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Focusing of Negative Ions by Vortices in Rotating He 3 − A
Abstract
Experiments with negative ions in rotating superfluid /sup 3/He-A show strong retardation and deformation of ion pulses drifting parallel to the angular velocity ..cap omega... This is caused by the interaction between the ions and the l texture in the soft cores of A-phase vortices. The interaction is mediated by the anisotropic ion mobility: v = ..mu../sub perpendicular/E-..delta mu..(Exl)l. The (..cap omega..,T) dependence of the retardation is explained by a model that assumes focusing of ions into the vortex cores, along which the mobility ..mu../sub c/ is lower than that in the bulk liquid, ..mu../sub perpendicular/. We find ..mu../sub c/ = ..mu../sub perpendicular/-0.6 ..delta mu...
... Indeed the anisotropy of the negative ion mobility was calculated by extending the scattering theory for the B-phase by Baym et al. 37 to scattering by an ion in 3 He-A, 22,49 and measurements of the mobility anisotropy, µ ⊥ − µ || were made via pulse-shape, time-of-flight experiments on vortex textures of superfluid 3 He-A. 50 Note that the drag force on the electron bubble is insensitive to the direction of the chiral axis, i.e. the drag force for +E||l and −E||l are the same. ...
Electrons embedded in liquid $^3$He form mesoscopic bubbles with radii large compared to the interatomic distance between $^3$He atoms, voids of $N_{bubble}\approx 200$ $^3$He atoms, generating a negative ion with a large effective mass that scatters thermal excitations. We develop scattering theory of Bogoliubov quasiparticles by negative ions embedded in $^3$He-A that incorporates the broken symmetries of $^3$He-A, particularly time-reversal and mirror symmetry in a plane containing the chiral axis $\hat{\bf l}$. Multiple scattering by the ion potential, combined with Andreev scattering by the chiral order parameter, leads to a spectrum of Weyl Fermions bound to the ion that support a mass current circulating the electron bubble - the mesoscopic realization of chiral edge currents in superfluid $^3$He-A films. A consequence is that electron bubbles embedded in $^3$He-A acquire angular momentum, ${\bf L}\approx -(N_{bubble}/2)\hbar\,\hat{\bf l}$, inherited from the chiral ground state. We extend the scattering theory to calculate the forces on a moving electron bubble, both the Stokes drag and a transverse force, ${\bf F}_{W} = \frac{e}{c}{\bf v}\times{\bf B}_{W}$, defined by an effective magnetic field, ${\bf B}_{W}\propto\hat{\bf l}$, generated by the scattering of thermal quasiparticles off the spectrum of Weyl Fermions bound to the moving ion. The transverse force is responsible for the anomalous Hall effect for electron bubbles driven by an electric field reported by the RIKEN group. Our results for the scattering cross section, drag and transverse forces on moving ions are compared with experiments, and shown to provide a quantitative understanding of the temperature dependence of the mobility and anomalous Hall angle for electron bubbles in normal and superfluid $^3$He-A. We also discuss our results in relation to earlier theoretical work on negative ions in superfluid $^3$He.
New techniques, both for generating and detecting turbulence in
the helium superfluids 3He-B and 4He, have recently given insight in
how turbulence is started, what the dissipation mechanisms are, and
how turbulence decays when it appears as a transient state or when
externally applied turbulent pumping is switched off. Important
simplifications are obtained by using 3He-B as working fluid, where
the highly viscous normal component is practically always in a
state of laminar flow, or by cooling 4He to low temperatures where
the normal fraction becomes vanishingly small. We describe recent
studies from the low temperature regime, where mutual friction
becomes small or practically vanishes. This allows us to elucidate
the mechanisms at work in quantum turbulence on approaching the
zero temperature limit.
The first measurements on vortices in rotating superfluid 3He have been conducted in the Low Temperature Laboratory at Helsinki University of Technology during the past five years. These experiments have revealed unique vortex phenomena that are not observed in any other known superfluids. In this review, the concept of broken symmetry is applied to investigate the quantized vortex lines in superfluid 3He. In the superfluid A phase, vorticity can be supported by a continuous winding of the order parameter; this gives rise to continuous "coreless" vortices with two flow quanta. Novel vortices with a half-integer number of circulation quanta may also exist in 3He-A due to a combined symmetry of the superfluid state. In the superfluid B phase, the vortices have a complicated core structure. The vortex-core matter is ferromagnetic and superfluid, and it displays broken parity. The ferromagnetism of the core is observed in NMR experiments due to a gyromagnetic effect. The calculated core structures exhibit an experimentally observed first-order phase transition. This vortex-core transition in rotating 3He-B may be understood in terms of a change in the topology for flaring-out of the vortex singularity into higher dimensions; the topological identification further suggests that the phase transition manifests a spontaneous bifurcation of vorticity—involving half-quantum vortices in 3He-B. These recent advances of interest in quantum liquids are also of general relevance to a wide range of fields beyond low-temperature physics.
