M Trassinelli

D F Anagnostopoulos

G Borchert

⋆⋆ A Dax

J.-P Egger

D Gotta

M Hennebach

Y.-W Liu

B Manil

N Nelms

L M Simons

A Wells

Measurement of the charged pion mass using a low-density target of light atoms

We present a new evaluation of the negatively charged pion mass based on the simultaneous spectroscopy of pionic nitrogen and muonic oxygen transitions using a gaseous target composed by a N 2 /O 2 mixture at 1.4 bar. We present the experimental set-up and the methods for deriving the pion mass value from the spatial separation from the 5g -4 f πN transition line and the 5g -4 f µO transition line used as reference. Moreover, we discuss the importance to use dilute targets in order to minimize the influence of additional spectral lines from the presence of remaining electrons during the radiative emission. The occurrence of possible satellite lines is investigated via hypothesis testing methods using the Bayes factor.

Introduction

The first estimation of the charged pion mass came with its discovery from cosmic rays traces in photographic plates [1,2]. Counting the photographic emulsion grains in the particle trajectory, Powell and his group estimated the pion-to-muon mass ratio to be about 1.5. With the pion production from accelerators [3], the charged pion mass has been measured from the deflection trajectory in a magnetic field [4][5][6] but also from the energy of the gamma-ray produced by the reaction π -+ p → n + γ [7] (see Fig. 1). The first mass measurement employing pionic atoms was performed in 1954 by using 4 f → 3d X-ray emission of several light pionic atoms with the critical absorption edge technique reaching an accuracy of 0.5% [8]. With the increase of intensity of pion beams, the use of crystal spectrometers has become possible. In 1967, the spectroscopic measurement of pionic calcium and titanium reached the relative accuracy of 10 -4 [9]. As shown in Fig. 1, in the ⋆ e-mail: martino.trassinelli@insp.jussieu.fr ⋆⋆ present address: TU Munich, D-85747 Garching, Germany ⋆⋆⋆ e-mail: d.gotta@fz-juelich.de ⋆⋆⋆⋆ present address: DAHER NUCLEAR TECHNOLOGIES GmbH, D-63457 Hanau, Germany † present address: Phys. Depart., National Tsing Hua Univ., Hsinchu 300, Taiwan ‡ present address: Lab. de Physique des Lasers, Université Paris 13, Sorbonne Paris Cité, CNRS, France § present address: ESA-ESTEC, PO Box 299, 2200 AG, Noordwijk, The Netherlands following years the pion mass has been re-measured several times using X-ray spectroscopy of pionic atoms with continuously improved accuracy [10][11][12][13][14][15][16]. In parallel, additional results for the charged pion mass were obtained from the measurement of the muon momentum in the decay π + → µ + ν µ [17][18][19].

With increasing experimental accuracy, in the '80s discrepancies between pionic atoms and pion decay results showed up. The disagreement was due to the ambiguous assumption on the remaining electrons in pionic atoms when produced in solid-state targets. In the πMg experiment of Jeckelmann and collaborators [13,14], where the 4 f -3d transition (25.9 keV) was measured with a transmission X-ray DuMond spectrometer, different assumptions on the K electron population lead to a difference in the pion mass of 16 ppm between the two possible interpretations, called A and B [15]. In particular, solution A is in complete disagreement with evaluations obtained from the pion decay measurement at rest [17][18][19].

To solve this dilemma, in the '90s our collaboration designed and realized a new experiment with a gaseous nitrogen target for having X-rays emitted from a purely hydrogen-like pion-nucleus system. In this experiment, the energy of the 4 keV X rays of the 5g → 4 f line was measured with a reflection spectrometer set up in Johann geometry. The spectrometer was previously developed for light antiprotonic atom spectroscopy [20]. This experiment's limitation was the calibration line, the Cu Kα fluo-Figure 1. Results of for the mass of the charged pion from various experimental methods since its discovery. The world average value "PDG 2014" [START_REF] Olive | Particle Data[END_REF], indicated by the magenta region, is calculated from "Lenz 1998" [16] and solution B of "Jeckelmann 1994" [14]. Other results are taken from Refs. 4-14, 17, 18. rescence radiation, whose large natural width (4 eV) prevents an energy determination at ultimate precision.

The present pion-mass value given by the Particle Data Group (PDG) [START_REF] Olive | Particle Data[END_REF] has an accuracy of 2.5 ppm and is the result of the average of solution B of the measurement of pionic magnesium [15] and the one obtained from pionic nitrogen spectroscopy [16].

The experiment described here resumes the strategy of gas targets, but exploits (i) the high precision of 0.033 ppm for the mass of the positively charged muon being m µ + = (105.6583715 ± 0.0000035) MeV/c 2 [START_REF] Olive | Particle Data[END_REF] and (ii) the unique feature that in πN and µO transition energies almost coincide (Fig. 2). When using a gas mixture, the simultaneous measurement of πN and µO lines becomes possible with the muonic transition serving as an on-line calibration.

