Threshold phenomena in the excitation spectrum of liquid 4He

The elementary theory of threshold phenomena in the excitation spectrum of superfluid ⁴He is presented. The multiphonon threshold in the initial part of the spectrum and the two-roton threshold at energy 2 are discussed.     

Three-phonon vertex corrections to the excitation spectrum and liquid-structure function of liquid4He

The energy spectrum of elementary excitations and the liquid-structure function of liquid⁴He are calculated by means of the method of correlated basis functions in the approximation of second-order perturbation theory. The procedure is based on (i) the construction of phonon functions in terms of collective coordinates and the optimum Bijl-Dingle-Jastrow type of ground-state wave function and (ii) the evaluation of leading correction terms to the Bijl-Feynman excitation spectrum, which are generated by two types of three-phonon vertices. Numerical results are obtained using the optimum liquid-structure function computed by Campbell and Feenberg in the paired-phonon analysis.Research supported in part by the University of Kansas General Research Fund under grant No. 3235-5038.     

Ground state of two-dimensional liquid 4He

The ground-state energy and liquid structure function of two-dimensional liquid ⁴He are calculated in the density range 0.028–0.075 Å^–2, using the Lennard-Jones 6–12 potential. The optimal Jastrow function is first determined by a self-consistent paired-phonon calculation and then the contributions of three-body factors the wave function are calculated. Tables of values are given at several densities for the radial distribution function, Jastrow function, and liquid structure function.Research supported by National Science Foundation Grants DMR 74-01237 A01 and DMR 78-09226.     

Variational calculation of the binding energy of one3He impurity in liquid4He

We present a variational microscopic study of the liquid⁴He with one³He impurity at zero temperature, within the framework of the average correlation approximation. In this approach all calculations can be referred to quantities of liquid⁴He. The chemical potentials of liquid⁴He and of a³He impurity are studied in the liquid range of densities and the results show an overall good agreement with the experimental data. Detailed comparisons with previous calculations are also presented. Due to the presence of the pressure term, the calculation of chemical potentials is a severe test to the capability of the wave function and the interaction to obtain a good saturation density. The calculation of the chemical potential of the impurity using the Lennard-Jones and the Aziz potentials, points out the different balance of kinetic and potential energies obtained in the variational minimization of these two potentials.Supported by a Fulbright Fellowship.     

Inhomogeneous liquid4He: A density functional approach with a finite-range interaction

The properties of liquid helium-4 are investigated in a mean-field approximation using a finite-range effective interaction. The long-range part (i.e.,r>2.556 Å) coincides with the standard Lennard-Jones potential and the hard core is phenomenologically renormalized. The model reproduces the static polarizability of bulk liquid, as well as the actual surface energy of helium at zero temperature. The surface width is found to be 5.8 Å. A comparison is made with the earlier work of Ebner and Saam.Unité de recherche de l'Ecole Normale Supérieure et de l'Université Paris 6, associée au CNRS.Unité de recherche des Universités Paris 11 et Paris 6 associée au CNRS.     

Transport properties for a dilute solution of3He in two-dimensional liquid4He

The temperature variations of the diffusion coefficientD(T), thermal diffusion ratio k _T (T) and thermal conductivity (T) in a dilute solution of³He atom in two-dimensional liquid helium are evaluated explicitly by solving the kinetic equations via phonon-phonon, phonon-roton, roton-roton, impurityelementary excitation and impurity-impurity scatterings. In the low-temperature region, the main contributions toD(T) and (T) come from the interactions between phonons and impurities, while in the high-temperature region the interactions between impurities and whole elementary excitations contribute more strongly toD(T) and (T) than those of only elementary excitations. For a dilute solution, the thermal diffusion ratio k _T (T), neglecting the internal mass counterflow, is much smaller than the effective thermal diffusion ratio k _T ^* (T), which is a function of thermostatic properties. The effective thermal conductivity _eff is much larger than the thermal conductivity and has different temperature dependence from the thermal conductivity. The behaviors of the two-dimensional diffusion coefficient and thermal conductivity are much like the bulk case, where they exhibit exponential decay with increasing temperature, although they are much smaller than those of the bulk case.     

