Jules Selles
email: jules.selles@gmail.com

Sylvain Bonhommeau

Selles Jules

Bonhommeau Sylvain

Guillotreau Patrice

Optimal bioeconomic management of the Eastern Atlantic Bluefin tuna fishery: where do we stand after the recovery plan?

Keywords: Bioeconomic modeling, Uncertainty, Optimal resource management, Atlantic Bluefin tuna, Fisheries management, International fisheries, Policy making

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Optimal bioeconomic management of the Eastern Atlantic Bluefin tuna fishery: where do we stand after the recovery plan?

Introduction

Highly migratory species, such as tunas and tuna-like species, represent both economically and biologically significant stakes for the world fisheries [START_REF] Munro | The conservation and management of shared fish stocks: legal and economic aspects[END_REF][START_REF] Galland | Netting billions: a global valuation of tuna, A report of The Pew Charitable Trusts[END_REF]. Tunas are spread and caught over all the oceans [START_REF] Miyake | Recent Developments in the Tuna industry: Stocks, Fisheries, Management, Processing, Trade and Markets[END_REF]. They sustain some of the most valuable fisheries and feed international supply chains [START_REF] Catarci | The world tuna industry: an analysis of imports and prices, and of their combined impact on catches and tuna fishing capacity[END_REF][START_REF] Majkowski | Global fishery resources of tuna and tuna-like species. Food and Agriculture Organization[END_REF]. Tunas fisheries are integrated in a global market involving many regional markets [START_REF] Jeon | Is there a global market for tuna? Policy implications for tropical tuna fisheries[END_REF][START_REF] Jiménez-Toribio | Global integration of European tuna markets[END_REF][START_REF] Mullon | Exploring future scenarios for the global supply chain of tuna[END_REF] under the influence of the worldwide demand for tuna commodities [START_REF] Catarci | The world tuna industry: an analysis of imports and prices, and of their combined impact on catches and tuna fishing capacity[END_REF][START_REF] Mullon | Exploring future scenarios for the global supply chain of tuna[END_REF]. High values on international market make species such as tuna and tuna-like species particularly vulnerable (Colette et al. 2011), and the status of a number of stocks are particularly worrying [START_REF] Maguire | The state of world highly migratory, straddling and other high seas fish stocks, and associated species. Food and Agriculture Organization[END_REF], Juan-Jorda et al. 2011).

The Eastern and Mediterranean stock of the Atlantic Bluefin tuna (EABFT) falls in this group of concerns. The end value was estimated to more than $700 million in 2014, out of a total end value exceeding $2 billion for the three major BFT species [START_REF] Galland | Netting billions: a global valuation of tuna, A report of The Pew Charitable Trusts[END_REF]. The EABFT fishery faces today a critical phase in its management. After a long period of over-exploitation [START_REF] Fromentin | The spectre of uncertainty in management of exploited fish stocks: The illustrative case of Atlantic Bluefin tuna[END_REF], signs of recovery are evidenced by stock assessment experts (ICCAT 2017). A stock rebuilding plan was launched in 2007 with a 60% likelihood of achieving sustainability by 2022. In 2017, under the majority of recruitment level scenarios1 , the stock had already recovered to the expected level. The new management stakes consist in defining quotas to keep the stock (biomass) above the maximum sustainable yield level (BMSY), despite high pressures from the fishing nations to increase their quotas drastically. The last estimates from the International Commission for the Conservation of Atlantic Tunas (ICCAT) indicate that the stock rebuilding could be achieved by 2022 with probabilities higher than 60% for all recruitment scenarios by setting catch limits up to 30,000 tons (ICCAT, 2017), i.e. more than twice the limits enforced in 2009.

In this context of high uncertainties, the new management scheme of ICCAT implies a TAC of up to 32,000 tons by 2020 2 . Is the new management policy based on MSY adapted to the productivity of the EABFT stock, or does it jeopardize the future of the fishery by going back to the crisis situation of the late 2000s? Shifting objectives from the traditional MSY target to the dynamic maximum economic yield should result in most cases a no-regret situation in which management promotes both larger fish stocks and higher profits (BMEY > BMSY 3 , Grafton et al. 2007, 2010, Clark, 2010). In a context of international trade, this dual benefit could be strengthened (or mitigated) by the market price response to changes in landings (Sun et al. 2015;Sun et al. 2017;[START_REF] Mullon | Exploring future scenarios for the global supply chain of tuna[END_REF]Tokunaga 2018). Previous bio-economic optimization models analyzed optimal harvest in age-structured frameworks compatible with stock assessments procedure. [START_REF] Bertignac | Maximizing resource rent in the Western and Central Pacific tuna fisheries[END_REF] and [START_REF] Kompas | Bioeconomic losses from overharvesting tuna: Bioeconomic losses[END_REF] applied stochastic dynamic programming to the Western and Central Pacific tuna fisheries to show that adopting the BMEY target leads to better conservation outcomes with larger fish stocks and higher economic profits than the business-as-usual scenario. Similarly, [START_REF] Kulmala | Reconciling economic and biological modeling of migratory fish stocks: Optimal management of the Atlantic salmon fishery in the Baltic Sea[END_REF] numerically solve their harvest optimization model for the agestructure population of the Atlantic salmon (Salmo salar) fishery in the Baltic Sea and demonstrate the economic benefits of the optimal solution without compromising the sustainability of the resource. Finally, Tahvonen et al. (2017) analyzed the optimal harvesting strategy of the Baltic cod (Gadus morhua) fishery including gear selectivity as a state variable in a stochastic model. They showed that endogenous selectivity strongly changes the MEY harvest pattern and increase substantially the profit obtained from the fishery. They also highlighted that the stochastic solution can be accurately approximated by the certainty equivalence principle.

The objective of this study is to develop tools for producing economically sound management advice for EABFT. To the best of our knowledge, the only study attempting to estimate economic optimal management of EABFT fishery was Bjorndal & Brasao 2 ICCAT Recommendation 17-07 (2017).

3 This result comes from the literature based on the surplus production model considering reasonable discount rates. ( 2006) who based their analysis on an age-structured, multi-gear model which lead to pulse fishing 4 as an optimal solution. In the present research, we propose to update and revisit their results by using a discrete age-structured population optimization model for the EABFT in line with the framework used for the stock assessment by ICCAT. This optimization framework based on the general age structured bio-economic model of Tahvonen et al. (2013) uses directly annual harvest, total allowable catch (TAC) as an optimized variable. We extend the model by including non-linear demand function affected by the supply of the three Bluefin tuna species on the global market and stockdependent harvesting costs. Furthermore, we analyse the effects of considering different recruitment levels according to ICCAT scenarios and different evolution of Bluefin tuna supply levels. Finally, we determine the optimal management when selectivity pattern is integrated as an endogenous variable in addition to the TAC in the model. Our work suggests that adopting the MEY target meets both conservation and economic objectives by keeping the stock to higher level than under the current MSY target and producing higher benefits for the fishery. This result is exacerbated when selectivity is defined endogenously, shifting to a more balanced fishing mortality over all age classes. Finally, we deal with high stock estimation uncertainties 5 underlying the EABFT stock assessment. The uncertainty surrounding the assessment of the exploited fish stock is a pervasive feature of fishery management. Uncertainties in EABFT stock assessment arise from several sources: our understanding of EABFT biology and population dynamics; the ability of assessment models to correctly reproduce population dynamics patterns and the quality/quantity of the data used (detailed in [START_REF] Fromentin | The spectre of uncertainty in management of exploited fish stocks: The illustrative case of Atlantic Bluefin tuna[END_REF]. Data used for the assessment of highly migratory species suffer from a lack of independent observations (i.e. not directly related to catches) to track down changes in stock abundance. The fitting quality of the stock assessment relies on catch per unit of effort (CPUE) indices produced by commercial activities which are, among other factors, affected by recent regulatory measures of the rebuilding plan. The recent development of aerial fishingindependent surveys is a good basis to improve abundance indices (Fromentin et al. 4 Periodic fishing.

5 Refers to observational uncertainty which arises when state variables cannot be directly observed (e.g. stock biomass). Tahvonen et al., (2017) we did not consider stochastic processes, but we fully acknowledge the presence of stock estimation uncertainty by extending the previous work of [START_REF] Sethi | Fishery management under multiple uncertainty[END_REF] to an age-structured framework specifying equilibrium assumptions. Using dynamic stochastic programming, our results indicate that stock estimation uncertainties strongly affect the optimal management of the EABFT fishery, resulting in a lower SSB steady state compared to the full information case.

2011). Based on the results from

Material and methods

The Age-Structured model

Following the formulation and notation for an age structured schooling fishery from Tahvonen et al., (2013), we define a discrete age-structured population model for the EABFT. We extend the work of Tahvonen et al. ( 2013) by integrating a non-linear demand function and a cost function which integrates a stock effect on harvesting cost.

We also focus our analysis on stock estimation uncertainties by extending the previous work of [START_REF] Sethi | Fishery management under multiple uncertainty[END_REF] using an age structured framework. This section shows the general age structured population model, then the parameterization of the EABFT fishery and finally the numerical analysis of the optimal management. We define xs,t as the number of fish (in 10 6 individuals) in each age class s = 1, …, n and each year t = 0, 1, ..., T. We determine the recruitment function by ϕ(ssbt) and the spawning biomass by 𝑠𝑠𝑏 𝑡 , considering an equal sex ratio, the first age class of the agestructured population model can be written as:

𝑥 1,𝑡 = 𝜙(𝑠𝑠𝑏 𝑡 ) Eq 1 𝑠𝑠𝑏 𝑡 = ∑ 𝑔 𝑠 𝑛 𝑠=1 . 𝑤 𝑠 . 𝑥 𝑠,𝑡 Eq 2
We denote the parameters gs and ws the constant age-specific maturities, and weight of fish (kg) respectively.

As Tahvonen et al., (2013), we assume that fishing activity takes place every year after recruitment but before natural mortality. We determine total catch (Ht) in biomass (kg) With c and ÷ the cost scale in euros and the schooling parameters respectively.

Revenues depend on the price of Bluefin tuna which is formulated as an overall isoelastic downward-sloping demand function 𝑃(𝐻 𝑡 ):

𝑅(𝐻 𝑡 ) = 𝑃(𝐻 𝑡 ). 𝐻 𝑡 Eq 7

Finally, the optimization problem is:

𝑚𝑎𝑥 {𝐻 𝑡 } ∑ 𝑈(𝐻 𝑡 ). ä 𝑡 𝑇 𝑡=0
Eq 8

With T the finite planning horizon, ä = 1 1+𝑟

, the discount factor and r ≥ 0 as the discount rate.

The objective function is subject to equations 1, 2, 3, 4, and the conditions:

𝑥 𝑠,0 given, 𝑥 𝑠,𝑡 ≥ 0 , 𝑓𝑜𝑟 𝑠 = 1, . . . , 𝑛 𝑎𝑛𝑑 𝑡 = 1, … , 𝑇;

𝐻 𝑡 ≥ 0, 𝑓𝑜𝑟 𝑡 = 1, … , 𝑇.

