G Baaklini

G Gbabode

S Clevers

Philippe Négrier

Denise Mondieig

G Coquerel
email: gerard.coquerel@univrouen.fr

Trimorphism of N-methylurea: crystal structures, phase transitions and thermodynamic stabilities

Introduction

N-Methylurea (NMU), molecular structure shown in Fig. 1, is a hygroscopic non-linear optical material. 1 Many co-crystals and salts of NMU are reported in the literature. [2][3][4][5] Up to this study, the only known crystal structure of NMU is a stable form that was first proposed in 1933, 6 the structure was solved in 1976 by Huiszoon and Tiemessen 7 and lastly refined at 8 K by Chęcińska et al. 8 NMU crystallizes in the orthorhombic system with the chiral (therefore non-centrosymmetric) space group P2 1 2 1 2 1 . Several studies investigated the enthalpies and entropies of fusion and solid-solid transition of urea, and its derivatives, including NMU where only the enthalpy of fusion of NMU has been reported. [9][10][11] Hence, one can notice that the thermal programs applied in these studies consisted in first cooling down the NMU sample then heating up to melting.

In the present work, our experiments provide evidence of two metastable polymorphic forms of NMU obtained by meltcrystallization at different quenching temperatures. Thereafter, we report the crystal structures and stability relationships between the three polymorphs of NMU by X-ray diffraction and DSC analyses.

Materials and methods

Materials

NMU was acquired from Sigma Aldrich and used without further purification.

Methods

X-Ray powder diffraction (XRPD).

Melt crystallization experiments were conducted in capillaries. Commercial NMU was introduced in 0.5 mm diameter Lindemann glass capillaries and heated up to fusion, then directly quenched at a predetermined temperature using a cryostat fed with liquid nitrogen (Oxford Cryosystems). The capillaries were mounted on an Inel CPS 120 diffractometer that works in a typical Debye-Scherrer transmission geometry (CuKα 1 radiation -λ = 0.15406 nmas incident beam) where diffracted rays are simultaneously collected on a 120°position sensitive curved counter by gas ionization (argon + C 2 H 6 ). The glass capillaries are rotated around their axis all along the experiment in order to minimize preferential orientation of crystallites. External calibration to convert the measured 4096 channels to 2θ degrees was applied using cubic Na 2 Ca 2 Al 2 F 4 (high angle calibration) 12 mixed with silver behenate (low angle calibration). 13 Measurements were performed on the quenched samples at different temperatures during the whole day or overnight in order to obtain reflections with exploitable intensities.

2.2.2 DSC analyses. Two types of DSC were used: A heat flux DSC that served to determine the melting points and the enthalpy of fusion of Form I and Form II. The thermal analyses were performed using a DSC 214 Polyma (Netzsch). Samples were weighed in 25 μL aluminum pans and a heating rate of 5 K min 1 was applied, under nitrogen atmosphere. Data treatment was performed using Netzsch Proteus® software v6.1.

A power-compensated DSC was also used in order to reach low temperatures down to 150 °C.

The thermal analyses were conducted in a powercompensated Perkin Elmer Pyris Diamond DSC equipped with a liquid nitrogen cooling system. Solid samples (mass ∼ 5 mg) were placed in a 30 μL pierced aluminum crucible. The atmosphere of the analyses was regulated by nitrogen flux (20 mL min 1 ), and heat runs were conducted at 5 K min 1 . Data treatment was performed using Pyris-Thermal Analysis Software.

Results

X-Ray analyses of the melt-crystallized NMU

The capillary containing the commercial NMU was heated up to the melting temperature and directly quenched at 173 °C and 50 °C. X-ray analyses were performed at these temperatures. The comparison between the XRPD patterns of NMU obtained after melt quenching and the stable form reveals the crystallization of new forms at different quenching temperatures (Fig. 2). Indeed, the XRPD pattern recorded at 50 °C exhibits new characteristic peaks different from those observed for the stable form. Hence, this new phase will be assigned to as Form II (CCDC number: 1464782) whereas the stable form will be assigned to as Form I. The XRPD pattern recorded at 173 °C also reveals new characteristic peaks dif-ferent from Form II and Form I. Hence, this new phase will be assigned to as Form III (CCDC number: 1464783). The 2θ positions of characteristic reflections of Forms I, II and III are summarized in Table 1.

Temperature-resolved (TR)-X-ray analyses on NMU forms. TR-X-ray analyses were performed between 163 °C and 113 °C with a heating rate of 1 °C min 1 (Fig. 3) between two subsequent measurements (a time lag of 4 minutes at the desired temperature was kept before each measurement). Each analysis lasted for 30 minutes at the defined temperature.

Upon heating from 163 °C, characteristic peaks of Form III start decreasing in intensity between 143 °C and 138 °C. These peaks completely disappear at 123 °C where the transition seems to be completed. On the other hand, characteristic peaks of Form II appear in the same range of temperature and are the only ones observed at 123 °C.

The transition between Form II and Form III was found to be reversible upon heating and cooling (data not shown).

