The density maximum in liquid He4 and its relationship to the lambda point

The lambda point in liquid He⁴ is a very well-established phenomenon. London, in 1938, suggested that the effect might be an example of Bose–Einstein condensation and it is this model that is generally accepted nowadays. There are, however, serious discrepancies between the theory and experimental results. Here we present an alternative model based on the existence of the density maximum in liquid He⁴ near 2.18 K, that explains the anomaly in the specific heat capacity but does not have the failings of the BEC model. In addition the theory explains the occurrence of the density maximum in liquid He⁴.     

Pippard relations and critical phenomena in binary liquid mixtures

The cylindrical approximation originally proposed by Pippard for the λ-transitions in one-component systems is extended to multicomponent systems. The application of the generalised Pippard relations to binary liquid mixtures is considered. The relations for He³ + He⁴ mixtures are shown to be equivalent to those obtained by earlier workers. The validity of the relation relating specific heat and thermal expansion for binary liquid systems is discussed on the basis of the measurements performed on the system methaonl +n-heptane. Lack of information on the behaviour of a few quantities prevents a quantitative test of the relation.     

A macroscopic theory of He3-He4 mixtures

A macroscopic theory of liquid He³-He⁴ mixtures is developed, analogous to the smoothed potential theory ofHeer andDaunt, when the energy spectrum has the generalized formE=Ap ^1/r. Takingr to be a function of atomic volumes of the two isotopes in the mixture, the observed data on the shift of lambda temperature, vapour pressure, heat of mixing etc. is accounted for satisfactorily. A quantitative agreement between the theory and measurements of specific heat, normal fluid density and second sound velocity in mixtures is reached by taking a temperature dependent effective mass of He⁴ particles below the lambda temperature, for which a physical picture is provided. Calculations on second sound velocity show that He³ moves with the same velocity as the normal component of He⁴, which is in agreement with the heat flush experiments.     

A three-body model of the <Superscript>11</Superscript>B nucleus

The binding energy and the rms charge and mass radii have been calculated in terms of the single-channel three-body ⁴He⁴He³H model of the ¹¹B nucleus with an expansion of the three-body wave function in a nonorthogonal Gaussian basis. Parameters of the wave function are presented and convergence of the three-body energy depending on the number of expansion terms is demonstrated.     

Discussion of the He-He interaction energy in the average energy approximation

Using the Unsöld average energy method a double perturbation theory expression for the interaction energy through second order is obtained which includes intra-atomic correction terms arising from the use of trial wavefunctions to represent the non-interacting molecules. These formal expressions are applied to the ground state He-He interaction and results for the interaction energy are obtained that compare favourably with recent semiempirical and ab initio calculations. The Unsöld calculations are used to investigate approximations that have been introduced into the average energy calculation by other workers, and as a model for discussing the relative importance of second-order charge overlap and exchange effects and the convergence of the multipole treatment of the interaction energy for this typical non-bonded interaction. The results illustrate the importance of knowing as many terms as possible in the R ^-1 expansion of the energy. Finally a portion of this work required the recalculation of the He-He repulsive energy obtained by using the Eckart wavefunction to represent the helium atom. The resultant energy is approximately 25 per cent lower than that previously obtained using this wavefunction for most values of R.     

Spectroscopic evidence for superfluidity in liquid He droplets

A beam of liquid He _N droplets (N≈5000) is prepared by a cryogenic free jet expansion, doped with single molecules and studied via laser spectroscopy. The spectra consist of sharp lines (Δν?0.003–0.1 cm^?1) and show rotational structure (for SF₆), providing the first information on the droplet temperature, which is 0.37 K for⁴He _N and 0.14 K for³He _N . The rotational constants give information on the microscopic interaction of the molecule with the He environment. Each of the sharp vibronic lines of the elecronic S₁←S₀ transition in glyoxal (C₂O₂H₂) is accompanied by a phonon sideband, showing a distinct gap followed by a sharp peak at the roton energy (8.1 K) and a second weaker maxon peak followed by a broad multiphonon wing. These provide the first evidence for superfluidity in⁴He _N predicted theoretically.     

He5 scattering states in the refined cluster function representation

As a first application of the refined cluster model reaction theory we calculated the scattering states of He⁵ using a specific ansatz for the nucleon-nucleon forces. He⁴-n and H³-d channels are considered only. Our results agree qualitatively with the experimental values if available; in particular we were able to explain the narrow resonance in theD _3/2 He⁴-n channel as a consequence of the strong coupling to theS _3/2 T-d channel.     