The biomass and harvest steady state solution of this problem commonly refers to the dynamic maximum economic yield (MEY) concept.

Based on this model, we derive the equilibrium age structured population considering the long term population under a constant fishing mortality Fmax (Supplementary materials, appendix 1). This leads to the definition of the MSY level which is the management target of EABFT. The MSY level will be the basis for the comparison of the optimal management, but it drastically depends on the selectivity pattern and the recruitment function parameters estimation which is highly variable throughout time.

As ICCAT, we consider 3 recruitment levels corresponding to the fitting of the Beverton and Holt relationship for a high recruitment period (1990-2010), a medium recruitment period which is our reference case and a low recruitment period (1970)(1971)(1972)(1973)(1974)(1975)(1976)(1977)(1978)(1979)(1980).

We also compare the outcomes from the recruitment level reference case with scenarios including a non-constant evolution of EABFT's substitutes global supply. We consider two cases: a linear 50% increase or decrease of BFT supply over the next 25 years to estimate the potential impact of exogenous variation of prices on the EABFT management. Finally, we integrate selectivity as an endogenous state variable in the optimization process to evaluate the impact of optimizing selectivity on the dynamic MEY.

The East Atlantic Bluefin tuna (EABFT) fishery

The Eastern Atlantic Bluefin tuna has been an archetype of the overexploitation and mismanagement of marine resources [START_REF] Fromentin | The spectre of uncertainty in management of exploited fish stocks: The illustrative case of Atlantic Bluefin tuna[END_REF]. Several countries, either coastal or distant water fishing nations, have contributed to a high level of depletion driven by the high market value of the tuna on the Japanese market. The decline in the EABFT has raised considerable concerns about its management in the 2000s (Hurry et al., 2008, ICCAT 2007, ICCAT 2009). Under the governance of ICCAT, a Regional Fishery Management Organization (RFMO), the fish population has suffered, at the same time, from its failure to follow scientific advice and a high level of illegal, unreported and unregulated (IUU) fishing. This situation occurred when the first management regulation based on quotas (TAC) appeared in 1999 and lasted until 2007 with the implementation of a recovery plan for the EABFT fishery. After 2009 and the strict management measures which have been implemented, the stock has showed signs of increase in the last years to peak a potential spawning stock biomass (SSB) value up to 610.106 t in 2015 (ICCAT 2017). A combination of a decrease in fishing pressure and potential high recruitment events resulted in a strong increase of the SSB. Presently, the stock is regarded as fulfilling the objective of the recovery plan (F<F0.1 and SSB>SSB0.1 6 ), depending nonetheless on the assessment scenarios and assumptions. The magnitude of the SSB recovery appears to be very sensitive to slight changes in the input data (notably catch data) and technical assumptions (ICCAT 2017).

Following the standard stock assessment for the EABFT by ICCAT ( 2014), we consider 10 age classes (n). Age-specific maturities (gs) and survival rates (á 𝑠 = 𝑒 -𝑚 𝑠 , with m the natural mortality at age) which are directly taken from the (ICCAT 2017) assessment report. For the age-specific weights (ws in kg per individual), we use the mean values of the period 2011-2014 (Table 1). Selectivities (sels) are estimated by the catch curve analysis method for the purse seiner [START_REF] Kell | A catch curve analysis for east Atlantic and Mediterranean Bluefin tuna[END_REF] for the period 2011-2015 and equal to 1 for the oldest age class by normalization. Recruitment is assumed to follow the [START_REF] Beverton | On the dynamics of exploited fish populations[END_REF] recruitment function:

𝑥 1,𝑡 = 𝜙(𝑠𝑠𝑏 𝑡 ) = 𝜙1.𝑠𝑠𝑏 𝑡 𝜙2+𝑠𝑠𝑏 𝑡 Eq 9
We use ICCAT (2017) data on spawning stock biomass and recruitment for the years 1970 to 2010 (corresponding to the medium recruitment scenario of ICCAT), and we estimate the parameters by maximizing the likelihood function assuming a lognormal error structure with the Fisheries library R ('FLSR' package in FLCore 3.0, [START_REF] Kell | FLR: An open-source framework for the evaluation and development of management strategies[END_REF]). We constrained the estimation by setting the steepness 7 at 0.99 following ICCAT (2017) parametrisation in order to specify a quasi-constant recruitment level. This yields the estimates of the asymptotic recruitment ϕ1 = 2,230,398 recruits (standard deviation 12,672 individuals) and the SSB needed to produce the half of the asymptotic recruitment ϕ2 = 1,155,983 kg (standard deviation 5,403 kg). EABFT is a large highly migratory schooling species which is the main target for purse seine commercial fisheries in the Mediterranean Sea. The low TAC level of the early 2010s as well as the good knowledge of seasonal migration and concentration patterns in the Mediterranean [START_REF] Fromentin | Atlantic Bluefin tuna: population dynamics, ecology, fisheries and management[END_REF], in combination with technologybased information on the fishing activities and prospecting equipment, result in minor 7 Steepness represents the fraction of the virgin recruitment expected when SSB has been reduced to 20% of its maximum [START_REF] Francis | Use of risk analysis to assess fishery management strategies: a case study using orange roughy (Hoplostethus atlanticus) on the Chatham Rise New Zealand[END_REF].

search costs within a short fishing period (usually less than a month, the purse-seiners remaining idle at ports for the rest of the year). We estimate the generalized cost function by mean of non-linear least square method (R package 'stats'). Using data on catch, efficient biomass and variable costs, we estimate the cost parameter (c) and the catch-stock elasticity parameter (÷) following the relationship:

𝑐 𝑡 = 𝑐. 𝐻 𝑡 . 𝐵 𝑡 -÷ + 𝜖 𝑡 , 𝑤𝑖𝑡ℎ 𝜖 𝑡 ~𝐼𝐼𝐷𝑁(0, σ 2 )
We only include variable costs (𝑐 𝑡 ) that directly depend on fishing activity including gear and vessel maintenance, fuel and labor (crew wages) costs. Assuming a normal error structure, we obtain the estimate of the cost scale parameter, c = 201.6 € (standard deviation € 2102.5) and the estimate of ÷ = 0.18 (standard deviation 0.53). This value is consistent with a schooling fishery parameter (÷< 1) and close to previous elasticity used in the literature (÷ = 0.2 in [START_REF] Bjorndal | The East Atlantic Bluefin Tuna Fisheries: Stock Collapse or Recovery?[END_REF].

The EABFT purse fishery is driven by the rising Japanese demand for fatty tuna intended to high-quality sushi/sashimi market. The quasi totality of EABFT caught by purse seiners is sold to fattening farms in the Mediterranean Sea [START_REF] Mylonas | Atlantic Bluefin tuna (Thunnus Thynnus) farming and fattening in the Mediterranean Sea[END_REF]. Since the 1990s selectivity mainly shifts toward large tuna caught in the Western Mediterranean (Fromentin & Bonhommeau., 2011). Tunas are fattened during a 3-5 month period and sold on the Japanese sashimi market when domestic demand is high (usually at the end of the year). Fattened EABFT is mainly sold frozen to the Japanese market, representing more than 80% of the market since 2006 [START_REF] Mylonas | Atlantic Bluefin tuna (Thunnus Thynnus) farming and fattening in the Mediterranean Sea[END_REF]. In the Eastern Mediterranean, a smaller part of EABFT are caught by Croatian purse seiners, generally smaller individuals (less than 60 kg), and reared for longer periods (around 2 years) during which initial weight can be doubled [START_REF] Mylonas | Atlantic Bluefin tuna (Thunnus Thynnus) farming and fattening in the Mediterranean Sea[END_REF].

Despite the relationship between the age, quality and price of tuna (Caroll et al., 2001[START_REF] Mylonas | Atlantic Bluefin tuna (Thunnus Thynnus) farming and fattening in the Mediterranean Sea[END_REF], we consider a constant price per age. This assumption is adopted because of a lack of price data per age or size, therefore we use the same price for all age groups. We further assume that the overall demand-function is iso-elastically downward-sloping. Thus, the price function 𝑃(𝐻 𝑡 ) is defined as 𝑃(𝐻 𝑡 ) = 𝑝. ( 𝐻 𝑡 + 𝐻𝑏𝑓𝑡 𝑡 ) -ö Eq 10

With p the theoretical price of the first sold kilos of Bluefin tuna, the price scale flexibility parameter and 𝐻𝑏𝑓𝑡 𝑡 is defined as the aggregated supply of all other Bluefin tuna species (including Pacific and Southern Bluefin tuna) which are considered as close substitutes on the sashimi-grade tuna market (Sun et al., 2017). Using catch data from the different RFMOs (ICCAT, IATTC, ISC, and CCSBT, Supplementary materials, appendix

3) in charge of the management of Bluefin tuna species and purse seine ex-vessel price from STECF we estimate the price parameter p = 91,983,194 € (standard deviation 8,947,638) using non-linear least square method.

𝑝 𝑡 = 𝑝. ( 𝐻 𝑡 + 𝐻𝑏𝑓𝑡 𝑡 ) -ö + 𝜖 𝑡 , 𝑤𝑖𝑡ℎ 𝜖 𝑡 ~𝐼𝐼𝐷𝑁(0, σ 2 )

As Japan is the main market for Bluefin tuna products, we integrate the price scale 34,636 tons.

Numerical analysis

We numerically analyze optimal management in a setting of no uncertainty by solving the dynamic optimization model as an open-loop 8 nonlinear programming problem. This is performed using the COBYLA algorithm of the NLopt optimization package [START_REF] Johnson | The NLopt nonlinear-optimization package[END_REF] with R (R core team 2017).

We . 𝑤 𝑠 . 𝑥 𝑠,𝑡 ), and H (the harvest level in biomass), represents the actions space which determined all possible actions that a theoretical manager could decide. V is the value function, U is the utility function (Equation 5) corresponding to the immediate reward and P represents the transition probability matrix between each state Xt to Xt+1 given all harvest level Ht.

Considering a discount factor ä < 1 , the mapping underlying the Bellman's equation is a strong contraction and thus, by the Contraction Mapping Theorem, possess a unique solution.

For computational effort reasons, we simplify the optimization problem by taking the equilibrium age structure 𝑥 𝑠,𝑡 for each value of the state Xt considering a constant fishing 8 In open-loop optimisation, once the optimal path has been defined the control action from the manager is independent of the resource state.

mortality Fmax (Supplementary materials, appendix 1 following the specification of

Tahvonen et al. 2009).

