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Journal of Physics B Atomic Molecular and Optical Physics

On the limit of existence of Borromean binding in three-particle systems with screened Coulomb interactions

Mariusz Pawlak, Mirosław Bylicki, Prasanta Kumar Mukherjee

Published Apr 15, 2014• 13 citations
Publisher

Abstract

Molecular-type systems, ( m 1 ) ± ( m 2 ) ± ( m 3 ) ∓ , consisting of three particles of masses m i and of unit electric charges with the Coulomb interactions weakened by the Debye screening are considered. Existence and range of Borromean binding in symmetric systems (of masses m 1 = m 2 ) is investigated with respect to the masses of their constituents. It is shown that such systems can exist in Borromean ground states if the mass ratio q = m 3 / m 1 is less or equal to 1.668. This improves considerably the lower bound to the limit of existence of the Borromean binding of symmetric systems suggested by Pont and Serra (2009 Phys. Rev. A 79 032508) as q ⩽ 1. Qualitative meaning of the improvement is that the Borromean binding occurs not only for systems where two identical particles are heavier than the third one but it is also possible for systems of the opposite mass relation. The range of screening parameter in which a system is Borromean known as a Borromean window is also determined for μ + μ + e − , π + π + μ − , π + μ − μ − and e + e − e − .

DOI
10.1088/0953-4075/47/9/095701

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Molecular-type systems, ( m 1 ) ± ( m 2 ) ± ( m 3 ) ∓ , consisting of three particles of masses m i and of unit electric charges with the Coulomb interactions weakened by the Debye screening are considered. Existence and range of Borromean binding in symmetric systems (of masses m 1 = m 2 ) is investigated with respect to the masses of their constituents. It is shown that such systems can exist in Borromean ground states if the mass ratio q = m 3 / m 1 is less or equal to 1.668. This improves considerably the lower bound to the limit of existence of the Borromean binding of symmetric systems suggested by Pont and Serra (2009 Phys. Rev. A 79 032508) as q ⩽ 1. Qualitative meaning of the improvement is that the Borromean binding occurs not only for systems where two identical particles are heavier than the third one but it is also possible for systems of the opposite mass relation. The range of screening parameter in which a system is Borromean known as a Borromean window is also determined for μ + μ + e − , π + π + μ − , π + μ − μ − and e + e − e − .
We show the existence of Borromean bound states in a one-dimensional quantum three-body system composed of two identical bosons and a distinguishable particle. It is assumed that there is no interaction between the two bosons, while the mass-imbalanced two-body subsystems can be tuned to be either bound or unbound. Within the framework of the Faddeev equations, the three-body spectrum and the corresponding wave functions are computed numerically. In addition, we identify the parameter-space region for the two-body interaction, where the Borromean states occur, evaluate their geometric properties, and investigate their dependence on the mass ratio.
The stability of the ground state of three-unit charge ${q}_{i}=\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}1\ensuremath{\mp}1$, interacting through a statically screened Coulomb potential, has been studied as a function of the values of the constituent masses for different values of the screening parameter. General conditions are given and accurate variational calculations have been performed to determine the region where the three-body ground state is Borromean. The possible existence of Efimov states when the screening parameter is equal to the two-body critical screening parameter is discussed. The critical exponent for the energy in different regions of the stability diagram has been also calculated.
Abstract We investigate the bound states, Borromean states and resonant states for the 3Po (J = 1) states of ppμpp\muppμ, ddμdd\muddμ andttμtt\muttμ molecular ions interacting with exponential-cosine-screened Coulomb potential. The bound 3Po state energies are calculated using the Rayleigh-Ritz variational method and the resonance parameters are calculated using the stabilization method. Highly correlated exponential wave functions are used. The bound 3Po state energies, the 3Po resonance parameters (energy and width), along with 1S and 2S threshold energies of the pμp\mupμ, dμd\mudμ andtμt\mutμ atoms are also reported for different shielding parameters. The critical values of the screening parameters for the 3Po states are also reported for which the three-body exotic molecular systems are stable while all the possible fragments are unbound, that is shows windows for Borromean binding.
Non-relativistic quantum chemical calculations of the particle mass, m±2, corresponding to the dissociation threshold in a range of Coulomb three-particle systems of the form m±1m±2m±3, are performed variationally using a series solution method with a Laguerre-based wavefunction. These masses are used to calculate an accurate stability boundary, i.e., the line that separates the stability domain from the instability domains, in a reciprocal mass fraction ternary diagram. This result is compared to a lower bound to the stability domain derived from symmetric systems and reveals the importance of the asymmetric (mass-symmetry breaking) terms in the Hamiltonian at dissociation. A functional fit to the stability boundary data provides a simple analytical expression for calculating the minimum mass of a third particle required for stable binding to a two-particle system, i.e., for predicting the bound state stability of any unit-charge three-particle system.
The question of stability of a given quantum system made up of charged particles is of fundamental interest in atomic, molecular, and nuclear physics. In this work, the stability for the negatively charged positronium (Ps)-like ions or the three-body system ( Z e + , e − , e − ) with Yukawa potentials is studied using correlated exponential wavefunctions based on the Ritz variational method. We obtained the critical screening parameter μ C as a function of the continuously varied nuclear charge Z , the critical nuclear charge Z C as a function of the screening parameter μ , and the ionization energies in terms of the screening parameter μ and Z . The critical nuclear charge for the bare Coulomb system ( Z e + , e − , e − ) obtained using 700-term correlated exponential wavefunctions is in accord with the reported results. The ionization energy, μ C , and Z C for the Yukawa system ( Z e + , e − , e − ) exhibit interesting behaviors. The present study describes the possible nonexistence of Borromean binding as well as Efimov states. The possible existence of quasi-bound resonances states for the negatively charged screened Ps-like ions is briefly discussed.
The study of quantum mechanical bound states is as old as quantum theory itself. Yet, it took many years to realize that three-body Borromean systems that are bound when any two-body subsystem is unbound are abundant in nature. Here we demonstrate the existence of Borromean systems of spin-polarized (spinless) identical fermions in two spatial dimensions. The ground state with zero orbital (planar) angular momentum exists in a Borromean window between critical two- and three-body strengths. The doubly degenerate first excited states of angular momentum one appears only very close to the two-body threshold. They are the lowest in a possible sequence of so-called super-Efimov states. While the observation of the super-Efimov scaling could be very difficult, the Borromean ground state should be observable in cold atomic gases and could be the basis for producing a quantum gas of three-body states in two dimensions.