1 + |z -ỹ|) ν dỹ ≤ ≤ ε -1 Q 0 Cε 2 dz (X 2 3 + |z| 2 ) 1/2 {2|ỹ|≤|z|} 1 (X 2 3 + |ỹ| 2 ) 1/2 C(ν) (1 + |z|) ν dỹ ≤ ≤ ε -1 Q 0 C(ν)ε 2 dz (X 2 3 + |z| 2 ) 1/2 (1 + |z|) ν |z|/2 0 rdr (X 2 3 + r 2 ) 1/2 ≤ ε -1 Q 0 C(ν)ε 2 dz (X 2 3 + |z| 2 ) 1/2 (1 + |z|) ν C 1 (ν)(X 2 3 + |z| 2 ) 1/2 ≤ C 2 (ν)ε 2 .
The contribution of Q 3 can be estimated in the same way if we exchange the order of integration in the variables ỹ and z. It remains to estimate the integral over Q 2 . We have

Q 2 Cε 2 dz (X 2 3 + |z| 2 ) 1/2 dỹ (X 2 3 + |ỹ| 2 ) 1/2 1 (1 + |z -ỹ|) ν ≤ ≤ Q 2 4Cε 2 dz (X 2 3 + |z| 2 ) dỹ (1 + |z -ỹ|) ν ≤ ε -1 Q 0 4Cε 2 dz (X 2 3 + |z| 2 ) 2ε -1 Q 0 dŷ (1 + |ŷ|) ν ≤ ≤ C(ν)ε 2 ε -1 Q 0 dz (X 2 3 + |z| 2 ) ≤ C 3 (ν)(1 + | ln(|x 3 |)|).
Combining the above estimates we conclude that

E U 2 (x, t) ≤ C(ν)ε 2 1 + | ln(|x 3 |)| .
This yields the desired statement if the assumption R2 is fulfilled with ν > 2.

To complete the proof, we use the fact that the condition M1 implies the condition R2 with ν = ν 1 /3.

Theorem 5. Assume that at least one of the conditions R1, R2 with ν > 2, or M1 with ν 1 > 6 is satisfied. Then the discrepancy (u ε -u 0 ) admits the estimate

E u ε -u 0 2 L 2 ((t,∞);L 2 (Q)) ≤ C exp(-κt)(ε 2 + ε 2γ ), κ > 0.
By following the line of the proof of Theorem 4 one can show that under the condition R1 the upper bound holds

E u ε -u 0 2 L 2 (0,T ;L 2 (Q)) ≤ C(ε 2 + ε 2γ );
and finally, combining the above local in time estimates and the statement of Lemma 3, and considering the presence of exponentially decaying factors in (13), ( 26) and (27), we arrive at the result. In order to improve the statements of Lemma 4 and Proposition 1, we will need an additional assumption. Let's denote

Q δ = {x ∈ Q : |x 3 | > δ} and Q - δ = {x ∈ Q : |x 3 | ≤ δ};
we will assume then: H1. there exists δ 0 > 0 such that Q - δ 0 = Σ × [-δ 0 , δ 0 ];i.e. in the vicinity of the hyperplane Σ the domain Q has a cylindrical shape. Lemma 5. Under the assumptions of Lemma 4 and the additional assumption H1, for any δ > 0 , and almost surely, the two relations hold:

lim ε→0 1 ε γ u ε -ûε L ∞ (Q δ ) = 0. ( 50
) lim ε→0 1 ε γ u ε -ûε L p (Q) = 0, p ∈ [1, +∞). (51)
Proof. In view of (35) the second relation is a consequence of the first one.

To prove (50) we make use of the representation for all x ∈ Q δ 0 , t > 0 and ε < δ 0 /2. This implies the desired statement. Now we proceed by studying the asymptotic behaviour of the normalized difference ε -1 (ǔ ε -ǔε ). Denote c(t, s) = lim

f ε (x, t) -F ε (x, t) = ε γ ∂ 2 ∂x 2 3 ϑ 1 x 3 ε γ 1I Bε (x ′ )f (T x'/ε ω, t),
N →∞ 1 N 2 [0,N ] 4
R(t, s, y -z)1I B (y)1I B (z)dy 1 dy 2 dz 1 dz 2 ; it is easy to verify that under condition R1 the above limit exists and admits the upper bound c(t, s) ≤ 4 ∞ 0 ∞ 0 R(y)dy 1 dy 2 .

3 Corrector's limit law Theorem 6. Assume that one of the conditions M1 or M2 is fulfilled with ν 1 > 2. Then for each t > 0 and x ∈ Q, x 3 = 0, the normalized difference ε -1 (ǔ ε -ûε ), with ûε solution of (29)-(30) and ǔε solution of (38)-(39), converges in law towards a Gaussian random variable with zero mean and covariance σ 2 (t, x) = t 0 t 0 Σ G(t -s, x, y ′ )G(t -r, x, y ′ ) c(s, r)dy ′ dsdr.

Proof. The proof is obtained by adapting the lines of the Central Limit Theorem, with ε -1 ≈ N ; N 2 ≈ numbers of local sources, for stationnary homogeneous random field in [ [START_REF] Jacod | Limit theorems for stochastic processes[END_REF]] to the random variable

X j (x, t) = t 0 Σ G(t -s, x, y ′ ) × φ(T j ω, t)1I Bj ε (y ′ )dy ′ dt; j ∈ Z 2 . ( 52
)
We only have to find the limit lim ε→0 E (ε -1 (ǔ ε (x, t) -ûε (x, t)) 2 .

It is straightforward to compute E (ε -1 (ǔ ε (x, t) -ûε (x, t))

with ϑ 1 ( 3 G∂y 2 3 G

 133 formula and due to the assumption H1, for sufficiently small ε > 0 we have|u ε (x, t)-û ε (x, t)| = ε γ t γ 1I Bε (y ′ )f (T y'/ε ω, s) dyds (t -s, x, y)ϑ 1 y 3 ε γ 1I Bε (y ′ )f (T y'/ε ω, s) dyds .Since the Green function satisfies the estimate∂ 2 (t, x, y) ≤ C(δ)for all x, y and t such that |x -y| ≥ δ, t > 0, then |u ε (x, t) -ûε (x, t)| ≤ C(δ 0 /2)ε 2γ

  -s, x, y ′ )G(t -r, x, z ′ )× R s, r, y ′ -z ′ ε 1I Bε (z ′ )1I Bε (y ′ )dz ′ dy ′ dsdrSince ν 1 > 2, then the assumption R1 is satisfied. Also, for x 3 = 0, the function G(t -s, x, y ′ ) is continuously differentiable in s and y ′ . Therefore,-s, x, y ′ )G(t -r, x, z ′ )R s, r, y ′ -z ′ ε 1I Bε (z ′ )1I Bε (y ′ )dz ′ dy ′ = = Σ G(t -s,x, y ′ )G(t -r, x, y ′ ) c(s, r)dy ′

Acknowledgement

The work of both authors has been partially supported by grant from the CNRS/ANDRA/BRGM/CEA/EDF GDR 2439 whose support is gratefully acknowledged. Also, the work of A. Piatnitski has been partially supported by RFBR, grants 00-01-22000 and 02-01-00868.

for all s and r, 0 ≤ s, r ≤ t. The relation lim ε→0 E (ε -1 (ǔ ε (x, t) -ǔε (x, t)) 2 = σ 2 (t, x) now follows from the Lebesgue theorem.

