Nonlinear NMR in 3He-B with arbitrary orientation

The high -field nonlinear NMR absorption signal of the B phase of ³He with an arbitrary orientation of the rotation axis n with respect to the field is calculated to third order in the amplitude of the perpendicular magnetization. An instability like those in ferromagnetic resonances is predicted, which changes its asymmetry with respect to the sense of sweeping at an orientation angle of 35 .On leave of absence from the Technical University of Munich. Postdoctoral Research Fellow of the DFG (German Science Foundation).     

An attractor regime for pulsed NMR in3He-B

We have found a new long-lived mode that can be generated in the regime of pulsed NMR in³He-B. The mode has a frequency different from both the Larmor one and that which was generated in the experiment of Corruccini and Osheroff by rf pulses longer than 104°. It does not correspond to stationary solutions of the Leggett-Takagi equations. The mode is due to the existence of an attractor that is a result of an interplay between nonlinear and dissipative effects in the spin dynamics of superfluid³He-B.     

Theory of Josephson flow oscillations in superfluid3He-B

Avenel and Varoquaux have proposed an experiment for observing pressure-induced Josephson flow oscillations across an orifice connecting two reservoirs of³He-B. The aim of the present work is to specify the conditions for the observability of this effect. First the six coupled Ginzburg-Landau (GL) equations for the order parameter components within the orifice are solved, subject to rigid boundary conditions at the ends (fixed phase difference between the bulk order parameters) and specular reflection boundary conditions at the walls. The longitudinal component is depressed and one of the transverse components is enhanced toward the middle of the orifice. From the order parameter functions the supercurrent through the orifice I(), and the GL energyF() are calculated for 0 2 and for different lengths and cross sections of the orifice. A new feature of the functionI() in comparison to the current-phase relationship for superconducting microbridges is its multivaluedness and the occurrence of two nontrivial zeros ofI(). The functionF() exhibits two local maxima besides the absolute maximum at =. The optimal length of the orifice for the observability of the ac Josephson effect turns out to be about seven GL coherence lengths . Then the frequency of the Josephson flow oscillations, which follow the sudden application of a force F on a diaphragm, depends on F in a way similar to that in which the frequency of the parallel ringing experiment in³He-A depends on the field step H.     

The reverse AC-Josephson effect in 3He-B

No Heading We have investigated ³He Josephson currents under an externally applied ac-pressure modulation. The ³He-flow was driven by the motion of a flexible membrane through an array of 65 by 65 holes of 150 nm size in a 50 nm thick SiN wall. The membrane was deflected electrostatically. If a sinusoidal modulated electrostatic force was applied to the membrane, we could observe phase locking of the Josephson oscillations to the external ac-drive. As a result, a zero net flow through the weak link was observed while the membrane was stuck for many seconds at a non zero pressure position. For this pressure the Josephson frequency matched the ac-drive frequency. We could resolve the dynamics which led to this trapped, phase locked state. The dynamics looked quite similar to dynamics which took place when the system ended up in Helmholtz oscillations at the zero pressure difference equilibrium state.PACS numbers: 67.57.De, 67.57.Fg, 74.50.+r, 85.25.Cp     

NMR experiments on rotating superfluid3He-A and3He-B and their theoretical interpretation