3.2 Crystal structures of Form I, II and III Fig. 4 and 5 display the comparison between the X-ray diffraction patterns calculated from the crystal structure solved from X-ray powder data and the measured ones for Form II and III, respectively. The agreement is very good for Form II (Rwp = 7.3%) as shown by the almost flat difference curve in Fig. 4 and a little less for Form III (Rwp = 10.6%) as shown in Fig. 5. The differences might be due to preferential orientation effects, which might be more severe at very low temperatures. Crystallographic data determined for Forms II and III are given in Table 2 together with those for Form I for the sake of comparison. Forms II and III both crystallize in a P2 1 /c space group, thus suggesting a different packing of NMU molecules in those forms compared to Form I. The cell parameters of Form II are quite different from those of Form I, yet a similar unit cell volume is preserved. Indeed, the a parameter of Form II is around half that of Form I while its c parameter is almost twice that of Form I. The cell parameters of Form III are close to those of Form II but with a much lower value of c (reduction from 12.9 Å for Form II to 10.5 Å for Form III). We will come back to this point later on in this manuscript.

Despite, the differences in unit cell dimensions and space groups between the two new polymorphs and Form I, the packing of the three forms appears to display a common feature based on the establishment of "V-shaped" (bifurcated) intermolecular hydrogen bonds (Table 3) that gives rise to molecular chains (Fig. 6) that spread along the a-axis for the three crystal structures. The hydrogen bond distances, as well as the angles, are relatively close in the three forms.

As shown in Fig. 6a, the molecules within the chains of Form I present different orientations, the methyl moieties are on both sides of the central axis of the molecular chain. Indeed, NMU molecules inside a chain are related by a 2 1 screw axis parallel to the a-axis in Form I (P2 1 2 1 2 1 space group); while in the case of Forms II and III, they are related by a glide plane parallel to the (a, c) plane (P2 1 /c space group). Furthermore, the chains have a zig-zag shape in Form I when viewed from the side (Fig. 6a, bottom) while they are fairly flat in Forms II and III (Fig. 6b andc, bottom).

These chains are connected with each other through another type of hydrogen bond (Table 3) leading to the final arrangement in a three dimensional network for Form I (Fig. 7) and in layers for Forms II (Fig. 8) and III (Fig. 9).

In the case of Form II and Form III, the differences are quite small. They are due to a slight shift between the packing of the layers, while the differences between Form I and the other two forms are obvious: the different orientation of the methyl moieties along the molecular chains, and the three dimensional hydrogen bonding network that arises from this alternating orientation.

To emphasize the latter for Form I, the molecular chains are depicted in different colors in Fig. 7 (intra-chain = dashed pink line and inter-chain = dashed cyan line).

The availability of hydrogen bonds involved in the connection of the molecular chains in Form I is increased due to the different orientation of the methyl moiety and the "brickwall packing" of NMU between adjacent molecular chains. Therefore, the density and the orientation of the hydrogen bonding network gives rise to an efficient three dimensional packing (Fig. 7).

Neighboring molecular chains in Forms II are also connected through a third type of hydrogen bond (the same color code as in Fig. 7 is kept for inter-chain (represented in dashed cyan lines) and intra-chain (represented in dashed pink lines)). The molecular chains in Form II are oriented in an anti-parallel way and generate layers in the (a, b) plane that pack along the c-axis through inter-chain hydrogen bonds on one side and methyl-methyl contacts on the other (Fig. 8).

A similar packing is observed for Form III (see Fig. S1 in ESI †). The major difference between Forms II and III lies in the packing arrangement of the layers along the c-axis. Hence, it can be noticed that the consecutive layers in Form III are shifted so that the steric hindrance generated by the methyl moieties is limited (Fig. 9b-d) while in Form II, the methyl groups are almost oriented face to face (Fig. 9a-c).

Stability relationship between the three forms of NMU

X-Ray results were completed by thermal analysis measurements. Thus, the stability behavior and the thermodynamic relationships between Forms I, II and III were investigated.

The melt quenching process was reproduced inside the crucible as follows: starting from the commercial batch of Form I, the sample was heated until melting at 5 K min 1 . The liquid melt was then rapidly cooled down to 50 °C or to 150 °C with a cooling rate of 20 K min 1 (fastest cooling rate accessible by the DSC apparatus). The sample was heated again at 5 K min 1 up to 130 °C.

Data obtained from the quenching experiment at 50 °C. During the first heating of the commercial batch, the DSC thermogram shows an endothermic peak at 98 °C (ΔH I F = 188.5 J g 1 ) that was assigned to the fusion of Form I. During the second heating, the DSC thermogram shows an endothermic peak at 95.2 °C (ΔH II F = 166.6 J g 1 ) that was assigned to the fusion of Form II (Fig. 10). This strongly indicates that Forms I and II have a monotropic relationship (Form I being the most stable form) because Form I has an enthalpy of fusion and temperature of fusion higher than that of Form II (this is in agreement with the Burger and Ramberger rules 14 ). Besides this, an X-ray diffraction follow-up of Form II revealed a spontaneous conversion of Form II to Form I within a few days of storage at room temperature. The kinetics of this return to thermodynamic equilibrium is greatly influenced by moisture adsorbed by the compound.

