Search

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.

In the Raman spectrum of superfluid 4He under saturated vapour pressure at 1.51K, fine structures have been observed at energy shifts closely corresponding to twice the energies of the extremal points of the phonon-roton dispersion curve. From the observed cusps, twice the energies of the maximum and plateau regions of the dispersion curve have been determined to be 2 Delta 1=27.5+or-0.5K and 2 Delta 2=34.8+or-0.5K respectively. The two-roton bound state has been resolved as a pronounced single peak, and twice the roton energy and the attractive binding energy of the two-roton state have been determined to be 2 Delta 0=17.1+or-0.5K and EB=0.5+or-0.3K respectively. The observed fine structures clearly show the existence of the interactions between two elementary excitations having opposite and equal wavevectors.

We report the first observation of objects moving through He II with equilibrium drift velocities v beyond the Landau critical velocity vc,for roton creation. With T about 0.4 K, P = 25 bar, F =2 kV/cm, (v - vc) for negative ions is larger than a theoretical prediction by a factor of about 10^5. The vortex-ring nucleation rate is found to decrease with F above 300 V/cm, thus resolving apparent inconsistencies between earlier experiments.

In continuation of our investigations on the cooling effect which occurs when He II is forced to flow under adiabatic conditions through a system consisting of a superleak, a chamber and a capillary in series measurements on the flow of He II under high overall pressures are reported. It is found that as expected the limiting temperature for cooling of about 0.75 K does indeed decrease with increasing overall pressure, occurring at a fixed roton density. The occurence of this limit appears to be a complicated phenomenon, in which the roton density as well as dissipation by the flow, either in the chamber or in the capillary, seems to play an essential role. In the present apparatus the velocities of the normal and superfluid components could be varied independently in a wide range of velocities, enabling us to study the nature of the all-important mutual-friction force. It is found that a mutual-friction force of the Gorter-Mellink type, i.e., Fsn = Aρsρn (vs - vn)3 gives an adequate description of the behaviour of the chamber temperature at sufficiently low velocities.

A Monte Carlo calculation of the Feynman-Cohen excitation spectrum is performed for helium confined to two dimensions. No approximations are used. Results for several different areal densities are compared with an exactly similar calculation in three dimensions. Among other results the roton gap is found to be significantly lower at corresponding densities in two dimensions, much smaller than previous estimates, and in substantial agreement with experiment.

The forces between HD molecules have been derived from those of H2. The damping of theJ=2 roton by its decay intoJ=1 rotons and phonons has been estimated. The result agrees with experiments of Silvera and McTague.

The energy excitation spectrum of a system of bosons interacting through a two-body potential composed of a hard core followed by a sum of two gaussians and having a density equal to that of liquid 4He at the lambda transition temperature (ϱλ = 2.06 x 1022 particles/cm3) is obtained using the reaction matrix formalism. The spectrum gives not only a qualitative agreement, showing the “phonon” branch and the “roton” dip clearly, but also a quantitative agreement when compared with the experimental energy excitation spectrum for liquid 4He. The maximum and the minimum of the theoretical energy excitation spectrum occur at k = 1.1 A -1 , E k /ℵ = 13.56 K , and at k = 1.9 A -1 , E k /ℵ = 8.692 K , respectively. These values compare very favourably with the experimental values which are maximum at k = 1.1 A -1 , E k /ℵ = 13.7 K , and minimum at k = 1.91 A -1 , E k /ℵ = 8.65 K . To obtain good experimental agreement, the other parameters are m ∗ /m = 1.2 and the hard-core diameter a = 2.1 A .

Various alternatives to the Balian-Werthamer model of the B phase of superfluid 3He are considered. The linear NMR properties of a certain f-wave state of axial symmetry are shown to be very similar to those required by experiment.

We suggest consistency checks on neutron scattering data on liquid helium under pressure. The theoretically calculated static structure functionS(k|P) leads to pressure dependence of the roton gap, momentum, and curvature in general agreement with experiment.

We study the Zawadowski-Ruvalds-Solana (ZRS) model of quasiparticle interactions in liquid helium both quantitatively and theoretically. We find that the model cannot account quantitatively for the multiphonon peak in the neutron-scattering spectrum even if we vary the quasiparticle spectrum phenomenologically. We find that the model can account for the experimental light-scattering spectrum if the effective roton-roton coupling ${g}_{4}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{\ensuremath{'}})$ is sharply peaked at the roton minimum ($k\ensuremath{\cong}{k}^{\ensuremath{'}}\ensuremath{\cong}{k}_{0}$). We suggest that the neutron-scattering results would be improved without changing the light-scattering prediction if the density operator contained nonlinear terms in the quasiparticle operator. We criticize the implicit assumption in the ZRS model that the zero-order ground state is nondegenerate. The relationship of this model to one devised by Jackson is discussed.

