Model for quantum scattering of rotons. II. Two-dimensional interaction

A speculative roton Hamiltonian is used to calculate scattering cross sections in the two-dimensional problem. For the hard-cylinder interaction, one gets the same result as that obtained by the usual wave scattering in the high-energy limit. The variational calculation for the roton-vortex interactionP ·v _s yields the temperature-independent momentum-transfer cross section _T 8 Å, improving the results obtained before by other methods.     

Model for quantum scattering of rotons. I. Hard-sphere interaction

A new model is proposed for the behavior of rotons. We postulate a linear Schrödinger equation with a hypothetical Hamiltonian fit to the dispersion relation and treat the three-dimensional scattering problem. All calculations are done in closed form. As an example, we study scattering by a hard sphere, which yields the same result as that obtained in the high-energy limit of the usual Schrödinger equation. The calculated mobility gives a reasonable fit to the radii of both positive and negative ions.Research sponsored in part under Contract No. F44620-68-C-0075.     

The Waves on the Vortex Ring in HeII

Using the equations of vortex dynamics the waves on a vortex ring are studied both analytically and by computer simulations. At small amplitudes the results coincide. The dispersion relations and the expressions of phase and group velocities are determined. The free and forced oscillations are considered. At large amplitudes the shape of a vortex is strongly distorted and the ring motion changes its direction. The second approximation calculations and computer simulations agree in order of magnitude.     

Scattering and binding of rotons

The interaction of rotons is studied by a direct examination of their orbital dynamics. It is found that upon representing rotons as point dipoles in the superfluid, obeying the Landau dispersion relation, negative energy states occur, some of which are stable and some unstable. The binding energy and effective mass of these states are found, but are too large to agree with experiment. Improvements in the Hamiltonian are explored which recognize the actual dispersion curve as well as higher moments in the interaction. Both modifications lead to binding energy and effective mass in better agreement with experiment. Scattering is also examined, and one finds that a representative roton-roton cross section may be obtained in the dipole approximation, again using Landau dispersion. The scattering cross section is shown to be in satisfactory agreement with neutron linewidths in the roton-dominated region, as well as the roton contribution to the shear viscosity.     

Variational theory of rotons in superfluid4He

We have extended the shadow wave function for excited states by including an explicit backflow term in the shadow variables. The variational computation gives a roton energy 9.18 K, only 6% above experiment. Similar agreement is found at freezing density. Also the strength Z_q of the single excitation peak greatly improves. The uniform agreement between theory and experiment at all wave vectors suggests that there is no qualitative difference in the wave function of phonon, maxon and roton excitations. Only the amount of backflow and short range correlations greatly varies with k.     

Bound states of rotons to impurities in He II

Rotons and impurities such as ions or solvated ³He particles can interact via dipolar forces from the flow fields of the roton and the motion of the impurity through the background fluid. Stable bound states of rotons to impurities can occur with near-zero total momentum. We give explicit results of bound-state calculations for solvated ³He, the positive helium ion, and the negative electron bubble.Research supported by the National Science Foundation and the Air Force Office of Scientific Research under grants NSF DMR 76-21814 and AFOSR 76-2880.     

Recent progress in the theory of rotons in superfluid4He

The excitation spectrum of superfluid ⁴ He is discussed on the basis of a shadow wave function. The potential and kinetic energies of the excitations are computed at different densities. The theory has been extended at finite temperature and we obtain the roton contribution to the radial distribution function, to the depletion of the condensate and to the dynamical structure factor S(q,). We present also the first realistic microscopic computation of the roton energy at finite T. Possible role of thermally excited vortices on rotons is considered. With a simple model we show that some discrepancies between the linewidth of the roton response in neutron and in Raman scattering can be resolved.     

On the Lamb Impulse Dynamics in Superfluid Turbulent HeII

Macroscopic dynamics of the hydrodynamic (Lamb) impulse J _V induced by the chaotic vortex filaments in the counterflowing superfluid turbulent HeII is studied. Both the Lamb impulse and its dynamics are considered on the basis of the Gaussian model of chaotic vortex tangle elaborated earlier by one of the authors (S.N.). The rate of change of the quantity J _V is shown to be equal to the force F _sn exerted by the normal component on the vortex tangle. That contradicts the obvious fact that in steady case the quantity J _V should have a constant value. Resolving that apparent paradox we draw some conclusions concerning the stochastic dynamics of the quantized vortex lines.     

Harmonics Generation in 4HeII by Femtosecond Laser Excitation

Non-resonant laser excitation of superfluid He by means of sub-picosecond pulses (=776 nm, 160 fs, 1 kHz) at intensities below the optical breakdown (I<4.5×10¹³ W/cm²) results in a strong emission at 258.7 nm corresponding to the third harmonic of the pump beam. Above a threshold of 0.75×10¹³ W/cm² the intensity and halfwidth of this collinear emission show a nearly linear dependence on the input pulse energy. Absolute UV energies up to 10 nJ and a conversion efficiency of about 10^–4 are observed. The emission intensity shows a significantg instability around 2.17 K.     

