Bruno Drouot

Jean Bonnet
email: jean.bonnet@unicaen.fr

Does organizational and local anchoring promote the resilience of fishing companies? A comparison of innovation behaviors in Brittany, Normandy and the Basque Country

Keywords: R11, Q22, Q57 resilience, territory, fishing

The objective of this article is to show that fishing companies in France that have succeeded in demonstrating resilience in a context of crisis retain economic and territorial characteristics making them still vulnerable to a new external shock (overexploitation of the resource, for example). Basing our research on resilience theory, we present the adaptive capacity of fishing companies through strong social organization, then their current weaknesses. The econometric model will highlight the explanatory factors of economic dependence by distinguishing fishing companies of three main regions: Brittany, Normandy and the Basque Country. Results make it possible to perceive the interest for each organization to maintain its economic trajectory or to modify it by developing new territorial diversification strategies.

Introduction

The number of fishing vessels in France has decreased by nearly 50% since 199050% since (Ifremer, 2019)). This situation mainly affects Brittany, the leading region for the maritime fishing sector, far ahead of Normandy. Fishing companies face international competition, falling selling prices for their production, rising production costs and changing consumer demand. In this context, fishing companies exploiting common sea bass have embarked on a voluntary and collective approach to better promote their production and better communicate their practices and commitments [START_REF] Charles | Coastal fishing: resource's enhancement and preservation[END_REF]. This profession, marked by its individualism and atomicity, has seen the birth of a strong social organization to defend its interests and communicate on the quality of their product. Their approach has resulted in the affixing of a quality label on their production, representing the species caught and the territory of capture.

Fishing companies have grouped together around "Line sea bass" quality labels, with their own organizational methods and logics, depending on their territory, Brittany, Normandy and the Basque Country. The appropriate scale for a label is the one to which the actors are linked by geographic and organizational proximity (Torre and [START_REF] Beuret | Proximités territoriales[END_REF]. This adaptation to the economic context has enabled them to continue their activity and observe an increase in the number of vessels exploiting this species [START_REF] Drouot | Effets d'une mesure de gestion sur les résultats économiques des exploitants d'une ressource naturelle[END_REF] in a sector in permanent decline. The creation of a "Line sea bass" label and the ability of companies to be part of a social and territorial organization has highlighted the organizational resilience of companies.

Local and organizational resilience as a theoretical framework

In 2007, Pendall et al. proposed a literature review of the concept of economic resilience based on different disciplinary fields (ecology, psychology, economics, geography, political science and archeology) and then underlined the interest of the concept for understanding regional economic dynamics [START_REF] Hassink | Regional resilience: a promising concept to explain differences in regional economic adaptability? Cambridge Journal of Regions[END_REF]. The concept is used to try to understand how regional or local economies react and recover in the face of shocks [START_REF] Crescenzi | The geography of the economic crisis in Europe: national macroeconomic conditions, regional structural factors and short-term economic performance[END_REF]. In the entrepreneurship literature, attention has been paid recently to the role of entrepreneurial ecosystems (EEs) in economic resilience thanks to innovation driven by new-firms startups [START_REF] Roundy | The resilience of entrepreneurial ecosystems[END_REF][START_REF] Roundy | The emergence of entrepreneurial ecosystems: A complex adaptive systems approach[END_REF][START_REF] Spigel | The relational organization of entrepreneurial ecosystems[END_REF]. In Germany, [START_REF] Fritsch | The long persistence of regional entrepreneurship culture: Germany 1925[END_REF] have proven the long persistence of regional entrepreneurial culture that favors some specific regions. In these regions, the entrepreneurial culture would involve attitude towards novelty that would avoid lock-in on unfavorable paths to regional economic development.

Organizational resilience is the concept of economic resilience at the level of an organization (company, association, etc.). It is the ability of this organization to face and overcome a hardship, incident or setback despite unfavorable conditions. Organizational resilience is an approach that can be described as proactive and which allows an organization to show its capacity to evolve in order to adapt to changes or threats in the environment [START_REF] Burnard | Organisational resilience : Development of a conceptual framework for organisational responses[END_REF]. Organizational resilience is seen as a learning process and no longer as a simple response to a disruptive element. The organization is thus able to develop a tolerance for risks but also an ability to adjust when faced with the uncertainties of its environment. Organizational resilience combines the interweaving of a shock absorption capacity with a capacity for renewal and a capacity for appropriation through organizational appropriation.

Case study for quantitative assessment: The "Line sea bass" quality label

The research method chosen will be the study of the three cases. According to [START_REF] Yin | Case Study Research[END_REF], the case study method can be used to explain, describe or explore events or phenomena in their real context. The case studies thus provide a detailed and in-depth analysis of the approaches promoting the sea bass valuation in Brittany, Normandy and the Basque Country (Drouot, Le Coroller, 2019).

Fieldwork was carried out through the collection of quantitative data and three "expert" interviews with the presidents of the "Line sea bass" associations. Quantitative information on fishing vessels is available and is grouped together in a database specific to the French system for collecting data on professional fishing. Thus, Ifremer's Fisheries Information System collects a set of information on fishing vessels, their activity, production and economic situation.

Organizational resilience that has been called into question in recent years

Statistical processing shows that fishing companies affixing the "Line sea bass" label continue to depend economically on sea bass, the other species caught by these vessels, in this case pollack and sea bream (Ifremer, 2020), only constituting a supplement of income. This economic dependence is closely linked to the territorial association the company belongs to. The strategies developed by fishing companies are different and territorialized. They more or less specialize in one species, while others have chosen to diversify by focusing on other species with lower added value.

The concept of organizational resilience is therefore ambivalent here because the success of adaptation and organizational learning of companies affixing the label "Line sea bar" is a success in a territory as long as a total depletion of the stock of common sea bass does not occur. Resilience is also linked to the capacity for adaptability, which designates the strategy of a company in its diversification into other economic trajectories. [START_REF] Hardin | The tragedy of the commons[END_REF] work on the tragedy of the commons remains relevant for this increasingly exploited, high-value resource. The organizational resilience of fishing companies located in Brittany and Normandy is challenging as well as their capacity to absorb a new external shock.

Bibliography:


D Chevallier

T Jonckheere

E Paladino

G Falci

T Martin

Detection of finite frequency photo-assisted shot noise with a resonant circuit

Keywords: numbers: 73.23.-b, 72.70.+m, 73.63.-b

Photo-assisted transport through a mesoscopic conductor occurs when an oscillatory (AC) voltage is superposed to the constant (DC) bias which is imposed on this conductor. Of particular interest is the photo assisted shot noise, which has been investigated theoretically and experimentally for several types of samples. For DC biased conductors, a detection scheme for finite frequency noise using a dissipative resonant circuit, which is inductively coupled to the mesoscopic device, was developped recently. We argue that the detection of the finite frequency photo-assisted shot noise can be achieved with the same setup, despite the fact that time translational invariance is absent here. We show that a measure of the photo-assisted shot noise can be obtained through the charge correlator associated with the resonant circuit, where the latter is averaged over the AC drive frequency. We test our predictions for a point contact placed in the fractional quantum Hall effect regime, for the case of weak backscattering. The Keldysh elements of the photo-assisted noise correlator are computed. For simple Laughlin fractions, the measured photo-assisted shot noise displays peaks at the frequency corresponding to the DC bias voltage, as well as satellite peaks separated by the AC drive frequency.

I. INTRODUCTION

The understanding of the transport properties of nanoscale conductors at low temperatures has known tremendous successes via experiments in a wide range of systems performed for the most past in the stationary regime. Correspondingly, theoretical modelling has allowed the description of these transport processes via scattering theory approaches as well as Hamiltonian formulations, in a fruitful dialogue with experimental investigations. Transport is first characterized by the average current flowing through conductors. But further information can be gained via the measurement and analysis of the current fluctuations [START_REF] Martin | Nanophysics : coherence and transport[END_REF][START_REF] Blanter | [END_REF] and more generally via the higher current moments. 3 Early investigations of quantum transport focused almost exclusively on the low frequency regime. Few recent experiments have probed quantum system on timescales comparable with the electron correlation time, where new physical effects are expected. The present work deals with the detection of quantum noise at such high frequencies, when both a DC and a AC bias is imposed between the source and the drain of the mesoscopic system.

Indeed, high frequency measurements can mean several things. First, if only a DC bias is imposed on the sample, a stationary current is generated and high frequencies refer to the Fourier component of the current-current correlation function in time. 4,5,6,7 Second, high frequencies can be injected as a drive on the mesoscopic circuit, 8,9,10,11 for instance when an additional AC drive is superposed to the DC bias. The later effect is called photo-assisted (PA) transport: electrons undergoing transmission from one lead to another are able to absorb/emit "photons" during this process. PA transport, and in particular PA noise has been studied theoretically and experimentally on several occasions for diffusive metals, 5 tunnel junctions, 12 normal metal/ superconductor junctions 13,14 as well as quantum point contacts. 6 The noise caracteristics then displays some structure at values of the DC bias which are multiple of the AC drive frequency.

