W Kang

J-P P Barbot

Discussions on observability and invertibility

In this paper, new definitions of observability are introduced. They include states and/or inputs estimation, as well as partial or global observability. From these definitions, problems of left invertibility, fault detection, observability, identifiability, unidirectional synchronization,... can be studied in the same context. Moreover, we propose the definitions for both nonlinear continuous-time systems and switched systems. Two examples are introduced to illustrate the efficiency of the proposed definitions and related propositions introduce in the paper.

Introduction

In this paper, the concept of observability for nonlinear systems and switched systems is investigated with new definitions in an extended context. Relative to the classical concept of observability, the proposed definitions are applicable to a much broader family of variables including unmeasured state, unknown input, unknown parameters, and even uncertainties and disturbances in a control system. Moreover, the proposed definition takes into account the notion of partial observability which may be useful, for example, in the context of networked systems. In addition, the first definition in this paper has a distinctive extension to switched systems and thus taking into account also the partial observability with respect to the time (i.e. some variable may be unobservable during some bounded time interval). It is well known that the definition of linear observability is universal for all linear systems, but there exist many different definitions of observability in the literature of nonlinear control systems. Without exhaustibility purposes, these 1 definitions include weakly local observability (Herman & al., 1977), algebraic observability [START_REF] Diop | On nonlinear observability[END_REF], infinitesimal observability [START_REF] Gauthier | Deterministic Observation Theory & Applications[END_REF], unboundedness observability [START_REF] Angeli | Forward completeness, unboundedness observability and their Lyapunov characterisations[END_REF],... It is a technical question at the crossway of several factors such as the generality of the concept, easy to check, and practical observer design. In this paper, the notation Z represents a variable or function associated with a control system. The first definition of observability proposed in this paper is the Z-observability. It is an effort to take into account the consideration of three factors that are not considered all together in existing definitions:

• Partial observability (observability of a part of the system variables, not all the state variables)

• Observability with partial model of a dynamical system, i.e. part of the equations in the system model is completely unknown (in contrast to the case with full known model with possibly unknown parameters);

• The observability of unknown input and unknown parameters.

This new definition is inspired by the observability definition adopted by moving horizon type of observers. However, the following definition has some fundamental differences from the existing ones. It defines the observability of individual variables, instead of the entire system. In addition to the classical case of observability of the entire system, this definition is applicable to the estimation of a subset of variables. It is also applicable to the unknown input and unknown parameters in a system. Consequently, this definition can be easily applied to fault detection and, more particularly, to some approaches using subsystems of connected or switched systems. Such definition can also be used in many observer based applications, such as cryptography of chaotic systems; identification problems; and general left invertible problems.

After the introduction of Z-observability, it is introduced the notion of Z(T N )-observability for switched systems. This new definition takes into consideration of five factors not considered all together in classical definitions:

• Partial state observability

• Observability with partial model of a dynamical system

• The observability of unknown input and unknown parameters

• Systems with algebraic constraints

• Partial time observability

One of the main advantage of this is to relax the assumption that requires all switched subsystems be observable (see for example [START_REF] Bemporad | Observability and Controllability of Piecewise Affine and Hybrid Systems[END_REF][START_REF] Boutat | About the observability of piecewise dynamical systems[END_REF][START_REF] Vu | Invertibilty of switched linear systems[END_REF]). The efficiency of both definitions Z-observability and Z(T N )-observability are highlighted with two examples: the first one is dedicated to Fault Detection in networked oscillator and the second one is dedicated to an academical switched system.

2 Z-observability for smooth system

Consider a general nonlinear control system Definition 1 The function z = Z(t, x, u) is said to be Z-observable in U with respect to the system 2.1 if for any two trajectories, (t, ξ i (t), u i (t)), i = 1, 2, in U defined on a same interval [t 0 , t 1 ], the equality

ξ = f (t, ξ, u), ξ ∈ R n , u ∈ R m Y = h(t, ξ, u) (2.1) Suppose z = Z(t,
h(ξ 1 (t), u 1 (t)) = h(ξ 2 (t), u 2 (t)), a.e. in [t 0 , t 1 ] implies Z(t, ξ 1 (t), u 1 (t)) = Z(t, ξ 2 (t), u 2 (t)), a.e. in [t 0 , t 1 ]
Now, suppose for any trajectory (t, ξ(t), u(t)) in U , there always exists an open set

U 1 ⊂ U so that Z(t, x, u) is Z-observable in U 1 . Then, z = Z(t, x, u) is said to be locally Z-observable in U .
For simplicity, "Z-observable in U " means "Z-observable in U with respect to the system 2.1". In linear control theory, the control input is assumed to be a known variable. So, Y = [y, u] T where y is the output in the classical definition. Then, the observability of a linear control system is equivalent to the observability of all state variables in x. For nonlinear systems, all the state variables in an uniformly observable system must be observable under the definition 1 with the function Z(t, x, u) equal to x and Y = [y, u] T . However, a system that is not uniformly observable may be observable under the definition 1 with particular Z(t, x, u). Thus, a system could be unobservable under all existing definitions, but part of the state variables can be observable. This partial observability is a new concept that is especially important for subsystems in a system of systems, in which the observability of the entire system is either impossible or unnecessary.

It is also important to notice that the input u is a variable treated equally as the state variable x in the definition 1. So, it automatically handles both the left invertibility problem and the parameter identification problem. Moreover, it will be proved later that z is Z-observable implies ż is also Z-observable (when ż exists), no matter we know or don't know the dynamic equations in the model that are not directly related to ż . The local Z-observability is defined differently from some typical local properties. In fact, the meaning of local in Definition 1 is not around a point in the state space. The local neighborhood U 1 is around a trajectory. A local observer around a single point would be too restrictive for many applications. Meanwhile, it is possible to design nonlinear observers applicable to non-local trajectories, but with local initial estimation (the observer converges if the initial estimation error is small), like the observer in [START_REF] Krener | Locally Convergent Nonlinear Observers[END_REF]. If a variable is observable, it can be estimated by Kalman type of filters (EKF or UKF), or moving horizon type of observables [START_REF] Gong | A Pseudospectral Observer for Nonlinear Systems, Discrete and Continuous Dynamical Systems, Series B[END_REF] and (Gong & al., 2007), sliding mode observer [START_REF] Levant | Robust Exact Differentiation via sliding mode terchnique[END_REF], high gain observer [START_REF] Gauthier | Deterministic Observation Theory & Applications[END_REF],... but the choice and design of observers are beyond the scoop of this paper.

In the next, we prove several sufficient conditions for the Z-observability. Some of them are numerically checkable. For the simplicity of notations, we denote ξ 0 = ξ, ξ 1 = u. We define the differentiation operator D by

D(ξ j ) = ξ j+1 , j ≥ 1 D(K(t, ξ 0 , ξ 1 , • • • , ξ j )) = ∂K ∂t + ∂K ∂ξ 0 f (ξ 0 , ξ 1 ) + ∂K ∂(ξ 1 , ..., ξ j ) ξ 2 , ..., ξ j+1 T Lemma 1 Suppose there exists a function g(•) so that z = Z(t, ξ, u) = g(t, h, Dh, • • • , D k-1 h)
for all (t, ξ, u) ∈ U (the equality implies that the value of ξ j+1 can be arbitrary for j ≥ 1). Then z(t, x, u) is Z observable.

Proof. Given two trajectories (ξ 1 (t), u 1 (t)) and (ξ 2 (t), u 2 (t)) in U with outputs Y 1 (t) and Y 2 (t). Suppose

Y 1 (t) = Y 2 (t), t ∈ [t 0 , t 1 ] (2.2) It is obvious that for i = 1, 2 Ẏ i (t) = D(h(t, ξ i , u i )) ξ0=ξ i (t),ξ1=u i (t),ξ2= ui (t) d j dt j (Y i (t)) = D j (h(t, ξ i , u i )) ξ0=ξ i (t),••• ,ξj+1=(u i ) (j) (t)
.

Because of (2.2), we know

d j dt j (Y 1 (t)) = d j dt j (Y 2 (t)) for 0 ≤ j ≤ k -1. Therefore, z(t, ξ 1 (t), u 1 (t)) = g(t, h, Dh, .., D k-1 h)| ξ 1 (t),..,(u 1 ) (k-1) (t) = g(t, Y 1 (t), Ẏ 1 (t), .., d k-1 dt k-1 Y 1 (t)) = g(t, Y 2 (t), Ẏ 2 (t), .., d k-1 dt k-1 Y 2 (t)) = z(t, ξ 2 (t), u 2 (t))
So, the variable z is observable.

Lemma 2 Let U ⊂ R × R n × R m be
an open set. Suppose z ∈ R is a scaler valued variable. Suppose there are k functions:

v = v 1 (t, ξ 0 , ξ 1 ) • • • v k (t, ξ 0 , ξ 1 )
T from the row components of the vector

(Y T , DY T , ..., D l-1 Y T ) T (2.3) so that (1) ∂v ∂ξ j = 0, for j ≥ 2, (t, ξ 0 , ξ 1 ) ∈ U (2) rank ∂v ∂(ξ 0 , ξ 1 ) = k, for (t, ξ 0 , ξ 1 ) ∈ U (3) for all (t, ξ 0 , ξ 1 ) ∈ U ∂ ∂(ξ 0 , ξ 1 ) Z(t, ξ 0 , ξ 1 ) ∈ row space of ∂v ∂(ξ 0 , ξ 1 )
,

Then z = Z(t, ξ, u) is locally Z-observable in U .
Proof. Given any point (t 0 , ξ 0 , u 0 ) in U .

There exists an open ball B(t 0 , ξ 0 , u 0 ) of (t 0 , ξ 0 , u 0 ) in R × R n × R m and a mapping

w(t, ξ, u) : R × R n × R m → R n+m-k so that (t, v(t, ξ, u) T , w(t, ξ, u) T )
T is a diffeomorphism on B(t 0 , ξ 0 , u 0 ). Therefore,

∂z ∂w = ∂z ∂(ξ, u)    ∂ξ ∂w ∂u ∂w   
However, there exists a row vector a

(t) ∈ R k so that ∂z ∂(ξ, u) = a(t) ∂v ∂(ξ, u) we have ∂z ∂w = a(t) ∂v ∂(ξ, u) ∂ξ ∂w ∂u ∂w = a(t) ∂v ∂w = 0 Therefore, z = Z(t, ξ(t, v, w), u(t, v, w)) is a function of v, which is in- dependent of w. Because v is from the vector (2.3), z is a function of Y, DY, • • • , D l-1 Y . Therefore, z = o(t, ξ, u) is Z-observable in B(t 0 , ξ 0 , u 0 ).
Given any trajectory (t, ξ(t), u(t)) in U . Then it is contained in some compact subset of U . There exist finite many open balls B(s 1 , ξ(s 1 ), u(s 1 )), B(s 2 , ξ(s 2 ), u(s 2 )), • • • , B(s p , ξ(s p ), u(s p )) so that the entire trajectory stays inside the open set

U 1 = ∪ p i=1 B(s i , ξ i , u i )
and z is Z-observable in every ball. There exists an open subset U 2 ⊂ U 1 containing the trajectory so that, for any two points (t, ξ i , u i ) ∈ U 2 with the same t, both points must lie in a same ball B(s i , ξ(s i ), u(s i )). In U 2 , z must be Z-observable because it is observable in all the balls B(s i , ξ(s i ), u(s i )). So, z is locally Z-observable in U .

Lemma 3 Consider a control system

ξ = f (t, ξ, u) Y = h(t, ξ, u) Assume z = Z(t, ξ, u) is Z-observable in U satisfying ∂Z(t, ξ, u) ∂u = 0
Then,

Dz = ∂Z(t, ξ, u) ∂t + ∂Z(t, ξ, u) ∂ξ f (t, ξ, u) is Z-observable in U .
Proof. Given two trajectories (t, ξ 1 (t), u 1 (t)) and (t, ξ 2 (t), u 2 (t)). Assume h((t, ξ 1 (t), u 1 (t)) -h((t, ξ 2 (t), u 2 (t)) = 0 for all t.

