Vortex Formation in Neutron-Irradiated Rotating Superfluid 3He-B

A convenient method to create vortices in meta-stable vortex-free superflow of ³He-B is to irradiate with thermal neutrons. The vortices are then formed in a rapid non-equilibrium process with distinctive characteristics. Two competing explanations have been worked out about this process. One is the Kibble-Zurek mechanism of defect formation in a quench-cooled second order phase transition. The second builds on the instability of the moving front between superfluid and normal ³He, which is created by the heating from the neutron absorption event. The most detailed measurements with single-vortex resolution have been performed at temperatures close to T_c. In the first half of this report we summarize the two models and then show that the experimentally observed vortices originate from the Kibble-Zurek mechanism. In the second half we present new results from low temperatures. They also weakly support the Kibble-Zurek origin, but in addition display superfluid turbulence as a new phenomenon. Below 0.6 T_c the damping of vortex motion from the normal component is reduced sufficiently so that turbulent vortex dynamics become possible. Here a single absorbed neutron may transfer the sample from the meta-stable vertex-free to the equilibrium vortex state. The probability of a neutron to initiate a turbulent transition grows with increasing superflow velocity and decreasing temperature. PACS numbers: 47.32, 67.40, 67.57, 98.80.     

Torsional Oscillations of a Rotating Column of 3He-B

No Heading We have analysed the axisymmetric and non-axisymmetric modes of a continuum of vortices in a rotating superfluid. We have investigated how changing the temperature affects the growth rate of the disturbances. We find that, in the long axial wavelength limit the condition q = /(1 – ) = 1, where and are temperature-dependent mutual friction parameters, is the crossover between damped and propagating Kelvin waves. Thus at temperatures for which q > 1, perturbations on the vortices are unlikely to cause vortex reconnections and turbulence. These results are in agreement with the recent discovery of Finne et al¹ of an intrinsic condition for the onset of quantum turbulence in ³He-B.PACS numbers: 67.40. Vs, 67.57.–z     

Superfluid 3He-B Vortex Simulations inside a Rotating Cylinder

No Heading We study numerically vortex dynamics in superfluid ³He-B by solving the full Biot-Savart equations inside a rotating cylinder. The initial vortex configuration seems to have an essential role whether the growth process starts or not. The growth process is, at least at the early stages of simulations, mostly governed by the reconnections with cylinder boundary. In order to see a large increase in vortex density one should go below 0.5T_c in temperature, somewhat lower than what is observed in the experiments.PACS numbers: 47.32, 67.57.     

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).     

Time-of-Flight Measurements on Quantized Vortex Lines in Rotating 3He-B

Vortex loops are injected into a long cylindrical sample of rotating vortex-free superfluid ³He-B. Using non-invasive NMR techniques, we measure the time required for them to expand along the sample axis to form rectilinear vortex lines. From the axial flight time we deduce the dissipative mutual friction parameter a which is found to agree with previously reported measurements. The flight time displays no anomaly at 0.6T _c, where the character of the vortex dynamics abruptly changes from regular over-damped flow to turbulence.     

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.     

NMR study of the effect of flow on3He-B

We have used a plastic diaphragm to generate flow in superfluid³HeB, through a cylindrical geometry, whilst monitoring the transverse NMR frequency. It was found that the critical flow velocity needed to induce NMR frequency shifts was dependent on the magnetic field. The results are explained by a numerical calculation of the texture in the³HeB, under the influences of magnetic field, fluid flow and surface energy terms. Also presented are measurements of the B phase NMR longitudinal frequency, derived from the magnitude of the flow-induced change in the transverse frequency. A flow-induced texture transition, which caused a dramatic narrowing of the NMR lineshape, is described.     

Precessing Vortex Motion and Instability in a Rotating Column of Superfluid 3He-B

In a rotating circular cylinder of superfluid ³He-B, an evolving vortex expands longitudinally such that its end point describes a helically spiralling trajectory along the cylinder wall. The spiral motion is found to give rise to a periodically oscillating NMR signal, which is brought about by the modulation in the superfluid counterflow and its influence on the “flare-out” order parameter texture. The new NMR signal becomes observable within a narrow temperature interval close to the onset temperature of turbulence, when new vortices are continuously generated by the single-vortex instability at the cylindrical wall at a slow rate, ～1 vortex/s. We use numerical vortex filament calculations to examine the precessing motion of the evolving vortices, while they expand towards their stable state as rectilinear line vortices.     