Unfortunately, we cannot claim that under conditions of the last theorem any finite dimensional distributions of ε -1 (ǔ ε -ûε ) converge in law to Gaussian vectors. This is due to the randomness dependence of temporal variable which is not specified. However, in some particular cases the limit field is Gaussian. For instance, if in [START_REF] Bourgeat | A homogenized model of an underground waste repository including a disturbed zone[END_REF],

with any Ψ(t) being a deterministic function; then the function (ε -1 (ǔ ε (x, t)ûε (x, t))) will converge in law to a Gaussian field.

, be of the form (53). Then for any finite set (x 1 , t 1 ), (x 2 , t 2 ), . . . , (x N , t N ) the random vector (ε -1 (û ε (x j , t j )ǔε (x j , t j ))) , j = 1, . . . , N , , with ûε solution of ( 29)-( 30) and ǔε solution of (38)-(39), converges in law towards a Gaussian vector with the covariance matrix {σ ij },

G(t i -s, x i , y ′ )G(t j -r, x j , y ′ ) c(s, r)dy ′ drds.

Proof. The proof is the same as that of Theorem 6.

Proposition 3 statement will remain valid for a R.H.S., in [START_REF] Bourgeat | A homogenized model of an underground waste repository including a disturbed zone[END_REF], of the form

with with all functions Ψ i (t) being deterministic . We proceed with estimating the difference (u 0 -ǔε ). Making use of the standard ergodic and homogenization as in [START_REF] Bourgeat | A homogenized model of an underground waste repository including a disturbed zone[END_REF] or [START_REF] Lewinski | Asymptotic analysis and homogenization[END_REF], one can prove the following assertion.

Lemma 6. For any t > 0 and x ∈ (Q \ Σ) the relation holds

Theorem 7. Let f ε (x, t) , in [START_REF] Bourgeat | A homogenized model of an underground waste repository including a disturbed zone[END_REF], be of the form (53). Then for any finite set (x 1 , t 1 ), (x 2 , t 2 ), . . . , (x N , t N ) the random vector (ε -1 (ǔu ε (x j , t j ) -u 0 ) , j = 1, . . . , N , converges in law towards a Gaussian vector with the covariance matrix {σ ij },

Proof. The convergence in law to a Gaussian r.v. for ε -1 (u ε -u 0 ) follows after combining statements of Theorem 6 and Lemmata 5 (hence with the additional assumption H1 when γ = 1, while for γ > 1 we don't need Lemma 5)and 6. Theorem 6 provides CLT for ε -1 (û ε (x j , t j ) -ǔε (x j , t j ))), Lemma 5 ensures that u ε -ûε = o(ε γ ), Lemma 6 -that ǔu ε -u 0 = o(ε).


domain: phys spi spi.opti

text:
R El Abdi
email: relabdi@univ-rennes1.fr

A Dumitrache Rujinski

M Poulain

MECHANICAL CHARACTERIZATION OF FLUORIDE OPTICAL FIBERS

Keywords: Optical fibers, mechanical tests, fluoride glasses, failure strength

Mechanical dynamic tests were performed on fluoride glass fibers and silica optical fibers to examine the mechanical strength. These fibers were tested on ambient atmosphere and after aging in a vacuum desiccator with silica gel used as the desiccant. A higher value of the ratio of Young's modulus to hardness indicates a greater tendency to plastic behavior in the case of fluoride fibers. For tensile tests, the strength of silica fiber is nine times greater than that of fluoride fiber and eight times greater in the case of bending tests. Aging in a desiccator decreases the residual water content and improves the fluoride fiber strength.

INTRODUCTION

Fluoride glass fibers were first intensively developed for long distance telecommunication applications due to their ultra-low theoretical optical loss (0.001 dB/km). However these glasses are very susceptible to crystallization. Past researchers have shown that crystallization of these glasses is suppressed in microgravity and enhanced in hyper-gravity, showing a crystallization dependency on gravity. The specific material used for this research was a fluorozirconate glass, which is a subset of the Heavy Metal Fluoride Glasses family known as ZBLAN (ZrF 4 , BaF 2 , LaF 3 , AlF 3 , NaF). Such fibers can be doped with a number of rare earth ions for application in fiber lasers and amplifiers.

The main interest of fluoride glasses is associated with their potential in developing a new generation of optical waveguides which are superior to traditional SiO 2 optical fibers in their transparency range and their optical characteristics [START_REF] Lucas | Fluoride glasses for modern optics[END_REF].

Comparing with silica fiber, the fluoride optical fiber has many unique characteristics, such as wide operating wavelength range (the wide range of infrared lights, visible lights and even ultraviolet lights) and high emission efficiency when rare-earth elements are doped into, especially excellent emission characteristic with Tm doped ZBLAN fiber [START_REF] Nazabal | Fluoride and oxyfluoride glasses for optical applications[END_REF][START_REF] Zhu | High power ZABLAN glass fiber lasers: Review and prospect[END_REF][START_REF] Méndez-Ramos | Rare-earth doped colour tuneable upconversion ZBLAN phosphor for enhancing photocatalysis[END_REF][START_REF] Su | Highly birefringent ZBLAN photonic quasi-crystal fiber with four circular air holes in the core[END_REF].

As a promising host material, ZBLAN glass used in fluoride fibers has been attracting great interest for up-conversion luminescence because it is relatively easy to prepare, and more importantly has a very low multiphonon relaxation rate [START_REF] Yang | Different dynamics of ultraviolet up-conversion in Tm 3+ : ZBLAN glass under blue laser excitation[END_REF][START_REF] Auzel | Experimental determination of exponential parameters for multiphonon residual absorption in a ZBLAN: Er 3+ fibre[END_REF][START_REF] Méndez-Ramos | Heavy rare-earth doped ZBLAN glasses for UV-blue up-conversion and white light generation[END_REF][START_REF] He | Enhanced ultraviolet up-conversion luminescence of Tm and Yb codoped ZrF 4 -BaF 2 -LaF 3 -AlF 3 -NaF Glass[END_REF].

Because of the highly ionic bond character, fluorides tend to have low phonon energies (~500 cm -1 ) compared to many oxides such as silica based glasses (~1200 cm -1 ). This low phonon energy causes the infrared (IR) absorption edge of fluorides to have a higher wavelength than silica glasses and allows fluorides to function as effective windows beyond the 4 mm IR cut-off observed in fused silica.

Silica glass, the most common optical fiber material is certainly robust, having high tensile strength and resistance to thermal shock [START_REF] Mcnamara | A large core microstructured fluoride glass optical fibre for mid-infrared single-mode transmission[END_REF].

However, silica has a spectral window of only 0.3 to 2 m and is virtually opaque to wavelengths longer than 2 m. It also suffers from laser-induced damage at quite low power [START_REF] Allison | Pulsed laser damage to optical fibers[END_REF][START_REF] Sudrie | Study of damage in fused silica induced by ultra-short IR laser pulses[END_REF].

