Analytic representation of the Efimov effect in the helium trimer

Abstract

Exact solutions for the low-temperature helium dimer and trimer, 4 He 2 and 4 He 3 , are derived, based on our ␦ function model for the interatomic potential. For the trimer, the Faddeev equations are shown to be separable in hyperspherical coordinates, with the S-wave alone giving an exact solution. The parameters 0 and r 0 are fitted to accurate computations on the dimer and trimer. Excited states of the trimer are shown to exhibit the Efimov effect, whereby artificially reducing the strength of the two-body potential causes an infinite number of weakly-bound levels to condense out of the continuum. All the features anticipated by Efimov are quantitatively reproduced within our model. Since short-range details of the intermolecular forces are not relevant, our results can be considered to be universally applicable.

Dirac bubble potential for He–He and inadequacies in the continuum: Comparing an analytic model with elastic collision experiments

“…35 has already been shown to reproduce several distinctive features of dimers 35 and trimers. [36][37][38] In this context, it is worth noting some relevant statement pointed out in Ref. 31 about the observation of "gigantic cross sections which increase with decreasing scattering energy, as expected for a potential with a possible bound state near the continuum."…”

“…One of the most insightful and fascinating potential models for pairwise interactions in helium consists of an attractive deltafunction in the shape of a spherical bubble of radius r 0 to mimic the interatomic potential. [35][36][37][38] At the origin of this model is the observation that 4 He- 4 He supports but a single bound state, a property that is highly evocative of a deltafunction (which has the same property). 35 However, it is still very surprising that such an oversimplified model can reproduce distinctive features of the helium dimers 35 and trimers.…”

“…35 However, it is still very surprising that such an oversimplified model can reproduce distinctive features of the helium dimers 35 and trimers. [36][37][38] In fact, the Dirac bubble potential (Dbp) is renowned for its capability to reproduce, without the need for elaborate computations, a wealth of stability properties in helium clusters, viz., the binding energies of 4 He 3 (96.1 mK) and 4 He 2 3 He (11.4 mK), 36,37 an Efimov state 39 for 4 He 2 3 He (Ref. 38), and the property of the isotopic variants 3 He 4 He, 3 He 2 , 4 He 3 He 2 , and 3 He 3 to be unstable.…”

“…[36][37][38] In fact, the Dirac bubble potential (Dbp) is renowned for its capability to reproduce, without the need for elaborate computations, a wealth of stability properties in helium clusters, viz., the binding energies of 4 He 3 (96.1 mK) and 4 He 2 3 He (11.4 mK), 36,37 an Efimov state 39 for 4 He 2 3 He (Ref. 38), and the property of the isotopic variants 3 He 4 He, 3 He 2 , 4 He 3 He 2 , and 3 He 3 to be unstable. 35,37 Even more remarkably, it reproduces the unique bound state of 4 He 2 "with an amazing overlap of 0.99 942 with the exact value" 35 and predicts correctly the average internuclear distance (52.6 Å) and delocalization (48 Å) for that state.…”

Semiempirical hyperspherical model for 4HeN clusters

“…In addition, Barletta and Kievsky [9] have carried out variational computations using hyperspherical harmonic functions. The present authors [10] have considered a delta function model for the trimer, mainly to explore the analytic properties of the Efimov and Thomas effects. Recently, the FaddeevYakubovsky formalism has been applied to the tetramer [11].…”