We include a random variable underpinning the uncertainty in period t, Zt, which affects stock estimation. This random variable is independent and identically distributed (IID) over years. Integrating a stock estimation uncertainty implies to define an observation variable of the stock Xobst, which is defined as follow:

𝑋𝑜𝑏𝑠 𝑡 = 𝑍 𝑡 . 𝑋 𝑡 Eq 12
Where 𝑋 𝑡 = ∑ 𝑥 𝑠,𝑡 𝑛 𝑠=1

. and 𝑍 𝑡 ~U(1-σ 𝑚 , 1 + σ 𝑚 ).

We consider a theoretical manager who only uses the current estimation Xobst to form beliefs about the current stock, Xt and select a total allowable catch (TAC) for the fishery based on the current rent U(Ht) from harvesting. Following [START_REF] Sethi | Fishery management under multiple uncertainty[END_REF], our assumption states that the manager only uses the current assessment when forming expectations. This method specifies the problem as a Markov decision process (MDP) in which the current estimation is the only state variable for the manager's problem. To keep the Markov property, we restrict the problem by defining the probability of a transition from state 𝑋 𝑡 to 𝑋 𝑡+1 conditionally independent on all past states and actions (𝑋 𝑡-1, 𝑋 𝑡-2 , 𝐻 𝑡-1, 𝐻 𝑡-2 and so on). We assume that the manager ignores past esstimations in forming expectations, mainly because of the modeling choice and practical considerations.

We solve this problem on a finite time horizon T = 80 years using value iteration algorithm (following the methodology of [START_REF] Boettiger | Optimal management of a stochastically varying population when policy adjustment is costly[END_REF], but we consider only the 50 first projection years for analysis. We also discretize the state and decision variable space over a regular sequence of a length out of 1000 steps from 0 to 12.10 6 individuals.

We simulate 500 times the optimal path following the optimal policy obtained from the solution to Bellman's equation (Equation 11) under the different uncertainty levels. The optimal solution of the deterministic closed-loop formulation has to be expressed as a feedback policy, which could refer to a harvest-control rule in fishery management. The feedback policy indicates the optimal harvest quantity Ht as a function of the state of the fishery Xt. As mentioned above, we simplified the optimization problem by assuming an n age structured population at equilibirum at each discretized state space. Despite this assumption, we control that the solution of the closed-loop formulation is close to the solution of the open-loop formulation. This shows that the approximation to discretize the state space and considering the age structure of the stock at the equilibrium leads to reasonably good results on the aggregated indicators level (Supplementary materials, appendix 4).

Results

Optimal management of Eastern Atlantic Bluefin tuna (EABFT)

In our deterministic model, long term equilibriums are fully determined by the biologic characteristic of the population and the selectivity pattern of the fishery. Harvest at the MSY level for the recruitment reference case reaches a level of 35,800 tons corresponding to a SSB of 197,400 tons (Figure 1). MSY varies between 18,700 tons corresponding to a SSB of 105,000 tons for the low recruitment case and 55,200 tons with a SSB of 302,700 tons for the high recruitment case. The optimal management of the EABFT targets the biomass level that maximizes the sum of the discounted (net present) values (NPV) from fishing (Equation 8), this level being called the BMEY. determined with a higher SSB and lower catch than MSY or on the contrary if the MEY converge to a level slightly above MSY for the catch and below for the SSB for lower cost and/or lower schooling behavior (χ tends to 1 and/or c tends to 0, see Supplementary materials appendix 5).

The optimal management of the EABFT could be also affected by an exogenous change of global Bluefin tuna supply. A 50% increase of overall Bluefin tuna supply negatively impacts the price from 8.4 to 6.8 € per kg and reduces the profitability of the fishery by 37.4% while a decreasing supply of substitutes increases the price to 11.4 € per kg and raises the profitability by 65.0 % (Figure 2). Furthermore, the recruitment levels affect the productivity of the stock and modify the equilibrium estimation (Figure 1).

Assuming a low recruitment level drastically reduces the profit from the fishery with a lower sustainable harvest level of 14,200 tons, increasing the price to 9.26 € per kg. On the contrary, a high recruitment level raises the profit expected from higher harvest level of 24,000 but lowered the price to 7.8 € per kg (Figure 2).

Including selectivity as an endogenous variable in the optimization produces optimal steady states very close to the current selectivity pattern. Considering a perfect and costless selectivity, the steady state harvest level stays around 20,000 tons with a more balanced fishing mortality over all age classes (Figure 3). This result is very sensitive to the estimation of the cost function parameters such as the optimal steady state discussed above (see Supplementary materials, appendix 5). For different combinations of scale cost and schooling parameters, the selectivity pattern switches toward the oldest age class. The optimal exploitation of fish populations also depends on interest rate levels [START_REF] Clark | Profit maximization and the extinction of animal species[END_REF]. The effect of increasing the discount rate up to 25% does not affect too much the optimal steady state. SSB steady state only decreases by 7% (Figure 4) while the steady state harvest increase of about 18%. Finally, it is worth noticing that the steady state SSB remains higher than MSY level even with high discounting.

Effect of stock estimation uncertainties

We observe that stock estimation uncertainty has an insignificant effect on the optimal harvest path even when the uncertainty level becomes high (Figure 5). The accuracy of information about the resource level does not affect the capacity of a manager to define a TAC-based policy. The MEY policy defined by a low harvest rate and a high population level is robust to stock estimation uncertainties. This result is also sensitive to the cost and schooling parameters. As the MEY becomes close to the MSY level, stock estimation uncertainties increase.

Discussion

This study provides a bioeconomic model of the EABFT fishery in line with the ICCAT stock assessment procedure. This work updates previous analysis of [START_REF] Bjorndal | The East Atlantic Bluefin Tuna Fisheries: Stock Collapse or Recovery?[END_REF] which found out pulse fishing as optimal management. We outline the optimal dynamic exploitation of the EABFT fishery by setting directly the harvest level as a fishing quota (TAC) and considering an age structured model which explicitly takes into account the schooling behavior of Bluefin tuna. Our model integrates the effect of the global supply of EABFT close substitutes on the price, analyses the effects of different recruitment levels and selectivity as an optimized variable. Using actual economic data of the European purse seine fishery, the model has been applied to determine the optimal economic policy for EABFT fishery.

Toward a new management target

After the success of the stock recovery plan launched in 2009, the definition of the pathway toward the optimal management target is crucial for the management of the EABFT fishery. The bio-economic optimal management strategy defines a pathway converging toward the steady state starting with high harvest rate to reach a SSB steady state slightly inferior to the SSB at the MSY level (even with zero discounting). This result contradicts what has been observed in Tahvonen et al. (2017), and supports the message from previous economic studies showing that the MEY management policy leads to a 'no regret' situation, increasing both the stock size and economic profits [START_REF] Kompas | Bioeconomic losses from overharvesting tuna: Bioeconomic losses[END_REF]. Moreover, the sharp decrease of the SSB steady state compared to the steady state harvest with an increasing discount rate widens the gap between the MEY and MSY outcomes. In face of a highly uncertain future, the EABFT fishing industry has strong incentives to harvest more and overexploit the stock beyond the MSY level.

A key issue in the assessment of the EABFT fishery is the confidence that scientists put in the magnitude of the recent SSB recovery, which mainly relies on a high estimate of recruitment levels from 2004 to 2007 (ICCAT, 2017). We integrate to our model the 3 recruitment scenarios used in the projection scenarios by ICCAT to explore the effect of a potential discrepancy in the recruitment level. Considering a constant selectivity pattern, the different levels of recruitment does not qualitatively affect the results, the SSB steady state remains higher than the SSB at the MSY level. However, quantitative results change substantially, considering a low recruitment instead of a high recruitment level modifies the steady state harvest level from around 24,000 to 14,000 tons. The precautionary approach [START_REF] De Bruyn | The Precautionary approach to fisheries management: How this is taken into account by Tuna regional fisheries management organisations (RFMOs)[END_REF] would suggest to deal with different scenarios and to implement a smooth pathway toward the target SSB which ensures to reach the MSY by 2022 for EABFT fishery (ICCAT, 2017). For each recruitment level, reaching MSY in the short run (by 2022) fulfill this condition. However, in the long run the uncertainty in the estimated productivity could jeopardize the conservation effort made during the stock rebuilding phase. Adopting a more cautious target, such as MEY, should smooth potential errors in the stock estimation and the productivity of the EABFT.

Fishing selectivity

When harvesting selectivity is costless and perfectly adjusted, the optimal path keeps the same qualitative property but harvest is balanced over all age classes. Under the assumption of a constant price per age class, this new selectivity pattern results in a slightly higher harvest in term of biomass and consequently a higher profit. Selective gear such as purse seine could potentially detect and select schools of tuna which are often organized according to their size [START_REF] Newlands | Atlantic Bluefin tuna in the Gulf of Maine, I: estimation of seasonal abundance accounting for movement, school and school-aggregation behaviour[END_REF][START_REF] Ottolenghi | Capture-based aquaculture of Bluefin tuna[END_REF][START_REF] Bauer | Aerial surveys to monitor Bluefin tuna abundance and track efficiency of management measures[END_REF]. However, at low fishing mortality a balanced harvesting selectivity is economically promoted, and serves both conservation and economic performance of the fishery, ensuring in the long run higher profits than the current selectivity pattern and conservation objective by maintaining stock levels above the MSY limit even if productivity is uncertain. The demographic structure of stocks is an important factor affecting the resilience of a population facing environmental variability [START_REF] Perry | Sensitivity of marine systems to climate and fishing: concepts, issues and management responses[END_REF]. Maintaining a high proportion of old individuals which potentially contribute the most to reproduction is also an important factor to decrease fluctuations in population abundance [START_REF] Anderson | Why fishing magnifies fluctuations in fish abundance[END_REF]. In addition, fishing alters life-history traits with an evolutionary side effect, affecting maturity, growth and leads to a loss of genetic diversity and sub-populations [START_REF] Planque | How does fishing alter marine populations and ecosystems sensitivity to climate[END_REF][START_REF] Garcia | Reconsidering the Consequences of Selective Fisheries[END_REF]. In an ecosystem context, under the synergic effect of fishing activity and climate change, community structure can shift toward new states with lowest productivity rates. Some Mediterranean fish stocks, hakes and small pelagic fishes, have already undergone such shifts (Hidalgo et al., 2011;[START_REF] Saraux | Small pelagic fish dynamics: a review of mechanisms in the Gulf of Lions[END_REF]. Our results support the new paradigm of balanced harvesting instead of the current increased selectivity objective. At the ecosystem level, balanced harvesting have been shown to produce better conservation result regarding both exploited and unexploited species which structured the ecosystem. Removing parts of the ecosystem can lead to unintended consequences even for exploited species managed under the MSY policy.

