Nuclear Instruments and Methods in Physics Research B 221 (2004) 129–133

www.elsevier.com/locate/nimb

Positronium formation in positron–noble gas collisions

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

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

Abstract

Distorted-wave Born approximation calculations for Ps formation in positron impact on He, Ne, Ar, Kr and Xe are

reported for the energy range up to 200 eV. Capture into the n ¼ 1, 2 and 3 states of Ps is calculated explicitly and 1=n3

scaling is used to estimate capture into states with n > 3. The calculations for the heavier noble gases allow for capture

not only from the outer np6 shell of the atom but also from the first inner ns2 shell. However, the inner shell capture is

found to be very small. Although by no means unambiguous, the calculations provide some support to the conjecture of

Larrichia et al. [J. Phys. B 35 (2002) 2525] that the double peak and shoulder structures observed experimentally for Ps

formation in Ar, Kr and Xe arise from formation in excited states.

� 2004 Elsevier B.V. All rights reserved.

PACS: 34.85.+x; 36.10.Dr

Keywords: Positron; Scattering; Noble gas; Positronium; Distorted-wave Born approximation

1. Introduction

In a recent paper Larrichia et al. [1] have re-

ported new measurements of positronium (Ps)formation in positron impact on Ne, Ar, Kr and

Xe. In the case of Ar a double peaked structure

was observed in the Ps formation cross section, for

Kr and Xe the second peak became a shoulder.

Although not able to make a direct verification,

Larrichia et al. presented strong arguments for

believing that the second peak/shoulder was a re-

sult of Ps formation in excited states, in quantitiesmuch greater than hitherto had been suspected.

Our purpose here is to see if this conjecture can be

supported by theory.

* 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.2004.03.043

A proper theoretical treatment of positron

scattering by the heavier noble gases poses major

computational difficulties [2]. Ideally, one would

like to perform a calculation comparable to that ofCampbell et al. [3] for eþ–He scattering. Campbell

et al. used a coupled pseudostate approach which

gives a complete dynamical description in the sense

that it incorporates representations of all the main

processes, i.e. elastic scattering, discrete excitations

of the atom, Ps formation, ionization. It is there-

fore able to balance the competition between dif-

ferent processes, e.g. between Ps formation anddirect ionization of the atom, to give quite accu-

rate results. The best that has been achieved for Ps

formation in the heavier noble gases is the trun-

cated coupled-static approximation of McAlinden

and Walters [4]. In this approximation only Ps

formation into the ground 1s state was considered.

Despite the simplicity of the approach, quite

ved.

130 S. Gilmore et al. / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 129–133

reasonable, if rough, agreement was obtained with

existing experiments, in particular, the approxi-

mation was in agreement with the absolute scale of

the measured cross sections.Our attitude here is that we wish to get some

idea of the relative importance of Ps formation

in the ground state compared with Ps formation

in excited states and that maybe this can be ac-

hieved in a comparatively simple approximation.

Accordingly, we have chosen to make calculations

in the distorted-wave Born approximation.

2. Method

Consider positron scattering by the ground

state of an N -electron noble gas atom. In atomic

units (a.u.) (�h ¼ me ¼ e ¼ 1), the Hamiltonian for

this system may be written

H ¼ � 1

2r2

p þ HAðr1; . . . ; rNÞ þ V ðrp; r1; . . . ; rN Þ;

ð1Þ

where rpðriÞ is the position vector of the positron(ith electron) relative to the atomic nucleus, HA is

the atomic Hamiltonian given by

HAðr1; . . . ; rN Þ ¼XNj¼1

�� 1

2r2

j �Zrj

�

þXN�1

j¼1

XNk¼jþ1

1

jrj � rkj; ð2Þ

where Z ¼ N is the nuclear charge and V is the

interaction between the positron and the atom,

V ðrp; r1; . . . ; rN Þ ¼Zrp�XNj¼1

1

jrp � rjj: ð3Þ

The Hamiltonian (1) may be alternatively cast as

H ¼ � 1

4r2

Riþ HPsðtiÞ

þ HIðr1; . . . ; ri�1; riþ1; . . . ; rNÞþ VPsðrp; ri; r1; . . . ; ri�1; riþ1; . . . ; rN Þ: ð4Þ

The Hamiltonian (4) represents the system fromthe viewpoint of Ps formation. Here Ri � ðrp þ

riÞ=2 and ti � rp � ri are respectively the centre of

mass of the Ps relative to the atomic nucleus and

the Ps internal coordinate when the Ps is com-posed of the positron and the ith electron,

