Showing papers in "Journal De Physique Lettres in 1981"

Spin-density wave ground state in the one-dimensional conductor (TMTSF) 2PF6: microscopic evidence from 77Se and 1H NMR experiments

Abstract: The vanishing of the 77Se nuclear resonance signal and the broadening by internal magnetic fields of the proton resonance line establish the existence of spin-density waves in the low temperature semiconducting state of (TMTSF) 2PF6. The thermally activated collective mode of the commensurate spin-density wave contributes significantly to the proton spin-lattice relaxation. It is also suggested that the very small amplitude of the spin-density wave is due to the coexistence of two diverging channels in the conducting state : the SDW and Superconductivity channels.

Ring polymers in solution : topological effects

Abstract: The effect of topological constraints on the properties of ring polymers in solution are studied. When the rings are short and rigid, the effects can easily be understood and a simple result is given here. When the rings are long and flexible, the situation is complex and a more subtle analysis is needed. Fortunately recent mathematical studies concerning the linking numbers of two curves lead to a significant result. This information is used to argue that the topological constraints produce essentially an increase of the local excluded volume interaction ; this topological effect could therefore be taken into account within the framework of current theories.

Intermittent behaviour in the Belousov-Zhabotinsky reaction

Abstract: 2014 One of the possible routes to turbulence is the intermittent transition. Below the onset of turbulence regular oscillations exist, and above the onset seemingly regular oscillations are interrupted randomly by large amplitude bursts. This behaviour shows up in the Belousov-Zhabotinsky reaction. Moreover an analysis of the measurements indicates that a so-called « type 1 » transition takes place in this case. Tome 42 No 13 ler JUILLET 1981 LE JOURNAL DE PHYSIQUE-LETTRES J. Physique LETTRES 42 (1981) L-271L-273 1 er 1UILLET 1981, Classification Physics Abstracts 82.20M 82.40 82.60 The possibility of turbulence in chemical kinetics, as suggested by Ruelle [1], has been verified recently [2, 3] for the Belousov-Zhabotinsky (B-Z) reaction. We report here the discovery of an intermittent behaviour in this reaction. Such a behaviour has been seen during the transition to turbulence in thermoconvection experiments [4] and in studies of ordinary equations [5]. In our experiment, the B-Z reaction takes place in an open well stirred tank reactor. Due to stirring, no spatial pattern can develop and the concentrations of the chemical species evolve according to the non linear equations of chemical kinetics, as derived from the mass action law ; this deterministic behaviour is sufficient to produce chaos. The experimental conditions are : reacting volume 28 ml, temperature 39.6 ~C, concentration of reagents before reaction in mol. 1 1 (standard analytical reagent grade without further purification) : A peristaltic pump feeds the reactor at constant adjustable rate. The chemical reaction in the reactor is monitored by the optical density at 340 nm. As this wavelength is absorbed [6] by the Ce4+ ion only, the signal is a sort of « pure quantity » and has been preferred to the redox potential that depends on the concentration of several chemical species. As shown in figure la, when the mean residence time of chemicals in the reactor is 100 min., the optical density oscillates regularly. At higher fluxes (residence time 76 min. in figure I b) the time record changes in a specific manner : seemingly stable oscillations exist which are interrupted from time to time and at random by large peaks. Such a transition from stable periodic behaviour to oscillations interrupted by random bursts denotes an intermittent transition to turbulence [5]. The present transition is well described by «type 1 intermittency » [7], shortly described hereafter. Although detailed modelizations of the B-Z kinetics have been proposed [8], we have not tried to relate them to our observations, since most features of the intermittent transition are model independent. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019810042013027100 L-272 JOURNAL DE PHYSIQUE LETTRES Fig. 1. [Ce4 +] oscillations recorded as a function of time : a) residence time 100 min. ; b) residence time 76 min. ; c) residence time 35 min.

Mean field theory for Heisenberg spin glasses

Abstract: The infinite-ranged spin glass model is studied in the general case of m-component spins. Results for the magnetization laws, the shape of the phase diagram, the nature of the transitions are presented.

