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16

Recent progress in the description of positron scattering from atoms using theConvergent Close-Coupling Theory

A S Kadyrov and I BrayCurtin Institute for Computation and Department of Physics, Astronomy and Medical Radiation Science,

Kent Street, Bentley, Perth, Western Australia 6102, Australia

(Dated: September 15, 2016)

Much progress in the theory of positron scattering on atoms has been made in the ten yearssince the review of Surko et al. [1]. We review this progress for few-electron targets with a par-ticular emphasis on the two-centre convergent close-coupling and other theories which explicitlytreat positronium (Ps) formation. While substantial progress has been made for Ps formation inpositron scattering on few-electron targets, considerable theoretical development is still required formultielectron atomic and molecular targets.

PACS numbers: 34.80.-i 34.80.Lx 34.80.Uv 34.10.+x

CONTENTS

I. Introduction 1

A. Positron scattering 1

1. atomic hydrogen 2

2. helium 4

3. alkali metals 6

4. magnesium 6

5. inert gases 7

6. molecular hydrogen 7

B. Antihydrogen formation 8

II. Theory 8

A. Atomic hydrogen 8

B. Helium 11

C. Alkali metals 13

D. Magnesium and inert gases 13

E. Molecular hydrogen 14

III. Recent applications of the CCC theory topositron scattering 14

A. Internal consistency 14

B. Atomic hydrogen 17

C. Helium 18

D. Alkali metals 21

E. Magnesium and inert gases 23

F. Molecular hydrogen 24

IV. Antihydrogen formation 25

A. Numerical treatment of the Green’s function 26

B. Analytic treatment of the Green’s function 26

V. Concluding remarks 27

Acknowledgments 27

References 27

I. INTRODUCTION

There have been many reviews of positron physics overthe years [1–9]. More recently Laricchia et al. [10] andChiari and Zecca [11] considered the subject with an em-phasis on experimental measurements involving noble gastargets. The related topic of antihydrogen formation hasalso been thoroughly reviewed [12–15]. Resonances andthe closely related bound states of positrons with atomsand molecules has also been extensively discussed [1–3, 16–18].

This work concentrates on the progress in applica-tion of theoretical methods to scattering processes in aquantum few-body system involving positrons as pro-jectiles and multi-electron atomic targets with explicittreatment of positronium formation. Particular empha-sis is on the developments taken place since the com-prehensive review of positron physics by Surko et al. [1].It begins by describing the currently available theoriesof low-energy positron collisions with atoms and sim-ple molecules. Then it describes the development andapplication of the two-centre convergent close-couplingmethod, which explicitly treats the Ps-formation pro-cesses.

A. Positron scattering

Developments in positron physics have resulted in sev-eral technologies in medicine and material science. Inmedicine, the use of the positron emission tomography(PET) scanners help to make diagnoses of cancer detec-tion and of certain brain function disorders. Materialscience uses positron annihilation lifetime spectroscopy(PALS), to analyze and design specific materials. Thecritical component is positronium (Ps) formation with itsannihilation providing the key signature of its origin. Psis a short-lived exotic atom of a bound positron-electronpair that has similar structure to atomic hydrogen.

Scattering experiments are the main tool of modernphysics to learn about the structure of matter. By ana-

2

lyzing collision products we can extract useful informa-tion about the objects being studied. Historically, or-dinary matter particles like electrons and protons werepredominantly used as scattering particles in experimen-tal atomic and molecular physics. With the developmentof positron and antiproton beams, studies of interactionsof these particles with matter became possible.

The last decade has seen significant progress in low-energy trap-based positron beams [1, 19]. New high-resolution experimental measurements have been per-formed for a range of atomic and molecular targets in-cluding He [20], Ne and Ar [21, 22], Xe [23], Kr [24, 25]H2 [26] and H2O [27]. The development of positronbeams motivated novel experimental and theoreticalstudies. Particularly important are the positronium-formation cross sections, see the recent recommendationsof Machacek et al. [28]. In addition, interest has beenmotivated by possible binding of positrons to atoms [3].

Theoretical description of electron-impact ionisation(and excitation) processes has seen significant progress inrecent years due to the development of various highly so-phisticated methods including the exterior complex scal-ing (ECS) [29–31], R-matrix with pseudo-states (RMPS)[32–35], time-dependent close-coupling (TDCC) [36] andconvergent close coupling (CCC) [37–39]. A review ofelectron-induced ionisation theory has been given byBray et al. [40]. Such problems are examples of a classwhere there is only one “natural” centre, namely theatomic centre. All coordinates are readily written withthe origin set at the atomic centre. Yet there are manyatomic collision systems of practical and scientific inter-est that involve at least two centres, such as the positron-hydrogen scattering system. This is a three-body systemwhere all the particles are distinguishable, and which al-lows for their rearrangement. Here we have two “natu-ral” centres, the atomic centre and the positronium (Ps)centre. For positron-hydrogen scattering ionisation nowsplits into two separate components: the rearrangementprocess of Ps-formation and the three-body breakup pro-cess. A proper formulation of Ps-formation processes re-quires a combined basis consisting of two independentbasis sets for each of the centres which makes theoreticalstudies considerably more challenging than for electron-scattering. Furthermore, the positron-atom system is anideal prototype of the ubiquitous collision systems such asproton-atom scattering, where charge-exchange processesalso require a two-centre treatment, see Abdurakhmanovet al. [41] for example.

Every positron-atom scattering system has an ionisa-tion (breakup) threshold above which an electron maybe freely ejected. At 6.8 eV below this threshold is thePs formation threshold. At higher energies excitation orionization of the target can take place. In addition, formulti-electron targets there could be many more reactionchannels such as multiple ionisation and ionisation-with-excitation. For molecular targets there could be rovi-brational excitation and dissociation. Another reactionchannel is the positron-electron pair annihilation, which

can occur at any scattering energy. In this process thepositron collides with one of the target’s electrons and an-nihilates into 2 or 3 gamma rays. Theoretical [42, 43] andexperimental studies [44] of the annihilation process haveshown that its cross section is up to 105 times smallerthan the elastic-scattering cross section. Therefore theannihilation channel is often omitted from scattering cal-culations. The elastic scattering, excitation, Ps forma-tion and ionisation are the dominant channels of primaryinterest.

1. atomic hydrogen

The first theoretical studies of positron scattering fromatoms date back to the 1950s, when Massey and Mohr[45] used the first Born approximation (FBA) to describePs-formation in e+-H collisions. The Born method isbased on the assumption that the wavefunction for thescattering system can be expanded in a rapidly conver-gent series. This approximation consists in using planewaves to describe the projectile and scattered particles.The Born approximation is reliable when the scatteringpotential is relatively small compared to the incident en-ergy, and thus is applicable only at high energies. There-fore this method is mainly focused on high-energy exci-tation and ionization processes.One of the most successful methods applied to the low-

energy e+-H elastic-scattering problem is the Kohn vari-ational method. The method was initially developed forscattering phase shifts in nuclear reactions by Kohn [46].Later this method was extended to positron-hydrogenscattering by Schwartz [47]. A detailed description ofthe method was given by Armour and Humberston [48].The method is based on finding the form of the func-tional (called Kohn functional) involving the phaseshiftas a parametric function of the total wavefunction. Re-quiring the functional to be stationary with respect tothe variations of the parameters, generates equations forthe linear parameters. The phaseshifts can be accuratelyobtained by performing iterative calculations and find-ing the values of the nonlinear parameters that make thefunctional stationary.The many-body theory of Fetter and Walecka [49] uti-

lizes techniques that originated from quantum field the-ory. Using the Feynman-diagram technique the pertur-bation series in the interaction between particles canbe written in an intuitive way. When it is applied topositron-atom scattering, however, difficulties arise dueto necessity to take into account (virtual) Ps formation.However, a finite number of perturbation-theory termscannot describe a bound Ps state. Gribakin and King [50]developed a sophisticated method based on the many-body perturbation theory. They used an approximationby considering virtual Ps formation only in the groundstate. The calculations with this method showed that forelastic scattering of positrons on hydrogen (and heliumatoms), the virtual Ps-formation contribution was almost

3

30% and 20% of the total correlation potential, respec-tively. Gribakin and Ludlow [51] have further improvedthe method by introducing the techniques for the exactsummation of the electron-positron ladder diagram se-ries. The method was applied to e+−H scattering belowthe Ps-formation threshold, and resulted in good agree-ment with accurate variational calculations.

The momentum-space coupled-channel optical (CCO)potential method was first developed for electron-atomscattering in the 1980s by McCarthy and Stelbovics[52, 53]. The method relies on constructing a complexequivalent local potential to account for the ionizationand the Ps-formation channels. The CCO method gaveexcellent ionization [54], total and Ps formation [55] crosssections for positron scattering on hydrogen. Ma et al.[56] used the CCO method to study excitation of atomichydrogen from the metastable 2s state.

Following Macri et al. [57] Jiao et al. [58] developedthe so-called two-center two-channel eikonal final state-continuum initial distorted-wave model to calculate Psformation in the ground and the lowest excited states.They also presented a n−s scaling law for formation ofPs(n) cross sections on the entire energy range with svarying as a function of the positron incident energy.

A hyperspherical hidden crossing (HHC) method hasbeen applied to positron-impact ionization of hydrogennear the threshold by Jansen et al. [59]. They have cal-culated the ionization cross-section for S, P and D-waves.The HHC has also been used to calculate partial-wavePs(1s)-formation cross sections for low-energy positroncollisions with H, Li and Na atoms [60].

One of the most sophisticated and commonly usedmethods is the close-coupling (CC) formalism, which isbased on the expansion of the total wavefunction us-ing the target-state wavefunctions. Substitution of thisexpansion into the Schrodinger equation yields coupleddifferential equations in coordinate space, or Lippmann-Schwinger integral equations for the T-matrix in momen-tum space. By solving these equations the transition am-plitudes are obtained for all open channels. Considerablepioneering work in this field has been done by Hewittet al. [61], Higgins and Burke [62], Mitroy [63] and Wal-ters et al. [64], who demonstrated the success of usingtwo-centre expansions consisting of Ps and atomic states.

Here and below when we discuss close-coupling calcu-lations we denote the combined two-centre basis, usedto expand the total scattering wavefunction, as (N,M),whereN is the number of atomic (negative-energy) eigen-states and M is the number of positronium eigenstates.We also use a bar to indicate (negative- and positive-energy) pseudostates. For instance, CC(N,M) refers toclose-coupling calculations with a combined basis madeof N pseudostates for the atomic centre supplemented byM Ps eigenstates.

Higgins and Burke [62] performed the first accuratetwo-state CC(1,1) calculation. This work known as thestatic-exchange approximation showed a giant spuriousresonance near 40 eV incident positron energy. Absence

of such a resonance was demonstrated by Kernoghanet al. [65] using a larger CC(9, 9) calculation that in-cluded s, p, and d-type pseudostates for both centres.However, the CC(9, 9) gave new spurious resonancesabove the ionization threshold. An energy-averaging pro-cedure was used to get smooth results for the cross sec-tions to get rid of the pseudoresonances.

Considerable progress in the description of e+-H hasbeen made by Mitroy [63] by using the close-couplingapproach. Mitroy and Ratnavelu [66] have performedconvergence studies for the full positron-hydrogen prob-lem at low energies. Below the ionization thresholdthey showed good agreement of sufficiently large pseu-dostate close-coupling calculations and the benchmarkvariational calculations of Ps formation by Humberston[67].

The convergent close-coupling (CCC) method was firstdeveloped for e−-H scattering by Bray and Stelbovics[68]. Its modification to positron scattering in a one-centre approach was trivial in that electron exchangewas dropped and the interaction potentials changedsign [69, 70]. The CCC method with a (N, 0) basis, i.e.a single atomic-centre expansion without any Ps states,gave very good results for the total, elastic, excitationand ionization cross sections at higher incident energieswhere the Ps-formation cross section is small allowing fordistinction between two experimental data sets [70]. TheCCC calculations showed no pseudoresonances so long asa sufficiently large basis was taken.

Following the success of the large single-centre CCCcalculations, Kernoghan et al. [71] and Mitroy [72] useda large basis for the atomic centre supplemented by afew eigenstates of Ps. Calculations with the (28, 3) and(30, 3) bases, made of a large atomic basis similar to thatof Bray and Stelbovics [70] and the three lowest-lyingeigenstates of Ps, gave results significantly better thanthose from the (9, 9) basis [65].

The CCC method with a (N,M) basis was devel-oped by Kadyrov and Bray [73] to study convergencein two-centre expansions. This was applied to positronscattering on hydrogen within the S-wave model retain-ing only s-states in the combined basis. This work forthe first time demonstrated the convergence of the (non-orthogonal) two-centre expansions. The convergence inall channels was only possible when two independentnear-complete Laguerre bases are employed on both ofthe centres. Interestingly, the total ionisation cross sec-tion had two independently converged components. Onecomponent was coming from the atomic centre and repre-sented direct ionisation of the hydrogen atom, the othercame from the Ps centre and represented Ps formation inthe continuum. The convergence in the case of the fullpositron-hydrogen scattering problem was demonstratedby Kadyrov and Bray [74]. The CCC calculations withsuch a combined two-centre basis have shown very goodagreement with the experimental measurements of Zhouet al. [75] and Jones et al. [76].

