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arXiv:physics/0310141 · Physical Review A

Finite-field evaluation of the Lennard-Jones atom-wall interaction constant C3 for alkali-metal atoms

W. R. Johnson, V. A. Dzuba, Marianna S. Safronova, Marianna S. Safronova

Published Feb 23, 2004• 17 citations
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Abstract

A finite-field scaling method is applied to evaluate the Lennard-Jones interaction constant ${C}_{3}$ for alkali-metal atoms. The calculations are based on the relativistic single-double approximation in which single and double excitations of Dirac-Hartree-Fock wave functions are included to all orders in perturbation theory.

DOI
10.1103/physreva.69.022508

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A finite-field scaling method is applied to evaluate the Lennard-Jones interaction constant C3{C}_{3}C3​ for alkali-metal atoms. The calculations are based on the relativistic single-double approximation in which single and double excitations of Dirac-Hartree-Fock wave functions are included to all orders in perturbation theory.
We present third-order many-body perturbation theory calculations of the Lennard-Jones C3{C}_{3}C3​ coefficient for the alkali-metal atoms lithium, sodium, potassium, rubidium, cesium, and francium. All-order singles-doubles calculations of C3{C}_{3}C3​ are also presented for lithium, sodium, and potassium. For lithium and sodium the present values of C3{C}_{3}C3​ are compared with other theoretical and semiempirical values.
The van der Waals coefficients, C6,C8,{C}_{6},{C}_{8},C6​,C8​, and C10{C}_{10}C10​ for the alkali-metal (Li, Na, K, and Rb) and alkaline-earth-metal (Be, Mg, Ca, and Sr) atoms are estimated by a combination of ab initio and semiempirical methods. Polarizabilities and atom-wall coefficients are given as a diagnostic check, and the lowest order nonadiabatic dispersion coefficient, D8{D}_{8}D8​ and the three-body coefficient, C9{C}_{9}C9​ are also presented. The dispersion coefficients are in agreement with the available relativistic many-body perturbation theory calculations. The contribution from the core was included by using constrained sum rules involving the core polarizability and Hartree-Fock expectation values to estimate the f-value distribution.
The van der Waals coefficients for the alkali-metal atoms from Na to Fr interacting in their ground states are calculated using relativistic ab initio methods. The accuracy of the calculations is estimated by also evaluating atomic static electric-dipole polarizabilities and coefficients for the interaction of the atoms with a perfectly conducting wall. The results are in excellent agreement with the latest data from studies of magnetic field induced Feshbach resonances in ultracold collisions of Na and of Rb atoms. For Cs we provide critically needed data for ultracold collision studies.
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.
The C3{C}_{3}C3​ coefficients and their ratios for the alkali-metal atoms are determined very accurately by taking into account the optical properties of the atoms and four distinct types of trapping materials such as Au (metal), Si (semiconductor), vitreous SiO2{\text{SiO}}_{2}SiO2​ (dielectric), and SiNx{\text{SiN}}_{x}SiNx​ (dielectric). Dynamic dipole polarizabilities are calculated precisely for the alkali-metal atoms that produce C3{C}_{3}C3​ coefficients in a perfectly conducting medium within 0.2% accuracy. Thus, uncertainties from the atomic polarizabilities in the evaluation of the material-dependent C3{C}_{3}C3​ coefficients are small and accuracies of these C3{C}_{3}C3​ values mainly rely on the used optical data which are not taken into account in the present work. Our findings are in very good agreement with the available measurements, but they differ by around 10% from the previously reported high-precision calculations in heavy atoms for the perfect conductors. We also revisit the dispersion coefficients for the alkali-metal dimers and evaluate them using the above dynamic dipole polarizabilities which provide slightly different error bars than the other reported precise results. These coefficients are fitted into a ready-to-use functional form to aid the experimentalists in finding out the interaction potential only with the knowledge of atom-wall distance.
I specialize the Hartree-Fock, random-phase, and configuration-interaction approximations to a system of interacting bosons in an external potential. This approach affords a more direct connection to the methods and language of atomic-structure calculations. The derivations and equations parallel those found in atomic structure and are contrasted with the more commonly used Bogoliubov approach for bosons. I present a numerical method for solving the Hartree-Fock and random-phase equations to obtain the low-lying excitation energies for trapped alkali-metal atoms.