Data obtained from the quenching experiment at 150 °C. The same thermal program was applied for these quenching experiments at 150 °C (Fig. 11).

Different thermal events are detected by DSC analysis upon cooling from the melt and the second heating that starts at 150 °C.

The following observations can be noted: Upon cooling from the melt, Form II is crystallized upon solidification of the melt. This crystallization is illustrated by a sharp exothermic peak with an onset at ∼51 °C. Three additional exothermic peaks are observed just after this recrystallization. These events are likely to be the successive solidifi-cations of scattered drops of the melt inside the crucible. Upon further cooling, another exothermic peak with an onset at circa 134 °C is detected. This peak would correspond to the reversible solid-solid transition Form II → Form III with hysteresis (Fig. 11a).

Upon heating, an endothermic peak with an onset at 118 °C, corresponding to solid-solid transition Form III → Form II, is detected (Fig. 11b). It can be inferred from the hysteresis phenomenon and the detection of a (weak) enthalpy that this transition is of a first order. The mechanism of phase transition between Form II and Form III should then proceed through nucleation and growth.

Upon further heating, two endothermic peaks close to each other are observed: the first one corresponding to the fusion of Form II and the second one corresponding to the fusion of Form I (Fig. 11b).

A rough approximation of the enthalpy of fusion of Form III can be calculated as follows:

The comparison between the enthalpies of fusion of the three forms shows that:

ΔH I F > ΔH III F > ΔH II F .
In order to verify that Form I was not formed during the quenching procedure, the same protocol of quenching of DSC at 50 °C and 150 °C was reproduced in second harmonic generation (SHG). No SHG signal was detected at these temperatures. This indicated that Form I is not present at these temperatures since this optical non-linear sensitive (ppm levels) technique gives rise to a signal only if the compound crystallizes in a non-centrosymmetric space group 15 (which is the case of Form I crystallizing in P2 1 2 1 2 1 and not the case for Form II and III that crystallize in P2 1 /c). All investigations presented above prove that Form I is the thermodynamically stable form whatever the temperature. Form II and III are monotropically related to Form I. A reversible solid-solid transition between Form II and III was found at 118 °C using DSC. The discrepancy between the solidsolid transition temperatures observed by DSC and X-ray diffraction is most likely due to kinetics. Indeed, it is observed by X-ray diffraction that the complete transition of Form II → Form III occurs in a range of more than 15 °C (between 143 °C and 128 °C, see Fig. 3), thus indicating the slow kinetics of the solid-solid transition. Moreover, the X-ray diffraction analyses are considered as static measurements (each X-ray analysis lasted 30 minutes at a given temperature with a lag time of 4 minutes and a heating rate of 1 °C per minute between two temperatures), while the DSC analyses are considered as dynamic measurements. Given the slow kinetics of the solid-solid transition, it is difficult to compare the temperatures obtained by both techniques. Anyway, we chose to consider the temperature obtained by DSC as the most accurate evaluation of the actual transition temperature since the precise regulation of temperature of a capillary is more difficult than that of a DSC crucible (see Fig. 11).

Mechanism of transition from Form II to Form III

It was interesting to suggest a mechanism of transition between Form II and Form III, especially since this transition occurs at low temperatures.

As shown previously, Form II and Form III present common features: they crystallize in the same space group (P2 1 /c), they have the same asymmetric unit (Z′ = 1) and they present the same packing of NMU molecules characterized by molecular chains arranged in layers through hydrogen bonds and methyl-methyl contacts.

However, some differences still exist between the two structures, particularly at the methyl-methyl interface. Indeed, the distance between adjacent layers connected through methyl-methyl contacts was found to be 3.18 Å for Form II (at 50 °C) and 2.82 Å for Form III (at 173 °C), see Fig. 9 c andd. This shortening is consistent with the reduction of the c parameter from Form II to Form III, shown in section 3.2, and whose evolution as a function of temperature is displayed in Fig. 12 together with that of the unit cell volume. It appears that, upon cooling Form II, the crystal lattice is monotonically compressed until a certain point (the II → III transition temperature) at which a discontinuous decrease of the unit cell volume occurs, indicating a transformation into Form III.

Interestingly, the II → III transition (or vice versa) is found to be fully reversible at moderate to high heating/cooling rates despite the poor diffusion at low temperature.

We hypothesize that, upon cooling Form II, the contraction of the lattice could generate steric hindrance on the methyl moieties. This issue is overcome by a cooperative slide of the layers along the (b, c) plane, as shown in Fig. 8 and9, giving rise to the stabilization of Form III. Hence, the methyl moieties are shifted in Form III, contrary to Form II, where methyl moieties are aligned (Fig. 9). Our hypothesis is also valid the other way around, that is, upon heating Form III.