4

2. helium

Theoretical investigations of positron scattering fromhelium has an additional challenge due to the complex-ity of the target structure. In multi-electron targets two-electron excitation (or ionization with excitation) chan-nels are usually excluded. This is a good approximationas the contribution of these channels is typically two or-ders of magnitude smaller than the corresponding one-electron excitation processes [77].First calculations of e+-He scattering have been per-

formed by Massey and Moussa [78] in the FBA. Theyused only the ground states for He and Ps and obtainedcross sections for elastic scattering and Ps formation inits ground state. Their study highlighted the importanceof the Ps-channel coupling with the elastic channel andthereby motivated further studies. Another extensivestudy based on the Born approximation was presented byMandal et al. [79], to estimate Ps-formation cross sectionin arbitrary S-states. From the FBA studies it becameclear that more sophisticated approaches to the problemwere required.The distorted-wave Born approximation (DWBA) re-

sults are obtained by using distorted wavefunctions infirst-order calculations. This method can give more accu-rate results than the FBA down to lower energies. Stud-ies utilizing the DWBA by Parcell et al. [80, 81] were ap-plied to the helium 21S and 21P excitations by positronsin the energy range from near the threshold up to 150 eV.Although the agreement with the experimental data wasnot very satisfactory, the method indicated the impor-tance of the inclusion of the polarization potential in theexcitation channels at low energies. The most system-atic study of the ionisation process within the frameworkof DWBA was carried out by Campeanu et al. [82, 83].They used Coulomb and plane waves and also includedexchange effects. They obtained good agreement withthe experimental results of Fromme et al. [84], Knudsenet al. [85] and Moxom et al. [86] over the energy rangefrom near-threshold to 500 eV. However, the most impor-tant and difficult channel, Ps formation, was not includedin the early DWBA studies.Srivastava et al. [87] calculated the differential and to-

tal cross sections for the excitation of the helium 21Sstate using the second-order DWBA method. AnotherDWBA method including Ps channels has been reportedby Sen and Mandal [88] for intermediate to high scat-tering energies. They have calculated Ps-formation crosssection and achieved good agreement with available ex-perimental data above 60 eV. However, their results werenot accurate for Ps formation below 60 eV. Consider-ing the fact that Ps formation starts at 17.8 eV andreaches its maximum around 40 eV, the applicability ofthis method is quite limited.Amusia et al. [89] applied the random-phase approx-

imation (RPA), based on many-body theory [49], topositron-helium scattering at low energies. By using anapproximate account of virtual Ps formation they ob-

tained good agreement with the elastic-scattering exper-imental data of Jaduszliwer and Paul [90] and Canteret al. [91] at the lowest energies. A similar RPA methodwas used by Varracchio [92] to calculate positron impactexcitation of He into 21S and 21P states. As mentionedabove, Gribakin and King [50] developed a more sophis-ticated method based on the many-body perturbationtheory. The calculations with this method showed thatthe contribution from virtual Ps formation was signifi-cant. Applications of the method to various atomic tar-gets were reported by Dzuba et al. [93] and Gribakin andKing [94].

The Kohn variational method was first applied topositron-helium collisions by Humberston et al. [95–98]. A comprehensive study of positron-helium scatteringwith the Kohn variational method was given by Reethand Humberston [99]. They obtained very accurate crosssections at low energies. However, agreement with theexperimental results of Mizogawa et al. [100] and Steinet al. [101] for Ps-formation cross section was qualitative,with a similar energy dependence but with almost 25%difference in magnitude. Nevertheless, very good agree-ment was obtained for the total cross section below thePs-formation threshold.

The CCO method, mentioned earlier, was applied topositron scattering on helium by Cheng and Zhou [102].They calculated the total and Ps-formation cross sectionsfrom the Ps-formation threshold to 500 eV. The calcu-lated results agreed well with the corresponding exper-imental data except for the data of Griffith and Hey-land [9] for the total cross section in the energy rangefrom 50 to 100 eV. McEachran et al. [103] applied apolarized-orbital approximation method to low-energyelastic positron-helium scattering and obtained goodagreement with the experimental results. Other calcu-lations using optical potentials were presented by Tancicand Nikolic [104] and by Gianturco and Melissa [105] forslow-positron scattering from helium. Elastic-scatteringcross sections of both reports were in good agreementwith experimental data. In general, the optical-potentialmethods proved to be useful for calculations of total crosssections. They are problematic, however, when appliedto more detailed cross sections like target excitation andPs formation in excited states.

Schultz and Olson [106] utilized the classical-trajectoryMonte-Carlo (CTMC) technique to model positron scat-tering. The method is described fully for ion-atomcollisions by Abrines and Percival [107] and by Olsonand Salop [108]. Using this technique, Schultz and Ol-son [106] calculated differential ionization cross sectionfor positron-helium and also positron-krypton collisions.The main advantage of this method is that it can de-scribe dynamic effects occurring in collisions. For in-stance, the CTMC calculations showed that the proba-bility of positron scattering to large angles after ionisingthe target, may be comparable, or even much greaterthan, the probability of positronium formation. Theysuggested that the disagreement between theory and ex-

5

periment above 60 eV might be resolved by accountingfor the flux in the experiments measuring positroniumformation due to positrons scattered to angles that allowthem to escape confinement.

Tokesi et al. [109] also applied the CTMC method tohelium ionization by positron impact. They obtainedgood agreement with experimental data of Fromme et al.[84]. Results of CTMC reported by Schultz and Ol-son [106] overestimate the recent experimental data byCaradonna et al. [110] for the Ps-formation cross sectionbelow 60 eV. This questions the applicability of the clas-sical trajectory approach to positron scattering at lowand intermediate energies.

A very comprehensive study of positron-helium scat-tering using the close-coupling method was carried outby Campbell et al. [111]. They used two kinds of ex-pansions, the first one consisting of 24 helium eigen- andpseudostates and the lowest three Ps eigenstates, andthe second one with only 30 helium eigen- and pseudo-states. The helium-target structure was modeled usinga frozen-core approximation, which can produce goodexcited states, but a less accurate ground state. Theatomic pseudostates were constructed using a Slater ba-sis. For the 27-state approximation, only results in theenergy range above the positronium-formation thresholdwere given. Results for lower energies were unsatisfac-tory, and it was suggested that this might be due to thelack of convergence from the use of an inaccurate heliumground-state wavefunction. The total cross sections fromboth the 27- and 30-state approaches agreed well with theexperimental results of Stein et al. [101], Kauppila et al.[112] and Mizogawa et al. [100] for the energy range abovethe threshold of positronium formation. For lower ener-gies, qualitative agreement was obtained in terms of theshape and the reproduction of the Ramsauer-Townsendminimum near 2 eV, while the theoretical results were afactor of 2 larger than the experimental data. The Ps-formation cross section from the 27-state approximationwas in good agreement with the experimental data ofMoxom et al. [86] up to about 60 eV and with the dataof Fornari et al. [113] and Diana et al. [114] up to 90 eV.Above 100 eV the calculations were much lower than theexperimental data of Diana et al. [114] and Fromme et al.[84] while being closer to the data of Overton et al. [115].

Another close-coupling calculation by Hewitt et al.[61, 116], using a few helium and positronium states witha one-electron description of the helium atom, showedless satisfactory agreement with the experimental data.Chaudhuri and Adhikari [117–119] performed calcula-tions using only 5 helium and 3 positronium states in theexpansion. Their results for Ps formation agreed wellwith the experimental results by Moxom et al. [86] atenergies near the Ps threshold and displayed a betteragreement with the data of Overton et al. [115] at thehigher energies. However, the theoretical results weremuch lower than the experimental data at energies nearthe maximum of the cross section.

Igarashi et al. [120] applied the hyper-spherical CC ap-

proach to the problem. However, they also consideredhelium as a one-electron target, and thus the excitationand Ps-formation cross sections were multiplied by fac-tor of 2. Satisfactory results were obtained for the totalPs-formation and He(21S) and He(21P) excitation crosssections. The method was not able to describe low-energyscattering mainly because a one-electron approach to he-lium is not realistic at low energies.

Despite the obvious advantages of the above-mentioned close-coupling calculations in handling manyscattering channels simultaneously, none was able to de-scribe low-energy elastic scattering. In addition, the pres-ence of pseudo-resonances in cross sections below the ion-ization threshold [111, 116] indicated that there was roomfor improvement. The use of the frozen-core He statesalso needed some attention as this yielded an inaccurateresult for the ground state of helium.

The first application of the single-centre CCC methodto positron scattering on helium was made by Wu et al.[121, 122], using very accurate helium wavefunctions ob-tained within the multi-core approximation. Very ac-curate elastic cross section was obtained below the Ps-formation threshold by using orbitals with very high an-gular momenta. It was suggested that the necessity forinclusion of very high angular-momentum orbitals was tomimic the virtual Ps-formation processes. The methodalso gave accurate results for medium to high-energyscattering processes except that it was unable to explic-itly yield the Ps-formation cross section. Interestingly,the method was not able to produce a converged re-sult at the Ore-gap region (where only the elastic andPs-formation channels are open), which we shall discusslater in sec. III A. Comparison of the frozen-core and themulti-core results showed that the frozen-core wavefunc-tions lead to around 10% higher cross sections. Wu et al.[121, 122] demonstrated that single-centre expansions cangive correct results below the Ps-formation threshold andat high energies where the probability of Ps formation issmall, but for the full solution of the problem inclusionof the Ps centre into expansions was required.

The initial application of the two-centre CCC approachto the problem was within the frozen-core approxima-tion [123, 124]. It was then extended to a multi-coretreatment [125]. While generally good agreement wasfound with experiment from low to high energies cer-tain approximations made need to be highlighted. Thekey problematic channels are those of the type Ps-He+.Electron exchange between Ps and He+ is neglected. Ex-citation of He+ is also neglected. While these may seemreasonable approximations for the helium target due toits very high ionisation threshold (24.6 eV for He and54.4 eV for He+), they become more problematic forquasi two-electron targets such as magnesium, discussedbelow.

Positron scattering from the helium 23S metastablestate has been theoretically studied for the first time byUtamuratov et al. [126] at low and intermediate ener-gies. Converged results for the total, Ps-formation and

6

breakup cross sections have been obtained with a high de-gree of convergence. The obtained cross sections turnedout to be significantly larger than those for scatteringfrom the helium ground state.

3. alkali metals

Alkali atoms have an ionisation threshold that is lowerthan 6.8 eV. Consequently, for positron scattering on al-kalis the elastic and Ps-formation channels are open atall incident-positron energies. For this reason, theory hasto treat appropriately the “competition” for the valenceelectron between the two positively charged centers, thesingly-charged ionic core and the positron.Positron scattering on the lithium target was investi-

gated by Ward et al. [127], Sarkar et al. [128], Khan et al.[129] using a one-center expansion. However, conver-gence was poor due to the absence of Ps-formation chan-nels [130]. Two-center expansion was employed by Guhaand Ghosh [131], Basu and Ghosh [132], Abdel-Raouf[133], Hewitt et al. [134], McAlinden et al. [130] and Leet al. [135]. As expected, these approaches gave bet-ter agreement with the experiment. For positron-lithiumcase the two-center CCC approach to positron collisionswith lithium was reported by Lugovskoy et al. [136]. Thisis the most comprehensive study of the problem on anenergy range spanning six orders of magnitude. Whileconvergence was clearly established, and agreement withexperiment for the total Ps-formation cross section [137]is satisfactory, smaller experimental uncertainties wouldbe helpful to provide a more stringent test of the theory.Positron scattering from atomic sodium has been in-

tensively studied for more than two decades. The firsttheoretical calculations relied on simple two-center de-composition of the system wavefunction with only theground states of sodium and Ps atoms taken into ac-count [133, 138, 139]. Then, Hewitt et al. [140] conductedmore complex close-coupling calculations, adding severallow-lying excited states for each positively-charged cen-ter. The obtained results turned out to be in reasonablygood agreement with experimental data for both total[141, 142] and Ps-formation [143] cross sections.Further enlargement of the number of channels in the

close-coupling calculations revealed that the theoreticalPs-formation cross sections [111, 144, 145] deviated sys-tematically from the experimental results for low impactenergies. The experiment showed that the Ps-formationcross section became larger with decreasing energy whilethe most refined theoretical calculations utilizing differ-ent methods of solution predicted consistently the oppo-site.To resolve the discrepancy Surdutovich et al. [137]

conducted the experimental study on Ps formation inpositron collisions with Li and Na atoms. This exper-iment confirmed the earlier results of Zhou et al. [143].The authors managed to extend the impact-energy rangedown to 0.1 eV where the discrepancy between the the-

ory and experiment was even larger for sodium. In strik-ing contrast, for lithium the reasonable agreement of themeasured cross section with the theoretical predictionswas obtained with the use of the same methodology. Keet al. [146] applied the optical-potential approach. Theyfound that their theoretical cross section increases withthe decrease in the impact energies below 1 eV, but fasterthan the experimental results. Unfortunately, this resultwas obtained with the use of some approximations, whosevalidity were not analyzed. It would be instructive if thesame optical-potential approach was applied to the caseof positron scattering on lithium.Le et al. [135] calculated Ps-formation in positron-

alkali collisions with the use of the hyper-spherical close-coupling method. Their results support the previous the-oretical data [111, 144, 145]. Large two-center CCC cal-culations of positron scattering by atomic sodium werereported by Lugovskoy et al. [147]. Despite being themost comprehensive to date there was no resolution ofthe discrepancy with experiment for Ps formation at lowenergies, which we will highlight in Sec III D.While the lighter alkali atoms are well-modeled by

a frozen-core Hartree-Fock approximation, or even anequivalent local core potential, the heavier ones becomemore problematic. With a reduced ionisation thresholdpositron interaction with the core electrons, either di-rectly or via exchange of the valence and the core elec-trons, becomes a more important component of the in-teraction. To the best of our knowledge this has notbeen addressed to a demonstrable level of convergenceby any theory. Nevertheless, assuming that such prob-lems are more likely to be a problem at the higher en-ergies, Lugovskoy et al. [148] considered threshold be-haviour of the elastic and Ps-cross sections, and theirconvergence properties at near-zero energies for Li, Naand K. This work confirmed the expected threshold lawproposed by Wigner [149], but was unable to resolve thediscrepancy with the positron-sodium experiment at lowenergies. Some earlier studies by Hewitt et al. [140] andMcAlinden et al. [150] at low to intermediate energieswere performed at a time when convergence was compu-tationally impossible to establish.Chin et al. [151] further developed the CCO method

to study positron scattering on rubidium at intermedi-ate and high energies. They calculated the Ps-formationand total cross sections. Their total cross section resultsappear to overestimate the experiment.Though outside the scope of this review we note

that the complex scaling method was recently used tostudy resonance phenomena in positron scattering onsodium [152] and potassium [153].