HIðr1; . . . ; ri�1; riþ1; . . . ; rN Þ is the Hamiltonian for

the atomic ion that remains whenever the ithelectron has been removed to form Ps, it is given

by (2) with the restriction that j and k cannot be

equal to i and VPs gives the interaction between the

Ps and the ion,

VPsðrp; ri; r1; . . . ; ri�1; riþ1; . . . ; rN Þ

¼ Zrp

�XN

j¼1;j6¼i

1

jrp � rjj

!� Z

ri

�XN

j¼1;j6¼i

1

jri � rjj

!:

ð5Þ

In a coupled state treatment we would expand

the collisional wave function for the system, W,as

W ¼Xa

FaðrpÞwaðx1; . . . ; xN Þ þ AXb;c

GbcðR1Þ/bðt1Þ

� aðs1Þwþc;�1=2ðx2; . . . ; xN Þbðs1Þwþ

c;1=2ðx2; . . . ; xN Þ� �

;

ð6Þ

where xi � ðri; siÞ, si being the spin coordinate of

the ith electron, wað/bÞ is an atom (Ps) state, wþcsz

is

a singly ionized state with z-component of elec-

tronic spin sz, a and b are the usual one-electronspin-up and spin-down functions and A is the

electron antisymmetrization operator. In a non-

relativistic treatment of the collision the total

electronic spin and the positron spin will be sepa-

rately conserved, the positron spin need not

therefore be explicitly specified. In writing (6) we

have used the fact that the initial state of the noble

gas atom is its ground state which is a spin singlet.Consequently, in (6), all of the atom states wa are

singlets and the ion states wþcsz

are spin doublets.

The states wa, /b and wþcsz

may be either eigenstates

or pseudostates. Under the assumption that they

diagonalise their respective Hamiltonians, substi-

tution of (6) in the Schr€odinger equation and pro-

jection with waðx1; . . . ; xNÞ and /bðt1Þðaðs1Þwþc;�1=2

ðx2; . . . ;xN Þ�bðs1Þwþc;1=2ðx2; . . . ;xNÞÞ leads to cou-

pled equations of the form

10 100Energy (eV)

0.1

0.2

0.3

0.4

0.5

0.6

Cro

ss S

ectio

n (π

a 02 )

(a)

10 100Energy (eV)

0.1

0.2

0.3

0.4

0.5

0.6

Cro

ss S

ectio

n (π

a 02 )

(b)

Fig. 1. Ps formation cross sections for eþ–He collisions: (a)

Ps(1s) formation; (b) total Ps formation. Solid curve, present

DWBA calculations · 0.75; dashed curve, results of Campbell

et al. [3].

S. Gilmore et al. / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 129–133 131

ðr2p þ k2aÞFaðrpÞ¼ 2

Xa0

Vaa0 ðrpÞFa0 ðrpÞ

þ 2NXb0c0

ZKa;b0c0 ðrp;RÞGb0c0 ðRÞdR; ð7Þ

r2R

�þ p2bc

�GbcðRÞ

¼ 4Xb0c0

Ubc;b0c0 ðRÞGb0c0 ðRÞ

þ 4Xa0

ZK�

a0;bcðrp;RÞFa0 ðrpÞdrp

þ 4Xb0c0

ZLbc;b0c0 ðR;R0ÞGb0c0 ðR0ÞdR0; ð8Þ

where

k2a2þ �a ¼

p2bc4

þ Eb þ �þc ð9Þ

and �a (Eb) (�þc ) is the energy of the atom (Ps) (ion)

state wa (/b) (wþcsz). In (7) and (8) Vaa0 (Ubc;b0c0) gives

the Coulombic interaction between the positron

(Ps) and the atom (ion), the kernels Ka;bc describePs formation, while the kernels Lbc;b0c0 represent

electron exchange between the Ps and the ion; �denotes complex conjugation.

Despite the formal simplicity of (7) and (8), the

kernels Ka;bc and Lbc;b0c0 are very complex objects to

evaluate. We have therefore made the following

simplifications: (i) we retain only the ground state,

w0, of the atom in (6); (ii) we drop the K-kernels in(7); (iii) we drop the L-kernels in (8); (iv) we drop

the direct potentials Ubc;b0c0 in (8). The coupled

equations then reduce to

ðr2p þ k20ÞF0ðrpÞ ¼ 2V00ðrpÞF0ðrpÞ; ð10Þ

ðr2R þ p2bcÞGbcðRÞ ¼ 4

ZK�

0;bcðrp;RÞF0ðrpÞdrp;

ð11Þ

which is now the distorted-wave Born approxi-

mation (DWBA) that we use for Ps formation.

Our DWBA allows the positron to be scattered by

the repulsive static potential V00 of the atom, Eq.