In the light of all these results, we can suggest that the transition between Form II and Form III is of first order as shown by the discontinuous changes in enthalpy (Fig. 11) and unit cell volume (Fig. 12) observed at the phase transition temperature. It should then proceed through nucleation and growth of crystals of Form II (resp. Form III) at the expense of crystals of Form III (resp. Form II) upon cooling (resp. heating). The significant temperature hysteresis observed between II → III and III → II transitions (Fig. 11), suggesting an activated process, is consistent with this postulate. The full reversibility experimentally observed might be related to the small amplitude of the movements necessary to switch from one form to another, which might be achievable even at low values of thermal motion.

Conclusion

Quenching experiments on NMU at two different temperatures allowed us to discover two polymorphs of NMU (designated as Form II and Form III). As far as we know, these have not been reported before. The crystal structure of the two forms were solved from X-ray diffraction powder data and it appears that both forms crystallize in the centrosymmetric space group P2 1 /c, contrary to the known commercially available NMU (Form I) crystallizing in the non-centrosymmetric space group P2 1 2 1 2 1 .

Comparisons between the crystal structures of Forms II and III reveal extensive similarities between them, including the hydrogen bonding network. Besides this, the "V-shaped" hydrogen bonds between NMU molecules observed in Form I also constitute the main building units at the origin of the packing arrangement of Forms II and III.

A mechanism is suggested to explain the reversible conversion at 118 °C and the transition is first order. Indeed, there is a discontinuity change in entropy and enthalpy with a small, but significant, hysteresis effect. The transformation could take place by nucleation and growth featuring limited shear movements and a self-templating effect. Form I is found to be the most stable phase (Forms II and III are monotropically related to Form I) as shown by DSC and X-ray diffraction measurements. This result is clearly consistent with the crystal structure of Form I that provides the most efficient packing arrangement of NMU molecules through a complete three dimensional hydrogen bonding network. However, upon fast crystallization (such as melt-quenching), this complex arrangement might not have time to settle and alternative structures with less efficient packing, such as Forms II and III, are kinetically favored.

The present article is another evidence that even for a small rigid molecule such as NMU with highly directional interactions, a rich polymorphic behavior can be revealed by suitably modifying the conditions of crystallization of this compound. Indeed, the three reported crystal structures of NMU can be viewed as the results of alternative crystal packing arrangements of the same building units (NMU molecules connected through V-shaped hydrogen bonds) or "supramolecular synthons" in the concept of crystal engineering. 16

Fig. 2

 2 Fig. 2 XRPD patterns of the NMU stable form at 25 °C (blue) and after melt quenching at -50 °C (red) and -173 °C (black).

Fig. 3

 3 Fig. 3 TR X ray diffraction patterns obtained upon heating Form III, represented in the [12 23°] 2θ range. Characteristic peaks of Form II are marked with arrows while those of Form III are marked with an "*".

Fig. 4

 4 Fig. 4 Experimental (red) and calculated (black) XRPD patterns for the crystal structure of Form II at -50 °C. Systematic existences calculated from the monoclinic unit cell (green ticks) and difference between calculated and experimental XRPD patterns (blue). The inset represents a zoom on the higher angle 2θ domain [45 63°].

Fig. 5

 5 Fig. 5 Experimental (red) and calculated (black) XRPD patterns for the crystal structure of Form III at -173 °C. Systematic existences calculated from the monoclinic unit cell (green ticks) and difference between calculated and experimental XRPD patterns (blue). The inset represents a zoom on the higher angle 2θ domain [40 63°].

Fig. 6

 6 Fig. 6 Molecular chains built from hydrogen bonds (dashed pink lines) established between consecutive molecules along the a axis in a) Form I, b) Form II, and c) Form III.

Fig. 7

 7 Fig. 7 Molecular chains of Form I along a) the a axis and b) the c axis.

Fig. 8

 8 Fig. 8 Molecular chains of Form II along a) the c axis and b) the a axis showing methyl moieties face to face from adjacent layers.

Fig. 9

 9 Fig. 9 Comparison between the crystal structures of Form II represented along a) the a axis and c) the b axis and Form III repre sented along b) the a axis and d) the b axis illustrating a shift of methyl moieties in Form III. Carbon atoms colored in yellow correspond to the lower layer. The red and green segments show the alignment of methyl groups.

Fig. 10

 10 Fig. 10 DSC curves of the commercial NMU upon first (upper curve) and second (lower curve) heating.

Fig. 11

 11 Fig. 11 DSC curves of NMU upon a) cooling from the melt (cooling rate: 20 K min -1 ) followed by b) heating up to the melt (heating rate: 5 K min -1 ).

Fig. 12

 12 Fig. 12 Evolution of the a) c parameter b) unit cell volume of NMU as a function of temperature (from -163 to -113 °C). The different values have been determined from the X ray diffraction patterns shown in Fig. 3. The dashed blue vertical line delimits the respective stability do mains of Forms II and III established from this data.

