G Loas

) M Alouini

M Vallet

Optical fiber Sagnac interferometer for sensing scalar directional refraction: application to magnetochiral birefringence

We present a set-up dedicated to the measurement of the small scalar directional anisotropies associated to the magnetochiral interaction. The apparatus, based on a polarizationindependent fiber Sagnac interferometer, is optimized to be insensitive to circular anisotropies and to residual absorption. It can thus characterize samples of biological interests, for which the two enantiomers are not available and/or which present poor transmission. The signal-tonoise ratio is shown to be limited only by the source intensity noise, leading to a detection limit of ∆φ = 500 nrad.Hz -1/2 . It yields a limit on the magnetochiral index n MC < 4 10 -13 T -1 at 1.55 µm for the organic molecules tested.

I. INTRODUCTION

 . These studies aim at a better understanding of the symmetry principles governing the interaction of light and matter and target varied applications. Here, we focus on the magnetochiral interaction, which is important in biochemistry as it has been shown to provide asymmetric photochemical reactions 7 and because of its potential role in the origin of homochirality of life 8 . This fundamental effect consists of a change in the optical index of a chiral media subject to a static magnetic field parallel to the direction of propagation of light 9 . It can be regarded as a cross effect between magnetic optical activity (MOA) and natural optical activity (NOA). As for MOA, its sign depends of the orientation of the magnetic field and, as for NOA, it has opposite sign for the two enantiomers. Moreover, it depends on the direction of propagation of light. But, contrary to MOA and NOA which are circular differential, i.e., different optical index for left-handed and right-handed circular polarization, the magnetochiral

interaction is scalar, that is, it does not depend on the polarization of light. The magnitude of this cross effect is weak. For a magnetic field of 5 T and a diamagnetic compound, the amplitude is predicted to be at least 10 -3 smaller than MOA 10 . The first observation of magnetochiral interaction was made in absorption, i.e., magnetochiral dichroism (MCD).

Using luminescence spectroscopy techniques, Rikken and Raupach proved that the photoemission of chiral media depends on the relative orientation of the light with respect to an external magnetic field 4 . This was recently followed by MCD detection in chiral ferromagnets 11 and in organic compounds 12 . For the detection of the refractive part of the magnetochiral interaction, named magnetochiral birefringence (MCB), to the best of our knowledge only two setups have been designed up to now. At Zürich, a folded single-pass Sagnac interferometer was used to detect the variation of directional phase shift associated to MCB 5 . In Rennes, we developed an active interferometer, based on a ring laser, in order to detect the frequency shifts due to intracavity samples 6 . However, for both setups, the response is polarization-dependent. They thus require the availability of the two enantiomers of the compound under study, in order to compensate to zero the circular anisotropies inherent to chiral samples. This drawback prevents from using most of all samples of biological interests, where the availability of the two enantiomers is scare. Moreover, such samples are usually diffusive or absorbing which forbid their insertion into an active interferometer as in Ref. 6. Besides, the experimental results reported in Ref. 5 and 6 do not agree with models based on molecular ab-initio calculations 13 . There is thus a need for additional measurements and consequently for the development of a passive and polarization insensitive interferometer. Fiber Sagnac interferometers are able to detect very small directional phase-shifts, induced by, e.g., external magnetic fields 14 or mechanical rotation of the interferometer. The latter phenomenon is the so-called Sagnac phase shift 15, and is the basis of the interferometric fiber-optic gyroscope (IFOG) 16 , which has evolved over the three last decades to industrial devices achieving navigation grade performance. Furthermore, in order to minimize the spurious contributions of the fiber anisotropies to the useful signal, a polarization independent technology based on the insertion of depolarizers in the fiber loop has been successfully demonstrated [START_REF] Laskoskie | Fiber optic and laser sensors X[END_REF][START_REF] Szafraniec | [END_REF] . One can then wonder if the detection of small high-order scalar effects such as MCB could benefit from a similar technique by circumventing the contributions of first-order circular anisotropies, here MOA and NOA.

In this paper, we report on the design of a modified fiber-optic depolarized Sagnac interferometer developed in view of investigating MCB at 1.55 µm. In Section II, we detail the setup. Section III is dedicated to the description of the noise sources and to the characterization of possible systematic effects, leading to an estimation of the instrument detection limit. Section IV reports the calibration by means of Fizeau effect. Measurements performed on several organic molecules are presented, leading to a new limit on the magnetochiral birefringence level. Finally, Section V is devoted to the conclusion.

II. EXPERIMENTAL SETUP

The experimental setup is schematized in Fig. 1. It consists in a table-top fiber-optic Sagnac interferometer operating at 1.5 µm. Part of the loop comprises polarizationmaintaining fibers (PM) (grey color in Fig. 1), followed by Lyot depolarizers and two sections of standard single-mode fiber (SMF) connected by two collimators. These collimators sandwich the sample under test which presents a non-reciprocal phase shift ∆φ. The perimeter of the loop is equal to P = 38 m, essentially determined by the length of the depolarizers. The so-called Y-coupler configuration of the interferometer is composed of a LiNbO 3 integrated optic circuit (IOC) provided by Photline Inc. This multifunction circuit, originally designed for IFOG, integrates a 50/50 coupler, a push-pull phase modulator and a 60 dB extinction ratio polarizer on a Lithium Niobate waveguide. The LiNbO 3 guides are fabricated using proton exchange technique. The optical source is a 2 mW superluminescent diode at λ = 1.55µm (Superlum Inc.). As discussed abundantly in the literature [START_REF] Lefèvre | The Fiber-optic gyroscope[END_REF][START_REF] Sanders | Agard/Nato Conf. Rep. on Optical Gyros and their applications[END_REF] , broadband emission is mandatory in order to avoid residual interferences that might be caused by coherent back-reflections and backscattering in the interferometer. It also permits to get rid of spurious non reciprocities due to Kerr nonlinearities, i.e., a non-reciprocal change of optical index due to slight power imbalance between the two counterpropagating waves. Moreover, as will be detailed later, a broadband source is required to efficiently depolarize the two counterpropagating optical beams travelling through the sample under test. As shown on Fig. 2(a), the spectral width of our source is measured to be equal to ∆λ = 60 nm, leading to a coherence time τ c = 130 fs and a coherence length of 26 µm in the fiber. Its emission spectrum presents a Gaussian shape which optimizes the flatness of the degree of coherence Γ(τ) for τ > τ c , as reported in Fig. 2(b). This explains why a superluminescent diode was preferred to a rare-earth doped fiber amplified spontaneous emission (ASE) source. Indeed, in preliminary experiments, we observed a lower depolarization level when the interferometer was seeded with a homemade Erbium doped fiber amplified spontaneous emission source. An optical circulator permits to direct the beam to the interferometer and extract the output beam for detection. The two in-loop home-made Lyot depolarizers 21 are the key elements for our Sagnac interferometer. They consist of two sections of Panda birefringent fiber spliced at 45° with respect to each other, with associated length L and 2L and 4L and 8L respectively (see Fig. 3). In our setup, L is equal to 1.5 m. As discussed by Szafraniec [START_REF] Szafraniec | [END_REF] , the depolarized interferometer drastically reduces the contribution of in-loop spurious non-reciprocal linear and/or circular birefringences ∆φ L and ∆φ C . Indeed, in the case of a linear birefringence, the effective birefringence

eff L φ ∆ , i.e.,
detected by the interferometer, is given by ( )

, n eff L L φ γφ ∆ = ∆ ∆ (1)
with ∆γ the misalignment angle between fiber splices and n the number of sections in the depolarizers. Assuming that the mean misalignment is lower than 1°, it yields

( ) 12 ( ) ( ) / 2. eff CC φ δτ δτ φ ∆ = Γ +Γ ∆ (2)
δτ 1 and δτ 2 are the group delay differences associated to the two depolarizers, denoting difference of time propagation between the proper axes of the birefringent fiber. The beat length of the panda fiber we used is of the order of two millimeters 22 , leading to δτ 1 and δτ 2 equal to about 3 ps and 12 ps, respectively. From Fig. 2(b), the associated degrees of coherence Γ(τ 1 ) and Γ(τ 2 ) are then shown to be lower than 10 -6 . This means that The mean power impinging on the photodiode is of the order of 10 µW. The photodiode is followed by a low-noise transimpedance amplifier whose electrical diagram is reported in Fig. 4(a). This assembly is positioned in a Faraday box in order to isolate it from parasitic signals. In order to detect the in-loop directional phase shift ∆φ, the interferometer must be biased at an operating point with a non-zero response slope. As in usual FOG, an asymmetric phase modulation is applied at a frequency matching the loop proper frequency defined by 1/ (2 )

pg f =
τ , with τ g the propagation time in the Sagnac ring. In our setup, f p is equal to 2.7 MHz. This phase modulation of amplitude φ p offers two major advantages. First, it permits a heterodyne phase measurement around the offset frequency f p , thus rejecting low frequency noises. Second, it is applied at the entrance of one arm of the interferometer only, leading to a phase bias at the detection side between the two contra-propagating waves [START_REF] Lefèvre | The Fiber-optic gyroscope[END_REF] .