The experiment and some aspects of the data analysis have been described in Ref. 22. Here, we present additional aspects. In particular, we discuss in detail the possibility of remaining electrons during the radiative dominated part of the cascade (Sec. 2) applying specific analy- ses and statistical tests to our data (Sec. 5). We present a brief general description of the experiment (Sec. 3) and the detailed formulas used for extracting the pion mass from the line positions (Sec. 4).

Pionic atoms production and atomic cascade

The experiment was performed by using the intense pion beam at the πE5 beam line of the Paul Scherrer Institut. A general overview of the set-up is presented in Fig. 3. The pions, with an initial momentum of 112 MeV/c, are captured and slowed down in the so-called cyclotron trap II.

The cyclotron trap consists of a superconducting split-coil magnet with a field perpendicular to the pion trajectories. In a set of plastic degraders, pions are slowed down in order to be stopped in the gas contained in a target cell at center of the trap. A first version of this trap had been initially developed to efficiently decelerate and trap antiprotons for antiprotonic atoms production [23]. Lenz et. al. [16] used this trap to decelerate and stop pions.

The cyclotron trap II used in the present experiment was specifically designed with a larger gap between the magnet coils. This allows a more efficient trapping of muons produced in the decay of slow pions, which have to be captured in the bottle field of the magnet before being stopped in the target [24]. 1 -3% of the initial pions are stopped in the gas inside the target cell. The stop rate of the decay muons is about 10% of the one of the pions.

For the simultaneous measurement of 5g -4 f transitions from pionic nitrogen and muonic oxygen, comparable count rates are required. This was achieved with a N 2 /O 2 mixture of 10%/90%. The gas was kept at a pressure of 1.4 bar and room temperature.

Pions and muons are captured where the overlap of the wave-functions of the outermost electrons and of the pion or muon is largest. Electronic quantum numbers n e correspond to highly exited states with an initial quantum number n i ∼ n e × 16 (14) for pions (muons), from where a quantum de-excitation cascade starts [25].

For exotic atoms with atomic number Z > 2, the formation and the first steps of the de-excitation proceed via Auger emission and self-ionization of the target atom or molecule. Acting at the femtosecond time scale, this process quickly leads to a high degree of ionization.

Auger emission determines the upper part of the atomic de-excitation cascade where the radiative emission is dominating for lower levels. The Auger transition probability Γ A is proportional to 1/ √ 2∆E + 37.8 eV, where ∆E is the transition energy, favouring ∆n = 1 transitions with the selection rule ∆ℓ = 0, ±1 [26,27]. The radiative transition probability Γ X is proportional to ∆E 3 with selection rules ∆ℓ = ±1 and, hence, maximal ∆n de-excitation steps are preferred. Such a dependence on ∆E efficiently populates |n, ℓ = n -1 circular states allowing subsequently only slow radiative (n, ℓ = n -1) → (n -1, ℓ = n -2) transitions after depletion of the electron shells. Therefore, the atomic cascade duration is essentially determined by the first radiative transitions.

If the target consists of molecules, the initial ionisation causes a repulsive Coulomb force between the atomic cores leading to an acceleration (Coulomb explosion) where the captured pion or muon is attached to one of the atoms. After Coulomb explosion, Auger emission continues until the complete depletion of the electron shells if capture from neighbouring target atoms is avoided.

If a low-density target (≤ 1 -2 bar) is used, electron recapture from external atoms is unlikely because the probability for having a collision with another atom of the target is low even in the presence of Coulomb explosion. This is proven by the appearance of X-ray lines at n ≥ 5, which otherwise would be converted into Auger transitions [26,[28][29][30][31]. In the case of solid targets, however, electron refilling is unavoidable.

The circular transitions (n, ℓ = n -1) → (n -1, ℓ = n -2) are by far the most intense X-ray lines and, therefore, to be used for the measurement. For these transitions, strong interaction effects in the case of pionic atoms are minimal but Coulomb explosion leads to a significant Doppler broadening of the X-ray lines [32]. More details on the atomic cascade in exotic atoms can be found in Refs. [25,[33][34][35][36][37][38].

In our experiment, electron recapture during the cascade is not expected when using a N 2 /O 2 mixture with a pressure of 1.4 bar at room temperature. This hypothesis is explicitly tested in the analysis presented here (Sec. 5).

X-ray spectrometer and data acquisition

The exotic atom transition energies are precisely measured using Bragg diffraction spectroscopy. The crystal spectrometer is set up in Johann geometry [39] and has been specifically designed for high-accuracy X-ray spectroscopy of light exotic atoms [40,41] and was initially used for antiprotonic hydrogen and deuterium spectroscopy at LEAR [20]. The Johann configuration allows the simultaneous measurement of two different energies. The limit of the acceptable energy difference is given by the extension of the X-ray source (target) and/or of the detector on the focus position.