Global supply and international market

The supply on the global Bluefin tuna market also plays a critical influence on the price and the profit of the EABFT fishery. The 50-year horizon net present value is increased by 20% when the supply of EABFT substitutes is reduced by 50% over the period. Even if the price flexibility does not qualitatively affect the optimal management, the price responsiveness substantially affects the profitability of the fishery. The scale flexibility smaller than unity evaluated by Sun et al., (2017) creates an incentive from the EABFT fishery to maintain high catch level. The potential increase of the EABFT TAC under the optimal management path will negatively affect other BFT fisheries and justify coordination and a consistent management across RFMOs (Allen et al., 2010;Sun et al., 2017). In our modeling framework, we exclude investment, technical change and overcapacity management issues which are nevertheless important in a fishery which has undergone an over 50% reduction of its activity within a couple of years. Selecting a smooth pathway strategy toward the MEY or MSY objective is a preferable option for managers to ensure a safe investment context, keeping the supply chain unchanged, avoiding overcapacity, maintaining confidence in the future and holding low discount rate values to avoid a detrimental ''race to fish'' [START_REF] Armsworth | The economic value of ecological stability[END_REF][START_REF] Patterson | Change and stability in landings: the responses of fisheries to scientific advice and TACs[END_REF][START_REF] Boettiger | Optimal management of a stochastically varying population when policy adjustment is costly[END_REF].

Stock estimation uncertainties

When considering only the current stock estimation information, results from [START_REF] Sethi | Fishery management under multiple uncertainty[END_REF] does not hold in the age-structured context when the dynamic MEY is defined for low harvest rate and high population level. High stock estimation uncertainty only affects the optimal policy by reducing the SSB steady state, and increasing the harvest while decreasing the profitability. We have based our model on specific assumptions reducing the information to the existing knowledge of the system. These assumptions lead to a counter intuitive policy which is less cautious than under a complete information setting. Conversely, the precautionary principle fosters more cautious harvest levels to ensure the conservation of the resource in the face of ambiguity [START_REF] De Bruyn | The Precautionary approach to fisheries management: How this is taken into account by Tuna regional fisheries management organisations (RFMOs)[END_REF]. This result exacerbates the need for a cautious target, such as MEY, in face of potential high stock estimation uncertainties affecting the EABFT.

Conclusion

As pointed out in this study, conservation and economic objectives are still aligned if we consider age structured models, especially when the considered species is a long-lived species. MEY as a new management reference point has the advantage to be robust to high stock estimation uncertainty and foster balanced harvesting. These characteristics are crucial if we consider the management of the EABTF at the scale of its ecosystem.

Keeping low catch rate has both the advantage to maintain ecosystem resilience, and smoothing stock variation over time. MEY policy has the potential to create confidence in the future of fishery and promote consistency between RFMOs to maintain a high price on the global market.

However, in face of strong individual incentives to increase TAC, ICCAT agreed on an increase of TAC up to 36,000 tons by 2020. A new approach should be considered to create new economic incentives, even when full property rights are defined, to allow the sustainability of fishery resources. Society benefits should be considered by including ecosystem services values to compensate losses from direct use of the resource (TACs).

between the optimal steady state SSB and the cost, price and recruitment function parameters are, after removing the linear effect of the other parameters. PRCCs show strong negative effects of the stock elasticity parameter (÷) and the price flexibility parameter (φ) on the optimal SSB steady state. We also notice a strong positive relationship between the cost scale parameter (c) on the SSB steady state. The dynamic MEY is fully determined and very sensitive to economic parameters.

  flexibility parameter φ=0.91 (standard deviation 0.034) estimated bySun et al., (2017) from an inverse demand analysis of Bluefin tuna auction price. This estimate is based on frozen Bluefin tuna (Pacific and Atlantic) products pricing on the wholesale market (Tsukiji) in Japan. The recent increase of TAC (from 13,500 tons in 2014 to 19,296 tons in 2016, i.e. an increase of 43%) has negatively impacted the global frozen BFT price (11% decrease, Sun et al., 2017) through the auction market in Japan which represents more than 80% of the Bluefin tuna sashimi market in the world (fresh and frozen,Sun et al., 2017). The effect of the EABFT supply on the global sashimi tuna product price is crucial in the analysis of the optimal management. In a near future, only the supply of EABFT is about to vary with an adjustment of the quotas (TAC) after the success of the recovery plan. Thus, we consider in our model a constant supply of BFT (Southern and Pacific Bluefin tuna) substitutes corresponding to the mean harvest from 2008-2015, i.e.

Figure 1

 1 Figure 1 Relationship between the steady state harvest (in tons) and spawning stock biomass (SSB in tons) for the Eastern Atlantic Bluefin tuna (EABFT). The solid line represents the references recruitment case corresponding to the medium recruitment scenario of ICCA which corresponds to data on spawning stock biomass and recruitment for the years 1970 to 2010. The upper dashed line refers to the high recruitment scenario corresponding to the period 1990 to 2010, and the lower dotted line shows the low recruitment scenario corresponding to the period 1970 to 1980.

Figure 2

 2 Figure 2 Historical and optimal management of Eastern Atlantic Bluefin tuna (EABFT) with a 2% discount rate for each recruitment level and supply scenarios. Historical data are collected from ICCAT (2017) for the period 1970-2014. Each panel presented the optimal economic management of the EABFT for 5 different scenarios: reference, high and

Figure 3

 3 Figure 3 Steady state stock number and harvest per age class for the reference dynamic MEY and dynamic MEY with endogenous selectivity.

Figure 4

 4 Figure 4 Dependence of the steady state spawning stock biomass (SSB in tons) on the discount rate. The dotted line represents the optimal steady state for the selected model parameters.

Figure 5

 5 Figure 5 Results of 500 independent simulations of the closed-loop optimization model with different level of uniformly distributed stock estimation uncertainties (ó 𝑚 = 0; 0.3; 0.6; 0.9 𝑎𝑛𝑑 1.2). On the top charts, the optimal path of spawning stock biomass (SSB in tons) from 2014 to 2064. On the bottom charts, the optimal path of harvest (in tons) from 2014 to 2064. The grey shaded region represents the standard deviations.

Figure A4. 1

 1 Figure A4.1 Histogram of the effects of stock recruitment and cost function on optimal steady-state spawning stock biomass (SSB in tons), harvest (tons) and profit (10 6 €) for 1000 randomly drawn parameter sets. The dotted line represents the optimal steady state for the selected model parameters.

Figure A4. 2

 2 Figure A4.2 Partial rank correlation of key functions parameters on the SSB optimal steady state level.

Table 1

 1 Parameters used in the model.

	Age-class	Survival	Maturity g	Weight	Selectivity	Xs,0
	[year]	rate á		w [kg]	sel	[individuals]
	1	0.61	0	4.33	0.002	6,541,127
	2	0.79	0	10.66	0.55	5,232,334
	3	0.79	0	24.00	0.38	2,185,608
	4	0.79	0	35.67	0.67	1,036,139
	5	0.79	1	52.33	0.69	595,821
	6	0.82	1	72.67	0.34	421,575
	7	0.84	1	96.00	0.32	551,023
	8	0.86	1	119.33	0.33	824,995
	9	0.88	1	144.33	0.35	566,447
	10	0.90	1	202.00	1.00	1,417,591

Economic data are limited for this fishery. To estimate the price and the harvesting cost function, we use the data from 2008 to 2015 describing the European purse seine fishery (Supplementary materials, appendix 2), which represents the majority of the TAC (more than 60% since 2008). We estimate the parameters based on the French and Spanish purse seine fleet segment data available from the Scientific, Technical and Economic Committee for Fisheries (STECF 2016) and data on spawning stock biomass from ICCAT (2017).

  𝑉 𝑡 (𝑋 𝑡 ) = 𝑚𝑎𝑥 0≤𝐻 𝑡,…,𝑇 ≤𝐵 𝑡,…,𝑇 { 𝑈(𝐻 𝑡 , 𝑋 𝑡 ) + ä . ∑ 𝑃(𝑋 𝑡+1 |𝐻 𝑡+1 , 𝑋 𝑡 ). 𝑉 𝑡+1 (𝑋 𝑡+1 )

	also perform the optimization problem by integrating stock estimation
	uncertainties. Stock estimation uncertainties represent an important feature of fishery
	management. To analyze the effects of stock estimation uncertainty on management, we
	consider a stylized representation where a uniformly distributed noise Z t with different
	errors level σm (0, 0.3, 0.6, 0.9 and 1.2) alters information on stock levels. The solution of
	this problem requires to solve the stochastic program in closed-loop formulation by
	backward recursion of Bellman's equation. Bellman's (1957) principle of optimality
	implies that the optimal policy must satisfy the functional equation:

𝑇 𝑡 } Eq 11

Where X (the number of individuals), represents the state space which determines all states attainable, B the resulting efficient biomass (𝐵 𝑡 = ∑ 𝑠𝑒𝑙 𝑠 𝑛 𝑠=1

  The program is solved in the deterministic open-loop formulation which displays a smooth pathway toward a constant steady state population level (Figure2). Considering an interest rate of 2%, the optimal steady-state SSB is around 564,600 tons (mean over the period 2055 to 2065), much higher than the SSB at MSY level representing only 35% of the steady state level 9 . Furthermore, this level is higher than the mean SSB level from 1970 to 2009 while the steady state harvest of around 20,100 tons (mean over the period 2055 to 2065) which is quite lower than the mean level of the 'over-harvested' period from 1999 to 2009. Moreover, the catch level during this period is likely to be higher than the reported catch because of IUU fishing (ICCAT 2014). Available information on fishing capacity showed that catches EABFT were under-reported from 1998 to 2007. Under the reference recruitment level, the steady state harvest, representing 56% of the MSY level, is reached after a short period of slightly higher harvest intensity and creates an annual profit of 64 million euros with a high proportion of old age class fish. Along the smooth transition to the steady state, the harvest level gradually decreases while the profit stay quasi-constant, as the market price stays around 8.3 € per kg of fish. However, this result is very sensitive to the cost function parameters. Schooling and cost scale parameters fully determine if the MEY level is

Three recruitment scenarios are considered in the stock assessment: low, medium and high mean recruitment levels.

F0.1 and SSB0.1 are used as proxies of FMSY and SSBMSY and are common biological reference points for management[START_REF] Deriso | Optimal F0.1 criteria and their relationship to maximum sustainable yield[END_REF], Hilborn & Walters 1992).

Refer to the supplementary materials appendix 5 for a sensitivity analysis of the key economic and biological parameters of an optimal policy.

Acknowledgments

We are grateful to Thomas Vallée for useful comments on the modeling approach. We also thank Jacqueline Boursier for English editing. Finally, we acknowledge the financial support from the University of Nantes and IFREMER for the funding of a PhD grant.

Guillotreau, P., Squires, D., Sun, C.-H.J., & Compeán, G. (2017). Local, regional and global markets: what drives the tuna fisheries? Reviews in Fish Biology and Fisheries, 27 (4): 909-929. Hidalgo, M., Rouyer, T., Molinero, J. C., Massuti, E., Moranta, J., Guijarro, B., & Stenseth, N. C. (2011). Synergistic effects of fishing-induced demographic changes and climate variation on fish population dynamics. Marine Ecology Progress Series, 426, 1-12. Hilborn, R., & Walters, C. J., (1992). Quantitative Fisheries Stock Assessment: Choice, Dynamics, and Uncertainty. Chapman and Hall, New York, p. 570. Hurry, G. D., M., Hayashi, & Maguire, J. J. (2008). Report of the independent review.