However, high requency noise detection requires special care: conventional (low) frequency noise detection setups are often inadequate for such measurements, and one must often resort to on-chip detection schemes, or alternatively/equivalently to schemes where a good connection to the measurement circuit is achieved through adapted impedence lines. 15 On chip detectors have allowed the detection of single electrons travelling through quantum dots. Such detectors and the device they probe are parts of the same quantum system and must be treated on the same footing. They bear the peculiarity that the noise which is measured is a non trivial combination of non-symmetrized noise correlators. For DC driven systems there are existing proposals to detect high frequency noise using either capacitive or inductive coupling with an on-chip circuit. 16 In a recent theoretical work, a LC resonant circuit, which was coupled inductively to the mesoscopic device circuitry, was employed as a detector of both noise and higher current moments (third moment). 17 The description of this generic detector included its electromagnetic environement, described at a bath of harmonic oscillators with the Caldeira Legett model 18 . Predictions were made on the role of such a dissipative environment and on the relevance of this harmonic detector to capture on high frequency current moments. However, this study considered the case of a mesoscopic device in a stationary regime (with a DC bias only). The hypothesis of a stationary regime greatly simplies the analysis of the detection process because of time translational invariance. The presence of an additional AC voltage drive breaks such a property.

Given the interest in the study of time driven mesoscopic systems and in particular PA noise, it seems necessary to address how detection with an auxiliary circuit can be achieved in such situations. The purpose of this work is to present a high frequency detection scheme for photoassisted noise, and to illustrate it with a calculation of photoassisted noise in a specific situation where signatures of photoassisted transport are most dramatic. For devices composed of normal metal junctions as well as superconducting/normal metal junctions, PA noise exhibits singularities at integer ratios of the DC voltage with respect to the AC frequency: the derivative of this noise exibits jumps at such locations. On the other hand, for a weakly pinched quantum point contact placed in the fractional quantum Hall effect regime (FQHE), 19,20,21,22,23 the PA noise diverges when the DC voltage -multiplied by the filling factor -is a multiple of the AC frequency. This much stronger singularity is a motivation for us to apply our measurement scheme to the FQHE situation. We will show that as in the DC case, the measured noise captures the response of the mesoscopic circuit at the resonant frequency of the LC circuit. It exhibits a central peak at the DC voltage, which is surrounded by satellite peaks shifted by the AC frequency. These predictions have the potential to be tested in experiments.

The paper is organized as follows. In Sec. II we present the model for the LC detector. We review the results for the charge correlator of the DC cirsuit in Sec. III and extend this discussion to the PA situation. Sec. IV is devoted to the presentation of the QPC in the FQHE regime and its calculation of PA noise. Plots of these quantities and of the measured noise are discussed in Sec. V. We conclude in Sec. VI.

II. MODEL

The proposed setup is the same as that presented in Ref. 17, except for the fact that the voltage source on the mesoscopic device is time dependent. A lead from such device is inductively coupled to a resonant circuit (capacitance C , inductance L, and dissipative component R ). The signal which contains information about the noise of the mesoscopic circuit is encoded in the time correlation function of the charge on the capacitor.

FIG. 1: Mesoscopic circuit is coupled to a resonant dissipative circuit

We start with the description of the detector. The basic Hamiltonian which describes the dissipative oscillator circuit reads:

H osc = H 0 + H L C en v ; (1)
where

H 0 = H L C + H en v (2)
is the Hamiltonian of the uncoupled system "LC oscillator plus environment", and H L C en v describes the coupling between the two. For dissipative quantum systems, it is convenient to use a path integral formalism. In the absence of dissipation and coupling to the mesoscopic device, the Keldysh action describing the charge of the LC circuit reads:

S L C [ q]= 1 2 Z dtdt 0 q T (t)G 1 0 (t t 0 ) z q(t 0 ); (3)
where

G 1 0 (t t 0 )= L[ (i@ t ) 2 2 ] (t t 0 ); (4)
is the (inverse) Green function of an harmonic oscillator (L is its "mass"), = (LC ) 1=2 is the resonant frequency of the circuit, q T = (q + ;q )is a two component vector which contains the oscillator coordinate on the forward/backward contour, and z is a Pauli matrix in Keldysh space. Dissipative effects are treated within the Caldeira-Leggett model, where the environment is modeled by a set of harmonic oscillators (bath) with frequencies f! n g; the coordinate qis coupled linearly to the bath oscillators:

H L C en v = q X n n x n ; (5)
with the coupling constants n . The partition function of the LC oscillator plus bath, Z = R D qD xe iS [q;x] , has an action:

S = S L C + 1 2 X n x T n D 1 n z x n q T z X n n x n ; (6)
where

D 1 n (t) = M n [ (i@ t ) 2 ! 2 n ] (t)
and the symbol stands for convolution in time. The bath degrees of freedom can be integrated out in a standard manner 24 . As a result, the Green function G of the LC circuit becomes dressed by its electronic environment,

G 1 = G 1 0 ; (7)
with a self-energy

(t)= z P n 2 n D n (t) z .
In the remainer of this paper, when we mention the LC circuit, it will also imply the presence of its surrounding electromagnetic environment.

Next, we introduce the inductive coupling between the mesoscopic device and the LC circuit,

H in t = q _ I ; (8)
where _ I is the time derivative current operator 25 . This interaction is interpreted here as an external potential acting on the oscillator circuit. To calculate correlation functions of the LC circuit coordinate q, we introduce the generating functional,

Z [ ;I]= Z D q exp i h 1 2 q T G 1 q q T z ( _ I+ ) i ; (9)
where T = ( + ; )is a two-component auxiliary field. Performing integration over the LC oscillator variables q results in Z [ ;I]= e iS ef f [ ;I] with an effective action (restoring integrals):

S eff [ ;I] = i 2 Z dt Z dt 0 ( (t)+ _ I(t)) T z G (t t 0 ) z ( (t 0 )+ _ I(t 0 )) i : (10)

III. CHARGE CORRELATOR

By taking double derivatives of the Kelysh partition function with respect to the components of the spinor , the charge correlator is obtained:

hq (t)q 0 (t 0 )i Z 1 [ I] @ 2 Z [ I] @ (t )@ (t 0 0 ) = 0 ; (11)
where ; 0 1 are indices specifying the upper/lower branch of the Keldysh contour. To leading order in the coupling constant between the mesoscopic circuit and the detector 17 this can be expressed in terms of the current derivative correlator:

K 1 2 ( 1 ; 2 )= D T K _ I( 1 ) 1 _ I( 2 ) 2 E m eso ; (12)
where the average h: : : i m eso represents a non equilibrium average containing information on the occupation of the reservoirs connected to the sample and on its scattering properties. The charge correlator consists then of a Keldysh matrix:

D T K q (t)q 0 (t 0 ) E = 2 Z d 1 d 2 X 1 2 G 2 (t 2 ) 2 2 z K 2 1 ( 2 ; 1 ) 1 1 z G 1 0 ( 1 t 0 ); (13)
where the integrand contains the Green function G 0 (t) of the LC circuit only. While this Green function is a function of a single time argument because of time translational invariance, the current derivative correlator K 1 2 ( 1 ; 2 )is not a function of the difference 1 2 if the bias voltage is time dependent.

A. DC Voltage only

We recall the results obtained previously for the detection of finite frequency noise in the presence of time translational invariance. The initial proposal of Ref. 25 for a dissipationless LC circuit was to operate repeated time measurements on the charge q. This allows to construct an histogram for zero voltage, yielding the zero bias peak position, its width, skewness,... In the presence of bias, this histogram is shifted, and acquires a new width, skewness,... Information about all current moments at high frequencies is encoded in such histograms. Here, however, we only focus on the detection of noise. In Ref. 17, the inclusion of dissipation due to the electromagnetic environment was shown to be essential to obtain a finite result for the measuring process. There, expressions for the off diagonal Keldysh component of the charge correlator hT K q (t)q + (t 0 )i = hq(t t 0 )q(0)iwere derived with the help of Eq. ( 13). Note that in this situation, the current derivative correlator of Eq. ( 12) is a function of the difference 1 2 , and the charge correlator is a convolution product, which explains its dependence on t t 0 only. Going to the rotated Keldysh basis (see Appendix B) allows to rewrite the charge fluctuations at equal time (t= t 0 ) as:

hq 2 i = 2 Z d! 2 G R (!)fG K (!)K + (!) (G R (!) G A (!))K + (!)g ; (14)
with the three Green function components given by:

G R =A (!)= [ L(! 2 2 ) isgn(!)J(j !j )] 1 ; (15)
and

G K = (2N (!)+ 1)(G R (!) G A (!)); (16)
where N (!)is the Bose occupation number of the oscillator and the bath spectral function is defined as:

J(!)= X n 2 n =(2M n ! n ) (! ! n ): (17)
This spectral function is at the origin of the broadening for the LC circuit Green function.

The time derivative correlators K + ;+ are related to the Fourier transform of the current current correlation functions as

K + (!)= ! 2 S + (!)and K + (!)= ! 2 S + (!), with S + (!)= Z dthI(0)I(t)ie i! t ; (18)
and S + (!) = S + ( !)corresponding to the response function for emission/absorption of radiation from/to the mesoscopic circuit 25,26 . With these definitions, the final result for the measurable excess noise reads:

hq 2 i = 2 2 Z 1 0 d! 2 ! 2 [ 00 (!)] 2 S + (!)+ N (!)(S + (!) S + (!)) ; (19)
where

00 (!)= J(j !j )=[ L 2 (! 2 2 ) 2 + J 2 (j !j
)is the susceptibility of Ref. 18, here generalized to arbitrary J(j !j ). Eq. ( 19) constitutes a mesoscopic analog of the radiation line width calculation 27 : a dissipative LC circuit cannot yield any divergences in the measurable noise. Dissipation is essential in the measurement process.