Some effects of long range forces on interfacial phenomena

Abstract: The conventional theory of interfaces (with a free energy term ∼ (∇ρ) 2 where p is the appropriate density) may be qualitatively incorrect, because of long range effects associated with Van der Waals forces. 1) The tail of the density profile on the gas side of a liquid gas interface is not exponential (as usually predicted) but is ∼ z-3 (where z is the distance from the midpoint). 2) In the Moldover Cahn experiment, the thickness of the wetting layer of a liquid (b) between two other liquids (a) and (c) can be controlled by long range forces, not by the exponential repulsion between two interfaces which is derived from the conventional theory.

Exact results for a two-dimensional one-component plasma on a sphere

Abstract: HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Exact results for a two-dimensional one-component plasma on a sphere J.M. Caillol

Evidence for nucleon pair breaking even in the coldest scission configurations of 234U and 236U

Abstract: Primary fragment mass yields, measured at the highest values of the kinetic energy released in the thermal neutron induced fission of 233U and 235U, do not reveal any preference for splits into two even-mass nuclei. Moreover our data suggest that the mechanism of pair breaking would take place before scission, namely that the low energy fission would be a strongly dissipative process.

Observation of long range magnetic order in the reentrant superconductor HoMo6S8

Abstract: 2014 High resolution small-angle neutron scattering experiments have been carried out on HoMo6S8 to investigate the magnetic behaviour in the vicinity of the reentrant superconducting transition. With decreasing temperature a single magnetic Bragg peak develops below TM = 0.75 K at a wavevector Qc = 0.030 Å-1, demonstrating that a transversely-polarized, periodic magnetic structure has formed with a characteristic period of 200 Å. With further decrease of temperature additional scattering develops at smaller wavevectors, and the Bragg peak intensity decreases, as the transition to the ferromagnetic state proceeds and superconductivity is destroyed. The spectrum of scattering below 0.71 K has no peak in the range