We have constructed a rotating nuclear demagnetization cryostat and used it for continuous-wave NMR experiments on superfluid³He-A and³He-B. The measurements were performed in a long cylindrical geometry of 5 mm diameter, with the cylinder axis parallel to the axis of rotation and with the external magnetic field H₀=284 or 142 Oe in the same direction. The angular velocity of rotation was varied between 0.2 and 1.5 rad/sec, and the experiments were done under 29.3 bar pressure at temperatures between T_c=2.72 and about 1.4 mK. As a guide to the new and esoteric field of superfluid³He in rotation, we first review the general theory at some length in relatively simple terms. Pictorial explanations are often given.In³He-A, a rotation-dependent NMR satellite was found; its intensity a rotation-dependent NMR satellite peak was discovered; its relative intensity increases linearly with . The position of the satellite is independent of and H, and does not depend on whether the sample was cooled from the Fermi-liquid region to the A phase while rotating or at rest. At temperatures 0.1<1–T/T_c<0.3, the frequency shift of the satellite can be described by the parameter R_t=0.86–1.1(1–T/T_c). Cooldown under rotation produced systematically larger satellite intensities than cooldown at rest. A second, metastable satellite, best seen at rest and disappearing in less than 30 min, was also discovered. Furthermore, the main NMR peak broadens during rotation, while the total NMR absorption remains the same. The behavior of the rotation-dependent satellite strongly supports the existence of vortices in³He-A, their number being proportional to ; the satellite is caused by localized spin wave modes trapped by vortex cores. Theoretical calculations agree quite well with our experimental data if continuous vortices, without a singularity in the order parameter, are assumed. Their presence is also responsible for the additional broadening of the main peak, due either to increased spin diffusion or to scattering of spin waves. The metastable satellite is caused by textural boundaries, probably by twist solitons in the superfluid, created by the rapid cooldown of the sample.In³He-B, a series of nearly equally spaced NMR satellites was found on the high-frequency side of the main peak with the cryostat at rest. Under rotation the separation between the satellites increases linearly with . The spacing displays a jump, proportional to , at 1–T/T_c=0.40. The discontinuity occurred only during start/stop experiments, not if the cryostat was continuously rotated while warming over the transition region. Immediately after rotation had been started the whole NMR spectrum shifted toward higher frequencies for about 30 sec; these transients were seen only at >0.25 rad/sec. In³He-B, the order parameter is strongly influenced by the wall of the container, producing the so-called flareout texture, with the angle between the vector andH equal to 63° at the walls. The satellites can be explained as spin wave modes arising from an almost harmonic potential well formed by the texture. The creation of vortices changes the texture and increases the steepness of the potential and therefore increases the satellite spacing during rotation. The vortices themselves perturb the texture due to the long-range orientating effect of their cores on the order parameter. The discontinuity in the satellite splitting at 1–T/T_c=0.40 is explained as being due to a first-order phase change in the vortex core at this temperature. The transient shift in the NMR spectrum, immediately after the start of rotation when vortices are not yet present, is caused by the large superfluid vs. normal liquid counterflow; this phenomenon thus gives an estimate for the time needed to create vortices in³He-B.     

NMR Line Shape of Rotating 3He-B at Large Counterflow Velocity

We have investigated the NMR line shapes of superfluid ³He-B in a rotating cylinder. In the vortex-free state at sufficiently large angular velocity of rotation, 1 rad/s, the main feature of the absorption spectrum is a large frequency-shifted peak above the Larmor frequency. The shape of this counterflow peak is both measured and calculated as a function of external magnetic field strength, angular velocity, temperature, and number of vortex lines. The NMR spectrum is derived from the calculated order-parameter texture. Reasonable agreement with the measured line shape is obtained by including line-broadening effects due to the external field inhomogeneity and Leggett–Takagi relaxation.     

Study of the orientational electric field effect in 3He-B

We have studied the orienting effect that an electric field has on the orbital motion of the pairs in superfluid ³He-B, using the orientation-dependent transverse NMR frequency shift as a probe. This method provides high resolution for studying the orienting effect for either sign of the orientational energy. Electric fields as high as 35,000 V/cm applied across the sample produced no measurable frequency shift at 0-, 10-, and 32-bar pressures. This gives upper limits to the orientational energy of 1.4 × 10^\s-3, 1.0 × 10^\s-3 and 4 × 10^\s-4 times the values expected from simple theory.     

A new NMR mode in the Landau field in superfluid3He-B

We report the experimental observation of the separate internal precession of the normal and superfluid magnetizations around the molecular Landau field in³He-B, the magnetic analog of second sound. The mode is detected by cross relaxation with conventional NMR precession when the external and Landau molecular fields have similar values. From NMR measurements down to 0.12T_c we conclude that the previously observed catastrophic relaxation phenomenon can be explained as a capture of the internal precession mode by the Larmor precession. This NMR mode provides an additional relation between the parameters F ₀ ^a , F ₂ ^a which taken with susceptibility data allows both parameters to be distinguished, giving for zero bar, F ₀ ^a =–0.713 and F ₂ ^a =0.4.     