Then, Z(t, ξ 1 (t), u 1 (t)) = Z(t, ξ 2 (t), u 2 (t)) for all t. Therefore,

Dz| ξ=ξ 1 (t),u=u 2 (t) = d dt Z(t, ξ 1 (t), u 1 (t)) = d dt Z(t, ξ 2 (t), u 2 (t)) = Dz| ξ=ξ 2 (t),u=u 2 (t)
So, Dz is Z-observable.

It is also interesting to deduce the Z-observability of variables constrained by algebraic equations. It is a result applicable to systems defined by DAEs.

Lemma 4 Consider a sequence of functions

z i = Z i (t, ξ, u), i = 1, 2, • • • , k defined in an open set U . Suppose z i is locally observable in U for i ≥ k 1 +1. Suppose g(•) is a C 1 function defined in the range of (z 1 , • • • , z k ) satisfying g(z 1 , z 2 , • • • , z k ) = 0 (2.4)
along all trajectories and

∂g ∂(z 1 , • • • , z k 1 ) = 0 Then, z 1 = Z 1 (t, ξ, u), • • • , z k 1 = Z k 1 (t, ξ, u) are locally Z-observable in U .
Proof. Given any point (t 0 , ξ 0 , u 0 ) in the closure of a trajectory (t, ξ(t), u(t)) (If u(t) is continuous, no need to consider the closure). Consider the point

(z 10 , • • • , z k0 ) = (Z 1 (t 0 , ξ 0 , u 0 ), • • • , Z k (t 0 , ξ 0 , u 0 )) in R k . Then there exists an open neighborhood V 1 around (z 10 , • • • , z k 1 0 ) in R k 1 ; and an open neighborhood V 2 around (z k 1 +1 0 , • • • , z k0 ) in R k-k 1 so that, for any point (z k 1 +1 , • • • , z k ) in V 2 , there exists a unique point (z 1 , • • • , z k 1 ) in V 1 satisfying g(z 1 , z 2 , • • • , z k ) = 0
There exists an open ball B(t 0 , ξ 0 , u 0 ) around (t 0 , ξ 0 , u 0 ) so that for any (t, ξ, u) in B(t 0 , ξ 0 , u 0 ),

(Z k 1 +1 (t, ξ, u), • • • , Z k (t, ξ, u)) ∈ V 2 (Z 1 (t, ξ, u), • • • , Z k 1 (t, ξ, u)) ∈ V 1
According to the assumption, we can assume

Z k 1 +1 (t, ξ, u), • • • , Z k (t, ξ, u)
are all observable in B(t 0 , ξ 0 , u 0 ). Given two trajectories (t, ξ 1 (t), u 1 (t)) and (t, ξ 2 (t), u 2 (t)) in B(t 0 , ξ 0 , u 0 ). If they have the same output, then for

i = k 1 + 1, • • • , k Z i (t, ξ 1 (t), u 1 (t)) = Z i (t, ξ 2 (t), u 2 (t)).
Because

z 1 = Z 1 (t, ξ, u), • • • , z k 1 = Z k 1 (t, ξ, u) satisfy (2.4
) and because the equation has a unique solution, we know that

Z i (t, ξ 1 (t), u 1 (t)) = Z i (t, ξ 2 (t), u 2 (t)) for i = 1, • • • , k 1 . So, z 1 , • • • , z k 1 are Z-observable in B(t 0 , xi 0 , u 0 ).
Because the entire trajectory (t, x(t), u(t)) can be covered by finite many ball in which

z i = Z i (t, ξ, u), i = 1, • • • , k 1 , are Z observable,
there exists an open neighborhood around the trajectory in which

z i = Z i (t, ξ, u), i = 1, • • • , k 1 , are Z observable. Therefore, z 1 = Z 1 (t, ξ, u), • • • , z k 1 = Z k 1 (t, ξ, u) are locally Z-observable in U .

Z(T N )-observability

Let us consider the following class of switched systems:

ξ = f q (t, ξ, u), q ∈ Q, ξ ∈ R n , u ∈ R m Y = h q (t, ξ, u) (3.5) Where Q is a finite index set, f q R × R n × R m → R m is sufficiently smooth, the dwell time [t i,0 , t i,1
] between two switches of the structure (i.e. change of q ) is such that ∃τ min > 0 /(t i,1 -t i,0 ) > τ min (this exclude Zeno phenomena).

For the input u in any time interval [t i,0 , t i,1 [⊆ [t ini , t end [, we assume that u(t) is bounded and C ∞ . For switched systems the observation concept is strongly linked to the dwell time and the sequence of switch, thus it is important to recall (in our context) the following definition of hybrid time trajectory [START_REF] Lygeros | Dynamical Properies of Hubrid Automata[END_REF] (see also (Goebel & al., 2004)).

Definition 2 A hybrid time trajectory is a finite or infinite sequence of intervals T N = {I i } N i=0 , such that

• I i = [t i,0 , t i,1 [, for all 0 ≤ i < N ;
• For all i < N t i,1 = t i+1,0

• t 0,0 = t ini and t N,1 = t end Moreover, we define T N as the ordered list of q associated to T N (i.e. {q 0 , ..., q N } with q i the value of q during the time interval I I ).

From these it is possible to define a new concept of observability:

Definition 3 The function z = Z(t, ξ, u) is said to be Z(T N )-observable in U with respect to system 3.5 and hybrid time trajectory Suppose for any trajectory (t, ξ(t), u(t)) in U , there always exists an open set U 1 ⊂ U so that (t, ξ(t), u(t)) is contained in U 1 and Z(t, x, u) is Z(T N )observable in U 1 with respect to system 3.5 and hybrid time trajectory T N and T N . Then, z = Z(t, x, u) is said to be locally Z(T N )-observable in U with respect to system 3.5 and hybrid time trajectory T N and T N .

Lemma 5 Consider system 3.5 and hybrid time trajectory T N and T N .

Let U be an open set in time-state-control space. Suppose Z(t, x(t), u(t)) is always continuous under admissible control input. Suppose there exist N + 1 linear projections {P i } so that

(1) given any 0 ≤ i ≤ N , P i Z(t, ξ, u) is Z-observable for t ∈ [t i,0 , t i,1 [ and (t, ξ(t), u(t)) ∈ U ;

(2) Rank{P T 0 , ..., P T N } = dim(Z) = n z ; and

(3) d Pi Z(t, ξ(t), u(t)) dt = 0 for t ∈ [t i,0 , t i,1 [ and (t, ξ(t), u(t)) ∈ U , where

{ P T i , P T i } has full rank in R nz×nz . Then, z = Z(t, ξ, u) is Z(T N )
-observable in U with respect to system 3.5 and hybrid time trajectory T N and T N Proof. By the assumption, within each I i the unobservable parts of the function Z are constant; and they are observable in some other time intervals. More specifically, the integration of condition (3) yields Pi Z(t, ξ(t), u(t)) = Constant for t in I i . However, P j Z(t, ξ(t), u(t)) is uniquely determined in the time interval I j . Because Rank{P T 0 , ..., P T N } = n z , Pi Z is unquely determined. Because { P T i , P T i } has full rank, lemma 4 ensures that Z(t, ξ(t), u(t)) is observable in I i for all 0 ≤ i ≤ N .

In the following, "Z(T N )-observable in U " means "Z(T N )-observable in U with respect to system 3.5 and hybrid time trajectory T N and T N ".

Remark 1 The Hybrid time trajectory T N and T N influence the observability property in the way similar to an input. Therefore, in the future work it is interesting to define the universal hybrid time trajectory as the time trajectory which preserves the observability (see [START_REF] Gauthier | Deterministic Observation Theory & Applications[END_REF] for the definition of universal input).

Remark 2 Thus if system 3.5 is Z(T N )-observable, then it is possible to design an observer for the system. But this is beyond the scoop of the paper.

It is important to mention that all lemmas of the section 2 are valid for each time interval I i . However, the generalization of the Z(T N ) observability with nonlinear projections is not so obvious because, in this case, the projection can be function of unknown parameters. In the next section the usefulness of the Z-observability is highlighted by a system of networked oscillators.

Network Systems

This section is dedicated to a networked system, more particularly to the influence of the small world effect with respect to the Z-observability. In [START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF], D.S. Watts and S.H. Strogatz have given an example of small-world effect for a network with twenty vertices (see figure 1). where ξ i,1 and ξ i,2 are the two variables state of the i th vertex. The dynamic at each vertex is similar. It is an oscillator coupled with four other ones. Moreover, only four a i,j = 0.25 all other one are equal to zero. And, m i is the fault. Only some variables ξ i,1 are measurable. First of all let us consider the following regular network (i.e. a i,i-2 = a i,i-1 = a i,i+1 = a i,i+2 = 0.25 all other a i,j = 0, where a i,21 = a i,1 , a i,22 = a i,2 , a i,0 = a i,19 and a i,-1 = a i,18 ). Consider the case of no fault in the system. With one output, the system is not observable, and the dimension of the observable space is only 20; with two successive outputs the dimension of the observable space is 36; with three successive outputs the dimension of the observable space is 38; and only for four and more successive outputs the regular system is observable. Moreover, if a fault detection problem is considered vertex by vertex, then the state of a vertex can be observable

Figure 1 :

 1 Figure 1: Networked of Watts and Strogatz

  x, u) is a variable to be estimated. In the following, U represents an open and connected set in the time-state-control space R × R n × R m . For the input in any time interval [t 0 , t 1 ], we assume that u(t) is bounded and C ∞ at all but finite many points in [t 0 , t 1 ]. At each discontinuous point t i ∈]t 0 , t 1 [, lim t→t +

	i	u(t) and lim t→t -i	u(t) exist. For the
	state, we assume ξ(t) absolutely continuous. If (ξ(t), u(t)) satisfies the dif-
	ferential equation in (2.1) for all t in [t 0 , t 1 ] except for finite many points,
	then (t, ξ(t), u(t)) is called a trajectory. In this note, equations involving
	u(t) always mean " equal almost everywhere," and the notation is "a.e. in
	[t 0 , t 1 ]."

  T N and T N if for any two trajectories, (t, ξ i (t), u i (t)), i = 1, 2, in U defined on the interval [t ini , t end ], the equality

		h(t, ξ 1 (t), u 1 (t)) = h(t, ξ 2 (t), u 2 (t)), a.e. in [t ini , t end ]
	implies	Z(t, ξ 1 (t), u 1 (t)) = Z(t, ξ 2 (t), u 2 (t)), a.e. in [t ini , t end ]

even if the entire system is not. Define Z(t, ξ, m) for the i th vertex as follows Z(t, ξ, m) = (ξ i,2 , m i ) T

The outputs are defined by ξ i-2,1 = y -2 , ξ i-1,1 = y -1 , ξ i,1 = y 0 , ξ i+1,1 = y 1 and ξ i+2,1 = y 2 the following differential algebraic equations are obtains:

Then the lemma 1 guaranty that

In this case, the entire system is not observable without information about all m i . However, the fault of a vertix is observable Now consider the same network (with the same vertex dynamics) but with small world effect (see figure 1). Assuming no fault, then with one output the dimension of the observable space is 36; with two successive outputs the system is observable. This seems a beneficial effect on observability of small world network architecture (i.e. less outputs are necessary in order to observe the full state of the system). Nevertheless, from this architecture and without assumption on the m i , it is impossible to detect fault on each vertex with only the knowledge of the two upper and lower outputs neighborhood (i.e. ξ i-2,1 = y -2 , ξ i-1,1 = y -1 , ξ i,1 = y 0 , ξ 1+1,1 = y 1 and ξ i+2,1 = y 2 ). This shows that the sensors repartition is of first importance for fault detection in network with small world structure. Moreover the choice of sensor localization must be done with the Z-observable concept because detection of m i is requested and the observation of the full state is not necessary.