Actually, the output signal from the transimpedance amplifier can be written as

( ) 0 ( ) 1 cos cos(2 ) , p p Vt V ft φφ π   = + ∆+   (3)
where V 0 depends of the transimpedance gain and φ p is the phase modulation amplitude. By using a standard Bessel decomposition and assuming (2 ) 2 cos 2 .

Pp p A f VJ VJ φφ φ = ∆≅ (4.b)
The non-reciprocal phase shift of interest can be easily deduced from the ratio of these two components as

( ) ( ) 2 1
( )/ ( ) ( )/ (2 ) .

p p P P J J Af A f φφφ ∆= (5)
Experimentally, A(2f P ) is first measured by a lock-in amplifier at 2f P whose integration time is set to 30 ms. In order to extract A(f P ), an analog 8 th order Butterworth low-pass filter [START_REF] Horowitz | The art of electronics, 2nd Ed[END_REF] (detailed in Fig. 4(b)) is used before lock-in amplifier detection, now at f P . The -3dB cut-off frequency of the filter is equal to 3.5 MHz, leading to a rejection level of 30 dB at 2f p . We have experimentally noticed that such a high rejection level was mandatory in order to avoid saturation of the input stage of the lock-in amplifier by the 2f P component.

As discussed in the next section, the amplitude of the directional phase-shift of interest has to be also modulated in order to increase the detection sensitivity. However, this modulation must be done at a very low frequency f m , that is, well below the time response of the lock-in amplifier which is ruled by its integration time (30 ms in our case). -When the non-reciprocal effect of interest is for example the magnetochiral birefringence, this modulation can be performed through the amplitude of the magnetic field. Now assuming that sin(2 ) mm ft ∆= φφ π , a second lock-in detection at f m will provide the amplitude of the components at f p ±f m that are given by ( )

01 ( )2 P m pm Af f VJ
±= φφ using Eq. (4.a). φ P is experimentally adjusted to 1.8 rad in order to maximize this signal. From Eq. ( 4) and Eq. ( 5), one gets 0.53 ( ) / (2 ).

m Pm P Af f A f = ± φ (6)
A double modulation-demodulation scheme will thus enable the detection of a modulated non-reciprocal effect. Here, the first lock-in amplifier (high frequency) is a Stanford Research SR844, whereas the second one (low frequency) is a 7220 from EG&G.

Let us mention that the selected output filter of the latest is a fourth-order low pass filter. For Gaussian noise, the equivalent noise bandwidth is given by the one associated to a perfect rectangular filter which would pass the same amount of noise; this leads to maximize the noise rejection to an equivalent bandwidth of 5/(64TI) with TI the integration time [START_REF]SR844 RF Lock-in amplifier User's manual[END_REF] .

Finally, we point out the fact that large phase dynamic range is here not required, since the amplitude of the non-reciprocal phase shift is supposed to be fairly constant during the measurement time. It is thus not necessary to apply a closed-loop signal processing, as in standard FOG for which large dynamic range is obtained by using feedback electronics driving a control element in the interferometer [START_REF] Lefèvre | The Fiber-optic gyroscope[END_REF] .

III. DETECTION LIMIT

In this section, we evaluate the noise floor of the detection and discuss the amplitudes of the spurious non reciprocal offsets that could add to the signal.

The noise floor ∆φ Ν associated to ∆φ is equal to the ratio of the current noise 1/ 2 2 N i with respect to the photocurrent ph i , both measured at the output of the Butterworth filter.

This ratio is reported in Fig. 5(a). At the detection frequency f P , the noise floor value of the interferometer is about 0.5 µrad/Hz 1/2 . This level is comparable to the noise floor obtained in state-of-the-art FOG [START_REF] Lefèvre | [END_REF] . At f p = 2.7 MHz and for an integration time of 1000 s, the detection limit of our setup is potentially of 5 nrad. This has to be compared to the photon and electron noise floor. This later is obtained from the root sum squaring of shot noise, relative-intensity noise (RIN), and Johnson noise of the load R, that is 20

( ) 1/ 2 1/ 2 1/ 2 2 2 2 1.38 / 4 / , N N ph ph ph B ph f i i ei i kT R i φ ν ∆ ∆ = = + ∆+ (7)
with ∆f the detection bandwidth, e the charge of the electron, ∆ν the source spectral width, k B the Boltzmann constant, and where the factor 1.38 stands for phase-to-intensity conversion in depolarized Sagnac interferometers. As can be seen on Fig. 5(b), its value corresponds to the experimental level. It shows that, here, the detection limit is ruled by the RIN of the broadband source.

Static non reciprocal phase shifts lead to a DC output voltage at the output of the first lock-in amplifier, which adds to the signal of interest at f m . Their contributions may spoil or even overcome the useful signal, depending on the rejection level of the second lock-in amplifier, and thus have to be minimized. First, the spurious directional linear and circular anisotropies should be reduced by four and six orders of magnitude respectively, as discussed in Section II. In particular, we have experimentally verified the sensitivity of our depolarized Sagnac interferometer to the Faraday effect. A 0.13 T magnetic field was applied on a 1 cm section of SMF fiber in the loop. The Verdet constant of silica being of about 0.6 m -1 T -1 , it gives a Faraday phase shift of 0.8 mrad. No signal was detected at f p . The effect is thus smaller than the detection noise given by (7). This confirms that our interferometer permits us to get rid of vectorial effects in the limit of a depolarization better than 10 -5 . Second, contrary to FOG, no particular effort has been performed on the winding of the fiber to minimize possible asymmetric thermal drift of the fiber. This effect, known as shupe effect 26 , is here negligible since the fiber loop is extremely short as compared to FOG. Then, Rayleigh backscattering (and back reflections on the collimators) might also provide non reciprocal noise. Its contribution can be circumvented by symmetrizing the loop 27 . We have therefore placed the two collimators at equal distance from interferometer entrance. Finally, the tiny Sagnac directional phase shift due to earth rotation was minimized by properly orienting the sensing loop plane perpendicular to the terrestrial rotation vector. In summary, from the DC component of the first lock-in output and using ( 5), the total static phase shift was measured to be lower than 1 µrad. No contribution of this spurious phase shift to the output signal of the second lock-in, i.e., at f m , was detected. Since our interferometer was specifically designed for sensing small modulated nonreciprocal phase shifts, that is scalar directional refraction, a possible small modulated scalar absorption associated to the effect under study may contribute to the signal of interest. In order to give an order of magnitude of this contribution and without loss of generality, we can write the counterpropagating optical fields E ± (see Fig. 1) as 2) sin( cos( 2))

2 1 2 sin( 2) cos( cos( 2)).

m m mp p mp p Vt V ft V ft ft V ft ft = +ε π -φ π φ π + +ε π φ π (10)
A straightforward calculation shows that both components A(f p ±f m ) and A(2f p ) depend neither on ε nor on ∆ε. Relation ( 6) thus still applies when φ p is adjusted to 1.8 rad. Consequently, at first order, the double modulation-demodulation scheme that we propose cancels any spurious contribution related to an absorption term modulated at the same frequency that the refraction term under study. This is confirmed by the measurements that are now detailed in the next section.