(10); Ps is formed in the state /b by a singleinteraction with the atom in which an electron is

captured leaving the ion in the state wþcsz, Eq. (11).

For the ground state w0 we have used the

Hartree–Fock wave functions of Clementi and

Roetti [5]. Our ion states wþcsz

are obtained by re-

moval of a spin-orbital from w0 corresponding tothe captured electron. For Ne, Ar, Kr and Xe we

have considered capture both from the outermost

np6 shell of the atom and the next inner shell, i.e.

ns2. We have calculated Ps formation in the n ¼ 1,

2 and 3 states and have used 1=n3 scaling to esti-

mate Ps formation in states with n > 3; our total

Ps formation cross section is the sum of all these.

3. Results

Fig. 1 shows the DWBA cross sections for

Ps(1s) and total Ps formation for eþ–He collisions.

In the figure these cross sections are compared

with the calculations of Campbell et al. [3]. We see

that the DWBA cross sections are generally largerthan those of Campbell et al. and need to be

renormalised downwards by a factor of about

10 100Energy (eV)

0.1

0.2

0.3

0.4

0.5

0.6

Cro

ss S

ectio

n (π

a 02)

(a)

DWBA x 0.57

10 100Energy (eV)

0

1

2

3

4

Cro

ss S

ectio

n (π

a 02)

(b)

DWBA x 0.5

10 100Energy (eV)

0

1

2

3

4

5

6

Cro

ss S

ectio

n (π

a 02)

(c)

DWBA x 0.37

10 100Energy (eV)

0

2

4

6

8

10

12

Cro

ss S

ectio

n (π

a 02)

(d)

DWBA x 0.36

Fig. 2. Ps formation cross sections for positron collisions with (a) Ne, (b) Ar, (c) Kr and (d) Xe. Solid curve, DWBA results (scaled) for

total Ps formation; dashed curve, DWBA results (scaled) for Ps(1s) formation; dotted curve, measurements of Larrichia et al. [1] for

total Ps formation.

10 100Energy (eV)

0

1

2

3

4

5

6

7

Cro

ss S

ectio

n (π

a 02)

Fig. 3. DWBA total Ps formation cross sections for eþ–Ar

collisions: solid curve, Ps formed by capture of 3p electron;

dashed curve, Ps formed by capture of 3s electron.

132 S. Gilmore et al. / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 129–133

0.75 to be on a comparable scale. We also note

that the DWBA cross sections peak at slightly

lower energies, that the Ps(1s) cross section seems

to rise slightly more rapidly from threshold and

that the total Ps cross section is slightly narrower

than the Campbell et al. results.Fig. 2 shows the DWBA cross sections for

Ps(1s) and total Ps formation for positron scat-

tering by Ne, Ar, Kr and Xe. Here comparison is

made with the experimental results of Larrichia

et al. [1] which, for clarity, are shown as a dotted

curve rather than the individual measured points.

As with He, the DWBA cross sections have had to

be renormalised downwards to put them on thesame scale as the measurements, the degree of

reduction being greater than in Fig. 1 and

increasing systematically on going from Ne to Xe.

The other features we have noted in Fig. 1 appear

again in Fig. 2 but are much more amplified:

compared with experiment the DWBA cross sec-

tions peak at a lower energy, rise more rapidly

from threshold and are narrower than the experi-

mental results. These failings aside, the DWBA

does predict no structures for Ne, a double peaked

structure for Ar and shoulder structures for Kr

and Xe, all arising from Ps formation in excited

states.

S. Gilmore et al. / Nucl. Instr. and Meth. in Phys. Res. B 221 (2004) 129–133 133

Finally, although we have taken account of

capture from the first inner ns2 shell in Ne, Ar, Kr

and Xe, in the DWBA this contribution turned out

to be very minor. We illustrate this in Fig. 3 for Arwhere we show the total Ps formation cross sec-

tions for capture from the 3p and 3s shells sepa-

rately.

4. Conclusions

It is clear that the DWBA has failings in rela-tion to overall magnitude, peak position, width

and rate of rise from threshold, but it does

reproduce structures not unlike those observed by

Larrichia et al. [1] and it does identify these

structures with excited state Ps formation. The

problem is that the failings of the DWBA are

sufficiently severe that it is not possible to say

unambiguously that it supports the interpretation

of Larrichia et al., but it does hint at it. A better

approximation is needed.

Acknowledgements

This research was supported by EPSRC grants

GR/N07424 and GR/R83118/01.

References

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Walters, Nucl. Instr. and Meth. 143 (1998) 41.

[4] M.T. McAlinden, H.R.J. Walters, Hyperfine Interact. 73

(1992) 65.

[5] E. Clementi, C. Roetti, At. Data Nucl. Data Tables 14

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