When the approximation of overlapping momenta, in calculating the phonon and roton contributions to the thermodynamic properties of liquid helium II, is removed and the resulting expression for the specific heat of the system is fltied to experimental data at low temperatures, seemingly improved values of the phonon dispersion ccefficient gamma are obtained. At saturated vapor pressure, gamma turns out to be - 5.6 x 1037 cgs units, which is in excellent agreement with the value resulting from x-ray-scattering measurements. Comparison with neutron-scattering data suggests that gamma may be strongly temperature dependent.

I discuss soundlike collective excitations in anisotropic superfluid Fermi liquids in both the collisionless and the hydrodynamic regimes. Expressions for the velocities of zero, first, second, and fourth sound are derived in terms of the Landau parameters of the normal state. The second- and fourth-sound velocities are found to vary appreciably with orientation.

From second-sound velocity measurements, superfluid densities near superfluid transition points were determined in 3 He- 4 He mixtures under saturated vapor pressure. The critical exponent of the superfluid density thus obtained increases about 15% with increasing 3 He concentration up to 40 mole percent, which contradicts the universality concept. Furthermore, the critical exponents obtained here and the exponents of the specific heat do not satisfy the scaling laws. A new concept of universality is introduced and discussed.

The problem of the propagation of second sound in superfluid helium is studied. The second sound wave is assumed to be described by a set of variables, the local energy and local energy flux densities. Relaxation function expressions are obtained for the thermal con­ ductivity X which is analogous to the normal thermal conductivity /CT, as is distinct from the abnormal thermal· conductivity due to the counterflow of the super and normal components, and also for the kinetic coefficients (1) (3, (4. It is pointed out that, to the thermal conduc­ tivity. x, a restoring force due to temperature is essential. 1 ) first recognized that another wave motion, in which the density and pressure remain practically in local equilibrium values but in which entropy (or temperature) fluctuates, is also possible. This leads to an essential condition fol,' its existence; There must be a sufficient number of interactions between excitations, so that such a local equilibrium can be established within distances which are small compared with the wavelength of second sound. That is, an inequality (1)f

The theory of Landau and Khalatnikov for the viscosity of liquid He II is examined in the neighbourhood of the lambda point. Arguments are advanced against the validity of the ideal gas approximation for rotons, and a qualitative roton liquid model is proposed. The temperature dependence of viscosity of the liquid model is found to correspond to the observed viscosity above 1-8°k, and with the discontinuity in the temperature derivative of viscosity at the lambda point.

Synopsis New measurements were made of the specific heat of liquid helium, employing a capillary-connected calorimeter and using the demagnetization method with a properly chosen paramagnetic salt for reaching temperatures below 1°K. It was found that the specific heat below 0.6°K satisfies a T3-law as was expected from theoretical predictions. Above this temperature, the measured values were compared with Landau's formula on the roton specific heat. These new values of the specific heat were used for the recalculation of the entropy of liquid helium II.

Feynman has recently advanced physical arguments to deduce an approximate form for the wave function of an elementary excitation in helium. Qualitatively, his theory leads to an excitation spectrum similar to that postulated by Landau; however, serious quantitative discrepancies remain in the ‘roton ’ region of the spectrum. It is shown here that improved wave functions may be constructed by using combinations of wave functions for single and double Feynman excitations—this amounts to a self-energy correction to the roton spectrum, arising mainly from roton-phonon interactions. The numerical value of the roton energy agrees remarkably well with Landau’s value. The treatment depends, however, on the use of perturbation theory, where the perturbation is not small. Little confidence should therefore be placed in the actual value found.

We calculate the roton-limited mobility of ions in superfluid ${\mathrm{He}}^{4}$ for temperatures below \ensuremath{\sim} 1.7\ifmmode^\circ\else\textdegree\fi{}K. Taking, as a first approximation, a constant ion-roton transition-matrix element, which we treat as an adjustable parameter, we find good agreement with experiment for both positive and negative ions. This model calculation indicates that the most mobile of the recently discovered "exotic" negative carriers is a very light ion.