IV. EXPERIMENTAL RESULTS

In order to check the calibration of our interferometer, we have first measured the tiny scalar directional phase shift associated to the well-known Fresnel drag effect. When two beams counterpropagate in a medium of length L 0 moving along the propagation axis, they experience a directional phase shift ∆φ FD given by the following formula 28

4v 1, o FD L n n c π φλ λλ ∂  ∆ = --  ∂  ( 11
)
where v is the displacement speed of the medium and n its refractive index. Here, we took as moving medium a 51 mm long cylinder of fused silica that was inserted between the two collimators. It was periodically translated back and forth at f m = 0.3 Hz by means of a motorized stage. Figure 6 reports the measured phase shifts obtained when v was varied from 0.25 mm.s -1 to 25 mm.s -1 . Perfect agreement is observed between the experimental data and the theoretical expectations from (11). These results evidence the good sensitivity and linearity of the interferometer response. Let us mention that the moving glass rod induces slight reciprocal losses that are periodically modulated at f m because of unavoidable small misalignments. The associated amplitude was estimated to be in the range of ε ≈ 10%.

Nevertheless, as expected from (10), it does not contribute to the signal, which evidences the insensitivity of the setup to modulated absorption. The rotation of this magnet induces along the optical axis an amplitude modulation of the magnetic field H of 0.13 T. When an achiral compound, such as acetone, is poured inside the cell, we detect a residual phase shift that follows the rotation of the magnet. We found that this systematic signal corresponds to a cross effect between the linear strain birefringence φ L of the cell windows and the rotation of plane of polarization θ F experienced by the optical beam through the sample, i.e., circular birefringence associated to the Faraday effect.

Experimentally, the amplitude of the phase shift varies from a few tens to a few hundred of nrad, depending on the spot position on the cell window, that is, on the value of the probed residual birefringence. If the input beam was linearly-polarized, the amplitude of this phase shift would have been of the order of φ L θ F and dependent of the direction of polarization with respect to the birefringence neutral axes 29 . The typical residual strain birefringence of a 1mmthick silica window is of the order of 5 mrad 30 , while the Faraday rotation angle θ F is estimated to be 6 mrad. Consequently, a phase shift of the order of 30 µrad is expected from theory for a linearly polarized beam. The measured systematic effect is well below this value which is consistent with the fact that the beam travelling through the cell windows is depolarized. This is confirmed by Fig. 8, which reports the systematic phase shift when a quarter wave plate, i.e., φ L = 1.5 rad, is inserted between the cell and one collimator for several orientations. Although φ L θ F is now equal to 9 mrad, the measured signal never exceeds 1.2 µrad. The drastic reduction of this systematic effect is made possible because the sensing beams are depolarized. This justifies the need of perfectly depolarized beams and consequently the use of a passive interferometer. Measurements on chiral compounds can be now conducted. To compare with previous measurements, we consider samples formerly tested at 488nm with an active interferometer 5 and at 633 nm with passive interferometers 6 . However, a rough λ -2 dependence for the magnetochiral index is predicted, both from dipole-dipole interaction model [START_REF] Ruchon | [END_REF] and from full quantum-mechanical considerations [START_REF] Barron | Molecular light Scattering and optical activity[END_REF] . The expected effect is thus lower at 1550 nm compared to visible wavelengths and a large rotatory dispersion is required in order to get a detectable magnetochiral effect. Three molecules comply pretty well with the previous requirements: limonene, 3-(trifluoroacetyl)-camphor and carvone. These three molecules are fairly transparent at 1550 nm, both enantiomers are available and the optical activity is rather large. The first measurement was done with racemic limonene. A systematic phase shift of 30 nrad was detected. We have then successively replaced the racemic mixture by R(+)-limonene and S(-)-limonene. Even with large integration times (1000s) on the lockin amplifier, corresponding to a noise rejection equivalent bandwidth ∆f = 0.1 mHz, we did not detect any magnetochiral directional phase shift within 10 nrad fluctuations. ( )

2 0 2 22 0 2 MC A eH mc λλ φ θ π λλ ∆= - , (12)
where e and m are the charge and mass of the electron and where the angle of rotation due to optical activity obeys the following wavelength dependence ( ) ( )

22 0 / A = - θ λθ λλ . For
Limonene, θ A and λ 0 are estimated to be respectively 600°/dm and 210 nm. Eq. ( 12) then leads to ∆φ MC = 500 nrad, that is n MC = 4.7 10 -11 T -1 at 1550nm. This is two orders of magnitude above the noise floor of our instrument. This can also be compared to the values measured at visible wavelength. For limonene, we measured in the past 6 n MC = 3.9(±1.3) 10 -10 T -1 at 488 nm, while at 633 nm, Kleindienst and Wagnière measured, for 3-(trifluoroacetyl)-camphor and for carvone, respectively n MC = 3(±0.2) 10 -8 T -1 and n MC = 1.3(±0.3) 10 -9 T -1 5 . Again, a λ -2 dependence for the magnetochiral index then leads to expected estimations at 1.55 µm two orders of magnitude above detection limit of our instrument. These unexpected results might mean that the λ -2 dependence has to be reconsidered or/and that the magnetochiral index is actually much lower than that expected from Bequerel model. Further theoretical investigations are thus required. For instance, we have assumed that optical index is only due to UV transitions [START_REF] Ruchon | [END_REF] . Contributions to residual absorption bands in the visible or infrared should modify the expected value. Furthermore, at 1550 nm, contributions to IR vibrational transitions have probably to be taken into account.

V. CONCLUSION

An apparatus combining a depolarized fiber-optic Sagnac interferometer and a double modulation-demodulation scheme has been designed to measure the non-reciprocal phase shifts associated to the magnetochiral index. We have validated a depolarization level better than 10 -5 . This permits to decrease the amplitude of systematic phase shifts, mainly due to residual birefringences of the cell windows, below 30 nrad. The good stability of the interferometer allows one to reach a measurement time of 1000 s with a noise-floor value of 0.5 µrad/Hz 1/2 comparable with the state-of-the-art. This yields an experimental detection level of 33 nrad with a confidence level of 99.6%. Tests on three different organic molecules have shown that the magnetochiral index is lower than 4 10 -13 T -1 at 1.55 µm. As compared to figures previously obtained for the same molecules in the visible 5,6 , it implies a significant discrepancy with the values expected from a λ -2 dependence for the magnetochiral index. To validate our results, the next step is consequently to compute accurately the magnetochiral index at 1.55 µm from ab-initio models 13 As the detection performance reaches the noise floor defined by the relative intensity noise of the superluminescent source, it seems difficult to drastically lower the limit of this setup below 33 nrad, the acquisition time being already set to its maximum. Increasing this acquisition time further would only slightly diminish the noise floor, at the expense of a larger sensitivity to thermal and mechanical long-term fluctuations. In order to detect magnetochiral index with our setup, a possibility would be to use larger alternated magnetic field 4 . Alternatively, the availability of samples that present larger magnetochiral

birefringence is an open question. Indeed, compared to resonant passive ring interferometers 3,[START_REF] Golub | [END_REF] , our single-pass setup is not sensitive to samples which present residual absorption or diffusion. One can then consider using samples made of chiral compounds mixed with ferrofluid [START_REF] Ruchon | [END_REF] , or chiral ferromagnets 11 . Organic compounds with a large optical activity in the infrared, due to, e.g., delocalized π-electrons such as helicene, could also be tested 34 . The apparatus could also be easily modified to test in-loop samples under reflexion at non-normal angle. This would permit to test solid samples, such as, e.g., photonics crystals 35 where giant magnetochirality is expected. Perspective for the setup would also include the extension to the detection of other magnetical directional anisotropies in crystals 36

Fig. 1 :

 1 Fig. 1: Scheme of the modified depolarized Sagnac interferometer. See text for details.

Fig. 2 :

 2 Fig. 2: (a) Optical spectrum of the broadband source and (b) associated coherence function.

Fig. 3 :

 3 Fig. 3: Schematic of the 2 pieces Lyot depolarizer inserted in the Sagnac interferometer.

  the case of a circular birefringence, the effective birefringence eff C φ ∆ measured by our Sagnac interferometer and related to the non-reciprocal circular birefringence reads

Fig. 4 :

 4 Fig. 4: (a) Transimpedance amplifier. (b) 8 th -order Butterworth filter.

Fig. 5 :

 5 Fig. 5: (a) Experimental power spectral density (PSD) versus frequency. (b) Estimated PSD versus optical power P impinging on the photodiode. The arrow indicates optical power in our experimental conditions (7.5 µW).