Low-energy excitations in quantum fluids are most directly encountered by ions. In the superfluid phases of (3)He the relevant elementary excitations are Bogoliubov quasiparticles which undergo repeated scattering off an ion in the presence of a divergent density of states. A quantum-mechanical calculation is given of the resonant (3)He quasiparticle-scattering-limited mobility for negative ions in the anisotropic bulk (3)He-A (A phase) and (3)He-P (polar phase) which is exact when the quasiparticles scatter elastically. A new numerical scheme is developed to solve the singular equations for quasiparticle-ion scattering in the A and P phases. The (3)He quasiparticle-ion cross sections are discussed, which allow one to account for the mobility data with essentially no free parameters. The calculated mobility thus facilitates an introduction of ion spectroscopy to extract useful information on fundamental properties of the superfluid state, such as the temperature dependence of the energy gap in (3)He-A.
Completely novel continuous vortices in He³-A and spontaneously magnetized singular vortices in He³-B are just two of the many interesting peculiarities of rotating superfluid He³.
First experiments suggesting the presence of singular vortices in rotating ³He-A are reported. Our studies on the interaction between negative ions and vortex textures verify the existence of the continuous vortices seen earlier by NMR, but evidence for a new vortex state with high ion mobility along the core is also found. The nucleation of the new vortex depends on the external magnetic field and the rotation history of the sample.
The flow of superfluid helium around a vibrating microsphere is investigated at temperatures between 1 K and 25 mK. At small oscillation amplitudes pure potential flow is observed, the linear drag force on the sphere being determined only by ballistic quasiparticle scattering below 0.7 K with phonons contributing exclusively below 0.5 K. At larger oscillation amplitudes a strongly nonlinear drag force gives evidence of stable turbulent flow if at least 0.6 pW are transferred from the sphere to the turbulent superfluid. In an intermediate range of amplitudes (or driving forces) both flow patterns are unstable and intermittent switching between both is observed below 0.5 K. We have recorded time series of this switching phenomenon at constant drives and temperatures lasting up to 36 hours. We have made a statistical analysis of the times series by means of reliability theory. The lifetime of the turbulent phases grows with increasing drive and diverges at a critical value (or at least becomes unmeasurably long). Stability of the laminar phases in the intermediate regime depends on the excess velocity of the sphere above the critical velocity. Metastable laminar phases are observed above the critical velocity having a mean lifetime limited to 25 minutes by natural background radioactivity which occasionally produces local vorticity due to ionization of the liquid. Finally, it is suggested that the breakdown of potential flow belongs to the class of “system failure” experiments which is well known in reliability testing and whose statistical properties are described by extreme-value theory.
PACS numbers: 67.40.Vs, 47.27.Cn.
We have constructed an experimental chamber permitting simultaneous observation of both the transverse cw-NMR spectrum and the time-of-flight of negative ions from a rotating sample of vortices in 3He. Our measurements in the A-phase confirm that the vortex state, causing the large delay and strong deformation of the ion pulses drifting parallel to the rotation axis, is also responsible for the satellite peak corresponding to continuos vortices in the NMR spectrum. Unfortunately, singular vortices, whose distinct ion response was detected earlier, could not be observed in the present setup.
Experimental results on vortices in rotating superfluid He-3 are reviewed. In He-3 A, vorticity may be supported by a continuous winding of the order parameter which gives rise to the observed continuous coreless vortices with two flow quanta. Exotic vortices with half-integer circulation quanta may also exist in this phase. In He-3 B, an abrupt change in the cortex-core structure has been theoretically explained as a topological phase transition involving half-quantum vortices. NMR experiments on rotating He-3 B have revealed that the cores of vortices possess spontaneously broken parity, manifested by their ferromagnetic superfluid cores.
