Collective modes and spin waves in superfluid3He-B

Making use of a model Hamiltonian, which includes, besides the kinetic energy and the BCS-like pairing energy, the spin exchange energy as well as the dipolar interaction energy, we study theoretically the collective modes of the Balian - Werthamer state in the collisionless limit. In the absence of the dipolar interaction energy, the collective modes are classified according to their total angular momentumJ. Among 18 distinct modes, one mode withJ=0 and three modes withJ=1 are gapless. The latter modes couple with the spin fluctuation and are identified with spin wave modes. The effect of the dipolar interaction on these modes is discussed.Supported by the National Science Foundation.     

Surface coupling to collective and single-particle spin modes in normal 3He

We propose a simple technique for probing the single-particle and collective excitations in normal ³He via a standing magnetic surface wave of arbitrary and fixed k generated by a meander-line coil. An analytic treatment of the Landau-Sillin equation enables us to calculate the power absorption spectrum, which displays singularities associated with the l = 0 spin mode and a Doppler-shifted spin resonance (DSSR) of the single-particle excitations. Experimental measurement of the spectrum would determine the Landau parameters (F ₀ ^a, F ₁ ^a) with spectroscopic precision. Furthermore, observation of the DSSR thresholds would provide an independent determination of the Fermi velocity and F ₀ ^a. Finally, a possibility of exciting higher, l 1, spin modes is briefly mentioned in the framework of an extended Leggett-Rice equation.     

Ultrasound attenuation peaks due to order parameter collective modes in impure superconductors with strong electron-hole asymmetry

First we develop the general theory of order parameter collective modes and their sound attenuation in dirty superconductors with even-parity pairing. For the BCS state the theory yields a peak in the attenuation of longitudinal ultrasound whose height is proportional to _o[J(s)]² and whose positiont _p 1 -T _p /T _c and width are proportional to _o/ = 2sql(s=v _s /v _F ). Here, is the particle-hole asymmetry parameter,J(s) a given function, _o the sound frequency, and =1/2 _N ^–1 the normal state scattering rate. For T _c /T _o 0.1,s=2/3, and _o/ ql « 1, the peak agrees qualitatively with the peaks found earlier for the polar and axial states. However, at low temperatures the attenuation goes like T⁴ instead ofT ³. We conclude that the observed longitudinal ultrasound peaks in the heavy-fermion superconductors UBe₁₃ and UPt₃ can be explained equally well by any anisotropic state having nodes in the gap, in particular, by a conventional state whose gap has the full symmetry of the Fermi surface. The nature of the state is reflected mainly in the dependence of the peak on the direction of sound propagation.     

Gap parameters and collective modes for 3He-B in the presence of a strong magnetic field

First we determine the 4 × 4 matrix Green's function for a p-wave pairing superfluid in a magnetic field where the order parameter is given by a real 3 × 3 matrix. For the B phase we take an order parameter which is equal to ₁ times a 3 × 3 matrix, yielding a rotation of angle about the z-axis and a dilatation ₂/₁ along the z-axis. Then the self-consistency equation for the 4 × 4 matrix self-energy reduces to three equations for ₁, ₂, and cos , and a fourth equation for the renormalized Larmor frequency. We find that, for increasing field, ₁ increases and ₂ decreases with respect to the zero-field gap _00. Above a (temperature-dependent) critical field we find ₂ = 0 and ₂ = 90° corresponding to the planar state of lowest dipole energy. The correlation functions for the order parameter collective modes are calculated with the help of a previous theory. The results can be expressed in terms of six universal functions describing internal magnetization and virtual excitations of pairs of quasiparticles with all spin orientations. The complete set of eigenfrequencies as functions of the field is calculated for T = 0 and q = 0. The longitudinal NMR frequency is found to be almost independent of the field. We find a splitting of the pair-vibration frequencies (8/5) ^1/2₀₀ and (12/5)^1/2₀₀ which is linear in the field. This splitting is caused by fluctuations involving the spin-singlet component of the anomalous propagator. The splitting of the pair-vibration frequency (12/5)^1/2₀₀ (of the order 6H MHz, H in kG) should be observable by sound absorption experiments in strong magnetic fields.     

The influence of superflow on collective excitations in3He-B

The influence of superflow on the order parameter collective modes in³He-B is investigated for temperatures 0T/T _c 0.5 by the continual integral method. The superflow induces a gap distortion, which splits the pair-breaking, squashing, and real squashing modes at zero momentumk. Furthermore, a crossing of the real squashing mode branches with different|J _z | occurs at nonzerok. In addition, the superflow makes it possible to observe the pair-breaking modes and causes a hydrodynamic shift of longitudinal NMR frequency.     