Variational method for liquid3He. I. The Fermi hypernetted-chain approach

We develop a systematic procedure for the computation of ground-state properties of dense Fermi liquids, which is based on a Jastrow trial wave function and forms a systematic extension of the hypernetted-chain technique to Fermi statistics. The derivation of the Fermi hypernetted-chain equations is described in detail, and preliminary numerical results are reported.     

A study of the neutral excitation current in liquid 4He above 1 K

Neutral excitations generated in a discharge near a sharp tungsten tip operated in either the field emission or field ionization mode in liquid helium were detected with a surface ionization detector submerged in the liquid. The measured times of flight of the neutral excitations from the tip to the detector are in excellent agreement with a model in which the neutral entities drift with the fluid which is set in motion by the force of the ion current extracted from the discharge. Times of flight as long as 30 sec were measured. These data, along with measurements reported by other workers, clearly identify the neutral excitation as an a ³ _u ⁺ metastable helium molecule. We report a lower limit of 10 sec on the lifetime of this state.Supported in part by the National Science Foundation.     

Excitations of inhomogeneous liquid He4: A density functional study

Excitations of the free surface, films and bounded channels of liquid⁴He are investigated with a non-local density functional theory at zero temperature in the Feynman approximation. The nature of the various branches is discussed. At the free surface, for large momentum (k2 Å^–1), a hybridization occurs between ripplons and bulk rotons. For low momentum, the ripplon dispersion relation in thick films and the third sound mode in thin films are recovered, including the expected oscillations of the third sound velocity as a function of film thickness due to the layering of the film close to the substrate. Low energy 2-dimensional (2D) rotons confined at the liquid-wall interface are found on most substrates, except on the weakest binding surfaces such as Cs, for which we consider a channel geometry. In thin films, non trivial coverage dependence of the heat capacity may result from the interplay between the contribution of the low momentum part of the spectrum, characterized by the third sound velocity oscillations, and that of the large momentum part, dominated by the 2D rotons.Unité de Recherche des Universités Paris XI et Paris VI associée au CNRS.     

Experimental studies of the excitation spectrum of3He-4He mixtures using neutron inelastic scattering

Neutron inelastic scattering techniques have been used to study the excitation spectrum of mixtures containing 6%, 12%, and 25% of³He in⁴He at 0.75 K. The 6% mixture has also been investigated at temperatures of 0.6, 1.0, and 1.5 K. The measurements have been made in the wave-vector transfer rangeQ=0.8–2.3 Å^–1. The energy spectra observed contain two principal features, which are interpreted as scattering from (a) phonon-roton excitations arising from the⁴He, and (b) quasiparticle-quasihole excitations arising from the³He. In the dilute³He-⁴He mixture, our results indicate a³He particle-hole continuum which deviates from the quadratic form, proposed by Landau and Pomeranchuk, and which lies below the⁴He phonon-roton dispersion curve. However, we are unable to say whether or not the continuum exhibits a minimum as a function of energy, as has recently been proposed by several authors, because we were unable to resolve the scattering from the particle-hole continuum forQ1.7 Å^–1. In this region ofQ space the energy spectra are dominated by the roton excitations. We have also determined the shifts in energy and increases in linewidth of the phonon-roton excitations relative to those of pure⁴He and compared these measurements to several theoretical predictions.Work performed under a grant from the Science Research Council of Great Britain.     

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.     

Liquid4He. I. Binding energy and excitation energy spectrum from two-body correlations

As a first step in a theoretical study of the properties of liquid ⁴ He we have calculated the binding energy from two-body correlations in the system. Using an effective interaction or reaction matrix obtained by a modified Brueckner theory, low-temperature properties such as the binding energy, the elementary excitation energy spectrum, and the velocity of first (ordinary) sound are calculated or estimated. For simplicity we use the approximation of a reference energy spectrum with a quadratic momentum dependence for the input single-particle energy spectrum, which in principle should be fitted to self-consistent single-particle energies. The intermediate-state potential energies are, however, chosen to be equal to zero. Hence, the three-body energy contribution and also higher order energy contributions must be estimated by separate calculations. A self-consistent solution is obtained through the depletion of the zero-momentum state, which is also calculated. The calculations are done for two different two-body potentials, an Yntema-Schneider potential given by Brueckner and Gammel, and a Frost-Musulin potential given by Bruch and McGee. The theoretical results are –3.1 to –4.0°K for the binding energy, 39–44% for the depletion, and 176–217 m/sec for the sound velocity. The corresponding experimental results are –7°K, 83%, and 238.3 m/sec, respectively, i.e., the difference is generally within a factor of two. The agreement with experimental results is reasonably good (or bad), especially since three-body and higher order cluster terms are not included in this first approximation.     