The first measurements on vortices in rotating superfluid ³He have been conducted in the Low Temperature Laboratory at Helsinki University of Technology during the past five years. These experiments have revealed unique vortex phenomena that are not observed in any other known superfluids. In this review, the concept of broken symmetry is applied to investigate the quantized vortex lines in superfluid ³He. In the superfluid A phase, vorticity can be supported by a continuous winding of the order parameter; this gives rise to continuous ''coreless'' vortices with two flow quanta. Novel vortices with a half-integer number of circulation quanta may also exist in ³He-A due to a combined symmetry of the superfluid state. In the superfluid B phase, the vortices have a complicated core structure. The vortex-core matter is ferromagnetic and superfluid, and it displays broken parity. The ferromagnetism of the core is observed in NMR experiments due to a gyromagnetic effect. The calculated core structures exhibit an experimentally observed first-order phase transition. This vortex-core transition in rotating ³He-B may be understood in terms of a change in the topology for flaring-out of the vortex singularity into higher dimensions; the topological identification further suggests that the phase transition manifests a spontaneous bifurcation of vorticity: involving half-quantum vortices in ³He-B. These recent advances of interest in quantum liquids are also of general relevance to a wide range of fields beyond low-temperature physics.
We report the first measurements of negative ion motion in the superfluid phases of 3He and in the normal phase below 17 mK. Refrigeration was achieved with nuclear demagnetization of copper and we used a pulsed NMR platinum powder thermometer immersed in the liquid. In the A phase the longitudinal resonance frequency provided an additional high-resolution thermometer. In the normal phase we observed a strictly temperature-independent mobility. In the superfluid phases we found two velocity regimes. For small applied electric fields the velocity is a linear function of the field and the corresponding mobility increases monotonically toward lower temperatures. At high electric fields the velocity is a nonlinear function of the field as a result of the pair-breaking effect of the moving ion. Available theoretical calculations are only in partial agreement with our results.
Four vortex textures in 3He-A, one analytic and three
singular, are proposed as possible candidates to explain recent
transverse NMR results in the Helsinki rotating cryostat. Although the
analytic texture, previously discussed by Seppälä and Volovik,
has a transverse NMR shift and absorption closest to the observed
values, the agreement with experiment is not wholly satisfactory.
Longitudinal NMR shifts and absorption are calculated for the four
textures; such measurements might determine which vortex is present.
The mobility of positive and negative ions has been studied as a function of temperature, electric drift field, and (in the A phase) magnetic-field direction in superfluid 3He; the A-phase mobility is observed to be anisotropic with μ⊥>μ∥ for both positive and negative ions.
A theory for the satellite peak in the NMR spectrum of rotating superfluid ³He--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.
New NMR measurements are reported on continuous /sup 3/He-A vortices in tilted magnetic fields. We introduce a symmetry classification of the continuous vortices with broken axial symmetry. It is found that the discrete internal symmetry may in addition be broken in two inequivalent ways, producing two different continuous vortices. Although NMR may not distinguish between these two vortices, the observed vortex satellite peak is well accounted for by spin waves localized in the soft cores of such vortices.
The free energy of rotating 3He-A in zero magnetic field and large cylinders (R≫10 μm) is calculated for several textural configurations at slow rotation speeds (Ω≈ℏ/2mR2). The sequence of equilibrium states with increasing Ω is studied near Tc. A transition from the Mermin-Ho state to one of higher circulation should occur for Ω≈9.6ℏ/2mR2. This transition may be delayed, however, by the onset of azimuthal distortion ("swirl") in the Mermin-Ho state. A new experimental method is proposed for studying various textural configurations, based on the anistropy of negative-ion mobility. Significant and characteristic focusing should occur when ions are swept through long cylinders of length L>10R. In addition, time-of-flight information can also help identify the textural state. Finally, the qualitative effects of weak magnetic fields and lower temperatures are discussed.
The mobility tensor of negative ions in the A phase of superfluid 3He is calculated for temperatures close to T
c
. In this regime the scattering of superfluid quasiparticles from an electron bubble is practically elastic and the mobility tensor is expressed in terms of momentum transfer cross sections for an ion at rest. These generalized transport cross sections are obtained from the quasiparticle-ion scattering T-matrix, which we evaluate in terms of the normal state scattering amplitude. The p-wave pairing correlations in the intermediate states result in important interference effects among all partial waves in the scattering process, and, in addition, for the low-energy quasiparticles they lead to resonant states below the gap edge. These phenomena modify the scattering amplitude in the superfluid in an essential way and the differential quasiparticle-ion cross section is found to display strongly anisotropic, energy-dependent variations on the scale of the superfluid energy gap. We find that, in contrast to simple approximations, for low quasiparticle energies the parallel and perpendicular momentum transfer cross sections are very different from one another. Close to T
c
, the calculated mobility remains rather isotropic, but at lower temperatures the anisotropy is considerably larger than predicted by simple approximations for the cross section. The computed results are compared with the available measurements.
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 Seppl and Volovik.
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 [^(n)]\hat n
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
[^(n)]\hat n
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
[^(n)]\hat n
texture due to the long-range orientating effect of their cores on the order parameter. The discontinuity in the satellite splitting at 1–T/Tc=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.