Switched System

Let us consider the following switched system without jump.

where y = x 1 is the output. The hybrid time trajectory T N is the set of following time interval [iτ, (i + 1)τ [ ∞ i=0 with τ = τ min > 0 and < T N > the order list of q i = (q i,1 , q i,2 ) is defined as follow:

q 1 = (1, -1) and for j = 3k + 1, q j = q 1 q 2 = (-1, 1) and for j = 3k + 2, q j = q 2 q 3 = (1, 1) and for j = 3k, q j = q 3

For a function z = Z(t, x) = (x 1 (t), x 2 (t), x 3 (t)) T , a choice of projections matrixes is:

Moreover, the P j and Pj for j > 3 are defined as follow:

P j = P 1 and Pj = P1 for j = 3k + 1 P j = P 2 and Pj = P2 for j = 3k + 2 P j = P 3 and Pj = P3 for j = 3k

From the lemma 5 it is clear that all state components are Z(T N ) observable (if the hybrid time trajectory ends at least after 3τ ), even if the function z is not observable for any couple (T i , q i ).


Rafaël Hernandez

Damasceno Dos Passos

Madjid Arab

Carlson Pereira

De Souza

Christine Leroux
email: leroux@univ-tln.fr

□ 1/7 WO 4 ; a new modulated ternary scheelite compound

Keywords: scheelite, ordering, vacancies, incommensurate, high resolution electron microscopy, simulations, x ray diffraction, band gap

Synopsis -A new ternary scheelite tungstate (Sr,Ce)WO 4 phase was synthesized, with a 2D incommensurately modulated structure linked to partial ordering of Sr, Ce cations and vacancies as was evidenced by high resolution electron microscopy. The bang gap of 3.2 eV makes it a promising violet emitting compound.

Introduction

Scheelite molybdates or tungstates are known to be chemically stable luminescent compounds as well as good ionic conductors. In the last decade, their potential use as light emitting materials was explored and many new scheelite compounds were synthesized, by adding lanthanide cations in the structure (Abakumov et al., 2014;Morozov et al., 2013, Nogueira et al.,2013, Giang et al.,2015). The binary scheelite tetragonal structure of general formula ABO 4 , (space groupe I4 1 /a) corresponds to (A 2+ ,B 6+ ) and the cation/anion ratio [A]/ [BO 4 ] is equal to 1. The structure, shown Fig. 1, can be described as a piling up of layers along the c axis built by [BO 4 ] tetrahedra and [AO 8 ] polyhedra sharing vertices. This structure accommodates easily cations with different oxidation states for A and B as in BiVO 4 (A 3+ ,B 5+ ), ZrGeO 4 (A 4+ , B 4+ ) or AgIO 4 (A + ,B 7+ ). When the cation/anion ratio is different from 1, the tetragonal structure of binary tungstate (B=W) or molybdate (B=Mo) scheelite can be distorted into a monoclinic structure as in Ce 2 W 3 O 12 (A 3+ ,B 6+ ) (Arab et al., 2013) or an orthorhombic one as in Ce 10 W 22 O 81 (Patout et al., 2014). Ternary scheelites refer to compounds with general formula (A,A') n (BO 4 ) m . Most of them are built with a charge difference of 2 between A et A' cations, as for example (Ag,RE)(MoO 4 ) 2 or (Li,RE)(MoO 4 ) 2 RE=rare earth elements (Abakumov et al., 2014). The charge difference is usually accommodated by cation vacancies on A sites, with or without cation ordering or incommensurate modulations (Abakumov et al., 2014). Ternary scheelite with a charge difference of 1 are uncommon, and apart from a lead-lanthanide based compound (Takai et al., 2004), only (Ca,Eu)(BO 4 ) 2 and (Sr,Eu)(BO 4 ) 2 were extensively studied (Abakumov et al., 2014;Iiang et al., 2015).

Among rare earth compounds with the scheelite structure, rare earth tungstates exhibit excellent thermal and chemical stability. Tungstates have been proven to be good host lattices for the luminescence of lanthanide ions (Zhou et al., 2014). Eu 3+ is a commonly used luminescent activator and Eu 3+ activated SrWO 4 are efficient light emitting diodes (LED) (Jiang et al., 2015). An interesting alternative to Eu 3+ is Ce 3+ because of a potential tunable light emission . The light emission of Ce 3+ varies from ultraviolet to orange, depending on the crystal field (Zhu et al., 2016;Li et al., 2008). The mechanism of Ce light emission is usually linked to the interaction of the incomplete 4f shell energy levels of Ce with the band state of the host lattice (E. Roy et al., 2010). The determination of the band gap is one way to obtain information of the potential optical properties of one compound. As the band gap value is also linked to the structural defects around the lanthanide ions, which are promoted by the distortions of angles and bond distances within different polyhedra clusters, (Cavalcante et al., 2012;Rosa et al., 2008;Lacomba-Perales et al., 2008), structural investigations have to linked to optical characterizations.

The aim of this work was to synthesize a (Sr,Ce)WO 4 ternary scheelite compound with different optical properties compared to those of the binary scheelite SrWO 4 . A complexing method was used to synthesized powders of binary SrWO 4 , and (Sr,Ce)WO 4 ternary tungstate scheelites. Rietveld refinements of the X ray diffraction patterns were performed to obtain accurate cell parameters and atomic distances. Electron diffraction and high resolution electron microscopy gave access to the incommensurate modulations and cation ordering. The chemical composition was determined by Energy Dispersive Spectroscopy (EDS) and the cation valence by Electron Energy Loss Spectroscopy (EELS). UV-Visible Diffuse Reflectance Spectra (DRS) was used to determine the energy band gap.

We showed that the (Sr,Ce)WO 4 has an incommensurate modulated structure linked to partial ordering of Sr 2+ and Ce 3+ . The influence of the structure on the band gap values was discussed in terms of bonds distances and angles splitting of the polyhedra WO 4 and Sr(Ce)O 8 .

Experimental procedures

Synthesis

The powders were synthesized by the so called EDTA-citrate complexing method, EDTA accounting for Ethylene-Diamine-Tetracetic-Acid. This method was already used to obtain binary tungstates SrWO 4 and Ce 2 W 3 O 12 , (Arab et al., 2013) and was adapted in this work to synthesize ternary tungstates. Strontium nitrate Sr(NO 3 ) 2 , cerium(III) nitrate hexahydrate Ce(NO 3 ) 3 .6H 2 O and tungsten oxide WO 3 were used as cation source. EDTA was diluted in ammonium hydroxide (27% vol.) NH 4 OH at 40°C with a constant stirring during 15 min. Solutions 0.3 M of strontium and cerium nitrates and WO 3 were added in the solution of the EDTA/ NH 4 OH in various proportions and stirred 15 minutes again. Then 0.45 M citric acid was added to initiate the complexation reaction at a temperature of 80°C and at pH =9. The reaction time lasted 3h until the appearance of a gel-like precipitate. This gel was subjected to a heat pretreatment at a temperature of 230°C for 3h to remove waste liquids and volatiles, leading to the formation of a black powder. The final annealing was performed at 1000°C for 5h, a temperature ensuring the crystallization of the gel, leading to yellowish-white powders.

Structural and chemical characterization

Transmission Electron Microscopy (TEM) and High Resolution Electron Microscopy (HREM) were performed with a conventional Tecnai 200 kV and with a Titan FEG 300 kV C s corrected, equipped with a Gatan Imaging Filter (GIF). Simulated HREM images were calculated using the multislice program JEMS. The parameters used for simulations of images taken with the 300 kV microscope were C s =-1.9µm, C c =2 mm, 1/2 α=1 mrad, ∆E=0.7eV. The chemical composition of the ternary tungstate was determined by Energy Dispersive spectroscopy (EDS). Binary compounds SrWO 4 and Ce 2 W 3 O 12 were used as standards for the determination of the K Sr,W and K Ce,W factors. The statistical study of powder's chemical homogeneity consisted in 20 EDS analyses on randomly chosen individual grains. Electron Energy Loss Spectroscopy (EELS) gave access to the valence of cerium in the ternary compound. Spectra were acquired in the diffraction mode, with a 0.03 eV/channel resolution .

X rays powder diffraction (XRD) patterns were collected in the classical θ-2θ mode, with an Empyrean Panalytical diffractometer, with a Cu anti cathode (λ Kα1 = 0.154056 nm),. Rietveld refinements with the software Fullprof were performed, starting from data cif files found in literature, and cell parameters, atomic positions, as well as site occupations, were extracted.

The experimental profiles were fitted with the most suitable pseudo-Voigt analytical function taking into account the line broadening function and the symmetric part of instrumental function.

Optical properties

The band gap values were obtained using the UV-visible diffuse reflectance spectroscopy (DRS). The spectra of the different samples were recorded using a Perkin Elmer LAMBDA 190s UV/Vis/NIR spectrometer along with a 150 mm integrating sphere. A given powder was compacted in a manual press into a 1.2 cm powder cup, clamped on the external port of the integrating sphere. The measurements were performed in the range of 250 to 800 nm at room temperature with a resolution of 0.08 nm. The baseline was determined using a calibrated reflectance standard to allow achievement of a reflectance accuracy of 0.5%.

Results

Chemical composition

The mean composition of the ternary compound was found to be Sr 0.50 Ce 0.35 WO 4 , which can also be written Sr 1/2 Ce 5/14 □ 1/7 WO 4, emphasizing the occurrence of cation vacancies.

Although the majority of ternary tungstate or molybdate are stoichiometric, ternary scheelite compounds with cations vacancies were already mentioned (Abakumov et al., 2014). Ce 3+ may partially oxidized into Ce 4+ , thus the valence state of cerium in Sr 0.50 Ce 0.35 WO 4 was determined by EELS. Ce 4+ and Ce 3+ M 4,5 core loss edges can unambiguously be distinguished due to differences in shape, intensities and energies (Garvie et al., 1999). The Ce 4+ M 4,5 edges consist in two main symmetrical peaks situated at 884 eV and 901.6 eV and two satellites peaks of lower intensities at 889.2 eV and 906.7 eV. On the contrary, the Ce 3+ M 4 edge is asymmetrical with features at 896.6, 898.4 and 899.7 eV and the Ce 3+ M 5 edge shows a broad maximum from 881.7 to 882.3 eV. Apart from the shape and energy positions of the energy loss peaks, the branching ratio, defined as the intensity ratio I(M 5 )/[I(M 4 )+I(M 5 )], can be used to distinguish between Ce 3+ (branching ratio 0.55) and Ce 4+ (branching ratio 0.49). Fig. 2 shows a characteristic EELS spectrum, acquired on a single grain of the ternary powder. The best fit was obtained with two peaks at 882 and 883 eV for the M 5 edge and three peaks at respectively 896.9, 898.5 and 899.8 eV for the M 4 edge, indicating the cerium is in a 3+ valence state in the ternary scheelite compound. However, the occurrence of two peaks around 883 eV, and a branching ratio of 0.53, indicate the presence of Ce 4+ . Assuming the linear relation between branching ratio and relative amount of Ce 4+ proposed by Gravie et al.

(1999), one can estimate the Ce 4+ amount to be around 30%.

Electron diffraction

The main spots of the electron diffraction (ED) pattern, taken under different zone axes, could all be indexed in the tetragonal scheelite structure. indices [START_REF] Van Smaalen | Incommensurate Crystallography[END_REF]. Spots, indexed (00l00) with l=2n, that are forbidden in the space group I4 1 /a of the tetragonal scheelite structure, can be observed in Fig 3b and3c.

