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Physical Chemistry Chemical Physics

Spatial dimensionality and the binding of small clusters

Ludwig W. Bruch

Published Nov 10, 2020• 0 citations
Publisher

Abstract

The effect of spatial dimensionality D on the near-threshold binding of small clusters of identical particles is shown. Estimates of the threshold coupling constants for 2 and N bosons are given for D equal to 1 to 5 and the relation to conditions for the Efimov effect is discussed. Variational trial functions for 4 identical spin-½ fermions in D = 2 are given.

DOI
10.1039/d0cp04998k

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The effect of spatial dimensionality D on the near-threshold binding of small clusters of identical particles is shown. Estimates of the threshold coupling constants for 2 and N bosons are given for D equal to 1 to 5 and the relation to conditions for the Efimov effect is discussed. Variational trial functions for 4 identical spin-½ fermions in D = 2 are given.
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The results of accurate variational calculations for the ground-state binding energies of several NNN-boson systems in two and three dimensions are reported; the systems specifically considered are small (N=2,3,4,and5N=2, 3, 4, \mathrm{and} 5N=2,3,4,and5) clusters of 4He^{4}\mathrm{He}4He atoms interacting through pairwise Lennard-Jones potentials. Evidence is found strongly suggesting each set of systems have a common functional relationship linking the binding energies and the De Boer quantum constant near threshold for self-binding. These calculations also confirm the existence of a unique threshold value of the interaction strength for all NNN-boson systems in 2D2D2D. In addition, the linear relation between the binding energies of helium clusters present in 3D3D3D holds ture in 2D2D2D as well.
Abstract The quantum mechanical three-body problem is a source of continuing interest due to its complexity and not least due to the presence of fascinating solvable cases. The prime example is the Efimov effect where infinitely many bound states of identical bosons can arise at the threshold where the two-body problem has zero binding energy. An important aspect of the Efimov effect is the effect of spatial dimensionality; it has been observed in three dimensional systems, yet it is believed to be impossible in two dimensions. Using modern experimental techniques, it is possible to engineer trap geometry and thus address the intricate nature of quantum few-body physics as function of dimensionality. Here we present a framework for studying the three-body problem as one (continuously) changes the dimensionality of the system all the way from three, through two, and down to a single dimension. This is done by considering the Efimov favorable case of a mass-imbalanced system and with an external confinement provided by a typical experimental case with a (deformed) harmonic trap.
Abstract We study a three-body system, formed by two identical heavy bosons and a light particle, in the Born–Oppenheimer approximation for an arbitrary dimension D . We restrict D to the interval 2 < D < 4, and derive the heavy–heavy D -dimensional effective potential proportional to 1/ R 2 ( R is the relative distance between the heavy particles), which is responsible for the Efimov effect. We found that the Efimov states disappear once the critical strength of the heavy–heavy effective potential 1/ R 2 approaches the limit − ( D − 2 ) 2 / 4 . We obtained the scaling function for the 133 Cs– 133 Cs– 6 Li system as the limit cycle of the correlation between the energies of two consecutive Efimov states as a function of D and the heavy-light binding energy E D 2 . In addition, we found that the energy of the ( N + 1)th excited state reaches the two-body continuum independently of the dimension D when E 2 D / E 3 ( N ) = 0.89 , where E 3 ( N ) is the N th excited three-body binding energy.
The existence of the Efimov effect is drastically affected by the dimensionality of the space in which the system is embedded. The effective spatial dimension containing an atomic cloud can be continuously modified by compressing it in one or two directions. In the present Rapid Communication we determine the dimensionality DDD for which the Efimov effect can exist for different values of the mass ratio A=mB/mA\mathcal{A}={m}_{B}/{m}_{A}A=mB​/mA​ for a general AABAABAAB system formed by two identical bosons AAA and a third particle BBB in the two-body unitary limit. In addition, we provide a prediction for the Efimov discrete scaling factor $exp(\ensuremath{\pi}/s)$ as a function of a wide range of values of A\mathcal{A}A and DDD, which can be tested in experiments that can be realized with currently available technology.