International Commission for the Conservation of Atlantic Tunas (ICCAT

Supplementary materials

Appendix 1 Equilibrium age structured relationship.

Following Tahvonen et al. (2009), the age structure equilibrium is defined as:

With ì 𝑠 = á 𝑠 . (1 -𝑠𝑒𝑙 𝑠 . 𝐹𝑚𝑎𝑥), for s = 1, ..., n-2 and ì 𝑛-1 = á 𝑛-1 .(1-𝑠𝑒𝑙 𝑛-1 .𝐹𝑚𝑎𝑥)

1-á 𝑛 +á 𝑛 .𝑠𝑒𝑙 𝑛 .𝐹𝑚𝑎𝑥

Considering ℎ 𝑠 = 𝑥 𝑠 . 𝑠𝑒𝑙 𝑠 . 𝐹𝑚𝑎𝑥, the equilibrium age structure for s=2,…,n can be written as:

Given the [START_REF] Beverton | On the dynamics of exploited fish populations[END_REF] recruitment function, we could write:

With 𝜙 1 , 𝜙 2 the Beverton & Holt stock recruitment parameters and 𝐷 = ∑ 𝑤 𝑠 .

𝑛 𝑠=1

𝑔 𝑠 . ö 𝑠 , the equilibrium spawning stock biomass becomes:

Appendix 2 Data on East Atlantic Bluefin tuna (EABFT) French and Italian purse seine 762 fishery from statistics of the Scientific, Technical and Economic Committee for Fisheries 763 [START_REF] Stecf | The 2016 Annual Economic Report on the EU Fishing Fleet[END_REF]. 764 Appendix 5 Sensitivity analysis.

We assess the effect of key parameters uncertainties on the steady state optimal levels based on the open-loop formulation of the model. For each parameter set, we generate 1000 observations assuming normal distributions with means given by their estimates and standard deviation derived from the estimation. We carry out the sensitivity analysis using the latin hypercube sampling (LHS) method from the 'pse' package in R (Chalom et al., 2017).

An important drawback of age structured population model is the necessity to specify a stock-recruitment relationship. Stock recruitment models are low explanatory because of the low availability of recruitment data and the variability of the recruitment process independently of the spawning biomass (Hilborn & Walters, 1982). EABFT is not an exception, and the existence of a density dependence mechanism has not been observed because of the lack of contrast in the available recruitment data (time series begin well after the stock has been reduced by exploitation, ICCAT 2017). The choice of the Beverton and Holt function is controversial and leads to the estimation of highly uncertain parameters (mean recruitment levels over different periods are used for projections in the EABFT stock assessment procedure, ICCAT 2017). We also analyse the effect of the variations of the catch-stock elasticity parameter (÷) which influences the hyperstabiliy of the harvest productivity through the cost function. Moreover, we jointly evaluate those 2 parameters, and the effects of economic parameters which are related to the performance of the fishery. We analyse the effects of changes in the cost function parameter (c) which is subjects to large uncertainty. We only have 8 observations of variable costs and aggregate landings from the STECF data, and consequently few degrees of freedom. We also consider the price function parameter (p), the price scale flexibility parameter (φ) and the estimation of EABFT substitutes' supply (𝐻𝑏𝑓𝑡).

The optimal SSB steady state is very sensitive to economic parameters variations and shows a skewed right distribution with a peak centered on 190,000 tons slight below the SSB at MSY level (Figure A4.1). We observe the inverse pattern for the optimal harvest steady state which shows a long tail to the left of a mode centered on the steady state of 40,000 tons slightly above the MSY level. However, the optimal steady state profit shows large variations on the right of its optimal steady state. The partial rank correlation coefficients (PRCCs, Figure A4.2) measure how strong the linear associations


Téa Skopak

Scott Kroeker

Kirill Levin

Marc Dussauze

Raphaël Méreau

Yannick Ledemi

Thierry Cardinal
email: thierry.cardinal@icmcb.cnrs.fr

Evelyne Fargin

Younès Messaddeq

Structure and properties of gallium-rich sodium germano-gallate glasses

Keywords: germano-gallate glasses, infrared, glass structure, vibrational spectroscopy, 71 Ga NMR spectroscopy

1-Introduction

Optical applications in the near to mid-infrared (up to 5 m) are highly desirable notably for the detection of molecules showing absorption (vibrational modes) in this specific region (for example CO2, CH4, NOx, SOx). Compared to common glass systems such as silicate or phosphate, gallate glass matrices present potential applications in such a spectral range due to their low phonon energy (~550 cm -1 for some alkaligallate glasses or alkali-germano-gallate glasses [START_REF] Fukumi | Raman spectroscopic study of the structural role of alkaline earth ions in alkaline earth gallate glasses[END_REF]- [START_REF] Skopak | Mixture experimental design applied to gallium-rich GaO3/2-GeO2-NaO1/2 glasses[END_REF]). Glass compositions such as tellurite or bismuthate are known for their extended transmission in the infrared (up to 7-8 m for a 2 mm thickness glass of Bi2O3-PbO [START_REF] Dumbaugh | Heavy-Metal Oxide Glasses[END_REF]). Nevertheless, their transmission windows are narrower due to absorption in the UV to visible region and their use can be limited due to the toxicity of their composition. Gallate and germanate glassy systems offer the advantages of lower toxicity, higher glass transition temperatures, and an extended IR transmission window.

Structurally, in the case of germanate glass, gallium oxide is assumed to act like aluminum oxide in a silicate network. In fact, it has been already shown that adding aluminum oxide to an alkali-silicate glass system leads to the substitution of Si 4+ by Al 3+ cations which requires charge compensation of the alumina units by alkali ions, implying a decrease of the non-bridging oxygen content [START_REF] Brawer | Raman spectroscopic investigation of the structure of silicate glasses (II). Soda-alkaline earth-alumina ternary and quaternary glasses[END_REF]- [START_REF] Maekawa | Si MAS NMR investigation of the Na2O-Al2O3-SiO2 glasses[END_REF]. Nevertheless, it may be noted that high-coordinate aluminum is sometimes discussed without clear experimental evidence of its presence [START_REF] Brawer | Raman spectroscopic investigation of the structure of silicate glasses (II). Soda-alkaline earth-alumina ternary and quaternary glasses[END_REF], [START_REF] Huang | Structure and properties of calcium aluminosilicate glasses[END_REF]- [START_REF] Hwa | Infrared and raman spectra of calcium alumino-silicate glasses[END_REF]. As gallia and germania are chemically similar to alumina and silica, respectively, assuming the same structural behavior is a good starting point. However, several parameters influence the structure of the glasses, such as the role and content of gallium oxide, as well as the presence of other glass formers.

Several studies on germano-gallate glasses have been reported in the literature [START_REF] Higby | Properties of barium gallium germanate glasses[END_REF]- [START_REF] Bayya | BaO-Ga2O3-GeO2 glasses with enhanced properties[END_REF] but only a few have been focused on gallium-oxide-rich compositions [START_REF] Skopak | Mixture experimental design applied to gallium-rich GaO3/2-GeO2-NaO1/2 glasses[END_REF], [START_REF] Hwa | Polarized Raman scattering in lanthanum gallogermanate glasses[END_REF], [START_REF] Murthy | Properties and structure of glasses int he systeme M2O-Ga2O3-GeO2 ( M=Li, Na, K)[END_REF]. Depending on the content of gallium oxide, its role can change from network modifier to network former [START_REF] Srinivasa Reddy | Structural influence of aluminium, gallium and indium metal oxides by means of dielectric and spectroscopic properties of CaO-Sb2O3-B2O3 glass system[END_REF]. In the presence of a glass former such as germanium oxide, gallium tends to reinforce the glass network, as can be seen in BaO-Ga2O3-GeO2 glass compositions, where GaO4 and GeO4 tetrahedra are bound by the corners [START_REF] Mckeown | Raman spectroscopic studies of BaO-Ga2O3-GeO2 glasses[END_REF]. Furthermore, the presence of high gallium content led to the suspected presence of highly coordinated gallium site in several studies [START_REF] Whichard | Glass formation and properties in the gallia-calcia system[END_REF]- [START_REF] Zhong | Determination of gallium coordination in cesium gallate glasses by highresolution pulsed NMR[END_REF].

This work focuses on the investigation of gallium-rich glass compositions belonging to the sodium germano-gallate ternary system with a fixed content of germanium oxide. In order to better understand the relationship between the properties and the structure of the glasses, thermal, optical and physical properties are investigated. A thorough structural study has been carried out by infrared (IR), Raman and 71 Ga Magic Angle Spinning Nuclear Magnetic Resonance (MAS-NMR) spectroscopies in order to provide new insights into the role played by gallium oxide with respect to germanium and sodium oxides.

2-Experimental details

2-1-Synthesis

Glasses in the ternary system studied here are presented with the GaO3/2, GeO2 and NaO1/2 components to highlight the role of cations. Glasses were prepared by traditional melting technique from gallium oxide Ga2O3 (99.9%), germanium oxide GeO2 (99.999%), and sodium carbonate Na2CO3 (99.99%). After weighing, the powders were mixed and placed in a platinum crucible for melting under nitrogen in an induction furnace at 1300-1450°C for 1 to 2 hours depending on the composition. The melted glasses were then quenched in ambient air by pouring in a mold or between two brass plates at room temperature for compositions that tend to crystallize during cooling. No annealing step was done. Finally, glasses were cut and polished on both parallel faces for further characterizations. Crystalline KGaGeO4 was synthesized by the total recrystallization of the stoichiometric glass: 33.33GaO3/2-33.33GeO2-33.33KO1/2 (mol%), and confirmed by powder x-ray diffraction (JCPDS -052-1595).

2-2-Physical and thermal characterizations

Glass experimental chemical composition was determined by electron-probe micro-analysis (Wavelength Dispersive Spectroscopy) with a Microprobe Cameca SX 100. The averaged values were deduced from 10 measurements with a deviation of 2 mol.%. Characteristic temperatures which include the glass transition temperature (Tg), the onset of crystallization (Tx) and the maximum of the exothermic peak corresponding to the crystallization (TC) were measured by differential scanning calorimetry on the Netzsch DSC Pegasus 404PC apparatus, on glass chunks in a Pt pan at a heating rate of 10°C/min up to 900˚C with a precision of ±2°C for Tg and Tx, the precision for TC is estimated at ±1°C.

The vitreous state of the samples studied was evaluated by Powder X-ray diffraction (XRD) patterns which were collected on a PANalitycal X'pert PRO MPD diffractometer in Bragg-Brentano θ-θ geometry equipped with a secondary monochromator and X'Celerator multi-strip detector. Each measurement was made within an angular range of 2θ = 8-80°. The Cu-Kα radiation was generated at 45 kV and 40 mA (λ = 0.15418 nm).

