Microscopic theory of superfluid3He-superfluid4He solutions I. Derivation of hydrodynamic equations

The hydrodynamic equations for mixtures of superfluid³He and superfluid⁴He are derived on the basis of the microscopic theory proposed by Bogoliubov.     

Microscopic theory of3He—4He solutions

The ring diagram contribution to the grand partition function of a mixture of fermions and bosons is obtained from a rigrous formula which extends the single component formula of Montroll and Ward. Together with the contributions from some other relevant graphs, it is applied to a dilute solution of³He with⁴He. For a small mass difference parameter, the ground state energy is shown to be expressed as a linear function of³He concentration. The coefficients of this energy expression are obtained explicitly for soft and hard-sphere potential models. From the energy, the differential volume coefficient is obtained also.This work was supported by ONR under Contract No. N00014-79-C-0451     

Microscopic theory of liquid 4He

A finite temperature theory is given of the temperature variation of the sound velocity and of the structure factor of liquid helium. The sound velocity is found to decrease slightly as the temperature is increased from absolute zero. The first peak of the structure factor is higher for higher temperature below the λ point. The internal energy is characterized by the quasiparticle distribution function with the Bogoliubov-Zubarev type excitation energy. The structure factor and the energy are determined by the effective interaction which depends on the average density, the density of the excited particles, and temperature in addition to the interaction potential.     

Towards a microscopic theory of superfluid 4He

A generic Hamiltonian, which contains the potential energies and the coulomb interactions of an assembly of N ⁴He nuclei and 2N electrons is studied using second quantisation techniques and different sets of boundary conditions for three related problems. For an infinite volume of fluid periodic boundary conditions are imposed. For a finite volume which fills a cube the wave functions are chosen to vanish on the boundaries and for a finite volume contained in a vessel of arbitrary shape similar boundary conditions are imposed on a macroscopic volume of the liquid which is sufficiently remote from the walls and surface. In each example the appropriate value of N is determined by using the known density of the liquid. Use is also made of several other experimentally determined properties; that, at the lowest temperatures, the liquid is almost entirely super-fluid and that as the temperature is raised it appears to be composed of two homogeneously mixed fluids, the super- and the normal fluid, with little change in overall density. The first of these properties is used as a guide to a possible form for the ground state of an unperturbed Hamiltonian, that it should be a many-particle function that ensures that the density of particles and charge is uniform in space. The second is used to reason that each low-lying excited state of the unperturbed Hamiltonian will contain a part in which the ground state appears to have been expanded and the overall density has been restored by replacing the particles which have been removed in the expansion back into states which have not been used in forming the expanded ground state. A detailed examination of such a possibility shows that there are many possible ways of constructing mutually orthogonal states with this property. So the way is open to using these states as basis states for a perturbation theory and characterising them with a momentum variable, k. Another property, that the super-fluid supports longitudinal phonon-like modes is then used to define some more low-lying states, so enlarging the number of mutually orthogonal many-particle states. It is then a relatively simple exercise in perturbation theory to show how, after the meaning of the momentum variable, k, has been slightly changed, the famous (, k) dispersion curve emerges.     

Propagation of a magnetic disturbance in superfluid 3HeB

A strong disturbance evoked magnetically has been observed to propagate in ³HeB at 20.7 bar over a distance of about one centimeter.     

Microscopic theory of soliton propagation in4He films

A microscopic theory has been provided for propagation of solitons in superfluid⁴He films at temperatureT=0°K.     

Statistical mechanics of dilute mixtures of 3He in superfluid 4He

A statistical mechanical theory of the dilute mixtures of ³He and ⁴He is presented. It is shown that the ground state energy is proportional to the ³He density. The chemical potential expressed in terms of the sound velocity assumes the form which has been suggested phenomenologically. From the chemical potential, the effective ³He⁴He interaction and osmotic pressure are derived. The effective mass of ³He is also calculated as a function of potential parameters.     

Microscopic theory of He II-He3 mixtures including dissipative processes

The conclusions following from the solutions of the linearized hydrodynamic equations are discussed. By using some of symmetry properties of the Green functions the kinetic coefficients coupled to the external fields are determined and some interesting informations concerning the thermodynamic derivatives of the condensate density arise. A number of the Green functions is calculated in the hydrodynamic approximation. The examination of their denominators leads to the expressions for damping of the acoustic modes in He II-He³. A set of Kubo-type formulae for the kinetic coefficients and certain sum rules are obtained.     

Microscopic modes in a Fermi superfluid. II. Dispersion relations

This is the second of two papers in which microscopic expressions for the amplitudes and dispersion relations for hydrodynamic modes in an isotropic Fermi superfluid are derived. In this paper we obtain approximate solutions to the linearized kinetic equations for the bogolon spin density and total density for the case of long-wavelength disturbances after long times when a fluctuating superfluid velocity is present. In so doing, we obtain microscopic expressions for the amplitude and dispersion relations for the spin diffusion mode, the two shear modes, and the four longitudinal modes (two first-sound modes and two second-sound modes).