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Ionization of positronium (Ps) in collision with atom Hasi Ray

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Phys. Status Solidi C 6, No. 11, 2281–2284 (2009) / DOI 10.1002/pssc.200982113

∗

Life-member, Indian Society of Atomic & Molecular Physics; S-407, B. P. Township, Kolkata 700094, India Received 9 February 2009, revised 30 June 2009, accepted 30 June 2009 Published online 12 October 2009 PACS 34.70.+e, 36.10.Dr, 82.30.Gg ∗

e-mail hasi [email protected], Phone: +91-9874756319

The theoretical studies on ionization of positronium (Ps) and the phenomenon of exchange are of fundamental interests. Again the singly energy differetial cross section (SDCS) is more informative than the total ionization

cross section (TCS). We present a brief comparative study on TCS and SDCS for Ps-ionization in Ps-H and Ps-He collision to provide a very basic knowledge about the system. © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction Theoretical studies on collision of Ps with atom [1–18] is a bit difficult and challenging due to composite structure of both projectile and target. In such a system of a Ps and an atom, the exchange between the Pselectron and atomic-electron is highly important and sensitive at lower incident energies [5, 6, 17]. It is an effect of fundamental interest [1, 2]. When ionization, the ionized or fragmented electron of Ps moves in the field of parent nucleus (i.e. positron) in the final channel. This fragmented electron should be treated as a continuum Coulomb wave [19] in the electric field of positron; the idea was originally prescribed by Geltman [20] for ionization of atom by electron impact. It makes the problem more rigorous. One needs to choose a simpler approximation to ease the mathematical computation and at the same time to include the important physics tactfully. In this sense the CoulombBorn approximation (CBA) is the simplest and the best. The author extended the theory in Ps and atom scattering for Ps-ionization, atom-ionization and the ionization of both the projectile and target [11–15]. Since at lower energies the exchange effect is important, the CBA with exchange named as Coulomb-Born-Oppenheimer approximation (CBOA) [5, 6] is most useful for reliable data. In case of Ps and H system, both the atomic wave functions are exact since they are one electron atoms. In case of multielectron target (e.g. He) the correlated wave function for the target atomic electrons could provide better accuracy of the results but it complicates the calculation. We borrow

the simplest wave function used by Winter et al. [21]: Ψ (r1 , r2 ) = u(r1 )u(r2 ); with Slater type orbital u(r) = (Z 3 /π)1/2 exp(−Zr) and Coulomb screening parameter, Z = 1.6875. At higher incident energies the projectile interacts with target field for a very short duration, so such approximate wave function is sufficient to produce reasonable result. But at lower incident energies since the projectile interacts with target field for a longer period, the electron-electron correlation between the target electrons can influence the result. We present the singly (energy) differential (SDCS) as well as total Ps-ionization cross sections (TCS) in PsHe scattering using the Coulomb-Born-Oppenheimer approximation (CBOA), the Coulomb-Born approximation (CBA), the first-Born approximation (FBA), the first BornOppenheimer approximation (BOA) and compare them with the available experimental [22–24] and theoretical data. We also present the theoretical TCS data for Ps-H system. 2 Theory The momentum transfer Q is defined as Q = ki − kf , so that Q2 = ki2 + kf2 − 2ki kf cosθ

(1)

if θ is the scattering angle; ki and kf are initial and final momenta of the projectile Ps. The direction of ki is © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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ˆ here ’k’ is the chosen as polar axis and dk = k 2 dkdk; ˆ is the angular part i.e. radial part of vector ’k’ and ’k’ ˆ = sinθk dθk dφk . According to our definition [17], the dk SDCS is defined as   dσ d3 σ ˆ ˆ = dkf dk (2) ˆ k dEk dkˆf dkdE

