www.elsevier.com/locate/nimb

Nuclear Instruments and Methods in Physics Research B 247 (2006) 75–78

NIMBBeam Interactions

with Materials & Atoms

Continuum distorted wave theory for positronium formation reactions

P.A. Macri *

Departamento de Fısica, FCEyN, Universidad Nacional de Mar del Plata, Dean Funes 3350, 7600 Mar del Plata, Argentina

Available online 20 March 2006

Abstract

The formation of ground-state positronium in collisions of positrons on hydrogen atoms is considered. In previous theoretical works,two-center distorted wave functions were employed to approximate either the initial or the final channel. Here we report results obtainedby means of the eikonal final state-continuum distorted wave (EFS-CDW) approximation for which asymptotically correct distortedwave functions are used for both the initial and final states of the scattering system. Comparison of the present theoretical total crosssections with experimental data reveals that distortion effects become important in both channels as the impact energy decreases. Thiswork also shows that the EFS-CDW model is suitable for capture process in the scattering system at unexpected low energies.� 2006 Elsevier B.V. All rights reserved.

PACS: 34.70.+e; 36.10.Dr

Keyword: Ps formation

1. Introduction

The positronium atom (Ps) is an exotic atom compoundof one electron and one positron and may be considered asan isotope of H in which the proton is replaced by a posi-tron. Ps atoms attract a great deal of interest in manyfields. In quantum electrodynamics, Ps atoms offer the pos-sibility of studying a fundamental bound system containingonly leptons [1] (i.e. particles not affected by the stronginteractions). In astrophysics, the study of annihilationradiation (direct annihilation, or 2 photon and 3 photonpositronium decay) from outer space gives informationabout otherwise inaccessible parts of the Universe [2,3].Moreover, positron annihilation can be used to probe theFermi surface of solids [4] as well as to obtain scans ofthe human brain through positron emission tomographytechniques [5].

Since the pioneering work of Massey and Mohr [6] usingthe first order of the Born (B) approximation for the initialand final states, the understanding of Ps formation reac-

0168-583X/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.nimb.2006.01.041

* Tel.: +54 223 4913192; fax: +54 11 7868114.E-mail address: macri@mdp.edu.ar

tions by impact of positrons on atoms and molecules hasbeen a formidable theoretical task. Various theories havebeen applied to this problem as reviewed in recent papersby Walters et al. [7] and Bransden and Noble [8]. Thecontinuum distorted wave (CDW) approximation wasoriginally proposed in 1964 by Cheshire [9] to deal withion–atom charge exchange. It has proved to be fairlysuccessful in describing experiments over a wide range ofcollision velocities and target–projectile combinations.However, the CDW total cross sections become unrealisticas the collision speed decreases and even fail to show theobserved maximum at intermediate energies. As Crothersrealized [10], one of the reasons for this behavior is thatthe CDW function is not properly normalized. In orderto avoid these undesirable features, the introduction ofnormalized eikonal states to describe the entrance or theexit channel was proposed. The continuum distortedwave-eikonal initial state (CDW-EIS) is, up to now, themore relevant example of such approximations [11,12]. Alight projectile, however, introduces some technical difficul-ties that arise because different Jacobi center-of-mass coor-dinates are required to describe the initial and final states.This is in contrast with the theoretical considerations for

76 P.A. Macri / Nucl. Instr. and Meth. in Phys. Res. B 247 (2006) 75–78

electron capture by heavy ions where, in addition, anotherimportant approximation is also possible: the relativemotion of nuclei can be treated classically at all but verylow energies and, for the computation of total crosssection, can be straight-forwardly ignored. No such simpli-fication occurs in positron–atom collisions since the posi-tron–nucleus interaction plays a decisive role even at highenergies. The unique antecedent of a two-center dis-torted-wave theory for both the initial and final channelsapplied to a light projectile collision was presented in thework of Jones and Madison [13], who successfullyemployed the CDW-EIS theory for an (e�, 2e�) process.For Ps formation collisions, two-center distorted wavefunctions were employed only to approximate either theinitial or the final channel [14]. In this work, we presentthe first calculations [15] for the charge transfer reactionby positron impact on hydrogen-like atoms using thetwo-channel eikonal final state-continuum distorted waveapproximation (EFS–CDW) theory.

