Structure of isolated vortex lines in rotating3He-A and3He-B

First we calculate the texture of a Mermin-Ho vortex in a field of 284 Oe and compare its energy with the energies of singular vortices in³He-A as a function of rotation speed and temperature. Then we calculate the order parameter components ψ _r , ψ_φ and ψ _z (r, ф,z are cylindrical coordinates) for an isolated vortex line in³He-B. We find that in the core region the transverse components ψ _r and ψ_φ are strongly reduced in comparison to the component ψ _z along the rotation axis. A strong magnetic field has the reverse effect: ψ _z is reduced in comparison to ψ _r and ψ_φ. Finally we calculate the corrections of order (1?T/T _c ) to the Ginzburg-Landau solutions. These yield an interesting structure of the vortex core. For radial distances large compared to the coherence length, ψ_φ is reduced as the temperature decreases.     

Vortex structures in rotating3He-A

In the hydrodynamic approximation, the free energy of high-field Seppälä-Volovik vortices in rotating³He-A is minimized with respect to an additional textural rotation angle. The transition from singular, singly quantized vortices to nonsingular, doubly quantized vortices is predicted to occur at a temperature-dependent angular velocity of order 1 rad / sec, in rough agreement with experimental observations.     

Magnetic textural deformation in rotating3He-A

An axial magnetic field is shown to deform the type I lattice of nonsingular vortices in rotating³He-A. At a critical field of orderH _D (30 G), this texture becomes unstable with respect to an array of doubly quantized vortices, seen in experiments at 284 G.     

NMR studies on vortices in rotating3He-A

NMR measurements are reported on rotating³He-A in a long cylindrical geometry of 5 mm diameter at a liquid pressure of 29.3 bar and in axial magnetic fields of 14.2, 28.4, and 56.9 mT. At 28.4 mT, NMR studies are also reported in fields inclined by 25° and 90° from the axis of rotation. The frequency shift, the width, and the intensity of the spin wave modes localized on the soft vortex cores, as well as the additional broadening of the main NMR line during rotation, were measured as a function of temperature, angular velocity , magnetic field intensity, and its inclination angle. Also observed were a critical angular velocity of vortex formation, hysteretic behavior in the number of vortices when comparing accelerating rotation to decelerating, and metastable vortex densities, presumably a vortex tangle after rapid oscillatory acceleration. The results can be understood in terms of the continuous 4 vortices first proposed by Seppälä and Volovik.     

Zero-sound attenuation in rotating and stationary3He-A and3He-B

We report on measurements of zero-sound attenuation in rotating and stationary³He-A and³He-B, in magnetic fields up to 350 mT. Strong and highly nonlinear rotation speed dependencies of sound amplitudes have been observed in both phases. The data gives information on vortex types and core sizes, although the analysis is not straightforward. The anomalous attenuation in³He-B at 200 mT near the AB transition, both in the stationary and in the rotating state, is interpreted to arise from the distortion of the energy gap of the B phase. Excess attenuation during the AB phase change was observed. Evidence for soft vortex cores in³He-B is presented. In addition, a critical velocity in the vortex free state, related to a textural transition, and the vortex creation times have been measured in³He-B. Furthermore, a metastable structure, possibly a new vortex state, has been observed in³He-B by rotating the sample through the A B transition.     

Ion trapping on vortex lines in rotating 3He-4He mixtures

The trapping lifetime and mobility of positive and negative ions on vortex lines in ³He-⁴He mixtures have been measured in the temperature range between 0.1 and 1.3 K. The binding energy of the positive ion is found to be reduced by the ³He, and the mobility of both ion species is lowered, providing evidence that the ³He condenses at the vortex core. The negative ion trapping lifetime is smaller than that of the positive ion in the region 0.3–0.5 K, indicating that the bubble structure may be modified by the presence of the condensed ³He. At 0.3 K the negative ion lifetime increases sharply, in qualitative agreement with theoretical models predicting an increase in the bubble radius at that temperature.Work supported by the National Science Foundation.     

Decay of vortex state and mutual friction in superfluid3He-A

The time dependence of decaying superfluid circulation is studied in³He-A in a singly connected cylindrical geometry. Comparison with experimental data is discussed.     

NMR signatures of topological objects in rotating superfluid3He-A

NMR spectrometry can be used to identify different topological objects in the order parameter field of rotating superfluid³He-A. We list their signatures in the cw NMR absorption line shape. Quantized vortex lines, domain walls, and their combination, the vortex sheet, all induce satellite peaks with specific intensities and frequency shifts in the NMR spectrum. Examples of spectra are presented to allow a comparison and to distinguish between different objects.     

Evidence for nonsingular vorticity in the Helsinki experiments on rotating3He-A

A theory for the satellite peak in the NMR spectrum of rotating superfluid³He-A is presented. Absorption at the frequency of the satellite is produced by spin waves localized on vortex lines. The peak frequency and total intensity of the satellite are calculated for both singular and nonsingular vortices; a comparison with the Helsinki NMR data strongly suggests that nonsingular vorticity was actually observed in the experiments.     

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.     

Anisotropy and flow in3He-A

Thermally driven counterflow in³He-A is investigated, using techniques that exploit the anisotropy of the fluid. The results are discussed in terms of both uniform-texture and helical-texture hypotheses. Anisotropy of heat flow is observed for the first time and a limitation of the superfluid velocity is strongly suggested.This work was supported by the U.S. Department of Energy under contract EY-76-S-03-0034, P.A. 143.     