Effect of a horizontal magnetic field on convective instabilities in mercury

Abstract: 2014 The effect of a horizontal magnetic field on the three-dimensional instabilities which occur in a Rayleigh-Bénard experiment with mercury is studied. When the magnetic field is parallel to the roll axis, the Busse oscillatory instability is inhibited, and the fluid motions become bidimensional. The high heat transport may be associated with an inertial convection mode. Tome 42 No 21 ler NOVEMBRE 1981 LE JOURNAL DE PHYSIQUE LETTRES J. Physique LETTRES 42 (1981) L-455 L-457 Classification Physics Abstracts 47.20 47. 65 1 er NOVEMBRE 1981, We report here some new results on the effect of an externally impressed magnetic field on the dynamical instabilities which occur in a Rayleigh-Benard experiment in mercury, a low Prandtl number fluid. (The Prandtl number is P = u/K, where u is the kinematic viscosity, and K the thermal diffusivity; for mercury at room temperature, P = 0.025.) It is known from the studies of Chandrasekhar that a magnetic field induces a tendency towards two-dimensionality of convective motions [1]. A horizontal uniform magnetic field, applied along the convective roll axis, does not influence the steady bidimensional motions, but inhibits three-dimensional disturbances. We have recently studied the three-dimensional instabilities, using a low aspect ratio cell of mercury heated from below, without impressed magnetic field [2]. (The aspect ratio is r = L/d, where L is the cell larger horizontal length, and d the cell height.) We have observed the onset of convection for a critical Rayleigh number R = R~, and the transition from steady twodimensional rolls to time dependent three-dimensional convection for R = 2.5 Re [3]. (The Rayleigh number is R = ocgd 3 Ar/uK, where a is the isobaric thermal expansion coefficients the acceleration of gravity, and AT the temperature difference across the cell.) This time-dependent motion, called by Busse the oscillatory instability [4], corresponds to a wave travelling in the direction of the roll axis, and consequently to a transverse time-dependent oscillation of the rolls. Its period corresponds to the characteristic circulation time of a roll d 2/K (d 2/K = 11 s for d = 0.7 cm). When a magnetic field is applied parallel to the roll axis, the oscillatory instability becomes inhibited, releasing its energy into an increase of the roll circulation velocity. This is verified by an increase of the oscillatory instability frequency before its disappearance, and an increase of the effective conductivity. For a magnetic field perpendicular to the roll axis, the oscillatory instability is also inhibited, but its frequency is essentially unaffected and the effective conductivity unchanged. The released energy goes Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019810042021045500 L-456 JOURNAL DE PHYSIQUE LETTRES into joule heating due to the large eddy currents which are present in this case. Further increase of the magnetic field generates a new convective pattern, with twodimensional rolls parallel to the magnetic field direction. This last result is in agreement with the theoretical predictions of Chandrasekhar [1], and the experiments of Lehnert and Little [5]. Let us recall that experiments on the inhibition of convection by a vertical magnetic field have been performed by Nakagawa [6]. 1. Experimental apparatus. The experimental set-up, described in our previous article [2], has been modified in order to visualize the convective patterns [7]. The cell is a parallelepiped of aspect ratio r = 4 (L = 2.8 cm,1= 0.75 cm, d = 0.7 cm; L larger horizontal length of the cell, I smaller horizontal length of the cell, d cell height). The lateral boundaries are made of plexyglass. The bottom plate consists of a copper block. Under the top boundary made of sapphire, a layer of cholesteric liquid crystal visualizes the temperature profile due to hot ascending fluid and cold descending fluid (refer to Fig. 1). The heating and temperature regulation systems are described in our previous article [2]. For time-dependent measurements we use a small N.T.C. (negative temperature coefficient) thermistor placed in a 2 mm diameter hole drilled in the copper block. The temperature signal given by the bolometer is analysed by a 5420 Hewlett Packard digital signal analyser. An electromagnet provides a uniform horizontal magnetic field ; the strength of the field could be varied up to 0.5 tesla. Fig. 1. Photograph of the cholesteric liquid crystal layer seen from above, and qualitative sketch of the convective rolls. The bright regions correspond to hot ascending fluid, the dark regions to cold descending fluid. 2. Effect of the magnetic field. The characteristic diffusion time scales are d 2/ u for vorticity, d 21 K for heat, and Jlo ud2 for magnetic field disturbances (7 is the electrical conductivity). For mercury at room temperature we have d 2/u = 890 s, d 2/K = 22 s, Jlo ad 2 = 10-4 s. The magnetic field diffusion time scale is the shortest. Thus, the magnetic field disturbances will not propagate as Alfven waves (see Ref. [1]). Besides, we deal with very low magnetic Reynolds numbers. Thus we can neglect the magnetic field due to eddy currents. Consequently, the magnetic field effect is only to add a term in the Navier-Stokes equation (p is the fluid density, d -1 represents the inverse of the Laplacian, v is the fluid velocity, and x 1 the direction of the magnetic field). This additional term represents some kind of anisotropic viscosity, inhibiting velocity variations along the magnetic field direction, and introduces a tendency towards two-dimensionality when it becomes more important than the velocity diffusion term 3. Transverse magnetic field. The dependence of the oscillatory instability amplitude and frequency on the magnetic field strength, in the case of a transverse magnetic field, is shown in figure 2. With this geometry, the field acts along the x axis and the oscillatory instability corresponds to a wave travelling in the y direction. The magnetic field dependent term of the Navier-Stokes equation, _ ~B 2 d _ ~ ax2 , a2v P B~/ vanishes for a velocity which depends only on y ; thus, the oscillatory instability frequency is almost unchanged. But the velocity amplitude, which depends Fig. 2. The dependence of the oscillatory instability amplitude and frequency on the magnetic field strength. The magnetic field direction is perpendicular to the roll axis. A is the oscillatory instability amplitude, in arbitrary units. L-457 MAGNETIC FIELD EFFECT ON CONVECTION IN MERCURY on x, is affected by the magnetic field, and decreases. The circulation time of convective rolls increases and consequently the oscillatory instability frequency slightly decreases. An equivalent point of view is to notice that a transverse field induces closed patterns of large eddy currents. The potential energy released by the buoyancy force must balance the energy dissipated by both viscosity and joule heating. Consequently, the field inhibits convective motions. 4. Longitudinal magnetic field. When the magnetic field is parallel to the roll axis, it mainly affects Fig. 3. The dependence of the oscillatory instability amplitude and frequency on the magnetic field strength. The magnetic field direction is parallel to the roll axis. the transverse oscillations of the rolls. As shown in figure 3, the oscillatory instability amplitude vanishes as the field is increased, and convection becomes steady and bidimensional if we neglect the velocity variations due to the lateral boundaries perpendicular to the field direction. In a bidimensional convection, (av/ay = 0), there are no electromagnetic forces and no joule dissipation. Thus the kinetic energy associated with oscillatory motion is transferred to convective bidimensional motion, and the convective velocity is increased. Consequently, before the complete attenuation of the oscillatory instability, we observe its frequency increase (refer to Fig. 3). The convective velocity increase is in good agreement with the observed effective conductivity increase (i.e. with the Nusselt number increase). 5. Conclusion. As theoretically predicted, a horizontal uniform magnetic field, applied along the roll axis, delayed the onset of the oscillatory instability by inhibiting three-dimensional motions, and increases the effective conductivity of the convective fluid. This higher heat transport may correspond to an inertial convection mode [8]. Besides, we expect that experiments with a magnetic field will allow us to observe new scenarios in the transition to weak turbulence. At low magnetic field values, its main effect is to add an additional damping term to the oscillators. This could affect the various period multiplication bifurcations to turbulence. At very high field values, by inhibiting any oscillatory instabilities, it may lead to new degrees of freedom in the bifurcation leading to turbulence.