4. magnesium

Magnesium can be thought of as a quasi two-electrontarget with the core electrons being treated by the self-consistent field Hartree-Fock approach. Positron scat-

7

tering on magnesium is particularly interesting due to alarge resonance in elastic scattering identified at low en-ergies by Mitroy et al. [154]. This was confirmed, thoughat a slightly different energy, by the one-centre calcula-tions of Savage et al. [155] which were able to be taken toconvergence in the energy region where only elastic scat-tering was possible. Minor structure differences are likelyto be responsible for the small variation in the positionof the resonance. Two-centre CCC calculations [156] alsoreproduced the resonance, but had to make substantialapproximations when treating the Ps-Mg+ interaction.This is even more problematic than in the case of he-lium discussed above since now we have a multi-electronHartree-Fock core. Agreement with experiment is some-what variable, but there are substantial experimental un-certainties, particularly in the Ps-formation cross section.Other theoretical studies of positron-magnesium scat-tering include those by Gribakin and King [94], Camp-bell et al. [111], Campeanu et al. [157], Bromley et al.[158], Hewitt et al. [159].

5. inert gases

Inert gases heavier than helium represent a particu-lar challenge for theory, which is unfortunate becausethey are readily accessible experimentally [11, 160]. Justthe target structure is quite complicated, but some goodprogress has been made in the case of electron scatter-ing by Zatsarinny and Bartschat [161, 162]. For positronscattering once Ps forms, the residual ion is of the open-shell type making full electron exchange incorporationparticularly problematic. The relatively high ionisationthresholds for such targets mean that there is always asubstantial Ore gap where the Ps-formation cross sectionmay be quite large, but unable to be obtained in one-centre calculations which are constrained to have onlyelastic scattering as the open channel. Nevertheless, out-side the extended Ore gap, formed by the Ps-formationand the ionisation thresholds, one-centre CCC calcula-tions can yield convergent results in good agreement withexperiment [163]. There are also first-order perturba-tive calculations by Campeanu et al. [83], Dzuba et al.[93], Chen et al. [164], Parcell et al. [165] and some basedon close-coupling with convergence not fully established,see McAlinden and Walters [166], Gilmore et al. [167].Green et al. [168] studied positron scattering and anni-

hilation on noble-gas atoms using many-body theory atenergies below the Ps-formation threshold. They demon-strated that at low energies, the many-body theory is ca-pable of providing accurate results. Fabrikant and Grib-akin [169] used an impulse approximation to describe Psscattering on inert gases and provided quantitative theo-retical explanation for the experimentally-observed sim-ilarity between the Ps and electron scattering for equalprojectile velocities [170]. According to Fabrikant andGribakin [169] this happens due to the relatively weakbinding and diffuse nature of Ps, and the fact that elec-

trons scatter more strongly than positrons off atomic tar-gets.

Poveda et al. [171] developed a model-potential ap-proach to positron scattering on noble-gas atoms basedon an adiabatic method that treats the positron as a lightnucleus. The method was applied to calculate the elasticcross section below the Ps-formation threshold.

6. molecular hydrogen

Positron collisions with molecular hydrogen have beenstudied extensively by various experimental groups overthe last 30 years [26, 85, 113, 114, 172–181]. Theoreti-cal studies of this scattering system are challenging be-cause of the complexities associated with the molecularstructure and its non-spherical nature. Rearrangementprocesses add another degree of complexity to the prob-lem. Until recently theoretical studies [48, 182–194] havebeen focused only at certain energy regions. In addition,there are few theoretical studies which include the Ps-formation channels explicitly. The first calculations ofPs-formation cross section [182–184, 186] were obtainedwith the use of the first Born approximation. Biswaset al. [187] used a coupled-static model, which only in-cluded the ground states of H2 and Ps. This simple modelwas until recently the only coupled-channel calculationavailable. Comprehensive review of the positron interac-tions with atoms and molecules has been given by Surkoet al. [1].

Fedus et al. [195] have recently reported the total crosssection for positron scattering from the ground state ofH2 below the Ps-formation threshold using density func-tional theory with a single-center expansion. Their re-sults are in good agreement with recent single-centreCCC calculations of Zammit et al. [196] below 1 eV. Fe-dus et al. [195] have also performed analysis of exper-imental and theoretical uncertainties using a modifiedeffective range theory (MERT). They concluded that apractically constant value of the total cross section be-tween 3 eV and the Ps-formation threshold is likely to bean effect of virtual Ps formation.

The recent single-centre CCC calculations of positronscattering on molecular hydrogen by Zammit et al. [196]and antiproton collisions with H2 by Abdurakhmanovet al. [197, 198] have shown that the CCC formalism canalso be successfully applied to molecular targets. In or-der to obtain explicit Ps-formation cross section a two-centre approach is required, with the first attempt pre-sented by Utamuratov et al. [199]. They found some ma-jor challenges associated with the Ps-H+

2 channel. Somesevere approximations were required in order to man-age the non-spherical H+

2 ion. Nevertheless, some goodagreement with experiment was found, see Sec. III F, butconsiderably more work is required.

8

B. Antihydrogen formation

A major application of positron-hydrogen scatteringis to provide a mechanism for antihydrogen formation.The idea is fairly simple, with some accurate calculationsperformed quite some time ago [200]. The basic ideais to time reverse the Ps-formation process to hydrogenformation from Ps scattering on a proton, and then touse the resultant cross sections for the case where theproton is replaced by an antiproton, and hence formingantihydrogen

e+ +H ↔ Ps + p+ ≡ e− + H ↔ Ps + p−. (1)

The advantage of antihydrogen formation via this pro-cess is that it is exothermic and so the cross sec-tion tends to infinity as the relative energy goes tozero [149]. This behaviour is enhanced in the case ofexcited states with degenerate energy levels [201]. Anti-hydrogen formation is presently particularly topical dueto several groups (AEgIS [202–204], GBAR [205–207],ATRAP [208], ASACUSA [209] and ALPHA [210]) at-tempting to make it in sufficient quantity in order to per-form spectroscopic and gravitational experiments. Kady-rov et al. [211] provided CCC results for Ps energy start-ing at 10−5 eV, which suffices for currently experimen-tally accessible energies of around 25 meV. Recent cal-culations of the cross sections for these processes will bediscussed in Sec. IV.In this section we gave a general historical overview

of various theoretical developments related to positronscattering on atomic targets and the H2 molecule. In thenext section we consider in some detail basic features ofthe coupled-channel formalism mainly in the context ofconvergent close-coupling method and discuss the latestresults.

II. THEORY

A. Atomic hydrogen

Here we describe basics of the close coupling approachbased on the momentum-space integral equations. Weconsider the simplest case of scattering in a system ofthree particles: positron (to be denoted α), proton (β)and electron (e). Let us also call α the pair of protonwith electron, β - positron with electron and e - positronwith proton. We neglect spin-orbit interactions. In thiscase spacial and spin parts of the total three-body wave-function separate. The latter can be ignored as it has noeffect on scattering observables. The spacial part of thetotal three-body scattering wavefunction satisfies

(H − E)Φ = 0, (2)

where

H = H0 + Vα + Vβ + Ve ≡ H0 + V (3)

is the full Hamiltonian, H0 is the three-free-particleHamiltonian, Vi is the Coulomb interaction between par-ticles of pair i (i = α, β, e). The total Hamiltonian canalso be expressed in the following way

H = Hγ +q2γ

2Mγ+ V γ , (4)

where Hγ is the Hamiltonian of the bound pair γ, qγ isthe momentum of free particle γ relative to c.m. of thebound pair,Mγ is the reduced mass of the two fragments

and V γ = V − Vγ is the interaction potential of the freeparticle with the bound system in channel γ (γ = α, β).Coupled-channel methods are based on expansion of

the total wavefunction Φ in terms of functions of allasymptotic channels. However, since the asymptoticwavefunction corresponding to 3 free particles has a com-plicated form [212, 213], this is not practical. There-fore, we approximate Φ by expansion over some negativeand discrete positive energy pseudostates of pairs α andβ especially chosen to best reproduce the correspondingphysical states. Suppose we have some NH pseudostatesin pair α and NPs in pair β satisfying the following con-ditions

〈ψα′ |ψα〉 = δα′α, 〈ψα′ |Hα|ψα〉 = δα′αεα (5)

and

〈ψβ′ |ψβ〉 = δβ′β , 〈ψβ′ |Hβ |ψβ〉 = δβ′βεβ , (6)

where ψγ is a pseudostate wavefunction of pair γ and εγis the corresponding pseudostate energy. Then we canwrite

Φ ≈NH∑

α=1

Fα(ρα)ψα(rα) +

NPs∑

β=1

Fβ(ρβ)ψβ(rβ)

≡NH+NPs∑

γ=1

Fγ(ργ)ψγ(rγ), (7)

where Fγ(ργ) is an unknown weight function, rγ is therelative position of the particles in pair γ, ργ is the po-sition of particle γ relative to the centre of mass (c.m.)of pair γ (γ = α, β), see Fig. 1. For convenience, here weuse the same notation not only to denote a pair and acorresponding channel, but also a quantum state in thispair and the channel. So, the indices of functions Fγ andψγ , additionally refer to a full set of quantum numbersof a state in the channel. In the case of vectors ργ andrγ , the indices still refer only to a channel and a pair inthe channel, respectively.In principle, at this formal stage one could keep the

continuum part only for one of the pairs in order not todouble up the treatment of the three-body breakup chan-nel. However, a symmetric expansion of the type (7),with the continuum states on both centres, was found togive fastest convergence in calculations with a manage-able number of states [73, 214, 215].Now we use the Bubnov-Galerkin principle [216] to find

the coefficients Fγ(ργ) so that the expansion (7) satisfies

9

r

r

β

α

α

e

ρα

re

βρ

ρe

β

FIG. 1. Jacobi coordinates for a system of three particles:positron (α), proton (β), and electron (e).

Eq. (2) the best possible way. Accordingly, we substitutethe expansion (7) into Eq. (2) and require the result tobe orthogonal to all (γ′ = 1, . . . , NH +NPs) basis states,i.e.

〈ψγ′ |H − E|NH+NPs∑

γ=1

Fγψγ〉ργ′= 0. (8)

In this equation subscript ργ′ indicates integration overall variables except ργ′. Now taking into account condi-tions (5) and (6) we can write Eq. (8) in the followingform:(E − εγ′ −

q2γ′

2Mγ′

)Fγ′(ρ

γ′) =

NH+NPs∑

γ=1

〈ψγ′ |Uγ′γ |ψγ〉

× Fγ(ργ). (9)

The potential operators Uγ′γ are given by

Uαα = V α, Uββ = V β ,

Uαβ = Uβα = H − E. (10)

The condition imposed above in Eq. (9) is a sys-tem of coupled equations for unknown expansion coef-ficients Fγ(ργ). These functions carry information onthe scattering amplitudes. We transform these integro-differential equations for the weight functions to a set ofcoupled Lippmann-Schwinger integral equations for tran-sition amplitudes Tγ′γ .To this end we define the Green’s function in channel

γ

Gγ(q2) =

(E + i0− q2

2Mγ− εγ

)−1

, (11)

to describe the relative motion of free particle γ andbound pair γ with binding energy ǫγ . We can now write

the formal solution of the differential equation (9) as

Fγ′(ργ

′) =δγ′γeiqγργ +

NH+NPs∑

γ′′=1

∫dq

(2π)3eiqργ′′Gγ′′(q2)

× 〈q| 〈ψγ′′ |Uγ′′γ |ψγ〉 |Fγ〉. (12)

The addition of positive i0 defines the integrationpath around the singularity point at q = qγ =

[2Mγ(E − εγ)]1/2, which is real for E > εγ , and corre-

sponds to the outgoing wave boundary conditions. Tak-ing Eq. (12) to the asymptotic region one can demon-strate [213] that

Tγ′γ(qγ′ , qγ) = 〈qγ′ | 〈ψγ′ |Uγ′γ |ψγ〉 |Fγ〉 (13)

represents the matrix element for transition from channelγ to channel γ′, where qγ and qγ′ are the initial and finalmomenta of the incident and scattered particles. Thenfrom (12) we get

Fγ′(ργ′) =δγ′γe

iqγργ +

NH+NPs∑

γ′′=1

∫dq

(2π)3eiqργGγ′′(q2)

× Tγ′′γ(q, qγ). (14)

Now using Eq. (14) in Eq. (13) we get a set of cou-pled Lippmann-Schwinger-type equations for the tran-sition matrix elements (γ, γ′ = 1, ..., NH +NPs)

Tγ′γ(qγ′ , qγ) =Vγ′γ(qγ′ , qγ) +

NH+NPs∑

γ′′=1

∫dq

(2π)3

× Vγ′γ′′(qγ′ , q)Gγ′′(q2)Tγ′′γ(q, qγ).(15)

The effective potentials are given by

Vγ′γ(qγ′ , qγ) = 〈qγ′ |〈ψγ′ |Uγ′γ |ψγ〉|qγ〉. (16)

Thus we have a set of three-dimensional momentum-space integral equations. Eq. (15) can be solved via par-tial wave decomposition. The latter is an ideal methodfor collisions involving light projectiles like positrons [217,218] or positronium [211, 219]. For heavy projectiles a di-rect three-dimensional discretization technique has beendeveloped [220–222].After partial wave expansion for each value of the total

angular momentum J one gets

T L′LJγ′γ (qγ′ , qγ) =VL′LJ

γ′γ (qγ′ , qγ) +

NH+NPs∑

γ′′=1

∑

L′′

∫dq

(2π2)

× VL′L′′Jγ′γ′′ (qγ′ , q)Gγ′′(q2)T L′′LJ

γ′′γ (q, qγ),

(17)

where L, L′ and L′′ are the relative angular momenta ofthe fragments in channels γ, γ′ and γ′′, respectively. The