Dispersion relations of the high-frequency modes in3He-B

The dispersion relations for the high-frequency modes in³He-B are obtained for finite temperature and including Fermi liquid corrections. The coupling between zero sound and the squashing mode is studied in detail. It is shown to have a more complicated structure than what was previously believed. Finally, it is shown that there are no modes with frequency around 2.     

A study of theJ=2 collective modes of superfluid3He-B using a dynamic pressure sweeping technique

We report measurements of the positions of theJ=2 collective modes in zero field using a new technique involving alternate pressure and temperature sweeps. Extensive measurements over a wide range of pressure, temperature, and frequency have been performed and are compared with previous data.     

Simultaneous spin and space rotation experiments in3He-B

Observation of enhanced magnetic relaxation due to counterflow and quantized vortices is reported in rotating superfluid³He-B. These phenomena were studied using a novel method based on the homogeneously precessing domain (HPD) mode, characteristic of the B-phase spin dynamics. The additional HPD absorption can be related to the equilibrium vortex structures and, therefore, this method yields information complementary to textural CW-NMR measurements, which are sensitive to the static magnetic properties of vortex cores. Primarily, the HPD method provides a simple and effective means for studying transient processes, e.g., vortex creation in³He-B; a clear difference between axisymmetric and non-axisymmetric vortices was observed.     

The spin oscillations of 3He-B in the presence of dipole forces

The problem of the spin oscillations of ³He-B is investigated in the presence of dipole forces under the assumption that only the zeroth parameter of the Fermi-liquid interaction is significant. The solution of the gap equation gives the new thermodynamical phases. The spin susceptibility tensor is obtained in the most general form. It is shown that in addition to the longitudinal and transverse components some extra circular components can appear. These components are analogous to the components of the dielectric tensor that are responsible for the circular dichroism of optically active substances.     

Collective modes and sound propagation in a magnetic field in superfluid3He-B

A high-resolution, ultrasonic (12–89 MHz) acoustic impedance technique has been used to investigate the order parameter collective modes in superfluid³He-B over a pressure range of 0–15 bar and in magnetic fields up to 180 mT. In agreement with earlier experiments, theJ=2 real squashing mode has been observed to split into five components in small magnetic fields. However, contrary to earlier theoretical estimates, the Zeeman shifts have been found to become extremely nonlinear as the magnetic field is increased. The extent of this nonlinearity is largest at low pressures and at temperatures close toT _c. In comparison with recent theoretical work, the nonlinear Zeeman shifts may be explained as a result of two effects. First, there is a significant distortion of the B-phase energy gap in large magnetic fields. Second, there is an important coupling between the sameJ _zsubstates of the differentJ modes. In this sense the nonlinear evolution of the real squashing mode constitutes the observation of the Paschen-Back effect in³He-B. A comparison of the observed Zeeman shifts with theoretical expressions has yielded information about particle-particle and particle-hole interaction effects in the superfluid. In the limitT 0 and above a threshold field, the real squashing mode has been found to possess additional structure. TheJ _z=0 substate has been observed to split into a doublet. The separation between the two components of the doublet is of the order of 100–200 kHz and remains independent of the magnetic field. The origin of the doublet may be understood in terms of a recent theory which postulates a texture-dependent collective mode frequency. Further, at extremely small fields the effects due to dispersion of the real squashing modes have been found to be important. The magnitude of the dispersion-induced mode splitting in zero field is found to be consistent with theoretical predictions. TheJ=2 squashing mode has also been studied in the presence of a magnetic field. TheJ _z=0 state of the squashing mode is observed to shift to lower temperatures in a magnetic field. An additional field dependence of the observed acoustic impedance is interpreted as the evolution of theJ _z=–1, –2 states, but appears to be inconsistent with theoretical predictions.     

Comment on the coupling of zero sound to theJ=1− modes of3He-B

Features in the zero sound attenuation near the pair-breaking edge in superfluid³He-B have been observed in large magnetic fields. Schopohl and Tewordt [J. Low Temp. Phys. 57, 601 (1984)] claim that theJ=1^–,M=±1 order-parameter collective modes couple to zero sound as a result of the distortion of the equilibrium order parameter by a magnetic field; they identify the new features with these modes. However, we show that, when the effect of gap distortion on the collective modes is properly taken into account, the collective modes equations of Schopohl and Tewordt yield no direct coupling of zero sound to theJ=1^– modes. Thus, the identification of the absorption features reported by Ling, Saunders, and Dobbs [Phys. Rev. Lett. 59, 461 (1987)] near the pair-breaking edge with theJ=1 ^– modes is not clearly established.     