Isospin projected FBCS wavefunction for light nuclei

Two equivalent isospin projection methods are proposed and applied to a conventional FBCS wavefunction, thus obtaining a F _NT BCS one. To test the effect of the projection the FBCS and F _NT BCS wavefunctions are compared with an exact shell model one. Practical calculations are carried out for He⁶ using the Volkov's semi-realistic potential and the first 3 major shells. Results of the calculations show that a) the F _NT BCS with 5 free parameters is capable of approximating the shell model wavefunction with 87 parameters and b) the FBCS results differ little from the F _NT BCS ones in this special case, what should be caused by the Pauli principle and the large energy spacing ?ω.     

On the issue of zero point energy in Bose–Einstein condensates

Following on our earlier work in this area, here we examine in some detail the physical mechanism involved in the Bose–Einstein condensation process. In particular we emphasise the significance of the zero value of the chemical potential at and below the critical temperature. The molar zero-point energy (ZPE) for an ideal gas of He⁴ atoms in our new analysis is estimated and found to be very close to that calculated for an ideal Fermi gas of He³ atoms under the same conditions. This gives numerical support to our theory. We also show how the theory is consistent with the presence of a density maximum in liquid He⁴.     

LOCV calculation for the uniform electron fluid at finite temperature

The free energy of the homogeneous electron fluid at finite temperature is obtained using the lowest order constrained variational (LOCV) method. In order to test the convergence of cluster expansion series the three-body cluster terms are calculated with the LOCV correlation functions. The results agree reasonably with those of Monte Carlo, coupled-cluster, perturbational expansion etc, techniques at zero temperature. The flashing and critical temperatures as well as the critical exponent are found to be about 0.6, 1.3 eV and 0.384 respectively. A similar liquid-gas phase transition to that of nuclear matter and liquid He³ is observed. Received 15 April 2002 / Received in final form 19 October 2002 Published online 4 February 2003 RID="a" ID="a"e-mail: modarres@khayam.ut.ac.ir     

Positron impact ionisation of He ion

Triple differential cross-sections (TDCS) of a hydrogenic (He⁺) ion has been studied by positron impact using coplaner geometry for both symmetric and asymmetric kinematics in the intermediate and medium high incident energy region. TDCS has also been studied of He⁺ ion by electron impact for symmetric kinematics taking account of the electron exchange effect. The final state wavefunction is chosen as the correlated 3-body Coulomb wavefunction satisfying the exact asymptotic boundary condition. The long range Coulomb interaction in the initial channel between the ionic target and the projectile has also been taken into account properly. For positron impact, the collision is found to be almost recoilless at lower incident energies, in contrast to the strong recoil peak noted in the case of electron impact ionisation. For electron impact, the exchange effect is found to be significantly high for equal energy sharing in the final channel. Received 10 July 1999 and Received in final form 7 December 1999     

A phase cell cluster expansion for Euclidean field theories

We adapt the cluster expansion first used to treat infrared problems for lattice models (a mass zero cluster expansion) to the usual field theory situation. The field is expanded in terms of special block spin functions and the cluster expansion given in terms of the expansion coefficients (phase cell variables); the cluster expansion expresses correlation functions in terms of contributions from finite coupled subsets of these variables. Most of the present work is carried through in d space time dimensions (for φ₂⁴ the details of the cluster expansion are pursued and convergence is proven). Thus most of the results in the present work will apply to a treatment of φ₃⁴ to which we hope to return in a succeeding paper. Of particular interest in this paper is a substitute for the stability of the vacuum bound appropriate to this cluster expansion (for d = 2 and d = 3), and a new method for performing estimates with tree graphs. The phase cell cluster expansions have the renormalization group incorporated intimately into their structure. We hope they will be useful ultimately in treating four dimensional field theories.     

