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Atomic collisions involving positrons H.R.J. Walters * , S. Sahoo, Sharon Gilmore Department of Applied Mathematics and Theoretical Physics, Queen’s University, Belfast BT7 1NN, UK Available online 29 April 2005 Abstract In this short review we look at bound states, positron-atom scattering, positronium-atom scattering, positronium– positronium scattering, cold antihydrogen and annihilation. Ó 2005 Published by Elsevier B.V. PACS: 34.50.s; 34.85.+x; 36.10.Dr; 71.35.y Keywords: Positron; Positronium; Bound states; Ps ; Ps 2 ; Antihydrogen; Protonium; Atom; Scattering; Ionization; Annihilation; Coupled-states; Pseudostate; Resonance; Bose–Einstein condensate; Exciton 1. Introduction In a short review like this a comprehensive treat- ment of the field is an impossibility. Rather, our aim is to give a flavour of areas of interest and to convey the spirit of the subject. Accordingly, our references will be driven by the narrative rather than by any pretence at completeness. To correct for this we list under [1] all of the proceedings of the biennial ‘‘Workshop on Low Energy Positron and Positronium Physics’’ over the past 10 years. These, together with a recent book [2] arising out of a meeting at ITAMP in Harvard in October 2000, give a good overview of most activities. Positronic atomic physics is interesting because of its very highly correlated nature. This correla- tion arises because of the competition between positrons and between positrons and nuclei for the ‘‘attention’’ of the electrons in the system and the fact that the positron, being a light particle, is able to weave and dodge its way through the sys- tem. It is also to be remembered that, if an electron and positron are in close proximity for a suffi- ciently long time, then annihilation of both parti- cles into c-rays will take place, i.e. positronic atomic systems have a finite lifespan, typically measured in nanoseconds (ns). We shall briefly cover the following areas: bound states, positron-atom scattering, positronium-atom 0168-583X/$ - see front matter Ó 2005 Published by Elsevier B.V. doi:10.1016/j.nimb.2005.03.089 * Corresponding author. E-mail addresses: [email protected] (H.R.J. Walters), [email protected] (S. Sahoo), [email protected] (S. Gil- more). Nuclear Instruments and Methods in Physics Research B 233 (2005) 78–87 www.elsevier.com/locate/nimb

Atomic collisions involving positrons

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Nuclear Instruments and Methods in Physics Research B 233 (2005) 78–87

www.elsevier.com/locate/nimb

Atomic collisions involving positrons

H.R.J. Walters *, S. Sahoo, Sharon Gilmore

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

Available online 29 April 2005

Abstract

In this short review we look at bound states, positron-atom scattering, positronium-atom scattering, positronium–

positronium scattering, cold antihydrogen and annihilation.

� 2005 Published by Elsevier B.V.

PACS: 34.50.�s; 34.85.+x; 36.10.Dr; 71.35.�y

Keywords: Positron; Positronium; Bound states; Ps�; Ps2; Antihydrogen; Protonium; Atom; Scattering; Ionization; Annihilation;

Coupled-states; Pseudostate; Resonance; Bose–Einstein condensate; Exciton

1. Introduction

In a short review like this a comprehensive treat-ment of the field is an impossibility. Rather, our

aim is to give a flavour of areas of interest and to

convey the spirit of the subject. Accordingly, our

references will be driven by the narrative rather

than by any pretence at completeness. To correct

for this we list under [1] all of the proceedings of

the biennial ‘‘Workshop on Low Energy Positron

and Positronium Physics’’ over the past 10 years.These, together with a recent book [2] arising out

0168-583X/$ - see front matter � 2005 Published by Elsevier B.V.

doi:10.1016/j.nimb.2005.03.089

* Corresponding author.

E-mail addresses: [email protected] (H.R.J. Walters),

[email protected] (S. Sahoo), [email protected] (S. Gil-

more).

of a meeting at ITAMP in Harvard in October

2000, give a good overview of most activities.

Positronic atomic physics is interesting becauseof its very highly correlated nature. This correla-

tion arises because of the competition between

positrons and between positrons and nuclei for

the ‘‘attention’’ of the electrons in the system and

the fact that the positron, being a light particle,

is able to weave and dodge its way through the sys-

tem. It is also to be remembered that, if an electron

and positron are in close proximity for a suffi-ciently long time, then annihilation of both parti-

cles into c-rays will take place, i.e. positronic

atomic systems have a finite lifespan, typically

measured in nanoseconds (ns).