Based on an empirically extended Landau roton formula and on the fact that3He atoms participate in the movement of the normal component of liquid4He, the variation of the λ temperature with3He concentration is obtained at the saturated vapor pressure. The results are in good agreement with experiment. The present approach can also be applied to higher pressures.

A closed set of hydrodynamic equations involving the macroscopic parameters of the 4He II condensate, rather than those of the superfluid, is derived from the exact quantum-mechanical equation of motion for the first-order reduced density matrix. The method is based on a Chapman-Enskogtype expansion about a local equilibrium together with the explicit use of the microscopic symmetry properties of the first- and second-order reduced density matrices.

A modified “ phonon-roton ” model for liquid Helium II is proposed and is shown to be connected smoothly with the theory of Bose condensation. The method used is the direct extension of the method of “ auxiliary variables “ used by Bogolyubov and Zubarev, and is equivalent to that used by Bohm and Pines in the theory of the electron gas in metals. The phonon excitation is described in terms of the density operator, and the roton part by the so-called individual coordinates. The elimination of the kinematical interaction leads to the following model: The excitations for long wave length are described by phonons, and for short wave length by rotons, and the total number of excitations cannot exceed the total number of the particles. The thermal properties of this model is seen to agree with the experimental fact qualitatively.

Assuming that London's formula, Δp = ϱSΔT , is also valid for the fountain effect in Helium II at high pressures, the entropy as a function of temperature and pressure has been evaluated from measurements of the fountain effect in the temperature range 1.15–2.00°K and at various pressures between the saturated vapour pressure and 25 atm. On extrapolating the entropy data as a function of pressure to the saturated vapour pressure good agreement has been found with the entropy values of Kramers e.a. , the deviations being smaller than 2%. The entropy data of Lounasma and Kojo at different densities have been fitted to the present entropy data to obtain an entropy diagram up to the λ-curve. As however, the various data on the density of liquid helium do not agree there is some doubt with respect to tis fit. Finally, values have been determined for the parameters of the roton spectrum at several pressures. The results have been discussed and compared with those obtained from experiments on inelastic neutron scattering.

In the perturbation method which is called " sound approximation " in Quantum Hydrodynamics, the invariancy of each same order term in the Hamiltonian is studied in detail by the gauge transformation. For a special case, where the fluid is nonviscous and incompressible, the eigen value problem of the Hamiltonian which includes only the rotational part of the fluid velocity is calculated. The energy spectrum of the lowest state of the roton is given by Ziman' s method for this special case. The expression of the energy spectrum of the roton does not diverge or contain any additional constant term while Ziman's result does. The discussion to explam the difference between these two results is given.

A wave function previously used to represent an excitation (phonon or roton) in liquid helium, inserted into a variational principle for the energy, gave an energy-momentum curve having the qualitative shape suggested by Landau; but the value computed for the minimum energy Δ of a roton was 19.1°K, while thermodynamic data require Δ=9.6°K. A new wave function is proposed here. The new value computed for Δ is 11.5°K. Qualitatively, the wave function suggests that the roton is a kind of quantum-mechanical analog of a microscopic vortex ring, of diameter about equal to the atomic spacing. A forward motion of single atoms through the center of the ring is accompanied by a dipole distribution of returning flow far from the ring. In the computation both the two-atom and three-atom correlation functions appear. The former is known from x-rays, while for the latter the Kirkwood approximation of a product of three two-atom correlation functions is used. A method is developed to estimate and correct for most of the error caused by this approximation, so that the residual uncertainty due to this source is negligible.

In the basis for current descriptions of the behavior of helium II lies the idea of the existence of two fundamental types of quantum thermal excitations, called phonons and rotons. A series of investigations in which the scattering of cold neutrons in liquid helium is interpreted in terms of scattering events in which a phonon or roton is created and takes up the recoil momentum is surveyed. Correlation of the neutron's energy and momentum changes occurring in the event yields a dispersion curve relating the energy and momentum of the thermal excitations. The experimental techniques and the analysis and interpretation of data are outlined. The results reproduce quite well a segment of the dispersion curve predicted by Landau. The high-momentum end of the curve (beyond the range of existing data) and its interpretation in terms of various possible forms of the behavior of unstable rotons is discussed. The possibility of the direct determination of the roton's chirality from the forward scattering cross section is discussed. Finally the temperature dependence of a series of thermodynamic functions predicted using an experimental dispersion curve is presented. The resulting curves agree remarkably well with direct measurements, except for a low-temperature discrepancy in the velocitymore » of second sound. (TTT)« less

It is shown within the framework of mean-field theory that the gas of roton excitations in superfluid helium becomes unstable at a critical temperatureT c if the roton-roton interaction is attractive. The instability is characterized by a singularity of the specific heat and signals the transition into the normal fluid state. The roton energy gap remains finite atT c .