Theoretical Study of Triatomic Systems Involving Helium Atoms

The triatomic ⁴He system and its isotopic species ${^4{\rm He}_2^3{\rm He}}$ are theoretically investigated. By adopting the best empirical helium interaction potentials, we calculate the bound state energy levels as well as the rates for the three-body recombination processes: ⁴He + ⁴He + ⁴He → ⁴ He₂ + ⁴He and ⁴He + ⁴He + ³He → ⁴He₂ + ³He. We consider not only zero total angular momentum J = 0 states, but also J > 0 states. We also extend our study to mixed helium-alkali triatomic systems, that is ⁴He₂ X with X = ⁷Li, ²³Na, ³⁹K, ⁸⁵ Rb, and ¹³³Cs. The energy levels of all the J ≥ 0 bound states for these species are calculated as well as the rates for three-body recombination processes such as ⁴He + ⁴He + ⁷Li → ⁴ He₂ + ⁷Li and ⁴He + ⁴He + ⁷Li → ⁴ He⁷Li + ⁴He. In our calculations, the adiabatic hyperspherical representation is employed but we also obtain preliminary results using the Gaussian expansion method.     

Exchange and nuclear symmetry interference in He+-He scattering

A method is described to determine the potential difference, V _g-V _u, that will accurately reproduce the exchange and nuclear symmetry oscillations observed by Aberth et al. in their experimental work on He⁺-He differential scattering. It is demonstrated that at higher energies and angles the experimental results for ⁴He⁺-⁴He, ⁴He⁺-³He and ³He⁺-³He appear incompatible with one another when investigated in terms of a two-state scattering theory. This is an indication that at these energies and angles the theory may require modification to include the effects of coupling to higher states.     

TDHF study of the He+ collision on atomic He targets at the8Be ground state energy

Experimentally the⁸Be ground state resonance has been studied in He⁺ collisions on atomic He atoms. The nuclear resonance manifests itself by satellite resonance lines corresponding to different electron configurations of the Be ion. Experimentally a large probability for the emission of one electron has been deduced. We study the atomic He⁺+He collision within a model in which the evolution of the electron wavefunction is treated dynamically in the TDHF scheme, and the motion of the nuclei is treated classically. In agreement with experiment we find a large probability for one electron to be emitted into the continuum during the lifetime of the⁸Be ground state resonance.     

Comparison of Brueckner theory with “exact” results for H3 and He4 nuclei

The consistency of the second-order Brueckner-Baranger approximation to the ground-state energyE of H³ and He⁴ nuclei is confirmed by the agreement ofE with the recent Faddeev and Faddeev-Yakubovsky equation results for Reid local potential, by the dependence ofE on different phase-equivalent transforms (= off-shell changes) of the Reid potential which agrees with Faddeev equation and nuclear matter calculations, and by the Coester line well reproduced for a set of realistic potentials in He⁴. The influence of approximations in solving Bethe-Goldstone equation is also discussed.     

Effective Field Theory Analysis of Three-Boson Systems at Next-To-Next-To-Leading Order

We use an effective field theory for short-range forces (SREFT) to analyze systems of three identical bosons interacting via a two-body potential that generates a scattering length, a, which is large compared to the range of the interaction, ?. The amplitude for the scattering of one boson off a bound state of the other two is computed to next-to-next-to-leading order (N²LO) in the ?/a expansion. At this order, two pieces of three-body data are required as input in order to renormalize the amplitude (for fixed a). We apply our results to a model system of three Helium-4 atoms, which are assumed to interact via the TTY potential. We generate N²LO predictions for atom-dimer scattering below the dimer breakup threshold using the bound-state energy of the shallow Helium-4 trimer and the atom-dimer scattering length as our two pieces of three-body input. Based on the convergence pattern of the SREFT expansion, as well as differences in the predictions of two renormalization schemes, we conclude that our N²LO phase- shift predictions will receive higher-order corrections of < 0.2 %. In contrast, the prediction of SREFT for the binding energy of the “deep” trimer of Helium-4 atoms displays poor convergence.     

The mobility of dilute solutions of impurities in He3 at low temperatures

The mobility of impurities in He³ is calculated using Kubo's formula and assuming a quasi-particle-like behaviour of the impurities. This is reasonable in the low temperature limit. If one has to deal with charged particles having a large cross-section for the scattering of He³ quasi-particles on account of their large effective radii, this limit is as low as presumably 0.01 °K. For uncharged particles the validity of the theory is not confined to such a small region of the temperature. The mobility behaves as 1/T ² and its dependence upon the frequency is almost that of a classical sphere experiencing a force of friction proportional to its velocity. The parameters occuring in the formula for the mobility (the effective mass and the scattering cross-section between He³ and the impurities) may in principle be determined by independent experiments.