The Bragg crystal was made from a silicon crystal disk cut along the 220 plane of 290 µm thickness with a diameter of 100 mm. The disk is attached to a high-quality polished glass lens defining a spherical segment. Its curvature was determined to be (2981.3 ± 0.3) mm. Spherical bending leads to a partial vertical focusing [42] which increases the count rate.

For the simultaneous detection of the pionic and muonic atoms lines, the spectrometer has been equipped with a large position-sensitive detector, which is composed of an array of charge-coupled devices (CCDs) corresponding to a total sensitive area of about 48×72 mm 2 [43,44].

The Kapton window of the target cell towards the crystal spectrometer had a diameter of 54 mm. For the geometry as given here, the overall efficiency of the crystal set-up is ≈ 5 • 10 -8 . About 85% of the reflected intensity is covered by the detector area.

After 5 weeks of data acquisition, about 9000 events were collected of the (5g -4 f ) transitions for each element. The final spectrum is obtained from the projection of the two-dimensional hit pattern on the CCD onto the axis of dispersion. The result of the projection is presented in Fig. 4.

The accuracy of the pion mass measurement essentially depends on the precision of the spectral lines position determination. This requires both the knowledge of the response function of the spectrometer F spectr. and the control of the effects that the atomic cascade can induce to the line shape via Doppler broadening. F spectr. depends on the characteristic crystal rocking curve but also to aberrations due to the spherical bending of the crystal and the positioning of X-ray source and detector. The imaging properties of the spectrometer were calculated by means of a Simultaneously measured spectra of 5g-4 f transitions in muonic oxygen (energy calibration) and pionic nitrogen. The parallel transitions 5 f -4d and possible additional transitions from the presence of one remaining electron in the K shell are indicated ("sat. pos."). For µO, the larger width of the 5g -4 f transition is due to the non-resolved fine structure splitting, which is resolved for the 5 f -4d transition.

Monte-Carlo simulation which has been validated by a series of experiments with highly charged ions where natural line widths are negligibly small and high X-ray intensities are available [41,[45][46][47][48].

Concerning the atomic cascade processes, the dominant effect on the line width for a molecular target is Coulomb explosion. The Doppler broadening and a possible additional broadening due to imperfections of the diffraction crystal are determined from the analysis of a dedicated measurement optimized for pion stops, where in total 60000 events were accumulated in the πN 5g -4 f transition. From this analysis, we found kinetic energies up to 146 +6 -7 eV, which corresponds to molecule fragments having about 3.3 elementary charges in agreement with previous results [32]. More details of this procedure can be found in Refs. [49,50].

Pion mass evaluation

The pion mass is deduced from the measurement of the 5g -4 f πN transition energy using the 5g -4 f µO transition as reference. The benefits of the choice of this reference are multiple: (i) the reference energy can be calculated very precisely due to the high accuracy of the muon mass value [START_REF] Olive | Particle Data[END_REF] and (ii) the two lines have almost coinciding energies, i. e. they can be simultaneously measured in the same order of reflection, which reduces drastically the effect of many systematic corrections. We present here in detail the formulas for deducing the pion mass from the position difference between the spectral line of a pionic atom transition with unknown energy E B and a reference spectral line with energy E A (in the same diffraction order). The basic formula is the Bragg law that relates the X-ray energies to diffraction angles Θ:

E A = n hc 2d sin Θ A , (1)
where h and c are the Planck and speed of light constant, d is the lattice spacing between crystal planes of the considered reflection direction with order n (an integer number) and Θ A is the Bragg angle.

The energy E B is related to E A by

E B = E A 1 cos ∆Θ AB -cot Θ A sin ∆Θ AB , (2)
where ∆Θ AB = Θ A -Θ B is the difference between the two Bragg angles, where a small corrections is applied because of the slightly different index of refraction. ∆Θ AB is deduced from the spectral line positions x A and x B on the detector plane (see Fig. 5), which is positioned at a distance D from the crystal. We have

∆Θ AB = arctan x B -x c D -arctan x A -x c D , (3)
where x c is the position corresponding to the center of the detector. From Eq. ( 2) we see that cot Θ A is the only quantity where the values of hc and d play a role according to Eq. ( 1).

The pion mass m is deduced from the relation of the transition energy

E B E B = µ πN c 2 (Zα) 2 2        1 n 2 f - 1 n 2 i        + O (Zα) 4 (4)
to the reduced mass of the pion-nuclear system

µ πN = m π 1 + m π M , (5)
with M being the nuclear mass of nitrogen, n i and n f the quantum number of the initial and final state of the transition and α the fine structure constant. The relationship between E B and µ πN contains corrections, not shown explicitly here, that include relativistic effects (additional terms due to the Klein-Gordon equation in our cases), quantum electrodynamics effects (vacuum polarization, self-energy, . . . ), recoil corrections, etc. and are summarized here by the term O (Zα) 4 .