The triply differential cross sections (TDCS) for the breakup of Ps in Ps-H and Ps-He scatterings are defined as d3 σ kf k 1 3 = { | Fk + Gk |2 + | Fk − Gk |2 } ˆ ˆ ki 4 4 dkf dkdEk (3a) d3 σ kf k He He 2 = | F k − Gk | (3b) ˆ k ki dkˆf dkdE

where Fk , Gk represent respectively the direct and exchange matrix elements for Ps-H scattering; similarly FkHe and GHe represent the direct and exchange matrix elek ments for Ps-He scattering. These are defined as  F ˆf ) = − 1 FkHe (k ] e−ikf .R1 ηk∗ (ρ1 )Φ∗f {r2 , r3 }[VHe π (4a) eiki .R1 η1s (ρ1 )Φi {r2 , r3 }dxdr1 dr2 dr3  1 G ˆ GHe ] e−ikf .R2 ηk∗ (ρ2 )Φ∗f {r1 , r3 }[VHe k (kf ) = − π

eiki .R1 η1s (ρ1 )Φi {r2 , r3 }dxdr1 dr2 dr3

Figure 1 Total Ps-ionization cross section in Ps-H system: solid curve, using CBOA [6]; dotted curve, using CBA [14]; big-dashed curve, using BOA; big-dashed-dotted curve, using FBA; small-dashed curve, using coupled-state R-matrix method of Belfast group [7]; small-dashed-dotted curve, quenching cross section [18].

(4b)

with F = VHe

Z Z 1 1 − − + | x | | r1 | | x − r2 | | r1 − r2 |

−

1 1 + | x − r3 | | r1 − r3 |

(5a)

and G = VHe

Z Z 1 1 − − + | x | | r2 | | x − r1 | | r2 − r1 |

−

1 1 + | x − r3 | | r2 − r3 |

(5b)

with Rj = 21 (x + rj ) and ρj = (x − rj ); j=1,2. x is the coordinate of positron and rj ; j = 1 to 3 are those of electrons in Ps and He respectively in the incident channel with respect to the center of mass (c.m.) of the system. η and Φ indicate the wave functions of Ps and He respectively. Subscript ‘i‘ and f ‘ represent the incident and final channels. ’k’ is the momentum of the fragmented electron in the final channel; ’Z’ is the nuclear charge of the target. In Eqns. (4a) and (4b), the direct and exchange scattering matrix elements are dependent on the incident energy E (∼ ki2 ), kf and k. Due to energy conservation, the available energy in the final channel (E − ) is shared between c.m. motion

© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Figure 2 Total Ps-ionization cross section in Ps-He system: solid curve, using CBOA [5]; big-dashed curve, using CBA [12]; dashed-dotted curve, using BOA; small-dashed curve, using FBA [10]; dotted curve, using coupled-state R-matrix method of Belfast group [8]; solid-square, experimental data [22].

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Contributed Article Phys. Status Solidi C 6, No. 11 (2009)

a) at incident energy 18 eV.

b) at incident energy 25 eV.

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c) at incident energy 33 eV.

d) at incident energy 60 eV.

Figure 3 The SDCS with respect to the longitudinal energy distribution of break-up electron and positron for Ps-He scattering. In a & b: solid curve, for positron and dotted curve, for electron using CBOA; dashed-dotted curve, for positron and big-dashed-small-dashed curve, for electron using CBA; big-dashed curve, for positron and small-dashed curve, for electron using IA [16]; the solid squares, the experimental data [24]. In c & d: solid curve, for positron and dotted curve, for electron using CBOA [17]; dashed-dotted curve, for positron and big-dashed-small-dashed curve, for electron using CBA; big-dashed curve, for positron and small-dashed curve, for electron using IA [16]; experimental data [24].