Novel liquid crystal structures in cyano bi-cyclohexanes

Abstract: 2014 The structures of liquid crystal phases of three members of the series have been determined. All have nematic phases with extremely weak pseudo-layer density fluctuations and the two lowest members have bilayer smectic B type structures in which tails and cyano end groups are interdigitated. There also exist in the SB structures pronounced local modulations probably of a polar antiphase nature associated with a tendency to end-to-end ordering. In contrast cyano compounds of the type where is either cyclohexane or bi-cyclooctane have a local bilayer packing similar to that of the analogous biphenyls. J. Physique LETTRES 42 (1981) L-135 L-139 15 MARS 1981,

A fluid smectic A antiphase in a pure nitro rod-like compound

Abstract: A transition between a smectic A phase and a smectic A antiphase (S A) is observed for the first time in a pure compound : the nitrobenzoyloxybenzoate of heptylphenyl. So far, the anomalies of periodicities which are connected to this type of transition were only detected in cyano systems. Moreover, the solid phase of the inferior homologue of this nitro series is surprisingly influenced by similar anomalies of periodicities.

Adhesive contact of a conical punch on an elastic half-space

Abstract: It is shown that the Sneddon general theory (1965) enables the contact of axisymmetric adhesive punches to be studied. The relation between load and penetration, and the adherence force for conical punch are given.

Acoustics of water saturated packed glass spheres

Abstract: 2014 We have measured the velocity and attenuation of sound in a pack of water saturated glass spheres. The frequency range spreads from 200 kHz to 10 MHz and the diameter of the spheres is varied from 50 03BCm to 500 03BCm. Our experimental data clearly confirm the validity of Biot’s theory of the propagation of sound in porous medium. J. Physique LETTRES 42 (1981) L-477-L-480 Classification Physics Abstracts 47.55M 62.60 62.65 15 NOVEMBRE 1981,