Fig. 6 :

 6 Fig. 6: Calibration of the interferometer with using Fresnel-Drag phase shift.

Fig. 7 :

 7 Fig. 7: Setup of the magnetochiral measurement. Scale: the cell is 1cm-long.

Fig. 8 :

 8 Fig. 8: Systematic phase-shift induced by an in-loop quarter wave plate.

  E 0 the amplitude of the fields and where ε and ∆ε correspond respectively to the reciprocal and non-reciprocal part of the absorption modulated at f m within the component

	( ) E t ± =	0 E	1  	( + ε±∆ε	)	sin(2	m ft π	) exp  	 	( sin(2 m ± φ j	m ft π	)	+φ	p	cos(2	p ft π	)) / 2 ,  	(8)
	under test. Instead of (3), the output voltage is now equal to
	[ ( ) 2 1 sin(2 0 Vt V = +ε	π	m ft	] 2 ) 1 cos( sin(2 m  ) cos(2 ) mp p ft ft + φ π +φ π 
	0 V 2 +	∆ε	22 sin (2	π	m ft	) 1 cos( sin(2 m  ) cos(2 ) . mp p ft ft × -φ π + φ π 	(9)
	At the first order with respect to ε, ∆ε and φ m , expression (9) simplifies to
	( ( 00 0 ) ) ( ) 2 1 2 sin(2 ) 2	sin(

with

Acknowledgments

The authors thank Elsa Rescan for preliminary works, Cyril Hamel and Ludovic Frein for technical assistance, Thierry Ruchon and Jeanne Crassous for fruitful discussions. This work was partly funded by Université de Rennes 1.


A Resmini
email: andrea.resmini@onera.fr

J Peter

D Lucor

HIGH DIMENSIONAL STOCHASTIC INVESTIGATION OF 2D RANS FLOW ABOUT AN HELICOPTER AIRFOIL

Keywords: CFD, Uncertainty Quantification, Detached Flow, RANS, Stochastic Collocation, Geometrical uncertainties, Operational uncertainties

Abstract. The use of Computational Fluid Dynamics (CFD) in the aeronautical research community is now well established. Due to the high costs that wind-tunnel tests require, the validation and verification of CFD simulations are of primary importance. Notwithstanding, several factors impact the solution quality. Physical modelling and numerical discretization are only two key examples on which most of the CFD researchers have been focusing on. Experiments and real flight conditions constantly show alterations of the observed quantities from nominal values, while, traditionally, CFD computations are performed for a given fixed condition. Since the last decade, more attention has been paid to the impact of input data lack of knowledge on simulations. Uncertainty Quantification (UQ) [START_REF] Le Maître | Spectral Methods for Uncertainty Quantification -With Application to Computational Fluid Dynamics[END_REF] deals with this aspect and, specifically, it quantifies the effect of data incertitude on the computed solution.

The first step in this UQ study is the identification of realistic uncertainties [START_REF] Evans | Identification and Quantification of Uncertainty Sources in Aircraft-Related CFD Computations -an Industrial Perspective[END_REF] both in wind-tunnel and real flight conditions. An extensive research has been performed where typical values of incertitudes have been quantified. Within the latter, the retain ones are the M a, incidence and several surface imperfections. Depending on the available experimental data, the probability distribution of these uncertainties has been inferred.

The second step is the choice of the appropriate stochastic approximation. A nonintrusive stochastic collocation method [START_REF] Xiu | High-Order Collocation Methods for Differential Equations with Random Inputs[END_REF] with sparse grids [START_REF] Smolyak | Interpolation and Quadrature Formulas for Tensor Products of Certain Classes of Functions[END_REF] built on Clenshaw-Curtis nodes [START_REF] Clenshaw | A Method for Numerical Integration on an Automatic Computer[END_REF] has been preferred. In this fashion, the ONERA finite-volume compressible elsA The present study addresses the prediction of 2D subsonic turbulent flow about NACA0015 airfoil in the presence of operational and geometrical uncertainties. The computations are carried out by means of RANS simulations at Re ≈ 2 • 10 6 .

The results at a pre-stall condition, angle of attack equals to 10 • and stochastic dimension equals to five, show that the joint uncertainties influence on the observed solution is not negligible in term of lift and drag counts. Figure 1 shows the C p distribution at the leading edge location. These observations with fully attached flow may be useful for robust airfoil design process.

Ongoing computations of higher stochastic dimension with adaptive sparse grids will better capture the uncertainties effect at detached condition.

A

  . Resmini, J. Peter and D. Lucor flow solver has been used without modifications. The statistics of the solution in terms of lift, drag and pressure coefficients have been computed.

Figure 1 :

 1 Figure 1: Stochastic pressure coefficient C p profiles at the suction side of NACA0015 airfoil at 0 ≤ x/c ≤ 0.2, A.o.A. = 10 • . Graycontour of non-uniform solution's PDFs at each x/c locations.


Aziz Elkechai

Farida Kias

Fazia Talbi

Abdou Boucekkine
email: abdou.boucekkine@univ-rennes1.fr

Redox properties of biscyclopentadienyl uranium(V) imido-halide complexes: a relativistic DFT study

Keywords: DFT, ZORA, uranium complexes, ionization energy, electron affinity, half-wave potentials, Hirshfeld analysis

published or not. The documents may come

INTRODUCTION

Organoactinide complexes often present structural arrangements and a particular reactivity not noticed in coordination chemistry of transition metals [START_REF] Boucekkine | f-element complexes[END_REF][START_REF] Ephritikhine | Recent Advances in Organoactinide Chemistry As Exemplified by Cyclopentadienyl Compounds[END_REF][3][START_REF] Burns | 3rd edn. The chemistry of the actinides and transactinides elements[END_REF][START_REF] Evans | [END_REF][6][7]. The knowledge of the Electron Affinities (EA) and Ionization Energies (IE) of organoactinide complexes is essential for the understanding of the redox properties of these species, especially as actinides can exhibit various oxidation states. In particular, the study of the redox behaviour of uranium complexes, carrying the ubiquitous cyclopentadienyl ligand, is of great interest. Because redox experimental studies for actinide systems could be difficult to achieve, computational quantum chemistry provides useful tools to access to this important property. As pointed out in a rather exhaustive review devoted to EA by Rienstra-Kiracofe et al [8], electron affinities obtained with calibrated Density Functional Theory (DFT) methods are rather reliable and in good agreement with experiment. The Kiplinger's group was pioneering in the theoretical determination of EA of fluoroketimide complexes [9].

The issues related to the scarcity, toxicity and handling of actinide complexes prevent experimental studies (to our knowledge, no direct experimental determination of EA of actinide complexes to date) which are limited to electrochemical measurements of the redox properties of uranium complexes [10][11][12][13][14][15][16]. Our previous studies of the redox behaviour of such species, concerning exclusively tetravalent organouranium complexes, namely the bis(cyclopentadienyl) series [Cp* 2 UX 2 ] [17] and [L 2 U(BH 4 ) 2 ] [18], and tri(cyclopentadienyl) [Cp 3 UX] [19],

showed that DFT computed EAs correlate very well with the experimental half-wave reduction potentials (R 2 ≥ 0.99).

In 2008, the Los Alamos group provided a number of experimental data (structures, electrochemical measurements, IR spectra, NMR chemical shifts, magnetic susceptibility) for a series of unprecedented pentavalent uranium complexes [20]. This availability of experimental data on such rare compounds, expected to be unstable because their easy disproportionation into uranium(IV) and uranium(VI) species, gives us the opportunity to apply our methodology to the study of this new class of pentavalent complexes of uranium.

The purpose of the present work is to perform a detailed relativistic DFT study of the redox behavior of neutral imido halide uranium(V) complexes [Cp* 2 U(=N-2,6-i Pr 2 -C 6 H 3 )(X)] (X = F, Cl, Br, I), through the determination of their electron affinities and ionization energies. In particular, we shall investigate what are the electronic factors which drive EAs and IPs of the U(V) complexes under consideration. Besides EAs and IEs, other molecular properties are discussed as molecular geometries, harmonic vibration frequencies and other electronic indices. In particular the bond-multiplicity indices and the Hirshfeld electronic charge analysis [22] will be used to characterize the bonding and electronic structure of selected organouranium complexes. To our knowledge, it is the first time that a theoretical investigation of the oxidation of uranium complexes is done at the planned level of theory.