Table 1

 1 Characteristic peaks of Form I, II and III

	16.6	13.7	14.6
	18.1	15.1	16.9
	20.9	19.2	20.8
	24.5	23.6	22.5
	25.5	24.7	25.1
	27.8	27.4	28.3

Form I (25 °C)

Form II (-50 °C) Form III (-173 °C)

2θ (°) 2 θ (°) 2 θ (°)

Table 2

 2 Crystal data for Forms I, II and III

					(III) at
	Form	(I) at 25 °C	( II) at -50	°C	-173	°C
	Crystal system,	Orthorhombic,	Monoclinic,	Monoclinic,
	space group	P2 1 2 1 2 1	P2 1 /c		P2 1 /c
	a (Å)	8.4767(6)	4.6147(6)		4.6604(3)
	b (Å)	6.9809(5)	6.5781(12)		7.4079(7)
	c (Å)	6.9227(6)	12.9004IJ16)	10.4566(6)
	β (°)	90	90.418(5)		92.007(4)
	Z	4	4		4
	Density (g cm 3 ) 1.201(1)	1.256(1)		1.364(1)
	Volume (Å 3 )	410(1)	392(1)		361(1)

Table 3

 3 Hydrogen acceptor bond distances (O⋯H) and angles of Forms I, II and III

	Form	I	II	III
	O 1 ⋯H 1 N 1 Distance (Å) 2.08 2.08(1) 2.26(1) Between molecular
	Angle (°)	167 168(1) 162(1)	chains
	O 1 ⋯H 2 N 1 Distance (Å) 2.22 2.17(1) 2.17(1) Establishment of
	Angle (°)	151 148(1) 147(1)	'V shaped'
	O 1 ⋯H 6 N 2 Distance (Å) 2.16 2.19(1) 2.12(1)	intermolecular
	Angle (°)	159 152(1) 155(1)	hydrogen bonds

Acknowledgements

This research was supported by the grant of Région Haute-Normandie. We would like to thank Prof. Yves Geerts for hosting S. C and for giving access to the power-compensated DSC. We also thank Dr. Valérie Dupray, Dr. Morgane Sanselme and Lina Yuan for their contribution.


Pierre David

Alexandre Escarguel

Yann Camenen

Romain Baude

) A Escarguel

Characterisation of coherent rotating modes in a magnetised plasma column using a mono-sensor tomography diagnostic

Keywords: numbers: 42.30.Wb, 52.35.Py, 52.70.Kz, 52.75.Xx

I. INTRODUCTION

Over the past few years, several linear devices have been investigating the physics of coherent rotating structures, helped with recent developments in both technology and theory. Plasmas with cross field configurations are present in many domains, including Hall effect thrusters, ion sources and magnetic fusion. These configurations can lead to various types of nonlinear instabilities among which coherent rotating structures have been observed frequently [1][2][3][4][5] . In this paper, we present a new way of monitoring the magnetized plasma of the Mistral device 6 sketched in figure 1. An Argon plasma created by a thermoionic discharge, with a typical pressure P ≃ 10 -2 Pa and electron density n e ≃ 10 15 m -3 , and temperature T e ≃ 4 eV is studied. The device is separated in two zones: the source chamber, and the linear plasma chamber. The source chamber consists of 32 heated tungsten filaments emitting electrons with an energy of a few tens of electron-volts (primary electrons), with randomized trajectories through a magnetic cusp configuration. The primary electrons are injected in the second chamber through an 80 mm diameter diaphragm. As a consequence, the central plasma column is slightly electronegative. The linear chamber, where we study the plasma, is a 1.2 m long and 0.4 m wide cylinder with a 16 mT axial magnetic field. The plasma is limited by a grid at each end of the column (called collector and separating grid in the figure 1) and is surrounded by a 20 cm wide cylinder, each of them can be polarized to control electrons confinement.

Global modes with azimuthal numbers m = 1 or m = 2 are routinely observed on Mistral [7][8][9][10][11] and can be kept steady for several hours. These modes have been interpreted [START_REF] Jaeger | Étude théorique et expérimentale des instabilités basses fréquences dans un plasma en champs magnétique et éléctrique croisés[END_REF] as Simon-Hoh [START_REF] Simon | [END_REF]14 instabilities with null axial numbers (k = 0). A simplified fluid model based on the azimuthal momentum equation in the presence of a a) pierre.david@univ-amu.fr b) alexandre.escarguel@univ-amu.fr radial electric field qualitatively reproduces the dependences of the mode rotation frequency on the pressure and the magnetic field 15 . Primary electrons play an important role in the apparition of those highly non-linear modes by creating a radial electric field between the negative plasma and the wall 8,9 . Consequently, the radial electric field is expected to be able to expel energetic electrons in the limiter's shadow that will ionise a plasma arm. Therefore the boundary conditions play a key role in the instability existence by balancing the axial and radial confinement in order to incite the electrons to be radially ejected from the column.