Two kinds of two-dimensional order : the SmF and SmI phases

Abstract: 2014 A X-ray study of TBDA, a homologous compound of TBBA, shows that the smectic F and I phases exhibit a quasi-two-dimensional order (2D). A quantitative analysis of the powder diagrams yields short range order in the SmF phase and long range order in the SmI phase though at higher temperatures. This phase could be an example of a 2D solid. J. Physique LETTRES 42 (1981) L-67 L-70 ler FÉVRIER 1981, Classification Physics Abstracts 61.30 In recent years, two-dimensional (2D) systems have been the subject of a growing interest. New « liquid crystal » phases, recently discovered, provide good examples of such systems. The synthesis of TBBA homologous compounds with the general formula first allowed us to establish the existence of the smectic F phase (SmF) in the Cs compound [1] then the existence of the smectic I phase (SmI) in the C9 and C10 compounds [2]. We have synthesized new homologous of the series according to the process used for TBBA [3]. These last compounds show the following polymorphism : (*) We follow the advice of D. Demus et al. [13] concerning the labelling of the different smectic phases (SmG =5mBc). The SmI phase appears between the SmF phase and the SmA or SmC phase, which consists of stacked layers of « liquid type ». (It should be noted that the C 15 compound does not have a SmF phase.) The purpose of this letter is to study the change in order at the SmF H SmI transition in the C 10 compound (TBDA). An accurate quantitative analysis of the order cannot be done on single domains, because such samples often exhibit a large and unknown Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:0198100420306700 L-68 JOURNAL DE PHYSIQUE LETTRES mosaic spread this has been avoided by recording Debye-Scherrer diagrams on a high resolution Guinier camera (Cokal, ~, = 1.788 9 A). Moreover, we must point out that the differential thermal analysis has shown a great enthalpy variation at the SmI H SmC transition and a weak variation at the SmF H SmI transition, which means that the main change of order takes place at the transition : SmI +--+ SmC. 1. Structures of the SmF and SmI phases. a) THE SmF PHASE. X-ray patterns previously recorded [4, 5, 6] enabled us to assert the following : i) the layered structure of this phase has a tilt angle of the molecules with respect to the layer normal; ii) there is a very weak coupling between adjacent layers (as a matter of fact there do not exist hkl spots with I =F 0 : so, the structure is two-dimensional kind); iii) locally there is pseudo-hexagonal packing of the molecules within the layers. (Indeed the hkO Bragg peaks which are on a nearly regular hexagon (distorded by one percent) in the SmG phase [7] become diffuse spots in the SmF phase, centred of the same points of the reciprocal space.) Though the in-plane order does not extend to a long distance, and though the layers are very weakly coupled, one defines a local cell which is monoclinic with centred C-faces and its corresponding reciprocal lattice (Fig. Ib). The cell parameters of TBDA (C 10) at T 120 °C

One-dimensional organic superconductivity : an investigation through electron Schottky tunnelling in N/GaSb-(TMTSF)2PF6 junctions under pressure

Abstract: Observation of a large pseudo-gap at the Fermi level by Schottky tunnelling of electrons from N doped-GaSb into the organic superconductor (TMTSF) 2PF6 under a pressure of 11 kbar (Tc = 1 K) establishes, on an unambiguous experimental basis the one-dimensional character of the low temperature superconducting divergence and so justifies the presence of 1-D superconducting fluctuations to high temperatures. The intrachain pairing energy 2 Δ(0) = 3.6 meV derived from tunnelling characteristics emphasizes the strength of the superconducting channel in the family of organic superconductors (TMTSF)2X.

The onset of dilatant behaviour in non-inertial flow of dilute polymer solutions through channels with varying cross-sections

Abstract: The apparent viscosity of dilute polymer solutions in non-inertial flows through channels consisting of successive short tubes or slits separated by expansions displays a marked divergence from shear-thinning power-law behaviour beyond a critical shear rate calculated for constricted sections. The product of this critical shear rate by the Rouse relaxation time was found to be nearly constant for various coil polymers, solvent viscosities and polymer concentrations, independently of the exact geometric shape of the channels. When the critical shear rate is reached, the maximum stretch rate in the converging sections is greater than the inverse of the Rouse relaxation time. An elongation of macromolecules is thus expected and the dilatant behaviour can be attributed to the increased viscous dissipation due to the elongated state of macromolecules in the converging parts of flow.