Fluoride optical fiber have a large-band transmission from 0.3 m to 4.3 m [START_REF] Lambin-Iezzi | Stimulated Brillouin scattering in SM ZBLAN fiber[END_REF][START_REF] Poulain | Verres fluorés au tétrafluorure de zirconium. Propriétés optiques d'un verre dopé au Nd 3+[END_REF][START_REF] Hewak | Low phonon-energy glasses for efficient 1.3 m optical fibre amplifiers[END_REF].

Fluoride glass fibers are less robust than silica but can nevertheless be made in multi-kilometer lengths, sufficiently strong to be spooled and handled. Some are attacked by water but the effect is minimized in suitable compositions. Fluoride glasses are generally resistant to radiation damage [START_REF] Yamashita | Recent advances in IR-transmitting fibers for laser power delivery[END_REF][START_REF] Doshida | Gain measurement and up-conversion analysis in Tm 3+ , Ho 3+ co-doped alumino-zirco-fluoride glass[END_REF][START_REF] Feng | Upconversion luminescence of ZBLAN: Tm 3+ , Yb 3+ glass pumped by a ~970 nm LD and its concentration effect[END_REF], though rare earth doping may lower the damage threshold [START_REF] Chandonnet | Photodegradation of fluoride glass blue fiber laser[END_REF].

The main problem with the fiber is its poor mechanical strength. Degradation may be due to attack by atmospheric moisture or stresses and cracks introduced into the fiber during drawing and spooling or to damage which was not removed after drilling. The intrinsic strength of a glass, its real strength, depends on minor flaws. Considerable effort was made to remove scratches.

Even one remaining crack is sufficient to precipitate the eventual complete rupture of the fiber.

Such cracks would probably be opened up and propagated by low level heating and applied stress [START_REF] Mcnamara | A large core microstructured fluoride glass optical fibre for mid-infrared single-mode transmission[END_REF].

Many studies were undertaken concerning optical and chemical properties of fluoride fibers, but very few studies have been carried out on their mechanical behavior when subjected to bending or tensile tests due to their poor mechanical strength. Two-point bending is the simplest technique for the measurement of failure strains of glass fibers under a variety of experimental conditions.

Tang et al [START_REF] Tang | Using the two point bending technique to determine failure stress of pristine glass fibers[END_REF] use two point bending technique to evaluate the failure strains, under both inert and ambient conditions, for a number of conventional glasses (commercial silica, soda-lime silicate, and E-glass). But no experimental mechanical study of very brittle fibers such as fluoride fibers was presented until then.

In this work, tensile and two point bending tests were performed on fluoride and silica fibers for different displacement velocities. A theoretical study is proposed to understand the mechanical behavior of each fiber. Some fibers were aged in a desiccator before undergoing mechanical tests.

OPTICAL FIBERS USED

Fluoride optical fiber

The multimode fluoride fiber has an operating wavelength between 0.3 and 4.3 m. The combined coating diameter is 220 m with a diameter clad of 150 m and a core diameter of 100 m (Fig. 1). The operating temperature is from -180°C to 150°C. This fiber has a numerical aperture of 0.2 (NA value). Attenuation was ≤ 0.1 dB/m [3.4 -3.6 m]. Elastic properties of cladding and core for the studied optical fiber of compositions (mol%) 53ZrF 4 -20BaF 2 -4LaF 3 -3AlF 3 -20NaF are given in Table 1. The elastic parameters of the glass below were obtained by measurement of sound velocities using the longitudinal and transverse velocities:

          T T V V V V V E L L T 2 2 2 2 2 . 4 . 3 .     (1)   T V G 2 .   (2)                  T T V V V V L L 2 2 2 2 . 2 . 2  (3)     3 . 4 . 3 . 2 2 T V V K L    (4)
where V

L and V T are the longitudinal and transverse velocities respectively. is the density obtained by electronic densimeter, E is the Young's modulus, G is the shear modulus, is the Poisson's ration and K is the bulk modulus.

Silica optical fiber

The used multimode silica optical fiber has two acrylate coatings (primary and outer coatings) (Fig. 2). This fiber has a numerical aperture of 0.2 (NA value) with an operating wavelength of 850/1300 nm. A soft, primary coating has a low module of elasticity, adheres closely to the glass fiber and forms a stable interface. It protects the fragile glass fiber against micro-bending and attenuation. The outer coating protects the primary coating against mechanical damage and acts as a barrier to lateral forces. It has a high glass transition temperature and Young modulus. It has good chemical resistance and serves as a barrier against moisture [START_REF] Severin | Reliability evolution of optical fibre subjected to chemical environment[END_REF][START_REF] Chean | Water effect on interfacial adhesion of an optical fiber embedded in a composite material[END_REF][START_REF] Abdi | Effects of immersion duration and temperature on mechanical properties of optical fibers aged in CTAC aqueous solution[END_REF]. The combined coating diameter is 245 m, the silica core has a diameter of 50 m and the clad diameter is 125 m (Fig. 2).

Fig. 2. Silica optical fiber used

Silica clad

Two acrylate coatings

TEST BENCHES USED

Bending test bench

While the bending method does not replace tensile testing as a fiber strength measurement technique, it presents attractive features and advantages, providing valuable information about flaw size distribution [START_REF] Mathewson | Strength measurement of optical fibres by bending[END_REF]. In our case, the ease and the duration of the testing and the small effective length of the fiber sample made the bending test one of the most appropriate choices for investigation.

The ''as-received'' fibers and those aged in a desiccator were subsequently put through dynamic tests using a two-point bending testing device (Fig. 3). The fiber package was cut into two 8 cm length parts.

Special care was required to prevent the fiber slipping during the faceplate displacement and to maintain the fiber ends in the same vertical plan.

The two points bending bench is made up of a displacement plate which is mounted on an aluminium plate (Fig. 3a). The first thrust block is movable and mounted on the displacement plate, while the second thrust block is fixed on a force sensor. The optical fiber is positioned between the two thrust blocks in such a way that it forms a "U". To avoid slipping, the fiber is positioned in the grooves of the thrust blocks (Figs. 3b,3c).

During the test, load and displacement are recorded, allowing the load/displacement curve to be obtained. At breaking point, the stress applied to the fiber was deduced using the distance d between the two faceplates (Fig. 3c). A non-linear relation defined by Proctor and improved by Griffioen [START_REF] Griffioen | Optical fiber reliability[END_REF] can give the evolution of the stress  (GPa) as a function of second polynomial order i.e.:

          2 . 1 . "     E (5) with 4 1 4 3 "     ( 6
)
where α is a non-linear elastic parameter (typical value of  is 6 for the most optical fibers [START_REF] Mallinder | Elastic constants of fused silica as a function of large tensile strain[END_REF][START_REF] Glaesemann | Effect of strain and surface composition on Young's modulus of optical fibers[END_REF]). The strain  was defined by:

           g c f d d d d 2 198 . 1  (7)
where E is the Young modulus, d c is the polymer coating diameter, d f is the fiber diameter, 2d g is the total depth of the 2 grooves and d is the distance between the two faceplates (Fig. 3c).