Most experimental works on linear devices have been performed with electrostatic probes assuming that the mode structure is constant over a period of rotation. Despite the fact that tomography diagnostics are seldom used in low temperature plasma experimental studies, due to price and dimensions, this technique can bring new insight on plasma physics. Tomographic analysis is non-invasive, temporally resolved acquisition is easily achievable and no hypothesis is required regarding the plasma shape and motion. The experimental set-up and the associated numerical tomographic inversion are presented in section II, before we analyse the results in section III. They are finally compared to 2D reconstructions of the plasma using a Langmuir probe in section IV.

II. MONO-SENSOR TOMOGRAPHY SET-UP

Taking advantage of the rotating mode long life span (up to 2-3 hours-limitation imposed by the source chamber increasing temperature), tomography is performed using conditional sampling with a single sensor moved to different positions (figure 2). The acquisition at each position is synchronized with the plasma rotation through the trigger probe (polarised at V > V p ) signal. The trigger occurs when an excess of current is detected, meaning that the probe is seeing the plasma rotating arm at a specific position. The sensor is an amplified photodiode with a 10 7 V/W gain, giving a sufficient sensitivity and a good signal/noise ratio. We separate the sensor from the plasma using an optical fibre in order to prevent any magnetic perturbation and to have a higher mobility of the optomechanical system.

The acquisition of signals for tomography with a single sensor requires its displacement to different calibrated positions. In order to limit associated errors, we only used a single movement, giving three possible chords setups. The first is a translation of the collimated optical fibre leading to parallel chords. The second is a translation of the bare fibre behind a pinhole, giving fan distributed chords. Finally, the third option is to reproduce the same fan distribution with a rotation of the collimated fibre. The fan distribution is known to lessen the effect of noise by providing more viewing angles 16 . However, the pinhole reduces the available luminosity by a factor 25, greatly decreasing the signal/noise ratio, and looking at the whole plasma with the rotative system would require the fibre holder to be placed through the vacuum chamber window. The parallel configuration has therefore been preferred. The plasma facing end of the fibre is collimated with a lens to a conical line of sight from 8 (at the lens) to 15 mm (at the opposite wall) wide.

The number of positions (tomography's lines of sight) is chosen so that the viewing cones do not intersect their neighbours before the wall, yielding 2 × 16 lines of sight through the optical apertures. The plasma's emissivity is recorded at each position during about 2 ms (about 10 mode rotations). For each position, the measurement is repeated 16 times and then averaged to increase statistics.

The numerical tomographic inversion is based on a finite-element scheme. The cross-section of interest is divided into N pixels whose emissivity E = (e j ) j∈ [1..N] contributes to the 32 integrated measured signals S = (s i ) i∈ [1..32] . E and S are related by the transfer matrix T = (t ij ) depending on the geometry of the system and the response of the detector. By inverting the system T.E = S, the local emissivity can be retrieved from the line integrated measurements. The transfer matrix is determined by calculating and discretising the power P received by the sensor from each plasma element d 3 r:

P = V chord Λ e(r, hν)E(hν) Ω lens (r) 4π d(hν) d 3 r
in which V chord is the volume of the chord, supposed here to be conical, Λ is the sensor spectral range, and Ω lens (r) is the solid angle of the lens seen from the plasma element d 3 r. In the integrand, e and E are the emissivity of the element d 3 r and the sensor spectral response respectively, with the three dimensional position vector r and the energy variable hν. For an optically thin plasma and lines of sight defined by a finite sensor behind a simple aperture 17,18 , each element t ij is given by the length l ij of the i th chord in the j th pixel multiplied by a calibration constant A: T = A.(l ij ). However, when a lens is used for the collimation, some simplifications cannot be made, and the transfer matrix elements become

t ij = l ij .I ij
where I is a function of the distance z between the lens and the pixel, depending on the lens size and focal length, the sensor (here the optical fibre) size, and the distance between the sensor and the lens. Figure 3 shows the variation of I(z) for several distances φ between the lens and the sensor, in the experimental configuration used here. With variation of only 0.5 mm ( 5 % of the focal length), there is an important variation of I(z), which is also different for either side of the focal point. In order to ensure which function had to be used for the transfer matrix, the optical fibre was placed at φ = f -ε f , with ε f = 0.3 mm. This way potential errors on the placement would not change the slope of I(z) for z in our range of interest (200 to 400 mm), so without an absolute calibration the result would remain unchanged.

To solve this large, ill-defined and underdetermined system, the Tikhonov regularisation is used, so solutions minimising

1 2 (T.E -S) 2 + αR,
with α a positive weighting parameter and R a regularising functional, are sought for. Several inversion methods were tested on arbitrary signals and on ghost images, having in mind future analyses on non-coherent modes using an upgraded tomography set-up being installed on Mistral. Among zero, first and second order, minimum Fisher information and SVD regularisations 19 , the second order spatial derivative proved to have the most consistent results and a low sensitivity to noise in our case. Figure 4 shows a ghost image (left), its discretisation (top-right) and the tomography result associated (bottom-right). The average error between each of their pixels is between 10% and 15% for different ghost images with up to 15% random noise.