Reentrant nematic and columnar phases in disc-like liquid crystals at atmospheric pressure

Abstract: 2014 We describe the first example of reentrant ND nematic and Dr columnar mesophases at atmospheric pressure in a pure disc-like compound. J. Physique LETTRES 42 (1981 ) L-417 L-419 15 SEPTEMBRE 1981, Classification Physics Abstracts 61.30 64.70E

The observation of a shear flow instability in a rotating system with a soap membrane

Abstract: 2014 A pattern of driven vortices can be observed in the shear layer created in a fluid by a rotation movement. We present such an experiment where a liquid membrane is used to close a thin experimental volume of air thus forming a wall with the following properties : 2014 The parallel component of the fluid velocity does not have to be zero at the boundary. 2014 The interference fringes provide a good visualization of the isobars of the flow. Tome 42 N° 19 ler OCTOBRE 1981 LE JOURNAL DE PHYSIQUE-LETTRES J. Physique LETTRES 42 (1981) L429 L-431 1 Classification Physics Abstracts 47.15 47. 80 1 er OCTOBRE 1981, ,

Polarization of calcium atomic fluorescence due to a coherence effect in the photodissociation of Ca2 molecules

Abstract: A strong degree of polarization (64 %) has been observed in the fluorescence from calcium atoms excited by the photodissociation of Ca2 molecules in their ground state. This large value, at variance with the predictions by Van Brunt and Zare, is explained by taking into account a coherence effect between the sublevels of the excited state.

Quadrupole transitions of the 1←0 band of N2 observed in a high resolution atmospheric spectrum

Abstract: Using a high resolution atmospheric absorption spectrum recorded at Kitt Peak with the solar Fourier transform spectrometer, it has been possible to observe the quadrupole transitions S7 to S16 of the 1 ← 0 band of N2. The analysis of the equivalent widths and of the central depths of these lines has enabled us to determine the derivative of the quadrupole moment (∂Q/∂r)e = (0.94 ± 0.05) ea 0 and to estimate the broadening coefficient by air γ0N2-air = 0.06 ± 0.02 cm-1 atm.-1 at 296 K. Finally the assignment of telluric lines between 2 391.5 and 2 467 cm-1 is also given.

Pressure dependence of the organic superconductivity in (TMTSF)2PF 6

Abstract: High pressure resistivity measurements reveal a very large decrease of the superconducting transition temperature of (TMTSF)2PF 6 with pressure. It is difficult to explain this phenomenon with current theories of narrow band superconductors. An explanation in terms of the soft mode of the low pressure metal to insulator transition is proposed.

Charge distribution and kinetic pressure in a plasma : a soluble model

Abstract: For a classical two-dimensional one-component plasma (with a logarithmic interaction), the onebody density can be calculated exactly when the coupling constant has the special value 0393 = 2. The behaviour of this density is studied near the wall of the circular box which encloses the system. If there is an excess charge, it concentrates near the wall. The density at the wall is related to the kinetic pressure, which differs from the thermal pressure. Tome 42 N~ 11 1 er JUIN 1981 LE JOURNAL DE PHYSIQUE LETTRES J. Physique LETTRES 42 (1981) L-223 L-226 Classification Physics Abstracts 05.20 64.10 50.00 1 er JUIN 1981,

Cascade of period doubling bifurcations and large stochasticity in the motions of a compass