Initially, the distance between the two plates (fixed block and movable plate), between which the optical fiber is placed, is 10 mm. The movable plate moves to the left and thereby compresses the fiber which breaks in its middle. The load gradually increases when the movable plate moves closer to the fixed block and the load decreases drastically after the fiber breaks. The distance d between the two parts of the fiber at rupture is equal to: 10 mm minus the distance covered by the movable plate before fiber breaking.

Tensile bench used

The dynamic tensile test consists of subjecting fibers to a deformation under a constant velocity until rupture. The fiber is rolled three times around two pulleys (Fig. 4); the lower pulley is fixed and the upper pulley is movable with different velocities (0.8, 2.6, 4.4, 6.2 and 8 mm/s). These strain rates, expressed as a percentage of the initial sample length (200 mm), correspond to 4 10 -3 s -1 , 13 10 -3 s -1 , 22 10 -3 s -1 , 31 10 -3 s -1 and 40 10 -3 s -1 . This represents the minimum and maximum deformation velocity for optical fibers subjected to stretching in telecommunication networks.

Tensile testing was performed in a controlled environment with 46-52% relative humidity with a maximum of 5 % humidity variation for each series of the tensile tests.

During the test, the tensile load was measured using a dynamometric cell (load sensor) while the fiber deformation was deduced from the displacement between the fixed lower pulley and the mobile higher pulley (Fig. 4).

The testing procedure used 20 samples for each velocity. It is noted that the failure distance between the two plates decreases according to their velocity.

Indeed, the fluoride fiber is very brittle and breaks for large curvature radii when placed between the two plates.

Two points bending test

As received fibers

Dessicated fibers

The stiffness k of the fiber is given as following [START_REF] Griffioen | Optical fiber reliability[END_REF]:

  4 . 2 / '. ' 1 . . . / 2 4 R r E r k     (8)
in which R is the fiber's bending radius and r is the radius of the fiber's cross-sectional area (Fig. 6 ).

Fig. 6. Schematic view of a bent fiber

The fiber breaks for large curvature radii R, and Eq. ( 8) shows that the fiber stiffness k is initially high at the beginning of the displacement of the faceplates and thus the rupture occurs quickly.

When placing the fluoride fibers in the desiccator for one week in order to maintain them in a dry atmosphere, some residual water was removed and this leads to partial micro-crack closures.

This allows the fiber to bend more and a smaller failure distance between the plates was obtained (Fig. 5). Thus, the failure fiber force for the desiccated fiber was a little higher than that of an 'as received fiber' (Fig. 7). Silica fiber tolerates more bending than the fluoride fiber. The failure distance for the silica fiber is on average 5 times smaller than that of fluoride fiber (Fig. 8). Silica fiber can endure smaller curvature radii than the fluoride fiber and therefore will break for loads which are 8 times higher (Fig. 9). The stress of an optical fiber subjected to bending between two plates at each part of the bent fiber is equal to [START_REF] Griffioen | Optical fiber reliability[END_REF]:

                                R r R r R R E 2 2 2 . 8 1 ' ' . 8 1 ' ' . 2 1 . ) (       (9)
The maximum stress  max was obtained for  = r (Fig. 6):

                                 R r R r R r R r E r 2 2 2 max . 8 1 ' ' . 8 1 ' ' . 2 1 . ) ( ) (      (10)                          R r R r R r E r 2 2 . 8 1 ' ' . 8 3 . ) (   (11)
For ' '  equal to 4.75 (using  = 6 in Eq. 6), Eq. ( 11) leads to:

                  R r R r E r 2 . 32 53 . ) (  (12)
One can note that for silica optical fibers under bending tests, when the plate velocity was small, the failure distance was large (Fig. 8), thus the movable plate (Fig. 3) had covered a small distance. That led to a large curvature radius R and then the maximum stress was small (Eq. 12).

For that reason, an increase of the failure force according to plate velocity was obtained (Fig. 9).

It was not the case for fluoride fibers where failure stresses decrease according to plate velocities (Fig. 7). As mentioned above, fluoride fibers contain many flaws, impurities and microcracks introduced during drawing and spooling [START_REF] Mcnamara | A large core microstructured fluoride glass optical fibre for mid-infrared single-mode transmission[END_REF] even if many efforts were made to remove these flaws. The fluoride fibers having defects with sizes a few nanometers cannot be represented by Eqs. 9-12. These equations are valid for a continuous medium without defects. The silica fibers have fewer intrinsic micro-defects than the fluoride fibers and the stresses inside silica fibers can be modeled by Eq. 11.

Tensile tests

Figure 10 shows the tensile failure strength for the fluoride fibers subjected to tensile tests at different faceplate velocities. The tensile bench determines the velocity to the upper pulley (movable pulley) (Fig. 4). It follows that the higher the velocity was, the higher the failure load.

The experimental results can be smoothed by a quadratic polynomial form.

Figure 11 shows that the higher the velocity was, the greater the tensile force, and therefore the failure distance was small. The fiber elongation had a linear change according to the movable pulley velocity. Table 2 gives some physical properties of fluoride and silica fibers. Young's modulus of silica fibers was higher than that of the fluoride fibers (Table 2). Therefore, failure strength of silica fibers was higher than that of fluoride fibers, even if the cladding diameter of fluoride fiber was greater than that of the silica fiber (Fig. 13). The failure load for the silica fiber was, on average, 9 times higher than that of the fluoride fiber (Fig. 12). The main problem with the fluoride fiber is its poor mechanical strength. Degradation may be due to attack by atmospheric moisture or stresses and cracks introduced into the fiber during drawing and spooling [START_REF] Mcnamara | A large core microstructured fluoride glass optical fibre for mid-infrared single-mode transmission[END_REF]. Whatever the intrinsic strength of a glass, its actual strength depends on minor flaws. It operates on the ''weakest link'' principle.

Coating

Core

Cladding

Fluoride fiber

Silica fiber

The Hillig and Charles model [START_REF] Hilling | Surfaces, stress-dependent surface reactions and strength[END_REF] considers that crack blunting can occur due to plastic or viscous deformation as well as corrosion (Fig. 14). When under strain, if the stress at crack tip is not enough to cause plastic flow, it is possible for the crack to sharpen due to corrosion until the plastic yield stress is reached [START_REF] Colaizi | Mechanical durability of ZBLAN and aluminum fluoride based optical fiber[END_REF]. On the other hand, for the fluoride fiber, a higher value of the ratio of Young's modulus to hardness (E/H) was obtained (Table 2) and indicates greater tendency to plastic behavior. So, a crack blunting (Fig. 14) can easily take place for fluoride fiber.

CONCLUSION

The difficulty for testing fluoride fibers was due to their fragility and their cost. Indeed, the fluoride fibers are more expensive than silica fibers and one must take a lot of care to minimize sample losses during testing. Furthermore, their fragility involves frequent breaking before the beginning of bending or tensile tests. Although the pulley diameters for the traction bench were quite large, the winding of fluoride fiber around the pulley must be done slowly to avoid handling or compression of another portion of the fiber already wound around the pulley.