Model formalism of liquid 3He-B at equilibrium

The approximate formal treatment of the nuclear spin system of normal liquid ³He given some time ago is extended to the ordered ³He phase. The formalism leads to the prediction of normal thermal behavior of ³He-B at lower pressures and at temperatures approaching its phase-boundary temperatures. In contrast to the disordered normal liquid phase, which is thermally anomalous, the entropy of the ³He-B decreases on isothermal compression, or its isobaric volume expansion coefficient is positive. The equilibrium thermal behavior of ordered ³He-B is thus qualitatively different from that of disordered liquid ³He. Experimental control of these aspects of the liquid ³He phase transformation is lacking at the present time. Both early and new ³He-B paramagnetic susceptibility data, extended recently over a wide reduced-temperature range, disclose a fundamental competition between the spontaneous ordering mechanism responsible for the existence of ³He-B and the specific ordering process imposed upon this phase on application of an external constant and uniform magnetic field. As a consequence, magnetized ³He-B will be shown to increase its entropy on isothermal magnetization and to cool on adiabatic magnetization. The magnetocaloric effect is, however, only moderate. The competition of the ordering process leads to the delay or possibly even to the suppression of the formation of the ordered phase, a state of affairs foreseen in our earlier work. At low or moderate magnetic field strengths, the zero-field phase-boundary temperatures are shown to shift toward lower temperatures while, simultaneously, the order of the phase change decreases, from second order, in the absence of the field, to first order. Although of model-theoretic character, involving limitations of various types, the rich physical content of ³He-B at equilibrium clearly emerges in the present work.Work performed under the auspices of the U.S. Department of Energy.     

Effective viscosity of3He-B at low temperatures

We show that there exists a nontopological soliton in the superfluid ³He-A phase near T _AB. The possibility that the soliton causes the ³He A-B transition is examined. Some properties of the soliton are considered.     

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.     

Acoustic attenuation in superfluid3He-B at low pressures

The attenuation of zero sound in superfluid³He-B has been measured up to 160 cm^–1, at pressures less than 4 bar and at frequencies 34.2, 44.2, and 54.0 MHz. The contribution of pair breaking to the attenuation has been measured for the first time. The gap (J=1 ^–) mode has been studied in magnetic fields up to 80 mT and the structure of its Zeeman components revealed. Coupling to the gap mode in the applied field allows a direct spectroscopic measurement of the energy gap. In zero magnetic field, the attenuation is well described by the theory of Wölfle, showing agreement with the magnitude of the attenuation and the frequency of the squashing mode resonance, for an appropriate choice of the parameterz=(c ₀–c₁)/c₁, wherec ₀, c₁ are the velocities of zero and first sound. This provides a determination of the Landau parameterF ₂ ^s and indicates that thef-wave interaction is negligible at these low pressures.     

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.     

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.     

The orientational phase transition at a vortex in3He-B

The orientational phase transition in the vicinity of a single vortex in³He-B is studied. It is the phase transition from a uniformn-texture withn parallel to the magnetic field and the vortex line to ann-texture that is nonuniform near the vortex. The problem of the instability of the the uniformn-texture is equivalent to the quantum mechanical two-dimensional problem of a bound state in a field with an attractive potential 1/r ². The orientational phase transition at a vortex array is also considered. In the limit of large vortex density the orientational phase transition transforms to the phase transition studied by Gongadze et al. The theoretical results are compared with the observed phase transition at a vortex in³He-B.     

Non-linear Stationary Spin-waves in Superfluid 3He-B at Ultra-low Temperatures

We have performed detailed CW NMR measurements on superfluid ³ He-B in order to investigate stationary spin-wave modes in a texture well. Our results at high temperatures are consistent with previous work. However, we find that the spin dynamics is profoundly modified, even for small excitation amplitudes, for temperatures on the order of 200 K, substantially lower than the limit reached in previous investigations. We report the observation of non-linear spin-waves at ultra-low temperatures and their characterization in terms of a third-order anharmonic oscillator.     

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.     

Depairing critical current of 3He-B at all temperatures including gap distortion

Gap functions and superfluid densities are calculated at all temperatures in the presence of superflow. The critical current undergoes a significant reduction due to gap distortion.Institut für Theoretische Physik, Freie Universität Berlin