For the limited range of pion-mass values, we can assume a linear dependency between E B and µ πN . The proportionality factor is calculated by the MCDF program [START_REF] Desclaux | Computational Chemistry[END_REF][START_REF] Santos | [END_REF][53] considering the charged pion mass value from the Particle Data Group m PDG π [START_REF] Olive | Particle Data[END_REF] and the corresponding computed transition energy E PDG B . In this approximation, the value of m π is given by

m = µ PDG πN E B E PDG B 1 - µ PDG πN M E B E PDG B . (6)
From the line positions and the evaluation of the systematic effects and uncertainties, a pion mass value of (139.57077 ± 0.00018) MeV/c 2 (±1.3 ppm) is determined. Additional details of the evaluation and of the estimation of the systematic uncertainties are given in Ref. 22. In the next section we specifically discuss the effect of possibly remaining electrons.

Determination of remaining electrons

The possible presence of remaining electron(s) in the exotic atom may induce an important systematic energy shift of the X-ray energies and, consequently, of the pion mass. One (or two) remaining electron(s) in the K shell in pionic nitrogen can generate satellite lines having energies 0.45 eV (0.81 eV) lower than the main transition 5g -4 f [START_REF] Desclaux | Computational Chemistry[END_REF][START_REF] Santos | [END_REF]. Such weak satellite lines cannot be resolved from the main transitions, in particular, as they are expected to be of very low intensity (see Fig. 4).

As discussed in Sec. 2, for light pionic atoms production in a gaseous target of moderate density, the possibility that electrons are present is small when X-ray emission starts. However, it cannot be excluded beforehand that small fractions of pions or muons may arrive at the 5g level by ∆n ≫ 1 transitions from low angular momentum states immediately after capture.

To estimate the probability for the occurrence of satellite lines we use the evaluation of the Bayes factor [START_REF] Jeffreys | Theory of Probability[END_REF][START_REF] Sivia | Data analysis: a Bayesian tutorial[END_REF][START_REF] Kass | [END_REF][57] relative to two hypothesis: the presence or not of satellite lines. The first hypothesis is associated to the model M 0 of the spectra without additional satellite lines. In the model M 1 associated to the second hypothesis, we consider the presence of additional satellite lines with fixed positions with respect to the main transition but keeping the intensity as a free parameter (two lines in the case of muonic oxygen, one per each fine structure main component).

For each model we sample the likelihood function for different values of parameters to evaluate the Bayesian evidence [START_REF] Sivia | Data analysis: a Bayesian tutorial[END_REF][START_REF] Jaynes | Probability Theory: The Logic of Science[END_REF][START_REF] Trotta | [END_REF], which corresponds to the integral over the parameter space of the likelihood function times the prior probability distributions of the parameters. The Bayes factor B 01 , the ratio of the Bayesian evidence of the two models, is then calculated as well as the relative probability of the two models. The evidence is calculated using an homemade code based on the nested sampling algorithm developed by John Skilling in 2004 [START_REF] Sivia | Data analysis: a Bayesian tutorial[END_REF]60] (see also Refs. 49, 61-63 for more details on the calculation method).

An important feature of the evidence calculation is, that contrary to maximum likelihood and minimum χ 2 methods which provide for each parameter only the most probable value and the standard deviation, a probability distribution is established for each model parameter (as well as joint probability distributions).

For the calculation of B 01 , two data sets were used: the high-statistics spectra of πN and the low-statistics spectra of µO (shown in Fig. 4). The results are summarized by Figs. 6,7 and Table 1. Due to the low statistics, the results from muonic oxygen cannot be used as test against one of the two models considering the associated uncertainty, because with ln B 01 being too close to zero the probability for the two models is similar. For the high-statistics pionic nitrogen spectra, the Bayes factor is significantly different to the unity and the relative probability of the two models can be reliably calculated. The value ln B 01 = 6.6 indicates a decisive support for the M 0 hypothesis for any Bayes factor scale considered ("decisive" for Jeffreys scale [START_REF] Jeffreys | Theory of Probability[END_REF], "very strong" for the Kass scale [START_REF] Kass | [END_REF] or "strong" for the Gordon-Trotta scale, equivalent to a p-value of about 10 -5 for M 1 [57]). Model M 0 and M 1 relative probabilities are 99.98% and 0.02%, respectively. Though being small, the effect of such a non-zero probability for M 1 on the pion mass can be evaluated.

When the model M 1 is considered, for both sets a satellite amplitude of about 1% of the main line (see Fig. 6) is found. As expected and clearly visible in Fig. 7, for πN the satellite amplitude is strongly correlated to the main line position. We found a shift of the main line of (δx) 1 = 0.08 pixels with respect to the case where satel- lite lines are not taken into account. This is equivalent to about 1 ppm of the pion mass. This shift cannot be considered in total but has to be weighted by the probability of the two different models using the values of Table 1 and where (δx) 0 = 0. The expected shift of the main line is then δx = 0 × P 0 + (δx) 1 × P 1 = 1.8 × 10 -5 pixels (7) corresponding to a systematic uncertainty of +0 -0.0002 ppm. When the Bayes factor accuracy is taken into account (Table 1) and the worst case is considered, the probability of M 0 drops to P min 0 = 99.86% and the systematic uncertainty increases to +0 -0.0014 ppm which is still completely negligi-ble with respect to the statistical error and other systematic effects.