In addition, fluoride fibers have many intrinsic defects that are not uniformly distributed. The number of testing experiments was quite high because many fiber failures occur around the pulley and not in the middle of the part of fiber between the two pulleys during the tensile test.

The bending test also required high accuracy. The setting of the fiber between two plates hed to be done slowly and a few seconds before the start of the dynamic test. A fiber which was left for a long time between the plates broke before the test starts.

The silica fiber, which is more flexible than the fluoride fiber, was easier to test.

Although the fluoride fiber has very interesting optical properties, it remains very sensitive to humidity. Even if the residual moisture was reduced, the fiber strength increased only very moderately. In addition, tests should be performed within a few minutes of the fiber being removed from the desiccator.

The tensile tests showed that fluoride fiber elongation decreased with increasing velocity. On the other hand, the failure loads remained much lower than that of the silica fiber.

Finally, fluoride fibers had a higher value of the ratio of Young's modulus to hardness (E/H) which led to crack blunting.

Fig. 1 .

 1 Fig. 1. Fluoride fiber: a): three components, b): Fiber crossed through light, c) fiber with removed coating

Fig. 3 .

 3 Fig. 3. (a) Bending bench used; (b) and (c) fiber between thrust blocks.

Figure 5

 5 Fig. 4. Tensile test bench used

Fig. 5 .

 5 Fig. 5. Failure distance for fluoride fiber under two point bending test

Fig. 7 .

 7 Fig. 7. Failure load for fluoride fiber under two point bending test

Fig. 8 .Fig. 9 .

 89 Fig. 8. Faceplates distance at failure for silica and fluoride fibers

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Failure load for fluoride fiber under tensile test for different faceplate velocities

Fig. 13 .

 13 Fig. 13. Schematic sizes of fluoride and silica fibers

Fig. 14 .

 14 Fig. 14. Crack blunting for fluoride fiber

Table 1 . Elastic constants of cladding and core of used fluoride fiber

 1

	Material Density	Young's	Shear	Poisson's	Bulk
		kg/cm 3	modulus	modulus	ratio	modulus
			GPa	GPa		(GPa)
	Cladding	4735	54.35 ± 6.10	20.56 ± 0.86 0.322 ± 0.039 50.8 ± 6.04
	Core	4375	53.39 ± 6.36	20.37 ± 0.29 0.311 ± 0.037	46.9± 5.59

Failure load for fluoride and silica fibers under tensile test for different faceplate velocitiesTable 2 . Physical properties of silica and fluoride glasses Parameters Silica Fluoride

 2

	Tensile test
		Fluoride fiber
		Silica fiber
	Fig. 12. Young's Modulus E (GPa)	72	55
	Knoop Hardness H (GPa)	7.7	2.2
	Shear modulus (GPa)	29.5	20.5
	E/H	9.4	24
	Glass Transition Temperature (°C)	1100-1700	257
	Melting Temperature (°C)	1721	450


domain: spi phys

text:
Rochdi El

Effects of immersion duration in water on strength of instrumented composite materials

Keywords: Water diffusion, optical fiber, interfacial adhesion, smart composite materials, pull-out test

The present paper reports on the study of the interfacial adhesion of an optical fiber embedded in a composite material aged in distilled water. Samples composed of optical fibers embedded in an epoxy vinylester resin or in resin/glass fibers systems were submitted to different aging duration in distilled water. Pull-out tests on optical fibers were carried out to measure the effect of water diffusion and glass concentration on fiber bonding. For resin/optical fiber samples, water diffusion leads to damage of the polymer/matrix interface from an aging duration of 15 days and the interfacial stress values present a low decrease up to an aging duration of 60 days. For the case of composite/optical fiber samples, the greater the glass fiber content, the less is the water damage at the polymer interface. On the other hand, a linear development of interfacial debonding stress with increasing glass fiber concentration is reported.

Introduction

The development of so-called smart structures, such as composite structures into services in which optical fibers such as sensors have been integrated, has led to characterization studies and analysis of the implementation of these fibers in composite materials that will be used in civil engineering work (bridges, footbridges, gateways, beams, etc.) [START_REF] Karbhari | Fiber reinforced composite bridge systems-transition from the laboratory to the field[END_REF]. The term "smart" comes from the dual functionality provided by the insertion of the optical fiber sensors into composite materials that provide information on the continuous evolution of damaged structures under mechanical stress [START_REF] Yashiro | Damage identification in a holed CFRP laminate using a chirped fiber Bragg grating sensor[END_REF].

But composite materials present some disadvantages, such as susceptibility to moisture absorption, and a non-negligible loss of tensile stiffness and strength can be obtained with immersion in water. For a long immersion period, the effect of water leads to significant damage to the interfacial shear strength [START_REF] Chen | Influence of moisture absorption on the interfacial strength of bamboo/vinylester composites[END_REF].

The diffusion of water depends on immersion time and on many other parameters [START_REF] Joliff | Experimental, analytical and numerical study of water diffusion in unidirectional composite materials -Interphase impact[END_REF] (temperature, composition of resin and curing agent,…) and the damage caused by the effect of water at the interface surface between an optical fiber embedded in the resin /reinforcement fiber composite is more complex to analyse and to quantify.

In this work, we developed composite specimens (epoxy vinylester resin + glass fibers) (with different fiber glass volumes) in which an optical fiber is embedded. These samples were immersed in distilled water (at laboratory temperature) for different durations (between 6 and 60 days).

Methodology a/ Sample preparation

The optical fibers used are silica fibers with one layer of acrylate polymer coating. The diameter of the cladding is 80 m and the coating diameter is 101.8 m. These fibers are designed to be used at elevated temperatures and pressures in aggressive chemical environments.

The polymer resin used in this study is a mixture of epoxy vinylester resin provided by Derakane 470-36, and a catalytic system composed of Styrene, Percadox 16 and Trigonoc C, provided by DC Pultrusion. The glass fibers used for the preparation of the composites come from "Roving Tex". The powdered Perkadox 16 is firstly diluted in the styrene to form the "styrene/Perkadox 16" system. Then the Trigonoc C and the epoxy vinylester resin were added respectively. Before characterizing the interfacial adhesion between the optical fiber and the epoxy vinylester/glass fiber composite material, a simple system composed of an optical fiber embedded in epoxy vinylester resin was studied. The system is a unidirectional composite, where glass fibers have the same sample length direction and optical fiber direction. The objective was to obtain basic data on a simple system. A typical sample preparation consisted of pouring the resin into a mould and introducing the optical fiber into the middle of the sample. In the case of composites, the impregnated glass fibers were successively superimposed and the optical fiber gently introduced. Special care was taken to ensure that the optical fiber was straight. The set of moulds were then inserted into an oven at 80°C for 2 hours to polymerize the sample.

b/ Used tensile set-up

The single fiber pull-out test is a direct method for measuring the fiber matrix interfacial properties by the evaluation of the interfacial shear strength. Single pull-out tests were performed on optical fibers. The samples used for pull out tests were carefully prepared. If we applied a gripping force (imposed by the jaws of the LLOYD LR 50K tensile testing set-up) (Fig. 1) to the right hand side of the tested specimen and if the optical fiber was pulled, the fiber breakage occurred at the bit. Thus, a notch was introduced into the specimen and the jaws clamped the right hand part of the specimen. The crosshead speed was set at 1mm/min, which corresponds to a strain rate of about 0.04 min -1 . The debonding force F d was taken as the maximum force preceding partial debonding. Different samples obtained with various glass fiber concentrations were tested.

c/ Water effect

The composite we studied will be a component of a civil engineering structure subject to bad weather. During manufacture, this structure receives protective layers to combat water diffusion. However, these protective layers are destroyed during its life and therefore the water diffuses through the composite material.