During tilt experiments, these spots do not vanish, excluding a double diffraction phenomenon. A reduction in symmetry must then be considered from I4 1 /a to I-4 (82). This symmetry loss was already observed for ternary scheelite molybdate like (Na,Gd)MoO 4 (Abakumov et al., 2012) or (Li1Yb)MoO 4 [START_REF] Volkov | [END_REF]. Thus, the superspace group for Sr 0.50 Ce 0.35 WO 4 was identified as I-4 (α,β,0, -βα0)00, with α= 0.54 and β=0.82. Fig. 4 is part of the [001] pattern, where the two incommensurate vectors q1 and q2 are drawn: all the weak spots in the ED pattern can be indexed using q1 and q2. Modulated structures are not uncommon in (A,A') n (BO 4 ) m scheelite compounds (B=Mo or W) and an exhaustive overview of them can be found in (Abakumov et al., 2014). This overview was realized in order to study the origin of these modulations, which were linked to the difference in cation sizes, the ordering of the A, A' cations and the presence of cations vacancies in the A sublattice. With a radius difference of ∆r=0.1 Å between Sr 2+ and Ce 3+ , and 1/7 of vacancies, Sr 1/2 Ce 5/14 □ 1/7 WO 4 belongs to the same group of partially ordered scheelites as Na 2/7 Gd 4/7 □ 1/7 MoO 4 (Abakumov et al., 2012). The same modulations vectors q1=0.54 a*+0.82 b*, q2= -0.82 a*+0.54 b* are observed for these structures, but the charge difference ∆q between A and A' cations is ∆q =2 (Na + ,Gd 3+ ) in one case, ∆q=1 in the other one (Sr 2+ ,Ce 3+ ). Another difference is that twodimensional incommensurate modulations were observed in molybdates with tetragonal structure but not in tungstate scheelites. On the contrary, ternary tungstate scheelites usually adopt a (3+1) D modulated monoclinic structure (Morozov et al., 2013). Thus, one can conclude that 2D modulations are linked to a small ∆r and vacancies on A sites, but not to ∆q, the charge difference between A and A' cations. Moreover, these modulations can occur in tungstates, not only in molybdates.

High resolution electron microscopy

High resolution electron microscopy (HREM) observations were performed along the [001] direction, where the projected structure corresponds to mixed columns of Sr, Ce and W cations, (see Fig5a) and along [111], where the projected structure corresponds to distinct atomic columns for the (Sr,Ce) cations and for W (Fig 5b).

Fig6 shows a HREM image taken along [001], along with a Fast Fourier Transform (FFT) and filtered images. The FFT image is similar to the diffraction pattern Fig4, with the same incommensurate spots. FFT performed on different regions of the HREM image do not always exhibit these spots, which indicate that modulations are localised in domains. The filtered image using the incommensurate spots confirms a 10-20 nm domain-type structure.

The filtered image, using Bragg spots 20000 and 02000, has an enhanced contrast compared to the original image, shows no defects and can be interpreted in terms of atomic columns of cations.

In order to study a possible ordering of Sr and Ce, HREM images were taken along [111], using a 300 KV microscope Cs corrected. This projection direction allows to interpret HREM images in terms of atomic columns of W, Sr, Ce (Fig5.b). In particular, along the trace of (112) planes, there will be successively two W columns, two (Sr,Ce) columns. Fig. 7 shows a HREM image taken along [111] were thickness variations give rise to different motifs. The two insets correspond to filtered images from a thin part and a thick part of the sample. In the thin part of the sample, the position of the bright spots as well as their intensity clearly varies along the trace of the (112) planes, suggesting different atomic occupation. Simulated HREM images were calculated using two structural models and compared to the filtered images. Both structural models were based on the I-4 space group deduced from the ED study. The first model (model 1) corresponds to a disordered scheelite structure; Sr, Ce cations occupy randomly the same 2(b) and 2(c) sites, and W is on the 2(a) and 2(d) sites. This structural model was already used for Na 2/7 Gd 4/7 MoO 4 (Morozov et al., 2012). In the second model (model 2), the Sr cations occupy the 2(b) site while the Ce cations are on 2(c) sites. The chemical composition was taken into account, thus vacancies were statistically distributed on 2(b) and 2(c) sites for both models. The atomic positions are summarized in Table 1.

The pattern observed in the thick part of the sample can solely be simulated with the structural model where Ce and Sr are ordered, for a thickness of 5-7 nm (Fig. 8). The pattern observed in the thin part can, at first sight, be simulated for both disordered and ordered model, but the intensity variation observed along one row corresponds strictly to the different site occupation for strontium and cerium, as can be seen in Fig. 8 where the projected structure is superimposed to a simulated one for a weak thickness of 3 nm. Thus, HREM allowed to prove experimentally the ordering of strontium and cerium atoms in Sr 0.50 Ce 0.35 WO 4 . Fig. 9 reports the XRD patterns of the binary SrWO 4 (Fig. 9a) and of the new ternary Sr 0.50 Ce 0.35 WO 4 (Fig. 9b) compounds. The diagrams correspond to well crystallized single phases without residual compounds. The phase identification showed that both tungstates exhibit XRD diagrams characteristic of the tetragonal scheelite structure. In particular, the splitting of the high angle peaks induced by a monoclinic structure, was not observed.

We performed different Rietveld refinements of the XRD patterns, according to electron microscopy results. The binary compound SrWO 4 was refined in the I4 1 /a space group. The ternary compound Sr 0.50 Ce 0.35 WO 4 was refined in the I4 1 /a space group and in the I-4 space group, with a random occupation of A sites by Ce 3+ and Sr 2+ (the so called disordered scheelite). The occupation rate of the A sites was fixed to 50% for Sr and 35% for Ce, according to EDS results. In both space groups, the cations occupy fixed Wyckoff positions (see Table 1) and only the coordinates of oxygen atoms were refined . The refinements were initiate using cif files already published (Pereira et al., 2015;Abakumov et al., 2012). The refined cell parameters, oxygen coordinates and reliability factors are reported in Table 2.

The refinements in the I4 1 /a space group provides a good agreement with the experimental data for the binary and the ternary compounds, with acceptable reliability factors. The cell parameters of the ternary scheelite Sr 0.50 Ce 0.35 WO 4 are smaller than those of the binary SrWO 4 , with a shrinkage of the unit cell of about 1,4% in volume. This is in agreement with the substitution of Sr 2+ by Ce 3+ , which has a smaller radius (∆r=-0.1 Å) and the induced cation vacancies. The loss of symmetry from the I4 1 /a space group to the I-4 one, corresponds to a splitting of each atomic position (see Table 1). Thus, the cations were distributed on the different sites according to model 1 (no ordering of Sr and Ce). Again, the reliability factors show a good agreement between the calculated and the experimental diffraction patterns. The differences in the reliability factors for the two refinements of the Sr 0.50 Ce 0.35 WO 4 structure are too small to discriminate between the two possible space groups (N° 82 and 88). This can be related to the polycrystalline character of our samples with heterogeneous size distribution in the nanometer range. Thus, investigations of a possible cation ordering through Rietveld refinements of XRD diagrams were meaningless.

The variation in the atomic coordinates observed for oxygen atoms, gives rise to different distortions of the polyhedra as can be seen in Table 3, 3. Thus, the structure refined with a space group I-4 is much more distorted than the one corresponding to I4 1 /a.

Optical properties

For potential optical applications (luminescence and photo-catalysis), the band-gap energies of the ternary scheelite structure, as well as those of the two binary compounds SrWO 4 and Ce 2 (WO 4 ) 3, were measured by UV-Vis diffuse reflectance spectra DRS at room temperature. This allowed us to discuss the contribution of the insertion of cerium in the scheelite structure Fig. 10 shows the UV-Vis spectra of Sr 0.50 Ce 0.35 WO 4 , SrWO 4 and Ce 2 (WO 4 ) 3 . Large bands were observed for all spectra after 3 eV. According to theoretical calculations reported in the literature, tungstates scheelite materials like SrWO 4 exhibit an optical absorption band due to direct electronic transition (Zhang et al., 1998), which occurs through excitation of electrons from the valence band to the conduction band (Hwang et al., 2003). UV-Vis spectra show that the absorption band obtained for Ce 2 (WO 4 ) 3 is more intense than that of Sr 0.50 Ce 0.35 WO 4 and SrWO 4 . The intense band observed in the visible region is attributed to the electronic transition of the cerium cation present in the sample.

The energy band gap is obtained by using the Kubelka -Munk method which consists in plotting F(R ∞ ) versus photon energy. The band-gap is deduced by extrapolating the F(R ∞ ) profile using a linear fitting method. R ∞ designates the reflectance of the sample at infinite thickness and F(R ∞ ) the absolute diffuse reflectance of the sample, which corresponds to the conversion of the reflectance measurements R ∞ according to the following equation:

F ( R ∞ )= (1-R ∞ ) 2 2 R ∞ = K S
where K is the molar absorption coefficient and S the scattering coefficient.

The absorption band gap values for Sr 0.50 Ce 0.35 WO 4 , SrWO 4 and Ce 2 (WO 4 ) 3 were found to be 3.22, 4.59 and 3.14 eV respectively. The energy gap of the SrWO 4 is well in accordance with the theoretical result of 4,51 eV (Song et al., 2009). In the case of Ce 2 (WO 4 ) 3 , the band gap energy of 3,14 eV is in agreement with the one obtained using DFT calculations in a complex system based on polyoxometalate [Ce(W 5 O 18 ) 2 ] 8-cluster with CeO 8 polyhedra (Roy et al., 2010). The calculated molecular orbital diagram shows that the electron density consists on the O2p atomic orbital for the HOMO and the Ce 4f orbitals for the LUMO, with a band gap of 3.28 eV. Variations of the gap values can be linked to different factors such as shape and crystallite size (Alivisatos et al., 1996), lattice parameters (Yang et al., 2015), degree of structural disorder in the lattice [Rosa et al., 2008;Lacomba-Perales et al., 2008). For SrWO 4 , the band gap value shows that the maximum of the valence band can be mainly attributed to O2p levels, while the bottom of the conduction band is mainly composed of W 5d levels. The Sr 5s levels are mainly distributed in the conduction band , thus do not contribute to the band gap. (Song et al., 2009 ;Arora et al., 2006). The insertion of cerium in the host structure of To further understand the optical properties evolution of the scheelite structures, the band gap is discussed in terms of microstructural parameters such as bond distance and angles of the polyhedra and the crystallite size.

Generally, it is admitted that the band gap of semiconductors increases when the particle size decreases (Lacomba-Perales et al., 2008). According to a statistical study of grain size distribution based on microscopy images (not reported here), the mean grain size is the same for the different powders, the low band gap value of the ternary compound cannot be attributed to a size effect. Concerning the influence of cell parameters on the band-gap value, the contraction of 1,4% in volume of the cell from SrWO 4 to Sr 1/2 Ce 5/14 □ 1/7 WO 4 should induce a small decrease of the band gap (Lacomba-Perales et al., 2008).

According to the data of Table 3,

Conclusion

In the present work, a ternary (Sr,Ce)WO 4 compound has been synthesized using a modified EDTA-Citrate method. The structure was identified as a tetragonal scheelite and the XRD patterns showed well crystallized single phases up to 35%at. of cerium. Electron diffraction and high resolution electron microscopy evidenced the incommensurate modulations linked to partial ordering of Sr 2+ and Ce 3+ , and the loss of symmetry from the I4 1 /a space group to the I-4 one. The Rietveld refinement results with the I-4 space group revealed distortions of the AO 8 polyhedra, ie. the local environment of Ce 3+ . The bang gap of the ternary compound is significantly lower than the band gap of SrWO 4 and this was related to the 4f atomic orbitals of Ce and the induced crystallographic distortions linked to the incommensurate modulations. The bang gap value of 3.2 eV makes Sr 1/2 Ce 5/14 □ 1/7 WO 4 a promising candidate for violet luminescence in the 380 nm range.