Conclusions

In conclusion, we present a new measurement of the negatively charged pion mass based on Bragg spectroscopy of pionic nitrogen and muonic oxygen using a gaseous target. The use of the low-density target with light atoms prevents recapture of electrons from neighbouring molecules during the atomic cascade after the formation of the exotic atom. Consequently, the X rays emitted in the last steps of the cascade stem from a purely hydrogen-like system without systematic effects from unresolved satellite lines due to remaining electrons. From our high-statistics pionic nitrogen measurement, the probability of presence of satellite peaks has been found to be about 0.02% confirming the hypothesis on the absence of electrons. Such a contribution introduces a negligible systematic uncertainty on the pion mass of less than one part per billion.

Figure 2 .

 2 Figure 2. Comparison of the intermediate parts of the cascade for pionic nitrogen and muonic oxygen.

Figure 3 .

 3 Figure 3. Experimental set-up with at the πE5 beam line of the Paul Scherrer Institut (Villigen, Switzerland). The pions enter the cyclotron trap from top perpendicular to its magnetic field.

Figure 4 .

 4 Figure 4. Simultaneously measured spectra of 5g-4 f transitions in muonic oxygen (energy calibration) and pionic nitrogen. The parallel transitions 5 f -4d and possible additional transitions from the presence of one remaining electron in the K shell are indicated ("sat. pos."). For µO, the larger width of the 5g -4 f transition is due to the non-resolved fine structure splitting, which is resolved for the 5 f -4d transition.

Figure 5 .

 5 Figure 5. Scheme of the X-ray detector position with respect to the Rowland circle (the focus location of the spectrometer) and two diffracted lines A and B with different energy E B > E A . The (small) displacements of the two lines from their individual focal conditions is taken into account in the calculation of the spectrometer response by means of a Monte-Carlo ray tracing calculation.

Figure 6 .

 6 Figure 6. Probability distribution of the amplitudes relative to the main line intensity of the possible satellite line due to the presence of one remaining electron in the K shell.

Figure 7 .

 7 Figure 7. Joint probability distribution of the relative satellite amplitude and the position of the main line 5g → 4 f in pionic nitrogen. The accumulation on the diagonal shows the correlation between satellite intensity and main line position. Assuming no satellite line, the position of the main line for this set of data is 872.58 ± 0.04 pixels.

Table 1 .

 1 Bayes factor (in logarithmic scale) and the average P 0 of the model M 0 from the analysis of the different spectra. We also present the possible range of P 0 tanking into account the uncertainty of ln B 01 .

	Spectrum	ln B 01	P 0	P min 0	P max 0
	high-stat. πN	6.6 ± 1.8 99.98% 99.86%	100%
	low-stat. µO	-0.3 ± 0.4 42.52% 32.70% 52.98%

We are grateful to N. Dolfus, H. Labus, B. Leoni and K.-P. Wieder for solving numerous technical problems. We thank the PSI staff for providing excellent beam conditions and appreciate the support by the Carl Zeiss AG, Oberkochen, Germany, which fabricated the Bragg crystals. We thank Prof. Dr. E. Förster and his collaborators at the University of Jena, and A. Freund and his group at ESRF, for the help in characterising the crystal material as well as A. Blechmann for a careful study of the CCD performance. We are indebted to PSI for supporting the stay during the run periods (D. F. A.). This work is part of the PhD thesis of B. M. (Université Pierre et Marie Curie, 2001), N. N. (University of Leicester, 2002) and M. T. (Université Pierre et Marie Curie, 2005).


Nathalie Mukolobwiez

Arnaud Chiffaudel

François Daviaud

Supercritical Eckhaus Instability for Surface-Tension-Driven Hydrothermal Waves

Keywords: numbers: 47.20. -k, 47.27. -i, 47.35. + i, 47.54. + r

We study the nonlinear dynamics of hydrothermal waves produced by a surface-tension-driven convective flow in a long and thin annular channel heated from the side. Above onset, the supercritical traveling wave pattern undergoes a secondary instability: a supercritical Eckhaus instability. This leads to a small-wave-number phase-modulated nonlinear mode, and shows the first experimental evidence of a nonlinearly saturated phase instability mode for traveling wave patterns. At higher forcing level, this secondary pattern is subject to a tertiary instability. This mode is an amplitude mode characterized by traveling hole patterns, i.e., space-time defects that change the wave number.