Eq. (19) indicates that for an infinitesimal line width, the integrand can be computed at the resonant frequency , and the measured noise takes the form of Ref. 25:

q 2 = 2 L 2 S + ( )+ N (S + ( ) S + ( )) ; (20)
where the prefactor is defined assuming a strict Ohmic or Markovian damping (J(!) = L !), which corresponds to a memoryless bath which is consistent with the adiabatic switching assumption, as discussed in Ref. 17.

As an alternative to the measurement of the width of the charge distribution, one can imagine that the capacitor itself is coupled to a measuring device (a single electron tunneling device) which directly detects the Fourier transform of the charge correlator. 28 Given the fact that the charge correlator matrix of Eq. ( 13) is a convolution product in this stationary situation, its Fourier transform take the simple form of a product of matrices:

Z dthT K q (t)q 0 (0)ie i! t = 2 h G (!) z K (!) z G (!) i 0 ; (21)
where G (!)and K (!)are respectively the matrix version of the LC Green's function and of the current derivative correlator. Naturally this will have substantial contributions when both K and G overlap significantly. This constitutes a rather compact way for describing the detection process in the case of a constant bias voltage.

B. AC drive and temporal invariance

We now turn to the main point of this section, which is to address how to deal with the presence of an AC voltage superposed to the DC one. The total bias potential V (t)which is applied to the mesoscopic device is thus a periodic function of time with period = 2 =! A C . We start by defining a correlator k(T;t 00 )from the current derivative correlator of Eq. ( 12):

K (t;t 0 ) k( t+ t 0 2 ;t t 0 ): (22)
Defining T = (t+ t 0 )=2, t 00 = t t 0 , the charge correlator of Eq. ( 13) is rewritten as:

D T K q (t)q 0 (t 0 ) E = 2 Z dt 1 dt 2 X 1 2 G 2 (t 00 t 2 ) 2 2 z k 2 1 (T + t 0 ;t 2 t 1 ) 1 1 z G 1 0 (t 1 ): (23)
where t 0 = (t 2 + t 1 )=2 t 00 =2. Next, we define the average of the charge correlator over the period of the AC drive 11 as follows:

1 Z 0 dT D q (t)q 0 (t 0 ) E = 2 Z dt 1 dt 2 X 1 2 G 2 (t 00 t 2 ) 2 2 z Z 0 dT k 2 1 (T + t 0 ;t 2 t 1 ) 1 1 z G 1 0 (t 1 ): (24)
Note that the last integral over the variable T is essentially a period average of the correlator k(T;t 00 )with the variable T shifted by t 0 . In the presence of an AC drive, this period average does not depend on the shift t 0 , because as a function of the variable T , k(T;t 00 )contains only harmonics of the drive frequency ! A C . This has been noticed in earlier works. 13,29,30 For our purposes, it means that we can safely replace t 0 by 0. As a result, the period averaged charge correlator takes the form of a convolution product as was the case for the constant DC bias, and it therefore depends only on the time difference t t 0 :

Q 0 (t t 0 ) 1 Z 0 dT D q (t)q 0 (t 0 ) E = 2 Z dt 1 dt 2 X 1 2 G 2 (t 00 t 2 ) 2 2 z K 2 1 (t 2 t 1 ) 1 1 z G 1 0 (t 1 ); (25)
where we defined the period averaged correlator:

K 2 1 (t) 1 Z 0 dT k 2 1 (T;t): (26)
Finally, the averaged charge correlator can be expressed in terms of the Fourier transfrom of both the LC circuit Green's function and the period averaged current correlator:

Q 0 (t 00 )= 2 X 1 2 Z d! 2 e i! t 00 G 2 (!) 2 2 z K 2 1 (!) 1 1 z G 1 0 (!): (27)
This result is the exact analog of the DC formula Eq. 21, extended to and AC drive. In addition, at t 00 = 0, this expression has the same form as the result of Ref. 17. We have therefore identified which quantity (K ) characterizes the influence of the mesoscopic circuit on the response of the LC circuit. Therefore, the protocol for measuring photoassisted shot noise is the same as in the DC case provided one averages the response over the frequency of the drive. This averaging procedure restores the temporal invariance of the charge correlator. In the following sections, we will compute the current derivative correlators and their period average for a specific system: a QPC placed in the conditions of the FQHE where the elementary transport process is the Poissonian transfer of Laughlin quasiparticles.

IV. NON SYMMETRIZED PHOTO-ASSISTED NOISE IN THE FQHE

The calculation of the symmetrized photoassisted noise has been carried out in Ref. 30. Here we use the same basic model and generalize the calculations of the noise correlator to the full Keldysh matrix elements of this correlator. Next, we extract from these the noise derivative correlators which are relevant for the measurement process. We use the Tomonaga-Luttinger formalism to describe the right and left moving chiral excitations. In the absence of tunneling between the two edges, the Hamiltonian reads:

H 0 = ( F 4 ) X r Z ds(@ s r ) 2 ; (28)
with r = + ; for right and left movers. Here, we focus solely on the weak backscattering regime because it is already known that the PA shot noise exhibits some strong singularities. The backscattering of quasiparticles is described by the Hamiltonian:

H B (t)= X " A (") (t)[ y + (t) (t)] (") ; (29)
where A (") (t) is a tunneling amplitude which depends on the applied voltage via the Peierls substitution. Here the notation = leaves an operator unchanged for ( = + ) or specifies its Hermitian conjugate ( = ). r is the quasiparticle operator which is expressed in terms of the bosonic chiral field r :

r (t)= 1 p 2 a e i p r (t) ; (30)
where a is a short distance cutoff and is the filling factor ( 1 is an odd integer to describe Laughlin fractions). Choosing a time dependent voltage in the form

V (t)= V 0 + V 1 cos(! A C t)
results in a tunneling amplitude:

A (") (t)= 0 e i"! 0 t exp(i" e V 1 ~!A C si n(! A C t)); (31)
where e = e and 0 is the bare tunneling amplitude. The backscattering current is deduced from the backscattering Hamiltonian:

I B (t)= ie ~X" "A (") (t)[ y + (t) (t)] (") : (32)

B. Non-symmetrized noise

The general expression for the Keldysh components of the noise correlator in the Heisenberg representation is:

S 0 (t;t 0 )= hI B (t)I 0 B (t 0 )i hI B (t )ihI B (t 0 0 )i: (33)
Since we are interested in Poissonian regime only, the product of current averages can be dropped out because it contributes to higher order in the backscattering Hamiltonian. [START_REF] Martin | Nanophysics : coherence and transport[END_REF] Moreover, in this second order calculation in the tunneling amplitude 0 , there is no difference between the Heisenberg and interaction picture. The noise then reads:

S 0 (t;t 0 )= (e ) 2 X "" 0 "" 0 A (") (t)A (" 0 ) (t 0 )hT K f[ y + (t ) (t )] (") [ y + (t 0 0 ) (t 0 0 )] (" 0 ) gi: (34)
This correlator is different from zero only when " 0 = " because of quasiparticle conservation. Replacing the quasiparticle correlators by their bosonized expression, the noise is then written in terms of a product over averages of bosonic fields:

S 0 (t;t 0 )= (e ) 2 4 2 a 2 X " "A (") (t)A ( ") (t 0 )hT K e i" p + (t ) e i" p + (t 0 0 ) ihT K e i" p (t ) e i" p (t 0 0 ) i: (35)
The final result for the real time noise correlator is then:

S 0 (t;t 0 )= (e ) 2 4 2 a 2 e 2 G 0 (t t 0 ) (A (t)A (t 0 )+ A (t)A (t 0 )) ; (36)
where we introduced the chiral green function of the bosonic fields:

G 0 (t;t 0 )= hT K f r (t ) r (t 0 0 )gi 1 2 hT K f r (t ) 2 gi 1 2 hT K f r (t 0 0 ) 2 gi: (37)
The double Fourier transform of this quantity, which will allow to relate it to the noise correlator, reads:

S 0 ( 1 ; 2 )= Z Z dtdt 0 e i( 1 t+ 2 t 0 ) S(t;t 0 ): (38)
We now specify the periodic voltage modulation, which allows to write the tunneling amplitude in terms of a series of Bessel functions J n :

A (t)= 0 + 1 X n = 1 e i(! 0 + n ! A C )t J n e V 1 ~!A C ; (39)
which gives the Fourier transform of non-symmetrised noise:

S 0 ( 1 ; 2 ) = (e ) 2 2 0 2 2 a 2 + 1 X n = 1 + 1 X m = 1 J n e V 1 ! A C J m e V 1 ! A C Z Z dtdt 0 e i( 1 t+ 2 t 0 ) e 2 G 0 (t;t 0 ) cos(! 0 (t t 0 )+ ! A C (nt m t 0 )): (40)
Next, it is convenient to perform a change of variable = t t 0 and 0 = t+ t 0 :

S 0 ( 1 ; 2 ) = 2 (e ) 2 2 0 2 2 a 2 + 1 X n = 1 + 1 X m = 1 J n e V 1 ! A C J m e V 1 ! A C Z Z d d 0 e i( 1 2 ) =2 e i( 1 + 2 ) 0 =2 e 2 G 0 ( ) cos ! 0 + n + m 2 ! A C + n m 2 ! A C 0 :
Using standard trigonometric identities, one can write this expression as a product of separate integrals over and 0 . Integrals over contain the (zero temperature) Green's function of the chiral fields and can be expressed in terms of Gamma function.