We shall briefly cover the following areas: bound

states, positron-atom scattering, positronium-atom

H.R.J. Walters et al. / Nucl. Instr. and Meth. in Phys. Res. B 233 (2005) 78–87 79

scattering, positronium–positronium scattering,

cold antihydrogen, annihilation.

Throughout we shall use atomic units (a.u.) in

which �h = m = e = 1; the symbol a0 will denote

the Bohr radius.

2. Bound states

Positronium (Ps) [3] is the simplest bound state

consisting of a single electron and a single positron.

Formally, it is the same as an H atom but with

reduced mass 0.5 a.u. rather than 1 a.u. Conse-quently, Ps bound states are classified in the same

way, Ps(nlm), and have half the energy of the corre-

sponding H states, Enlm = �0.25/n2 a.u. Ps exists

only for a short time, the electron and the positron

eventually annihilating. The lifetime of the Ps

depends not only upon its spatial state, nlm, but

also upon its overall spin state. Positronium in the

spin singlet state is called para-positronium (p-Ps)and that in the triplet state, ortho-positronium

(o-Ps). Thus, p-Ps(1s) (o-Ps(1s)) annihilates pre-

dominantly into two (three) c-rays with a lifetime

of 0.125 ns (142 ns) [4].

In 1946 Wheeler [5] showed that Ps could bind

an electron to form the negative ion Ps�, an ana-

logue of H�.1 A year later Hylleraas and Ore [6]

showed that two Ps atoms could combine to formthe ‘‘molecule’’ Ps2, while in 1951 Ore [7] demon-

strated the binding of Ps and H to form

positronium hydride, PsH. Recent values [8] for

the binding energies of Ps�, Ps2 and PsH are

0.3267, 0.4355 and 1.067 eV, respectively, and

for their lifetimes 0.477, 0.225 and 0.410 ns,

respectively.

Until 1997 only eight positronic bound stateshad been shown, in a convincing way, to exist.

They included the four states mentioned above,

Ps, Ps�, Ps2 and PsH, plus PsF, PsCl, PsBr and

PsOH [9]. Despite a search over a large number

of years for bound states of a positron with an

atom, no definitive results had been obtained. In

1997 the picture changed, with definite proof that

1 Equivalently, it could bind a positron to form Ps+. Note

that there is no analogue of this for H, i.e. H cannot bind a

positron to form a positive ion.

a positron could bind to lithium (binding energy

0.067 eV) [10,11]. The fact that it took so long to

establish this result confirms our introductory re-

marks that positronic systems are very highly cor-

related; this makes it very difficult to give anadequate theoretical treatment. Since 1997 more

than 50 bound states have been positively identi-

fied [12,13].

3. Positron-atom scattering

When a positron scatters off an atom, A, thefollowing processes are possible:

eþþA!eþþA Elastic scattering ð1aÞeþþA� Excitation ð1bÞeþþAnþþne� Ionization ð1cÞPsðnlmÞþAþ Ps formation ð1dÞPs�þA2þ Ps� formation ð1eÞ

PsþAðnþ1Þþ þne� Transfer

ionization ð1fÞ

Ps�þAðnþ2Þþ þne� Transfer

ionization with Ps�

formation ð1gÞAþþc-rays Annihilation ð1hÞ

This is a very much richer, and therefore much

more interesting, set of possibilities than is avail-

able under electron impact where only (1a)–(1c)

apply.

The most powerful theoretical method presentlyin use to treat positron-atom scattering is the cou-

pled-pseudostate approach. To illustrate the ideas

and to demonstrate the power of the method, let us

briefly consider positron scattering by atomic

hydrogen [14,15].