Understanding the transition from order to spatiotemporal chaos in extended nonlinear systems remains an important problem. Physicists first started to study systems unstable to stationary spatial patterns such as, e.g., Rayleigh-Bénard convection [1] and, later on, oscillatory instabilities and nonlinear traveling-wave patterns. For 1D waves, two closely related mechanisms-Eckhaus [START_REF] Eckhaus | Studies in Nonlinear Stability Theory[END_REF] and Benjamin-Feir (BF) [START_REF] Benjamin | [END_REF]-could be invoked. They have been intensively searched and studied in several physical systems [4][5][6][7][8]. This research has also been strongly stimulated by numerical experiments on the complex Ginzburg-Landau equation (CGLE) [9,10]. Remarkable predictions concern the existence of nonlinear phase regimes, presumably only in a tiny region close to the BF stability limit of the CGLE phase diagram: In this region, BF unstable patterns may show phase turbulence [9,10], whereas the Eckhaus unstable pattern could bifurcate on a supercritical way [4,[START_REF] Fauve | Instabilities and Nonequilibrium Structures[END_REF] leading to nonlinearly saturated phase modulations. In this Letter, an annular wave pattern is produced by a supercritical oscillatory instability. It behaves as a 1D traveling wave with periodic boundary conditions and should be described by a cubic CGLE. At higher order parameter, this pattern undergoes a nonlinearly saturated phase instability-a supercritical Eckhaus instability-in the form of a persistent smallwave-number traveling modulation. This is, as far as we know, the first report of a nonlinear phase pattern on the way to spatiotemporal chaos for a large system, and confirms the above predictions.

Experimental setup.-An annular convection channel (Fig. 1) with a glass bottom and vertical copper walls (cf. Ref. [START_REF] Daviaud | [END_REF] for rectangular geometry) is filled with a thin layer of silicon oil of viscosity n 0.65 cS and Prandtl number P 10. The fluid surface is free. The channel is 1 cm wide and its mean radius is L 8 cm. The perimeter is 50 cm which corresponds to an aspect ratio G 2pL͞h 300 for h 1.7 mm. The outer copper wall is cooled by a thermoregulated fluid circulation at 293 K while the inner copper block is heated electrically. Thermocouples allow accurate measurements of the temperature difference DT across the channel. This difference is typically established with a 620 mK stability. Convective patterns are observed through the glass bottom by shadowgraphy. Images are digitized and processed numerically as spatiotemporal diagrams of 512 data points extracted from a mean circle and plotted along time.

Basic flow.-The convective flow created by this horizontal thermal gradient was described by Daviaud and Vince [START_REF] Daviaud | [END_REF]. The thermal gradient across the cell induces a surface tension gradient on the free surface of the fluid. Because of the Marangoni effect, this gradient generates a surface flow towards the cold side, with a bottom recirculation: The basic flow is a long annular roll. Above a given threshold DT 0 , and for low fluid thickness h , 2.8 mm, this flow becomes unstable with respect to horizontal traveling waves (TW) which propagate along the channel (Fig. 1), i.e., the roll axis. This behavior characterizing hydrothermal waves, was predicted by Smith and Davis [13], detected in several experiments [START_REF] Daviaud | [END_REF]14], and recently studied numerically in conditions close to our experimental situation by Mercier and Normand [15].

In this Letter, we report the observation of hydrothermal traveling waves near the minimum of the stability curve (see Fig. 3 of Ref. [START_REF] Daviaud | [END_REF]), i.e., for h ͑1.7 6 0.1͒ mm. At this point, the first bifurcation to hydrothermal waves occurs at DT 0 3.5 K. This value is locally almost independent of h. This stability of the onset simplifies the experimental task because the fluid height decreases typically by 0.01 mm 3 h 21 , due to evaporation and capillarity. We note also that the thickness of the fluid layers varies along the channel owing to a slight nonhorizontality of the cell and even more negligible curvature of the bottom plate. For the present set of experiments, h was kept in a 60.04 mm range along the channel. This small imperfection induces a small (typically 4%) modulation of the local wave number of the pattern around its mean value. This modulation can be easily subtracted during the data processing phase.

Just above onset DT 0 , the waves invade the cell leading generally, as reported for the rectangular cell, to a classical source/sink pattern. However, after a relatively long transient, the defects generally annihilate, leading to a single homogeneous plane wave rotating along the annulus in either direction. The wave number selected by the source for a source/sink system is typically k 0 61͞L, whereas defectless wave patterns select between 50 and 57 wavelengths in the cell. This first bifurcation is supercritical, as shown in Fig. 2: The amplitude of the wave pattern behaves like ͑DT 2 DT 0 ͒ 1͞2 . Thus this system is a very good prototype for the study of supercritical homogeneous traveling wave patterns with periodic boundary conditions in a large aspect ratio container. It should be compared to binary-fluid convection for a subcritical TW bifurcation [5,6], to oscillatory instability studies in argon with small aspect ratio [4] or nonhomogeneous patterns [16], and to oscillatory rotating convection [7].

