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arXiv:1311.3132

The collision between two hydrogen atoms

Hasi Ray

Published Nov 13, 2013• 0 citations
arXivHTMLPDF

Abstract

The electron-electron correlation term in two-atomic collision is the most important, most difficult term to obtain the effective interatomic potential. Generally the H and H collision is a four center problem. It is extremely difficult to compute the electron-electron correlation term to include the effect of exchange or antisymmetry between two system electrons exactly. All the two-atomic collision related theoretical data differ from each other due to its difference in approximating the electron-electron correlation term. I invent a trick to evaluate the term exactly. Earlier the positronium (Ps) and H system was easily approximated as a three center problem due to the light mass of Ps. My new code for H-H collision using the ab-initio and exact static-exchange model (SEM) can reproduce exactly the same data of Ps and H system just by using the appropriate atomic parameters. The success of the present trick makes the foundation of a big monument in cold and low energy atomic collision physics. The Feshbach resonances are observed in H(1s)-H(1s) elastic scattering in both the singlet and triplet channels. One can use the trick to solve any four center problem including excitation and ionization processes. Again one can use the idea in larger systems too.

DOI
10.48550/arxiv.1311.3132

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The electron-electron correlation term in two-atomic collision is the most important, most difficult term to obtain the effective interatomic potential. Generally the H and H collision is a four center problem. It is extremely difficult to compute the electron-electron correlation term to include the effect of exchange or antisymmetry between two system electrons exactly. All the two-atomic collision related theoretical data differ from each other due to its difference in approximating the electron-electron correlation term. I invent a trick to evaluate the term exactly. Earlier the positronium (Ps) and H system was easily approximated as a three center problem due to the light mass of Ps. My new code for H-H collision using the ab-initio and exact static-exchange model (SEM) can reproduce exactly the same data of Ps and H system just by using the appropriate atomic parameters. The success of the present trick makes the foundation of a big monument in cold and low energy atomic collision physics. The Feshbach resonances are observed in H(1s)-H(1s) elastic scattering in both the singlet and triplet channels. One can use the trick to solve any four center problem including excitation and ionization processes. Again one can use the idea in larger systems too.
The collision between two positronium (Ps) atoms is a four-body Coulomb problem with all the particles of equal masses. It is very difficult to compute the Born-Oppenheimer (BO) scattering amplitude involving the nine-dimensional integrals with four Coulomb interaction terms between the atoms. It is extremely difficult in the electron-electron correlation term to include the exchange or antisymmetry between two system electrons exactly. Earlier the Ps and H system was easily approximated as a three-body problem, due to the light mass of Ps the center of mass of the system was confined in the H-nucleus. A simple substitution of variables using no approximation has enabled to evaluate the electron-electron Coulomb exchange-correlation term exactly in such a four-center problem in the center of mass frame involving a nine dimensional integral. The present code of Ps-Ps collision using an ab-initio and exact static-exchange model (SEM) that uses the BO amplitude as input, can reproduce exactly the same data of Ps and H system just by using the appropriate atomic parameters and reduced mass of the system. The s-wave elastic phase shifts and the corresponding partial cross sections are calculated for both the singlet and triplet channels. In addition, we calculate the integrated /total elastic cross section ( ) and the quenching cross section ( q  ); an augmentedBorn method is used to evaluate  more accurately. The effective range theory is applied to get the scattering length and the effective range. The SEM can include only the non-adiabatic short-range effects but no long-range interaction. Now it is easy to extend the basis of a coupled-channel method to include the excitation and continuum channels exactly for better accuracy following the present prescription and avoiding the approximation tools. PACS 03.65.Nk – Scattering theory PACS 36.10.Dr – Positronium PACS 34.50.Cx – Elastic; ultracold collisions