Abstract: 2014 We propose a very simple mechanical device, consisting of a compass placed in a periodic oscillating field, which renders the observation of large scale stochasticity in Hamiltonian systems possible. We show that this device also exhibits a cascade of period doubling bifurcations. We determine experimentally the threshold of the first doublings. J. Physique LETTRES 42 (1981) L-537 L-539 15 DÉCEMBRE 1981, Classification Physics Abstracts 46.90 If we place a compass, initially neglecting friction, in an oscillating magnetic field, perpendicular to the compass axis, we have a simple mechanical device, which models a well-known system : the synchronous bipolar motor. We might think that the possible motions of this compass are naturally a clockwise or a counterclockwise rotation. In fact, by doing so, we have not considered the non-linear property of this device, and if these two kinds of motion really exist, a multitude of different motions are also possible, some periodic, others chaotic. For this device, the angle of rotation 0 is governed by the equation : with F = (M. Bo/2 J ) and where M and J are respectively the magnetic and inertial momentum of the compass. Bo is the magnitude of the magnetic field, oscillating at the frequency (co/2 n). A detailed study of this kind of equation (1) has been performed by D. Escande and F. Doveil [1] and also in [6, 7]. Let us first consider the compass placed in a single rotating field. In the rotating frame, equation (1) becomes : 81 = F [sin 61 ] (2) (*) La version francaisc de cet article a été proposée pour publication aux Comptes Rendus de 1’Academie des Sciences. e) Equation (1) also models a phase lock loop in electronics. which is exactly the completely-integrable equation of the pendulum [2]. It has two kinds of solutions : oscillations in which the compass is locked with and oscillates around the rotating field with its own frequency coo ( = F when the magnitude of these oscillations is small) and rotations in which the compass is rotating at a rate different from that of the field. In phase space (0, 0), the lock-in or the resonance is associated with a set of closed trajectories (ellipsoidal for small oscillations), the largest closed trajectory being the separatrix. Similar results will be obtained with a second rotating field whose resonance will be symmetric with the first one about the axis 0=0. If we now consider the two fields simultaneously, we easily understand that two fundamental resonances will appear in phase space, symmetric about the axis 0=0. However, this is only true as long as the perturbation level is small, the system is no longer integrable, the phase space becomes three dimensional and stochastic trajectories appear. It is possible to return to the former phase space (0, 0), through a Poincare transformation. The K.A.M. theorem [3] means that the two resonances that we have already described remain stable as long as the perturbation level is not too high. The perturbation level is defined by the stochasticity parameter s = (2~/F/(D) which measures the width, in phase space, of the resonances compared with their separation. When s = 1, the separatrix of the two unperturbed resonances touch each other, the resonances overlap, this is the criterion Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019810042024053700 L-538 JOURNAL DE PHYSIQUE LETTRES ,proposed by Chirikov [2], for the occurrence of large scale stochasticity. The trajectories in the phases space at s = 0.68 are given in figure 1; we find a number of resonances of which the larger A and A’ are just those we have described. The resonances B and B: correspond to oscillations of the compass with ~0~=0. On the other hand, the phase space also contains diffuse areas, growing like s at the expense of the resonances, which also combine so allowing the system to diffuse into a larger and larger areas of phase space. When the system passes from resonance A to resonance A’, large scale stochasticity occurs. Fig. 1. Calculated shape of the phases space at s = 0.68. (We thank D. Escande and F. Doveil for this document.) 1. Experimental lay-out. A compass, fixed on an axis guided by two bearings, is placed between two coils in a Helmholtz configuration fed by an alternating current. The stochasticity parameter s is fixed by adjusting the magnitude of this current. The data acquisition system consists of a pick-up coil placed near the compass perpendicular to the excitation coils. The signal so induced is then Fourier analysed. An important feature of our device is that it is a dissipative system. However, the friction is small, and the contraction, in phase space, slow so that we will be able to draw conclusions on the phase space of the associated Hamiltonian system, provided that the « trajectories » of the compass are considered over a short time interval. It is possible to model (2) our device with the equation : with (oc/6o) ~ 3 x 10-2. 