10

effective potentials in partial waves are given by

VL′LJγ′γ (qγ′ , qγ) =

∑

m′mM ′M

∫ ∫dqγ′dqγY

∗L′M ′(qγ′)

× CJKL′M ′l′m′Vγ′γ(qγ′ , qγ)CJKLMlm

× YLM (qγ), (18)

where CJKLMlm are the Clebsch-Gordan coefficients of vec-tor addition, YLM (qγ) is the spherical harmonics of unitvector qγ . l (l′) is the angular momenta of pair γ (γ′)and M , m, K are the projections of L, l, J , respectively,with K =M +m =M ′ +m′.We introduce a K-matrix according to

T = K/(1 + iπK) [68]. This reduces Eq. (17) to aset of equations for the K-matrix amplitudes

KL′LJγ′γ (qγ′ , qγ) =VL′LJ

γ′γ (qγ′ , qγ) +

NH+NPs∑

γ′′=1

∑

L′′

P∫

dq

(2π2)

×VL′L′′Jγ′γ′′ (qγ′ , q)

E − q2/2Mγ′′ − ǫγ′′

KL′′LJγ′′γ (q, qγ),

(19)

where P indicates a principal value integral. These arethe major CCC equations to be solved numerically, inthe form of linear equations Ax = b utilising only realarithmetic [68]. Upon solution the reconstruction of theT-matrix and obtaining cross sections is a much simplercomputational task.Details of calculations of the matrix elements are given

by Kadyrov and Bray [74]. Here we present the final re-sults in order to highlight the difference in complexitybetween the matrix elements for direct and rearrange-ment transitions. The effective potentials (16) for directtransitions α→ α′ (transitions between states within thesame arrangement) read

Vα′α(qα′ , qα) =

∫dραdrαe

−iqα′ραψ∗α′(rα)Uαα

× ψα(rα)eiqαρα . (20)

In partial waves, after some algebra, we have

VL′LJα′α (qα′ , qα) = (4π)2(−1)J+(L′+L+l′+l)/2[L′l]

×∑

λ

(2)CL0L′0λ0Cl′0l0λ0

{L′ λ Ll J l′

}

×Iλα′α(qα′ , qα), (21)

where [l] =√2l+ 1, [lL] is a shorthand for [l][L], and

the braces denote the 6j-symbol. In the sum over λ onlyterms preserving the parity of l′ + l survive, leading toincrements of 2. The radial integral is defined as

Iλα′α(qα′ , qα) =

∫ ∞

0

dραρ2αjL′(qα′ρα)jL(qαρα)

×∫ ∞

0

drαr2αRn′l′(rα)Uλαα(ρα, rα)Rnl(rα),

(22)

whereRnl(r) are the radial parts of the pseudostate wave-functions ψ and

Uλαα(ρα, rα) =

δλ0

ρα− ρλα

rλ+1α

if ρα < rαδλ0

ρα− rλα

ρλ+1α

otherwise.(23)

Matrix elements for β → β′ transitions are defined as

Vβ′β(qβ′ , qβ) =

∫dρβdrβe

−iqβ′ρβψ∗β′(rβ)Uββ

× ψβ(rβ)eiqβρβ . (24)

The corresponding partial-wave amplitudes VL′Lβ′β (qβ′ , qβ)

have the same form as expression (21) but withUλββ(ρβ , rβ) in Eq. (22) for Iλβ′β(qβ′ , qβ) defined as

Uλββ(ρβ , rβ) =(1− (−1)λ

)

2λ+1ρλβ

rλ+1

β

if ρβ < rβ/2

rλβ

2λρλ+1

β

otherwise.

(25)

In the case of rearrangement α → β transitionsEq. (16) for the effective potentials reads as

Vβα(qβ , qα) =

∫dρβdrβe

−iqβρβψ∗β(rβ)(H − E)

× ψα(rα)eiqαρα . (26)

Calculation of the matrix elements for the rearrangementtransitions is significantly more complicated. Using theGaussian representation for the wavefunctions and theinteraction potentials Hewitt et al. [61] calculated thePs-formation matrix elements for transition amplitudesbetween arbitrary H and Ps states. Mitroy [63] suggesteda straightforward way of calculating the matrix elementswithout using additional expansions. Following Mitroy[63], Kadyrov and Bray [74] further reduced the resultsfor the case of the Laguerre-type pseudostates analyt-ically calculating integrals for momentum-space pseu-dostate wavefunctions and formfactors. The final resultfor the matrix elements in the rearrangement channels iswritten as

VL′Lβα (qβ , qα) =[l′lL′L][l′!l!](−1)J+L

′∑

l′1

[l′1l′2]

[l′1!l′2!]

2−l′

1−1

×∑

l1

[l1l2]

[l1!l2!]ql′1+l1β

∑

l′′1

(2)Cl′′1 0

l10l′10

×∑

l′′2

(2)Cl′′2 0

l20l′20

∑

λ

(2)[λ]2Cl′′1 0L′0λ0C

l′′2 0L0λ0

×

l1 l J l′

l2 L L′ l′1l′2 l′′2 λ l′′1

× Iλβα(qβ , qα), (27)

11

where [l!] =√(2l+ 1)!, l′1 + l′2 = l′ and l1 + l2 = l, the

braces denote the 12j-symbol of the first kind [223],

Iλβα(qβ , qα) =ql′2+l2α F (I)

λ (qβ , qα) +1

πqα

∫ ∞

0

dqql′

2+l2+1

×QL

(q2 + q2α2qqα

)F (II)λ (qβ , q), (28)

and where QL is a Legendre function of the second kind.For Legendre polynomial Pλ and z = qβ · qα,

F (I,II)λ (qβ , qα) =

∫ 1

−1

dzF (I,II)(qβ , qα)Pλ(z) (29)

with

F (I)(qβ , qα) =

(q2α2

+p2α2

− E

)R∗n′l′(pβ)Rnl(pα)

pl′

βplα

+R∗n′l′(pβ)unl(pα)

pl′

βplα

+u∗n′l′(pβ)Rnl(pα)

pl′

βplα

,

(30)

and

F (II)(qβ , q) =R∗n′l′(p

′β)Rnl(p

′α)

p′βl′p′α

l, (31)

where Rnl(p) and unl(p) are the pseudostate wavefunc-tions and pseudostate formfactors in momentum space,respectively. Here pγ is the relative momentum of theparticles of pair γ:

pβ = qβ/2− qα and pα = qβ − qα. (32)

In addition, p′α and p′

β are the relative momenta of theparticles of the corresponding pairs immediately beforeand after the rearrangement:

p′β = qβ/2− q and p′

α = qβ − q, (33)

where q is the relative momentum of the fragments inchannel e.As mentioned before, 2 sets of pseudostates are ob-

tained by diagonalising the H and Ps Hamiltonians. Theradial parts of pseudostates are taken as

Rnl(r) =N∑

k=1

Blnkξkl(r), (34)

where pseudostate basis ξkl(r) is constructed from theorthogonal Laguerre functions

ξkl(r) = Nkl(2r/a)l+1e−r/aL2l+2

k−1 (2r/a), (35)

with

Nkl =

[2(k − 1)!

a(2l + 1 + k)!

]1/2. (36)

Here L2l+2k−1 (2r/a) are the associated Laguerre polynomi-

als. Expansion coefficients Blnk are found by diagonal-izing the two-particle Hamiltonian of the relevant pair.

As to the choice of the exponential fall-off parameter,practice shows that the highest rate of convergence isachieved when aα and aβ are chosen to best reproducethe ground state energy of hydrogen and positronium, re-spectively, for a given size of the bases NH and NPs. Thechoice of aα = 1 and aβ = 0.5 fulfils this requirement.Associated pseudostate wavefunctions and form factorsin momentum space used in Eq. (30) and Eq. (31) havebeen calculated in analytic form.The set of equations (19) is solved using standard

quadrature rules. The singular kernel is discretised us-ing a Gauss quadrature. A subtraction technique isused to handle moving singularities. Very recently Brayet al. [224, 225] have proposed a new approach to solvingEq. (19), which handles the Green’s function analytically,see Sec. IVB.Cross section from initial state α to final state α′ (or

β) is calculated as a cross section for excitation of stateα′ (β). The total cross section is obtained as a sum ofall partial cross sections for each state included in theclose-coupling expansion, or utilizing the unitarity of theclose-coupling formalism by applying the optical theo-rem. We calculate the total cross section in both ways(which should give the same result) to check that theoptical theorem is satisfied. The total ionization crosssection is calculated as a sum of the integrated cross sec-tions for positive energy states (of both atom and Ps).The total Ps-formation cross section is defined as a sumof the cross sections for electron capture into Ps boundstates.

B. Helium

A distinct feature of the CCC formalism is that theresulting set of coupled equations (19) is essentially inde-pendent of the choice of the target, and is similar to thatgiven for hydrogen. Therefore, the basic formalism of thetwo-centre CCC method described above can be appliedto positron collisions with helium. However, in this casespins play an important role. There are two target elec-trons that can form positronium. The electrons can be inspin-singlet or spin-triplet states. Depending on the spinprojections of the electron and the positron that formpositronium, the latter can be formed in para (p-Ps) orortho (o-Ps) states . The target spin and Ps spin shouldcouple into the total spin that stays unchanged duringthe scattering process. However, separation of the spa-tial part of the total scattering wavefunction from thespin part is a non-trivial task.Here it is more convenient to adopt a coordinate sys-

tem, where r0, r1, and r2 denote the positions of thepositron, electrons 1 and 2, respectively (see Fig. 2), rel-ative to the helium nucleus, while R = (r0 + r1)/2 isthe position of the Ps centre relative to the He nucleusand ρ = r0−r1 is the relative coordinate of the positronand electron 1. Fig. 2 depicts one of the two possiblesystems of coordinates (r0, r1, r2) and (R,ρ, r2). There

12

are two sets of Jacobi coordinates corresponding to thetwo electrons that can form positronium. When neces-sary we will refer to them explicitly as (R1,ρ1, r2) and(R2,ρ2, r1). Fig. 2 shows the one where Ps is formed byelectron 1.

electron 1

electron 2

positron

He-nucleus

r1

r2

r0

R

ρ

FIG. 2. Coordinate systems for positron-helium scattering,where we assume that positronium is formed with electron 1.In this case the generic R and ρ become R1 and ρ1. For thecase where positronium is formed with electron 2 we use R2

and ρ2.

The Schrodinger equation for the total scattering wave-function Ψ of the positron-helium system with the totalenergy E is written as

(H − E)ΨsSM (x0,x1,x2) = 0, (37)

where x0,x1 and x2 are the full sets of coordinates ofthe particles including their spin. When spin-orbit inter-actions are neglected it should be possible (at this stagethis is merely an assumption; we will see later if this isindeed the case) to separate the radial and spin partsaccording to

ΨsSM (x0,x1,x2) = ΦsS(r0, r1, r2)χsSM (0, (1, 2)), (38)

where ΦsS(r0, r1, r2) is the wavefunction that depends onspatial coordinates and χsSM (0, (1, 2)) is the spin stateof the two electrons (with combined two-particle spin s)and the positron with the total spin S and its projectionM . We expand the total wavefunction ΨsSM in termsof states of all asymptotic channels and require the re-sult to be antisymmetric against permutation of the twoelectrons:

ΨsSM ≈ 1√2(1− P12)

[ NHe∑

α=1

F (s)α (r0)ψα(r1, r2)

× χsSM

(0, (1, 2))

+∑

s′

NPs∑

β=1

G(s′)β (R1)ψβ(ρ1)φ1s(r2)

× χs′SM ((0, 1), 2)], (39)

where ψα are the helium wavefunctions and ψβ are the Ps

ones. Expansion coefficients F(s)α andG

(s′)β depend on the

spins s and s′ of He and Ps, respectively. The NHe andNPs are the numbers of the He and Ps pseudostates. Thesecond term represents Ps formation by both electronsand the sum over s′ corresponds to Ps formation in para(s′ = 0) and ortho (s′ = 1) states. The wavefunction φ1sdescribes the residual ion of He+. The latter is consideredto be in the 1s state only. The operator 1√

2(1 − P12),

where P12 is a permutation operator that interchangesthe electrons 1 and 2, insures that the wavefunction isanti-symmetric. The spin wavefunctions are written as[125]

χsSM (0, (1, 2)) =∑

µ0,µ

CSM12µ0 sµ

χ 12µ0(0)χsµ(1, 2) (40)

for the e+ −He channel, and

χs′SM ((0, 1), 2) =∑

µ2,µ′

CSM12µ2 s′µ′χs′µ′(0, 1)χ 1

2µ2(2)

(41)

for the Ps−He+ channel.As the spin-orbit interactions are neglected, the initial

spin of the target s, the total spin S and consequentlythe total spin function must be conserved. Therefore thespatial part in Eq. (38) can be written as

ΦsS(r0, r1, r2) = 〈χsSM (0, (1, 2)) |ΨsSM 〉. (42)

Spin algebra to find the right-hand side of Eq. (42) wasperform by Utamuratov et al. [126]. They showed thatthe spin wavefunctions satisfy the following relations

χsSM (0, (2, 1)) = (−1)s+1χsSM (0, (1, 2)), (43)

χs′SM ((0, 1), 2) =∑

s′′

cs′s′′Sχs′′SM (0, (1, 2)) (44)

and

χs′SM ((0, 2), 1) =∑

s′′

(−1)s′′+1cs′s′′Sχs′′SM (0, (1, 2)),

(45)

where the overlap coefficients are given by the 6j-symbol

cs′s′′S = (−1)S−12 [s′s′′]

{1/2 1/2 s′

S 1/2 s′′

}. (46)

Taking into account Eqs. (43)-(45) and writing the sumover s′ explicitly (the Ps spin s′ can be 0 or 1) we have

ΦsS ≈NHe∑

α=1

F (s)α (r0)ψ

sα(r1, r2)

+1√2

NPs∑

β=1

{G(sS)β (R1)ψβ(ρ1)φ1s(r2)

+(−1)sG(sS)β (R2)ψβ(ρ2)φ1s(r1)}, (47)

13

where

ψsα(r1, r2) =1√2{ψα(r1, r2) + (−1)sψα(r2, r1)} (48)

is the antisymmetrized helium wavefunction and

G(sS)β (R) = c0sSG

(0)β (R) + c1sSG

(1)β (R). (49)

We emphasize that the superscript s of Gβ is the spinof the target, while that of Gβ is the spin of the formedpositronium. Note that

G(1)β =

√3G

(0)β . (50)

This relationship is often assumed and used in the liter-ature, however, its origin remained unclear. It was rig-orously derived for the first time by Utamuratov et al.[126].The anti-symmetrised wavefunction ψsα(r1, r2) for he-

lium is built using the configuration interaction (CI) ap-proach [226]. Two types of approximations are used: afrozen core (FC), where one of the electrons is describedby the He+ 1s orbital and a multi-core (MC), where thecore electron is described by any number of orbitals asnecessary to yield as accurate He wave functions as de-sired. However, as mentioned earlier the residual ion inthe Ps-He+ channels is considered to be always in the1s-state.Eqs. (39-47) show that the spin part of the total wave-

function is factorized and, therefore, can be removed fromthe equations. Consequently, for a given total spin S andtarget spin s, the Schrodinger equation (37) transformsto an equation for the spatial part of the total wavefunc-tion ΦsS :

(H − E)ΦsS(r0, r1, r2) = 0, (51)

where ΦsS is given by Eq. (47).Direct scattering and Ps-formation matrix elements

have been given by Utamuratov et al. [125]. Due tothe two-centre expansion the system of equations (19) ishighly ill-conditioned. The ill-conditioning makes it im-possible to use arbitrarily high basis sizes, and requiresthe matrix elements to be calculated to high precision.