Study of rotons in superfluid4He by Raman scattering

There is a long standing disagreement between neutron and Raman scattering from rotons in superfluid⁴He near T. In neutron scattering the linewidth becomes very large and the roton signal seems to disappear at T. A substantially smaller linewidth is observed with Raman scattering and the roton signal is present even at T. We have interpreted this difference (J. Low Temp. Phys. 93, 879 (1993)) as due to a modulation of the roton energy by a fluctuating local superfluid velocity due to proliferation of vorticity as T is approached. This gives rise to an extra contribution to the roton linewidth in neutron scattering but not in Raman scattering in which two rotons with almost opposite momenta are excited. We propose a test of this explanation with evanescent field Raman scattering. It has been suggested (A. Kuklov, A. Bulatov and J.L. Birman, Phys. Rev. Lett. 72, 3855 (1994)) that in such a measurement the scattering by a single roton should be measurable. In this case the presence of a fluctuating local superfluid velocity should show up as in neutron scattering and corroborate our proposal.     

A comment on rotons in two-dimensional liquid helium II

The spectrum of rotons in two-dimensional liquid helium II is estimated within the Bijl-Feynman theory. Due to stronger short-range order and more effective back flow, the roton gap is found to be about half as big as the bulk roton energy.Supported by grants of the Deutsche Forschungsgemeinschaft and, in part, by the U.S. National Science Foundation through Grants DMR73-02363 and DMR72-02977 A03.     

Unsteady Heat Transfer in HeII in Cylindrical and Spherical Geometries

The problem of evolution of strong heat pulses in superfluid helium interacting with quantum vortices induced by these pulses is investigated numerically on the basis of equations of hydrodynamics of superfluid turbulence. The “discontinuity decay” method (Godunov method) is used for the calculations. In order to study nonlinear effects, the initial equations are expanded to the second order in the amplitudes of pulses. The influence of the main flow parameters—heat flux and rest time—on temperature evolution is reported. The numerical investigations are carried out at the bath temperature of 1.4 K. The obtained results are compared with experimental data.      

The evaporation of atoms from the surface of superfluid4He by rotons

A beam of rotons is directed toward the liquid surface at normal incidence. At the surface some rotons will evaporate atoms, with a wave-vector dependant probability p(q). The kinetic energy of the evaporated atom is equal to the difference in energy between the energy of the roton and the binding energy of a⁴He atom to the surface (7.16 K). The atoms which are evaporated are detected by a bolometer above the surface. The path length of the roton in the liquid, and the atom in the vacuum, are varied by changing the liquid level in the cell. By thus changing the liquid level, we change the flight time between heater and bolometer for a given wavevector of roton. We analyse the time-of-flight spectra received by the bolometer to reveal the product n(q)p(q) where n(q) is the distribution of rotons. If we assume that n(q) reflects the temperature of the phonons in the heater used to inject the rotons, we can deduce p(q). The resulting p(q) shows an unexpected decrease towards roton minimum at q < 2.1 Å^–1.     

Energy Spectrum of the Random Velocity Field Induced by a Gaussian Vortex Tangle in HeII

Using the Gaussian model of the vortex tangle (VT) arising in the turbulent superfluid Hell, we calculate the energy spectrum E(k) of the 3D random velocity field induced by that VT. If the VT is assumed to be a purely fractal object with Haussdorf dimension H_D, the E(k) is a power-like function E(k)k^-2+HD. A more realistic VT in HeII is a semi-fractal object, behaving as smooth line for small separations R ( is the label coordinate, R is mean curvature) and having a random walk structure for large with H_D=2. For that case calculations give a spectrum E(k) that is k-independent for k smaller than 1/R (but larger than the inverse size of the system) and that scales as k^–1 for larger k. The latter reflects the fact, that for small, scales a vortex filament behaves as a smooth line. Our results agree with recent numerical simulations.     

Simple thermo-level meter for HeI and HeII by a dynamic method

A simple liquid helium level monitoring device has been developed using pairs of carbon resistors mounted on a stainless steel ‘dipstick’. The device described has a resolution of 50 mm and negligible power dissipation. The level of both superfluid and normal helium, and the liquid temperatures, can be determined. The device has no moving parts and after some years of use has proved to be sturdy and durable. A similar device is proposed with a resolution of 12 mm and a microcomputer data recording system.     

Thermomechanical correction to the drag force on a sphere and a cylinder moving in HeII

The drag force on a sphere and a cylinder moving in HeII is considered on the basis of the full system of the two-fluid dissipative hydro-dynamical equations. It is shown, that due to the thermomechanical effect in superfluid there exists a specific correction to the drag force on a body. The value of this thermomechanical correction F has the order of the ratio1 _T/R or1 _T ² /R² (R is the size of the body, and1 _T is the length of the formation of the convective heat flow). The value of F grows at low temperatures, where1 _T is proportional to the characteristic time of the phonon-roton scattering process, and in the region near the -point, where1 _T has the same temperature dependence as the order parameter correlation length 1_T (T–T)^–2/3. The effects under discussion can be investigated experimentally by measurements of the mobility of small objects. A comparison is given with the data of the mobility of positive and negative ions in HeII near the -point.     

The crack-inclusion interaction problem

With the application to weld defects in mind, the interaction problem between a planar crack and a flat inclusion in an elastic solid is considered. The elastic inclusion is assumed to be sufficiently thin so that the thickness distribution of the stresses in the inclusion may be neglected. The problem is reduced to a system of four integral equations having Cauchy-type dominant kernels. The stress-intensity factors are calculated and tabulated for various crack-inclusion geometries and the inclusion to matrix modulus ratios, and for general homogeneous loading conditions away from the crack-inclusion region.     

The interaction of phase-slip centers in superconducting filaments

Theoretical step structures for ideal homogeneous filaments based on the interactions implicit in the simple phase-slip-center model of Skocpol, Beasley, and Tinkham are displayed explicitly.Research supported in part by NSF and ONR.Senior U.S. Scientist awardee of Alexander von Humboldt Foundation.