In the coupled-pseudostate method the colli-

sional wavefunction for the system, W, is expanded

in atom states wa and Ps states /b according to

W ¼Xa

F aðrpÞwaðreÞ þXb

GbðRÞ/bðtÞ. ð2Þ

Here, for atomic hydrogen, rp(re) is the position

vector of the positron (electron) relative to the

80 H.R.J. Walters et al. / Nucl. Instr. and Meth. in Phys. Res. B 233 (2005) 78–87

proton, R � (rp + re)/2 is the position vector of the

center of mass of the Ps, and t � (rp � re) is the Ps

internal coordinate. The states wa and /b consist

not only of bound eigenstates but also of so-called

‘‘pseudostates’’. The pseudostates are a way of giv-ing a discrete representation of the atom/Ps contin-

uum. They are constructed by diagonalizing the

atomic/Ps Hamiltonian (HA and HPs) in some suit-

able basis, e.g. a basis of Slater orbitals,

hwajHAjwa0 i ¼ eadaa0 ;

h/bjHPsj/b0 i ¼ Ebdbb0 . ð3Þ

For a more complete discussion of pseudostates,

see [14–17]. Substitution of (2) into the Schroding-

er equation and projection with wa and /b leads to

coupled equations for the Fa and Gb.

Fig. 1 shows results calculated by Kernoghanet al. [15] in a 33-state approximation. It is

Fig. 1. Positron scattering by atomic hydrogen: (a) total Ps formati

approximation [15]; experimental data from [18,19].

seen that the agreement with experiment is very

good.

Besides atomic hydrogen, coupled-state calcula-

tion have been performed on the ‘‘one-electron’’

alkali metal systems Li, Na, K, Rb and Cs[16,17,20,21] and on the ‘‘two-electron’’ systems

He, Mg, Ca and Zn [21,22].

The inspiration behind the theoretical advances

in positron-atom scattering has been experiment

[1,16]. At present there are two interesting exper-

imental developments that are challenging theory

and therefore worth highlighting. The first is a

pioneering coincidence experiment on positronimpact ionization, the first (e+, e+e�) experiment

[23,24]. The second is a recent measurement of

Ps formation in the heavier noble gases which

suggests that excited state Ps formation may be

significantly larger than had been anticipated

[25,26].

on; (b) ionization; (c) total cross section. Solid curve, 33-state

Fig. 2. Cross sections for Ps(1s) + He(11S) scattering [74] in the

22-state frozen target approximation of [37]: solid curve, total

cross section; short-dashed curve, elastic scattering; dash-dot

curve, Ps ionization; long-dashed curve, Ps(n = 2) excitation;

solid circles, total cross section measurements of Garner et al.

[27].

H.R.J. Walters et al. / Nucl. Instr. and Meth. in Phys. Res. B 233 (2005) 78–87 81

4. Positronium-atom scattering

The development of an energy-tunable Ps beam

at University College London [27,28] has opened

up a whole new area of interest. The beam consistsof o-Ps(1s), p-Ps(1s) is too short lived to be trans-

portable. Measurements have been made of total

cross sections for o-Ps(1s) colliding with He, Ne,

Xe, H2 and N2. Additional information on Ps-atom

scattering comes from lifetime studies or from

observation of annihilation radiation [29–33] but

is limited to very low energies and to the momen-

tum transfer cross section. A very interesting newdevelopment has been the first measurement of

Ps fragmentation and the longitudinal energy dis-

tribution of the residual positrons [34,42].

From a theoretical viewpoint, one of the diffi-

culties with Ps-atom scattering is that both part-

ners have an internal structure whose dynamics

in a collision must be described. Early calculations

by McAlinden et al. [35] used the first Bornapproximation to study collisions in which the

atom is excited or ionized and a pseudostate close

coupling approximation, neglecting electron ex-

change between the Ps and the atom, for collisions

in which the atom remains in its ground state. This

way, however crudely, they were able to get some

idea of what happens to the atom and to the Ps.

Recent theoretical work has been concentratedupon the more interesting low energy end of the

impact energy scale and, once again, has profited

from the powerful coupled-pseudostate approxi-

mation, now including electron exchange. To be

specific, consider Ps scattering by atomic hydro-

gen. In the coupled-pseudostate approximation

we expand the collisional wavefunction as

W ¼Xa;b

½GabðR1Þ/aðt1Þwbðr2Þ

þ ð�1ÞSeGabðR2Þ/aðt2Þwbðr1Þ�; ð4Þ

where rp(ri) is the position of the positron (ith elec-

tron) relative to the proton, Ri � (rp + ri)/2,

ti � (rp � ri), and the sum is over Ps states /a and

H states wb, these being eigenstates and pseudo-

states. In a non-relativistic treatment the total elec-

tronic spin Se(=0 or 1) must be conserved. The

positron spin is separately conserved. Formula (4)

reflects the appropriate symmetrization in the spa-

tial coordinates of the electrons. It is clear from (4)

that the size of the calculation expands, roughly, as

the product of the number of Ps states times the

number of H states. To ease the magnitude of thecalculations, first attempts restricted the atom to

its initial state (frozen target approximation)