COMPUTATIONAL DETAILS

In reference 8, it was shown, in the case of large polycyclic aromatic hydrocarbons, that standard DFT quantum chemical calculations with relatively small basis sets could reproduce experimental values of the electron affinities with average error of about 0.15 eV. Our previous studies, using Relativistic DFT [23] based on the Zeroth Order Regular Approximation (ZORA) [24] to the Dirac equation, showed that calculations carried out on bis (or tri)cyclopentadienyl uranium complexes lead to computed EAs in good agreement with experimental half-wave reduction potentials. So, in this work, we shall use the same methodology which is detailed below. The Vosko-Wilk-Nusair functional (WVN) [25] for the local density approximation (LDA) and the gradient corrections for exchange and correlation of Becke and Perdew [26] respectively, i.e. the BP86 functional have been used. Solvents effects have been taken into account using the Conductor-like Screening Model (COSMO) [27]. Geometry optimizations which have been carried out at the scalar relativistic level were followed by single point computations including spin orbit coupling. The calculations were performed using the Amsterdam Density Functional (ADF2010.02) program package [28c]. In this study, we used for all atoms a Triple-ζ Slater-type valence orbitals (STO) augmented by one set of polarization functions, i.e. the triple-zeta polarized (TZP) basis set, taken from the ADF/ZORA/TZP database. The spin unrestricted DFT scheme is used.

The frozen-core approximation where the core density is obtained from four-component Dirac-Slater calculations has been applied for all atoms. For carbon C.1s, fluorine F.1s and nitrogen N.1s, the 1s core electrons were frozen. The 1s/2s/2p cores were frozen for the chlorine Cl.2p, while the 1s/2s/2p/3s/3p/4s/3d and 1s/2s/2p/3s/3p/4s/3d/4p//5s/4d cores were frozen respectively for the bromine Br.3d and iodine I.4d. The U.5d valence space of the heavy element includes the 5f/6s/6p/6d/7s/7p shells (14 valence electrons). Several studies have shown that the ZORA/BP86/TZP approach reproduces the experimental geometries and ground state properties of f-element compounds with a satisfying accuracy [29][30][31]. Molecular geometry and molecular orbital plots were generated, respectively, by using the MOLEKEL 4.3 [32] and the ADFVIEW [28c] programs.

Electron affinities (or ionization energies) are often not easily reachable experimentally; moreover, their computational evaluation is a difficult task owing to the fact that it implies odd-electron systems (spin contamination and SCF convergence problems). It should be noted that in our case, spin contamination was found negligible, owing to the fact that the computed values <S 2 > of the squared spin operator are very close to the exact values for all the studied species (deviation less than 3%).

F I Br Cl

In the present work, the electron affinities were calculated as the energy difference between the neutral and anionic (or cationic) forms of the complexes at their respective optimized geometries, i.e. the "∆E method".

In terms of the energies E at optimized geometries, EA and IE (Ionization Energy) are computed as follows:

EA = E(neutral) -E(anion) for the reduction reaction, and IE = E(cation) -E(neutral) for the oxidation reaction.

The ADF program that we use produces Total Bonding Energies (TBE) rather than total energies, so that EA is computed in our case as the TBE(neutral) -TBE(anion) difference for the reduction process and IE as

TBE(cation) -TBE(neutral).

RESULTS AND DISCUSSION

Structural Properties. First, the full geometry optimizations of the neutral complexes 4) and their anionic and cationic forms were carried out in the gas phase, at the spin unrestricted level of the theory. We considered the highest spin state for the anionic U IV (5f 2 ) complex, i.e. a triplet one, whereas the spin states are respectively a doublet state for the neutral U V (5f 1 ) species, and a singlet state for the cationic U VI (5f 0 ) one. All compounds have been taken in the C 1 symmetry. Finally, in order to check that the optimized geometries are minima on the potential energy surfaces and to simulate IR spectra, we also performed the calculation of the vibration frequencies of the neutral U V species under consideration. Then, the geometries were further re-optimized in solution, using the COSMO approach, considering two different solvents (toluene which is the solvent used for the synthesis of the complexes, and tetrahydrofuran THF, used for the electrochemical measurements). The non-default Delley type of cavity was used, the solvents being considered with their dielectric constant respectively equal to of 2.38 and 7.58 for toluene and THF, whereas the cavity radius have been set equal to 3.48 Å and 3.18 Å for toluene and THF respectively. Finally, single-point variational spin-orbit ZORA calculations using the previously optimized geometries have been carried out, in both the gas phase and solution, in order to compute the spin-orbit coupling contributions to the energies [28b, 33]. In our case of open-shell systems, the non collinear approximation has been used [33b].

[Cp* 2 U(=N-2,6-i Pr 2 - C 6 H 3 )(F)] (1), [Cp* 2 U(=N-2,6-i Pr 2 -C 6 H 3 )(Cl)] (2), [Cp* 2 U(=N-2,6-i Pr 2 -C 6 H 3 )(Br)] (3) and [Cp* 2 U(=N-2,6- i Pr 2 -C 6 H 3 )(I)] (
First of all, it is necessary to study the structures of the different species involved in the redox processes.

In Table 1 are listed the most relevant computed geometrical parameters, i.e. metal-ligand distances and bond angles for the three U IV , U V and U VI species in the gas phase as well as in solution at the ZORA/BP86/TZP level (the optimized structures and coordinates are given in Supporting Information (SI)).

Table 1 Computed Distances (Å) and Angles (deg.) for the U IV /U V /U VI Cp* 2 U(=N-2,6-i Pr 2 -C 6 H 3 )(X)

Complexes at the ZORA/BP86/TZP Level in the Gas Phase and in Solution (toluene), and the Available Experimental X-Ray Values for the Neutral U V Compounds (in Square Brackets) [20] Complex 172. 2/173.2/172.2 (173.0/173.9/173.3) [170.7( 6 The computed geometrical parameters are globally found in good agreement with the available crystallographic data, namely those of U V complexes 1, 2, 3 and 4 [20]. In particular, distances between central metal and the halide element U-X (2.110 Å for U-F, 2.618 Å for U-Cl, 2.823 Å for U-Br and 3.100 Å for U-I) reproduce well the experimental data i.e. 2.122, 2.621, 2.789 and 3.039 Å, within the limit of the experimental error. Values of U-X bonds which vary as follows U-F < U-Cl < U-Br < U-I reflects well the ordering of the ionic radii of these elements. It is noticed that all halide U V complexes exhibit short U=N imido bond distances The electron capture by the neutral species leading to the U(IV) anionic complexes causes a lengthening of the distances between the central metal and its neighbors; this is mainly due to the increase of the ionic radius of uranium from U(V) to U(IV); on the contrary, the oxidation process induces a shortening of the metal-toligands bond lengths (ca. 0.3 Å). Moreover, it should be noted that oxidation affects rather clearly the angle between the Cp* ring centroids and the central metal (it passes from 136° in the neutral species to 149° in the cationic species) whereas the Cp*centroid-U-X angles are not affected by the reduction or oxidation of the central metal (ca. 3° for the angles). We also found that the optimized geometrical parameters in solution are close to the values obtained for the isolated molecules (values for the THF solvent in the SI).

The optimized molecular geometries of the anionic U IV , neutral U V , and cationic U VI complexes for X=Cl (up) and X=F (down), are depicted on Fig. 2.

U

We consider now Nalewajski and Mrozek (NM) bond orders [21], which have been successfully used as a useful tool for the analysis of bonding and of the electronic structure of organometallic complexes [34].

Generally, the calculated NM bond orders correlate very well with experimental properties like bond lengths and vibrational frequencies. The imido-halide complexes present a single U-X bond (bond order equal to 0.999 for F, 1.037 for Cl, 0.995 for Br and 0.97 for I) whereas the bond order between the central metal and the nitrogen of the imido group is double of character (values between 1.83 and 2.25). On the other hand, this method predicts an increase in the U-X and U-N bond multiplicity in the cationic species, and a decrease in the anionic ones.

These variations of NM bond orders increase with the atomic number of the halogen element; thus, the variation of these indices between the U IV and U V species is 0.0938 for F, of 0.1970 for Cl, 0.2236 for Br, whereas it reaches 0.2758 for I (see detailed NM bond orders in SI). As expected, the changes in bond multiplicity are reflecting the changes in the bond-length, a higher bond order corresponding to a shorter bond length. Thus, an almost perfect linear correlation is obtained between the NM bond orders of the U-N bonds and the corresponding U-N bond lengths (R 2 = 0.99). The U-X bonds present also such correlation but with R 2 = 0.97 (in SI).