Notation and data processing.-Following the usual framework of nonlinear patterns, we will try to extract from the spatiotemporal images the quantities which could be directly written in a nonlinear model equation such as CGLE. For that purpose, the real shadowgraphic data u r ͑x, t͒ related to the thermal field can be written as

u r ͑x, t͒ u c ͑x, t͒ 1 c.c 1 . . . , (1)
u c ͑x, t͒ A͑X, T͒ exp͓i͑v 0 t 2 k 0 x͔͒ , (2)
A͑X, T ͒ R͑X, T͒ exp͓iF͑X, T ͔͒ , (3)
where u c is called the complexified signal, A͑X, T ͒ is a complex amplitude depending on the slow variables X and T , R͑X, T ͒ is its modulus, c.c. stands for the complex conjugate, and the dots for harmonics. In order to be demodulated, real data are complexified by the three following operations [17]: a Fourier transform of u r , a wide bandpass filtering around the positive fundamental in Fourier space, and then the reverse Fourier transform which creates the complexified signal u c . The complex spatiotemporal image of u c will be viewed as follows:

(i) Its modulus.-R͑X, T͒, i.e., the local amplitude of the wave.

(ii) The two gradients of its phase w.-≠ t w v 0 1 ≠ T F͑X, T ͒ and ≠ x w k 0 1 ≠ X F͑X, T ͒, i.e., the local frequency and local wave number of the pattern.

Results.-The first perturbations of the TW field observed above DT 0 consists of the appearance of small phase modulations of the pattern. The system appears to be very sensitive to such modulations: Once e DT ͞DT 0 2 1 is positive, due to noise, we observe very small nonhomogeneities of the local wave number traveling slowly along the cell. The velocity of these structures is typically 6 times slower than the phase velocity. Their amplitude is close to our noise detection level, i.e., of order 0.1͞L for the wave-number modulation ≠ X F. To get more information we forced these modulations by dropping a few drops of oil into the channel. This perturbation is done extremely fast with respect to the thermal time.

The relaxation of this forced perturbation is illustrated in Fig. 3 which shows the local wave number ≠ X F along time for the first 2000 s. After two rounds, the spatial harmonics of the initial pulse-shaped pattern have relaxed. The remaining pattern is a sine modulation at the size of the box. This corresponds to a compression-dilatation mode of the wavelength of the basic TW mode, which will be called carrier, by analogy with the radio-frequency domain. However, this phase modulation is also slowly damped. To measure this damping rate, we demodulate this ≠ X F signal. After a spatial bandpass filtering at about K M 1͞L-the size of the box-we extract the amplitude q͑T ͒ of the local wave-number modulation:

≠ X F͑X, T͒ q͑T ͒ exp͓i͑V M T 2 K M X͔͒ 1 . . . . (4)
This amplitude is plotted in Fig. 4. It is exponentially damped with a characteristic time t 4500 s. This explains why the very small spontaneous modulations, due to experimental noise, are regularly observed over several hundred seconds. In fact, this very low rate is the signature of a secondary bifurcation for slightly higher temperature gradient.

Once e is set greater than e 1 0.66 we observe the wave-number modulation at K M 1͞L to globally grow, saturate and persist for several hours (Fig. 5). This transition has no hysteresis. It's a supercritical Hopf bifurcation. The bifurcated modulation amplitude is quite small compared to the forced modulation: typically 2-3 times the noise level. This secondary instability is a phase instability because the local amplitude R of the carrier wave is almost unaffected by the modulation. It is also an Eckhaus instability because it affects the FIG. 4. Amplitude q͑T ͒ of the wave-number modulation for its first mode-of the size of the box-for 20 000 s after perturbation of the carrier TW. The noise level 0.12͞L is achieved after 12 000 s, for this very strong perturbation of order 2͞L, i.e., 4% of the mean wave number k 0 55͞L. The first 2048 s corresponds to the data of Fig. 3. longitudinal variable. In fact, by increasing the constraint above e 2 0.9 we recover the usual Eckhaus behavior.

Figure 6 represents the evolution of a k 0 52͞L modulated carrier pattern when e is increased above e 2 . We observe the spontaneous growth of high wavenumber modulations leading to space-time defects or amplitude holes [16], where the amplitude goes to zero and the phase jumps by p (see caption for details). After six phase jumps, the mean wave number of the carrier TW has decreased to k 0 46͞L and the patterns relax towards a nonmodulated state. Note that this example corresponds to a strong increase of the control parameter e. For smaller steps above e 2 , the transient state lasts much longer: Many more traveling amplitude holes are observed, whereas only five or six of them reach the zero-amplitude level and change the carrier pattern wave number. This recalls the slow chaotic state [5] for the Eckhaus transition of subcritical TW. Further increase of the thermal constraint produces a new Eckhaus modulation followed again by a loss of one or two wavelengths. The scenario reproduces itself several times until, for k 0 of order 40͞L, weak turbulence arises, consisting mainly of amplitude hole defects.