[36,37]. Fig. 2 shows such a calculation of Ps(1s)

scattering by He(11S). We see that, except at

10 eV, the calculated total cross section slightly

underestimates the measured total cross section

of Garner et al. [27], the down-turn in the measure-

ments at 10 eV is not reproduced by the theory.Fig. 2 also shows what happens to the Ps. Beyond

about 20 eV the main outcome of the collision is

ionization of the Ps, hence the necessity of repre-

senting the Ps continuum channels by using

pseudostates. Fig. 3 shows the same frozen target

approximation but now for the momentum trans-

fer cross section. It is in good agreement with the

measurement of Nagashima et al. [32] but in dis-agreement with the other experimental data.

The frozen target calculations threw up a num-

ber of interesting questions [16]. The obvious one

was ‘‘how important is target excitation/ionization

in the low energy domain where the excitation/

ionization is virtual rather than real?’’ The work

10 2Energy (eV)

0

5

10

15

20

Cro

ss S

ectio

n (π

a 02 )

Fig. 3. Momentum transfer cross section for Ps(1s) + He(11S)

scattering [74]. Solid curve, 22-state frozen target approxima-

tion of [37]. Dashed curve, approximation of [38] allowing for

excitation/ionization of the Ps and of the He. Experiment: up

triangle, Canter et al. [29]; down triangle, Rytsola et al. [30];

square, Nagashima et al. [32]; circle, Skalsey et al. [33].

0Energy (eV)

0

10

20

30

40

Cro

ss S

ectio

n (π

a 02 )

2 4 6

Fig. 4. Total cross section for Ps(1s) + H(1s) scattering in the

9Ps9H + H� approximation of [38]. Note: It is assumed that the

H target is spin unpolarised and that final spin states are not

resolved. The result is then independent of whether the Ps is in

the ortho or para state and, if the former, is independent of its

polarization, see [39].

82 H.R.J. Walters et al. / Nucl. Instr. and Meth. in Phys. Res. B 233 (2005) 78–87

of McAlinden et al. [35] had certainly shown that

real excitation/ionization was important at highimpact energies. For Ps(1s)–H(1s) scattering this

question was eventually answered by Blackwood

et al. [39]. The answer was ‘‘very important’’. That

is not to say, however, that the frozen target

approximation is not a reasonable approximation

at higher energies for transitions in which the atom

remains in its initial state. Fig. 3 also shows a cal-

culation which allows for (the virtual) excitation/ionization of the He atom as well as of the Ps. This

reduces the momentum transfer cross section be-

low the frozen target value by up to 30% in the en-

ergy range shown. However, this is not a definitive

result, questions concerning the use of approxi-

mate He wave functions, the importance of He

triplet states, and of mechanisms such as virtual

Ps� formation remain [16,38].The role of negative ions such as Ps�, H�, Li�,

etc. is a very interesting one [38,40] and is spectac-

ularly illustrated by the Ps–H system. The H� ion

has zero total electronic spin Se and so can be

formed in Se = 0 Ps–H scattering in the reaction

PsþH ! H� þ eþ: ð5ÞFig. 4 shows a calculation of Ps(1s)–H(1s) scat-

tering incorporating the H� channel. The specta-

cular Rydberg resonance structure converging on

to the H� formation threshold at 6.05 eV comes

from unstable states of the positron orbiting H�

[40,41]. The role of the Ps� ion in Ps scattering is

yet to be investigated. Interestingly, unlike the

Ps(1s)–H(1s) system, Ps� formation in Ps(1s)–al-

kali systems occurs before alkali ion formation

[38].

Further frozen target calculations on Ne, Ar,

Kr and Xe may be found in [43].

5. Positronium–positronium scattering

This may seem to be an esoteric topic of interest

only to theorists. But it is not.