C. Alkali metals

We model an alkali atom as a system with one activeelectron above a frozen Hartree-Fock core [227]. Accord-ingly, the positron-alkali collision is treated as a three-body system of the incoming positron, the active (outershell) electron and an inert core ion. The interactionbetween the active electron and the inert core Vα is cal-culated as a static part of the Hartree-Fock potential Vstsupplemented by an exchange potential Vex between theactive and the core electrons. Thus the interactions inthis model system are the electron-ion Vα, positron-ion

Ve, and electron-positron Vβ potentials defined as follows

Vα(r) = Vst(r) + Vex(r) + Vpol(r), (52)

Ve(r) = −Vst(r) + Vpol(r), (53)

Vβ(r) = −1/r. (54)

The static term is calculated by

Vst(r) = −Zr+ 2

∑

ψj∈C

∫d3r′

|ψj(r′)|2|r − r′| , (55)

where Z is the charge of the target nucleus and ψj arethe states of the ion core C generated by performingthe self-consistent-field Hartree-Fock calculations [227].The summation in Eq. (55) is done for all core states.The equivalent local-exchange approximation [228–230]is used to take into account the exchange between theactive electron and core electrons:

Vex(r, Eex) =1

2

[(Eex − Vst(r))

−√(Eex − Vst(r))2 + ρ(r)

], (56)

where

ρ(r) =∑

ψj∈C

∫dr|ψj(r)|2 (57)

and Eex is an adjustment parameter. The core polariza-tion potential Vpol in Eq. (52) is

Vpol(r) = − αd2r40

· 1− exp[−x6]x4

, (58)

where x = r/r0, αd is the dipole polarizability and r0are adjustable parameters to fit some physical quantities(e.g., energies of the valence electron).

D. Magnesium and inert gases

Mg is modelled as a He-like system with two activeelectrons above a frozen Hartree-Fock core [155, 231,232]. The interaction between an active electron andthe frozen Hartree-Fock core is calculated as a sum ofthe static part of the Hartree-Fock potential and an ex-change potential between an active and the core electronsas described in the previous subsection. Details of thetwo-centre CCC method for positron-Mg collisions aregiven in [156].Wavefunctions for the inert gases of Ne, Ar, Kr, and

Xe are described by a model of six p-electrons above afrozen Hartree-Fock core. Discrete and continuum tar-get states are obtained by allowing one-electron excita-tions from the p-shell in the following way. Taking Neas an example, self-consistent Hartree-Fock calculationsare performed for the Ne+ ion, resulting in the 1s,2s,2porbitals. The 1s and 2s orbitals are treated as the inertcore orbitals, while the 2p Hartree-Fock orbital is used asthe frozen-core orbital to form the target states. A set of

14

Laguerre functions is used to diagonalize the quasi-one-electron Hamiltonian of the Ne5+ ion, utilising the non-local Hartree-Fock potential constructed from the inertcore orbitals. The resulting 2p orbital differs substan-tially from the Hartree-Fock 2p orbital. A one-electronbasis suitable for the description of a neutral Ne atomis built by replacing the 2p orbital from diagonalisationwith the Hartree-Fock one, orthogonalized by the Gram-Schmidt procedure. The six-electron target states are de-scribed via the configuration-interaction (CI) expansion.The set of configurations is built by angular momentumcoupling of the wavefunction of 2p5 electrons and theLaguerre-based one-electron functions. The coefficientsof the CI expansion are obtained by diagonalization ofthe target Hamiltonian. The target orbital angular mo-mentum l, spin s, and parity π are conserved quantumnumbers and diagonalization of the target Hamiltonian isperformed separately for each target symmetry {l, s, π}.Full details of the single centre CCC calculations for no-ble gas atoms are given in Ref. [163]. A two-centre ap-proach to inert gases has not yet been attempted.

E. Molecular hydrogen

Positron-H2 scattering can be treated somewhat simi-lar to the helium case. We consider H2 within the Born-Oppenheimer approximation where the two protons areconsidered to be at a fixed internuclear distance denotedas d. Expansion for the total scattering wavefunction(after separation of the spin part) is similar to Eq. (47)for He. However, the wavefunctions for the target in thefirst term and the residual ion in the Ps-formation chan-nel depend on d. The residual ion of H+

2 , with sameinternuclear distance d, is considered to be only in itsground state. We only use a few Ps eigenstates so asto take advantage of their analytical form. The targetstates are obtained by diagonalizing the H2 Hamiltonianin a set of antisymmetrized two-electron configurations,built from Laguerre one-electron orbitals, for each tar-get symmetry characterized by the projection of orbitalangular momentum, parity and spin. To calculate H2

states, we use the fixed-nuclei approximation. Calcula-tions are performed at the ground-state equilibrium in-ternuclei distance taken to be d = 1.4 a0. When d is setto 0 one should obtain the He results. We used this testfor both structure and scattering calculations. Details ofH2 structure calculations can be found in [196–198].The derivation of the rearrangement matrix elements

are somewhat more difficult than for He because of theirdependence on the nuclear separation and target orienta-tion. Another difference is that partial wave expansion isdone over the total angular momentum projection K. Itis convenient to choose the z-axis to be along the d vec-tor (body-frame). Then it is possible to transform theobtained results with this choice of z-axis to any givenorientation of the molecule. To facilitate the calcula-tions only the spherical part of the nuclear potential is

considered when calculating the rearrangement matrixelements:

Vp(r0,d) =1

|r0 − d/2| +1

|r0 + d/2| ≈2

r>, (59)

where r> = max{r0, d/2}. Then the momentum spacerepresentation of the above positron-nucleus potentialcan be shown to be

Vp(p) =4π2sin(dp)

dp3. (60)

With these one further follows the procedure used forpositron-He calculations [125].For positron collisions with the ground state of H2 only

states with zero total spin are required and so S = 1/2.T -matrix elements are used for obtaining body-framescattering amplitudes. These are then rotated by Eu-ler angles to transform them to lab-frame scattering am-plitudes. Orientationally-independent cross sections arecalculated by averaging over all rotations of the molecule[233]. An orientationally averaged analytic Born sub-traction method [233] is employed for H2 direct transi-tion channels. This helped reduce the number of partialwaves requiring explicit calculation.

III. RECENT APPLICATIONS OF THE CCC

THEORY TO POSITRON SCATTERING

A. Internal consistency

Fundamentally, in order for a theory to be useful itneeds to be predictive. In the close-coupling approach toelectron/positron/proton scattering on relatively simpletargets, where the structure is readily obtained, there aretwo computational problems that need to be overcome.The first, is that for a given set of states used to expandthe total wavefunction the resulting equations need to besolved to an acceptable numerical precision. The secondis to systematically increase the size of the expansion anddemonstrate that the final results converge to a uniqueanswer that is independent of the choice of the expan-sion, so long as it is sufficiently large. Only once this isachieved can we be in a position to claim that the resultsare the true solution of the underlying Schrodinger equa-tion and hence predictive of what should happen in theexperiment.In the case of electron scattering there are only one-

centre expansions because electrons do not form boundstates with the electrons of the target. Electron exchangeis handled within the potential matrix elements all basedon the coordinate origin at the nucleus. Accordingly, es-tablishing convergence in just the one-centre approach isall that needs to be done, though historically this was amajor challenge [37, 68]. Though convergence was shownto be to a result that disagreed with experiment [68],subsequent experiments showed excellent agreement withthe CCC theory [234, 235], which lead to reanalysis of

15

the original data [236] yielding good agreement with theCCC theory and new experiments. A similar situationoccurred in the case of double photoionisation of he-lium [237, 238], which in effect is electron scattering onthe singly charged helium ion [239, 240].

For positron and proton scattering the issue of con-vergence is even more interesting due to the capacity ofthe projectile to form a bound state with a target elec-tron. This leads to a second natural centre in the prob-lem which also requires treatment to convergence. Forpositron scattering on atomic hydrogen the Ps-formationthreshold is at 6.8 eV, while the ionisation threshold is13.6 eV. However, Ps formation is also a form of ion-isation of the target except that the electron is cap-tured to a bound state of the projectile. Any expan-sion of the Ps centre using a complete basis will resultin negative- and positive-energy states, with the lattercorresponding to three-body breakup. However, expan-sion of the atomic centre will also generate indepen-dent positive-energy states corresponding to three-bodybreakup. Hence, expansions using a complete basis oneach centre, will yield independent, non-orthogonal statescorresponding to the same physical three-body breakupprocess. While this may appear to be a fundamentalproblem, in practice it is an interesting strength of themethod which allows to check internal consistency of theresults. By this we mean that the same results must beobtained from a variety of calculations utilising indepen-dent one- and two-centre expansions as detailed below.

We begin by considering two extremes: the first at-tempts to obtain convergence using only the atomic cen-tre, while the second attempts convergence using twocomplete expansion on both centres. Will either con-verge, and if they do, will the convergence be to the sameresult?

In Fig. 3 typical energies arising in two-centre calcula-tions are given. We see a similar spread of negative- andpositive-energy states on both the H and the Ps centres.A single centre expansion based on the atomic centrewould not have any Ps states included in the calculation.

The results of the two types of calculations may bereadily summarised by Fig. 4. On the left we have one-

centre cross sections σ(1)fi , where i, f = 1, . . . , N

(1)H . Tak-

ing the initial state to be the ground state of H (i = 1)

then σ(1)11 is the elastic scattering cross section, and σ

(1)f1

corresponds to excitation whenever εf1 < 0 and ioni-sation whenever εf1 > 0. The elastic and excitation

cross sections need to converge with increasing N(1)H in-

dividually. However, the ionisation cross sections con-verge as a sum, yielding the total ionisation cross section

σ(1)ion =

∑f σ

(1)f1 for ε

(1)f > 0.

Convergence of the one-centre CCC calculations,where there are no Ps states, has been studied exten-sively [70, 121, 122]. Briefly, at energies below the Ps-formation threshold the important contribution of vir-tual Ps formation is adequately treated via the positive-energy atomic states of large angular momentum lmax ≈

0.1

1

10

100

1000

H(S) Ps(S) H(P) Ps(P) H(D) Ps(D) H(F)

-0.01

-0.1

-1

-10

-100H(S) Ps(S) H(P) Ps(P) H(D) Ps(D) H(F)

Energy levels (eV)

FIG. 3. Energies ε(H)nl and ε

(Ps)nl , arising upon diagonalisation

of the respective Hamiltonians, in the CCC(123,122) positron-hydrogen calculations. Here 12 − l states were obtained foreach l with lmax = 3 for H states and lmax = 2 for Ps states,see Kadyrov et al. [211].

10. This allows for convergence of elastic scattering crosssection to the correct value. At energies above the ioni-sation threshold, the positive-energy atomic pseudostatestake into account both breakup and Ps-formation crosssection in a collective way yielding the correct electron-loss and excitation cross sections. However, in the ex-tended Ore gap region between the Ps-formation andionisation thresholds no convergence is possible due toall positive-energy pseudostates being closed.

