Nuclear Instruments and Methods in Physics Research B 221 (2004) 124–128

www.elsevier.com/locate/nimb

Electron/positron collisions with positronium

Sharon Gilmore *, Jennifer E. Blackwood, H.R.J. Walters

Department of Applied Mathematics and Theoretical Physics, Queen’s University, Belfast BT7 1NN, UK

Abstract

We report results for e�–Psð1sÞ scattering in the energy range up to 80 eV calculated in 9-state and 30-state coupled

pseudostate approximations. Cross-sections are presented for elastic scattering, ortho–para conversion, discrete exci-

tation, ionization and total scattering. Resonances associated with the Psðn ¼ 2Þ threshold are also examined and their

positions and widths determined. Very good agreement is obtained with the variational calculations of Ward et al. [J.

Phys. B 20 (1987) 127] below 5.1 eV.

� 2004 Elsevier B.V. All rights reserved.

1. Introduction

Electron scattering by positronium (Ps) is fun-

damental to atomic collision physics, being one of

the simplest three-body systems which interacts via

Coulomb forces. The system consists of two elec-

trons and a positron. These three particles are

capable of binding together to form a negative ion,

Ps�, with total electronic spin zero. This was pre-dicted by Wheeler [1] in 1946, and confirmed

experimentally by Mills in 1981 [2]. With the

exception of Ps itself, Ps� is so far the only elec-

tron–positron complex to have been confirmed

experimentally.

The system is therefore of fundamental interest,

and has been examined theoretically by a number

of authors. Ward et al. [3] obtained phaseshiftsand resonances for elastic scattering, using a

variational approach. Basu and Ghosh [4] used

eigenstate close coupling to obtain a number of

* Corresponding author.

E-mail address: s.gilmore@qub.ac.uk (S. Gilmore).

0168-583X/$ - see front matter � 2004 Elsevier B.V. All rights reser

doi:10.1016/j.nimb.2004.03.042

cross-sections and resonances for scattering up tothe first excitation threshold. Ho [5], Bhatia and

Ho [6] and Usukura and Suzuki [7] used the

complex coordinate rotation method to locate the

positions and widths of a number of resonances in

the Ps� system.

In the present work we use pseudostate close-

coupling to obtain a picture of e�–Ps scattering

over a much wider energy range than has previ-ously been studied, up to an impact energy of

80 eV. Our approach enables us to investigate

not only elastic scattering and resonances but also

discrete excitations and ionization of the Ps, in a

dynamically consistent way.

While we shall describe our formalism and

present our results for electron–Ps scattering, by

charge conjugation invariance we get identicallythe same results for positron–Ps scattering. We use

atomic units (au) in which �h ¼ me ¼ e ¼ 1 and

denote by a0 the Bohr radius. To express energies

derived in atomic units in electron volts (eV), e.g.

the calculated position and width of a resonance,

we use the conversion 1 au¼ 27.21138344(106) eV

[8].

ved.

S. Gilmore et al. / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 124–128 125

2. Method

Our approach is based on pseudostate closecoupling. Assuming that the projectile is an elec-

tron, the Hamiltonian for the system may be

written

H ¼ � 3

4r2

T1þ HPsðt2Þ þ

1

jr1 � r2j� 1

jr1 � rpj;

H ¼ � 3

4r2

T2þ HPsðt1Þ þ

1

jr1 � r2j� 1

jr2 � rpj;

ð1Þ

where rpðriÞ is the position vector of the positron

(ith electron) relative to a fixed origin, ti � rp � ri,T1 � r1 � R2, T2 � r2 � R1, Ri � ðrp þ riÞ=2 and

HPs is the Ps Hamiltonian given by

HPsðtÞ ¼ �r2t �

1

t: ð2Þ

We expand the collisional wave function W for

the system as

W ¼Xb

ðFbðT1Þ/bðt2Þ þ ð�1ÞSFbðT2Þ/bðt1ÞÞ; ð3Þ

where /b is a Ps eigenstate or pseudostate and Srepresents total electronic spin. The set of states /b

includes both eigenstates and pseudostates of the

Ps atom. The purpose of the pseudostates is pri-

marily to represent the Ps continuum. They are

chosen so that, together with the eigenstates, theydiagonalize the Ps Hamiltonian, i.e.

