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Physical Review A

Stability diagrams of plasma-embedded three-unit-charge systems: Borromean states and the Efimov effect

Federico M. Pont, Pablo Serra

Published Mar 16, 2009• 24 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.79.032508

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The stability of the ground state of three-unit charge qi=\ifmmode±\else\textpm\fi1,\ifmmode±\else\textpm\fi1\ensuremath∓1{q}_{i}=\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}1\ensuremath{\mp}1qi​=\ifmmode±\else\textpm\fi1,\ifmmode±\else\textpm\fi1\ensuremath∓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.
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.
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 − .
The stability of weakly bound ground states, with zero total angular momentum, in three-body Coulomb systems with unit charges, is studied by means of highly accurate variational calculations. Weakly bound ground states are predicted for a number of systems, including d+{\mathit{d}}^{+}d+t+{\mathit{t}}^{+}t+${\mathit{p}}^{\mathrm{\ensuremath{-}}}$, t+{\mathit{t}}^{+}t+K+{\mathit{K}}^{+}K+${\mathrm{\ensuremath{\pi}}}^{\mathrm{\ensuremath{-}}}$, ${\mathrm{\ensuremath{\pi}}}^{+}$${\mathrm{\ensuremath{\mu}}}^{+}$${\mathrm{\ensuremath{\mu}}}^{\mathrm{\ensuremath{-}}}$, and ${\mathrm{\ensuremath{\pi}}}^{+}$${\mathrm{\ensuremath{\mu}}}^{+}$${\mathrm{\ensuremath{\pi}}}^{\mathrm{\ensuremath{-}}}$.
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.
We present an approach to analyze recent experimental evidences of Efimov resonant states in mixtures of ultracold gases, by considering two-species three-body atomic systems bound in a Borromean configuration, where all the two-body interactions are unbound. For such Borromean three-body systems, it is shown that a continuum three-body s -wave resonance emerges from an Efimov state as a scattering length or a three-body scale is moved. The energy and width of the resonant state are determined from a scaling function with arguments given by dimension-less energy ratios relating the two-body virtual state subsystem energies with the shallowest three-body bound state. The peculiar behavior of such resonances is that their peaks are expected to move to lower values of the scattering length, with increasing width, as one raises the temperature. For Borromean systems, two resonant peaks are expected in ultralow-temperature regimes, which will disappear at higher energies. It is shown how a Borromean-Efimov excited bound state turns out to a resonant state by tuning the virtual two-body subsystem energies or scattering lengths, with all energies written in units of the next deeper shallowest Efimov state energy. The resonance position and width for the decay into the continuum are obtained as universal scaling functions (limit cycle) of the dimensionless ratios of the two and three-body scales, which are calculated numerically within a zero-range renormalized three-body model.
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.