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arXiv:1904.04295 · Journal of Physics Conference Series

Low-energy scattering properties of ground-state and excited-state positronium collisions

Michael D. Higgins, K. M. Daily, Chris H. Greene

Published Jul 29, 2019• 4 citations
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Abstract

Low-energy elastic and inelastic scattering in the Ps($1s$)-Ps($2s$) channel is treated in a four-body hyperspherical coordinate calculation. Adiabatic potentials are calculated for triplet-triplet, singlet-singlet, and singlet-triplet spin symmetries in the spin representation of coupled electrons and coupled positrons, with total angular momentum $L=0$ and parity equal to $+1$. The $s$-wave scattering lengths for the asymptotic Ps($1s$)-Ps($2s$) channel are calculated for each spin configuration. Results obtained for the $s$-wave scattering lengths are ${a}_{\mathrm{TT}}=7.3(2){a}_{0}\ensuremath{-}i0.02(1){a}_{0}, {a}_{\mathrm{SS}}=13.2(2){a}_{0}\ensuremath{-}i0.9(2){a}_{0}$, and ${a}_{\mathrm{ST}}=9.7(2){a}_{0}$ for each spin configuration. Spin recoupling is implemented to extract the scattering lengths for collisions of Ps in different spin configurations through properly symmetrized unitary transformations. Calculations of experimentally relevant scattering lengths and cross sections are carried out for Ps atoms initially prepared in different uncoupled spin states.

DOI
10.1088/1742-6596/1412/12/122023

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Low-energy elastic and inelastic scattering in the Ps(1s1s1s)-Ps(2s2s2s) channel is treated in a four-body hyperspherical coordinate calculation. Adiabatic potentials are calculated for triplet-triplet, singlet-singlet, and singlet-triplet spin symmetries in the spin representation of coupled electrons and coupled positrons, with total angular momentum L=0L=0L=0 and parity equal to +1+1+1. The sss-wave scattering lengths for the asymptotic Ps(1s1s1s)-Ps(2s2s2s) channel are calculated for each spin configuration. Results obtained for the sss-wave scattering lengths are aTT=7.3(2)a0\ensuremath−i0.02(1)a0,aSS=13.2(2)a0\ensuremath−i0.9(2)a0{a}_{\mathrm{TT}}=7.3(2){a}_{0}\ensuremath{-}i0.02(1){a}_{0}, {a}_{\mathrm{SS}}=13.2(2){a}_{0}\ensuremath{-}i0.9(2){a}_{0}aTT​=7.3(2)a0​\ensuremath−i0.02(1)a0​,aSS​=13.2(2)a0​\ensuremath−i0.9(2)a0​, and aST=9.7(2)a0{a}_{\mathrm{ST}}=9.7(2){a}_{0}aST​=9.7(2)a0​ for each spin configuration. Spin recoupling is implemented to extract the scattering lengths for collisions of Ps in different spin configurations through properly symmetrized unitary transformations. Calculations of experimentally relevant scattering lengths and cross sections are carried out for Ps atoms initially prepared in different uncoupled spin states.
The stochastic variational method is used in conjunction with stabilization ideas to compute the low-energy s-wave phase shifts and scattering lengths for Ps-Ps scattering. The scattering lengths obtained were 8.44a0{8.44a}_{0}8.44a0​ for the singlet-singlet and 3.00a0{3.00a}_{0}3.00a0​ for the triplet-triplet states (the spin state refers to the coupling of the two electrons and the coupling of the two positrons in the system). The positive scattering length in the triplet-triplet channel means a stable Bose-Einstein condensate of triplet Ps atoms is physically possible. The large value of the ${}^{3}{\mathrm{Ps}}^{3}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{Ps}}}^{1}{\mathrm{Ps}}^{1}\mathrm{Ps}$ cross section, $7.41\ensuremath{\pi}{a}_{0}^{2}$ suggests the use of spin-polarized positrons as advanced by Platzmann and Mills [Phys. Rev. B 49 454 (1994)].
We investigate the elastic and inelastic scattering of positrons by sodium atoms in both the ground state, Na(3s3s3s), and excited states, Na*(3p3p3p, 4s4s4s, 3d3d3d), using the hyperspherical coordinate method with a model potential to represent the atomic core. The threshold behavior of positronium (Ps) formation cross sections is analyzed as the positron impact energy EEE approaches zero. Within this framework, we derive a generalized expression for partial-wave Ps-formation cross sections at low positron energies, which applies to both ground-state and excited-state sodium targets. Our results confirm that the total threshold behavior follows the expected power-law dependence: σPs∝E−1/2 σ_{\text{Ps}} \propto E^{-1/2} σPs​∝E−1/2 for exothermic reactions and σPs∝Ea σ_{\text{Ps}} \propto E^{a} σPs​∝Ea for endothermic reactions, where a>0 a > 0 a>0. Furthermore, we find that Ps-formation cross sections for positron scattering from excited Na states are significantly larger than those from the ground state in the low-energy region. A notable enhancement of Gailitis-Damburg oscillations is observed above the Ps(n=2n=2n=2) threshold, which may account for the increase observed in experimental data. Incorporating contributions from excited sodium targets improves agreement with experimental results and may help resolve discrepancies between theoretical predictions and measurements.
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.
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
Scattering of ortho positronium (Ps) by cesium and rubidium atoms has been investigated employing a three-Ps-state coupled-channel model with Ps(1s,2s,2p) states using a time-reversal-symmetric regularized electronexchange model potential. We find a narrow S-wave singlet resonance at 5.057 eV of width 0.003 eV in the Ps-Rb system and at 5.067 eV of width 0.003 eV in the Ps-Cs system. Singlet P-wave resonances in both systems are found at 5.3 eV of width 0.4 eV. Singlet D-wave structures are found at 5.4 eV in both systems. The pronounced P- and D-wave resonances in these systems lead to easily detectable local minima in the low-energy elastic cross sections. We also report results for elastic and Ps-excitation cross sections for Ps scattering by Rb and Cs. PACS Number(s): 34.10.+x, 36.10.DrRecent measurements of ortho positronium (Ps) scattering cross section by H2, N2, He, Ne, Ar, C4H10, and C5H12 [1–7] have initiated new theoretical investigations [8–14] in this subject. We suggested [15–17] a regularized, symmetric, nonlocal electron-exchange model
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