Nonlinear motion of magnetization in superfluid3He-B. The Leggett-Takagi spin relaxation effect in the pulsed NMR configuration

The recovery motion of the magnetization and the associated motion of the order parameter for the pulsed NMR geometry in superfluid³He-B are investigated by solving the Brinkman equations including the Leggett-Takagi spin relaxation effect. It is found that the relaxation of longitudinal magnetization strongly depends upon the initial tip angles according to both the numerical and adiabatic perturbation methods. For large tip angles, these equations yield square-root and linear types of recovery. For small tip angles near the magic angle ₀=cos^–1 (–1/4), the longitudinal magnetization does not recover to the full equilibrium value, and the transverse magnetization shows time-dependent resonance shifts arising from the relaxation effect.     

Collective modes of Larmor precession in3He-B. Transverse NMR on Homogeneously Precessing Domain

The Homogeneously Precessing Domain (HPD) and the stationary state (Non-Precessing Domain, NPD) are particular cases of the more general states of the coherent Larmor precession in the³He-B. The symmetry is discussed between NPD and HPD which connects the properties of the dynamical HPD state with that of the conjugated stationary state of the³He-B. This symmetry allows us to obtain the spectrum of the NMR absorption due to excitation of the collective modes of Larmor precession, arising on the background of HPD.     

NMR and axial magnetic field textures in stationary and rotating superfluid3He-B

We have performed NMR measurements on the flare-out texture of superfluid³He-B in a cylindrical container of 5 mm diameter in axial magnetic fields of 28.4 and 56.9 mT. The transverse cw NMR spectra have been analyzed both with respect to their overall shape and the spin-wave absorption peaks close to the Larmor frequency. Our analysis of the stationary state spectra, based on texture computations, yields the longitudinal resonance frequency v _L (T), the magnetic healing length _H (T), and the dipolar length _D (T), which we report for pressures below 29 bar. A lattice of quantized vortex lines appears in the rotating state, and two additional textural free energy terms have to be included in the analysis. One of the terms is linear in the applied magnetic field and arises from the spontaneous magnetization of the vortex cores. The second term is quadratic in magnetic field; it is generated both by the superflow field v _s (r) about the vortex core and the difference in the induced magnetizations of the vortex-core and the bulk superfluids. The rotational orienting effects have been studied for rotation speeds up to 2red/sec.     

Experiments on nonlinear acoustics in3He-B

Observations and applications of nonlinear acoustic phenomena in superfluid³He-B are reported. Two-phonon absorption (TPA) by the real squashing (rsq) mode has been detected under several experimental conditions below p = 3.5 bar, using two coincident sound pulses. The attenuation peak height has been investigated as a function of the energy densities of the two sound waves. We discovered the five-fold Zeeman splitting of TPA by parallel sound pulses in an applied magnetic field and the two-fold dispersion splitting due to the finite wave vector of the mode when the two sound pulses are mutually perpendicular. The dispersion relation of the real squashing mode has been investigated at zero pressure and in zero magnetic field by exciting the mode with two parallel, perpendicular, or antiparallel sound waves. Experimental values for the parameters that determine the collective-mode velocities have been extracted from the positions of the observed attenuation maxima. An anomalous structure has been observed in the attenuation and phase velocity spectra of a single high-intensity sound wave near the threshold for pair breaking by two phonons; in an applied magnetic field the phase velocity anomaly splits into a triplet.     

Experiments on the Twisted Vortex State in Superfluid 3He-B

We have performed measurements and numerical simulations on a bundle of vortex lines which is expanding along a rotating column of initially vortex-free ³He-B. Expanding vortices form a propagating front: Within the front the superfluid is involved in rotation and behind the front the twisted vortex state forms, which eventually relaxes to the equilibrium vortex state. We have measured the magnitude of the twist and its relaxation rate as function of temperature above 0.3T _c. We also demonstrate that the integrity of the propagating vortex front results from axial superfluid flow, induced by the twist.      