Convergence in two-centre calculations is potentiallyproblematic at all energies due to two independent treat-ments of the breakup processes. In practice this man-ifests itself as an ill-conditioned system which requireshigh-precision matrix elements and limits the size of the

calculations i.e. N(2)H and N

(2)Ps are typically substantially

smaller than N(1)H . It is for this reason that we have

drawn the one-center matrix to be substantially largerin Fig. 4. Furthermore, the H-Ps matrix elements takeat least an order of magnitude longer to calculate due tothe non-separable nature of the radial integrals [74]. Ac-cordingly, even with much smaller number of states (withsmaller lmax) the two-centre calculations take consider-ably longer to complete. Nevertheless convergence is ob-tained for individual transitions involving discrete states,

16

N N N Ps

NNPs

ε=0 ε=0 ε=0

ε=0

Nε=0

ε=0

H

H

H

i i

f f

σfi

(2)σfi

(1) H(1) (2) (2)

(2)

(2)

(1)

FIG. 4. Matrix overview of the cross sections σfi arisingfrom one-centre (left) and two-centre (right) CCC positron-

hydrogen calculations. N(1)H is the number of H states in the

one-centre calculation which has no explicit Ps states. N(2)H

is the number of H states in the two-centre calculation withN

(2)Ps explicit Ps states. First presented by Bray et al. [241].

explicit Ps formation, and explicit breakup [218, 241].Internal consistency is satisfied if at energies outside theextended Ore gap, for discrete (εAf , ε

Ai < 0) atomic tran-

sitions the two approaches independently converge suchthat

σ(2)fi = σ

(1)fi . (61)

Furthermore, at energies above the breakup threshold

σ(2)eloss = σ

(2)Ps + σ

(2)brk

= σ(1)ion, (62)

where for some initial state i

σ(2)Ps =

∑

f :εPsf<0

σ(2)fi , (63)

σ(2)brk =

∑

f :εPsf>0

σ(2)fi +

∑

f :εAf>0

σ(2)fi , (64)

σ(1)ion =

∑

f :εAf>0

σ(1)fi . (65)

In the extended Ore gap only the two-centre calculationsare able to yield convergent results.The great strength of the internal-consistency check is

that it is available (outside the extended Ore gap) forevery partial wave of the total orbital angular momen-tum. Checking that Eq. (62) is satisfied for every partialwave provides confidence in the overall results of the twocompletely independent calculations, which will typicallyhave very different convergence properties with increas-ing N and l. Due to the unitarity of the close-couplingtheory agreement for Eq. (62) suggests agreement forother channels, and so Eq. (61) will also hold.In Fig. 5 we give the example of an internal-consistency

check, presented by Bray et al. [241]. We see that outsidethe extended Ore gap the two calculations are generally

0.0

1.0

2.0

3.0

4.0

5.0

6.0

1 10 100

L=0 Total CCC(309,0) CCC(202,202)

Ps-formation

0.00

0.05

0.10

0.15

0.20

10 100

electron-loss CCC(309,0) CCC(202,202)

Ps-continuum

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

1 10 100

L=1

0.0

0.5

1.0

1.5

2.0

10 100

0.0

1.0

2.0

3.0

4.0

5.0

6.0

1 10 100

L=2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

10 100

0.0

1.0

2.0

3.0

4.0

5.0

1 10 100cross section (a.u.)

L=3

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

10 100

0.0

0.5

1.0

1.5

2.0

2.5

3.0

1 10 100

L=4

0.0

0.5

1.0

1.5

2.0

10 100

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1 10 100

L=5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

10 100

0

5

10

15

20

1 10 100

all L

0

2

4

6

8

10

12

10 100

positron energy (eV)

FIG. 5. Total and electron-loss (Ps formation plus breakup)cross sections for positron scattering on atomic hydrogen forspecified partial waves L obtained using the one- and two-centre CCC calculations, see text. The indicated points cor-responding to the energies at which the calculations were per-formed are connected with straight lines to guide the eye.The vertical lines are the Ps-formation and breakup thresh-olds, spanning the extended Ore gap. The experimental datain the bottom left panel are due to Zhou et al. [178]. Firstpresented by Bray et al. [241].

17

in very good agreement. One systematic exception is atjust above the ionisation threshold. Here the breakupcross section is almost zero, but the Ps-formation cross

section is near its maximum. Even with N(1)H = 30 the

one-centre calculation does not have enough pseudostatesof energy just a little above zero which would be neces-sary to reproduce the what should be step-function be-haviour in one-centre CCC calculations. Due to explicitPs-formation in the two-centre calculations there are nosuch problems here or within the extended Ore gap. Hav-ing checked the individual partial waves, and summingover all to convergence, excellent agreement is found withexperiment. Having performed the internal consistencychecks we remain confident in the theoretical results evenif there is potentially a discrepancy with experiment atthe lowest energy measured.

B. Atomic hydrogen

Establishing convergence in the cross sections with asystematically increasing close-coupling expansion placesa severe test on the scattering formalism. This is as rele-vant to positron scattering as it is for electron scattering.Pseudoresonances must disappear with increased size ofthe calculations, and uncertainty in the final results canbe established via the convergence study.One of the earliest successes of the two-centre CCC

method for positron-hydrogen S-wave scattering was toshow how the Higgins-Burke pseudoresonance [62] disap-peared utilising a (N,M) basis of only s-states on eachcentre [73]. The cross sections for all reaction channelswere shown to converge to a few % with a (16, 16) ba-sis of s-states. Interestingly, the symmetric treatment ofboth centres was particularly efficient in terms of reach-ing convergence and eliminating pseudoresonances, withno double-counting problems.The question of convergence in the case of the full

positron-hydrogen scattering problem was investigatedby Kadyrov and Bray [74]. Setting M = N , states ofhigher angular momentum were increased systematically.The same level of convergence as in the S-wave modelcase was achieved with the (34, 34) basis made of ten s-,nine p-, eight d- and seven f -states for each centre, forscattering on the ground state. The largest calculationsperformed had a total of 68 states, 34 each of H and Psstates. The convergence was checked for the total andother main cross sections corresponding to transitionsto negative-energy states. Reasonably smooth cross sec-tions were obtained for all bases with l-convergence beingrather rapid. For the three cases considered f -states con-tribute only marginally.Figs. 6-8 show the CCC results in comparison with

other calculations and experimental data of Detroit [178]and London [76] groups. The CCC results agree well withexperiment. So do CC(30, 3) calculations of Kernoghanet al. [71] and CC(28, 3) calculations of Mitroy [72]. Notethat in these calculations the n−3 scaling rule was used

0

1

2

3

4

5

6

0 20 40 60 80 100

cros

s se

ctio

n (π

a 02 )

positron incident energy (eV)

CCCKernoghan et al 96Kernoghan et al 95

Mitroy 96Zhou et al 97

FIG. 6. Total cross section for e++ H scattering. The experi-mental results are due to Zhou et al. [178]. The CCC result isfrom [74]. The other theoretical results are due to Kernoghanet al. [65, 71] and Mitroy [72].

0

1

2

3

0 20 40 60 80 100

cros

s se

ctio

n (π

a 02 )

positron incident energy (eV)

CCCKernoghan et al 96Kernoghan et al 95

Mitroy 96Zhou et al 97

Zhou et al 97 LL

FIG. 7. Total Ps-formation cross section for e++ H scattering.The experimental results are due to Zhou et al. [178] (LLindicated lower limit). The CCC result is from [74]. Theother theoretical results are due to Kernoghan et al. [65, 71]and Mitroy [72].

to estimate the total Ps formation. Also, an energy-averaging procedure was used to smooth the CC(9, 9)calculations of Kernoghan et al. [65]. In the CCC methodconvergence is established without such procedures beingused.For the breakup cross section the CCC results have

two comparable contributions, one from the excitation

18

0

0.5

1

0 20 40 60 80 100

cros

s se

ctio

n (π

a 02 )

positron incident energy (eV)

CCCCCC: direct ioniz.

Kernoghan et al 96Kernoghan et al 95

Mitroy 96Jones et al 93

FIG. 8. Total breakup cross section for e++ H scattering.The experimental results are due to Jones et al. [76]. TheCCC result is from [74]. The other theoretical results are dueto Kernoghan et al. [65, 71] and Mitroy [72].

of the positive-energy H pseudostates (shown in Fig. 8as direct ionization), and the other from excitation ofpositive-energy Ps pseudostates. This was also noted byKernoghan et al. [65] using the CC(9, 9) calculations.By contrast, in CC(N,M)-type calculations the contri-bution to breakup comes only from from direct ionisa-tion. At the maximum of the cross section the sep-arately converged indirect contribution to the breakupcross section is approximately a third of the total. How-ever, the CC(30, 3) cross section of Kernoghan et al. [71]is only a marginally smaller, indicating that absence of Pspositive-energy states is absorbed by the positive-energyH states.

Fig. 9 shows the CCC results of Fig. 8, but againstexcess (total) energy to emphasize the lower energies.The full CCC(H+Ps) results with breakup cross sec-tions coming from both the H and Ps centres are con-trasted with those just from H and twice H (labeled asCCC(H+H)). We see that below about 20 eV excess en-ergy the CCC(H+Ps) and CCC(H+H) curves are muchthe same, indicating that the Ps and H contributions tobreakup converge to each other as the threshold is ap-proached.

Utilising the CCC method Kadyrov et al. [217] re-ported calculations of positron-hydrogen scattering nearthe breakup threshold in order to examine the thresholdlaw. The results are given in Fig. 10. The Wannier-like threshold law, derived by Ihra et al. [242], is ingood agreement with the CCC results below 1 eV ex-cess energy. This law was derived for the L = 0 par-tial wave, and Rost and Heller [243] showed the sameenergy-dependence holds in all partial waves. As for the

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1 10 100

cross section (units of

πa02)

excess energy E (eV)

total breakup CCC(H+Ps) CCC(H) CCC(H+H) Jones et al.

FIG. 9. Total e+−H breakup cross section as a function ofexcess energy calculated using the two-center CCC method,first presented by Kadyrov et al. [217]. The argument to theCCC label indicates which center’s positive-energy states wereused, see text. The experiment is due to Jones et al. [76].

full problem the contributions from both centres to thebreakup cross section converge to each other with de-creasing excess energy, without any over-completenessproblems.

10-5

10-4

10-3

10-2

10-1

1 10 100

cross section (units of

πa2 0)

excess energy E (eV)

∝ E2.64exp(-0.73√E) (Ihra et al) CCC(H+Ps)

CCC(Ps+Ps)

FIG. 10. Total e+−H S-wave model breakup cross sectionas a function of excess energy calculated using the two-centerCCC method, from Kadyrov et al. [217]. As in Fig. 9, theargument to the CCC label indicates which center’s positive-energy states were used. The Wannier-like threshold law isdue to Ihra et al. [242]

.

C. Helium

Helium in its ground state is the most frequently usedtarget in experimental studies of positron-atom scatter-ing. First measurements on positron-helium scattering

19

were carried out by Canter et al. [91] in 1972. Sincethen many other experimental studies have been con-ducted [84, 86, 101, 112–114, 244, 245]. Further devel-opments of positron beams in terms of energy resolutionand beam intensities have recently motivated more ex-perimental studies [20, 110, 246–250]. In general, theresults from the experiments agree well with each other.A complete theoretical approach from low to high ener-gies had been lacking until the development of the CCCmethod for the problem by Utamuratov et al. [125].

0.0

0.4

0.8

1.2

0 20 40 60 80 100 120 140

cros

s se

ctio

n (u

nits

of

10-1

6 cm

2 )

incident energy (eV)

total cross section Sullivan et al. Caradonna et al. Stein et al. Kauppila et al. Campbell et al. Wu et al. FC CCC MC CCC

FIG. 11. Total positron-helium scattering cross section. Ex-perimental data are due to Stein et al. [101], Sullivan et al.[20], Kauppila et al. [112], and Caradonna et al. [250]. Thecalculations are due to Campbell et al. [111] and Wu et al.[122]. The FC CCC and MC CCC results are from Utamura-tov et al. [125].

A vast amount of experimental data is available for in-tegrated cross sections for positron scattering from thehelium ground state. The total CCC-calculated crosssection is shown in Fig. 11 in comparison with experi-mental data and other calculations. Considerable discus-sion on the topic has been presented by Utamuratov et al.[125]. It suffices to say that so long as an accurate groundstate is used, obtained from a multi-core (MC) treat-ment, agreement with experiment is outstanding acrossall energies. We expect the frozen-core (FC) treatmentof helium to result in systematically larger excitation andionization cross sections because it understimate the ion-ization potential by around 0.84 eV. Generally, a largerionisation potential leads to a smaller cross section. Inthe CCC method we are not free to replace calculatedenergies with those from experiment, as this leads to nu-merical inconsistency. Consequently, there is no way toavoid the extra complexity associated with the MC cal-culations if high accuracy is required.

The total Ps-formation, breakup and electron-losscross sections are given in Fig. 12(a), (b) and (c), respec-tively. Beginning with Ps formation, given the minor

0.0

0.2

0.4

0.6

0.8

20 25 30 40 50 75 100 150

incident energy (eV)

electron loss

(c)

Fromme et al. Murtagh et al. Campbell et al. Wu et al. FC CCC MC CCC

0.0

0.2

0.4

0.6

cros

s se

ctio

n (

units

of

10-1

6 cm

2 )

breakup

(b)

Jacobsen et al. Murtagh et al. Campbell et al. FC CCC MC CCC FC CCC:Ps FC CCC:He

0.0

0.2

0.4

0.6

Ps-formation (a) Caradonna et al.

Fornari et al. Diana et al. Murtagh et al. Campbell et al. FC CCC MC CCC

FIG. 12. Positron-helium (a) Ps-formation, (b) breakup, and(c) electron-loss (sum of the other two) cross sections. Themeasurements are due to Caradonna et al. [250], Fornari et al.[113], Diana et al. [114], Murtagh et al. [246], Fromme et al.[84] and Knudsen et al. [85]. The calculations are due toCampbell et al. [111], Wu et al. [122]. The FC CCC and MCCCC results are from [125].

variation in the measurements agreement between thevarious theories and experiment is satisfactory. However,turning our attention to the breakup cross section we seethat (MC) CCC appears to be substantially higher thanexperiment. Yet when these cross sections are summed toform the electron-loss cross section, the agreement withthe experiment of Fromme et al. [84] which measured this

20

directly, is good. Given the complexity of the problemand the experimental uncertainties, the agreement withexperiment is very satisfying.

0.00

0.02

0.04

0.06

0.08

0.10

20 25 30 35 40

incident energy (eV)

cros

s se

ctio

n (

units

of

10-1

6 cm

2 )

21S

(b)

Caradonna et al. Hewitt et al. Adhikari and Ghosh Iragashi et. al. FC CCC MC CCC

0.02

0.04

0.06

0.08

0.10

0.12

0.14

21P

(a)

Caradonna et al. Hewitt et al. Adhikari and Ghosh. Iragashi et. al. Campbell et. al. FC CCC MC CCC

FIG. 13. Integrated cross sections for (a) He(21S) and (b)He(21P) excitations by positron impact. Experiment is dueto Caradonna et al. [250]. The FC CCC and MC CCC resultsare from [125], other calculations are due to Hewitt et al. [116],Adhikari and Ghosh [251], Igarashi et al. [120] and Campbellet al. [111].