The elastic collision between two ortho-positronium (e.g. S=1) atoms is studied using an ab-initio static-exchange model (SEM) in the center of mass (CM) frame considering the system as a four-body Coulomb problem where all the Coulomb interaction terms in the direct and exchange channels are treated exactly. A coupled-channel methodology in momentum space is used to solve Lippman-Schwinger equation following the integral approach. A new SEM code is developed in which the Born-Oppenheimer (BO) scattering amplitude acts as input to derive the SEM amplitude adapting the partial wave analysis. The s-, p- and d- wave elastic phase shifts and the corresponding partial cross sections for the spin alignment S=0, i.e. singlet (+) and S=2, i.e. triplet (-) states are studied. An augmented-Born approximation is used to include the contribution of higher partial waves more accurately to determine the total/integrated elastic cross section ( ), the quenching cross section ( ) and ortho to para conversion ratio ( ). The effective range theory is used to determine the scattering lengths and effective ranges in the s-wave elastic scattering. The theory includes the non-adiabatic short-range effects due to exchange.
Elastic collision between two H-like atoms using the ab initio static-exchange model (SEM) and a modified static-exchange model (MSEM) at cold energies are investigated in the center of mass frame considering the system as a four-body Coulomb problem where all the Coulomb interaction terms in the direct and exchange channels are treated exactly.The importance of an exact calculation to find basic physics is highlighted.In addition, the dependence of scattering length on the van der Waals interaction between the atoms and the dependence of scattering length on reduced-mass of the system are derived which is completely new information in the field of science.
We have continued our investigation into the elastic scattering of positronium atoms by hydrogen atoms, extending the work to include the spin-one (triplet) state of the two electrons. Once again we have extracted the SSS-wave phase shifts from normalizable trial functions by examining the spatial behavior of the center-of-mass wave functions. An effective-range expansion linear in the positronium energy fits the results very well and leads to a scattering length ${a}^{(\ensuremath{-})}\ensuremath{\simeq}2.36{a}_{0}$, compared with the static result ${a}^{(\ensuremath{-})}=2.476{a}_{0}$ of Hara and Fraser. The SSS-wave total and ortho-para conversion cross sections are calculated by making use of both 1S^{1}S1S and 3S^{3}S3S results. In the 3S^{3}S3S case, we see no indication of any resonances below the excitation threshold at 5.1 eV.
Positronium-hydrogen (Ps-H) scattering is of interest, as it is a fundamental four-body Coulomb problem. We have investigated low-energy Ps-H scattering below the Ps(n=2) excitation threshold using the Kohn variational method and variants of the method with a trial wavefunction that includes highly correlated Hylleraas-type short-range terms. We give an elegant formalism that combines all Kohn-type variational methods into a single form. Along with this, we have also developed a general formalism for Kohn-type matrix elements that allows us to evaluate arbitrary partial waves with a single codebase. Computational strategies we have developed and use in this work will also be discussed.With these methods, we have computed phase shifts for the first six partial waves for both the singlet and triplet states. The 1S and 1P phase shifts are highly accurate results and could potentially be viewed as benchmark results. Resonance positions and widths for the 1S-, 1P-, 1D-, and 1F-waves have been calculated.We present elastic integrated, elastic differential, and momentum transfer cross sections using all six partial waves and note interesting features of each. We use multiple effective range theories, including several that explicitly take into account the long-range van der Waals interaction, to investigate scattering lengths for the 1,3S and 1,3P partial waves and effective ranges for the 1,3S-wave.
We have investigated the elastic scattering of positronium atoms by hydrogen atoms below the excitation threshold at 5.1 eV, limiting the investigation to the spin-zero (singlet) state of the two electrons. Nonresonant SSS-wave phase shifts have been obtained from normalizable trial functions by extracting the center-of-mass wave function and examining its spatial behavior. An effective-range expansion is fitted to the phase shifts and the previously obtained positronium-hydride bound-state energy to yield a scattering length $a\ensuremath{\cong}5.3{a}_{0}$ compared with Fraser's corrected exchange result, a=7.28a0a=7.28{a}_{0}a=7.28a0​ (a0{a}_{0}a0​ is the Bohr radius). The stabilization method suggests that a resonance occurs at a scattering energy of 4.5 eV, about 0.6 eV below the excitation threshold. Using the complex rotation method, we have obtained evidence confirming the existence of the resonance and an estimate of its width of about 6 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}2}$ eV.