2. Theory

We consider a general system formed by an incomingpositron of mass MP and charge ZP colliding with anhydrogen-like atom composed by one electron and anucleus target of charge ZT and mass MT. In the centerof mass system, the problem can be described by any ofthe pairs of Jacobi coordinates (rT,RP), (rP,RT), or (R, r)as shown in Fig. 1. The coordinates RT,RP, r indicate theposition of each particle to the center of mass of the othertwo. The electronic coordinates rT and rP and their associ-ated momenta kT, kP are referred to the target nucleus andthe positron projectile, respectively, while the distance R

and the momentum K are considered between positronand the target nucleus. We use throughout atomic units,unless explicitly otherwise stated.

It is possible to construct a theory for Ps formation reac-tions which includes the distortion of both the entrance andthe exit channel choosing for the functions A+,B+ andC�,D� in the initial and final states

ZT

e+

e–

RT

r

RP

rP

R

rT

Fig. 1. Set of Jacobi coordinates used in this work.

Wi ¼ WBi ðRP; rTÞAþðaP;�vi=2; rPÞBþðai; vi;RÞ;

Wf ¼ WBf ðRT; rPÞC�ðaT; vf ; rTÞD�ðaf ; vf ;RÞ;

from the following possibilities which guarantee the correctasymptotic Coulomb boundary condition of the three-body problem

F �ða; p; rÞ ¼ exppa2

� �Cð1� iaÞ;1F 1ð�ia; 1;�ikr � ik � rÞ;

E�ða; p; rÞ ¼ exp½�ia lnðpr � p � rÞ�;

where aP = �ZP/vi, aT = ZT/vf and ai,f = �ZTZP/vi,f areSommerfeld factors and vi, vf are the positron and thePs velocities. Here we denoted with WB

i;f the undistortedBorn initial and final states which are given by WB

i;f ¼Ui;fðRP;TÞwi;fðrT;PÞ, where U and w represent a plane waveand a bound state wave functions, respectively.

In this context, the election of C� = F�,D� = F�,A+ = 1,B+ = 1, gives the continuum distorted wave–Bornapproximation (CDW–B) and the election of C� =1,D� = 1, A+ = F+,B+ = F+, gives the Born–continuumdistorted wave approximation (B-CDW) which are theone-channel distorted wave theories introduced by Brans-den [14] to calculate Ps formation reactions. In the sameway, we can write the two-channel continuum distortedwave-eikonal initial state (CDW-EIS) theory used by Jonesand Madison [13] for the (e�, 2e�) process choosingC� = F�,D� = F� for the final state and A+ = E+,B+ = E+ for the initial state.

Our goal is to find, from all the possibilities, the besttwo-channel first-order distorted wave theory for Ps forma-tion reactions. We can avoid the try and discard methodover the finite number of options considering the followingphysical analysis. First, we have to mention that at veryhigh impact energies all possible combinations of the func-tions A+,B+ and C�,D� are expected to give the sameresult for the total cross section (TCS). On the other hand,for heavy ion projectiles it is well known that continuumdistortions functions F must be used in the entrance or inthe exit channel to take into account the Thomas multiplescattering mechanism [16] which generates a prominentpeak in the intermediate and high energy differential crosssections (DCS). But if only continuum distortions func-tions F are employed for both the initial and the finalstates, the CDW–CDW theory thus constructed introducesan artificial deep at the center of the Thomas peak [17].Therefore, we have to select F functions in one channeland E functions in the other channel. CDW-EIS theory sat-isfies this condition. Why not to use it to compute Ps for-mation reactions? After all, it is the most extendeddistorted wave theory used for charge exchange in colli-sions by heavy ions [12]. The reason is that we want todescribe not only the behavior of the DCS at intermediateand high energies but also the TCS for energies as low aspossible where experiments are available [18,19]. It wasdemonstrated [11] that for heavy ion–atom collisions, con-tinuum distorted waves are not normalized. This defect ismore severe as the impact energy diminishes provoking in

P.A. Macri / Nucl. Instr. and Meth. in Phys. Res. B 247 (2006) 75–78 77

general an overestimation of total cross sections. For Psformation reaction, we have at all impact energiesvf ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2

i =2þ ð�i � �fÞp

, so the final collision velocity issmaller than the initial one, being this condition more evi-dent close to the threshold of the process. As a conse-quence, the failure in the normalization of distorted wavefunctions will be more noticeable using Coulomb contin-uum F functions in the exit channel rather than in the entrychannel. Therefore, we choose to distort the final wavefunction with eikonal functions C� = E�,D� = E�, assur-ing its normalization for all interparticle velocities and dis-tances. In order to describe the appearance of the Thomaspeak at intermediate and high energies, we have to selectthe continuum F functions A+ = F+,B+ = F+ for the initialstate. In this way, we construct the eikonal final state-con-tinuum distorted wave (EFS-CDW) theory, where ourchoice of the distortion functions was based totally onphysical grounds. In this context, we can readily write theEFS-CDW transition amplitude as

T EFS-CDW ¼ hWEFSf jW ijWCDW

i i; ð1Þwhere the residual interaction Wi, which is the sum of twomass-polarization terms (MT� 1) is defined by

W iWCDWi ¼ ðH � EÞWCDW

i ;

¼ UiðRPÞ1

MP

rR � rrP�rrT

� rrP

� �

� wiðrTÞF þðaP;�vi=2; rPÞF þðai; vi;RÞ.