  Fig.3 presents high symmetry ED patterns, with respectively [001], [100], [1-10] and [111] zones axes, indexed in the tetragonal structure. In the ED pattern taken along [001] (Fig.3a), the main spots exhibit the characteristic four fold symmetry of the tetragonal structure along c, but the weaker spots cannot be indexed with integer indices. They correspond to two incommensurate vectors, q1=0.54 a*+0.82 b*, q2= -0.82 a*+0.54 b*, thus to a two dimensional structure modulation, which led to an indexation of the different ED patterns in a (3+2)D superspace group with 5

  which summarizes the A-O and W-O bond lengths and angles. The cerium insertion in the SrWO 4 scheelite structure leads to a contraction of WO 4 tetrahedra and an expansion of AO 8 polyhedra. Basically, the SrWO 4 binary compound presents a main W-O distance of 1,8408 Å and two characteristic O-W-O angles values of 107° and 114°. The W-O distance decreases to 1.7903 Å in the Sr 0.50 Ce 0.35 WO 4 structure with I4 1 /a space group, but the O-W-O angles remain the same, 107.7°and 113°. The A-O bond length increases, showing the WO 4 contraction is accompanied by a AO 8 expansion, and the O-A-O angles do not change significantly. On the contrary, the refined structure with the I-4 space group shows much more WO 4 and AO 8 distortions. In this space group there are two W-O distances of 1.7964 and 1.7554 Å. The O-W-O angles deviate much more from the ideal tetrahedral angle of 109.9°, with values of 104°, 105°, 111°, 120° respectively. Similarly, the AO 8 polyhedron has four different A-O bond lengths and different O-A-O angles, as reported in Table

  SrWO 4 can promote different energy distribution at intermediate levels, linked to the 4f, 5d and 6s atomic orbitals, in the band gap near the valence and conduction bands. Combining all these elements, the band structure of Ce 0.35 Sr 0.50 WO 4, would consist in a valence band with mainly O2p states, and the bottom of the conduction band mainly composed with Ce 4f levels.

  the band-gap decreases with decreasing W-O bond lengths and increasing of A(Sr,Ce)-O bonds lengths. In the same way, all O-W-O and O-A-O angles are much more distorted in the ternary compound refined in the I-4 space group. These changes in bonds and angles correspond to the W-O and A(Sr, Ce)-O stretching and twisting motions. They can be described as a combination of the symmetric and antisymmetric stretches of the oxygen atom bonds in the polyhedra WO 4 and AO 8 . In addition to the vacancies, the presence of cerium ions Ce 4+ in the structure should induce even more distortions, favouring the appearance of intermediate levels in the band gap. As a consequence, the Ce 0.35 Sr 0.50 WO 4 scheelite structure gap energy decreases from 4.59 to 3.22 eV, a value very close to those of Ce 2 (WO 4 ) 3 .

Figure 1

 1 Figure 1 ABO4 scheelite structure, a) drawing of the unit cell showing the piling up along the c axis of edges-linked AO8 polyhedra. b) drawing of the unit cell cell along [001] showing the BO4 tetrahedra and AO8 polyhedra are linked by vertices.

Figure 2

 2 Figure 2 Cerium M4,5 core loss edge of Sr0.50Ce0.35WO4 .

Figure 3 Figure 4

 34 Figure 3 High symmetry electron diffraction patterns of the incommensurately modulated Sr0.50Ce0.35WO4 ,along the zone axes [001] (a), [1-10] (b), [100] (c), [11-1] (d)

Figure 5

 5 Figure 5 Projection of the ABO4 structure along two different crystallographic a) [001] and b) [111]. A and B cations are drawn as light blue (A) and dark blue (B), and the A-O and B-O links are drawn as sticks.

Figure 6

 6 Figure 6 HREM image taken along [001], along with FFT and filtered images.

Figure 7

 7 Figure 7 HREM image taken along [111]. The inset show two characteristic patterns.

Figure 8 Figure 9

 89 Figure 8 Map of images simulated (model 2). The defocus d varies from 2.5 nm to -1.5 nm, the thickness t from 1.4 nm to 7 nm. Projected atoms are represented as dots, Sr = blue,Ce = red, W= green

Table 1

 1 atom coordinates of Sr, Ce , W and O in I41/a and I-4 space groups

	Space group	atome	site	coordinate
		Sr,Ce	4b	0,0,1/2 0,1/2,3/4
	I41/a	W	4a	0,0,0	0,1/2,1/4
		O	16 f	x,y,z
		Sr,Ce	2b	0,0,1/2
	I-4	Sr,Ce	2c	0,1/2,1/4
		W	2a	0,0,0
		W	2d	0,1/2,3/4
		O1 O2	8 g	x,y,z

Table 2

 2 Refinement results for the binary SrWO4 and the ternary Ce0.35Sr0.50WO4 compounds using two different space groups.

	Compound	SrWO4	Sr0.50Ce0.35WO4	Sr0.50Ce0.35WO4
	Space group	I41/a	I41/a	I-4
	Cell parameters (Å)	a=5.416(8)	a=5.394(2)	a=5.394(2)
		c=11.975(0)	c=11.884(7)	c=11.884(7)
	Cell volume (Å 3 )	351.363	345.809	345.814
	Oxygen coordinates	O (16f)	O (16f)	O1 (8g)	O2 (8g)
	x	0.2504	0.3559	0.2533	0.6492
	y	0.1106	0.0135	0.1395	0.7892
	z	0.04121	0.2919	0.9250	0.1601
	Reliability factors
	R p	9.48	6.22	6.68
	R wp	13.4	7.32	7.53
	R exp χ² R Bragg	7.88 2.91 3.59	5.07 2.08 3.12	5.26 2.05 2.76

Table 3

 3

	Compound	SrWO4	Sr0.50Ce0.35WO4	Sr0.50Ce0.35WO4
	Space group	I41/a	I41/a	I-4
	Polyhedra		WO 4
	Bond (Å)	d: W-O	1.840(8)	1.790(4)	1.755(4) 1.796(4)
	Angle (º)	O-W-O	106.9	107.7	105.0	104.2
			114.6	113.1	111.7	120.5
				AO 8
		A-O 1	2.533(4)	2.561(4)	2.519(5)	2.581(2)
	Bond (Å)	A-O 2	2.571(6)	2.585(0)	2.588(9)	2.675(1)
			68.8	67.9	68.3	65.4
			74.8	73.5	70.2	72.6
			75.6	77.5	79.5	76.8
	Angle (º)	O-A-O	76.1	78.5	85.38	77.7
			98.6	98.4	97.2	99.2
			128.3	126.8	122.7	130.5
			134.5	135.1	138.5	132.9
			150.5	151.0	152.8	148,9

Acknowledgements -This study was done in the general framework of the french-brazilian cooperation project CAPES COFECUB PHC 77-113.


Olivier Cazorla
email: cazorla@montp.inserm.fr

Y Aït Mou

L Goret

G Vassort

M Dauzat

A Lacampagne

S Tanguy

P Obert

EFFECTS OF HIGH-ALTITUDE EXERCISE TRAINING ON CONTRACTILE FUNCTION OF RAT SKINNED CARDIOMYOCYTE

Introduction

Aerobic exercise induces cardiac morphological remodelling (increase in myocardial mass and ventricular chamber dimensions) and functional adaptations (for review see {Moore, 1995 #43}) leading to increased maximal stroke volume and subsequently cardiac output. Part of the training effect on the cardiac performances is due to increased myocyte contractile function possibly through changes in their intrinsic active properties {Moore, 1995 #43}. Various cellular parameters have been investigated to account for the beneficial effect of exercise on contraction. Training does not appear to modify intracellular Ca 2+ transient {Natali, 2001 #2; Palmer, 1998 #11; Laughlin, 1992 experimental conditions that could lead to some heterogeneity in the results. Cardiomyocyte properties are not uniform within the myocardium. In the rat heart, there are regional differences in electrical {Aimond, 1999 #45} as well as in mechanical properties. We have described previously that intact {Cazorla, 2000 #33}, and skinned {Cazorla, 2005 #1} myocytes isolated from the sub-endocardium (Endo) are more responsive to stretch than those from the sub-epicardium (Epi), with higher passive and active tensions developed in Endo at the same sarcomere length (SL). Natali et al. {Natali, 2002 #12} showed that voluntary exercise increases the steepness of the active tension-SL relationship of attached intact cardiomyocytes exclusively in Endo but not in Epi. However, Diffee et al. reported that training increases Ca 2+ sensitivity in both Endo and Epi skinned myocytes {Diffee, 2003 #5}. The latter experiments were performed at 2.35 µm SL, while the intact cells in the former work could be stretched experimentally to 2.05 µm SL at the best.

In their constant search to improve performance, many athletes train at high altitude because hypoxia exposure enhances erythropoiesis and subsequently oxygen carrying capacity. However, there is now clear evidence in both animal models and human that training at high altitude does not provide advantages over training at sea level as regards maximal oxygen uptake and aerobic performance {Henderson, 2001 #30;Reboul, 2005

Methods

Training procedure

Male Wistar rats (300-350 g) were randomly assigned to one of the four groups (7 animals per group): rats living continuously either in normoxic conditions without or with aerobic training sessions (N and NT, respectively) and rats living continuously in hypoxic conditions without or with aerobic training sessions (H and HT, respectively). High altitude was simulated by a hypobaric hypoxia environment obtained with a vacuum pump as described previously {Reboul, 2005 #7}. Rats were maintained for 5 weeks at a barometric pressure of 760 mmHg (PIO2 ≈ 159 mmHg, altitude ≈ 80 m) for N and NT, or at a pressure of 475 mmHg (PIO2 ≈ 90 mmHg, altitude ≈ 4000 m) for H and HT. Some were trained under sea-level (NT) or high-altitude (HT) conditions for 5 weeks in a driven wheel as described previously {Reboul, 2005 #7}. Briefly, the training program consisted of 5 sessions (45-min) per week at 80% of the maximal aerobic velocity (MAV) determined in each environmental condition. MAV was evaluated using a driven wheel during a maximal exercise test {Gonzalez, 1993 #22} in each environment and for each rat before the study, during the 3 rd week, and at the end of the protocol. All procedures were performed in agreement with the Guide for the Care and Use of Laboratory Animals CVR-2006-181-R2 4 published by the US National Institutes of Health (NIH Publications No. 85-23, revised 1996) and with the approval of the French Ministry of Agriculture.

Echocardiography and heart weighing.

Rats were anesthetized with intraperitoneal ketamine HC1 (75 mg/kg) and xylazine (10 mg/kg). A twodimensional short-axis view of the left ventricle was obtained at the level of the papillary muscles using an echocardiograph (5-8 MHz micro-convex transducer HDI 3000, ATL, Phillips Company, Bothell, USA) as reported previously {Reboul, 2005 #7}. End-diastolic anterior and posterior wall thicknesses (AWT and PWT, respectively), and left ventricular end-diastolic (LVEDd) and end-systolic (LVESd) diameters were measured. The diastolic relative wall thickness (RWT= AWT+PWT/LVEDd) was calculated as an index of cardiac remodelling. Left ventricular stroke volume (LVSV) and heart mass were evaluated as described previously {Reboul, 2005 #7}. Measurements from at least three consecutive cardiac cycles were averaged.

Single cardiomyocyte sarcomere length and force measurements

The Ca 2+ -activated force of single skinned myocyte was measured as described previously {Cazorla, 2005 #1; Cazorla, 2006 #56}. Myocytes were isolated by mechanical dissociation from pre-skinned left ventricular strips. Some cells were kept on ice and used within a day for mechanical experiments; the others were solubilized for gel electrophoresis. Skinned myocytes were attached to a piezoresistive strain gauge (AE801 sensor, Memscap, Crolle, France) and to a stepper motor driven micromanipulator with thin needles and UV-polymerized optical glue (NOA 63, Norland products Inc, North Brunswick, NJ). Sarcomere length (SL) was determined online throughout the experiment (Ionoptix Boston, USA). Force was normalized to cross-sectional area measured from the imaged cross-section as described previously {Cazorla, 2000 #9}. Slack SL was measured before attachment to serve as zero reference and did not differ significantly between myocytes from normoxic (1.87±0.01 µm SL, n = 45) and hypoxic (1.86 ±0.01 µm SL, n =4 3) rats. The pCa (=-log[Ca 2+ ]i)-force relationships were established at two SL: 1.9 then 2.3 µm, at 22°C. Cells that did not maintain 80% of the first maximal tension or a visible striation pattern were discarded. We continued with cells that were well attached with minimal SL changes (< 0.1 µm) during activation.

