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Iranian Journal of Physics Research

The ground state energy of 3He droplet in the LOCV framework

Majid Modarres, Shahrooz Motahari, Amin Rajabi

Published Jun 15, 2012• 0 citations
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Abstract

The (extended) lowest order constrained variational ((E) LOCV) method was used to calculate the ground state energy of liquid helium 3 (3He) droplets at zero temperature. Different types of density distribution profiles, such as the Gaussian, the Quasi-Gaussian and the Woods-Saxon were used. It was shown that at least, on average, near 20 3He atoms are needed to get the bound state for 3He liquid droplet. Depending on the choice of the density profiles and the atomic radius of 3He, the above estimate can increase to 300. Our calculated ground state energy and the number of atoms in liquid 3He droplet were compared with those of Variational Monte Carlo (VMC) method, Diffusion Monte Carlo (DMC) method and Density Functional Theory (DFT), for which a reasonable agreement was found.

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arXiv:cond-mat/0604034•Physical Review B•Mar 30, 2006•24 citations
The (extended) lowest order constrained variational ((E) LOCV) method was used to calculate the ground state energy of liquid helium 3 (3He) droplets at zero temperature. Different types of density distribution profiles, such as the Gaussian, the Quasi-Gaussian and the Woods-Saxon were used. It was shown that at least, on average, near 20 3He atoms are needed to get the bound state for 3He liquid droplet. Depending on the choice of the density profiles and the atomic radius of 3He, the above estimate can increase to 300. Our calculated ground state energy and the number of atoms in liquid 3He droplet were compared with those of Variational Monte Carlo (VMC) method, Diffusion Monte Carlo (DMC) method and Density Functional Theory (DFT), for which a reasonable agreement was found.
Helium systems at zero temperature are strongly correlated neutral systems which can only be described by means of quantum mechanics. The existence of the two isotopes 4He and 3He allows the comparison between the quantum effects due to different statistics, bosonic and fermionic. In this Thesis, an analysis of Helium droplets has been done. Throughout this work, both energetic and structural properties of systems with different number of particles N are obtained by means of the Quantum Monte Carlo techniques (Variational and Diffusion Monte Carlo - VMC and DMC-). Several studies on Helium droplets have been published in the last years, most of them using the mean field theory (as the Density Functional) and variational approaches (as VMC). In the case of 4He, also exact DMC results have already been reported, actually, a great number of works have described with high accuracy the ground state of this bosonic system. The appearance of magic numbers from experimental point of view has motivated the study of 4He droplets, and one chapter of this Thesis is focused on this bosonic system. We have carried out a one-by-one energetic study in the range from 20 to 30 particles using the DMC technique to discuss the existence of magic numbers. On the other hand, a brief study about vortices is also done; using out code, one can calculate the energy of a droplet with one vortex inside (EV) and, therefore, the energy of a vortex (EV -E0). About 3He droplets, the sign problem makes their study quite more difficult, for this reason, the number of works is considerably smaller than in the bosonic case. One aim of this thesis was to write a DMC code using both Fixed and Released Node approximations (FN and RN) able to perform high accurate calculations of energetic and structural properties of these finite fermionic systems.
It is shown that the method of lowest order constrained variational (LOCV) which is based on the cluster expansion theory is a reliable many-body technique to calculate the nuclear matter equation of state. In this respect, the state dependent correlation functions and the effective interactions which have been produced by the LOCV calculation with the Reid and \ensuremath{\Delta}-Reid soft core interactions are used to estimate the size of higher order cluster terms such as the effect of three-body cluster energy on the nuclear matter ground state energy. Finally it is shown that the LOCV normalization constraint plays a major role in the convergence of the cluster expansion and the result of LOCV calculation can be as good as more sophisticated approaches which go beyond lowest order.
A lowest-order constrained variational (LOCV) method, with modified conditions of healing on the two-body Jastrow wave function, is investigated in calculations for the ground-state energy levels of many-body spin-polarized atomic deuterium. Results are obtained for the , and phases, corresponding to equal occupations of one, two or three nuclear spin states. Estimates for the optimum healing conditions are obtained by comparison of LOCV results with current Monte Carlo benchmarks. The nature of the phases, i.e., quantum gas or quantum liquid, is discussed, the energy of the phase in our calculations always occurring above the gas-liquid interphase for healing conditions within a physically acceptable range.
The description of the properties of liquid Helium is a challenge for any microscopic many-body theory. In this context, we study the ground state and the excitation spectrum of one 3 He impurity in liquid 4 He at T=0 with the aim of illustrating the power of the correlated basis function formalism in describing heavily correlated systems. The strong interatomic interaction and the large density require the theory to be pushed to a high degree of sophistication. A many-body correlation operator containing explicit two- and three-particle correlation functions is needed to obtain a realistic ground state wave function, whereas a perturbative expansion including up to two phonon correlated states must be enforced to study the impurity excitation energies. The theory describes accurately the experimental spectrum along all the available momentum range. As empirically shown by the experiments, a marked deviation from the quadratic Landau-Pomeranchuck behavior is found and the momentum dependent effective mass of the impurity increases of ~50% at q~1.7 Å-1 with respect to its q=0 value. Although the main emphasis is given to the correlated basis function theory, we present also comparisons with other methods, as diffusion Monte Carlo, variational Monte Carlo with shadow wave functions and time dependent correlations.
Below Tλ = 2.17 K, bulk 4He is a superfluid and has a non-zero Bose-Einstein condensate fraction. This work will focus primarily on how phenomena such as superfluidity, Bose condensation and superfluid vortices are manifested in microscopic, inhomogeneous helium systems. Path Integral Monte Carlo is a powerful method for calculating the equilibrium properties of quantum systems at finite temperature. We have achieved linear scaling of computer time with number of particles through the use of neighbor lists, allowing us to simulate systems of several thousand atoms. We have derived a local superfluid estimator and used it to examine the microscopic superfluid response around a molecule rotating in a helium nanodroplet. We found that the first solvation layer is well-described by a two dimensional superfluid, with thermal excitations occuring at a lower temperature than in bulk helium. The effective moment of inertia of a linear impurity in a helium droplet is calculated, and compared with experimental scattering results. In addition, we calculated the vortex formation energy for both pure droplets and droplets doped with linear impurities, and found that the linear impurities had a negligible impact on the formation energy. A possible spectroscopic signature of vortices in doped helium droplets was suggested. After deriving a local estimator, we calculated the condensate fraction throughout the free helium surface of a semi-infinite slab. These results, along with densitydensity correlation functions, were used to characterize the surface excitations and calculate the extent to which ripplons are present. In addition, the ripplon dispersion relation was calculated using imaginary-time correlation functions, and found to be lll in good agreement with experimental results. Finally, we have calculated the Dey be-Waller factor in solid helium for a range of temperatures and densities, and compared the scaling behavior with the predictions of harmonic theory. The first non-Gaussian contribution to the density distribution was calculated.
Small and stable drops of 3He^{3}\mathrm{He}3He atoms can only exist above a minimum number of particles, due to the combination of the 3He^{3}\mathrm{He}3He atom Fermi statistics and its light mass. An accurate estimation of this minimum number using microscopic theory has been difficult due to the inhomogeneous and fermionic nature of these systems. We present a diffusion Monte Carlo calculation of 3He^{3}\mathrm{He}3He drops with sizes near the minimum in order to determine the stability threshold. The results show that the minimum self-bound drop is formed by N=30N=30N=30 atoms with preferred orbitals for open shells corresponding to maximum value of the spin.