The partly crystallized Ga40Ge26Na34 was analyzed on the same apparatus with the same conditions. The density, ρ, was obtained by Archimedes' method with a Mettler Toledo XSE204 densimeter by using distilled water at room temperature as buoyant liquid. The averaged values (from 4 measurements) are estimated with an error of 0.001g/cm 3 .

The refractive index was measured by the M-line prism coupling technique (Metricon 2010/M) at 532, 633, 972, 1308 and 1538 nm.

The transmission spectra in the UV-Visible-NIR and the mid-IR ranges were respectively obtained from an Agilent Cary 5000 and a Perkin Elmer Frontier FTIR spectrometers on polished glass discs of about 2.5 mm (for glasses poured in a mold) and 0.7 mm (for glasses pressed between two brass plates). The short wavelength and infrared multiphonon cut-offs are taken from normalized spectra at an absorption coefficient of 10 cm -1 .

2-3-Structural characterizations

The Raman spectra were obtained using a Renishaw inVia spectrometer coupled to a Leica DM2700 microscope equipped with a 633 nm laser. No interaction between the sample and the laser beam was observed. IR spectra were recorded on a Vertex 70V (Bruker) vacuum spectrometer equipped with a DTGS detector and a MID/FIR range beam splitter. Infrared spectra in reflection mode were recorded using an external reflection attachment (Graesby, Specac) with a 12° incidence angle. The complex refractive index of glasses was obtained through Kramers-Kronig analysis of the measured specular reflectance spectra, and this allowed for the calculation of the absorption coefficient spectra [START_REF] Kamitsos | Thin Film Amorphous Electrolytes: Structure and Composition by Experimental and Simulated Infrared Spectra[END_REF]. 71 Ga magic-angle spinning (MAS) nuclear magnetic resonance (NMR) spectra of ground glass samples were collected on a Bruker Avance II 900 (B0 = 21.1 T) with a 1.3 mm Bruker MAS probe. Quadrupolar echo [START_REF] Massiot | Modelling one-and two-dimensional solid-state NMR spectra[END_REF] experiments were done using 90°-90° excitation (νrf = 250 kHz) synchronized with a single rotor period during sample spinning at 62.5 kHz. A recycle delay of 1 second was used -verified to be sufficient for full relaxation -to collect 8k co-added transients. Chemical shifts were referenced to a 1M Ga2(SO4)3 solution at 0 ppm. Spectral lineshapes were fit using DMFit2015 [START_REF] Massiot | Modelling one-and two-dimensional solid-state NMR spectra[END_REF] employing the "CZSimple" model, which is a simplified version of the Czjzek distribution of chemical shifts and quadrupolar parameters. The critical exponent in the Czjzek distribution was set to 5, which corresponds to the Gaussian Isotropic Model [START_REF] Neuville | Al environment in tectosilicate and peraluminous glasses: A 27Al MQ-MAS NMR, Raman, and XANES investigation[END_REF] implying a statistical distribution of charges around the observed nucleus. First-order spinning sidebands were approximated using Gaussian peaks separated by twice the spinning frequency. Uncertainties were conservatively estimated by altering a given parameter and manually adjusting other parameters through chemically sensible ranges to try to compensate for the observed peak shape changes.

Density functional theory (DFT) calculations have been performed on small sized clusters. After geometry optimizations of the clusters, harmonic vibrational frequency calculations were achieved in order to obtain the IR and Raman spectra of the so obtained stable structures. Prior to any geometry optimizations, the dangling bonds originating from singly bonded oxygen atoms were neutralized by adding hydrogen-like atoms with masses equal to germanium to provide better agreement between calculated and experimental vibrational frequencies. Qualitative vibrational assignments were made by visualizing the atom displacement vectors for each mode likely responsible for experimental IR and Raman bands. In addition, a potential energy distribution (PED) analysis was performed to also quantify the contribution of internal coordinates to a specific vibrational mode. Both geometry optimization and harmonic vibrational frequency calculations were carried out using the three-parameter hybrid B3LYP exchange-correlation functional [START_REF] Becke | Density-functional thermochemistry. III. The role of exact exchange[END_REF][START_REF] Lee | Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density[END_REF] associated with the 6-311++G(3df,2p) basis set. The Gaussian09 software package [30] was used for all computations.

3-Results

3-1-Physico-chemical properties

In order to have a better understanding of the evolution of the properties and structure of the Ga-rich GaO3/2-GeO2-NaO1/2 glasses, their Na/Ga ratio was varied while their GeO2 content was fixed at 25 mol.%. The theoretical and experimental compositions as well as the composition labels are reported in Table 1. It is worth noting that the labels employed here are based on the experimental compositions to enable a better accuracy for interpreting and discussing the properties-structure relationship. The compositions can be represented on a ternary diagram (Figure 1) following a specific line toward an increase of the GaO3/2 content. One can notice in Table 2 that the material aspects differ depending on the quenching technique (and thus the cooling speed). The appearance of the glasses is different specifically for the composition with the highest Na/Ga ratio (which corresponds to a low gallium oxide content -Ga30Ge23Na47) that has a clouded aspect on the surface and for the composition in the lowest Na/Ga ratio (corresponding to a high GaO3/2 content -Ga40Ge25Na34) which shows crystallized areas.

The characteristic temperatures (transition temperature, onset of crystallization, maximum of crystallization and the thermal stability against crystallization) of the glasses are presented in Table 3. It can be seen that the glass transition temperature shows a minimum when Na/Ga ratio is the highest 1.54. Then, the glass transition temperature remains fairly constant with a stable T.

The density of the glasses depending on their gallium oxide content are presented in Table 3 and plotted in Figure 2. One can see a clear increase of the density with an increase of the gallium content and the decrease of the sodium amount. An increase is also well observed for the refractive index at different wavelengths, as listed in Figure 3 and presented in Table 3. The short wavelength and infrared multiphonon cut-offs were both calculated for an absorption coefficient of 10 cm -1 , and reported in Table 3. It can be seen that all the obtained glasses transmit in the infrared up to 5.9 µm while the short wavelength cut-off is minimal for a Na/Ga ratio of 1.05. The transmission in the midinfrared range is not significantly affected by the variation of the gallium content.

3-2-NMR

The 71 Ga MAS-NMR spectra collected at ultrahigh magnetic field and very fast spinning are shown in Figure 4. Dominating each spectrum is a central peak which can be assigned to 4-coordinated gallium, [4] Ga. The shape of this peak is asymmetric due to a distribution in the second-order quadrupolar interaction, which "smears" the intensity to lower frequency for the larger CQ values within the distribution [START_REF] Neuville | Al environment in tectosilicate and peraluminous glasses: A 27Al MQ-MAS NMR, Raman, and XANES investigation[END_REF]. This characteristic shape can be modeled using a Czjzek distribution [START_REF] Neuville | Al environment in tectosilicate and peraluminous glasses: A 27Al MQ-MAS NMR, Raman, and XANES investigation[END_REF] applicable to half-integer nuclei in isotropically disordered environments. The glass samples having the largest sodium content (i.e., Ga30Ge23Na47, Ga36Ge23Na41 and Ga37Ge24Na39) can be satisfactorily fit by a single tetrahedral gallium site with NMR parameters typical for tetrahedral gallium in oxides (e.g., δiso = 198 ppm, CQ = 11 MHz, see Table 4) [START_REF] Vosegaard | Crystal structure studies by single-crystal NMR spectroscopy. 71Ga and 69Ga single-crystal NMR of beta-Ga2O3 twins[END_REF]. Even at the fastest accessible spinning rates there is some overlap between the spinning sidebands and the central peak. Because the fitting software does not account for spinning sidebands, these were modelled using Gaussian peaks of approximately the same width as the central peak, separated by twice the spinning frequency, as shown in Figure 5. However imperfect this approach may be, it does not interfere with the most important parts of the lineshape, namely the slope on the low-frequency side of the central peak. For the sodium-poor glass samples (Ga41Ge26Na33 and Ga40Ge26Na34), a single gallium site does not yield a satisfactory fit to the experimental data (Figure 4 and Figure 5a). Despite extensive spectral overlap, reasonable parameters representing high-coordinate gallium produce a very good fit to the data (Figure 5b).

Consistent with previous NMR data of gallate glasses, and considering the relationship between 27 Al and 71 Ga isotropic chemical shift [START_REF] Massiot | 71Ga NMR of reference Ga IV , Ga V , and Ga VI compounds by MAS and QPASS, extension of galliumraluminum NMR parameter correlation[END_REF], the putative 5-coordinate Ga should appear in the 100-120 ppm range.

Excellent fits can be obtained by including high-coordinate Ga fractions in glass Ga40Ge26Na34 ranging from 15-30% by subtle modifications of the peak positions and widths. By fixing the 4-coordinate Ga NMR parameters from the glasses having the largest sodium content and adjusting only the high-coordinate Ga peak, a best-fit value of 22% is obtained for Ga40Ge26Na34. Considering that 5-and 6-coordinate Ga are often found together in amorphous environments, the same spectrum was also fit using reasonable shift and quadrupolar values for [5] Ga and [6] Ga, resulting in amplitudes of 17 and 6% respectively, bringing the total from the three-site fit into agreement with that of the two-site fit (Table 4 and Figure 5c) [33]. While these numbers are not independently reliable due to extensive spectral overlap, they illustrate that a plausible combination of two high-coordinate Ga species leads to comparable fits as a single high-coordinate site. A similar approach to the analysis of Ga41Ge26Na33 revealed 8% high-coordinate Ga (not shown).