In recent years, many studies have focused on the analysis of moisture absorption in composite materials [START_REF] Karimt | Interdiffusion of polymers at short times[END_REF]. It has been shown that absorbed water can modify the elastic-plastic behaviour of the resin and lead to decohesion of the matrix/ fiber interface, and composite performance degradation may occur during use [START_REF] Hanson | Incorporating multiregion crack growth into mechanical reliability predictions for optical fibres[END_REF].

When composite material was aged in water, water diffusion had a noticeable effect on material properties and a study of the mechanical behaviour change of the interface between the matrix and optical fiber surface was undertaken.

Results and analysis a/ Change of water content versus immersion time

The studied samples were either made of pure resin in which an optical fiber was embedded or made of resin and glass fibers with an embedded optical fiber. The composite samples contained different volume fractions of glass (0%, 40%, 50%, and 70%). The different samples were 30 mm in length, 10mm in width and 3 mm in thickness. For each case, four identical samples were used to obtain an average value.

The most used model to explain water recovery in polymers is classically based on the Fick's diffusion law [START_REF] Popineau | Free/bound water absorption in an epoxy adhesive[END_REF] which gives water content changes versus the square root of time. Figure 2 gives the water content change for resin samples and for resin samples with an embedded optical fiber.

During the first 15 days, the water content increased in a linear manner with versus the square root of immersion time. After a transition period extending up to 50 days, the sample was saturated and water weight reached 1.25%. Composite samples with various glass fiber concentrations were submitted to immersion in distilled water (Fig. 3). Composite samples with 50% and 70% glass fiber content have similar behaviour. During the first 9 days, the water content had a linear behaviour, followed by a transition period up to 36 days, and then the saturation was obtained.

For the sample with 40% glass fiber concentration, the water content linearly increased up to 10 days then instantly stabilized.

For all the glass fiber contents, the same final water content of 1.7% (Fig. 3) was obtained. There was less water diffusion in resin samples (Fig. 2) than in composite samples (Fig. 3) where micro-voids and micro-cracks exist.

b/ Tensile tests analysis

Tensile tests on the optical fiber embedded in samples were undertaken: a/ for resin samples immersed in distilled water for 10 days, 29 days and 60 days, b/ for composite samples (with 40%, 50 %, and 70% glass fiber content) for 60 days.

*Aging of resin samples

Figure 4 gives the force-displacement curves for different aging durations. All the force-displacement curves have the same appearance. The tensile deformation of the free part of the optical fiber gives the elastic behaviour. After the maximum value of the applied force, a harsh decrease is obtained and is indicative of decohesion of the optical fiber polymer. The second part of the curve indicates the optical fiber sliding in the resin. At first, this slippage is unstable (stick-slip), and then stabilizes at the end of the fiber pulling. The mean interfacial stress decreased versus aging time. The water diffusion produced tensile radial stresses at the optical fiber/ resin interface and led to the interface decohesion. On the other hand, the water diffusion was near saturation after 15 days and the sample was completely saturated after 50 days. A decrease of resin/fiber adhesion was then obtained.

*Aging of composite samples

Figure 5 gives the tensile tests results for composite samples with different glass fiber contents aged in distilled water for 60 days. All the force-displacement curves have the same appearance with a first zone corresponding to the elastic deformation of the free part of the optical fiber. After this phase, the interface decohesion was obtained. During the fiber slip in the composite, a decrease of applied force occurred until complete fiber extraction.

Conclusion

Using the tensile test procedure, the interfaces resin/optical fiber and composite/optical fibers were characterized after different aging durations in distilled water.

There was less water diffusion in resin samples than in composite samples where micro-voids and micro-cracks exist. The damage of the interface between the polymer coating and the composite surface was due to the chemical-physical action of diffused water.

For resin/optical fiber samples, water diffusion led to polymer/matrix interface damage from an aging duration of 15 days and force values presented a low decrease up to an aging duration of 60 days. For the case of composite/optical fiber samples, the greater the glass fiber content, the lower the water damage at the polymer interface. We can also mention that glass fibers are stiffer than vinylester resin; therefore this effect can seriously affect the stiffness of the optical fiber surrounding the composite material.

Figure 1 :

 1 Figure 1: Description of the dynamic tensile test bench and detail of used composite sample

Figure 2 :Figure 3 :

 23 Figure 2: Water content versus immersion time for resin samples

Figure 4 :

 4 Figure 4: Force-displacement curves for different immersion durations in water for optical fiber embedded in resin samples and submitted to pull out test.

Figure 5 :

 5 Figure 5: Force-displacement curves from pull-out test for composite samples aged in water for 60 days


C Monrozeau

E Khan

Y Blumenfeld

C E Demonchy

W Mittig

P Roussel-Chomaz

D Beaumel

M Caamaño

D Cortina-Gil

J P Ebran

N Frascaria

U Garg

M Gelin

A Gillibert

D Gupta

N Keeley

F Maréchal

A Obertelli

J-A Scarpaci

First Measurement of the Giant Monopole and Quadrupole Resonances in a Short-Lived Nucleus: 56 Ni

Keywords: numbers: 21.10.Re, 24.30.Cz, 21.60.Jz, 24.50.+g, 21.60.Ev

The Isoscalar Giant Monopole Resonance (GMR) and Giant Quadrupole Resonance (GQR) have been measured in the 56 Ni unstable nucleus by inducing the 56 Ni(d,d') reaction at 50 A.MeV in the Maya active target at the GANIL facility. The GMR and GQR centroids are measured at 19.3 ± 0.5 MeV and 16.2 ± 0.5 MeV, respectively. The corresponding angular distributions are extracted from 3 to 7 degrees. A multipole decomposition analysis using Distorted Wave Born Approximation with Random Phase Approximation transition densities shows that both the GMR and the GQR exhaust a large fraction of the energy-weighted sum rule. The demonstration of this new method opens a broad range of giant resonance studies at intermediate energy radioactive beam facilities.

Over the past few years the focus of nuclear structure has moved towards short-lived exotic nuclei. Major discoveries of unexpected new phenomena have been recorded, such as neutron haloes and skins [1] and the modification of magic numbers far from stability [2]. Giant resonances are collective excitations of nuclei involving a large fraction of the constituent nucleons. They have proved to be a major source of information on collective and single-particle properties of atomic nuclei [START_REF] Harakeh | Giant Resonances[END_REF]. The reproduction of their properties represents a strong constraint for nuclear models, in particular modern microscopic approaches based on the density functional theory [START_REF] Kohn | [END_REF]. Fundamental nuclear properties such as the equation of state of nuclear matter or the effective nucleonnucleon interaction have benefited from detailed measurements of giant resonances (GR) [5]. The study of GR's in exotic nuclei is thus expected to enhance our understanding of exotic nuclear structure. Experimentally such studies encounter major hurdles and have until now been limited to the isovector giant dipole resonance in neutron-rich Oxygen [6], Neon [7] and Tin isotopes [8]. They provided the first evidence for soft E1 modes in unstable nuclei.