The result has the form:

S 0 ( 1 ; 2 ) = 2 (e ) 2 2 0 2 2 a 2 + 1 X n = 1 + 1 X m = 1 J n e V 1 ! A C J m e V 1 ! A C h I 1 ( 1 + 2 ;!)I 0 2 ( 1 2 ;! 0 ;!) I 3 ( 1 + 2 ;!)I 0 4 ( 1 2 ;! 0 ;!) i : (41)
The integrals I 1 , I 0 2 , I 3 , I 0 4 are defined and computed in the Appendix. The final result for the 4 Keldysh matrix elements of the noise correlator is:

S ( 1 ; 2 )= 2 (e ) 2 2 0 4 2 a 2 + 1 X n = 1 + 1 X m = 1 J n e V 1 ! A C J m e V 1 ! A C (2 ) ( a F ) 2 h (1 sgn ( 1 + ! 0 + n! A C ))j 1 + ! 0 + n! A C j 2 1 ( 1 + 2 + (n m )! A C ) + (1 sgn( 1 ! 0 n! A C ))j 1 ! 0 n! A C j 2 1 ( 1 + 2 (n m )! A C ) i ; (42)
S ( 1 ; 2 )= 2 (e ) 2 2 0 4 2 a 2 + 1 X n = 1 + 1 X m = 1 J n e V 1 ! A C J m e V 1 ! A C (2 ) ( a F ) 2 e i cos( ) h j 1 + ! 0 + n! A C j 2 1 ( 1 + 2 + (n m )! A C )+ j 1 ! 0 n! A C j 2 1 ( 1 + 2 (n m )! A C ) i : (43)
We recognize that since we are dealing with simple Laughlin fractions of the FQHE, is the inverse of an odd integer and all Keldysh component exhibit power law singularities when the quantity 2 (! 0 + n! A C )vanishes. As a check, it is possible to recover from these components the previous result for the symmetrized noise: 30

S sym ( 1 ; 2 ) = 1 2 (S + ( 1 ; 2 )+ S + ( 1 ; 2 )) = 2 (e ) 2 2 0 4 2 a 2 + 1 X n = 1 + 1 X m = 1 J n e V 1 ! A C J m e V 1 ! A C (2 ) ( a F ) 2 h j 1 + ! 0 + n! A C j 2 1 ( 1 + 2 + (n m )! A C ) + j 1 ! 0 n! A C j 2 1 ( 1 + 2 (n m )! A C ) i : (44)
It is also useful to know that the standard property of Keldysh Green's functions:

S + + ( 1 ; 2 )+ S ( 1 ; 2 )= S + ( 1 ; 2 )+ S + ( 1 ; 2 ) (45)
applies as it should for the double Fourier transform expressions.

C. Current derivative correlators

The relation between the Fourier components of the noise correlator computed in the previous section and the current derivative correlator introduced in Sec. III reads:

K 0 ( 1 ; 2 )= 1 2 S 0 ( 1 ; 2 ): (46)
Yet, we need to relate the noise correlator S 0

( 1 ; 2 )to the correlator k(T;!)and ultimately, to its time average K 2 1 (!). This is achieved using the relation:

k 0 (T;!)= Z d! 1 2 e i! 1 T K 0 ( ! 1 2 + !; ! 1 2 !): (47)
So the final result for the four averaged noise derivative correlators reads:

K (!)= 1 Z 0 dT k + (T;!)= (e ) 2 2 0 4 2 a 2 + 1 X n = 1 J 2 n e V 1 ! A C 1 (2 ) ( a F ) 2 ! 2 h (1 sgn (! + ! 0 + n! A C ))j ! + ! 0 + n! A C j 2 1 + (1 sgn (! ! 0 n! A C ))j ! ! 0 n! A C j 2 1 i (48) K (!)= 1 Z 0 dT k + + (T;!)= (e ) 2 2 0 4 2 a 2 + 1 X n = 1 J 2 n e V 1 ! A C 1 (2 ) ( a F ) 2 e i cos( ) ! 2 h j ! + ! 0 + n! A C j 2 1 + j ! ! 0 n! A C j 2 1 i : (49)
To be complete, we can compute all the Keldysh element in the rotated basis. This is performed in Appendix B. While the advanced and retarded contribution do not bear information on the non equilibrium nature of the transport processes taking place in the mesoscopic devices and therefore in the detector, the Keldysh component:

Q K = 2 Z d! 2 h G R (!) K R (!) G K (!)+ G R (!) K K (!) G A (!)+ G K (!) K A (!) G A (!) i ; ( 50
)
summarizes such an information in a compact way. We recall that as an alternative to the measurement of the equal time charge correlator, Eqs. ( 27) and (50) are also likely to be measured directly (resolved in frequency) by an SET device. 28 This completes the calculation of the current derivative correlators. In the following section we continue with the same analysis as with the DC case 17 . That is we use the contour ordered elements of the charge correlator, in particular the + component evaluated at equal time: in Sec.V we insert the expressions for Eqs. ( 48) -(49) and discuss the results.

V. RESULTS

We now discuss the formulas obtained in Sec. IV. In all of the results below, we have checked that when the AC drive frequency is set to 0, we recover the DC results for the finite frequency noise at = 1 [1] and = 1=3 [31]. For the QPC we focus on the voltage dominated regime where the temperature is taken to zero in the current correlator, but it nevertheless enters the detector response.

A. Excess noise in the quantum Hall effect

We start with a discussion of the results for the non symmetrized excess PA noise. We show in Fig. 3 the curves for the current derivative correlator K + ( )(see eq. 48), which is the quantity which enters the expression of the measured noise (the charge correlator). This is displayed for two different values of the filling factor . We choose for our main interest the Laughlin fraction = 1=3, which is in principle the easiest attainable Laughlin fraction of the FQHE in experiments, and = 1, the integer quantum Hall effect case, which here also corresponds to the noise caracteristics of a single channel normal tunneling junction.

Here we have chosen the DC voltage so that the central frequency ! 0 eV 0 =~is larger than the drive frequency ! A C , and the amplitude of the AC voltage (!

1 = eV 1 =~) is such that ! 1 =! A C = 1
For = 1=3 we find divergences for K + ( ) located at ! 0 and at sidebands ! 0 + n! A C . Sidebands with n = 1; 2 are visible. K + ( )vanishes at zero frequency. For frequencies larger than = ! 0 + 2! A C , this noise derivative correlator seems to be negligible. While the formulas for K + ( ) show a power law divergence, here one has to add a regularization procedure because strickly speaking, the calculations have been performed in the weak backscattering regime. This means that the differential conductance associated with the tunneling current has to be lower than the conductance quantum (otherwise, one should examine the case of the crossover to the strong back scattering regime). The validity condition of our results have been previously derived in Eq. ( 24) of Ref. 30. For our purpose, it just implies that the finite frequency PA noise saturates at locations

= ! 0 + n! A C .
In the integer quantum Hall case = 1, no divergences are found for K + ( ). Instead, singularities in the derivative occur for ! 0 + n! A C , and the current derivative correlator seems again to be negligible again beyond

= ! 0 + 2! A C .
!0 = 3!A C , !1 = !A C and K + ( )is normalized by e IB ! 2 0 )
However it is also interesting to plot K + ( )= 2 ; in this way we have access to an "averaged" current correlator (noise) because the term 2 in K + ( )is in fact due to derivative operators acting on the current correlator. This is depicted in Fig. 4. For = 1=3 we again find divergences (at the same locations as for K + ( )). The only noticeable difference with the latter curves is that the averaged noise does not vanish at zero frequency. If one ignores the side bands, the central peak reminds us clearly of the finite frequency non symmetrized noise computed recently for a QPC in the FQHE. 31 For = 1 the finite frequency noise again exhibits jumps in its derivative with respect to frequency, but its behavior is linear between two successive singularities. Thus, for this ratio of frequencies ! 0 =! 1 > 1, the excess noise characteristics resembles the finite frequency noise in the absence of an AC drive: the later is (essentially) linear for < ! 0 and vanishes beyon this. Yet the PA noise does not vanish at = ! 0 , it shows a singularity in its derivative at its location, together with singularities at 4) and in the following section, we use the back scattering current to zero order in the amplitude of ! 1 of the modulation wich corresponds to the pure stationnary regime 30 :

I (0) B = e 2 0 2 2 a 2 (2 ) ( a F ) 2 sgn(! 0 )j ! 0 j 2 1 : (51)

B. Measured PA noise

In this section, we display curves for the charge correlator at equal times. We consider excess quantities. By excess, we mean that the charge correlator at zero voltage has been subtracted from the charge correlator at V 0 ;V 1 6 = 0.

In Figs. 5, 6, 7 and 8 we plot these quantities:

Q + (0)= 2 X 1 2 Z d! 2 G 2 (!) 2 2 z K 2 1 (!) 1 1 z G 1 + (!): (52)
with weak and strong dissipation, low and high (detector) temperature for two different values of the ratio ! 1 =! A C , which corresponds to the argument of the Bessel functions in the expression of the charge correlator. In the following curves Q + is always normalized by L 2 =( 2 e I B ! 2 0 )and dissipation and temperature are in unit of ! 0 . The frequency = ! 0 corresponds to the positions of the central peak.