Effect of dipole interaction on collective modes in3He-A

A general theory for the correlation functions of superfluid ³ He which takes into account rigorously the magnetic dipole interaction is developed. The resulting equations are solved for the Anderson-Brinkman-Morel (ABM) state and for wave vectorsq oriented parallel to the energy gap axis. Then the dispersion relations of low-frequency modes, including Fermi liquid corrections and damping due to pair breaking, are calculated in the zero-temperature and zero-field limit. There are two real frequency modes arising from each of the longitudinal and transverse spin density correlation functions: a spin wave and an orbit wave,both exhibiting a frequency gap where that of the spin wave is somewhat modified in comparison to the unperturbed longitudinal nuclear magnetic resonance frequency _L ^/ABM .The orbit wave is damped much more strongly than the spin wave. Further, there are two real frequency modes arising from the density correlation function: the sound wave, having a frequency gap of the order _L ^/ABM , and an orbit wave, exhibiting a gap in wave number of order _L ^/ABM /v _F.—The NMR frequency undergoes a small splitting, which is the result of the splitting of the energy gap due to the dipole interaction. One of the two gaps still has nodes.—In addition to these low-frequency modes our equations yield resonances at frequencies of the order of the gap frequency ₀ /, i.e., at =1.22 ₀ / and at =1.58 ₀ /. The damping and the oscillator strengths of these resonances are calculated.     

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.     

Nonlinear Dynamics of Superflow in 3He-B: The Coupling of Sound and Helmholtz Oscillations

The nonlinear relation between supercurrent and superfluid velocity:j_s(v_s)=_sv_s (1-(v_s/v_c)²/3)provides a coupling mechanism between the sound and Helmholtz modes of a short cylindrical cavity closed by a superleak at the centre of one end. Appropriate nonlinear equations have been formulated and solved numerically. The simulations reveal an interesting coupling of sound and Helmholtz oscillations at large amplitude. If the driving frequency of the sound is above the (large amplitude) resonance by twice the (large amplitude) Helmholtz frequency, only an initial Helmholtz drive is required to set up combined Helmholtz and sound oscillations, which can then be sustained by off-resonance sound drive alone. The combined motion, in which Helmholtz oscillations are amplified by sound drive, has been observed experimentally, but it decays when the Helmholtz drive is switched off, although it is sustained for a while by Helmholtz drive alone. Modified simulations that allow for quasiparticle relaxation show that slow relaxation is the probable reason that the combined motion decays without Helmholtz drive even at the lowest temperatures where the resonance Q is high. Comparison of simulations and experiment also gives evidence for larger dissipation at large amplitudes and associated memory effects. The simulations show that in the combined motion the negative slope region of j_s(v_s) for v_s>v _c is being explored.     

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.     

The collective excitations of the order parameter in HTSC and heavy fermion superconductors (HFSC) underD-pairing

Within the path integral model ofd-pairing for HTSC and HFSC developed recently by P. N. Brusov and N. P. Brusova¹, the whole collective mode spectrum has been calculated for the first time for five states of HTSC that arise in their symmetry classification as well as for three states of HFSC. For HTSC the calculations have been made for the state which is a candidate for the superconducting state of HTSC in light of numerous recent experiments as well as for , d_xy, d_xz, d_yz states. For HFSC we considered three states, among themdγ and Y₂₋₁ states treated by Hirashima and Namaizawa² who used the theory of response. The number of the collective modes in each phase of both superconducting systems is equal to 10, among which five are high frequency modes while five others are Goldstone or Goldstone-like ones. The spectrum can be used to identify the superconducting states through ultrasound and microwave absorption experiments as well as to interpret these experiments.     

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.     

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     

Coupling of rayleigh-like waves with zero-sound modes in normal3He

The Landau kinetic equation is solved in the collisionless regime for a sample of normal³He excited by a surface perturbation of arbitrary ω andk. The boundary condition for the nonequilibrium particle distribution is determined for the case of specular reflection of the elementary excitations at the interface. Using the above solution, the energy flux through the boundary is obtained as a function of the surface wave velocity ω/k. The absorption spectrum and its frequency derivative are calculated numerically for typical values of temperature and pressure. The spectrum displays a sharp, resonant-like maximum concentrated at the longitudinal sound velocity and a sharp maximum of the derivative concentrated at the transverse sound velocity. The energy transfer is cut off discontinuously below the Fermi velocity. An experimental measurement of the energy transfer spectrum would permit a determination of both zero-sound velocities and the Fermi velocity with spectroscopic precision.