The isoscalar Giant Monopole Resonance (GMR) is of major importance because its properties are related to the nuclear matter incompressibility K ∞ . The interest in determining K ∞ stems from its impact on the nuclear matter equation of state. Some significant progress in our understanding of how K ∞ can be constrained has been achieved in recent times. Namely, the procedure generally adopted to extract the incompressibility from the measurement of the GMR energy [9] is based on energy functionals which allow one to calculate finite nuclei and nuclear matter on the same footing [10]. The first step consists in constraining the energy functional in order to reproduce the experimental value of the monopole energy in a given nucleus [11,12]. Then using this constrained energy functional, the value of K ∞ is extracted.

Within the relativistic and non-relativistic frameworks, K ∞ can be fixed with an accuracy of ∼ 10% [12]: K ∞ ≈ 230-250 MeV. Part of the uncertainty is due to our poor knowledge of the symmetry energy [12]. Therefore, the investigation of the GMR along isotopic chains should be relevant to probe the role played by the symmetry energy in the determination of the nuclear matter incompressibility. Moreover, the incompressibility of asymmetric matter is a basic parameter in calculations describing neutron stars or supernovae [START_REF] Glendenning | Compact Stars[END_REF]. Measurement of the GMR in neutron-rich systems may be a milestone towards testing the energy functionals to predict the equation of state in asymmetric infinite matter.

It is therefore tantalizing to achieve a pioneering method to measure the GMR in unstable nuclei. The 56 Ni nucleus plays a major role in stellar nucleosynthesis: it was observed in the ejecta from supernova 1987A [START_REF] Bethe | [END_REF] and is related to the so-called iron peak. This provides strong motivation to investigate the structure of this nucleus, as stressed in Ref. [15]. The measurement of GMR and GQR in 56 Ni also represents a first step in the study along the Ni isotopic chain.

The best probes for the investigation of the GMR and the Giant Quadrupole Resonance (GQR) are isoscalar probes such as deuterons or alpha particles at energies between a few tens and a few hundreds A.MeV [START_REF] Harakeh | Giant Resonances[END_REF]. No such experiment has been performed for unstable nuclei up to now, due to the difficult conditions in reverse kinematics. Indeed, the GMR cross section peaks at 0 deg. in the center of mass frame which gives rise to very low recoil velocities for the light probe. To measure the excitation energy range between 0 to 30 MeV in reverse kinematics, it is necessary to detect the recoiling particle (d or α) with energies ranging from 100 keV to 2 MeV at angles from 0 to 40 degrees in the laboratory frame. A standard set-up with a recoiling particle telescope would necessitate a very thin target (∼ 100 µg/cm 2 ) to minimize straggling and thus require an intensity of over 10 7 pps which is prohibitive for current radioactive beam facilities.

With respect to the above experimental constraints an active target such as Maya [16] is the key to measuring the GMR and GQR in unstable nuclei. In an active target, the detector gas also acts as target. Such a set-up has in principle an angular coverage close to 4π, a low energy threshold and a large effective target thickness. In this Letter we report on the first measurement of the GMR and the GQR with a radioactive beam using this new experimental technique.

In the domain of secondary beams, the active target archetype is the detector IKAR [17] which was used at the GSI facility. Maya, developed at GANIL for the lowerenergy domain [16], can be characterized as a 28x26x20 cm 3 Time and Charge Projection Chamber. The electrons from the ionization of the gas by particles drift down the electric field to amplifying wires set parallel to the beam. For a two body reaction, scattered and recoiling particles are in a plane which can be determined by the drift time to the wires. The amplified signal is induced on the anode, a matrix of 35 by 34 pads connected to Gassiplex [18] chips. A hexagonal structure was chosen for these pads in order to have the best conditions for the reconstruction of the projected recoil trajectory, independent of the recoil direction.

The secondary 56 Ni beam at 50 A.MeV (± 0.5% energy spread) was produced at the GANIL facility by fragmentation of 58 Ni at 75 A.MeV on a 70.5 mg/cm 2 C target located in the SISSI [19] device and purified by passing through a 135 mg/cm 2 Al degrader placed between the two dipoles of the alpha fragment separator before being sent to the SPEG area. Maya was placed on the focal plane of the SPEG spectrometer [20] which was used to purify the beam, so only 56 Ni 27+ was transmitted. Two plates were added above and below the beam trajectory in Maya to prevent the highly ionizing beam particles from inducing charges. However, due to the angular spread of the beam, some non-interacting 56 Ni ions were detected. This limited the beam intensity that could be used to 5.10 4 pps, while 10 6 pps of 56 Ni were available.

The Maya active target was filled with deuterium gas, at a pressure of 1050 mbar, forming a pure deuterium target of 1.6 mg/cm 2 (equivalent to a target of 6.3 mg/cm 2 of CD 2 ). It should be noted that alpha particle scattering, which would be the preferred probe for the GMR, could not be undertaken because the detector, like any gaseous detector, sparked when filled with pure He. As pointed out above, reverse kinematics generate recoil particles in a large energy domain. High energy light particles such as deuterons with E ≥ 2 MeV were not stopped in the Maya gas volume. For such escaping particles, we added ancillary 500 µm Si detectors outside the active volume covering from 10 to 60 degrees around the beam direction for events occurring at the center of the detector. For normalization purposes the beam was counted in a polycrystalline diamond detector of 1 cm 2 surface and 100 µm thickness, after traversing Maya.

For each event two physical observables are determined in order to reconstruct the kinematics of the reaction : the energy and the angle of the recoil. The recoiling angle in the horizontal plane is reconstructed by a linear least squares fit of the positions of the pads hit with a weight proportional to the amount of charge deposited. In order to obtain a reliable trajectory, at least 10 pads with non-zero charges were required corresponding to 5 cm long trajectories, which introduced an effective deuteron threshold of 700 keV. The reaction plane is determined from the drift times of the electrons towards the anode wires. The intersection of the deuteron trajectory with the beam axis gives the vertex of the interaction. The recoil stopping point is determined by projecting the charges on the direction of the trajectory whose maximum corresponds to the Bragg peak. For each event, the recoil angle and the range in the deuterium gas are therefore determined. The energy of the recoil is then deduced from its range in the gas through energy-loss tables [21]. The 3D-reconstruction of scattering trajectories in Maya provides the kinematics of the 56 Ni(d,d') reaction, shown in figure 1. The decrease of intensity to small recoiling deuteron angles is mainly due to the shielding of the beam, leading to an acceptance cut.