In order to perform the tomographic inversion, the following hypotheses are made: (H1) The chords are cone-shaped, defined in cylindrical coordinates (r, θ, z) by

{r ∈ [0, r max (z)], θ ∈ [0, 2π], z ∈ [0, z max ]}
with z the distance to the lens, z max the distance between the lens and the wall, and r max (z) the radius of the line of sight at position z.

(H2) In any given chord, the emissivity is constant in a (r, θ) disk: e(r, hν) ≃ e(z, hν).

(H3) The solid angle of the lens seen from any point of those disks is constant: Ω lens (r) ≃ Ω lens (z). (H4) The emissivity is constant inside a pixel.

(H5) The emissivity is zero where there is no pixel.

(H6) Pixels are only linked by the minimisation of the second order spatial derivative, present in R.

Some comments on those hypotheses are in order. (H1) only assumes no irregularity in the circular symmetry of the optical system. (H2) supposes that the chords' spread is small enough compared to the pixels size. If the characteristic sizes of the lens (radius and focal length) and its line of sight's radius are much smaller than the distance at which the plasma is seen, (H3) is respected. (H4) requires to have small enough pixels, but the number of pixels (and consequently their size) is limited by the number of measurements plus the regularisation method. Finally, the combination of (H2), (H3) and (H4) points out the need for an accurately collimated system. For the experimental conditions considered in this paper, the plasma spectrum is largely dominated by neutral argon emission lines 9 . With the low plasma densities in Mistral (< 10 15 m 3 ), the coronal model can be used [START_REF] Mc Whirter | Plasma diagnostic techniques[END_REF] . In these conditions, the intensity of the neutral emission lines is proportional to the thermal electrons' density n e,th and the primary electrons' density n e,p , so the calculated local emissivity E is a linear combination of n e,th and n e,p : E = a.n e,th + b.n e,p , with a and b functions of the energy of the thermal and primary electron respectively and the excitation cross sections of Argon by collision with electrons.

III. EXPERIMENTAL TOMOGRAPHY RESULTS

In this section, we will use the system described previously to study the behaviour of the coherent mode during its rotation.

The time evolution of the raw measurements at every positions is shown in figure 5, with the positions number 1 to 16 from the top of the plasma and 17 to 32 from the side, as displayed in figure 2. The sinusoidal oscillation of the mode is clearly seen, with a phase shift of +π/4 between the top and the lateral measurements indicating a anticlockwise rotation (looking at the plasma from the left side on figure 1). The mode frequency is f ≃ 5.8 kHz. An intensity asymmetry between the rising edge and the falling edge of the oscillation, i.e. when the sensor sees the plasma arm is in front of and behind the central plasma respectively, is noticeable, especially for the lateral aperture (positions number 17 to 32). This asymmetry is due to the geometrical factor I discussed in the previous section. The plasma being between 200 and 400 mm from the lens, the decrease in I exactly compensates for the asymmetry of the signal and the different asymmetries between the top and lateral positions come from the top measurements being made a couple centimetres further from the plasma, where I(z) is flatter.

Tomographic inversion is applied at each time step to reconstruct the evolution of the local plasma emissivity. To increase readability, we separate the emissivity into the temporal average and fluctuations component:

E = E t + δE,
and show in 6-(a) E t normalised to the value at the plasma center, and in 6-(b) the emissivity variations relatively to the time average. The frames are separated by 45.6 µs, for a total of about 80% of a whole period. The time evolution of the emissivity profile visually confirms that the mode remains constant during its rotation. In order to compile the spatial evolution during the whole acquisition into a single figure, the azimuthal emissivity profiles are extracted at several radii and phase shifted to have their maxima aligned for all time steps, and the average over all time steps is calculated and displayed in figure 7. The information about the time evolution of the coherent mode is contained in the error bars, representing the standard deviation. The figure 7 shows the small standard deviation of the azimuthal shape of the mode, confirming that the mode remains constant during its rotation. Additionally, the same dis- play method can be applied to ghost images whose time evolution is simulated by a numerical rotation, in which case the same error bars appear, indicating that those variations mainly come from numerical noise.

IV. COMPARISON BETWEEN TOMOGRAPHY AND LANGMUIR PROBE MEASUREMENTS

In addition to tomography, a radially movable cylindrical Langmuir probe has been used to characterise the coherent rotating structures. The probe acquisitions were triggered using a second probe polarized above the plasma potential, as in the previous section. There are two main downsides to this diagnostic: it is intrusive, and a radially movable probe can a priori only give information on the time evolution at different radii. Yet, reconstructing the azimuthal density profile using a delayed trigger is possible under the hypothesis of the plasma uniformity during its rotation, which was validated by the tomography results of the figure 6. Two-dimensional probe measurements (azimutal and radial) were therefore performed. The electron density and temperature, and the plasma potential, respectively n e , T e and V p , were computed from the probe measurements. However, the magnetic field and the primary electrons lead to a highly anisotropic and bi-Maxwellian plasma. Therefore, we used the Druyvestein formula [START_REF] Druyvesteyn | [END_REF] to reconstruct the electron energy distribution function f from the second derivative of the I(V ) curve (current versus polarization). Densities and temperatures then come from the first and second moments of f respectively. This method has the advantages of being independent of the probe shape (as long as it is convex) and being weakly sensitive to the details of the electron velocities distribution 22 .