The cross sections of 21S and 21P excitation of heliumare presented in Fig. 13a and b, respectively. The MCCCC result for 21S is in good agreement with the data ofCaradonna et al. [250] while the 21P result is somewhatlower than experiment. The fact that the FC CCC 21Presults agree better with the experimental data is for-tuitous. Other available theories show some systematicdifficulties for these relatively small cross sections.The cross sections for the rather exotic Ps formation

in the 2s and 2p excited states are presented in Fig. 14(a)and (b) respectively. The cross sections are particularlysmall. Nevertheless, agreement with the sole availableexperiment of Murtagh et al. [247] is remarkable.Thus we have seen that there is good agreement be-

tween theory and experiment for the integrated cross sec-tions for positron scattering on helium and hydrogen intheir ground states. In both cases the ionisation thresh-olds are well above 6.8 eV, and so a one-centre calculationis applicable at low energies where elastic scattering is the

0.00

0.01

0.02

0.03

20 40 60 80 100 120

incident energy (eV)

cros

s se

ctio

n (

units

of

10-1

6 cm

2 )

Ps(2p) - excitation

(b)

Murtagh et. al.:2p Campbell et. al.:2p Iragashi et. al.:2p FC CCC:2p MC CCC:2p

0.00

0.01

0.02

0.03 Ps(2s) - excitation

(a)

Iragashi et. al.:2s FC CCC:2s MC CCC:2s

FIG. 14. Integrated cross sections for Ps formation in the (a)2s and (b) 2p states. Experimental data for Ps(2p) are due toMurtagh et al. [247]. The FC CCC and MC CCC results arefrom [125] and other calculations are due to Campbell et al.[111] and Igarashi et al. [120].

only open channel. Though experimentally challengingpositron scattering on either H or He metastable excitedstates results in Ps formation being an open channel atall energies. Taking the example of positron scatteringfrom the 23S metastable state of helium the Ps thresh-old is negative (-2.06 eV). This collision system has beenextensively studied by Utamuratov et al. [126]. As faras we are aware, no experimental studies have been con-ducted for positron scattering on metastable states ofhelium. Given the experimental work on electron scat-tering from metastable helium [252], in a group that alsohas a positron beam, we are hopeful that in the futurethere might be experimental data available for such sys-tems. As there are still unresolved discrepancies betweentheory and experiment regarding electron scattering frommetastable states of He [253–255], using positrons insteadof electrons may assist with their resolution.

21

D. Alkali metals

Just like for H and He in metastable states, for positroncollisions with alkali-metal atoms in their ground state,both elastic and Ps(1s) formation channels are open evenat zero positron energy Accordingly, we require two-centre expansions even at the lowest incident energies.Lugovskoy et al. [136] conducted two-centre calculationswith different basis sets to achieve results that are inde-pendent of the Laguerre exponential fall-off parameter λl,and convergent with the Laguerre basis size Nl = N0 − lfor target orbital angular momentum l ≤ lmax.

CCC: tr. basisCCC: spd basisMcAlinden et al

Le et al

Surdutovic et al: exp 2Surdutovic et al: exp 1

impact energy (eV)

cros

sse

ctio

n(1

0−16

cm2)

10110010−110−2

40

30

20

10

0

FIG. 15. Total positronium-formation cross section for e+-Li along with the experimental points by Surdutovich et al.[137] and theoretical calculations by McAlinden et al. [130], Leet al. [135]. The truncated basis (CCC tr) calculation is anattempt to reproduce the states used by McAlinden et al.[130], see Lugovskoy et al. [136] for details.

Figure 15 shows the positronium-formation cross sec-tion. Agreement between the various calculations is quitegood, while comparison with the experimental data ofSurdutovich et al. [137] is rather mixed. The key featureis that the Ps-formation cross section diminishes with de-creasing energy, supported by all theories, and consistentwith experiment. Overall, it appears there is no majorreason to be concerned.Unfortunately, changing the target to sodium, substan-

tial discrepancies between theory and experiment arise,and remain unresolved to date. One of the motivationsfor extending the CCC theory to two-centre calculationsof positron-alkali scattering was to address this prob-lem. Lugovskoy et al. [147] performed the most extensivestudy of this problem that included one- and two-centrecalculations. Despite establishing convergence and con-sistency of the two approaches no improvement on pre-vious calculations was found.We begin by considering the total cross section for

positron-sodium scattering, presented in Fig. 16. Wesee good agreement between various two-centre calcula-tions, with the one-centre CCC(217,0) calculation behav-ing as expected: agreeing with CCC(116,14) only above

CCC(217,0)

CCC(116,14)

Campbell et al

Ryzhikh-Mitroy

Kauppila et al

Kwan et al

impact energy (eV)

cross

section(10−16cm

2)

INa

100101

140

120

100

80

60

40

20

0

FIG. 16. Total cross section for positron-sodium scattering.The two- and one-centre CCC results of Lugovskoy et al. [147]are compared with the calculations of Ryzhikh and Mitroy[145] and Campbell et al. [111]. The experimental points aredue to Kwan et al. [141] and Kauppila et al. [142].

CCC(116,14)

Campbell et al

Ryzhikh & Mitroy

Le et al

Surdutovich et al

Zhou et al

impact energy (eV)

cross

section(10−16cm

2)

INa

1010.1

100

80

60

40

20

0

FIG. 17. Total positronium-formation cross section in e+-Na scattering. Experiment is due to Surdutovich et al. [137],Zhou et al. [143], and the calculations due to Le et al. [135],Ryzhikh and Mitroy [145] and Lugovskoy et al. [147].

the ionisation threshold, and not being valid (or evenconvergent) below the ionisation threshold. All of thetwo-centre calculations are considerably above the exper-iment at low energies.

Curiously, the situation is reversed for the total Ps-formation component of the total cross section, presentedin Fig. 17, where now all of the theories are considerablybelow the experiment. Given that the total cross sectionat the lowest energies considered is the sum of elastic andPs-formation cross sections, the presented discrepancieswith experiment imply that the theoretical elastic scat-tering cross sections are overwhelmingly high [147]. Whythis would be the case remains a mystery.

Faced with the problems identified above, Lugovskoyet al. [148] considered threshold behaviour in positronscattering on alkali atoms. Following Wigner [149] we

22

10-5

10-4

10-3

10-2

10-1

10-5 10-4 10-3 10-2 10-1 10+0

cross section (a.u.)

k0σPs CCC(254,1) CCC(254,2) CCC(258,1)

10-1

10+0

10+1

10+2

10+3

10-5 10-4 10-3 10-2 10-1 10+0

cross section (a.u.)

positron energy (eV)

e+-Li(2S)

σel CCC(254,1) CCC(254,2) CCC(258,1)

FIG. 18. Elastic σel (lower panel) and scaled Ps-formationk0 σPs (upper panel) cross sections for positron-lithium scat-tering, as a function of energy (13.6k2

0), for the zeroth par-

tial wave calculated with the indicated CCC(N(Li)lmax

, N(Ps)lmax

)Laguerre basis parameters, as presented by Lugovskoy et al.[148].

expect exothermic reactions such as Ps formation inpositron-alkali scattering to result in a cross section thattends to infinity as 1/k0 as the positron energy (k20) goesto zero. This is not yet evident in Fig. 15 or Fig. 17 for theconsidered energies. Nevertheless, Lugovskoy et al. [148]did obtain the required analytical behaviour, but only inthe zeroth partial wave. In Fig. 18 the Ps-formation crosssection is presented for the zeroth partial wave multipliedby k0 so as to demonstrate the expected threshold be-haviour. The convergence study of Lugovskoy et al. [148]is also presented, where the effect of adding the Ps(2s)state was able to be reproduced by atomic pseudostatesof high orbital angular momentum. Lugovskoy et al. [148]found that higher partial waves J become major contrib-utors to Ps formation at energies above 10−3 eV, andhave a threshold behaviour as k2J−1

0 , and so rise rapidlywith increasing energies. It is the contributions of thehigher partial waves that is responsible for the behaviourin the Ps-formation cross section seen in Fig. 15.

We find the same is the case for positron-sodium scat-tering. In Fig. 19 we present k0σPs and σel for the zerothpartial wave, demonstrating the expected threshold be-

10-2

10-1

10+0

10-5 10-4 10-3 10-2 10-1 10+0

cross section (a.u.)

e+-Na(3S)

k0σPs CCC(251,1) CCC(251,101) CCC(258,1)

10-2

10-1

10+0

10+1

10+2

10+3

10-5 10-4 10-3 10-2 10-1 10+0cross section (a.u.)

positron energy (eV)

σel CCC(251,1) CCC(251,101) CCC(258,1)

FIG. 19. The same as Fig. 18 but for positron-sodium scat-tering.

haviour. Details of the convergence study are discussedby Lugovskoy et al. [148]. It suffices to say that thereis a range of combinations of atomic and Ps states thatshould yield convergent results, with such studies beingconsiderably easier at the lower energies where there areonly two open channels. As in the case of lithium the ze-roth partial wave is dominant only below 10−3 eV, withthe results presented in Fig. 17 being dominated by thehigher partial waves.Lastly, having been unable to explain the discrepancy

between experiment and theory for positron-sodium scat-tering Lugovskoy et al. [148] also considered the positron-potassium scattering system. However, much the samebehaviour as for the lighter alkalis was found, see Fig. 20.Consequently, the discrepancy between experiment andtheory for low-energy positron scattering on sodium re-mains unresolved.One interesting aspect of the presented elastic cross

sections for positron scattering on the alkalis are theminima. In the case of Li and Na they are at justabove 0.001 eV, whereas for K the minimum is at around0.04 eV. Given that in all cases we have just two channelsopen, elastic and Ps formation, we have no ready expla-nation for the minima, or their positions. The generallysmaller Ps-formation cross section shows no structure inthe same energy region, which is surprising given the sub-

23

10-2

10-1

10+0

10+1

10+2

10-5 10-4 10-3 10-2 10-1 10+0

cross section (a.u.)

e+-K(4S)

k0σPs CCC(254,1) CCC(254,51) CCC(258,1)

10+0

10+1

10+2

10+3

10+4

10-5 10-4 10-3 10-2 10-1 10+0

cross section (a.u.)

positron energy (eV)

σel CCC(254,1) CCC(254,51) CCC(258,1)

FIG. 20. The same as Fig. 18 but for positron-potassiumscattering.

stantial minima in the elastic cross section.

E. Magnesium and inert gases

For experimentalists the transition from say sodium tomagnesium for the purpose of positron scattering is rela-tively straightforward, not so in the case of theory. Twovalence electrons on top of a Hartree-Fock core makes fora very complicated projectile-target combination to treatcomputationally. However, with an ionization energy of7.6 eV the single-centre approach is valid below 0.8 eV al-lowing for a test of the two-centre method in this energyregion. This is an important test because in the singlecentre approach there are no approximations associatedwith explicit Ps formation, and the core is fully treatedby the Hartree-Fock approach rather than an equivalentlocal core potential approximation.One-centre positron-magnesium CCC calculations

were presented by Savage et al. [155], and two-centre onesby Utamuratov et al. [156]. The results confirmed theexistence of a low-energy shape-resonance predicted ear-lier by Mitroy et al. [154]. The results are presented inFig. 21. Given that the resonance is at a very low energyand that a slight energy difference in the target struc-ture may affect its position the agreement between the

theories is very encouraging. As explained in Sec. III A,one-centre calculations are unable to yield convergent re-sults within the extended Ore gap (presently between0.8 eV and 7.6 eV). The unphysical structures displayedwithin this energy region in the calculations of Savageet al. [155] depend on the choice of basis with just oneexample presented.Unfortunately, the agreement with the experiment of

Stein et al. [256] for the total cross section has the unex-pected feature of being good above the ionisation thresh-old and poor below, see Fig. 22. Given that the validityof the two-centre CCC approach should be energy inde-pendent we are unable to explain the discrepancy.

10

100

1000

0.01 0.1 1 10

cros

s se

ctio

n (a

.u.)

incident energy (eV)

CCC Savage et al. Mitroy et al.

FIG. 21. e+-Mg elastic-scattering cross section.The first ver-tical line is at the Ps-formation threshold, the second one isat the Mg-ionization threshold. The (two-centre) CCC calcu-lations are due to Utamuratov et al. [156], the one-centre dueto Savage et al. [155], and the variational ones due to Mitroyand Bromley [257].

The Ps-formation cross section is presented in Fig. 23,where there are large experimental uncertainties. Theexperimental data of Surdutovich et al. [258] are prelim-inary estimations for the upper and the lower limits ofthe Ps-formation cross section. The (two-centre) CCC re-sults are compared with the previous calculations. Whilethere is substantial variation the theories tend to favourthe upper limit estimates.Positron scattering on inert-gas atoms has been stud-

ied using the single-centre CCC method [163]. As dis-cussed in the introduction the complexity of adding thesecond (Ps) centre is immense, and has not yet been at-tempted within a convergent close-coupling formalism.The large ionisation thresholds mean that the single cen-tre calculations are valid for elastic scattering on the sub-stantial energy range below the Ps-formation threshold,as well as above the ionisation threshold, but no explicitPs-formation cross section may be determined. This isparticularly unfortunute in light of the intriguing cusp-like behaviour observed by Jones et al. [21] across thePs-formation threshold. Due to the small magnitude of

24

0

200

400

600

800

1000

1 10

cros

s se

ctio

n (a

.u.)

incident energy (eV)

CCC Savage et al. Stein et al.

FIG. 22. e+-Mg total scattering cross section. The verticalline indicates the Mg-ionization threshold. Experimental dataare due to Stein et al. [256], and the calculations are as forFig. 21.