The data are then transformed into the center of mass frame using two-body relativistic kinematics. The corresponding excitation energy spectrum of 56 Ni corrected for the geometrical efficiency of the detector, is shown in figure 1. This spectrum presents a peak centered at 0 MeV corresponding to elastic and inelastic scattering towards low-lying states. It must be noted that due to the accessible deuteron energy range, the elastic scattering is measured over only a very small angular domain in the center of mass frame, precluding the extraction of the corresponding angular distribution. For the elastic peak the FWHM is 3 MeV and is mainly induced by the trajectory reconstruction uncertainties, especially the vertex localisation due to the beam shielding. At energies ranging from 12 to 25 MeV a bump in the excitation energy spectrum points to the presence of isoscalar giant resonances. Only deuterons corresponding to excitation energy below 25 MeV are stopped in Maya, whereas the Si detectors placed 70 mm behind the Maya detector allow the energy spectrum between 28 MeV and 35 MeV to be measured. Since the pressure and the bias used in the experiment were optimized to detect very low deuteron energies, it was not possible to separate protons from deuterons through range vs. charge measurements [16]. Consequently some proton contamination may be expected in the data. Protons can be produced essentially by two mechanisms : deuteron break-up and one neutron transfer (d,p) reactions. The latter will not be considered since the lowest energy protons are scattered at backward angles. The contribution of the deuteron break-up was estimated from direct kinematic measurements for 58 Ni at 50 A.MeV at larger angles (θ CM > 8 deg.) [22]. These data are extrapolated to lower angles and transformed to the reverse kinematics frame, corrected for the geometrical acceptance of Maya and normalized to the experimental data (Fig. 1). The shape of the deuteron breakup does not exhibit any bump in the giant resonance region but rather corresponds to a smooth background. In order to extract the GMR and GQR centroid location and the corresponding angular distributions, a Multipole Decomposition Analysis (MDA) is performed [START_REF] Harakeh | Giant Resonances[END_REF]. After subtraction of the background, experimental angular distributions were extracted for five excitation energy intervals and fitted by a linear combination of theoretical distributions for L=0 and L=2 excitations. The theoretical angular distributions are calculated using the Distorted Wave Born Approximation (DWBA). Both real and imaginary optical potentials are microscopically determined using 56 Ni and deuteron densities within the double folding model with the M3Y interaction as in [23]. The deuteron density is obtained from the Hulthen deuteron wave function [START_REF] Hulthén | The two-nucleons Problem[END_REF]. The 56 Ni nucleus densities are calculated with the Hartree-Fock model and the SGII (K ∞ =217 MeV) [5] Skyrme functional without including pairing since it is a doubly magic nucleus. The corresponding L=0 and L=2 strength functions are well concentrated, similar to those of Fig. 3 and 28, respectively, of Ref. [START_REF] Terasaki | [END_REF]. No data are available for deuteron elastic scattering from the unstable 56 Ni nucleus therefore the optical potentials are renormalized in order to reproduce 58 Ni(d,d) at 120 MeV [26] : the real part of the optical potential is increased by 7 %. The GMR and GQR transition potentials are calculated with the same procedure, using microscopic RPA transition densities [27] obtained self-consistently from the same Skyrme functional as in the HF calculation. It should be noted that it is the first time that the GMR is measured for such a light nucleus using the (d,d') reaction. Consequently, the validity of reaction models cannot be thoroughly tested on stable nuclei for this reaction.

Fig. 2 shows the angular distributions whose shapes are well reproduced by the calculations. As expected, the L=2 mode dominates below 17 MeV whereas the L=0 mode is the largest above 17 MeV. The percentage of the exhausted Energy-Weighted Sum Rule (EWSR) is determined for each energy interval for both multipolarities. It is then straightforward to obtain the centroid of the giant resonances and the total exhausted EWSR (Table I). Error bars are purely statistical and do not take into account uncertainties on the background or on the reaction models. In order to cross-check the results, another independent analysis of the data was undertaken by fitting the excitation energy spectrum with the background shape and two Gaussians corresponding to the GMR and the GQR, for different angles between 3 deg. The 56 Ni results are comparable to those obtained on 58 Ni and 56 Fe with (α,α') scattering [28,29] (Table I).

The FWHM obtained by the Gaussian fits on the 56 Ni data (5.2 ± 0.5 MeV both for the GMR and the GQR) are also consistent with neighbouring stable nuclei where FWHM between 5.5 MeV and 7.7 MeV are found [28]. With only 15 hours of effective data taking and an average beam intensity of 5.10 4 pps, isoscalar GMR and GQR resonances were measured in the 56 Ni unstable nucleus. It is the first measurement of these resonances in a short-lived nucleus. This makes the method relevant with respect to predicted production rates of a large domain of exotic nuclei at current and future facilities. A redesign of the detector in order to increase the amplification gain would allow to access to smaller angle measurements and improve particle identification. Measurements in neutron-rich Nickel isotopes up to 68 Ni could be undertaken as of now in order to probe the evolution of giant resonances along an isotopic chain. Also, measurements on heavier nuclei such as 132 Sn would complement the systematics already existing for stable tin isotopes [30].

FIG. 1 :

 1 FIG. 1: Top (color online): Scatterplot of recoiling deuteron energy versus scattering angle in the laboratory frame for the 56 Ni beam. The lines correspond to elastic scattering and the 14 MeV-22 MeV excitation energy of 56 Ni. The CM angles ares denoted by squares along these lines, with 1 deg. step, starting from 0 deg. for the lowest deuteron energy. Bottom left: geometrical efficiency of the detection setup in the CM frame. Bottom right: 56 Ni excitation energy spectrum deduced from the deuteron kinematics and corrected for geometrical efficiency. The background which was subtracted is shown by the solid line. The insert shows the background subtracted inelastic data fitted with Gaussians located at 16.5 MeV and 19.5 MeV for the GQR and the GMR, respectively (see text).

FIG. 2 :

 2 FIG. 2: GMR and GQR inelastic angular distributions of 56 Ni at 50 A•MeV and DWBA calculations using microscopic double folded RPA form factor (see text). The data (squares) are also displayed. Bottom right: analysis of the data obtained by using two Gaussians located at 16.5 MeV and 19.5 MeV for the GQR and the GMR, respectively, with FWHM=5.2 MeV (see text).

  and 7 deg. The angular distributions of the Gaussians are shown in the bottom right part of Fig. 2 and are well reproduced by DWBA calculations. The percentage of EWSR extracted by normalisation are in close agreement with the MDA results.

TABLE I :

 I GMR GQR Ec (MeV) % EWSR Ec (MeV) % EWSR 56 Ni 19.3 ± 0.5 136 ± 27 16.2 ± 0.5 76 ± 13 58 Ni [28] 19.20 +0.44 GMR and GQR centroids and EWSR measured in 56 Ni, 58 Ni and 56 Fe

	-0.19	85 +13 -10	16.31 +0.17 -0.10 82 ± 10
	58 Ni [29] 19.9 +0.7 -0.8	92 +4 -3	16.3 +0.8 -0.9	73 ± 3
	56 Fe [28] 18.35 +0.33 -0.19	98 +14 -10	16.35 +0.19 -0.10 77 ± 10

This work has been supported in part by CNRS/IN2P3, CEA/DAPNIA and by the U.S. National Science Foundation (Grant No. PHY04-57120).