CVR-2006-181-R2

5 When required, cell length was varied during contraction in order to keep SL constant. Active tensions at submaximal activations were normalized to maximal isometric tension (classically obtained at pCa 5) at the same SL. The relation between force and pCa was fitted to the following equation:

[ ] [ ]       + = + + H H n n Ca K Ca force 2 2
where nH is the Hill coefficient and pCa50, pCa for half-maximal activation equals -(log K)/nH.

Solutions

Solutions were similar to those used previously {Cazorla, 2005 #1}. For cell dissociation, the Ca 2+ -free Hanks-HEPES buffered solution contained (in mM): NaCl 117, KCl 5.7, NaHCO3 4.4, KH2PO4 1.5, MgCl2 1.7, HEPES 21, glucose 11.7, taurine 20, 2,3-butanedione monoxime 15, pH 7.15 adjusted with NaOH, and was bubbled with 100 % O2.

For myocyte force measurements, activating solutions were prepared daily by mixing relaxing and maximal activating solutions containing (in mM): Na-acetate 10, phosphocreatine 12, imidazole 30, free Mg 2+ 1, EGTA 10, Na2ATP 3.3 and dithiothreitol 0.3 with either pCa 9.0 (relaxing solution) or pCa 4.5 (maximal activating solution), protease inhibitors (PMSF 0.5; leupeptin 0.04 and E64 [transepoxysuccinyl-1-leucine-guanidobutylamide] 0.01), pH 7.1 adjusted with KOH. Ionic strength was adjusted to 180 mM with K-acetate.

Biochemistry

Myosin heavy chain (MHC) isoforms were separated following a procedure adapted from Warren et al.

{Warren, 2003 #41}. The myocytes were dissolved in a SDS lysis buffer (TRIS-HCl 50 mM, 2% (w/v) SDS, urea 8 M, EGTA 1 mM, EDTA1 mM, DTT 80 mM, 10% (v/v) glycerol, pH 6.8, protease inhibitors) and heated at 60°C for 5 min. Proteins ( 0.1 µg) were separated on a 6% SDS-PAGE at 16 mA constant current for 20 h at 4°C (Shelton Scientific gel system). The gels were silver stained and analysed with an imaging system (Kodak Image Station 2000R). β-MHC content was expressed relative to the total amount of the MHC protein.

Citrate synthase (CS) activity was assayed by measuring absorbance over 3 min at 412 nm at 25°C in soleus muscle homogenate. Muscle was first homogenized in buffer (in mM: 210 sucrose, 2 CVR-2006-181-R2 6 EGTA, 40 NaCl, 30 HEPES, 5 EDTA, and 2 PMSF, pH 7.4) and then mixed with 15 µM acetylCoA, 0.5 mM oxaloacetate, 0.1 mM DNTB {Srere, 1969 #48}.

Statistics

Data are presented as mean ± SEM. The effects of hypoxia and training were examined by using a 2way analysis of variance (factor 1 : hypoxia vs normoxia, factor 2 : sedentary vs trained) with or without repeated measures (i.e. time or sarcomere length) depending on variables. The effect of the cellular localization on the stretch-dependent Ca 2+ sensitization in the different conditions was tested by a 3-way ANOVA (Statview, Abacus concepts, Inc., Berkeley, CA). Post-hoc tests of Fisher's Protected Least Significant Difference were used when appropriate. Linear regression analyses performed on raw data examined relationships between ΔpCa50 and β-MHC content or passive tension. Comparison of linear regressions (Y intercepts and slopes) was examined using Statgraphics Centurion XV. Statistical significance was defined as P<0.05.

Results

Training features and cardiac morphological parameters.

Efficiency of the training program in NT and HT rats was evaluated from maximal aerobic velocity (MAV) and skeletal muscle citrate synthase activity. Both increased significantly after training, irrespective of environmental conditions (Fig. 1A). LVSV increased significantly in NT but was unchanged in HT (Table 1). No changes were reported for N and H rats for all parameters.

Right ventricular mass increased significantly with hypoxic exposure. Left ventricular mass did not differ between the two sedentary groups, but it increased significantly to the same extent in the two groups after exercise training performed at sea level or high altitude (Table 1).

Echocardiographic morphological parameters were affected differently by training depending on environmental conditions (Fig. 1B). Namely, LVEDd increased in NT but decreased in HT rats.

AWT and PWT increased significantly with training conducted under normoxic as well as hypoxic conditions. The cardiac remodelling was harmonious in NT rats (unchanged RWT values) but CVR-2006-181-R2 7 disharmonious (marked increase in RWT highlighting a concentric LV hypertrophy) in HT rats. LV dimensions were not affected throughout the study period in N and H rats.

Stretch-dependent contractile properties

The Ca 2+ sensitivity of myofilament activation (pCa50) at 1.9 µm SL was similar across the ventricular wall in N, H and NT rats. However Ca 2+ sensitivity decreased significantly at this SL in HT rats for both Epi and Endo cells (Fig. 2). Stretching cells to 2.3 µm SL induced a leftward shift of the tension-pCa curve in all groups, indicative of increased myofilament sensitivity to Ca 2+ . For both Epi and Endo cells, Ca 2+ sensitivity at 2.3 µm SL increased with training in normoxic conditions but decreased when training was performed in hypoxic conditions (Fig. 2C).

The stretch-dependent modulation of activation or Ca 2+ sensitization (∆pCa50 , difference in pCa50 obtained at 2.3 and 1.9 µm SL) is an important factor accounting for the Frank-Starling mechanism of the heart. It increased significantly with chronic exercise at sea level in Endo and in Epi cells (Fig. 3). Chronic hypoxia exposure did not modify significantly ∆pCa50 although a slight decrease was observed in Endo H cells compared to Endo N cells (p=0.09). High-altitude endurance training had no beneficial effect on stretch-dependent Ca 2+ sensitization. Rather it even decreased in Endo cells from HT when compared to N and NT (Fig. 3 and Table 2). Maximal active tension did not differ between the various groups and myocardium localization (Table 2). The Hill coefficient increased significantly only in Epi from the HT group at both SL, but this effect was not investigated further.

Involvement of MHC isoform

The heart co-expresses two isoforms of MHC (α-or β-MHC). The change in α/β-MHC ratio influences cardiac function and occurs under various stresses including exercise (see for review {Schiaffino, 1996 #47}). As shown previously rat ventricles used in our study contained mostly α-MHC (Fig. 4A). After training in normoxic conditions, β-MHC expression was reduced significantly in both Epi and Endo cells. Chronic hypoxia exposure increased β-MHC expression particularly in Endo cells. β-MHC expression was similar in N and HT rats in both Endo and Epi cells. Thus, chronic exercise affected β-MHC expression in both myocardial layers and by chronic hypoxia mostly in Endo cells. Accordingly, CVR-2006-181-R2 8 we investigated the relationship between the stretch-dependent Ca 2+ sensitization and β-MHC expression level. A significant negative correlation was obtained in Endo and Epi when data from the 4 groups were pooled (Fig. 4B). When groups were analysed separately, the correlation between ΔpCa50 and β-MHC expression levels was still evident in N and NT rats (Fig. 4C). After normoxic training, ΔpCa50 was higher and associated with lower β-MHC expression in both myocardial layers.

However under hypoxic conditions, the correlation was no longer obtained in Endo cells. Values in Epi cells from sedentary and trained rats were mixed and spread similarly from 20 to 40 % of β-MHC, resulting in similar average values.

Passive tension-dependent contractile properties.

In the relaxing solution Endo cells developed significantly higher passive tension than Epi cells after a stretch from slack length up to 2.3 µm SL (Table 2). Chronic hypoxic exposure or exercise training did not affect the transmural gradient of passive tension. ∆pCa50 is closely related to passive tension rather than to sarcomere length {Cazorla, 2001 #13; Cazorla, 1999 #57} and there is a transmural gradient of stretch-sensitization {Cazorla, 2005 #1}. We thus established the relationships between ∆pCa50 and the passive tension developed at 2.3 µm SL by individual cells (Fig. 5). A positive correlation was found across the free wall in N and NT rats. Higher passive tensions and ∆pCa50 were obtained in NT when compared to N rat cells but individual values from the 2 groups were distributed along the same regression lines (Fig. 5A). A positive correlation between passive tension and ∆pCa50 was also observed for H rat cells. The regression line (slope and intercept) of H rats was not significantly different to the ones observed in N and NT rats although its slope was slightly lower (Fig. 5B). When rats were trained under a hypoxic environment, the relationship between passive tension and ∆pCa50 disappeared, mostly due to a diminished ∆pCa50 of Endo cells. When the stretch-dependence of Ca 2+ sensitivity was indexed by ∆[Ca 2+ ]50 (i.e. difference in [Ca 2+ ]i required for half-maximal activation at 1.9 and 2.3 µm SL, Table 2) similar correlations were obtained between the groups (data not shown).

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Discussion

To gain insight into the effects of exercise training at high altitude on cardiac function, we subjected rats to chronic hypobaric hypoxia and endurance training. Hypoxic stress had a moderate negative impact on the contractile properties of the myofilaments, affecting preferentially cardiomyocytes from the endocardium. The transmural gradient of cardiomyocyte contractile properties, classically observed within the normal ventricular wall, was improved by sea-level training but disappeared after high-altitude training. This depressed cardiac myofilament calcium responsiveness may contribute to the lack of improvement in ventricular function observed after high-altitude training.

Effect of training at sea level

Effect of training at altitude

High-altitude training did not improve LVSV (Table 1). From the present results, we suggest that this is partly due to specific cardiac remodelling (reduced LVEDd) and deteriorations of the myofilament contractile properties. The marked wall thickening may be a compensatory mechanism for the depressed contractile function, allowing maintenance of the cardiac output and a response to the exercise-induced increase in energetic demand. Chronic hypoxia exposure alone did not alter significantly Ca 2+ sensitivity of the myofilaments.

However high-altitude exercise training reduced the Ca 2+ sensitivity of the myofilaments in both Epi and Endo cells whatever the sarcomere length. Moreover, the transmural gradient of stretch-induced Ca 2+ sensitization of the myofilaments was abolished completely in HT rats. Whether haemodynamic factors and/or hypoxia per se are responsible for this effect cannot be ascertained. On one hand, assuming similar LV end-diastolic pressure in NT and HT rats and considering the cardiac remodelling in these 2 groups, lower end-diastolic wall stress is to be expected in HT when compared to NT rats.

On the other hand, we could postulate that episodes of severe myocardium hypoxemia occurred as a result of exercise at high-altitude thus affecting the contractile machinery. In this context, it is interesting to note that similar alterations in cardiomyocyte contractile properties were obtained previously by our group in a rat model of chronic myocardial ischemia {Cazorla, 2005 #1}.

Molecular basis of the stretch-dependent Ca 2+ sensitivity

The stretch-induced Ca 2+ sensitization of the myofilaments relates to passive tension rather than to SL {Cazorla, 2001 #13;Cazorla, 1999 #57;Cazorla, 2005 #1}. This relationship across the wall was enhanced in NT rats but no longer existed in HT rats. Over the physiological range of SL, titin is the major component of the cellular passive properties {Cazorla, 2001 #13}. No change in titin content or isoform expression was observed in the present study (data not shown). Although not significant, the opposite passive tension changes observed in NT and HT rats could be due to other mechanisms that may have participated in the fine tuning of passive tension-dependent modulation of activation across the wall such as PKA phosphorylation, titin-actin interaction or Ca 2+ binding on titin {Granzier, 2004 #1004} that may directly and/or indirectly affect the properties of the contractile and/or regulatory proteins.