3. Results

We have obtained differential and total cross sectionsfor the process e+ + H! Ps + H+ employing the one-channel and two-channel distorted-wave models discussedabove. In Fig. 2, we present differential cross sectionsfor Ps formation in the framework of our best choice

0 20 40 60 80 100 120 140 160 18010-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

2650 eV

1000 eV

500 eV

250 eV

150 eV

75 eV

25 eV

10 eV

DC

S (

a.u.

sr -1

)

Angle (deg)

Fig. 2. EFS-CDW DCS for Ps formation reactions on H, at positronimpact energies between 10 eV and 2650 eV.

EFS-CDW approximation for a wide range of positronimpact energies between 10 eV and 2650 eV. The resultsshow two outstanding features in the angular spectra. First,there is a peak in the forward direction for all DCSs. Whileat higher energies, the scattering of the positron at 0� dom-inates the angular distribution, at the lower energies itspreads over a wider angular domain, as expected from col-lision dynamics. Second, a shoulder appears at low energiesin the vicinity of 45�, which progressively becomes a per-fectly well-defined peak as the impact energy increases.The presence of this peak can be understood as a classicaltwo-step Thomas mechanism in the process of Ps forma-tion [16]: in the first encounter the positron transfers halfof its energy to the electron, and the two particles go outunder ±45�. In the second encounter either the electronor the positron is deflected through 90� in the field of theproton. In this way, both particles leave with equal speedand direction allowing the formation of Ps.

In Fig. 3, we compare our results for total cross sections(TCS) with experiments measured by Weber et al. [18] andby Zhou et al. [19]. As experiments do not discriminatefinal Ps bound state, we have multiplied all the theoretical1s–1s TCS by the factor 1.202 to estimate roughly the cap-ture to excited states at high energies. We should note thatthis estimation may not be valid at low energies. We can seethat the B approximation gives a rough sketch of the Psformation with an overestimation of the TCS maximumaround 15 eV. On the other hand, the comparison ofthe one-channel distorted wave theories B-CDW andCDW–B with experiments confirms the generally acceptedsentence that distorted-wave theories are not expected to bevalid at low energies. As previously discussed, the failure inthe normalization of the distorted wave function is moredramatic if the CDW is used in the final channel. Thisbehavior is put in evidence in the proximity of the thres-hold where vf annulates, while vi remains always different

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170

0

1x10-16

2x10-16

3x10-16

4x10-16

5x10-16

TC

S (

cm2)

Positron energy (eV)

Fig. 3. TCS for positronium formation reaction on H. In (a) results forthe EFS-CDW (full line) model are compared with the CDW–CDW(short-dotted line), CDW-EIS (short-dashed line), B-CDW(dashed line),CDW–B (dot-dashed line) and B (dotted line) theories and withexperiments of Weber et al. [18] (circles) and of Zhou et al. [19] (triangles).

78 P.A. Macri / Nucl. Instr. and Meth. in Phys. Res. B 247 (2006) 75–78

from zero for the studied reaction. Comparing, for exam-ple, B–CDW and CDW–B approximations at 10 eV (notshown in the figure) we can observe a discrepancy greaterthan five orders of magnitude. The question is if includingdistortion in both the initial and the final states can give abetter description through, for example, the CDW–CDWor the CDW-EIS theories. As we anticipate in the previousdiscussion the presence of the CDW function in the finalstate leads, as in the case of the CDW–B theory, to anunderestimation of the experiments. Fortunately, we showhere that this shortcoming can be remedied including thenormalized and also boundary correct EFS through theEFS-CDW approximation. In the figure it is possible tosee how the EFS-CDW theory represent precisely the posi-tion, width and intensity of the capture peak and describethe general behavior of TCS being able to extend thedomain of distorted wave theories to unexpected low ener-gies. However, the agreement of EFS-CDW and experi-mental data must be taken with care in the proximities ofthe threshold due to the perturbative character of theEFS-CDW model. On the other hand, coupled channelmethods which successfully describe TCS at low energies[20], present important differences with CDW theories forthe DCS at intermediate and high impact energies wherethe last models are expected to be the most reliable of theexisting results. In this way, we find in the EFS-CDWmodel a theory accurate at high energies which can beextrapolated to a lower energy regime where contact tocoupled channel results are reliable.