The quantum-mechanical Hamiltonian obtained by Ziman and by Thellung for an ideal liquid is diagonalized in the approximation of small compressibility. On taking account of the interaction terms between ‘phonon' and ‘roton’ variables, the roton rest energy is found to be proportional to the inverse of the sound velocity and to the ninth power of the wave number cut-off.

A canonical transformation which takes into account both pair correlations and the single-particle condensate is made on the full Hamiltonian for a system of interacting bosons, and the quasi-particle interactions are neglected. The theories of Bogoliubov, Valatin and Butler, Brueckner and Sawada, and Etters can be obtained as special cases of the theory under certain approximations which are critically examined. A delta function potential which describes a soft spherically symmetric shell around a particle is used to reduce the complicated coupled integral equations to coupled equations for parameters. The formula for the quasi-particle energy is used to fit the experimental excitation spectrum, and good agreement in both the phonon and roton regimes is obtained with three parameters. An effective mass approximation is made to derive the empirical formula used by Larsson and Otnes to fit their neutron scattering data.

Many entities in physics have both particlelike and wavelike properties. In representing the particlelike properties physicists usually use words ending in -ON. The originators of the concept underlying the particle nature, and the coiners of the name are discussed for the following “particles”: boson, electron, exciton, fermion, magnon, neutron, phonon, photon, plasmon, polariton, polaron, proton, and roton.

A new potential, which accounts for roton-roton interaction is proposed. Using this potential without any adjustable parameters, the calculated roton-roton contribution to the viscosity of He II in the temperature range between 1.35 °K and 1.9 °K is found to be in qualitative agreement with the experimental value.

Measurements of the spin-diffusion coefficient and the nuclear magnetic susceptibility of ${\mathrm{He}}^{3}$-He II solutions of molar concentrations between 0.030 and 3.0% have been made at temperatures above 0.4\ifmmode^\circ\else\textdegree\fi{}K. The diffusion measurements show that the reciprocal of the spin-diffusion coefficient varies linearly with ${\mathrm{He}}^{3}$ concentration. The constant term is identified as the reciprocal of the binary diffusion coefficient. Results for the binary diffusion coefficient are in agreement with the results of other measurements where comparison can be made. Between 0.8 and 1.6\ifmmode^\circ\else\textdegree\fi{}K, the binary diffusion coefficient varies exponentially with the reciprocal of the temperature, in a manner consistent with the roton excitation energy of 8.65\ifmmode^\circ\else\textdegree\fi{}K. The susceptibility exhibits only small deviations from perfect-Fermi-gas behavior. A calculation of the effect on the susceptibility of the direct interaction between ${\mathrm{He}}^{3}$ quasiparticles made by utilizing an effective potential indicates that deviations from perfect-Fermi-gas behavior are small. There is reasonably good agreement between the measured and the calculated temperature dependence of the susceptibility.

Velocity distribution of atoms evaporating from superfluid He II at low temperatures, noting roton shifting and multiexcitation processes

The mobility of positive and negative charges in liquid helium II has been measured as a function of the pressure, for pressures ranging from the vapour pressure to the melting point, in the temperature range 1.9 to 1.1 °K. The mobility of the positive charges was found to decrease with the increasing pressure, but this decrease cannot be accounted for in-terms of the known change in the roton density. The negative charges behave in a peculiar way which can be easily accounted for by the bubble model. For both charges an unexplained sharp break is found at about 1.3 °K especially at the higher pressures.

Interactions between excitations in superfluid helium are investigated by Green's function techniques. In our model, a sharp peak appears in the two-excitation spectrum corresponding to a resonance of two rotons. Interaction of the resonance with the single-particle spectrum results in a hybridization and splitting of the one-particle spectrum into two distinct branches. These theoretical results are in agreement with recent neutronscattering and Raman-scattering experiments.