Calculations of the analytical vibrational frequencies of the neutral species reveal practically similar spectra for all the complexes. The simulated DFT spectra show two distinct bands, the first going from 0 to 1500 cm -1 with rather low IR intensities, and the second ranging between 2500 and 3250 cm -1 . The latter, containing in particular two localized transitions around 2900 cm -1 , corresponds to vibrational modes within the

[Cp 2 *-U=N-(2,6)-i Pr 2 -C 6 H 3 ]
moiety, whereas the U-X vibration is localized in the first band. The stretching frequencies of the U-X bonds are respectively equal to 512, 543, 581 and 592 cm -1 for 1(F), 2 (Cl), 3 (Br) and 4

(I).

Redox Properties. We consider first the oxidation of the neutral U V imido-halide complexes, half-wave oxidation potentials from electrochemical measurements, relative to the U V /U VI redox system being available. In all cases, the IEs were computed according to the "∆E method", that is the difference between the TBEs of the cationic U VI and neutral U V species at their optimized geometries. In Table 2, are given the computed IEs, in the gas phase as well as in solution (toluene and THF) for all complexes, at the ZORA/BP86/TZP level of theory;

the row with the acronym SO corresponds to the values of the IE taking into account spin-orbit coupling. In the last row of this Table, are displayed the experimental half-wave oxidation potentials in volts (E 1/2 vs. [Cp 2 Fe] +/0 ) of the neutral uranium(V) complexes, in THF) [20]. The computed TBE values are given in SI. Considering the complexes in the gas phase, it can be noted in Table 2 that all the computed IEs are positive, their values being between 5.507-5.659 eV and between 5.568-5.728 eV when including spin-orbit coupling. The iodide compound, which has the highest half-wave oxidation potential (+0.11 V) exhibits the highest IE (5.728 eV) and is therefore the most difficult to oxidize. On the contrary, the fluoride congener exhibits the lowest IE (5.568 eV) and probes to be the easiest to oxidize. In addition, it can be seen that spinorbit coupling affects similarly the TBE of the neutral and cationic species, in their doublet and singlet state respectively, leading to an energy lowering of 2.17 to 2.35 eV and a smaller increase of IE (0.07 eV on average).

The effect of a solvent leads to an important variation of IEs, even if their ordering is maintained, i.e. IE(F) < IE(Cl) < IE(Br) < IE(I).

Moreover, it can be seen that the IEs decrease with the solvent polarity; in our case, THF leads to the lowest IEs.

A very nice linear correlation is obtained between the computed IEs (in THF at the TZP level including the spinorbit correction) and the experimental half-wave oxidation potentials E 1/2 , the R 2 correlation coefficient of the linear regression being equal to 0.99 (Fig. 3). We note that neglecting solvation worsens slightly the IE-E 1/2 correlation (R 2 =0.98). All halide complexes present the same FMO diagram; the LUMO and SOMO exhibit mainly a uranium 5f character with a small contribution of the Cp* rings; if the weight of nitrogen is zero in these orbitals, SOMO-1 exhibits a very strong U=N π bonding character (17% of nitrogen contribution), while the X ligand contribution appears only starting from the SOMO-2 (3% on average). Finally, in our case, oxidation can be considered as the loss of an electron by the metal ion which changes consequently its oxidation state (from U V to U VI ), as expected.

In Figure 5, is displayed the frontier MO diagram of the cationic U(VI) species, which are a closed-shell complexes. ). It is also interesting to note the large value of the HOMO-LUMO gap (1.44 eV on average) thus conferring to these species a relative thermodynamic stability. Now, we examine the reduction of the neutral U V complexes. The EAs are computed as the difference between the TBEs of the neutral U V and anionic U IV species at their optimized geometries. The computed TBEs and EAs, in the gas phase as well as in solution (THF), for all complexes at the ZORA/BP86/TZP level of theory, are given in Table 4; the rows with SO correspond to the values of the TBE and EA taking account of spin-orbit coupling. In the last row, are displayed the experimental half-wave reduction potentials in volts (E 1/2 vs. [Cp 2 Fe] +/0 ) of the neutral uranium(V) complexes, measured in THF) [20].

Table 4 Computed TBEs (eV) and EAs of the U IV /U V redox couple of Cp* 2 U(=N-2,6-i Pr 2 -C 6 H 3 )(X)

Complexes at the TZP Level in the Gas Phase and in THF (in parentheses).

Species

Complex As expected, the computed electron affinities calculated in gas phase are all positive since the TBEs of the anionic species in their triplet state are lower than those of their neutral parents in their doublet state. The fluoride complex, which has the lowest half-wave reduction potential (-1.81 V) exhibits the lowest EA (1.665 eV) and is therefore the most difficult to reduce. On the contrary, its iodide congener exhibits the highest EA (1.921 eV) and should be the easiest to reduce.

1 (F) 2 (Cl)
While the effects of the spin-orbit coupling affect similarly the TBEs of U V and U VI complexes, the corrections brought by the solvent are relatively significant and affect differently the TBE of the U V and U IV species inducing a substantial energy lowering of the order of 1.6 eV for the anionic compounds and 0.35 eV in average for the neutral precursors; this causes an important variation of the EAs. In the same way as for oxidation, we found an excellent linear correlation (R 2 = 0.998) between the computed EAs (calculated in THF at the TZP level including the spin-orbit correction) and the experimental half-wave reduction potentials E 1/2 (Fig. 6). If the solvent effects are not taken into account the above correlation worsens significantly (R 2 = 0.940). The SOMO energies of the anionic species are positive in the gas phase, but become negative in THF, due to the stabilizing effect of the solvent. All the frontier MOs (Figure 7) i.e. the empty LUMO and the singly occupied SOMO and SOMO-1 exhibit a major 5f metallic character with a minor contribution of the Cp* rings, while the contribution of the X and N atoms are zero. In addition, the variation of the SOMO energies of the U IV species as well as the LUMO ones of the U V compounds, follows that of the EAs or of the reduction potentials, the lowest SOMO corresponding to the highest EA. Energies of the U IV -SOMO and U V -LUMO decrease according to the order: I > Br > Cl > F and a good linear correlation is found between the U V LUMO energies and the experimental reduction potentials E 1/2 of the U V /U IV pair (R 2 = 0.98, see SI). A final observation on the redox properties of these imido-halide U(V) uranium complexes is that the energy range between the metal-based oxidation (U(V)/U(VI)) and reduction (U(V)/U(IV)) is found almost the same for all the studied complexes : ∆(TBE redox ) = IE + EA ≈ 7.8 eV. This trend was first observed experimentally by Kiplinger's group [20] who noted that the difference of the half-wave potentials of the U(VI)/U(V) and U(V)/U(IV) redox processes of eight imido-halide uranium complexes was practically constant:

∆E 1/2 = E 1/2 (U VI /U V ) -E 1/2 (U V /U IV ) = 1.51 V.
-2,5 series bring light on some other aspects of the metal-ligands interaction. Despite its well know limitations, MPA permits to describe qualitatively the evolution of charge transfers and bonding interactions occurring in a series of homologous molecular systems. In Table 5, are collected the computed Mulliken metal spin populations and Hirshfeld charges of U, X atoms and Cp* 2 + N-Aryl ligands of the anionic, neutral and cationic Cp* 2 U(=N-2,6i Pr 2 -C 6 H 3 )(X) species. The numbers -1, 0 and +1 indicate respectively the anionic U(IV), the neutral U(V) and the cationic U(VI) species. Metal spin population is calculated as the difference between the total α and β electronic populations of the metal. By net charges of Cp* 2 and N-Aryl, one understands the global charge of the two Cp* rings and the imido-aryl moiety respectively and not only that of the atom connected to uranium. The Mulliken analysis (detailed in SI) indicates that the net charges of the metal are largely smaller than its oxidation state (+6, +5 and +4 for the cationic, neutral and anionic species respectively) due to the ligands to metal donation, the latter effect being highlighted by the weak negative charges carried by the X -, Cp* 2 and Naryl ligands. In particular, the Cp* 2 net charge increases considerably with the oxidation state of the metal; the oxidation processes (U IV U V U VI ) induce a significant transfer of electron density from the Cp* ligands to the metal, the charge carried by these rings, being largely positive (of +1.1 to +1.3) in the cationic species. In addition, for all neutral and anionic complexes, it is worth to note a weak delocalization of the unpaired electrons. The MPA metal spin population is indeed slightly larger than 1 for the neutral 5f 1 U(V) species, larger than 2 for the anionic 5f 2 U(IV). Thus, the DFT computations indicate 2.22 unpaired electrons on central metal for the fluoride complex in solution with minor spin populations on the Cp* 2 and N-Aryl ligands while the spin density is 1.28 for the anionic species. The spin population data for the isolated molecules are practically similar to those computed in solution.