β-MHC is characterized by lower adenosine triphosphatase activity and lower filament sliding velocity, but can generate cross-bridge force with a higher economy of energy consumption than α-MHC {Holubarsch, 1985 #37}. In accordance with the present data, previous {Pagani, 1983 #50;Jin, 2000 #51;Hwang, 2005#53} but not all {Farrar, 1988 #52;Diffee, 2003 #49} studies have shown increased α-MHC isoform expression after sea-level training. Therefore, increased in α-MHC isoform expression could be at least in part responsible for the enhanced stretch-dependent Ca 2+ sensitization of the myocytes in NT rats. In the present study, we did find a correlation between the level β-MHC and the stretch-dependent mechanism at sea level. However, Endo cells from N and HT rats had similar β-MHC content but different ΔpCa50. This suggests that other proteins are very likely responsible for the changes in Ca 2+ sensitivity observed in our study. In a previous publication {Cazorla, 2005 #1} we observed in normoxic sedentary rats that Endo cells had a higher ΔpCa50 than Epi cells due to a higher phosphorylation level of myosin-light chain 2 (MLC2*) after stretch. In post-myocardial infarcted rats, the MLC2* was no longer phosphorylated after stretching CVR-2006-181-R2 12 and was associated with reduced ΔpCa50 in Endo cells. It is thus possible that the level of phosphorylation of MLC-2 had been increased in NT rats and decreased in HT rats, contributing thus to the Ca 2+ sensitivity results of the present study. The slight decrease in passive tension and the rightward shift of the tension-pCa curves in HT rats may be due to an increase in PKA-dependent stimulation {Cazorla, 2006 #56}. The levels of phosphorylation of MLC2 and other regulatory proteins such as TnI and cMyBP-C before and after stretch have to be explored.

In conclusion, nonuniformity is a major characteristic of the normal adult left ventricle that is partly due to heterogeneous contractile properties of the myocytes across the wall. CVR-2006-181-R2

  #29}. However, it increased Ca 2+ sensitivity of the myofilaments in intact {Wisloff, 2001 #4} and skinned cardiomyocytes {Diffee, 2001 #25}. The aforementioned experiments used various training and/or

  #7}. This altitude-training paradox is in part due to changes in cardiac morphological and functional parameters determining stroke volume {Reboul, 2005 #7}. We showed previously that rats trained under hypobaric hypoxic conditions presented a specific cardiac remodelling characterised by increased left ventricular wall thickness without chamber CVR-2006-181-R2 3 enlargement. Moreover, contrary to sea-level training, cardiac function was not improved when training was conducted under these experimental conditions. In particular resting and maximal stroke volumes were not improved, leading to the assumption that high-altitude training depressed heart and especially myocyte contractile properties. This study was specifically designed to test the hypothesis that high-altitude training limits the beneficial effects classically observed after training at sea level. Considering that cardiomyocyte properties, including the response to exercise training, are not uniform within the myocardium, we investigated the effects of training under normoxic and hypobaric hypoxic conditions on myofilament Ca 2+ sensitivity and its stretch-dependency in single skinned cardiomyocytes, taking into account their localization in the sub-endocardium and sub-epicardium.

  Sea-level training improved the contractile properties of the myofilaments in both Endo and Epi layers. This is in accordance with previous studies reporting an increase in contraction and Ca 2+ sensitivity in cardiomyocytes isolated from endurance-trained rats {Wisloff, 2001 #4; Diffee, 2001 #25}. Diffee & Nagle {Diffee, 2003 #6} reported an increase of the length-dependence of maximal and submaximal tension in LV cardiomyocytes, mostly due to increased Ca 2+ sensitivity at long sarcomere length. Furthermore, we reported recently a gradient of contractile properties across the LV wall. After stretch, Endo cells developed more passive tension than Epi cells and the stretch-dependent Ca 2+ sensitization of the myofilaments (ΔpCa50) was higher {Cazorla, 2005 #1}. Sea-level training shifted the stretch-induced Ca 2+ sensitization-passive tension relationships of Epi and Endo cells to higher values. Cells from NT rats were more responsive to stretch than those from N rats but the reasons for this improvement are still unclear (see below). These results may however account for the steeper slope of the active tension-SL relationship observed by Natali et al. {Natali, 2002 #12} in attached intact cardiac myocytes from trained rats . The changes in cellular properties observed in the present study might contribute to the enhanced ventricular function (i.e. higher LV stroke volume) reported in NT rats. This is in line with CVR-2006-181-R2 10 Reboul et al {Reboul, 2005 #7} who observed a greater increase in LVSV following volume overloading in endurance-trained rats compared to sedentary ones, indicative of an exercise-induced improvement of the Frank-Starling mechanism of the whole heart. In accordance with previous results from our group {Reboul, 2005 #7} and others {Wisloff, 2001 #4}, sea-level training induced LV hypertrophy, accounted for increased wall thickness and internal dimensions, with also important significance for cardiac performance. Thus, the enhanced LVSV in NT rats was likely due to greater ventricular filling associated with improved contractile properties of the myofibrils across the ventricular wall.

  Long-term exposure to high-altitude hypoxemia of foetal sheep has also been reported to decrease contraction of intact papillary muscle {Browne, 1997 #19}. Finally, 4 weeks of chronic hypoxia has been shown to reduce Ca 2+ transient in association with impaired Ca 2+ release and uptake in rat cardiomyocytes {Pei, CVR-2006-181-R2 11 2003 #16}. The loss of the contractile transmural gradient in HT rats should contribute to the lack of improvement in LVSV after training at simulated high altitude. Similarly the present cellular data can account for the lack of improvement after high-altitude training in the Frank-Starling mechanism investigated with a volume overloading protocol {Reboul, 2005 #7}.

  The major mechanical adaptations to sea-level training improved the transmural gradient of stretchdependent Ca 2+ sensitization of the myofilaments. This cellular mechanism should improve the Frank-Starling relationship solicited particularly during exercise. High-altitude training induced a reduction of Ca 2+ sensitivity of the myofilaments and stretch-dependent Ca 2+ sensitization mostly in the Endo layer, which might have contributed to the lack of improvement of the ventricular function observed in vivo.

Fig. 1 .Fig. 2 .Fig. 3 Fig. 4 .

 1234 Figures and Legends

Fig. 5 .

 5 Fig. 5. Correlation between stretch-induced Ca 2+ sensitization (ΔpCa50) and passive tension across the wall. ΔpCa50 was plotted against mean passive tension for each tissue area. Symbols and error bars correspond to each group of cell (Endo: open symbols, Epi: close symbols) in sedentary (squarefor N and triangle for H) and trained (circle for NT and star for HT) rats living in normoxic (A) or hypoxic (B) conditions. Linear regressions were calculated on a scatter plot including all raw data points from both myocardial layers in each condition (line for sedentary, dash line for trained). Exercise increased ΔpCa50 and passive tension but values are distributed over the same linear relationship in N (y=0.036+0.017x, r=0.67 p=0.001) and NT rats (y=0.069+0.016x, r=0.78 p=0.001). Stretch-dependent sensitization was decreased by chronic hypoxia exposure (y=0.071+0.010x, r=0.47 p=0.02). No correlation was found in HT cells (y=0.120+0.003x, r=0.21 p=0.38).

Acknowledgments

This work was supported by the « Association Française contre les Myopathies », « Fondation pour la Recherche Médicale », and « Région Languedoc-Roussillon ». OC and AL are established investigators of CNRS. Thanks are due to Patrice Bideaux and Guillermo Salazar for technical assistance. We are thankful to Guillaume Py and Cyril Reboul for helpful expertise in citrate synthase activity and echocardiography, respectively. CVR-2006-181-R2

 CVR-2006-181-R2 CVR-2006-181-R2 CVR-2006-181-R2 CVR-2006-181-R2


Rosario D ' Esposito
email: rosario.desposito@gmail.com

Suresh Balanethiram

Jean-Luc Battaglia

Sébastien Frégonèse

Thomas Zimmer

Prof Nanver Lis

Suresh Rosario D'esposito

Sébastien Balanethiram

Thomas Frégonèse

Keywords: SiGe HBT, thermal penetration depth, thermal impedance, BEOL metals thermal impact, thermal capacitance

In this letter we present a detailed investigation on how dynamic thermal phenomena take place in state-of-the-art SiGe HBTs when excited by a sinusoidal power dissipation. To give a better insight into the mechanisms leading to the thermal impedance (Zth) decay, we introduce the concept of thermal penetration depth; then, with the help of 3D thermal simulations, we illustrate its effect on the spatial distribution of the temperature variations within the transistor structure, according to the frequency of operation. In order to experimentally analyze the impact on a real device, dedicated HBT structures are designed; they consist of multi-finger SiGe HBTs realized in B55 technology from STMicroelectronics, for which modifications are made in the Back-End-Of-Line (BEOL) metallization or in the transistor layout, increasing its Deep Trench Isolation (DTI) enclosed area. For these transistors, Zth measurements are carried out in the frequency range 10kHz-1GHz; the results show that the metal connections configuration in the BEOL or layout modifications can considerably impact the Zth decay at low frequencies. An identical Zth trend is instead measured above 1-2 MHz, demonstrating that at higher frequencies just the region close to the heat source is concerned by dynamic thermal phenomena.

I. INTRODUCTION

hermal effects are one of the key factors limiting the performance and reliability of devices and integrated circuits realized using SiGe HBT technologies, in which the continuous trend to shrink the device dimensions leads to critically high power densities [START_REF] Schröter | Compact Hierarchical Bipolar Transistor Modeling with HiCuM[END_REF], [START_REF] Chevalier | Si/SiGe:C and InP/GaAsSb Heterojunction Bipolar Transistors for THz Applications[END_REF]; in addition, the adoption of shallow and deep trench isolations, filled with low thermal conductivity materials, limits the lateral heat diffusion, and thus contributes to increase the devices Rth. Due to the strong relationship between electrical and temperature effects in these components [START_REF] Kamrani | A Deterministic and Self-Consistent Solver for the Coupled Carrier-Phonon System in SiGe HBTs[END_REF], which is worsened by their positive feedback [START_REF] Rinaldi | Theory of electrothermal behavior of bipolar transistors: Part I -single-finger devices[END_REF], a realistic and physics-based thermal compact modeling is mandatory; a reliable compact model must ensure accurate circuit simulations during the design phase [START_REF] Ozalas | The Impact of Electro-Thermal Coupling on HBT Power Amplifiers[END_REF], [START_REF] Alessandro | On the safe operating area of bipolar cascode amplifiers[END_REF], which take into account the operating temperatures of the HBTs. In stateof-the-art SiGe HBTs thermal phenomena arise when a static power dissipation (Pdiss) is applied, but also when the devices are operated in dynamic conditions, since their thermal bandwidth is within few hundreds of MHz [START_REF] Hasnaoui | Self-Heating Characterization of SiGe:C HBTs by Extracting Thermal Impedances[END_REF], [START_REF] Rafei | Experimental Characterization and Modeling of the Thermal Behavior of SiGe HBTs[END_REF]. Many works can be found in literature about DC self-heating effects [START_REF] Rinaldi | Theory of electrothermal behavior of bipolar transistors: Part I -single-finger devices[END_REF], [START_REF] Balanethiram | An improved scalable self-consistent iterative model for thermal resistance in SiGe HBTs[END_REF]- [START_REF] Alessandro | Influence of Scaling and Emitter Layout on the Thermal Behavior of Toward-THz SiGe:C HBTs[END_REF], but less attention is given to dynamic thermal modeling. It must be considered, though, that when the HBTs are driven by a signal having a bandwidth within their thermal constant, due to the intrinsic transistor nonlinearities, third-order intermodulation products (IMD3) arise and can trigger low frequency (LF) variations of the junction temperature, which can significantly affect the electrical behavior of the component [START_REF] Ishikawa | Analytical Design Method for a Low-Distortion Microwave InGaP/GaAs HBT Amplifier Based on Transient Thermal Behavior in a GaAs Substrate[END_REF], [START_REF] Camarchia | Self-Consistent Electrothermal Modeling of Class A, AB, and B Power GaN HEMTs Under Modulated RF Excitation[END_REF]. A deep understanding of the thermal mechanisms taking place in sinusoidal operation and a precise characterization of the Zth are therefore necessary to obtain a realistic thermal model, which can result very helpful in certain applications, like the design of pre-distortion linearizers for power amplifiers [START_REF] Takahashi | Precise modeling of thermal memory effect for power amplifier using multi-stage thermal RC-ladder network[END_REF]- [START_REF] Bosch | Measurement and simulation of memory effects in predistortion linearizers[END_REF].