The Ps–Ps system has a high degree of symme-

try. Denoting by r1 and r2 (r3 and r4) the positions

of the two positrons (two electrons) relative tosome fixed origin O, we may write the Hamilto-

nian for the Ps–Ps system as

H.R.J. Walters et al. / Nucl. Instr. and Meth. in Phys. Res. B 233 (2005) 78–87 83

H ¼ � 1

2r2

1 �1

2r2

2 �1

2r2

3 �1

2r2

4

þ 1

jr1 � r2j� 1

jr1 � r3j� 1

jr1 � r4j

� 1

jr2 � r3j� 1

jr2 � r4jþ 1

jr3 � r4j: ð6Þ

It is clear that H is unchanged by the permutations(12) (interchange of positrons), (34) (interchange

of electrons), (12)(34) (simultaneous interchange

of positrons and interchange of electrons),

(13)(24), (14)(23), (1324) and (1423) (four possi-

ble interchanges of positrons with electrons, i.e.

charge conjugation symmetry). These seven opera-

tions together with the identity operation form the

symmetry group of Ps–Ps. Until the work of King-horn and Poshusta [44] it appears that the full

symmetry of the Ps–Ps system was not widely

appreciated.

The symmetry group of Ps–Ps is isomorphic

with the point symmetry group D2d [45] and con-

sequently the symmetry may be classified accord-

ing to the irreducible representations of D2d.

These consist of 4 one-dimensional representa-tions, labelled A1, A2, B1, B2 and a single two-

dimensional representation, labelled E. Bound

states of Ps–Ps, i.e. Ps2, must be classified accord-

ing to these representations. In Section 2 we

pointed out that, in 1947, Hylleraas and Ore [6]

had proved that a bound state did exist. It was

another 40 years before it was shown that there

were other bound states, the most recent one beingfound in 1998 [46]. Table 1 summarises the known

bound states [47]. The A1 state is the ground state

and the original state found by Hylleraas and Ore.

The excited states are prevented by symmetry from

breaking up into Ps(1s) + Ps(1s); they are bound

relative to the next highest threshold, Ps(1s) +

Ps(n = 2), see Table 1. The S-states can only decay

Table 1

Ps–Ps bound states

SLp Symmetry Lowest accessible threshold Binding en

1Se A1 Ps(1s) + Ps(1s) 0.43541Se B2 Ps(1s) + Ps(2p) 0.05413Se E Ps(1s) + Ps(2s,2p) 0.48051P0 B2 Ps(1s) + Ps(2p) 0.5961

by electron–positron annihilation, primarily into

two c-rays. However, the 1P0 state can also decay

by an electric dipole transition to the A1 ground

state with a branching ratio of 17% [48] and emit-

ting a photon of 4.94 eV.Besides bound states of Ps–Ps, resonance states

have also been studied theoretically [49]. As with

Ps–H, Fig. 4, these should provide some interest-

ing collision physics to study.

There are two interesting areas of practical

application. The first concerns a suggestion of

Platzman and Mills [50] for producing a Bose–Ein-

stein condensate (BEC) of Ps, i.e. a matter–anti-matter condensate, a project which is actively

under consideration [51]. The second application

is to exciton–exciton processes in solids, e.g.

[52,53]. An exciton is a bound state of a conduc-

tion band electron with a valence band hole in a

semiconductor. Exciton–exciton systems therefore

bear a similarity to Ps–Ps except that, in general,

the hole will have a different mass from the elec-tron and so the exciton–exciton system will have

a lower symmetry. Considerable interest also exists

in Bose–Einstein condensation of excitons [52,53].

6. Cold antihydrogen

The end of 2002 saw the announcement of thefirst production of cold ([15 K) antihydrogen

ðHÞ by two experimental groups, ATHENA

[54,55] and ATRAP [56–58]. In both cases the H

had been formed by mixing positrons and antipro-

tons in a nested Penning trap. Two mechanisms

are possible for the formation, radiative recombi-

nation and three-body recombination [59,60].

The relative contribution of these two mechanismsis presently unclear [60] but estimates suggest that

the H is mainly formed in Rydberg states with

ergy (eV) Lifetime against two c-rays annihilation (ns)

0.23

0.48

0.43

0.45

84 H.R.J. Walters et al. / Nucl. Instr. and Meth. in Phys. Res. B 233 (2005) 78–87

n J 48; this together with the high production

rate of H is consistent with the three-body recom-

bination mechanism [57].