Another quantitative approach of the major electron transfers occurring in molecular systems is provided by Hirshfeld's analysis (HA) [22], which is supposed to give more realistic net charges than MPA, agreeing well with chemical intuition [35]. As already noticed with MPA, the ligand-to-metal donation increases with the uranium oxidation state; this is highlighted by the decrease of the net charges of the ligands. For example, in the fluoride complex, the sum of net charges carried by the three bulky ligands (i.e. the two Cp* rings and the imidoaryl moiety), which is definitely negative (-1.29) in the anionic species, passes to -0.43 in the neutral molecule, and become positive (+0.46) in the cationic species. Thus, the HA reproduces the same tendency than MPA (-1.30, -0.34 and +0.71 respectively with MPA, values in SI), but leads to a relatively lower net charge for the U VI species. On the other hand, HA is more realistic than MPA concerning the net charges carried by the uranium; logically, these uranium charges increase slightly with the oxidation state of the central metal.

Table 6 Hirshfeld charge variation of U, X and N atoms after the U V U IV reduction process calculated for an isolated molecule and in THF (between parentheses) Finally, as showed by the values in Table 6, it is worth noting that the variations of the uranium Hirshfeld charges of the different complexes when passing from the U(V) to the U(IV) complexes correlates very well with the half-wave reduction potentials of these species, the R 2 correlation coefficient being equal to 0.999 (see Fig. 8). Indeed, the differences between the uranium Hirshfeld charges in the U V and U IV species (Table 6) are the following, 0.1022, 0.0836, 0.0778 and 0.0726 respectively for X = F, Cl, Br and I. A rather good correlation (R 2 =0.971) is also obtained between the halide Hirshfeld charge variation and the experimental reduction potential. A similar result was obtained in previous study of L 3 UCl complexes [36].

Ligand/Complex ) ( X q δ ) (U q δ { } Ar N Cp q + * 2 ) ( δ 1. Fluoride -0.

CONCLUSION

The redox behavior of a series of biscyclopentadienyl imido-halide uranium(V) complexes Cp* 2 U(=N-2,6-i Pr 2 -C 6 H 3 )(X) (X= F, Cl, Br and I) have been investigated for the first time, using relativistic ZORA/BP86 computations including spin-orbit coupling, solvent effects being taken into account using the COSMO solvation model. A very good linear correlation (R 2 = 0.99) was found between the computed ionization energies (IE) and the experimental half-wave oxidation potentials E 1/2 ; in addition, as expected, a similar good correlation was found between the computed electron affinities (EA) and the measured electrochemical reduction potentials (R 2 = 0.99). These main results confirm thus the reliability of the ZORA/DFT technique to study the redox processes of uranium complexes. Our study brings to light the importance of spin-orbit coupling and solvent effect which must be taken into account in order to achieve the best agreement between theory and experiment. Moreover, MO diagrams and population analyses permitted to understand the evolution of the computed IE and EA with the nature of the halide X -ligand. Indeed, considering the frontier MOs of the U V species, it can be seen that energetically the highest SOMO and LUMO are those of the fluoride complex, leading to the lowest IE and AE for this species. Finally, the Nalewajski-Mrozek bond order and the Hirshfeld charges analyses permit also to well rationalize the effect of the redox processes on structural properties. Indeed, on one hand, the NM methodology explains well the bond distances variations, while the HA approach leads, on the other hand, to an excellent correlation (R 2 = 0.998) between the Hirshfeld charges variation of uranium atom following the U V /U IV process and the half-wave reduction potentials E 1/2 .

Fig. 1

 1 Fig. 1 The studied Cp* 2 U(=N-2,6-i Pr 2 --C 6 H 3 )(X) halide complexes

  all the mononuclear molecules Cp* 2 U(=N-2,6-i Pr 2 -C 6 H 3 )(X) complexes are found in the familiar pseudo tetrahedral bent sandwich configuration with an usual geometry of the UCp* 2 fragment. The uranium coordination in these complexes features classic bent-metallocene geometry, with the imido-aryl and halide ligands contained within the metallocene wedge.

( 1 .

 1 974 to 1.985 Å) which correlate very well with the experimental values. One also finds the usual values of various angles like 135° for Cp*-U-Cp* and 101 to 107° for N=U-X, with a better agreement for the Cp*-U-X angles (see Supporting Information). In addition, it is noted that the geometry of the Cp* 2 U(=N-2,6-i Pr 2 -C 6 H 3 ) moiety is not influenced by the nature of the X halide ligand; indeed, the U-Cp centroid (2.510 Å), U-C and N-C Ar distances are practically identical for the 4 studied compounds and are very close to the X-ray values; one notices also the almost linear arrangement of the atoms U=N-C Ar (angles 171.6 to 173.3°).

Fig. 2

 2 Fig. 2 Optimized geometries of the U IV /U V /U VI Cp* 2 U(=N-2,6-i Pr 2 -C 6 H 3 )(X) (X=F, Cl) complexes computed at the TZP level (isolated molecule). Hydrogen and carbon atoms of the Cp* rings are hidden for clarity (Geometries for X= Br, I are in SI)

Fig. 3

 3 Fig. 3 Correlation between the computed ionization energies at the ZORA/BP86/TZP level in THF and the experimental half-wave oxidation potentials E 1/2 .

Fig. 4

 4 Fig. 4 Frontier α-MO diagram of the imido-halide U V complexes (scalar ZORA/BP86/TZP computations in THF). The dots connect the SOMO energy levels.

Fig. 5 Frontier

 5 Fig. 5 Frontier MO diagram of the cationic U VI complexes (scalar ZORA/BP86/TZP computations in solution). The dots connect the LUMO energy levels.

Fig. 6

 6 Fig. 6 Correlation between the computed electron affinities at the ZORA/BP86/TZP level in THF and the experimental half-wave potentials E 1/2 for the Halide complexes

Fig. 7

 7 Fig. 7 Frontier α-MO Diagram of the anionic U(IV) species computed at the ZORA/BP86/TZP level in solution. The dots connect the SOMO energy levels.

  (MPA) and Hirshfeld Charge Analysis of the Cp* 2 U(=N-2,6-i Pr 2 -C 6 H 3 )(X)

Fig. 8 Uranium

 8 Fig. 8 Uranium Hirshfeld charge variation following the U V /U IV reduction vs. experimental half-wave reduction potentials (scalar ZORA/BP86/TZP computations in THF)

1 : 2 : 3 : 4 : 5 : 6 : 8 :

 1234568 Optimized coordinates for isolated and solvated molecules SI.Optimized structures of 3 and 4 complexes SI.Nalewajski-Mrozek Bond Order Analysis SI.Total Bonding Energies SI.SOMO Energy vs. Oxidation Potential Correlation SI.HOSpinor of U V vs. Oxidation Potential Correlation SI.7: Frontier MO interaction diagram of the U V complexes in THF; Correlation LumoUV-E1/2 red SI.Mulliken Populations Analysis and Nalewajski-Mrozek Bond Orders

Table 2 Ionization energies (eV) of the U V Cp* 2 U(=N-2,6-i Pr 2 -C 6 H 3 )(X) Complexes at ZORA/BP86/TZP Level in the Gas Phase and in Solution, i.e. in toluene and in THF (in parentheses), and the Experimental Half-Wave Oxidation Potentials E 1/2 (V)

 2

			1	2	3	4
			(F)	(Cl)	(Br)	(I)
		Gas phase	5.507	5.587	5.619	5.659
	TZP	Toluene	4.851	4.941	4.972	5.120
		THF	(4.531)	(4.618)	(4.653)	(4.691)
		Gas phase	5.568	5.657	5.690	5.728
	TZP/SO	Toluene	4.907	5.009	5.046	5.076
		THF	(4.586)	(4.687)	(4.723)	(4.753)
	Exp. E 1/2 (V)	-0.14	0.03	0.07	0.11

Table 5 Mulliken metal spin density and Hirshfeld Charges of Cp* 2 U(=N-2,6-i Pr 2 -C 6 H 3 )(X) Complexes (scalar ZORA/BP86/TZP computations in THF).