In this letter we present a detailed investigation on how dynamic thermal phenomena take place in state-of-the-art SiGe HBTs, by introducing the concept of thermal penetration depth. 3D thermal simulations are reported in section II, which gives an alternative insight into these physical mechanisms; in section III different sets of dedicated transistor structures are presented and their Zth(f) are compared; to our knowledge no previous study on the BEOL has been supported by Zth measurements.

II. THERMAL PENETRATION DEPTH SIMULATIONS IN 3D TCAD

To have an alternative insight into the dynamic spreading of the heat within the transistor structure, we can consider a very simplified case of study: let us assume a semi-infinite silicon block having an initial uniform temperature T0. When a uniformly distributed Pdiss is applied at one side, eventually the whole block of material will tend to reach a higher temperature T1. But before this happens, a thermal gradient can be observed within a certain distance from the heat source, while beyond this boundary the temperature of the block is still T0. The thermal penetration depth (δ) is defined as the distance that the heat diffuses through the material (under a sinusoidal stimulus) during a time 1/f and is approximated with the expression [START_REF] Salazar | Energy propagation of thermal waves[END_REF]:

f c k     (1)
where f is the Pdiss frequency, whereas k c and ρ are, respectively, the thermal conductivity, specific heat and density of the material. Note that (1) is only valid in 1D heat transfer by conduction; moreover it must be considered that a state-of-theart SiGe HBT has a complex structure, made of multiple material layers, having very different thermal properties.

Nevertheless, [START_REF] Schröter | Compact Hierarchical Bipolar Transistor Modeling with HiCuM[END_REF] indicates the general trend that should be Thermal penetration depth analysis and impact of the BEOL metals on the thermal impedance of SiGe HBTs expected; in particular, since δ is inversely proportional to , the volume of the silicon that effectively contributes to Zth reduces if the frequency increases. To show this behavior and understand how the induced temperature variations will penetrate in the transistor, 3D thermal simulations using Sentaurus TCAD have been performed on the transistor structure in Fig. 1 representing an HBT from the Infineon B11HFC technology (AE=5x0.2µm²). In these simulations the thermal conductivity degradation due to high doping is taken into account, according to experimental data in [START_REF] Liu | Modeling and Data for Thermal Conductivity of Ultrathin Single-Crystal SOI Layers at High Temperature[END_REF] and [START_REF] Mcconnell | Thermal conductivity of doped polysilicon layers[END_REF]. A planar heat source (HS) is placed at the base-collector (BC) junction and isothermal conditions are applied at the back surface of the wafer and at the top, above a 15µm thin air layer acting as a laminar sub-layer, in which just conduction can be considered [START_REF] Kaka | Convective Heat Transfer[END_REF]; due to the very low k of air (0.024 W/mK) this boundary condition is very close to be adiabatic. Sinusoidal power stimuli are applied to the HS, at frequencies ranging from 10kHz to 1GHz.

In Fig. 2 are plotted the normalized amplitude and the phase of the simulated temperature oscillations, for two different positions P1 and P2 (respectively 80nm and 6.38µm below HS). It can be observed that their amplitude will tend to reduce if the frequency is increased, due to the intrinsic thermal capacitance (Cth) of the structure. Moreover if the distance from the HS is higher, oscillations will tend to disappear; in fact in P1 significant oscillations (20% of the temperature variations simulated at DC) are visible till a frequency of 100MHz, whereas the same relative amplitude is already reached at 1MHz in point P2. The absolute value of the thermal oscillations is 20°C in P1 and 0.9°C in P2 at a frequency of 10kHz.

It can be also noticed that the phase shift will increase if the frequency of the Pdiss increases; furthermore in P2 the phase shift becomes more evident, since the distance from the heat source is higher than in P1 and thus a longer time is needed for the thermal variations induced from the HS to travel through the HBT structure. If instead we fix a frequency and have a look at the distribution of the amplitude of the thermal oscillations within the transistor, we will have a direct image of the spatial distribution of δ. As reported in Fig. 1, at 100MHz the thermal variations are only visible in a small volume around the BC.

III. THERMAL PENETRATION DEPTH EXPERIMENTAL STUDY

To investigate how the thermal oscillations induced by a sinusoidal power dissipation affect the behavior of a real transistor and to estimate the penetration depth inside the component, dedicated test structures need to be designed. If a sinusoidal Pdiss is applied, as shown in the previous section, temperature oscillations are induced in the lower part of the HBT, but, of course, they are also present in the BEOL. It is therefore possible to study these thermal penetration phenomena by characterizing and comparing the thermal behavior of transistor structures in which the lower part of the device is kept unaltered, whereas modifications are made in the BEOL region (during the design there is in fact a certain degree of freedom on the geometry of the different metal layers).

The HBTs under study in this work consist of five-fingers SiGe HBTs realized in STMicroelectronics B55 technology, of which a complete thermo-electrical characterization has been reported in [START_REF] Esposito | Innovative SiGe HBT Topologies With Improved Electrothermal Behavior[END_REF]; in this section we will further extend our study, showing the thermal impedance extracted from LF S-parameter measurements in the range 10kHz-1GHz. In particular, we will now focus on the set of test structures named VM, for which copper metal dummies, acting as heat spreaders, are present upon the emitter contacts till metal-3 (VM3), metal-6 (VM6) and metal-8 (VM8); in a reference transistor named VM1 the metallization on the emitter contacts stops instead at metal-1 level (refer to Fig. 3). It must be pointed out that the Rth value of these transistors is lowered as higher levels are reached for the metallization in the BEOL [START_REF] Esposito | Influence of the BEOL Metallization Design on the Overall Performances of SiGe HBTs[END_REF]- [START_REF] Alessandro | Advanced thermal simulation of SiGe:C HBTs including back-end-ofline[END_REF], since the heat flux can find an alternative path in the copper of the BEOL (the results for the Rth of these HBTs are detailed and reported in [START_REF] Esposito | Innovative SiGe HBT Topologies With Improved Electrothermal Behavior[END_REF]).

An accurate design of the metal connections can thus lead to improved SOA specifications [START_REF] Esposito | Innovative SiGe HBT Topologies With Improved Electrothermal Behavior[END_REF] and better RF performances [START_REF] Esposito | Influence of the BEOL Metallization Design on the Overall Performances of SiGe HBTs[END_REF]. Moreover, as it is suggested by equation (1), δ is inversely proportional to the square root of the Pdiss frequency, which means that, at low frequencies, the thermal oscillations can significantly penetrate the structure, both downwards (towards the back of the wafer) and upwards, till eventually reaching the higher levels of the metal in the BEOL. In this scenario, the different Zth trend observed at LF among the HBTs under study (Fig. 4), is due to the fact that δ has reached a distance within the abscissas of the metal dummies, and so transistors having heat spreaders till higher levels of metal will exhibit a lower Zth. In the high frequency range, instead, all the VM transistors show an identical Zth decay, in fact the thermal oscillations will tend to be confined in close proximity to the heat source as the frequency increases; since within this region all the transistor structures are identical (Fig. 3), their Zth will be the same.

From a physical modelling point of view, at low frequencies, temperature oscillations will interest a considerable volume of silicon around the HS, so it can be considered that high valued Cth are thermally charged and discharged. It is in fact possible EDL-2017-07-1234 to take into account the presence of the copper dummies in the VM transistors by adding big thermal capacitances in parallel to the thermal network of the lower part of the transistor [START_REF] Esposito | Dedicated test-structures for investigation of the thermal impact of the BEOL in advanced SiGe HBTs in time and frequency domain[END_REF]. These added Cth determine the thermal behavior at LF, but, as the frequency increases, the overall big volume of copper in the BEOL cannot participate to the temperature oscillations, due to its intrinsic thermal inertia; the AC dynamic phenomena can be rather approximated to happen in a small portion of silicon surrounding the BC junction.

It is interesting to notice, observing Fig. 4, that going from the highest frequencies (where all the VM transistors exhibit an identical Zth trend) towards the lowest, we can identify a frequency f1 (around 2MHz) starting from which the Zth of VM1 starts to deviate from the others. This means that, at f1 the thermal penetration depth has reached the abscissa corresponding to metal-1; the same assumption can be made as the frequency decreases, and it is also possible to identify f3 and f6, where δ reaches respectively the levels 3 and 6 of metallization. The set of test structures under study in [START_REF] Esposito | Innovative SiGe HBT Topologies With Improved Electrothermal Behavior[END_REF] also allows an investigation of the penetration depth in the horizontal plane: the thermal diffusion phenomena under study have in fact an isotropic nature. Among the structures presented in [START_REF] Esposito | Innovative SiGe HBT Topologies With Improved Electrothermal Behavior[END_REF], the transistor named HL1 contains metal connections reaching the same level as VM3, but has an increased Deep Trench Isolation (DTI) enclosed area. HL1 thus allows a wider angle for the lateral heat spreading, which in turn ensures a better evacuation of the heat generated from the HS towards the silicon substrate, yielding a lower Rth [START_REF] Esposito | Innovative SiGe HBT Topologies With Improved Electrothermal Behavior[END_REF].

Comparing the Zth of the HL1 and VM3 transistors shown in Fig. 5, it is again noticed that increasing the DTI enclosed area yields a decrease in the Zth at low frequency. At higher frequencies, though, it is just the volume in close proximity of the HS that determines Zth, and it can be supposed that, in this situation, the thermal variations do not reach the DTI, which restricts the heat flux. The condition that thermally favors HL1 does not hold any more, and the two structures tend to behave the same way for frequencies higher than around 1MHz (fDTI in Fig. 5), where we can assume that δ has reached, in the horizontal plane, the adiabatic wall represented by the DTI.

IV. CONCLUSIONS

In this letter we have shown, by means of thermal 3D simulations and reporting Zth measurements on specially designed transistor structures, how dynamic thermal mechanisms take place in SiGe HBTs from state-of-the-art technology nodes; in principle, though, similar thermal phenomena also take place in devices implemented in other technologies.

We demonstrated how, at different operating frequencies, the thermal impedance influences the spatial distribution of the temperature fluctuations within the transistor structure: at high frequencies, sinusoidal temperature variations are still present, even if they tend to be confined in the region close to the HS.

In the experimental part it has been shown how a specially conceived set of DUTs can allow to identify the frequencies at which the temperature oscillations imposed by a sinusoidal Pdiss reach a certain distance from the HS. This can be of interest in the development of physical thermal models for the Zth; the measured Zth(f) decays reported, in fact, clearly show a distributed nature. It is therefore strongly recommended to use at least a three poles network (instead of a single pole normally used in compact models) to accurately take into account dynamic thermal effects in the low frequency range. This is particularly important in circuit designs where IMD3 products arise within the thermal bandwidth of the HBTs.

Besides it is experimentally demonstrated that the configuration of materials in the BEOL, although far from the HS, has a non-negligible thermal impact: in particular the presence of metal connections can considerably affect the Zth decay at LF. In the same way, if the transistor layout is altered, (e.g. enlarging the DTI enclosed area) Zth variations should be rather present in theLF range, but it must be pointed out that instead at frequencies starting from around 1-2 MHz the same Zth trend has to be expected, regardless of any modifications of the HBT structure in the vertical or horizontal plane.

V. ACKNOWLEDGMENTS

The authors would like to thank STMicroelectronics for providing the B55 wafer used for the experimental characterization of the HBTs. f DTI

Fig. 1 .

 1 Fig. 1. Simulated 3D structure of the B11HFC HBT (zoom on the transistor), showing the thermal penetration depth (δ) profile. A sinusoidal power having a frequency of 50kHz (a) and 100MHz (b) are applied at the BC junction.

Fig. 2 .

 2 Fig. 2. Normalized amplitude (a) and phase (b) of the temperature oscillations in P1 (red) and P2 (blue) versus Pdiss frequency (PDC=40mW, PAC=2mW).

Fig. 4 .Fig. 3 .Fig. 5 .

 435 Fig. 4. Extracted Zth versus frequency for the VM test structures.

This work is funded by the French Public Authorities through NANO 2017 and by the EU, through Project Taranto (No. 737454).