We start with ! 0 > ! A C . In order to resolve these peaks, it is necessary that the width of the resonance level is smaller than the spacing ! A C between peaks. We observe that by varying ! 1 =! A C , the relative amplitudes of the peaks can be modulated.

In Fig. 6, the curves correspond to a ratio ! 1 =! A C = 2: we can clearly identify the central peak at = ! 0 but it is smaller than in the case ! 1 =! A C = 1 (in Fig. 5). In this situation we identify very clearly the first and the second satellite peaks, while the third one (n = 3) is visible but with a lesser intensity. The relative amplitude of the central peak and its satellite is tied to the oscillatory behavior of the Bessel function.

When ! 1 =! A C = 1, the 0th order Bessel function, which corresponds to the central peak has a large amplitude( 0: 6). The 1st order Bessel function which corresponds to the first satellite peak, has a smaller amplitude ( 0: 2). The third Bessel function which corresponds to the second satellite peak is almost zero. On the other hand for ! 1 =! A C = 2, the 0th and the 3rd order Bessel function are small compared to its 1st and 2nd order counterparts, thus the central peak is smaller than the satellites.

Next, we choose ! 0 < ! A C in Fig. 7 and8. The finite frequency spectrum of charge fluctuations does not seem to display any longer a central peak with equally spaced satellites.

In Fig. 7, the curves correspond to a ratio ! 1 =! A C = 1 and in Fig. 8 the curves correspond to a ratio ! 1 =! A C = 2. In Fig. 7, the curves have a central peak at frequency ! 0 , a secondary one at ! 0 + ! A C and a third one at ! 0 + 2! A C . However there appear peaks at frequencies ! 0 + ! A C and ! 0 + 2! A C : this corresponds to the satellites peaks of the negative frequency ! 0 . We can explain this phenomena as the overlapping of two combs, centered at ! 0 . In Fig. 8, the curves exhibit the same phenomena but the peaks have different relative amplitude, which can again be explained from the argument of the Bessel functions. In these differents plots, we can see on the one hand the effect of dissipation wich reduces the noise and smoothes the peaks. On the other hand we see the effect of temperature; the measured noise become negative at higher temperature (as in the case = 1 and D C applied voltage 17 ) because of S + S < 0 wich is larger than S + and because of the large population of LC oscillator states.

VI. CONCLUSION

The central point of this paper has been the presentation of a measurement scheme for detecting finite frequency photo-assisted noise of a mesoscopic conductor on which both a DC and an AC bias is imposed. This scheme uses a dissipative resonant circuit which is inductively compled to the mesoscopic circuit, in the same manner as some of the author's previous work 17 , which we reviewed at the begining of the paper. The major hurdle in analyzing PA shot noise lies in the lack of time translational invariance which results from the presence of the AC drive. We have shown that by considering the average of the charge correlator of the LC over the period of the AC drive, time translational invariance can be restored, and an extension of our previous detection scheme can be envisioned.

We illustrated our detection scheme by applying it it to a concrete situation where PA noise features are most visible. We therefore considered the PA noise generated from a point contact in the weak-backscattering regime, placed in the regime of the FQHE. While the symmetrized PA noise at zero frequency was previously derived by some of us, no derivation of its full Keldysh components at finite frequency was available to this date. The PA noise contains singularities at frequencies corresponding to the bias voltage, with satellite singularities separated by the AC drive frequency. These sharp features in the noise are the main motivation for the application of our detection scheme. After deriving in this situation the current derivative correlator, we were able to compute explicitly the response of the LC circuit via the charge correlator, and to display the results for a variety of parameters.

Coupling of the detector to an electromagnetic environment, here modelled by an ohmic bath of oscillators, smoothens the anomalies of the detected signal. The damping parameter ought to be smaller than either the DC frequency or the AC drive frequency in order that the desired effects are observed. This observation is crucial for experiments, and broadens the scope of the results since the electromagnetic environment may also model other backaction effects on the detector.

The second important and non trivial effect is that the measured noise become negative if we increase the temperature of the detector. Remember that we are considering excess measurements; negative noise thus means that the noise for non-zero DC and AC voltages is smaller that the noise for zero voltage.

Given the fact that the AC modulation gives rise to satellite peaks at ! 0 + n! A C , we distinguished two limits: ! 0 > ! A C where the central peak at the DC voltage is surrounded by its satellites, and ! 0 < ! A C where the satellites of the negative DC voltage frequency can lie in the positive frequency domain of the charge correlator. Both situations can be realized in practice. This brings us to the question about optimizing the detection of the location of the central peak and its satellites. Upon varying the ration between the AC drive amplitude and the AC frequency, we have shown that one can modify the respective amplitude of such peaks. This constitutes an additional knob for detection.

The present results constitute a step in the direction of fundamental aspects of mesoscopic physics of detection in the time domain. This is an area of growing importance in mesoscopic physics when conventional detection machinery has to be abandoned, and novel detection schemes for high frequencies adapted to the type of experiments on wishes to perform. Granted, from the experimental point of view, the LC circuit setup which we have presented here may appear a bit naive. In the long run, indeed one should attempt to describe more precisely the connection between the output signal of the mesoscopic circuit and the the transmission lines which are connected to it. This will be the topic of further investigations.

X n = 1 + 1 X m = 1 J n e V 1 ! A C J m e V 1 ! A C h I 1 ( 1 + 2 ;!)I
I 2 = 2 (2 ) ( a F ) 2 " 1 sgn 1 2 2 ! 0 n + m 2 ! A C 1 2 2 ! 0 n + m 2 ! A C 2 1
+ 1 sgn

1 2 2 + ! 0 + n + m 2 ! A C 1 2 2 + ! 0 + n + m 2 ! A C 2 1
# : (A9)

I 2 = 1 2 ( a F ) 2 (2 )
e i cos( )

1 2 2 ! 0 n + m 2 ! A C 2 1 + 1 2 2 + ! 0 + n + m 2 ! A C 1 (2 ) ( a F ) 2 ! 2 h ( itan( ) sgn (! + ! 0 + n! A C ))j ! + ! 0 + n! A C j 2 1 + ( itan( ) sgn (! ! 0 n! A C ))j ! ! 0 n! A C j 2 1 i : (B4) K K (!)= 1 Z 0 dT kK (T;!)= (e ) 2 2 0 4 2 a 2 + 1 X n = 1 J 2 n e V 1 ! A C 1 (2 ) ( a F ) 2 ! 2 2 h j ! + ! 0 + n! A C j 2 1 + j ! ! 0 n! A C j 2 1 i : (B5)
Turning now to the charge correlator at equal time, its matrix expression yields in the time ordered basis:

Q (0)= 2 Z d! 2 Ĝ (!) z K (!) z Ĝ (!): (B6)
or in the rotated basis:

z L 1 Q L = Q Q = 2 Z d! 2 G (!) K (!) G (!): (B7)
This allow to obtain the Keldysh rotated elements of the charge correlator:

Q R =A = 2 Z d! 2 G R =A (!) K R =A (!) G R =A (!): (B8) Q K = 2 Z d! 2 h G R (!) K R (!) G K (!)+ G R (!) K K (!) G A (!)+ G K (!) K A (!) G A (!) i : (B9)

FIG. 2 :

 2 FIG. 2: Quantum point contact

1 FIG. 3 :

 13 FIG. 3: Current derivative correlator for a QPC in the fractional (left) and integer (right) quantum Hall effect (!0 = 3!A C , !1 = !A C and K + ( )is normalized by e IB ! 2 0 )

2 Ν 1 FIG. 4 :

 214 FIG.4: Averaged current correlator (same parameters as in Fig.3except the fact that K + ( )= 2 is normalized by e IB )

FIG. 5 :FIG. 6 :

 56 FIG. 5: (Color online) PA noise for constant temperature T = 0: 01!0 (left) and constant dissipation = 0: 01!0 (right):!A C = !0=3, !1=!A C = 1. Q + is normalized by L 2 =( 2 e IB ! 2 0 ).

FIG. 7 :FIG. 8 :

 78 FIG. 7: (Color online) PA noise for constant temperature T = 0: 01!0 (left) and constant dissipation = 0: 01!0 (right):!A C = 3!0, !1=!A C = 1.The renormalization is the same that in Fig.5.

( 1 ; 2

 12

I 1 ( 1 + 2 ;!) = Z + 1 1 d 0 e i( 1 + 2 )I 3 1 1dI 1 1 + 2 + 1 + 4 depend

 1121123111214 ( 1 + 2 ;!) = Z + 0 e i( 1 + 2 ) 0 =2 si n of the Keldysh Green's function for the chiral field: and I 3 are expressed in terms of delta functions:(n m )! A C )+ ( 1 + 2 (n m )! A C )) : 2 + (n m )! A C ) ( 1 + 2 (n m )! A C)) : explicitly on the Keldysh indices and 0 . Here, we need two tabulated integrals:

Acknowledgments

We thank Alex Zazunov, Per Delsing and Tim Duty for useful discussions. T.M. and T.J. acknowledge support of an ANR grant "Molspintronics" from the French ministry of research. T.M., T.J., E.P. and G.F. acklowledge support from a Gallileo project of the "Partenariat Hubert Curien". E.P. thanks CPT for its hospitality. T.M. thanks University of Catania for its hospitality.