h/bjHPsj/b0 i ¼ Ebdb0 : ð4ÞSubstituting (3) into the Schr€odinger equation,

using (1), projecting with /b and using (4) gives a

set of coupled equations for the functions Fb,

ðr2T1þ k2bÞFbðT1Þ

¼ 4

3

Xb0

Vbb0 ðT1ÞFb0 ðT1Þ

þ 4

3ð�1ÞS

Xb0

ZKbb0 ðT1;T2ÞFb0 ðT2ÞdT2;

ð5Þ

where the potentials Vbb0 give the direct Coulombic

interaction between the projectile and the Ps tar-

get, and Kbb0 is the electron exchange kernel.

In the calculations presented here we use two

sets of states /b. The first is a 9-state set, 1s, 2s, 2p,

3s, 3p, 3d, 4s, 4p, 4d (where a ‘bar’ denotes a

pseudostate) derived from the hydrogen states WHb

of Fon et al. [9] according to

/bðtÞ ¼1ffiffiffi8

p WHb ðt=2Þ; ð6Þ

and used earlier by Kernoghan et al. [10]. The

second is a 30-state set, 1s, 2s, 3s to 9s, 2p, 3p to

9p, 3d to 9d, 4f to 9f, again derived from hydrogen

states according to (6), this time the hydrogen

states being those employed by Kernoghan et al.

[11]. As we shall see, the 9-state approximationgives very good agreement with the variational

calculations of Ward et al. [3] below the Ps(n ¼ 2)

threshold at 5.1 eV. Above this energy, however,

the 9-state cross-sections can be somewhat uneven

on account of giving too coarse a representation of

the Ps continuum. By contrast, the 30-state

approximation gives smooth cross-sections above

5.1 eV but suffers from linear dependence prob-lems at lower energies. Above 5.1 eV the 30-state

cross-sections lie close to an average through the

uneven 9-state results which adds to our confi-

dence in the 30-state approximation in this energy

region. The results we shall present at the lower

energies are therefore calculated in the 9-state

approximation, those at the higher energies in the

30-state approximation.The coupled equations (5) have been solved by

conversion to partial wave form and application of

the R-matrix technique [12]. Partial waves for total

angular momentum from 0 up to 20 have been

included and corrections applied for higher partial

waves using the first Born approximation for

s ! p transitions and a geometric progression

procedure for the other cases, as was done byKernoghan et al. for positron–hydrogen scattering

[10,11].

3. Results

Our calculations give not only information

about scattering but also about the Ps� boundstate. In our 9-state approximation we get a

binding energy of 0.3238 eV for Ps� which agrees

within 1% with the accurate result of 0.3267 eV of

Frolov [13].

Table 1

Resonances associated with the Ps(n ¼ 2) threshold

State Ho/Bhatia

and Ho [5,6]

Usukura and

Suzuki [7]

9-state

approximation

1Se(1) 4.733853 4.733972 4.73943

(1.17) (1.153) (1.204)

1Se(2) 5.0708 5.07038 5.08086

(0.27) (2.45) (0.322)

1Se(3) 5.097751

(0.00223)

3Se(1) 5.0741 5.07390628 5.0825888

(0.14) (0.00022) (0.00004217)

1Po(1) 5.08429 5.094450

(0.03) (0.0693)

3Pe(1) 4.807507 4.811719

(3.478) (3.548)

3Pe(2) 5.08589 5.098854

(0.44) (0.484)

3Pe(3) 5.1103 No

(7.3) evidence

1De(1) 4.954846 4.961945

(0.054) (0.0479)

Positions are given in eV and widths in meV in parenthesis.

126 S. Gilmore et al. / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 124–128

Fig. 1 shows our results for the elastic scattering

cross-section for energies below the first excitation

threshold at 5.1 eV. In non-relativistic e�–Ps

scattering the total electron spin (¼ 0 or 1) and thepositron spin are separately conserved. Fig. 1

shows the electronic spin singlet and triplet cross-

sections as well as the spin averaged cross-section.

This latter corresponds to the cross-section aver-

aged over initial electron spins and summed over

final electron spins, as remarked above the posi-

tron spin is unchanged in the collision.

