Spectra of helium clusters with up to six atoms using soft-core potentials

Abstract

In this work we investigate small clusters of helium atoms using the hyperspherical harmonic basis. We consider systems with A = 2, 3, 4, 5, 6 atoms with an inter-particle potential which does not present a strong repulsion at short distances. We use an attractive gaussian potential that reproduces the values of the dimer binding energy, the atom-atom scattering length, and the effective range obtained with one of the widely used He-He interactions, the LM2M2 potential. In systems with more than two atoms we consider a repulsive three-body force that, by construction, reproduces the trimer binding energy of the LM2M2 potential. With this model, consisting in the sum of a two-and three-body potential, we have calculated the spectrum of clusters formed by four, five and six helium atoms. We have found that these systems present two bound states, one deep and one shallow close to the threshold fixed by the energy of the (A − 1)-atom system.Universal relations between the energies of the excited state of the A-atom system and the ground state energy of the (A − 1)-atom system are extracted as well as the ratio between the ground state of the A-atom system and the ground state energy of the trimer. PACS numbers: 31.15.xj, 31.15.xt, 36.90.+f, 34.10.+x Systems composed by few helium atoms have been object of intense investigation from a theoretical and experimental point of view. The existence of the He-He molecule was experimentally established using diffraction experiments [1-4]. Its binding energy E 2b has been estimated to be around 1 mK and its scattering length a 0 around 190 a.u. This makes the He-He molecule one of the biggest diatomic molecules. On the theoretical side, several He-He potentials have been proposed; in spite of different details and derivations, all of them share the common feature of a sharp repulsion below an inter-particle distance of approximately 5 a.u..Another important characteristic of the He-He interactions is their effective range r 0 ≈ 13 a.u.. Accordingly, the ratio a 0 /r 0 is large enough (> 10) to place small helium clusters into the frame of Efimov physics [5,6]. As shown by Efimov, when at least two of the two-body subsystems present an infinitely large scattering length (or zero binding energy) an infinite sequence of bound states (called Efimov states) appear in the three-body system; their binding energies scale in a geometrical way and they accumulate at zero energy. The scaling factor, e −2π/s 0 ≈ 1/515.03, is universal and depends only on the ratio between particle masses (for three identical bosons s 0 ≈ 1.00624), not on the details of the two-body interaction (see Ref.[7] for a review). For a finite a 0 /r 0 ratio, the number of the Efimov states has been estimated to be N = (s 0 /π)ln|a 0 /r 0 | [6]; for example, the (bosonic) three 4 He system presents an excited state just below the atom-dimer threshold that has been identified as an Efimov state.Triggered by this interesting fact, several investigations of the helium trimer have been produced, establishing that its ex...

Convergence of the hyperspherical-harmonics expansion with increasing number of particles for bosonic systems. II. Inclusion of the three-body force

“…[1], it has been shown that the reason for the improved convergence with increasing boson number, observed in Ref. [2], originates from the decrease of the ratio of nondiagonal coupling potential to the lowest-order diagonal potential, associated with a single Gaussian potential, as N 1/2 or faster at N → ∞. The improved convergence has been further confirmed by numerical calculations, performed with the two-body potential from Ref.…”

“…The convergence of the hyperspherical expansion for six atoms in Ref. [2] was better than that for four atoms both when only the two-body force was used and when the three-body force was added. This observation is important since it suggests that in the limit of a large scattering length a fast solution of the many-body problem could be achieved by using a simple phenomenological soft two-body potential plus a repulsive three-body force accounting for missing correlations between the constituents of the many-body system.…”

“…[2]. These calculations used a simple twobody interaction and predict the correct value for the He-He scattering length and is represented by only one Gaussian, and a three-body He-He-He potential of the hypercentral form with the radius and the depth fitted to reproduce the realistic calculations of binding energies for four, five, and six helium atoms from other many-body methods.…”

Convergence properties of the effective theory for trapped bosons

“…Here the number in parentheses gives the difference in the last digit between the extrapolation in N and ǫ which can serve as an estimate of the numerical error. The spectra are much more compressed than the corresponding free space results [4][5][6][7][8][9]. However, a similar pattern can be observed: starting from A = 3, every A-body state is accomponied by a corresponding A + 1-body state.…”

“…In particular, there is a pair of universal four-body states associated with every Efimov state [4,5]. Moreover, universal cluster states have been predicted in five-, six-and possibly higher-body systems as well [6][7][8][9]. Using Feshbach resonances, the two-, three-, four-and five-body states have been confirmed in a ultracold atomic gases in a variety of experiments with different atom species [10][11][12].…”