The reconstructed section of the electron density and plasma potential is shown in figure 8. The time delay between two azimuthal positions is 30 µs and the plasma rotation period is 270 µs, hence 9 azimuthal measurements for each of the 5 radial positions. The plasma potential and density measurements also display both radial and azimuthal variations in the central column. The plasma potential shifts toward the direction in front of the arm, creating a local electric field inside the limiter in addition to the one at its boundary. Figure 9 shows the radial variation of the density azimuthal average n e θ and its associated variations δn e for both tomography and probe measurements, knowing that under the uniformity hypothesis we have n e θ ≃ n e t . The density variation δn e (r) is taken along the axis passing through the maximum density. Both diagnostics reveal a peaked plasma at the center and a positive and negative shift on each side of the center. However, the tomographic measurements display a narrower averaged density and a lower, but more symmetrical, radial shift than the probe equivalent. Such a difference may be explained by the emissivity being much more sensitive to n e,p than the total density (measured by the electrostatic probe), since n e,th ≫ n e,p . Also, primary electrons have a much higher velocity in the parallel direction, so the characteristic times of their perpendicular transport is less important than the parallel transport one, especially compared to the thermal electrons. In order to quantify the contribution of each electron population to the emissivity, the radiative theory would need to be detailed, or the tomography diagnostic would need to be coupled to a spectrometer.

V. CONCLUSION

In this work, we show that mono-sensor tomography gives the possibility to reconstruct the local emissivity of the plasma with a simple (regarding both configuration and calibration), low cost and low space demanding diagnostic, especially compared to a full scale tomographic instrument. Contrary to probe diagnostics, it has the advantage to be non intrusive. However, its use is limited to the study of periodic or reproducible events.

It was here used to characterise the time evolution of the emissivity inside the core plasma during coherent rotating modes, demonstrating its steady rotation. A single Langmuir probe could then be used to reconstruct a circular section of the plasma, giving more information on its density and temperature.

Preliminary intensified pictures taken outside of the rotating regime show potential evidence of small scale turbulence which could be at the origin of the instability when the external parameters (mainly boundary conditions and pressure) meet some specific requirements. A full scale tomographic diagnostic is under implementation to study the growth of the rotating modes.

FIG. 1 .

 1 FIG. 1. The Mistral linear plasma device. The blue underlined elements can be individually polarized, controlling the longitudinal (collector and separating grid) and the radial (inner cylinder) boundary conditions of the plasma.

FIG. 2 .

 2 FIG. 2. Single sensor tomography set-up (mid-plane section of the column).

FIG. 3 .

 3 FIG. 3. Variation of the coefficient I(z) weighting the transfer matrix, for different positions of the sensor (here the optical fibre) φ regarding the lens' focal length f . Here f = 10.9 mm.

FIG. 4 .

 4 FIG.4. Ghost plasma image (left) and its discretisation (top right) compared to its tomographic reconstruction (bottom right). On the ghost image the pixels used in tomography are shown in white, the lines of sight for the artificial measurements arein green, and the red circle represents the diaphragm.

1 FIG. 5 .

 15 FIG. 5. Raw time evolution (horizontal axis) of the signal from each position (vertical axis) during 1 ms. See figure 2 for the position numbering. The signal is normalized to the highest value of the whole acquisition.

2 FIG. 6 .

 26 FIG. 6. Tomographic inversion results: (a) temporal average emissivity over the whole acquisition normalized to the central point of the plasma. (b) Emissivity variation at four different times of a single period. The black circle represents the limiter. The spatial scale is in millimetres.

FIG. 7 .

 7 FIG. 7. Averaged emissivity over the whole rotation of the tomographic reconstruction of the experimental measurements as a function of the azimuthal angle at different radii. The errorbars represent the standard deviation during the whole rotation of the plasma.

8 BFIG. 8 .

 88 FIG.8. Two-dimentional probe measurements of the plasma potential Vp and electron density ne. Each area contains the value measured at its center. The red arrow shows the rotation direction of the plasma (counter clockwise). The black circle represents the limiter and the outer circle (visible at each corner) is the polarised wall.

FIG. 9 .

 9 FIG. 9. Radial evolution of the averaged density (top) and density perturbation through the density maximum (bottom) from probe measurements (red) and tomography (green). Values are normalized to the maximum averaged density. Vertical lines show the central region of the plasma.

ACKNOWLEDGMENTS

The authors would like to thank to L. Couëdel for several practical ideas and useful discussions.

This work has been carried out thanks to the support of the A * MIDEX project (n o ANR-11-IDEX-0001-02) funded by the "Investissements d'Avenir" French Government program, managed by the French National Research Agency (ANR).