The Eckhaus instability has been carefully studied for TW patterns in Rayleigh-Bénard convection [4][5][6][7] or wakes of bluff bodies [8]. In those experiments, this instability always appears to be subcritical, even if not explicitly stated, with exponential growth of phase modulations up to the point where an amplitude hole breaks the underlying carrier TW. No stable nonlinear phase mode was ever reported. In the present experiment, the Eckhaus instability is encountered for increasing control parameter which means that the stability domain, unlike the classical parabola for Rayleigh-Bénard convection, is somewhat slanted. Nevertheless, the existence of stable nonlinear solutions for the phase equation-the modulated pattern -has been predicted analytically by Fauve The K M 1͞L modulation of the initial pattern is thus not visible. Suddenly, near t 250 s, higher spatial frequencies start growing and generate six traveling wave-number peaks. The velocity of these structures is the same as the phase velocity of the previous modulation. The wave-number peaks correspond to growing phase jumps: once they reach p, with the local amplitude simultaneously attaining zero (not shown) and the local wave number changing sign, they create six spacetime defects which each annihilate one wavelength of the TW pattern.

[11] and calculated by Janiaud et al. [4]. The latter also observed supercritical pulses in numerical CGLE in the vicinity of the BF instability curve, a feature which is very similar to the one reported here. We did not try to measure the CGLE coefficients relevant to our experiment but, owing to the very long relaxation time observed just above the first bifurcation threshold (Fig. 4), we estimate the BF line to be close. Finally, note that our observations are limited to a small window in the DT range: Once the phase modulation amplitude grows, it triggers an amplitude instability and its train of correlated spacetime defects. This behaves like the growing long-lived transients for subcritical Eckhaus [4][5][6][7][8], where the K M 1͞L mode grows until a defect is created. However, in our case, the defects are nucleated by the growth of higher wave-number modes, e.g., harmonics number 6 or 7. We may thus suspect that, whereas the low wave-number modes are nonlinearly damped, the spatial harmonics which arise further above Eckhaus onset, are not. These defects are traveling hole patterns whose evolution is qualitatively similar to the Nozaki-Bekki hole solution of CGLE [16,[START_REF] Bekki | [END_REF] or homoclinic holes [19]. These holes precede "amplitude" or "defect" turbulence for higher DT , which is currently being studied.

In conclusion, we wish to emphasize the qualitative similarity of this experimental observation with numerical studies of the phase diagram for BF unstable regimes of supercritical CGLE (Fig. 3 of Ref. [9]). As CGLE presents a continuous transition from pure waves to phase turbulence and then to defect turbulence, our experiment transits from pure waves to nonlinearly saturated modulation-a pure nonlinear phase mode-and then to an amplitude regime consisting of space-time defects.

We wish to thank S. Fauve, B. Janiaud, J. Lega, and O. Dauchot for fruitful discussions, and C. Gasquet for technical assistance. Laboratoire de Physique Statistique is Laboratoire associé to the Centre National de la Recherche Scientifique.

FIG. 1 .

 1 FIG.1. Schematics of the annular channel with sketch of traveling hydrothermal waves and acquisition circle (---).

FIG. 2 .

 2 FIG. 2. Critical behavior of the first instability of the channel flow: the onset of traveling waves. We have plotted the squared amplitude jAj 2 for various values of the control parameter DT , and also various wave patterns. Squares correspond to homogeneous left or right TW, whereas circles represents source/sink patterns. Data for different wave-number patterns are collected together and are coherent with threshold DT 0 3.50 K and critical exponent 1͞2.

FIG. 3 .

 3 FIG. 3. Evolution of the perturbation ( just before t 0) of a stable TW pattern for DT 4.0 K or e 0.15. The carrier TW pattern consists of 55 wavelengths ͑k 0 55͞L͒ traveling at frequency v 0 ͞2p 0.51 Hz. This space-time picture with periodic spatial boundary conditions represents the local wave number ≠ X F of the pattern: dark (clear) regions show low (high) wave numbers, i.e., dilatations (compressions) of the TW carrier. For noise reduction this image has been filtered in space: It shows only the first four modes. The period of the first mode, at wave number K M 1͞L, is t M 2p͞V M 485 s, i.e., typically 300 times the period of the carrier.

FIG. 5 .

 5 FIG. 5. Space-time image of the local wave number of the k 0 L 51 and v 0 ͞2p 0.61 Hz hydrothermal wave for DT 6.6 K or e 0.88. This wave-number modulation is nonlinearly saturated and propagates steadily along time. According to Eq. (4), we measure qL 0.20 6 0.04, 2p͞V M 332 s, and K M L 1. Higher spatial frequencies are filtered and the contrast is enhanced 8 times with respect to Fig.3.

0031-9007͞98͞80(21)͞4661(4)$15.00 © 1998 The American Physical Society