 5

			MPA		Hirshfeld Charges
	Complex	Complex
		charge	Metal spin	U	X	[Cp* 2 + N-Aryl]
			density
		-1	2.22	+0.5672	-0.2828	-1.2843
	1 Fluoride	0	1.28	+0.6694	-0.2421	-0.4273
		+1	0.00	+0.7391	-0.1977	+0.4585
		-1	2.25	+0.5349	-0.2856	-1.2490
	2 Chloride	0	1.30	+0.6185	-0.1962	-0.4221
		+1	0.00	+0.6791	-0.1022	+0.4232
		-1	2.26	+0.5363	-0.3014	-1.2347
	3 Bromide	0	1.31	+0.6141	-0.1952	-0.4188
		+1	0.00	+0.6702	-0.0790	+0.4083
		-1	2.22	+0.5288	-0.3261	-1.2029
	4 Iodide	0	1.33	+0.6014	-0.1902	-0.4112
		+1	0.00	+0.6455	-0.0325	+0.3872

(F)

(Cl)

(Br)

(I) U-X

ACKNOWLEDGMENTS

The authors acknowledge the financial support from the Algerian Project CNEPRU n° D00520130051.

Computing facilities were provided by GENCI-IDRIS and GENCI-CINES. The COST CM-1006 action is also acknowledged.

In Table 3, are reported the calculated energies of frontier orbitals, the SOMOs (Singly Occupied Molecular Orbital) of the U V complexes and the LUMOs (Lowest Unoccupied Molecular Orbital) of U VI ones, in the gas phase and in THF, as obtained in the scalar relativistic case, as well as spinors energies when taking into account the spin-orbit coupling. In the gas phase, the SOMO energies of the neutral U V complexes are negative, from -3.413 to -3.651, and these energies undergo roughly the same slight reduction (-0.15 eV) in THF. The iodide complex exhibits the lowest SOMO energy (-3.651 eV) corresponding to the largest IE whereas its fluoride homolog exhibits the highest SOMO (-3.413 eV) and the weakest IE of the considered series. On the other hand, the LUMO energies values of the cationic compounds are definitely negative (below -6.5 eV), making these U VI species able to undergo a reduction process. Nevertheless, the MO energies of the cations are strongly destabilized by the presence of the THF solvent, their energies undergoing a significant increase (2.1 eV approximately). As expected, we observe a rather good linear correlation (R 2 =0.97, see SI) appears between the U V -SOMO energies in solution and the experimental oxidation potentials E 1/2 .

The effect of spin-orbit coupling is more pronounced for the SOMO energies of the neutral U V species (an average reduction of 0.25 eV) than for the LUMO energies of U VI complexes (0.6 eV). Moreover, the spinor energies vary in the same way as the frontier orbitals energies and a good linear correlation is found between the U V HOS energies and the experimental oxidation potentials E 1/2 (SI).

The frontier molecular orbital (FMO) diagrams of both the neutral Cp* 2 U(=N-2,6-i Pr 2 -C 6 H 3 )(X) complexes and the corresponding cations are now considered in order to study the changes undergone by the electronic structures upon oxidation.

One will be particularly interested by the orbitals involved in this electronic loss, namely the SOMO of the neutral complex. Fig. 4 shows the FMO diagram of the neutral U V imido-halide complexes composed of two highest singly occupied orbitals, namely the SOMO and SOMO-1, and the empty LUMO, as computed at the


M Hamelin
email: michel.hamelin@latmos.ipsl.fr

A Le Gall

W Schmidt

F Simões

R Grard

H Laasko

V Ciarletti

K Seidensticker

Simulation of in-flight calibrations and first cometary permittivity measurements by PP-SESAME on Philae

A Permittivity Probe (PP) instrument is now on board the Philae lander of the ROSETTA mission, en route towards comet Churyumov-Gerasimenko. The objective is to deduce the complex permittivity of the comet after touchdown from mutual impedance measurements between two pairs of transmitting and receiving electrodes. The full set of electrodes, also taking advantage of the instruments APX and MUPUS, will be deployed only some time after landing and then nominal operations will start. However, considering all other elements including the lander body as extra electrodes, it is possible to derive the complex permittivity from early measurements when only the landing feet would be deployed. The geometry is less suited due to symmetry constraints but in-flight calibration of the reduced system can be performed before landing in this configuration. A numerical model of the instrument on Philae has been developed for simulating both free flight calibrations and first measurements on the comet. We discuss the results with respect to different crude models of cometary material.

Introduction

The internal structure and dynamics of comets are poorly known. Their surfaces are exposed to fast spinning variations of illumination as well as deep variations with the distance to the sun. Assessment of complex permittivity variability along the orbit is a valuable approach to investigate comet characteristics at shallow depths, and related sublimation and outgasing phenomena. The mutual impedance method to measure the complex permittivity of the ground was first introduced by Wenner for geophysical prospection [START_REF] Wenner | A method of measuring earth resistivity[END_REF], and it was proposed for planetary investigations by Grard using a four-electrode array [START_REF] Grard | A quadrupolar array for measuring the complex permittivity of the ground: application to Earth prospection and planetary exploration[END_REF]. On Philae the probe consists of a system of 5 electrodes, 2 receivers and 3 transmitters, and works in principle as a quadrupolar probe utilizing only two receivers and two transmitters at a given time, as shown in Fig. 1. The 3 feet of the lander hold the 2 receivers and one transmitter, whereas the two other transmitters are hosted by the MUPUS and APX sensors. However, the system of electrodes is strongly influenced by the vicinity of the lander body and other grounded items that work as additionnal electrodes.

The three feet reduced model 2.1 Model

With two receiving electrodes and only one transmitter the usual quadrupole cannot be formed, except if we consider the return current flowing through the body of the lander and other connecting parts. As this ground electrode is geometrically distributed in the main body and several items as harpoons, it is not possible to consider them small or pin points. The PP electronics allows for measuring either the differential voltage between the two receivers or the voltage with respect to the lander potential; then it is possible to use the lander body and connected parts as an extra electrode. The quasi-static approximation can be used because the working frequencies are lower than 10 kHz and, for an angular frequency ω, the currents I and potentials V of the electrodes are linked by the equation:

I= iω [K] V (1)
where

Calibration

The free-flight of Philae after release from ROSETTA provides a unique opportunity to perform in-flight calibration of the reduced system in deployed configuration, except for harpoons and ice screws.. However, their influence is small and can be evaluated. As no calibration of PP in fully deployed configuration was possible before flight, the calibration during the release sequence is really necessary, even in the descent configuration. Due to operational constraints, the calibration will be repeated several times during the descent of Philae to the comet, but at the single frequency of 1 kHz.

In-flight calibration provides corrections to the capacity-influence matrix for the case of a vacuum.

In the case of a non negligible plasma environment, corrections would be evaluated from the MIP-ROSETTA instrument plasma measurements.

Simulation

For simulations of the first measurements after landing we assumed H2O ice with various values of porosity. In this model, Philae is assumed to have landed on a flat, homogeneous substrate. On the comet, the permittivity will be deduced by matching the potential measurements with a grid of simulated values computed with the numerical model. It is shown that these first measurements could provide a first estimate of the ground permittivity. Combined with other experiments such as MUPUS that measures the thermal properties, this approach would provide a first characterization of the cometary material.

Figures

Summary

The success of such early permittivity measurements of the cometary surface is expected if -as foreseenthe medium is not too conductive. On Earth, measurements would be generally difficult, except for low conductivity terrains such as glaciers.

This work defines a strategy for instrument calibration in idealized cases, with a nominal landing on a flat surface. This has to be confirmed by further modelling and laboratory experiments with spare models of the sensors. We also expect that in-flight calibrations of the reduced instrument would improve the data analysis of the nominal functioning with the two standard quadrupolar geometries.

Figure 1 :

 1 Figure 1: Sketch of the five electrodes used by PP.

Figure 2 :

 2 Figure 2: A model for the reduced geometry to calculate the 3D potential distribution.

  [K] is the electrostatic capacity-influence matrix of the electrode array computed with the finite element technique and a more or less representative geometrical model as shown in Fig.2. Currents and

	EPSC Abstracts Vol. 7 EPSC2012-666 2012 European Planetary Science Congress 2012 c Author(s) 2012	EPSC European Planetary Science Congress

potentials measured by the instrument are supposed to be well known, and [K] holds the information about the permittivity environment.