The first notable feature is the large spin singletcross-section at low energies. This is a result of the

spin singlet bound state of Ps� which gives a pole

in the scattering matrix at an energy of E ¼�0:3238 eV (in the 9-state approximation). As

there is no spin triplet bound state, the corre-

sponding spin triplet cross-section has no such rise,

rather it falls with decreasing impact energy to-

wards threshold. As a result of the opposite trendsof the singlet and triplet cross-sections, the spin

averaged cross-section possesses a dip-bump

structure near threshold. The results shown in Fig.

1 are in good agreement with the variational cal-

culations of Ward et al. [3].

The small spike in the elastic cross-section just

below 5 eV is a result of resonances associated with

the Ps(n ¼ 2) threshold. We have examined thesein more detail on a finer energy mesh and have

fitted the lower numbers to a Breit–Wigner form,

our results are shown in Table 1. Generally, we get

reasonable agreement with the calculations of Ho

[5], Bhatia and Ho [6] and Usukura and Suzuki [7]

although our resonance positions are slightly

0 1 2 3 4 5Energy (eV)

0

100

200

300

400

500

600

Cro

ss S

ectio

n (π

a 02) 9-state

approximationSpin singletSpin tripletSpin averaged

Fig. 1. Elastic scattering cross-section.

higher presumably on account of a more refined

representation of correlation between the three

particles in these other works. Exceptions are: the1Se(2) resonance where Usukura and Suzuki pre-

dict a width 10 times larger than calculated here or

by Ho [5], we think this to be a tabulation error;the 3Se(1) resonance, where the present work and

that of Usukura and Suzuki predict a width about

1000 times smaller than that calculated by Ho [5];

the 3Pe(3) resonance of which we have not been

able to find any evidence, according to Ho and

Bhatia [6] this is a shape resonance, unlike the

other resonances of Table 1 which are Feshbach

resonances.It is also of interest to see the impact of the

resonances upon the cross-section. This is illus-

trated in Fig. 2 for the S-wave resonances reported

in Table 1.

Fig. 3 shows the ortho–para conversion cross-

section for e� + o-Psð1sÞ ! e� + p-Ps (1) as

calculated in the 9-state approximation. The cross-

section shown assumes that the incident electron

4.71 4.72 4.73 4.74 4.75 4.76Energy (eV)

5

10

15

20

Cro

ss S

ectio

n ( π

a 02)

1S

e(1) resonance

5.06 5.07 5.08 5.09 5.1 5.11Energy (eV)

5

10

15

20

Cro

ss S

ectio

n ( π

a 02)

1S

e(2) and

1S

e(3) resonances

5.08 5.081 5.082 5.083 5.084 5.085Energy (eV)

13.4

13.6

13.8

14

Cro

ss S

ectio

n ( π

a 02)

3S

e(1) resonance

Fig. 2. Singlet and triplet S-wave elastic cross-sections calcu-

lated in the 9-state approximation, illustrating the S-wave res-

onances of Table 1.

0 5 10 15 20Energy (eV)

0

5

10

15

20

Cro

ss S

ectio

n (π

a02) 9 state approximation

Fig. 3. Cross-section for the ortho–para conversion e� + o-

Ps(1s) ! e� + p-Ps(1s).

0 20 40 60 80 100Energy (eV)

0

1

2

3

4

5

6

Cro

ss S

ectio

n (π

a 02)

1s-2p excitation1s-2s excitationother discrete transitions

30 state approximation

Fig. 4. Discrete excitation cross-sections.

0 20 40 60 80

Energy (eV)

0

1

2

3

4

5

6

7

Cro

ss S

ectio

n (π

a 02) 30 state

approximation

Fig. 5. Ionization cross-section.

S. Gilmore et al. / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 124–128 127

and the target o-Ps (1s) are spin unpolarized and

that the spin of the electron in the final state is not

measured. The result of Fig. 3 is in very good

agreement with the variational calculation of

Ward et al. [3] which only goes up to 5.1 eV. The

structures near 5.1 eV in Fig. 3 are the resonancesdiscussed above.

Figs. 4 and 5 show spin averaged discrete exci-

tation cross-sections and the ionization cross-sec-

tion respectively as calculated in the 30-state

approximation.