APPENDIX A: KELDYSH NOISE CORRELATOR CALCULATION

From Eq. (41) we use a standard trigonometric identity in order to factorize the noise into contributions with and 0 :

S APPENDIX B: TIME AVERAGE CURRENT DERIVATIVE CORRELATORS IN THE ROTATED KELDYSH BASIS

Here, for completeness we compute the components K R , K A and K K in the rotated Kelsysh basis. We recall that if the time ordered Keldysh components of any correlator (such as the LC Greens function) read:

then the rotated Kelysh matrix is defined as:

where L is the unitary transformation:

We obtain from the expressions of the previous section:


Frédéric Nguyen
email: frederic.nguyen@cea.fr

Didier Tarabelli

Hervé Marteau

Frédéric Dacquait

Nadia Perot

Gilles Ranchoux

Luc Guinard

Antoine Long

Céline Viala

First Numerical Simulations of Contamination of the PWR Primary Circuit by Activated Corrosion Products with the PACTOLE V3.0 Code

First numerical simulations of contamination of the PWR primary circuit by activated corrosion products

with the PACTOLE V3.0 code

I. INTRODUCTION
The PACTOLE code, developed by the CEA, in cooperation with EDF and Framatome-ANP, aims at predicting the Pressurized Water Reactor (PWR) primary circuit contamination.

In a challenging industrial environment, the development of the PACTOLE code is still of major concern for CEA, EDF and Framatome-ANP.

Taking into account recent development relevant to corrosion product behavior, the latest PACTOLE code version, namely V3.0.f, has been released in 2003. The V3 version of the PACTOLE code has been completely reengineering and is intended to evolve as soon as new developments are available. Written in a fully objectoriented design, the PACTOLE V3 code is modular and easily evolving.

The aim of this paper is to present the PACTOLE V3.0 code and a comparison of numerical simulations with measurement of deposited and volume activities in two types of nuclear power plants (900 MWe and 1300 MWe PWRs).

In the first part of this paper, the modeling of the corrosion product behavior is described. It focuses on improvements that take into account surface treatment impact on corrosion/release processes a .

Typical applications of the PACTOLE code are reactor fuel cycle simulations (one fuel cycle corresponds approximately to one year). The consequence on the PACTOLE V3 development was the implementation of a numerical resolution for the instationary corrosion product transport equations. So, the second part of this paper presents the implicit time scheme used for the discretization of the equations and the numerical methods used for the resolution of the discretized equations. The comparison of the numerical simulations with experimental measurements is discussed in the next section. The last part of the paper gives an overview of the foreseen PACTOLE developments.

II. PACTOLE V3 GENERAL OUTLINES

The aim of the PACTOLE code is to predict the primary circuit contamination caused by activated corrosion products (volume and deposited activities), in nuclear reactors. PACTOLE has been used in the past either to help in reducing the contamination of operating PWR [START_REF] Beslu | A Computer Code PACTOLE to Predict Activation and Transport of Corrosion Products in a PWR[END_REF] or to help at designing new reactor concept such as tokamak, cooling loops of ITER for instance [START_REF] Pace | Development of the PACTITER Code and its Application to the Assessment of the ITER Divertor Cooling Loop Corrosion Products[END_REF] .



The former PACTOLE code version, named V2, showed some lacks of modeling [START_REF] Tarabelli | Status and Future Plans of the PACTOLE Code Predicting the Activation and Transport of Corrosion Products in PWR[END_REF] . Therefore, a new code version, namely PACTOLE V3.0, has been designed to account for recent developments relevant to the behavior of corrosion products and to help in understanding and modeling the physico-chemical processes.

II.A. Primary Circuit Representation

Based on a control volume approach, the primary circuit is represented by an arrangement of several volumes in which mass balance equations are solved. As shown in Fig. 1, displaying a typical modelization of a primary circuit, the loops are gathered into one because a homogeneous contamination is assumed for the loops. Nevertheless, the loops can be individually computed in order to study, for instance, the effect of Steam Generator (SG) composition or surface treatment on contamination repartition between loops. The primary circuit is composed of one mainstream line to which derived circuits (like the CVCS) are connected. The reactor core can be multifold (e.g. threefold or fourfold), depending on the fuel management b .

II.B. General Input

For each control volume, representing a part of the primary circuit, input parameters are relevant to geometrical aspects, thermohydraulics considerations and material properties.

Since the PACTOLE code is based on control volume approach, the only requested geometrical parameters are hydraulics diameter and wet surface. The  b 1/3 or 1/4 core assemblies are removed at every fuel cycle.

later is defined as the surface that can be oxidized and on which corrosion products can be deposited. Concerning thermohydraulics data, classical parameters such as fluid velocity, flow rate and fluid temperature have to be provided. Furthermore, wall temperature is of major concern, since corrosion product equilibrium concentrations depend on the wall temperature gradient along the primary circuit.

Concerning material properties, the metal composition of each part of the primary circuit has to be given. In addition, surface roughness and surface treatments that appear to have a large influence 4 , 5 are provided in term of release and corrosion rates.

Obviously, corrosion product activation in reactor core is directly proportional to the neutron flux. Thus the power fraction, relative to the nominal power, has to be specified with care in every part of the core.

Reactor operating conditions, which may impact primary circuit contamination, are also provided in the PACTOLE input data. These data can be summarized as follows: power history, water chemistry (Li, B(pH), dissolved hydrogen) and reactor shutdown.

II.C. Form of Corrosion Products in a Control Volume

Although the PACTOLE code is based on a control volume approach, different physical states can be found in one control volume, also called region. These states, called media in what follows, have different behaviors, different corrosion products concentrations and can interact. Fig. 2 displays the different media that can exist in a control volume.

Directly produced by metal oxidation c , the inner oxide is an enriched chromium layer. Dense and relatively thin, it is a low soluble oxide and acts as a protective barrier against metal oxidation.

The outer oxide, more soluble in reactor operating conditions than the inner oxide, is mainly composed of nickel ferrite. As a result of metal oxidation combined with ion precipitation, the outer oxide becomes contaminated by incorporating activated corrosion products.

Released from base metal, ions are carried by the fluid in the different part of the primary circuit where they can precipitate in the bulk, producing particles, which form another medium. Both particles and ions, which can be activated when crossing a core region, can respectively  c Corrosion of zirconium base metal is not accounted for in the PACTOLE code.

deposit on the piping walls and precipitate either on the deposit or on the outer oxide. At last, a peculiar medium, which aims at clearing the primary circuit, can be represented: the filter and ion exchanger resin devices.

II.D. Isotopes and Nuclear Reactions

The chemical elements and derived isotopes are declared by the user. In addition, the nuclear characteristics such as reaction rates or half-life and the filiations between active and stable isotopes can be specified by the input data.

II.E. General Mass Balance Equation

In every control volume, constituting the primary circuit, corrosion product behavior is described by mass balance equations, which are formulated as follows:

( ) ∑ ∑ - = - + ∂ ∂ Source Sink m m e s i J J m m t m
Where m i is the mass of the i th isotope in a considered medium (see Fig. 2), t is the time, e s m m ¡ ¡ - is the convective term (balance between input and output). Except for nuclear reactions, which are mass production/loss within the considered medium, J m is an exchange mass rate between two different media.

II.F. Computer Aspects

Combined with a fully object programming language (C++), the PACTOLE V3 code has been written and designed using the Unified Method Language (UML).

Thanks to its code architecture, and confirmed by the development of a mock-up, the latest PACTOLE version is an easy evolving computer tool in which new media can be added and in which new models can be tested. Moreover, the class management process, introduced in the PACTOLE code, allows a non-limited number of control volumes, isotopes, fuel cycles… The only restriction comes from the memory of the computer on which the code runs.

The PACTOLE V3 code version is available on UNIX workstation and on personal computers using Windows operating system.

III. PACTOLE V3 MAIN MODELS

Major phenomena, which are involved in corrosion product transport, are accounted for in the PACTOLE code. The transfer mechanisms between all considered media are summarized in Fig. 3.

Fig. 3 : main mass transfers in the PACTOLE code

III.A. Corrosion, Ion Release and Ion Incorporation in Oxides

Experimental investigations and on-site deposited activity measurements have demonstrated the impact of surface treatment on corrosion and ion release in the fluid [START_REF] Coulet | Etude et Modélisation de l'Influence des Matériaux sur la Contamination du Circuit Primaire des Réacteurs à Eau Pressurisé par les Produits de Corrosion[END_REF] . Even though the experimental evidences are nowadays clearly identified, their mathematical expressions are up to now still under questions [START_REF] Guinard | Influence of the Initial Surface Condition on the Release of Nickel Alloy in the Primary Circuit of PWRs[END_REF] . In a first step, an empirical approach for corrosion and ion release processes has been chosen for the PACTOLE code modeling.

In the proposed modeling, corrosion is the process by which the base metal becomes oxidized. This oxidation leads to several mass transfers, which are: generation of an inner oxide, generation of an outer oxide and release of metallic ions directly in the fluid. These transfers are shown in Fig. 4. In the PACTOLE V3 code, corrosion and

III.B. Ions Incorporation in the Oxide Layers

Various authors 8 , 9 have highlighted that activated ions, which are present in the fluid, are incorporated either to the inner oxide or to the outer oxide while growing.