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(kf ) and relative motion of electron (k); so vector k and ˆ f are mentioned on the left hand side. We avoid to angle k mention E dependence considering the fact that it is well known; is the binding energy of Ps. The target atomic wave functions are at ground states in both the incident and final channels. If we remove the third electron from the PsHe system which is represented by the position coordinate r3 , the expressions (4a), (4b), (5a), (5b) should fit to Ps-H system [6]. The continuum Coulomb wave function ηk (ρ) for the ionized electron of Ps is chosen from Ref. [6] which was made orthogonal to the ground state Ps wave function η1s (ρ) to satisfy the orthonormality criterion of electronic states of Ps [12]; here ρ is the position vector of electron with respect to positron in Ps. 3 Results and discussion In Figs. 1 and 2, the TCS data for Ps-ionization using the first-Born without exchange (FBA) and with exchange (BOA), the CoulombBorn without exchange (CBA) and with exchange (CBOA) are compared with R-matrix data by Belfast group [7,8] and the available experimental data [22] for Ps-H and PsHe systems. One can get a good idea of exchange effects and the difference between FBA and CBA as well as BOA and CBOA from above comparisons. It is seen that the effect of exchange is important at lower incident energies. The quenching cross section due to ortho to para conversion of Ps in Ps-H collision [18] using close-coupling approximation is also compiled in Fig. 1. It provides the information that at very low energies when ionization channels are closed, the annihilation due to ortho to para quenching dominates in Ps-H scattering. In Figs. 3(a, b, c, d) our SDCS data using the CBA and CBOA are plotted with respect to the longitudinal energy distribution of break-up positron and electron. We compare them with the available experimental [23, 24] and theoretical impulse approximation (IA) [16] data at different incident energies e.g. 18 eV, 25 eV, 33 eV and 60 eV. The agreement of the CBOA data with experiment is gradually improving with the increase of the incident energy as well as the increase in the longitudinal energy of the break-up positron. This finding is consistent with the existing physics because the CBA is a first order approximation theory in which the effect of channel-coupling is totally neglected and we have used the simplest Slater type orbitals with a screening parameter Z instead of taking exact correlation. Both the effects are important at lower incident energies when the projectile interacts with the target field for a longer duration. There is no experimental data for SDCS to compare with respect to the longitudinal energy distribution of break-up electron.

the first-Born without exchange (FBA) and with exchange (BOA), the Coulomb-Born without exchange (CBA) and with exchange (CBOA). These compiled data should be useful to get the very basic information about the system. It is expected that the use of correlated wave function for target He and inclusion of channel coupling will improve the results at lower incident energies. Acknowledgements Author acknowledges the research facilities of Saha Institute of Nuclear Physics, Kolkata, India.

References [1] H.S.W. Massey and C.B.O. Mohr, Proc. Phys. Soc. 67, 695 (1954). [2] R.J. Drachman, Can. J. Phys. 60, 494 (1982). [3] A.P. Mills, Science 218, 335 (1982). [4] A.E. Ruark, Phys. Rev. Lett. 68, 278 (1945). [5] H. Ray, Euro. Phys. Lett. 73, 21 (2006). [6] H. Ray, PRAMANA 66, 415 (2006). [7] C.P. Campbell, M.T. McAlinden, F.G.R.S. MacDonald, and H.R.J. Walters, Phys. Rev. Lett. 80, 5097 (1998). [8] J. E. Blackwood, C.P. Campbell, M.T. McAlinden, and H.R.J. Walters, Phys. Rev. A 60, 4454 (1999). [9] H.R.J. Walters, C. Starrett, and M.T. McAlinden, NIMB 247, 111 (2006). [10] P.K. Biswas and Sadhan K. Adhikari, Phys. Rev. A 59, 363 (1999). [11] H. Ray, PRAMANA 63, 1063 (2004). [12] H. Ray, J. Phys. B 35, 3365 (2002). [13] H. Ray, NIMB 192, 191 (2002). [14] H. Ray, Phys. Lett. A 299, 65 (2002). [15] H. Ray, Phys. Lett. A 252, 316 (1999). [16] C. Starrett and H.R.J. Walters, J. Elecron. Spectroscopy 161, 194 (2007). [17] H. Ray, Phys. Lett. A 373, 759 (2009). [18] H. Ray and A.S. Ghosh, J. Phys. B 31, 4427 (1998). [19] M.R.C. McDowell and J. P. Coleman, Theory of Ion-atom Collisions (North Holland, Amsterdam, 1970), pp. 239243. [20] S. Geltman, Topics in Atomic Collision Theory (Academic Press, NewYork/ London, 1969), p.142. [21] T.G. Winter and C.C. Lin, Phys. Rev. A 12, 434 (1975). [22] S. Armitage, D.E. Leslie, A.J. Garner, and G. Laricchia, Phys. Rev. Lett 89, 173402 (2002). [23] S. Armitage, D.E. Leslie, J. Beale, and G. Laricchia, NIMB 247, 98 (2006). [24] G. Laricchia at UCL, London (2005/2006), private communication.

4 Conclusion We have presented a brief comparative study on Ps-ionization in Ps-H and Ps-He collision. We discussed on total Ps-ionization cross section as well as single (energy) differential cross section. The comparisons are made with experimental and all the available theoretical data using impulse approximation (IA), R-matrix theories,

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