Following Kernoghan et al. [10] the ionizationcross-section has been extracted from the pseudo-

states using the formula

0 20 40 60 80Energy (eV)

0

10

20

30

40

Cro

ss S

ectio

n (π

a 02)

Fig. 6. Spin averaged total cross-section and its components.

128 S. Gilmore et al. / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 124–128

rion ¼Xb

fbrb; ð7Þ

where rb is the cross-section for exciting the state

/b and fb is the fraction of /b lying in the con-

tinuum. Correspondingly, the cross-section for

discrete excitations other than Ps(n ¼ 2) shown in

Fig. 4 has been calculated from

rdisc ¼X

b 6¼1s;2s;2p

ð1� fbÞrb: ð8Þ

Fig. 6 illustrates the spin averaged total cross-

section and its components. Above 17 eV ion-

ization and Psð1sÞ ! Psðn ¼ 2Þ excitation are

dominant and comparable, but unlike elastic

scattering none of the inelastic cross-sections ever

exceed 5.5pa20 (see Figs. 4 and 5). It is clear from

Figs. 1 and 6 that elastic scattering at low energiesis by far the major event in the e�–Psð1sÞ system.

4. Conclusions

We have reported results for e�–Psð1sÞ scatter-ing in the energy range up to 80 eV. The coupled

pseudostate approach has enabled us to present acomprehensive and internally consistent picture of

the main processes, i.e. elastic scattering, Ps(n ¼ 2)

excitation, other discrete excitations (lumped to-

gether), ionization, ortho–para conversion, and

resonance structures. The very good agreement

obtained with the variational results of Ward et al.

[3] below 5.1 eV, the good agreement with the Ps�

bound state energy [13], and the generally accept-

able accord with other calculations of resonance

structure [5–7] suggest that our results are, if notabsolutely perfect, at least of a very high quality.

The resonances present a very stringent test of our

theory. That we seem to predict a slightly higher

position for each resonance than in more refined

calculations of resonance properties [5–7] suggests

that we have not got the correlation between the

three particles, e� eþ e�, absolutely perfect. We

attribute this to a certain lack of symmetry in ourformulation (3). Here we see that, because of the

product structures FbðT1Þ/bðt2Þ and FbðT2Þ/bðt1Þ,one of the electrons, say that in Fb, is always in a

different relationship to the positron than the other

electron, in /b. That we believe is the heart of the

problem.

Acknowledgements

This work was supported by EPSRC grants

GR/N07424 and GR/R83118/01.

References

[1] J.A. Wheeler, Ann. N.Y. Acad. Sci. 48 (1946) 219.

[2] A.P. Mills Jr., Phys. Rev. Lett. 46 (1981) 717.

[3] S.J. Ward, J.W. Humberston, M.R.C. McDowell, J. Phys.

B 20 (1987) 127.

[4] A. Basu, A.S. Ghosh, Europhys. Lett. 60 (1) (2002) 46.

[5] Y.K. Ho, Phys. Lett. 102A (1984) 348.

[6] A. Bhatia, Y.K. Ho, Phys. Rev. A 42 (1990) 1119;

A. Bhatia, Y.K. Ho, Phys. Rev. A 45 (1992) 6268;

A. Bhatia, Y.K. Ho, Phys. Rev. A 48 (1993) 264.

[7] J. Usukura, Y. Suzuki, Phys. Rev. A 66 (2002) 010502.

[8] P.J. Mohr, B.N. Taylor, Rev. Mod. Phys. 72 (2000)

351.

[9] W.C. Fon, K.A. Berrington, P.G. Burke, A.E. Kingston, J.

Phys. B 14 (1981) 1041.

[10] A.A. Kernoghan, M.T. McAlinden, H.R.J. Walters, J.

Phys. B 28 (1995) 1079.

[11] A.A. Kernoghan, D.J.R. Robinson, M.T. McAlinden,,

H.R.J. Walters, J. Phys. B 29 (1996) 2089.

[12] P.G. Burke, W.D. Robb, Adv. At. Mol. Phys. 11 (1975)

143.

[13] A.M. Frolov, Phys. Rev. A 60 (1999) 2834.