For the outer oxide, the ions incorporation mechanism is described as a function of the release velocity. This modeling allows to take into account the impact of the material surface treatment on the oxide contamination.

III.C. Ion and Particle Transport

After being released from base metal, metallic ions are transported in the primary circuit. Since solubility law depends on temperature and on water chemistry [START_REF] You | Dissolution and Solubility of Cobalt and Nickel Ferrites in PWR Primary Conditions[END_REF] , metallic ions can reach regions where their concentration exceed the saturation and therefore precipitate on wall surfaces or in the bulk, forming particles. When the ion concentration is below the saturation, the dissolution of the oxide and deposit occurs.

Transported by the fluid under particulate form, the corrosion products deposit on wall surfaces when submitted to the Brownian and turbulent diffusions and to thermophoresis effect. This last effect is important in steam generator tubes in which the thermal gradient between the fluid and the wall is high and positive [START_REF] Beal | Deposition of Particles in Turbulent Flow on Channel or Pipe Walls[END_REF] . Moreover, the particles assume to follow a lognormal probability function in the PACTOLE modeling.

Both deposited particles and oxides can be eroded when hydrodynamics forces become higher than those from adhesion. Erosion produces new particles in the bulk fluid, which can be transported by convection in the other regions of the primary circuit. Concerning erosion process, the PACTOLE modeling assumes that above a threshold deposit thickness limit, deposit and oxide may be eroded.

The key point of these interactions, between the wall surfaces and the fluid, occurs when the deposit/oxides are submitted to a neutron flux (in-core regions). Indeed, deposited corrosion products become activated and therefore can be again transported in out-of flux regions of the primary circuit where they participated to the dose rate.

III.D. Reactor outages

During refueling shutdown of French PWR equipped with nickel based steam generator alloys (Inconel 600 or 690), due to controlled oxygenation of the primary circuit which increases the solubility by a factor of 100 to 1000, most of the 58 Co, along with other isotopes, deposited on the zircaloy rods, is released in the primary water [START_REF] Dacquait | Corrosion Product Transfer in French PWRs during Shutdown[END_REF] . Even though the present PACTOLE modeling is simple for these operating conditions, it accounts for the global behavior of the corrosion products during reactor shutdown. Nevertheless when data concerning corrosion product solubility in oxidizing condition are available, the PACTOLE V3 version will be easily improved thanks to its new architecture.

IV. PACTOLE V3 NUMERICAL RESOLUTION

In order to obtain the isotope mass values, the global equations system must be solved with appropriated numerical methods [START_REF] Marchetto | Modélisation et Simulation Numérique du Transport des Produits de Corrosion dans le Circuit Primaire des Réacteurs à Eau Pressurisée[END_REF] .

First, the equation system has been discretized in time. Two different time schemes have been used. In order to obtain large time-step, an implicit time scheme has been chosen for the equations of ions, particles, deposit, outer and inner oxides as follows:

Discretized equations for ions and particles:

( ) ( ) ¢ £ ¤ + + - + + =           + - + + - - + q ¥ ¥ 2 1 C ¥ ¥ 2 1 C d ¦ ¨ § m m 1 i i i 1 i f f 1 n 1 i f f 1 n i m n i 1 n i
where ∆t is the time-step defined as follows

t t t n 1 n ∆ + = + , 1 n i m + (respectively n i m
) is the i th isotope mass at t n+1 (respectively t n ) in the considered media. Sink and source terms are gathered in q i . C is the concentration of the isotope and ρ is volume mass of water in a defined control volume.

Discretized equations for inner and outer oxide and deposit:

1 n i n i 1 n i q J m m + + = -
Since the metal and filter mass balance equations are not coupled with the others and no convective terms appear in these equations, an explicit time scheme has been chosen as follows:

Discretized equations for metal and filter:

n i n i 1 n i q J m m = - +
The resolution of the discretized system is performed in three steps. In a first step, the metal masses are determined directly thanks to the explicit time scheme. In a second step, equations of ions, particles, outer oxide, inner oxide and deposit are linearized with a Newton method and then solved with a preconditioned GMRES method. In a third step, isotope masses in the filter and resin are determined thanks to the explicit time scheme.

In addition, a time-step auto-adaptive management has been implemented based on the Newton method iterations. This code feature allows the time-step to be larger when equilibrium states are reached. On the other side, when the code deals with transient behavior of corrosion products, such as reactor shutdown and initial cycle, the time-step is automatically reduced.

V. PARAMETRIC STUDY WITH PACTOLE V3.0

Considering the large number of interactions between the different media, lots of physical models had to be implemented in the code, meaning that a great deal of parameters have to be given. Some of these parameters could be found in the literature, some could be determined experimentally on analytical experiments and others have to be provided by the user. It is thus very important to determine the parameter that produces the greatest effects on the final results in order to know where to put the emphasis on physical modeling and experimental measurements. A parametric study has then been achieved on the parameters of the physical models: 35 parameters have been chosen and the simulation results considered are the surface activity of the deposit, the total surface activity and the volume activity of nuclides 58 Co, 60 Co, and 59 Fe in six regions (spacer grid made of Inconel 718, core, hot leg, SG hot side, SG cold side, crossover leg and cold leg). The four most important parameters are finally the following (in decreasing order): enthalpy, activation reaction rate, release velocities for spacer grids and SG tubing, fraction of elements governing the release of spacer grid and SG regions.

VI. SIMULATIONS WITH THE PACTOLE V3 CODE

The objective of this section is to present the PACTOLE capabilities by comparing simulation results with experimental data. Two simulations are presented below: simulations of the contamination of running 900 MWe PWR (reactor A with alloy 600) and 1300 MWe PWR (reactor B with alloy 690). These simulations are compared to experimental data of volume and deposited activities [START_REF] Dacquait | Status of French PWR Primary System Contamination[END_REF] . Comparing PACTOLE simulations to on-site measurements is an important step for the code validation.

Reactor A (respectively B) is a PWR with a nominal power of 900 MWe (resp. 1300 MWe) that contains three (resp. four) individual loops. The following table gives an overview of the main characteristics, which may have an influence on corrosion products behavior, of the reactor A. Made of Inconel 600, the steam generator tube bundles are the largest source of nickel. Indeed, their total wet surface is more than 13000 m 2 (representing approximately 60 % of the total surface) and nickel represents 70 % of the metal alloy.

The second largest surface in contact with water, is the fuel assemblies made of zircaloy. As previously mentioned, the zirconium base metal does not participate to the corrosion product inventory but this surface is directly submitted to neutron flux and is therefore responsible to the activation of the corrosion products transported and deposited on fuel rods.

It is also important to mention that in the reactor A, the assembly spacer grids were made of Inconel 718 d for the 5 first fuel cycles, and that even though their surface is small, the spacer grids are nonetheless submitted to the neutron flux.

Concerning operating conditions, reactors A and B were operating under load follow as almost all French nuclear power plants. Fig. 5 shows the evolution of the power fraction, as used by the PACTOLE code, for the first seven fuel cycles of reactor A and five first cycles of reactor B. Nowadays, all French units are running with a socalled coordinated water chemistry [START_REF] Bretelle | EDF Strategy for the PWR Primary Coolant pH[END_REF] at pH 300°C 7.2. When the reactor A started operating, its water chemistry was set to former specifications: the water pH 300°C was 7.0 for the first part of the fuel cycle and increased up to 7.4 at the end of the fuel cycle. Fig. 6 shows the pH evolution of the reactors A and B fuel as used by the PACTOLE code. Up to fuel cycle 3, the reactors A and B were running with the coordinated water chemistry at pH 300°C 7.0, from cycle 4, they have been operated with the coordinated chemistry at pH 300°C 7.2.  d Inconel 718 is approximately made of 52 % nickel. It can be seen from these figures, that the PACTOLE code provides a reasonable estimation of the 58 Co and 60 Co deposited activity for the reactor B (1300 MWe PWR with alloy 690). On the other hand, the 58 Co and 60 Co deposited activity calculated by the PACTOLE code is over estimated for reactor A (900 Mwe PWR with alloy 600). Fig. 9 and Fig. 10 respectively represent the 58 Co and 60 Co deposited activities on the cold legs. Both computed results and on-site measurements are represented in these figures. Slightly different conclusions can be drawn from these two figures: the PACTOLE code also gives reasonable estimation prediction for 58 Co and 60 Co deposited activities but with a tendency to underestimation compared to on-site measurements. One should note however that the simulation presents a much better matching with experimental results at the end of the simulation (fifth shutdown for reactor A and seventh shutdown for reactor B). The new PACTOLE code is able to compute the global deposited activities in the different regions of the primary circuit even though some discrepancies can be found. On the other hand, the problem of computing in an accurate way the ion and particle concentration in the fluid remains: this problem is intended to be alleviated in a near future with a new solubility modeling as it will be seen in the last section of this paper.

Considering that some physical models are different in the V3 version of the PACTOLE code (in particular for the corrosion and release velocities) compared to the previous one (V2), a strict comparison of numerical results between the two code versions is then difficult in the sense that the input data could not be strictly identical.