3-3-Vibrational Spectroscopy

3-3-1 IR and Raman spectra analysis

Infrared and Raman spectra of three representative compositions with a Na/Ga ratio ranging from 0.8 to 1.54 are shown in Figure 6. All spectra can be decomposed in three main domains: 200-400, 420-600 and 700-900 cm -1 . The high frequency region is composed of two main contributions one at 770 cm -1 mainly active in IR and observed as a shoulder in Raman, the second is more active in Raman and peaking in the range 830-860 cm -1 . In comparison with vibrational responses of tetrahedrally connected glassy networks such as germanate or silicate, one can attribute the high frequency component to localized symmetric and antisymmetric stretching modes of tetrahedral units Ge-O or Ga-O bonds. At lower wavenumbers, the two envelopes can be attributed to bending modes involving mainly T-O-T bridges (T used for Ge or Ga in a tetrahedral coordination). These bending motions centered at 500 and 300 cm -1 can be attributed to oxygen motions respectively in the plane and out of the plane formed by a bent T-O-T bridge (similar to scissoring and wagging modes, respectively). When the Na/Ga ratio increases, the main spectral variations observed in Raman occurs in the envelope around 800cm -1 which operates a gradual increase and a shift from 860 to 830 cm -1 of its high frequency contribution. In IR, the main peak slightly shifts towards the higher wavenumbers from 760 to 785 cm -1 . For the lowest Na/Ga ratio, a new contribution appears observed around To explain these spectral changes, we first refer to the spectra reported in the literature for a germanate network [START_REF] Furukawa | Raman spectroscopic investigation of the structure and crystallization of binary alkali germanate glasses[END_REF][START_REF] Kamitsos | Raman and Infrared Structural Investigation of xRb2O•(1 -x)GeO2 Glasses[END_REF], for which both in IR and Raman, one can recognize all the vibrational modes described above. Nevertheless, for a pure GeO2 glass, the high frequency stretching modes are almost inactive in Raman [START_REF] Galeener | Longitudinal Optical Vibrations in Glasses: GeO2 and SiO2[END_REF]. The Raman activity in this spectral range is clearly observed upon insertion of modifier elements in the matrix forming non-bridging oxygens (NBOs) [START_REF] Furukawa | Raman spectroscopic investigation of the structure and crystallization of binary alkali germanate glasses[END_REF]. Raman spectroscopy is then commonly used to determine the presence of NBOs in a germanate or silicate glassy network. As an example, for the system GeO2-K2O, the signatures of Q3 and Q2 tetrahedral units (Qn represents tetrahedral units with n the number of bridging oxygens) are expected at 870 and 770 cm -1 respectively. If in a second step one takes as reference Raman spectra reported for germano-gallate glasses [START_REF] Henderson | Raman spectra of gallium and germanium substituted silicate glasses: variations in intermediate range order[END_REF], the stretching modes in the region 700-900 cm -1 are always active. The spectra reported by Henderson et al. [START_REF] Henderson | Raman spectra of gallium and germanium substituted silicate glasses: variations in intermediate range order[END_REF] also indicate that the aluminum to gallium substitution, keeping a ratio Na/(Al+Ga)=1, does not affect the Raman activity in the stretching mode region.

3-3-2 DFT modelling of glass vibrational response

Before interpreting the spectral variations observed upon changing the Na/Ga ratio, we would like to clarify the differences concerning IR and Raman respective activities between germanate and a sodo-germanogallate system for a ratio Na/Ga=1. In Figure 7, we have compared the IR and Raman responses of two representative glasses (a pure GeO2 glass and a germano-gallate of composition Ga37Ge24Na39) with (i) the Raman response of a crystalline reference: KGaGeO4 and (ii) results from DFT calculations conducted on four clusters depicted in Figure 8. Few investigations have been devoted to crystalline KGaGeO4 [START_REF] Barbier | Investigation of Structural States in the Series MGaSi04, MAIGeO, MGaGeO[END_REF], but several studies have been reported on isostructural compounds such as KAlSiO4 exhibiting the same beryllonite-type structure [START_REF] Creighton | Computer simulation and interpretation of the infrared and Raman spectra of sodalite frameworks[END_REF]. It consists of [GaO4] -and [GeO4] tetrahedra; each [GaO4] -unit shares corners with four [GeO4] forming ring arrangements in which alkali cations (K + or Na + ) are inserted and compensate for the negative charge of gallate tetrahedra. For the DFT calculations, we have started from clusters inspired from the structure of our crystalline reference KGaGeO4: the two first clusters (Figure 8) consist of rings formed of four GeO4 (cluster (A)) or of a mixed arrangement of two GeO4 and two GaO4 (cluster (B)) with the addition of two sodium ions to compensate for the charge of the [GaO4] -tetrahedra. Similarly, clusters (C) and (D) are made of the chaining of 6 tetrahedral units (respectively germanate and mixed germanogallate with the addition of sodium).

Comparing the Raman signatures of crystalline KGaGeO4 with the germano-gallate glass of composition Ga37Ge24Na39, a very similar spectral profile is observed, differing mainly in the peak breadth. The localized stretching modes in the range 700-900 cm -1 have a similar Raman activity in both glassy and crystalline materials. Focusing on the DFT calculations, the results corroborate the spectral assignment of the three main categories of vibrational modes described above. The frequency correspondence is satisfactory: symmetric and antisymmetric stretching modes of Ge-O or Ga-O bonds forming tetrahedral units are calculated between 780-905 cm -1 which is slightly overestimated for the germano-gallate system and underestimated for GeO2; the bending modes involving oxygen motions in T-O-T bridges show even better agreements: the modes calculated for clusters (A) and (C) describe very well both IR and Raman activities of GeO2 glass in the domain 400-600 cm -1 . Similarly, good agreement between cluster (B) and the KGaGeO4 crystal Raman spectrum may be observed. Finally, one should notice that the respective Raman/IR activities in calculated spectra for the four clusters are in very good agreement with the experimental observations. It confirms that for a fully interconnected network formed of only germanate tetrahedra, the localized stretching modes are almost inactive in Raman but they largely dominate the IR response, whereas for a mixed GaO4 and GeO4 fully interconnected network all modes appears both IR and Raman active. Changes of local structural geometries are at the origin of these selection rules variations. Results from DFT calculation evidence strong differences on the atomic charges (Mulliken type) attributed to oxygens involved in the T-O-T chains. For the "GeO2 type" clusters (A) and (C), all oxygen atomic charges are equivalent within a range of ±5%. For the germano-gallate clusters we observe a relative increase of 56% of the atomic charge for the oxygens in the vicinity of Na + cations. In addition, the calculated Mulliken charges of oxygens for the germanate clusters ((A) and (C)) are intermediate as compared to those obtained for the germano-gallate clusters ((B) and (D)) which are either 24% lower or 17% higher depending on the sodium location.

Finally, these spectral comparisons provide a link between the entire vibrational response of the glass Ga37Ge24Na39 (ratio Na/Ga=1) and a structure formed of corner-shared interconnected tetrahedra. If compared to a germanate system, the variations observed are mainly due to a non-uniform charge distribution along the T-O-T chains to compensate for the Na + charge which locally modifies chains and tetrahedral geometries at the origin of a global redistribution of the vibrational activity on both IR and Raman responses. If the ratio Na/Ga>1 (see spectrum of the Ga30Ge23Na47 glass in Figure 6, Na/Ga=1.54), the increase in the Raman band at 830 cm -1 should be linked to the appearance of NBOs in the structure, however as the rest of the spectral profiles do not vary significantly, a majority of the glassy structure should be considered as an interconnected tetrahedral network. On the other hand, for a ratio Na/Ga<1 (see spectrum of the Ga41Ge26Na33 glass in Figure 6, Na/Ga=0.8), the decrease of the high frequency stretching mode Raman intensity could be explained by a decrease of the sodium content which tends to homogenize the charge distribution along the T-O-T chains decreasing the Raman activity of the localized stretching modes.

Interestingly for the IR response, the new contribution observed around 670 cm -1 is not expected for a network based on tetrahedral units. However, by analogy with the IR response of the rutile GeO2 crystalline compound exhibiting a large band centered at 715 cm -1 [START_REF] Kamitsos | Raman and Infrared Structural Investigation of xRb2O•(1 -x)GeO2 Glasses[END_REF], this spectral variation might be linked to highcoordinated gallium ions.

4-Discussion

The optical properties, including the cut-off wavelengths and refractive indices of the samples studied here, are only weakly influenced by their structure. The short wavelength cut-off is indeed governed by electronic transitions and the presence of UV-absorbing transition metals impurities, which are expected to be similar in all the prepared samples. On the other hand, the long wavelength cut-off is governed by multiphonon absorption, which is also globally expected to be same in all the studied samples, considering their relatively similar chemical compositions (see Figure 1). The slight increase of the linear refractive indices observed in Table 3 and Figure 3 (from Ga30Ge23Na47 to Ga41Ge26Na33) can then be directly and simply related to the increase of gallium with respect to the sodium content, and its consequent increase in glass density. By contrast, the thermal properties are more sensitive to the structural arrangement of the vitreous network as gallium is introduced into the glass.

The key to understanding these structural changes lies in the gallium insertion into the germania network.

For the high-sodium glasses (corresponding to a high Na/Ga ratio), 71 Ga MAS NMR shows that the gallium is in four-fold coordination (Figure 4). These spectra are comparable to those typically recorded for AlO4 units, where no NBOs are thought to exist [START_REF] Creighton | Computer simulation and interpretation of the infrared and Raman spectra of sodalite frameworks[END_REF]. In accordance with this, vibrational spectroscopy shows that a fully interconnected structure comprising tetrahedral units corresponds well to the IR/Raman signatures for glasses with a Na/Ga close to 1, in which sodium compensates for the negatively charged [GaO4 -] tetrahedral units along the T-O-T skeleton. Thus, one may consider sodium to be acting mainly as a charge compensator for the gallium sites, and Tg remains unchanged for Na/Ga ratios ranging from 0.8 to 1.13 (Table 3). When the Na/Ga ratio increases further (#1.5), the appearance of Q3 units is expected from the vibrational spectroscopy analysis. By analogy with aluminosilicates, any NBOs are likely to reside on germanate tetrahedra (i.e., Q3 = [GeØ3O -]; Ø corresponds to a bridging oxygen). Although the 71 Ga NMR spectral signature of NBO-bearing GaO4 -species has not been documented, the NMR data can be satisfactorily modeled without considering any additional species, providing partial support for this assumption. This structural description correlates with the variation of the glass transition temperature, which is mainly related to the content of sodium ions in its role as network modifier. This function begins to appear for a Na/Ga ratio above 1.13, and up to 1.5, for which Tg decreases from 620°C to 561°C.

On the other side, once the sodium content is too low to compensate for the charges of all the gallium tetrahedra, the NMR data suggest the formation of high-coordinate (5 or 6) gallium sites, as shown in Figure 5. As it is necessary for the network to stabilize these gallium species, some degree of Ga clustering might be involved, which could lead to gallium-rich regions, and possibly phase separation and crystallization of Ga2O3-like crystalline structures containing octahedral and tetrahedral units [START_REF] Geller | Crystal structure of beta-Ga2O3[END_REF], [START_REF] Bermudez | The structure of low-index surfaces of beta-Ga2O3[END_REF]. This is consistent with our XRD data for Ga40Ge26Na34, which revealed crystallinity when quenched between metal plates at room temperature (see Table 2, XRD data not shown). Two phases (Ga2Ge2O7-ICDD® 00-050-0354 and (Ga2O3)3-(GeO2)2 ICDD® 00-035-0387) were identified on the crystallized portions of the Ga40Ge26Na34 sample, both involving the presence of both 4-fold and higher coordination number Ga ions [START_REF] Agafonov | COMMUNICATION : Crystal structure of new digermanate : Al2Ge2O7[END_REF], [START_REF] Meinhold | The system Ga2O3(Al2O3)-GeO2(SiO2) studied by NMR, XRD, IR and DTA[END_REF].