Hydrostatic Theory of Superfluid 3He-B

The determination of the texture of the order parameter is important for understanding many experiments in superfluid ³He. In addition to reviewing the theory of textures in superfluid ³He-B we give several new results, in particular on the surface parameters in the Ginzburg–Landau region and bulk parameters at arbitrary temperature. Special attention is paid to separate the results that are valid at all temperatures from those which are limited to the Ginzburg–Landau region. We study the validity of a trivial strong-coupling model, where the energy gap of the weak-coupling theory is scaled by a temperature dependent factor. We compare the theory with several experiments. For some quantities the theory seems to work fine and we extract the dipole–dipole interaction parameter from the measurements.     

Optical interferometry in superfluid3He-B

We report interferometric measurements on 0.1 ... 1 mm thick films of superfluid ³He-B. The menisci of three different rotational states of the superfluid were observed and analyzed theoretically using two-fluid hydrodynamics: These are (i) the equilibrium vortex state in which the superfluid and the normal components corotate (solid body rotation), (ii) the vortex-free state (the Landau state), in which only the normal component rotates, and (iii) the quasistationary vortex state in which only the superfluid fraction rotates (pure superfluid rotation). The Landau state manifested itself by a reduced parabolic meniscus at rotation speeds below the critical angular velocity 0.2 rad/s for vortex formation. Transition from the Landau state to the equilibrium vortex state yielded a sudden deepening of the meniscus when _c was exceeded. After a rapid halt of the cryostat, we observed a novel meniscus which was produced by the superfluid rotation while the normal component was at rest. The enhanced depth of this meniscus is governed by the reactive mutual friction parameter B'.By employing laser light, both for imaging and for thermomechanical excitation, we measured the response of a thin superfluid layer to a heat pulse and analyzed it within the theory of two-fluid hydrodynamics. The data were employed, using the dispersion relation for thin film oscillations, to deduce the second viscosity coefficient ₃ close to T_c.     

Domain Walls in Superfluid 3He-B

No Heading We consider domain walls between regions of superfluid ³He-B in which one component of the order parameter has the opposite sign in the two regions far from one another. We report calculations of the order parameter profile and the free energy for two types of domain wall, and discuss how these structures are relevant to superfluid ³He confined between two surfaces.PACS numbers: 67.57.Np     

Transverse waves in superfluid3He-B

We examine the theory of collisionless transverse current waves in bulk superfluid³He-B, including the coupling to the order parameter collective modes. At low frequencies, Ω ≪ δ(T), the order parameter modes do not contribute to the restoring force for a transverse current, and the quasiparticle contribution drops rapidly as the gap in the spectrum develops. Thus, low-frequency transverse sound becomes overdamped at temperatures nearT _c . However, at low temperatures (T ≲0.3T _c ) the off-resonant coupling to the J = 2⁻,M = +-1 modes stabilizes a propagating transverse current mode, with a large phase velocity and low damping for frequencies above a critical frequency that is approximately that of theJ = 2 ⁻ mode. We also discuss the similarities and differences of longitudinal and transverse sound in the superfluid phases. For example, in zero field, right- and left-circularly polarized waves are degenerate. A magnetic field, with , lifts this degeneracy, giving rise to the analog of circular dichroism and birefringence of electromagnetic waves. Thus, transverse waves may be more easily detected in the B-phase than in normal³He.     

Textural singularities in superfluid3He-B

The absorption signal occasionally found missing in the middle of resonance absorption lines in NMR is explained on the basis of a textural singularity of the anisotropy axis in the B phase of ³ He. The singularity combines with the field gradient present to give rise to an asymmetric hole in the signal with a singularity on the high-frequency side. The line shapes and temperature dependence are predicted and agree well with experiment. Spin waves are found to be trapped by the potential well formed by the texture and the field gradient. The frequency intervals predicted for the spin waves also agree well with experiment.Work supported in part by National Science Foundation Grant No. NSF DMR 74-18030.