The variation principle proposed by N. N. Bogolubov and the mathematical methods developed in the theory of superconductivity are employed to study the properties of the atomic nucleus. Weak interactions between protons (or neutrons) located in the same shell are treated on the basis of a nuclear model in which nucleons of the closed internal shells are assumed to produce a centrally symmetrical field which is somewhat distorted by nucleons in the outer shell. It is shown that the interaction between protons in an outer shell give rise to a superfluid state of the nucleus, the main role being played by interactions between protons with equal and opposite projections of the angular momenta on the nuclear symmetry axis. The energy of the ground superfluid state and first excited state is evaluated and it is shown that in the case of an even number of protons in the shell, an energy gap between the superfluid and first excited states should exist, whereas no such gap exists if the number of protons is odd. This permits one to explain the energy gap in heavy even nuclei. Conditions for the appearance of a superfluid state in an atomic nucleus have been obtained. They reduce to the requirement that attractive forces predominate between interacting protons located in the same shell near the Fermi surface energy. The critical temperature for the phase transition of a nucleus from the superfluid to normal state has been determined.

Precise measurements of the depolarization properties and intensity of laser light Raman-scattered from roton pairs in liquid4He are described and compared with several existing theories.

The problem of the appearance of quantized vortex lines and rings in superflow, and the modification of that flow by the vortices, is examined in the light of a suggestion by Iordanskii, namely that thermal fluctuations may play a vital role in the nucleation process. The view is taken that there exists a distribution of quantized vortex rings in liquid helium and that the smallest of these is a roton. The evolution of these rings in a counterflow is examined by means of a Langevin equation. A Fokker-Planck equation is developed, and a number of examples are presented which illustrate the features of such processes. These include nucleation of vortices in an unbounded fluid, nucleation in a finite channel, the thermally activated vortex mill, nucleation of vortices by ions and phonon-roton relaxation times.

The Raman scattering from liquid helium has been calculated by use of the measured dispersion relation for the excitations and a model for the coupling between the light and these excitations. A comparison of the results with experimental measurements suggests that the attractive interaction between the excitations is much weaker than presently supposed, and that the binding energy of a two-roton bound state, if it exists at all, is less than 0.1 K.

We have extended the theory of the roton lifetime to include two-roton resonances in the final-state interaction. When resonances are taken into account, the final density of states ${\ensuremath{\rho}}_{2}$ exhibits a complicated energy dependence and bears little resemblance to the noninteracting roton density ${\ensuremath{\rho}}_{2}^{(0)}$ used in the Landau-Khalatnikov theory. Using a point-interaction model for the roton coupling and including only direct roton-roton scattering, we obtain a single-roton linewidth which is consistent with the upper bound given by other authors; we then obtain a linewidth for the two-roton resonance which is smaller than the experimental value. In contrast to previous work, we have proceeded to include roton hydridization in the lifetime theory. The effect of hybridization is twofold: (a) The final density of states ${\ensuremath{\rho}}_{2}$ is markedly altered, and (b) the hybridization opens new scattering channels involving a single quasiparticle in the final state. Although the net effect of mechanism (a) on the roton lifetime seems to be small, the additional scattering (b) gives a non-negligible contribution. Furthermore, we discuss the importance of self-consistency and roton scattering in nonzero angular momentum channels. Our theory predicts an unusual pressure dependence for the roton linewidth which may be checked experimentally. We conclude that further experiments, such as light and neutron scattering, are needed to resolve the strength of the various contributions to the linewidth and thereby provide a truly quantitative analysis of the roton lifetime.

A review is given of the scattering of light by superfluid helium. A brief discussion is given first of the theory of Brillouin and Rayleigh scattering from an ordinary simple pure liquid. It is shown how this discussion must be modified in the case of the superfluid phase of purc 4He, and experimental observations of Brillouin scattering from first sound in liquid helium are described. The discussion is extended to the case of a superfluid mixture of 3He and 4He, and a recent observation of Brillouin scattering from second sound in such a mixture is described. Recent work on Raman scattering from superfluid helium (two roton production) is then discussed, and in a final section a brief review is given of the special situation that obtains in the neighbourhood of the &point.

In a superfluid liquid-helium cavity packed with a fine powder, only the supercomponent is free to move and pressure waves transmitted by the supercomponent are called “fourth sound.” We have observed that the resonance frequency of the powder-filled chamber may be shifted by introducing vortices. Both static and dynamic measurements were made on the resonance frequency and signal height in order to explore the effect of exceeding the critical velocity of the superfluid in the powder-filled chamber.

A purely macroscopic and unambiguous derivation of the hydrodynamical equations for superfluid helium is presented. The existence of a condensate wave function is inserted in the beginning as an essential in the description of the thermodynamic state of superfluid helium, and the derivation is carried out in the frame in which the normal velocity is zero.