5-Conclusion

The gallium-rich region of the ternary GaO3/2-GeO2-NaO1/2 vitreous system was explored through glass compositions varying as a function of the Na/Ga ratio (ranging from 0.80 to 1.54) for a fixed GeO2 content.

The thermal, physical and optical properties of the obtained glasses were characterized. Vibrational (IR/Raman) and 71 Ga MAS-NMR spectroscopic techniques were used to propose a structural model of these Ga-rich glasses for different Na/Ga ratios. In the compositions studied, the glass network is formed of mixed GeO4 and GaO4 units in which the charge is balanced by sodium ions. For Na/Ga below unity, there are not enough sodium ions to charge balance all the tetrahedral gallium, which leads to high-coordinate gallium (5-or 6-fold). For Na/Ga above unity, all gallium is in 4-fold coordination, with short-range order consisting of [GeO4], [GeØ3O -] and [GaO4] -units, and sodium ions charge balancing the anionic tetrahedral species.

These structural configurations help explain the variations in some of the bulk properties.

Table Captions

Table 1. Studied glass theoretical and experimental compositions in cationic percent (mol.%) with the Na/Ga and the Ga/Ge ratios (corresponding respectively to the ratio of NaO1/2/GaO3/2 and the ratio GaO3/2/GeO2).

Table 2. Studied glass compositions used quenching technique and observed aspects of the resulting materials.

Table 3. Main characteristics of the Ga-rich GaGeNa glasses studied -Onset of vitreous transition temperature (Tg), Onset of crystallization (Tx), Maximum of crystallization (TC), Thermal stability (ΔT = Tx-Tg), Density (ρ), Short wavelength cut-off (λUV), IR cut-off edge (λIR) and the refractive index at 532nm (n532).

Table 4. parameters obtained from lineshape fitting of sodium germano-gallate glasses. Figure 2. Density of the studied glasses as a function of their gallium oxide content. Figure 3. Refractive index at different wavelengths of the studied glasses as a function of their gallium oxide content.

Figure 4. 71 Ga MAS-NMR spectra of the sodium germanogallate glasses with overlaid fits (in red) based on a single 4-coordinate Ga site.

Figure 5. 71 Ga MAS-NMR spectrum of Ga40Ge26Na34 with three modeling scenarios (see text for details): a) single [4] Ga site, b) a [4] Ga and [5] Ga site, c) [4] Ga, [5] Ga and [6] Ga sites. Red traces represent the overall fit; blue, green and purple indicate four-, five-and six-coordinate Ga, respectively. See Table 4 for fitting parameters and uncertainties.

Figure 6 a) Raman spectra normalized to the principal band at about 500 cm-1 and b) Infrared spectra of the studied glasses.

Figure 7. Right part, Raman and IR spectra of a GeO2 glass and Raman and IR calculated spectra from DFT clusters (A) and (C) depicted in Figure 8. Left part, Raman spectra of the crystallized phase KGaGeO4 (blue), Raman and IR spectra of the glass Ga37Ge24Na39 glass and Raman and IR calculated spectra from DFT clusters (B) and (D) depicted in Figure 8.

Figure 8: DFT cluster defined to compare germanate and sodo-germano-gallate vibrational response and charge distribution within the structure. Cluster (A) and (B). are formed four GeO4 or of a mixed arrangement of two GeO4 and two GaO4 with the addition of two sodium ions. Similarly, cluster (C) and (D) are made of the chaining of 6 tetrahedral units (germanate or mixed germano-gallate with the addition of sodium). At the periphery of the clusters, hydrogen-like atoms with masses equal to germanium have been used.

Figure 1 .

 1 Figure1. Representation of the studied glass compositions (each composition has one shape), in purple the theoretical compositions, in yellow the resulting experimental composition. The vitreous domain (in light green) determined by Murthy et al.[START_REF] Agafonov | COMMUNICATION : Crystal structure of new digermanate : Al2Ge2O7[END_REF] is as well represented in the GaO3/2-GeO2-NaO1/2 ternary diagram with the delimited Ga>Ge border.Figure2. Density of the studied glasses as a function of their gallium oxide content. Figure3. Refractive index at different wavelengths of the studied glasses as a function of their gallium oxide content.Figure4. 71 Ga MAS-NMR spectra of the sodium germanogallate glasses with overlaid fits (in red) based on a single 4-coordinate Ga site.Figure5. 71 Ga MAS-NMR spectrum of Ga40Ge26Na34 with three modeling scenarios (see text for details): a) single[4] Ga site, b) a[4] Ga and[5] Ga site, c)[4] Ga,[5] Ga and[6] Ga sites. Red traces represent the overall fit; blue, green and purple indicate four-, five-and six-coordinate Ga, respectively. See Table4for fitting parameters and uncertainties.Figure6a) Raman spectra normalized to the principal band at about 500 cm-1 and b) Infrared spectra of the studied glasses.Figure7. Right part, Raman and IR spectra of a GeO2 glass and Raman and IR calculated spectra from DFT clusters (A) and (C) depicted in Figure8. Left part, Raman spectra of the crystallized phase KGaGeO4 (blue), Raman and IR spectra of the glass Ga37Ge24Na39 glass and Raman and IR calculated spectra from DFT clusters (B) and (D) depicted in Figure8.Figure8: DFT cluster defined to compare germanate and sodo-germano-gallate vibrational response and charge distribution within the structure. Cluster (A) and (B). are formed four GeO4 or of a mixed arrangement of two GeO4 and two GaO4 with the addition of two sodium ions. Similarly, cluster (C) and (D) are made of the chaining of 6 tetrahedral units (germanate or mixed germano-gallate with the addition of sodium). At the periphery of the clusters, hydrogen-like atoms with masses equal to germanium have been used.

Figure 1 .Figure 2 . 3 )Figure 3 .

 1233 Figure 1. Representation of the studied glass compositions (each composition has one shape), in blue purple the theoretical compositions, in yellow the resulting experimental compositions. The vitreous domain (in light green) determined by Murthy et al. [45] is represented in the GaO3/2-GeO2-NaO1/2 ternary diagram with the delimited Ga>Ge border.

Figure 8 .

 8 Figure 8. DFT clusters defined to compare germanate and sodo-germano-gallate vibrational response and charge distribution within the structure. Clusters (A) and (B) are respectively formed from four GeO4 or from a mixed arrangement of two GeO4 and two GaO4 with the addition of two sodium ions. Similarly, clusters (C) and (D) are made of the chaining of 6 tetrahedral units (respectively germanate and mixed germano-gallate with the addition of sodium). At the periphery of the clusters, hydrogen-like atoms (in white) with masses equal to germanium have been used.

  Téa Skopak, Scott Kroeker, Kirill Levin, Marc Dussauze, Raphaël Méreau, et al.. Structure and properties of gallium-rich sodium germano-gallate glasses. Journal of Physical Chemistry C, American Chemical Society, 2018, 123 (2), pp.1370-1378. 〈hal-02048674〉

  Téa Skopak, Scott Kroeker, Kirill Levin, Marc Dussauze, Raphaël Méreau, et al.. Structure and properties of gallium-rich sodium germano-gallate glasses. Journal of Physical Chemistry C, American Chemical Society, 2018, 123 (2), pp.1370-1378. 〈hal-02048674〉

Table 1 . Studied glass theoretical and experimental compositions in cationic percent (mol.%) with the Na/Ga and the Ga/Ge ratios (corresponding respectively to the ratio of NaO1/2/GaO3/2 and the ratio GaO3/2/GeO2).

 1

	Sample	GaO3/2 (mol.%) Theo. Exp. (±2)	GeO2 (mol.%) Theo. Exp. ( ±2)	NaO1/2 (mol.%) Theo. Exp. (±2)	Na/Ga Theo. Exp.	Ga/Ge Theo. Exp.
	Ga30Ge23Na47	31	30	25	23	44	47	1.42	1.54 ±0.17	1.24	1.30 ±0.20
	Ga36Ge23Na41	37.5	36	25	23	37.5	41	1	1.13 ±0.12	1.50	1.57 ±0.22
	Ga37Ge24Na39	38	37	25	24	37	39	0.97	1.05 ±0.11	1.52	1.58 ±0.21
	Ga41Ge26Na33	44	41	25	26	31	33	0.70	0.80 ±0.09	1.76	1.61 ±0.20
	Ga40Ge26Na34	50	40	25	26	25	34	0.50	0.84 ±0.09	2	1.56 ±0.20

Acknowledgements

This research was supported by the Canada Excellence Research Chair program (CERC) in Photonics Innovations and the Aquitaine Region. The authors are also grateful to the Fonds de Recherche Québecois sur la Nature et les Technologies (FRQNT, N° 097991), the Natural Sciences and Engineering Research Council (NSERC) of Canada, and the Canada Foundation for Innovation (CFI, N° 29191 and 32523) for the financial support. Access to the 21.1 T NMR spectrometer was provided by the National Ultrahigh-Field NMR Facility for Solids (Ottawa, Canada), a national research facility funded by a consortium of Canadian universities, supported by the National Research Council of Canada and Bruker Biospin, and managed by Téa Skopak, Scott Kroeker, Kirill Levin, Marc Dussauze, Raphaël Méreau, et al.. Structure and properties of gallium-rich sodium germano-gallate glasses. Journal of Physical Chemistry C, American Chemical Society, 2018, 123 (2), pp.1370-1378. 〈hal-02048674〉 the University of Ottawa. The research program was also supported by financial support from the French National Research Agency (ANR) in the frame of "the Investments for the future" Programme IdEx Bordeaux -LAPHIA (ANR-10-IDEX-03-02), as well as in the frame of the ANR program N°ANR-17-CE08-0042-01 and the French Region Nouvelle Aquitaine, with the program n° 2016 -1R10107. We are grateful to Dr Victor Terskikh for valuable assistance with the NMR experiments. Computer time was provided by the "Pôle Modélisation" HPC facilities of the Institut des Sciences Moléculaires UMR 5255 CNRS -Université de Bordeaux, co-funded by the "Nouvelle Aquitaine" region. Mobilities were supported by the CNRS (LIA LuMAQ) and by grants of the French Consulate in Québec (Frontenac Program), the association of Campus France and Mitacs Globalink.

Table 2. Studied glass compositions used quenching technique and observed aspects of the resulting materials.

Sample

Quenching technique Observations

Ga30Ge23Na47

Pressed between two plates Mainly glassysome clouded portions

Ga36Ge23Na41

Poured in mold at ambient air Glassy

Ga37Ge24Na39

Poured in mold at ambient air Glassy

Ga41Ge26Na33

Pressed between two plates Glassy

Ga40Ge26Na34

Pressed between two plates Glassy portionsmainly crystallized [4] Ga site, b) a [4] Ga and [5] Ga site, c) [4] Ga, [5] Ga and [6] Ga sites. Red traces represent the overall fit; blue, green and purple indicate four-, five-and six-coordinate Ga, respectively. See Table 4 for fitting parameters and uncertainties. Raman Shift (cm-1) Raman Shift (cm-1)