An experiment to measure the diffusion of4He in4He in the Fermi liquid region

The diffusion coefficient D and the thermal diffusion factor T, for⁴He in liquid³He at low temperatures T, have recently been calculated by solving the Boltzmann equation derived from Fermi-liquid theory. As T 0, T approaches a constant and D varies as 1/T. Both transport coefficients are determined solely by a ₀ ³⁴ , the l = 0 forward scattering amplitude. Because a ₀ ³⁴ can be found from thermodynamic data, e.g. the solubility of⁴He in liquid³He, X ₄ ^sat , a strict test of the theory is possible. We have built an apparatus to measure D and X ₄ ^sat as functions of pressure and temperature. The ratio of the capacitances of two parallel plate capacitors measures the number of⁴He atoms diffusing out of the liquid into the superfluid film. A bellows initiates the diffusion by changing the pressure in the liquid and thus the equilibrium concentration. A change of 10^–8 in the⁴He concentration can be detected.     

Excited states of alkali atoms in liquid4He from density functional calculations

Calculations are presented, based on a Density Functional approach, of the first excited states (l=1) of alkali metal atoms (Cs and Na) in liquid ⁴He. We find large departures from the spherical bubble in the liquid⁴He structure when the alkali atom is excited. The shifts from the free atom values of the excitation and emission transition lines are calculated and compared with the experimental results. According to our calculations, the so far unseen radiative deexcitation transition for light alkalis in⁴He may lie in an unexplored infrared region of the spectrum.     

Elastic reflection of4He atoms from the surface of liquid4He

The reflection of incident⁴He atoms from the surface of liquid⁴He can yield information on the microscopic features of the liquid at the surface. Edwards et al. have measured this reflectivity and have developed a theoretical model which successfully described the measurements. However, the density profile of the liquid⁴He surface has recently been determined experimentally, and this can now be used to model the reflectivity without any freely adjustable parameters. We use this density profile and the experimentally-determined interatomic pair potential to derive an effective potential for the incoming atom, and then solve the single-particle Schroedinger equation to find the reflectivity as a function of wavevector perpendicular to the surface. Within the uncertainty in the density profile and interatomic potential, we have chosen values that give good agreement between the measured and calculated reflectivities.     

Variational Study of a 3He Impurity Near a 4He Vortex Core

The system of kinetic equations for distribution functions of impurity excitations in solid 3He–⁴ He mixtures in the presence of phonons is solved. The results obtained allows us to calculate the spin diffusion coefficient of ³ He–⁴ He quantum crystals. The found expression differs from the results of the previous theoretical studies. A comparison of the obtained diffusion coefficient with experimental data makes it possible to determine the numerical values of the energy band width of impurity excitations.     

Ground state of liquid4He

A variational calculation of the ground-state energy of liquid⁴He is performed under the assumption that the ground-state wave function is a product of pair functions and the potential energy is a sum of pair potentials. Results obtained from the Percus-Yevick (PY) and Hypernetted Chain (HNC) theories are compared with both Monte Carlo (MC) studies and with experiment. It is found that in the case studied the PY equation is considerably less accurate than the HNC and that the PY is not useful for this variational calculation but that the HNC is. A comparison of three model pair potentials indicates that the uncertainty in the pair potential produces an uncertainty in the ground-state energy of about 2°K. One of the pair potentials used leads to the best agreement yet obtained with experimental ground-state energies.Supported in part by A.E.C. Contract No.AT(11-1)1569.Supported in part by grants from the National Research Council of Canada and the U.S. Department of the Interior, Office of Saline Water.