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Few-Body Systems

The $$^{\textbf{3}}{} \textit{P}^{{\varvec{o}}}$$ States of Exotic Molecular Ions with Exponential-Cosine-Screened Coulomb Potentials

Xin Yan Fu, Zishi Jiang, Sabyasachi Kar

Published Apr 1, 2023• 5 citations
Publisher

Abstract

Abstract We investigate the bound states, Borromean states and resonant states for the 3Po (J = 1) states of \(pp\mu\), \(dd\mu\) and\(tt\mu\) 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 \(p\mu\), \(d\mu\) and\(t\mu\) 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.

DOI
10.1007/s00601-023-01793-1

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Abstract We investigate the bound states, Borromean states and resonant states for the 3Po (J = 1) states of (pp\mu), (dd\mu) and(tt\mu) 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 (p\mu), (d\mu) and(t\mu) 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.
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 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.
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 − .
Abstract We have investigated the resonances in the e + − He ( 3 S e ) system interacting with screened potentials by using the stabilization method. Two types of screened potential have been considered: static screened Coulomb potential and exponential cosine-screened Coulomb potential. A model potential has been used to describe the interaction between the outer electron and the He + ionic core. We have found two resonances below the Ps ( n = 2 ) − He + threshold. For the unscreened case, our results are in nice agreement with some of the most accurate results available in the literature.
We investigate the bound state properties of the hydrogen-like atoms in the radial screened Coulomb potential (RSCP). using three complementary analytical approaches - expectation values with Coulomb and Kratzer reference states, variational optimization with a scaled Kratzer basis, and the Hellmann-Feynman theorem - we derive approximate energy eigenvalues as function of the screening parameter c. Benchmarked against high-precision generalized pseudospectral data, the expectation value-approach with the Kratzer basis achieves relative errors of 0.63% for the first ten s-states at c=0.1c=0.1c=0.1, while the variational method improves this further. The formalism extends naturally to Positronium, demonstrating its generality for arbitrary reduced-mass systems. The complementary biases of the methods provide robust error estimation for plasma-embedded atoms.
Exact analytical expressions for the matrix elements of the Uehling potential in a basis of explicitly correlated exponential wave functions are presented. The obtained formulas are then used to compute with an improved accuracy the vacuum polarization correction to the binding energy of muonic and pionic molecules, both in a first-order perturbative treatment and in a nonperturbative approach. The first resonant states lying below the n=2n=2n=2 threshold are also studied, by means of the stabilization method with a real dilatation parameter.