2. Study of the destabilization of resonances. In this study, the dissipative property plays a fundamental role : it allows an automatic centring on the elliptical point associated with each resonance to be achieved, so that it will be possible to follow the bifurcation cascade associated with the destabilization of these resonances. Let us consider, to begin with the fundamental resonances A and A’. They correspond to a synchronous rotation with one of the rotating fields. The second rotating field acts as a perturbation at the frequency (2 w/2 n). Experimentally we observed that if s is smaller than 1.74 ± 0.02, this perturbation does not destroy the elliptic point in the phase space. Nevertheless, a great number of lock-ins between the oscillations of the compass at 60o and the perturbation at 2 w can occur. They appear when 60o = (m.2 cv/n). In this case the system has to be acted on in order that they may set in. When s is increased these lock-ins appear with a well-defined hierarchy corresponding to increasing 60o; for instance, when s >, 0.88, or s ~ 1.05 or s > 1.36 we find that 60o is respectively 2 60/5, 2 w14, 2 w/3 t. In the case of the lock-in with (2 60/2) which appears when s >, 1.74, the elliptic point splits and the compass, which until then was oscillating at the frequency (2 wl2 n) from side to side of the rotating field, now oscillates with the frequency (60/2 n). The experimental evidence for this phenomenon is slightly complicated by the rotational motion of the compass : this rotation induces a signal whose frequency itself is (w/2 n). We have, in fact, to monitor the magnitude of the second harmonic whose sudden growth reveals the beating of the rotation at 60 and the oscillation also at w. From this bifurcation, the same scenario repeats itself on a smaller scale and possible lock-ins will now affect the oscillation motion at 60. The two elliptic points are stable until s = 1.93, when a new bifurcation with period doubling occurs. It simply corresponds to the occurrence of the subharmonic (2 w/4) (Fig. 2a) on the same scenario, but this time the scale of the motion becomes very small and the corresponding parameter variations become extremely restricted. Experimentally it is difficult to increase this parameter without getting a lock-in 1/3 (2 c~/12), which rapidly leads the motion of the compass to large scale chaos (for s ~ 1.94). One fascinating pro-\" perty of this device is that it seems that the destabilization of all the resonances follow the same scenario. e) In fact, dissipation results from solid friction, so this model is somewhat crude. L-539 CHAOTIC BEHAVIOUR OF A COMPASS Fig. 2. Pick-up coil signal spectra. a) After two period doublings; b) Chaotic phase; c) After a new lock-in. For example with resonance B, M/2 appears for s = 0.92, w/4 for s = 1.04. 3. Study of chaotic motion. The occurrence of chaos from a lock-in appears in a first phase during which the system is nearly locked but exhibits abnormal amplitudes. It slowly escapes, from the lock-in, after about ten periods. This phase is followed by a phase of large scale chaos where the motion of the compass appears totally erratic. The compass rotates a few times along with one of the fields, then suddenly reverses its rotation, stops, starts again and so on. We think that such a motion illustrates the large scale stochasticity of Hamiltonian systems since the rotation inversions indicate that the chaotic motions are of large extension in the phase space. Further, these inversions may occur after a few periods of excitation only, which is a very short time compared with the time related to the dissipation (typically 100 periods). Beside this, the nature of this chaos seems to be Hamiltonian as shown by the spectra of figure 2b where it can be seen that the noise merges at once in all the spectra, which is contrary to the usual experimental observations in dissipative systems [4]. Nevertheless the dissipation plays a fundamental role : this kind of chaos does not seem to be stable and the system finally finds a new lock-in which may be very complex as figure 2c shows. This illustrates the attractive property of the elliptic points in dissipative systems. However such chaotic phases exhibit a great variety of durations, under the same conditions, from five minutes to a few hours ! We also notice that new resonances appear at a finite s value. In the case of figure 2c : s ~ 1.92. We think that, even though the large scale stochasticity is a property of Hamiltonian systems, it also occurs in a transient way in slightly dissipative systems. The experimental device that we describe allows quantitative comparison with new theories in both dissipative and Hamiltonian systems. Thus we hope, that in a near future, we will be able to extract an experimental value for the 6 exponent of Feigenbaum [5] in both cases. Acknowledgments. We are grateful to D. Escande, F. Doveil, S. Aubry, Y. Pomeau, P. Berge and M. D