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Physical Review Letters

Application of Scattering Theory to the Calculation of Alkali Negative-Ion Bound States

David W. Norcross

Published Feb 4, 1974• 104 citations
Publisher

Abstract

Binding energies of the ground $^{1}S$ states of the alkali negative ions are calculated using the coupled equations of scattering theory, based on semiempirical effective potentials for the neutral atoms. An additional true bound state with configuration $^{3}P$ lying just below the first excited state of the neutral atom is predicted for every species except ${\mathrm{Li}}^{\ensuremath{-}}$.

DOI
10.1103/physrevlett.32.192

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Binding energies of the ground 1S^{1}S1S states of the alkali negative ions are calculated using the coupled equations of scattering theory, based on semiempirical effective potentials for the neutral atoms. An additional true bound state with configuration 3P^{3}P3P lying just below the first excited state of the neutral atom is predicted for every species except Li\ensuremath−{\mathrm{Li}}^{\ensuremath{-}}Li\ensuremath−.
Abstract We study theoretically interaction potentials and low-energy collisions between different alkali atoms and alkali ions. Specifically, we consider systems such as X + Y + , where X( Y + ) is either Li(Cs + ) or Cs(Li + ), Na(Cs + ) or Cs(Na + ) and Li(Rb + ) or Rb(Li + ). We calculate the molecular potentials of the ground and first two excited states of these three systems using a pseudopotential method and compare our results with those obtained by others. We derive ground-state scattering wave functions and analyze the cold collisional properties of these systems for a wide range of energies. We find that, in order to get convergent results for the total scattering cross sections for energies of the order 1 K, one needs to take into account at least 60 partial waves. The low-energy scattering properties calculated in this paper may serve as a precursor for experimental exploration of quantum collisions between an alkali atom and an alkali ion of a different nucleus.
Atomic many-body theory methods are used to calculate the fine structure of negative ions formed by binding a p electron into an open shell, ${\mathrm{Ca}}^{\mathrm{\ensuremath{-}}}$, ${\mathrm{Sr}}^{\mathrm{\ensuremath{-}}}$, ${\mathrm{Ba}}^{\mathrm{\ensuremath{-}}}$, and ${\mathrm{Ra}}^{\mathrm{\ensuremath{-}}}$. This binding is due to a strong correlation potential acting between the electron and the neutral atom. Comparison with experimental data shows that the second order many-body perturbation theory calculation overestimates the correlation potential by 10% to 15%. Scaling factors are introduced in the correlation potential to reproduce experimental binding energies of the lower p1/2{\mathrm{p}}_{1\mathrm{/}2}p1/2​ components. This procedure yields fine-structure intervals in excellent agreement with experiment for ${\mathrm{Ca}}^{\mathrm{\ensuremath{-}}}$, ${\mathrm{Sr}}^{\mathrm{\ensuremath{-}}}$, and ${\mathrm{Ba}}^{\mathrm{\ensuremath{-}}}$, and allows us to predict that in ${\mathrm{Ra}}^{\mathrm{\ensuremath{-}}}$ the p1/2{\mathrm{p}}_{1\mathrm{/}2}p1/2​ state is bound by 100 meV, and p3/2{\mathrm{p}}_{3\mathrm{/}2}p3/2​ is a resonance at 16 meV in the continuum.
The method of polarized orbitals is modified to treat low-energy scattering of electrons from highly polarizable systems, specifically alkali-metal atoms. The modification is carried out in the particular context of the eee-Li system, but the procedure is general; it consists of modifying the polarized orbital, so that when used in the otherwise orthodox form of the method, it gives (i) the correct electron affinity of the negative ion (in this case ${\mathrm{Li}}^{\ensuremath{-}}$), (ii) the proper (i.e., Levinson-Swan) number of nodes of the associated zero-energy scattering orbital, and (iii) the correct polarizability. A procedure is devised whereby the scattering length can be calculated from the (known) electron affinity without solving the bound-state equation. Using this procedure we adduce a 1S^{1}S1S scattering length of 8.69a08.69{a}_{0}8.69a0​. (The 3S^{3}S3S scattering length is $\ensuremath{-}9.22{a}_{0}$.) The above modifications can also be carried out in the (lesser) exchange adiabatic approximation. However, they lead to qualitatively incorrect 3S^{3}S3S phase shifts. The modified polarized-orbital phase shifts are qualitatively similar to close-coupling and elaborate variational calculations. Quantitative differences from the latter calculations, however, remain; they are manifested most noticeably in the very-low-energy total and differential spin-flip cross sections.
In order to investigate the binding mechanism of weakly bound states of positronic alkali atoms, we calculate the energies and wavefunctions using the Gaussian expansion method (GEM) where a positronium (Ps)–alkali ion channel and a positron–alkali atom channel are explicitly introduced. The energies of the bound states are updated using a model potential that reproduces well the observed energy levels of alkali atoms. The binding mechanism of the positronic alkali atom is analyzed by the wavefunctions obtained. The structure of the positronic alkali atom has been regarded as a Ps cluster orbiting the alkali ion, which is described by the Ps–alkali ion channel. We point out that the fraction having the positron–alkali atom configuration is small but plays an indispensable role for the weakly bound system.
The relativistic correlation-potential method was used to calculate binding energies and fine-structure intervals for Pd, Ba, and Yb negative ions and to investigate low-energy electron scattering by Yb, Hg, and Ra atoms. The results for the binding energies are the following: 540 meV for the 5s state of ${\mathrm{Pd}}^{\mathrm{\ensuremath{-}}}$, 190 and 133 meV for the 6p1/2{\mathit{p}}_{1/2}p1/2​ and 6p3/2{\mathit{p}}_{3/2}p3/2​ states of ${\mathrm{Ba}}^{\mathrm{\ensuremath{-}}}$, and 36 meV for the 6p1/2{\mathit{p}}_{1/2}p1/2​ state of the ${\mathrm{Yb}}^{\mathrm{\ensuremath{-}}}$. A number of prominent p and d resonances are revealed in the scattering phase-shift calculations. These p or d resonances lead to a phenomenon of 100% polarization of the scattered electron beam at appropriate electron energy and scattering angle. A criterion is proposed to measure the strength of the nonlocal correlation potential and to evaluate its ability to create a bound state: FG(r',r)\ensuremath{\Sigma}(r,r')dr dr'>1 is the necessary condition for the formation of a bound state. Here \ensuremath{\Sigma} is the correlation potential and G is the electron Green's function at zero energy.
We report a theoretical calculation of the relativistic corrections to the binding energy of positronic alkali-metal atoms. The ground state for the positronic alkali-metal atom is a loosely bound state with the structure of an alkali-metal ion surrounded by a positronium cloud. The correlation between the valence electron and positron in the field of a residual core ion is taken into account using a three-body model. Relativistic corrections to the binding energy for positronic alkali-metal atoms are evaluated based on the Breit-Pauli perturbation theory up to the second order of the fine structure constant. The relativistic corrections caused by the interaction between the valence electron and core ion was evaluated from the decomposed expectation values of the perturbation Hamiltonian. We found that the importance of relativistic corrections to the binding energy, namely the ratio of the net relativistic correction in the total binding energy, is remarkably enhanced in the loosely bound states of positronic alkali-metal atoms, where the charge of the valence electron is screened by the positron as positronium in the outermost region, while the electron is released from the positron near the nucleus. The results imply that positronic atoms with a dominant positronium can serve as an apt testing ground for relativistic quantum mechanics.