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Physical Review A

Shell structure in mixed 3He−4He droplets

Jesús Navarro, A. Poves, M. Barranco, Martí Pi

Published Feb 18, 2004• 9 citations
Publisher

Abstract

Due to the immiscibility of ${}^{3}\mathrm{He}$ into ${}^{4}\mathrm{He}$ at very low temperatures, mixed helium droplets consist of a core of ${}^{4}\mathrm{He}$ atoms coated by a ${}^{3}\mathrm{He}$ layer whose thickness depends on the number of atoms of each isotope. When these numbers are such that the centrifugal kinetic energy of the ${}^{3}\mathrm{He}$ atoms is small and can be considered as a perturbation to the mean-field energy, a novel shell structure arises, with magic numbers different from these of pure ${}^{3}\mathrm{He}$ droplets. If the outermost shell is not completely filled, the valence atoms align their spins up to the maximum value allowed by the Pauli principle.

DOI
10.1103/physreva.69.023202

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Due to the immiscibility of 3He{}^{3}\mathrm{He}3He into 4He{}^{4}\mathrm{He}4He at very low temperatures, mixed helium droplets consist of a core of 4He{}^{4}\mathrm{He}4He atoms coated by a 3He{}^{3}\mathrm{He}3He layer whose thickness depends on the number of atoms of each isotope. When these numbers are such that the centrifugal kinetic energy of the 3He{}^{3}\mathrm{He}3He atoms is small and can be considered as a perturbation to the mean-field energy, a novel shell structure arises, with magic numbers different from these of pure 3He{}^{3}\mathrm{He}3He droplets. If the outermost shell is not completely filled, the valence atoms align their spins up to the maximum value allowed by the Pauli principle.
We have studied the structure of 3He{}^{3}\mathrm{He}3He droplets at zero temperature using a density functional approach plus a configuration interaction calculation in an harmonic oscillator major shell. The most salient feature of open shell drops is that the valence atoms couple their spins to the maximum value compatible with Pauli's principle, building a large magnetic moment. We have determined that 29 atoms constitute the smallest self-bound droplet.
The diffusion Monte Carlo technique is used to calculate and analyze the excitation spectrum of 3He atoms bound to a cluster of 4He atoms by using a previously determined optimum filling of single-fermion orbits with well-defined orbital angular momentum L, spin S, and parity quantum numbers. The study concentrates on the energies and shapes of the three kinds of states for which the fermionic part of the wave function is a single Slater determinant: maximum L or maximum S states within a given orbit, and fully polarized clusters. The picture that emerges is that of systems with strong shell effects, whose binding and excitation energies are essentially determined by averages over configuration at fixed number of particles and spin, i.e., by the monopole properties of an effective Hamiltonian.
Diffuse clusters of arbitrary numbers of helium atoms are treated in a mean-field theory of the many-body system. The nonlinear Hartree self-consistent-field equations are solved numerically to give their energy eigenvalues and eigenfunctions. Both isotopes 4He and 3He are treated according to their bosonic and fermionic statistics and symmetry requirements. Clusters of all sizes of 4He atoms are stable, but there is a minimum threshold of ∼70 atoms for the formation of 3He clusters. All the excited states in addition to the ground state were found. The wave functions are seen to have interesting differences from electronic wave functions in atoms.
A stability chart of mixed 4He{}^{4}\mathrm{He}4He and 3He{}^{3}\mathrm{He}3He clusters has been obtained by means of the diffusion Monte Carlo method, using both the Aziz HFD-B and the Tang-Toennies-Yiu atom-atom interaction. The investigated clusters contain up to eight 4He{}^{4}\mathrm{He}4He atoms and up to 20 3He{}^{3}\mathrm{He}3He atoms. One single 4He{}^{4}\mathrm{He}4He binds 20 3He{}^{3}\mathrm{He}3He atoms, and two 4He{}^{4}\mathrm{He}4He bind 1, 2, 8, and more than 14 3He{}^{3}\mathrm{He}3He atoms. All clusters with three or more 4He{}^{4}\mathrm{He}4He atoms are bound, although the combinations 4He33He9,10,11{}^{4}{\mathrm{He}}_{3}^{3}{\mathrm{He}}_{9,10,11}4He33​He9,10,11​ and 4He43He9{}^{4}{\mathrm{He}}_{4}^{3}{\mathrm{He}}_{9}4He43​He9​ are metastable. Clusters with 2, 8, and 20 3He{}^{3}\mathrm{He}3He atoms are particularly stable and define magic 3He{}^{3}\mathrm{He}3He numbers.