4. Summary

In this work we present a new two-channel distortedwave model, which is formulated on physical grounds,where the importance of distorting both the initial and finalchannels has been shown. The success of the use of eikonalphases instead of Coulomb waves to distort the final boundstate is supported by a correct normalization which leads tothe proper scale and shape of the TCS even at low energy.Coulomb distortion affects less the results when are used inthe initial state and, on the other hand, they must be takeninto account to describe the Thomas multiple scatteringmechanisms at high energies.

We find that EFS-CDW theory is able to describe Psformation experiments on H from high (non relativistic)energies to unexpected low energies close to the reactionthreshold. The model is simple enough to be straightfor-wardly extended to more complex targets. At present, cal-culations for Ar and Ne are being carried out.

References

[1] R.S. Vallery, Phys. Rev. Lett. 90 (2003) 203402.[2] R.L. Kinzer, P.A. Milne, J.D. Kurfess, M.S. Strickman, W.N.

Johnson, W.R. Purcell, Astrophys. J. 559 (2001) 282.[3] J. Kndlseder1, P. Jean1, V. Lonjou1, G. Weidenspointner1, N.

Guessoum, W. Gillard1, G. Skinner, P. von Ballmoos, G. Vedrenne,J.-P. Roques, S. Schanne, B. Teegarden, V. Schnfelder, C. Winkler,Astron. Astrophys. 441 (2005) 513.

[4] Zs. Major, S.B. Dugdale, R.J. Watts, G. Santi, M.A. Alam, S.M.Hayden, J.A. Duffy, J.W. Taylor, T. Jarlborg, E. Bruno, D. Benea, H.Ebert, Phys. Rev. Lett. 92 (2004) 107003.

[5] A. Braem, M.C. Llatas, E. Chesi, J.G. Correia, F. Garibaldi, C.Joram, S. Mathot, E. Nappi, M. Ribeiro da Silva, F. Schoenahl, J.Sguinot, P. Weilhammer, H. Zaidi, Phys. Med. Biol. 49 (12) (2004)2547.

[6] H.S.W. Massey, C.B.O. Mohr, Proc. R. Soc. London, Ser. A 67(1954) 695.

[7] H.R.J. Walters, A.A. Kernoghan, M.T. McAlinden, in: P.G. Burke,C.J. Joachin (Eds.), Photon and Electron Collisions with Atomsand Molecules, Plenum, New York, 1997.

[8] B.H. Bransden, C.J. Noble, Adv. At. Mol. Opt. Phys. 32 (1994) 19.[9] I.M. Cheshire, Proc. Phys. Soc. 84 (1964) 89.

[10] D.S.F. Crothers, J. Phys. B 15 (1982) 2061.[11] D.S.F. Crothers, J.F. McCann, J. Phys. B 16 (1983) 3229.[12] A.E. Martınez, J.A. Bullrich, J.M. Maidagan, R.D. Rivarola, J. Phys.

B 25 (1992) 1883.[13] S. Jones, D.H. Madison, Phys. Rev. Lett. 81 (1998) 2886.[14] B.H. Bransden, C.J. Joachain, J.F. McCann, J. Phys. B 25 (1992)

4965.[15] P.A. Macri, J.E. Miraglia, J. Hanssen, O.A. Fojon, R.D. Rivarola, J.

Phys. B 37 (2004) L111.[16] L.H. Thomas, Proc. R. Soc. London, Ser. A 114 (1927) 561.[17] R.D. Rivarola, J.E. Miraglia, J. Phys. B 15 (1982) 2221.[18] M. Weber, A. Hofmann, W. Raith, W. Sperber, F. Jacobsen, K.G.

Lynn, Hyperfine Interact. 89 (1994) 221;W. Sperber, D. Becker, K.G. Lynn, W. Raith, A. Schwab, G.Sinapius, G. Spicher, M. Weber, Phys. Rev. Lett. 68 (1992) 3690.

[19] S. Zhou, H. Li, W.E. Kauppila, C.K. Kwan, T.S. Stein, Phys. Rev. A55 (1997) 361;S. Zhou, W.E. Kauppila, C.K. Kwan, T.S. Stein, Phys. Rev. Lett. 72(1994) 1443.

[20] J. Mitroy, J. Phys. B 29 (1996) L263.