Orbital motions in 3He-A

Orbital motions in ³He-A in very low magnetic field and driven by heat flow are studied using crossed detecting zero-sound fields. A precessional type of motion of the % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GabmiBayaajaaaaa!3C6D!\[\hat l\] field is frequently observed, though a rich variety of phenomena are also observed. There is no evidence for phase slip due to motion of nonsingular vortices along the surface. Motion of the % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GabmiBayaajaaaaa!3C6D!\[\hat l\] field following sudden parallel magnetic field turn-off is also observed above a threshold field change which is related to coherent dipolar energy. The time scale for these % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GabmiBayaajaaaaa!3C6D!\[\hat l\]-field changes is, however, a factor of about 10² larger than expected for the formation of composite solitons.Work supported by the U.S. Department of Energy under contract EY-76-S-03-0034, P.A. 143.     

Vortex sheet in superfluid3He-A

A new state of rotating superfluid³He-A has been found recently. Usually superfluids respond to rotation by creating an array of vortex lines, which are parallel to the rotation axis, and the circulation around them is quantized. In the new state the vorticity is located on a 2 dimensional sheet instead of 1 D lines. The sheet is parallel to the rotation axis z but in the x — y plane it folds to equidistant layers. The distance between the layers is larger but on the same order of magnitude as the distance between vortex lines. In contrast to other superfluids, the sheet is stable in the A phase because of its internal structure. The sheet has as a backbone a topologically stable domain wall called soliton, to which non-singular vorticity is bound. Thus it can exist in spite of its presumably higher energy. The vortex sheet is distinguished by its NMR response, in particular because of its higher absorption at a characteristic frequency. Experiment and theory on the vortex sheet are in good agreement.     

Stable circulation in superfluid3He-A

We present evidence of stable nonzero circulation in superfluid³He-A around a 16 µm diameter wire. The measurements are made on a rotating cryostat at T0.15T_c. Circulation offects the splitting of the wire's normal modes, which changes the beat frequency in the emf across the wire as it vibrates in a magnetic field. The different beat frequencies at 14 and at 21 bar suggest that circulation is present at 21 bar. Furthermore, at 21 bar the beat frequency is a linear function of rotation velocity, as is expected if circulation is trapped around the wire. At 14 bar no effect of rotation is observed. We propose that at 21 bar the ground state texture has nonzero circulation of order h/2m₃.     

Planar textures in superfluid 3He-A

Making use of the Ginzburg-Landau free energy, singularities such as disgyrations and vortex lines and stable textures in the planar (or two-dimensional) texture are studied. Here both % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaaieqaceWFSbGbaK% aaaaa!3883!\[{\hat l}\] and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaaieqaceWFKbGbaK% aaaaa!387B!\[{\hat d}\] vectors, characterizing the texture in ³He-A, are assumed parallel to each other and lie always in a plane. The dynamics of disgyrations is also considered.Work supported by the National Science Foundation under grant number DMR76-21032.     

Cutting of bent vortex lines

One of the major problems in the application of type II superconductors is the appearance of resistivity in cases where a current-carrying specimen is in a longitudinal magnetic field. This is explained by the onset of flux-line cutting events, followed by cross-joining of the line parts. The calculation given here shows the amount of repulsive force and energy between two curved vortex lines and examines the general stability of the vortex-vortex system. First, the actual interaction potential between curved vortices is computed. It includes all electromagnetic and core overlap terms of interaction and self-interaction, and allows computation of the system energy under all curved vortex-line configurations. A computer program is used to find the form of lowest free energy. To do this, special trial functions are established to describe the three-dimensional form of the vortex-vortex system. In these functions parameters determine the qualitative and quantitative form. The asymptotic boundary conditions are built into the nature of the trial functions. The computer program now minimizes the free energy with respect to these parameters. The resulting repulsive energy and force are more than ten times less than the known results for straight flux lines, especially for small asymptotic cutting angles. There is no sharp maximum in the plot of repulsive force versus flux-line separation. A remarkable result is the loss of general stability below a separation distance of several London penetration depths, depending on the cutting angle and the Ginzburg-Landau parameter. The explanation lies in the local attraction of central sections of the vortices as a result of configurational adaptation. This explains the onset of resistance at small currents and small magnetic fields.     

Equilibrium vortex density in a rotating two-dimensional superfluid

The equilibrium state of a two-dimensional superfluid in rotation is discussed. It is shown that, in equilibrium, the presence of polarizable vortex-antivortex pairs does not affect the critical angular velocity at which free vortices enter the system, nor does it affect the free vortex density at higher angular velocities.     

Energies of line singularities in 3He-A

We investigate a class of topologically equivalent line singularities in ³He-A to determine which of this class has the lowest free energy density. Most of our results concern the Landau-Ginzburg regime near T _c ; we offer a speculation on what might occur further below T _c .This research was supported in part by the National Science Foundation under Grant No. 77-18329 and through the Materials Science Center of Cornell University, Technical Report No. 4043.     

Theory of sound propagation in superfluid 3He-A

The theory of sound propagation in pair-correlated Fermi liquids developed previously by Wölfle is applied to the Anderson-Brinkman-Morel state, believed to describe ³He -A. The effect of quasiparticle collisions is included. Expressions for the anisotropic sound absorption and velocity valid at arbitrary frequency, temperature, and pressure are derived in terms of averages in momentum space of certain simple functions. Numerical results are in good agreement with lower pressure data but deviate by 10–20% at the melting pressure. The discrepancies are attributed to strong coupling effects not included in the theory.