One should note that the numerical simulations presented above are the first achieved with the new version of the PACTOLE code, meaning that the input data have not been ultimately adjusted so that the matching between calculations and on-site measurements would be even better. Considering the number of interactions between the different media, and thus of physical models, the number of physical parameters is also quite large: this explains the interest of the parametric study presented in section V. This parametric study has considered physical parameters, excluding operation parameters (power fraction, activation rate, water, chemistry, fluid velocities, wall and fluid temperatures, geometrical parameters, etc.) that are supposed to be known. Of course, the physical parameters are identical for the two simulations when they have to be, meaning that only parameters related to different alloys (Inconel 600 or 690) are different. As already mentionned, the parametric study will allow to put more emphasis on the most influent parameters: a workprogram including modeling and experiments on analytical loops is being defined. In addition, a multicriteria calibration, using optimisation methods with or without constraints, is also foreseen afterwards to determine relevantly the remaining unkown parameters: it will required the simulation of a large number of different PWRs to take into account the great variability in the contamination of different PWRs.

VII. PACTOLE V3 FURTHER DEVELOPMENTS

Major improvements have been achieved in the V3.0 PACTOLE code version for the data processing architecture (design using UML combined with a fully object programming language, C++) and numerical analysis for solving instationnary equations. This new architecture will allow the PACTOLE code of being an integration tool for the complex physics involved in contamination transfert studies. Some physical models have been already modified (among them corrosion and release velocities), but there are still both existing models to improve and new models to implement. In consequence, the three partners (CEA, EDF, Framatome-ANP) have decided to make a large effort to improve the physical modeling and every future version of the PACTOLE will present a significant appreciation of the physics. A particular attention will again be devoted to the validation process that includes verification of elementary models, of model couplings, comparison between simulations and experiments on system loops and finally comparison between simulations and measurements in reactors. One will undertake for every new version of the code this rigorous validation process that ends by a series of calculations of a list of validation cases (that remains identical for all versions of the code): this allows to identify the physical models that should be developped and completed for the next version. As mentionned in the previous section, the making of a wilful workprogram for improving the physical models and the data is thus under way and performed by CEA, EDF and Framatome-ANP. One could already identify some of the future important steps in the PACTOLE code development:

-Improvement of the corrosion/release mechanism. Based on the present empirical mathematical expression, the impact of macroscopic parameters such as solubility or temperature will be accounted for. This work will be performed on the basis of EDF and CEA experimental results obtained recently.

-Improvement of the oxide composition determination. A new approach for oxide determination and solubility calculation has been developed in the CEA. This model based on a survey of oxides thermodynamic behavior is aimed to predict in a more accurate way solubility values. It is designed, for instance, at predicting with a better confidence the corrosion product ion concentration that can be measured in operating PWRs.

-Addition of a specific module for shutdown. In the present code modeling, only reducing conditions can be treated: a real simulation of oxygenation phase is therefore impossible. Due, for a large part, to the lack of available data, an important effort is devoting to the data collection of oxide characteristics in oxidizing conditions.

VIII. CONCLUSIONS

Modeling the behavior of corrosion products in the PWR primary circuit is still of major concern as regards to present and future industrial challenges.

The latest code version, PACTOLE V3.0, is a major step forward for the prediction of reactor contamination by activated corrosion products. The main improvements are related to surface treatment impact on corrosion and release mechanisms and to a numerical scheme for solving transient behaviors. In addition, the code has been developed using oriented object methods (programming language, architecture design method) in order to get a flexible, and easily evolving computer tool.

The comparison of the simulations made with the PACTOLE code with on-site measurements of deposited activities demonstrates a reasonable agreement between numerical simulations and experimental data. Nevertheless, further calculations have to be performed with other reactors and with experimental loop results to get a good picture of the code capabilities.

Further code developments are already planned. Along with the improvements of existing models, new code capabilities will be available in a close future.

Fig. 1 :

 1 Fig. 1: Typical representation of a PWR primary circuit

Fig. 2 :

 2 Fig. 2 : Media in a Control Volume for a PACTOLE representation (Fluid means ions)

Fig. 4 :

 4 Fig. 4 : PACTOLE local modeling for inner and outer oxide generations and ion release

Fig. 5 :

 5 Fig. 5 : Nominal reactor power fraction of the reactor A first seven fuel cycles and first five fuel cycles of reactor B.

Fig. 6

 6 Fig. 6 Reactors A and B water pH (PACTOLE representation) Fig. 7 and Fig. 8 respectively show the 58 Co and 60 Co deposited activities calculated by the PACTOLE code and the experimental gamma-spectrometry data on the Steam Generator (SG) tubing. The experimental uncertainties, which are about 20 %, are represented on the figures.

Fig. 7 :

 7 Fig. 7 : 58 Co deposited activity on SG tubing from PACTOLE results (continuous line with the squares for the times of measurements) and on-site measurements (circles with error bars))

Fig. 9 :Fig. 10 :

 910 Fig. 9 : 58 Co deposited activity on the cold legs results from numerical simulations (continuous line with the squares for the times of measurements) and on-site measurements (circles with error bars)

Fig. 11 :

 11 Fig. 11 : 58 Co and 60 Co volume activities simulated for reactor A (900 MWe) (continuous lines with the circles for the times of shutdown) and on-site measurements (dotted lines).

Table 1 :

 1 Main characteristics of reactor A

	In core	Fuel cladding	Zry-4 -5020 m 2
	surfaces Spacer grids	In 718 / Zry-4 -513 m 2
		Other	Stainless Steel -182 m 2
		Steam Generator Inconel 600 -13413 m 2
		CVCS flow rate	2.7 10 -2 %
		Chemistry	pH 300°C = 7.0 / 7.2
		Remarks	Fluid chemistry was
			changed at fuel cycle #4
			Grid assembly
			composition changed at
			fuel cycle #6


Sameut Bouhaik
email: carole.bedos@grignon.inra.fr

A MODEL OF VOLATILIZATION AND SUBSEQUENT DRY DEPOSITION OF PESTICIDES TO SURFACE WATERS

INTRODUCTION

• Contamination of surface waters by pesticides is widespread near agricultural areas (as in lakes, rivers, etc.).

• Dry deposition of volatilized pesticides (gaseous) may be higher than the drift mainly due to longer emission periods.

• Volatilization from soil depends on the thermodynamical properties of the pesticide (e.g. Henry's law constant), its persistence in the soil, environmental conditions (soil and air temperature, soil water content and soil organic matter) and agricultural practices.

• Dry deposition depends on atmospheric concentrations of the pesticide and hence on the volatilization rate, environmental conditions, distance from the source and pesticides characteristics, as well as the nature of the surface (water, plant, soil).

• In this study we use 2 models to estimate 1) the volatilization and 2) the local dispersion and dry deposition of pesticides to water bodies at the local scale (<1 km).

MAIN RESULTS

• The drift at 250 m amounted around 0.15% of the applied dose (Rautmann et al., 2001).

• The modelled dry deposition of Acetochlor, Chlorothalonil, and Fenpropimorph roughly amounted to 0.5%, 1% and 2%. => The modelled gaseous dry deposition to water bodies was larger than the drift for 3 of the 5 pesticides (Figure 2). • Under simulated conditions, and for the selected pesticides, the surface resistance to water Rsurf was negligible when compared to the water boundary layer resistance Rb (Table 1). => Under these conditions, the main driver of the dry deposition was therefore the atmospheric concentration and hence the volatilization rate.

PERSPECTIVES

• This study should be extended to a larger range of conditions to check when R surf is negligible against R b .

• Scenarios should be build to cover a large range of environmental conditions and pesticides.

• Running the model on these scenarios will allow quantifying surface water contamination by gaseous pesticides.

• These scenarios will be used to identify good agricultural practices to limit surface water contamination.

LIMITATIONS

• Such a modelling study needs to be validated against experimental datasets.

• Volatilization from crop surfaces should also be considered in the future.

MATERIAL AND METHODS

The approach couples 2 models (Figure 1 A set of five pesticides were chosen to cover a wide range of chemical and physical properties, especially H, as a key property for the theoretical volatilization and deposition potential (Table 1).

Figure 2 :

 2 Figure 2: Cumulated dry deposition after 13 days and cumulated drift deposition of pesticides onto water. Curves are normalized by the amount of pesticides applied (1 kg ha -1 ). The drift curve is from Rautmann et al. (2001).

  ): a mechanistic model for pesticide volatilization following application on bare soil (Volt'Air-Pesticides, Bedos et al., 2009) which gives the source strength and the surface boundary layer characteristics to a local-scale dispersion and deposition model (FIDES-2D, Loubet et al., 2001), adapted for pesticides(Sameut, 2010). Dry deposition to water is modeled using the resistance analogy approach (Liss and Slater, 1974 further applied to pesticides byAsman et al. 2003)

  The model was run over 13 days, from March 1, 2002 to March 13, 2002. The soil and climatic conditions of the Picardie region in France was chosen. The field was 100 m long and the application dose was 1 kg ha -1 .

Figure

  Figure 1: Coupling Volt'Air-Pesticides and FIDES

ACKNOWLEDGEMENTS

The authors wish to thank the "Conseil Régional de Picardie" (regional council) and the French Ministries in charge of Environment and Agriculture who founded the PHAR project as well as the French National Institute for Industrial Environment and Risks (Ineris), especially A. Gouzy, who managed the project. They also wish to thank the Office National de l'Eau et des Milieux Aquatiques, Onema (Onema -Inra 2010) for their contribution to the CEPAG Project.