0

50

100

150

200

250

300

1 10

cros

s se

ctio

n (a

.u.)

incident energy (eV)

CCC Gribakin et al. Hewitt et al. Cheng and Zhou Walters lower limit upper limit

FIG. 23. e+-Mg Ps-formation cross section. Experimentaldata for upper and lower limits of the Ps-formation cross sec-tion are due to Surdutovich et al. [258]. The vertical line indi-cates the position of the Ps-formation threshold. The calcu-lations are due to Utamuratov et al. [156], Gribakin and King[94], Hewitt et al. [159], Walters (data taken from Ref. [258])and Cheng and Zhou [259].

the structures a highly accurate theoretical treatment isrequired, but does not yet exist. The experimental andtheoretical situation for positron-noble gas collision sys-tems has been recently reviewed extensively by Chiariand Zecca [11].

F. Molecular hydrogen

If we allow the internuclear separation of the two pro-tons in H2 to be fixed then the extra complexity, relative

to the helium atom, is somewhat manageable.Various cross sections for e+−H2 scattering have been

recently calculated by Utamuratov et al. [199] using thetwo-center CCC method. This represents the most com-plex implementation of the two-centre formalism to date.Issues regarding convergence with Laguerre-based molec-ular and Ps states have been discussed in some detail.Calculations with only up to three Ps(1s), Ps(2s) andPs(2p) states on the Ps centre were presented above10 eV. At lower energies the current implementation ofthe two-centre formalism fails to pass the internal consis-tency check with the single-center calculations of Zammitet al. [196]. The low energies are particularly sensitive tothe approximations of the treatment of the (virtual) Psformation in the field of the highly structured H+

2 ion.Here we just present the key two-centre results in com-parison of experiment and theory.

5

10

15

20

10 100

cros

s se

ctio

n (i

n a.

u.)

incident energy (eV)

GTCSCCC(142,3)CCC(108,0)

Macachek et al.Hoffman et al.Charlton et al.Karwasz et al.

Zecca et al.

FIG. 24. The grand total cross section (GTCS) for e+−H2.Experimental data are due to Machacek et al. [181], Hoffmanet al. [172], Charlton et al. [173], Karwasz et al. [180] andZecca et al. [26]. The single centre CCC(108 , 0) results aredue to Zammit et al. [196]. The two-centre CCC calculationsare due to Utamuratov et al. [199].

In Fig. 24 the theoretical results are compared with theavailable experimental data for the grand total cross sec-tion. Good agreement between the two- and one-centrecalculation above the ionisation threshold is very satisfy-ing, even if in this energy region the theory is somewhatbelow experiment. Good agreement with experiment ofthe two-centre CCC results below the ionisation thresh-old, dominated by the elastic and Ps-formation cross sec-tions, is particularly pleasing.Fig. 25 shows the Ps-formation cross section which is

a substantial component of the GTCS, particularly nearits maximum around 20 eV. There is a little variationbetween the three CCC calculations particularly at thelower energies, with the largest CCC calculation beinguniformly a little lower than experiment.In Fig. 26 the CCC results for the direct total ioniza-

tion cross section (TICS) are compared with the available

25

0

2

4

6

8

10

10 100

cros

s se

ctio

n (i

n a.

u.)

incident energy (eV)

Ps-formationCCC(141,1)CCC(141,3)CCC(142,3)

Biswas et al.Zhou et al.

Fromme et al.Machacek et al.

FIG. 25. The Ps-formation cross section in e+−H2 collisions.Experimental data are due to Zhou et al. [178] (shaded regionincorporates upper and lower limits with their uncertainties),Fromme et al. [175] and Machacek et al. [181]. Coupled staticmodel calculations are due to Biswas et al. [186]. The CCCcalculations are due to Utamuratov et al. [199].

0

1

2

3

4

5

6

10 100 1000

cros

s se

ctio

n (i

n a.

u.)

incident energy (eV)

direct-ionizationCCC(141,1)CCC(141,3)CCC(142,3)

Fromme et al.Knudsen et al.Jacobsen et al.

FIG. 26. The total direct-ionization cross section for e+−H2

collisions. Experimental data are due to Fromme et al. [175],Knudsen et al. [85] and Jacobsen et al. [177]. The CCC cal-culations are due to Utamuratov et al. [199].

experimental data. The experimental data of Frommeet al. [175] and Knudsen et al. [85] are in agreementwith each other but differ from measurements of Ja-cobsen et al. [177] between 30 and 100 eV. The largestCCC calculation CCC(142,3), which contains s-, p- andd-atomic orbitals together with the three lowest-energyPs states, is in better agreement with the measurementsof Jacobsen et al. [177]. The CCC(141,1) and CCC(141,3)are systematically lower, primarily due to the absenceof the d-atomic orbitals. Due to the unitarity of theclose-coupling formalism, larger l-orbitals are likely toonly marginally increase the TICS. Once we have con-

0

2

4

6

8

10

12

10 100 1000

cros

s se

ctio

n (i

n a.

u.)

incident energy (eV)

electron-lossCCC(141,1)CCC(141,3)CCC(142,3)CCC(108,0)

Moxom et al.Fromme et al.

FIG. 27. The total electron-loss cross section for e+−H2 col-lisions. Experimental data are due to Moxom et al. [176] andFromme et al. [175]. The CCC calculations are due to Za-mmit et al. [196] (one-centre) and Utamuratov et al. [199](two-centre).

vergence in the elastic scattering (large l-orbitals not re-quired) the GTCS is set, with the distribution betweenelastic, electron-excitation, Ps-formation and total ioni-sation cross sections thereby being constrained [260].Finally, the total electron-loss cross section is given in

Fig. 27, and is the sum of TICS and Ps-formation crosssections. It is useful because the one-centre CCC ap-proach is able to yield this cross section at energies abovethe ionisation threshold. This provides for an importantinternal-consistency check. We see that around 50 eV thelargest two-centre CCC calculation is somewhat lowerthan the one-centre calculation. Given the increase inthe cross section due to the inclusion of d-orbitals addingf -orbitals would go someway to reduce the discrepancy.Increasing the Laguerre basis Nl = 10 − l in the one-centre calculations would increase the cross section justabove the ionisation threshold due to an improving dis-cretisation of the continuum.Despite the immense complexity of the positron-

molecular-hydrogen scattering problem considerableprogress in obtaining reasonably accurate cross sectionshas been made.

IV. ANTIHYDROGEN FORMATION

As stated earlier, via Eq. (1), antihydrogen formationis effectively the time-reverse process of Ps formationupon positron-hydrogen scattering. Accordingly, it onlytakes place at positron energies above the Ps(n) forma-tion threshold, where n indicates the principal quantumnumber. Furthermore, we require two-centre calculationsbecause only these have explicit Ps and H states. Giventhat we are interested in as large cross sections as pos-sible, and that the exothermic transition cross sections

26

go to infinity at threshold [149, 201] the primary energyrange of interest is within the extended Ore gap, as givenin Fig. 5.

100

101

102

103

104

105

106

107

108

10-5 10-4 10-3 10-2 10-1 100 101

cros

s se

ctio

n (a

.u.)

Ps energy ε (eV)

(anti)hydrogen formation CCC Ps(1s) CCC Ps(2s) CCC Ps(2p) CCC Ps(3s) CCC Ps(3p) CCC Ps(3d) UBA Ps(2s) UBA Ps(2p) UBA Ps(1s) Variational Ps(1s) Merrison 1997

FIG. 28. Total cross sections for (anti)hydrogen formationupon positronium, in the specified initial state nl, scatteringon (anti)protons calculated using the CCC method, first pre-sented by Kadyrov et al. [211]. For Ps(1s), the variationalcalculations [200, 261, 262] are for (anti)hydrogen formationin the 1s state only (CCC-calculated unconnected points pre-sented for comparison), while the UBA calculations of Mitroy[263] and Mitroy and Stelbovics [264], and the CCC calcula-tions generally, are for (anti)hydrogen formation in all openstates. The three experimental points are due to Merrisonet al. [265].

There are few calculations of antihydrogen formationthat are accurate at low energies. As far as we are aware,apart from the CCC method [211, 266], only the varia-tional approach of Humberston et al. [200] has yieldedaccurate results. The unitarised Born approximation(UBA) calculations [263, 267, 268] are not appropriateat low energies, and neither are first order approxima-tions [269].

Large cross sections for antihydrogen formation oc-cur for transitions between excited states of Ps and H.To have these states accurate the Laguerre bases Nl forboth H and Ps need to be sufficiently large. Kadyrovet al. [211] used Nl = 12− l, with the resultant energiesgiven in Fig. 3. While it is trivial to have larger Nl asfar as the structure is concerned, the primary limitingfactor is to be able to solve the resultant close-couplingequations. In Fig. 3 the positive energies correspond tothe breakup of H and also of Ps, and yet the two rep-resent the same physical process. This leads to highlyill-conditioned equations. Bailey et al. [218] considerednon-symmetric treatments of the two centres by drop-ping the Ps positive-energy states. While these also sat-isfy internal-consistency checks they have not proved tobe more efficient in yielding convergence, with increasingNl, in the cross sections of present interest.

A. Numerical treatment of the Green’s function

A comprehensive set of antihydrogen formation crosssections for low-energy Ps incident on an antiproton hasbeen given by Rawlins et al. [266]. They used the originalnumerical treatment of the Green’s function in Eq. (19).In Fig. 28, we present the summary of the total antihy-drogen formation cross sections for specified initial Ps(nl)as presented by Kadyrov et al. [211]. Excellent agreementwith the variational calculations [200, 261, 262], availableonly for the ground states of H and Ps, gives us confidencein the rest of the presented CCC results.

As predicted by Wigner [149] the ground state crosssection goes to infinity as 1/

√ε at threshold, whereas

for excited states this is modified to 1/ε at thresholddue to the long-range dipole interaction of degeneratePs n ≥ 2 states, as explained by Fabrikant [201]. Thecross sections increase steadily with increasing principlequantum number of Ps, with the transition to the high-est available principal quantum number of H being themost dominant contribution [266]. Consequently thereis considerable motivation to push the calculations evenfurther to larger Nl.

B. Analytic treatment of the Green’s function

There is a second source of ill-conditioning of the close-coupling equations to that discussed previously. Giventhat we are particularly interested in low energies i.e.,just above the various thresholds, the singularity in theGreen’s function of Eq. (19) occurs very close to zeroenergy. A typically used symmetric treatment on eitherside of the singularity results in very large positive andnegative values which contribute to the ill-conditioningvia precision loss.

Very recently Bray et al. [224, 225] showed that theGreen’s function of the coupled Lippmann-Schwingerequations (19) can be treated analytically. The methodthat was previously used in calculating the optical poten-tial in the coupled-channel optical method [270, 271] wasapplied directly to the Green’s function in the Lippmann-Schwinger equation (19). In doing so, they showed thatthey could improve on the above Nl = 12 − l basis toreach Nl = 15− l, and thereby consider transitions fromPs(n = 4) states. In Fig. 29 we compare the “old” andthe “new” numerical formulations and find that the latteris quite superior. In fact, for Nl = 15− l Bray et al. [225]were unable to obtain numerical stability in the originalformulation with variation of integration parameters overthe momentum in Eq. (19), and presented just a com-bination that yielded somewhat reasonable agreementwith the results of the “new” numerical method. Thenew technique also proved to be particularly advanta-geous in studying threshold phenomena in positronium-(anti)proton scattering [272].

27

10-610-510-410-310-210-110+010+1

cross section (a.u.)

H(1s) new old

10-310-210-110+010+110+210+3

H(2s)

10-410-310-210-110+010+110+210+3

H(3s)

10-410-310-210-110+010+110+210+3

H(4s)

10-310-210-110+010+110+210+310+410+5

10-5 10-4 10-3 10-2 10-1 100 101

incident Ps(4s) energy (eV)

H(5s)

FIG. 29. H(ns) formation cross sections for n ≤ 5 in p-Ps(4s)scattering for the zeroth partial wave. The “old” results arisefrom solving the original CCC equations (19). The “new”results are due to the solution with the Green’s function beingtreated analytically, see Bray et al. [225].

V. CONCLUDING REMARKS

We have presented an overview of recent developmentsin positron-scattering theory on several atoms and molec-ular hydrogen, with particular emphasis on two-centrecalculations that are able to explicitly treat positroniumformation. While considerable progress has been madethere remain some major discrepancies between theoryand experiment such as for low-energy positron-sodiumscattering. Considerable technical development is stillrequired for complicated atoms such as the heavier inertgases to incorporate Ps formation in a systematically con-vergent way. A general scheme for doing so for moleculartargets is also required.It is also important to state that there are still fun-

damental issues to be addressed in the case of breakup.While no overcompleteness has been found in two-centrecalculations with near-complete bases on both centres,determining the resulting differential cross sections re-mains an unsolved problem. Kadyrov et al. [273] consid-ered the simplest energy-differential cross section whichdescribes the probablity of an electron of a certain en-ergy being ejected. If we have contributions to this pro-cess from both centres, how do we combine them? Fur-thermore, diagonalising the Ps Hamiltonian yields pseu-dostates of positive-energy Ps rather than the energy ofthe individual electron or positron. It may be that theonly practical way to obtain differential ionisation crosssections is to restrict the Ps centre to just negative-energyeigenstates, and obtain differential cross sections solelyfrom the atomic centre positive-energy pseudostates, asis done for electron-impact ionisation [40]. However, thiskind of approach may not be capable of describing thephenomenon of Ps formation in continuum as seen byArcidiacono et al. [274]. This is currently under investi-gation.

ACKNOWLEDGMENTS

Support of the Australian Research Council and theMerit Allocation Scheme on the National Facility ofthe Australian Partnership for Advanced Computing isgratefully acknowledged. This work was supported byresources provided by Pawsey Supercomputing Centrewith funding from the Australian Government and theGovernment of Western Australia. A.S.K. acknowledgespartial support from the U.S. National Science Founda-tion under Award No. PHY-1415656.

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