A primary motivation for the production of H

is that it offers the opportunity of making veryhigh precision tests of the Weak Equivalence Prin-

ciple of General Relativity for antimatter and of

the CPT invariance of relativistic quantum

mechanics [55,59]. However, to make these tests,

the H is required to be in a low lying quantum

state, preferably the 1s ground state. An important

question now is, can the highly excited H that has

been formed be de-excited in sufficient numbers?Assuming that H has been formed in the

ground state, can it be further cooled (tempera-

ture < 1 K is desired) by elastic collisions with cold

background gas and what is the chance of it being

destroyed in such collisions? The cold background

gas may be deliberately introduced to facilitate

cooling or it may simply be trace impurities that

are difficult to eliminate from the trap. The mostlikely candidates are H2 and He. However, these

targets are difficult for theorists, particularly H2,

and so theoretical studies of cooling and destruc-

tion of H have initially concentrated upon

H(1s) + H(1s) collisions. Here, destruction may

take place either through rearrangement into pos-

itronium (Ps) and protonium (Pn) (a bound state

of the antiproton ð�pÞ and the proton (p)),

Hð1sÞ þHð1sÞ ! PsðnlmÞ þ PnðNLMÞ ð7Þor electron–positron, or �p–p in-flight annihilation.

At the temperatures of interest ([15 K) the max-

imum value of N for Pn is 24, the Ps then being

formed in its 1s ground state.

Calculations have initially focussed upon a

Born–Oppenheimer treatment in which the groundstate potential energy curve for the H–H molecule

is constructed and the H and H then allowed to

move on this curve. There is one problem though.

Unlike H–H where the two electrons remain

bound at all internuclear separations, in H–H the

electron and the positron become unbound, form-

ing a free Ps(1s) atom, when the distance, R, be-

tween the �p and p is less than a critical value Rc.The present best estimate gives Rc < 0.744a0 [61].

To continue the potential curve below Rc for the

purpose of scattering calculations, the leptonic en-

ergy is fixed at its value at Rc, i.e. at the binding

energy, �0.25 a.u., of Ps(1s). So constructing the

potential energy curve, calculations of H(1s)–

H(1s) elastic scattering have been made, from

which estimates of �p–p in-flight annihilation havebeen obtained, and, using the distorted-wave

approximation, cross sections for the rearrange-

ment process (7) have been calculated [62–64]. A

more recent work by the same authors [65] has

introduced a non-local complex optical potential

to more consistently handle the effect of the re-

arrangement channel (7).

A more dynamical treatment has been given byArmour and Chamberlain [66] who have used the

Kohn variational method with four open channels,

viz., H(1s) + H(1s) and Ps(1s) + Pn(Ns), N = 22,

23, 24. They found that the N = 23 channel is dom-

inant for the rearrangement process (7).

While there are differences between results [64–

66], a typical set of cross sections in the cold colli-

sion regime would be rel ¼ 908a20 [66], rrear ¼0.67E�1=2a20 [66], r�pp ¼ 0.14E�1=2 a20 [63], where

rel, rrear, and r�pp are the elastic, rearrangement

(7), and in-flight �p–p annihilation cross sections,

respectively, and E is the impact energy in the cen-

ter-of-mass frame. In-flight electron–positron

annihilation seems to be negligible in comparison

to that of �p–p [63]. Note that rel is effectively con-

stant at low energies while rrear and r�pp diverge asE�1/2. Consequently, as E reduces, i.e. as the tem-

perature falls, the destruction processes begin to

dominate the cooling from elastic scattering. Using

rel and rrear from above, Armour and Chamber-

lain [66] calculate that 90% of the H(1s) would

be lost in cooling from 10 to 0.43 K.

Attention is now turning to the more relevant

H(1s)–He(11S) system [67].

7. Annihilation

In-flight annihilation of positrons in collision

with atoms and molecules is a unique signature

of positron collisions and therefore a subject of

much interest [68]. The rate of annihilation, C,may be written as

C ¼ pr20cNZeff ; ð8Þ

H.R.J. Walters et al. / Nucl. Instr. and Meth. in Phys. Res. B 233 (2005) 78–87 85

where c is the speed of light, r0 � e2/mc2 is the clas-

sical electron radius, N is the number of atoms/

molecules per unit volume and Zeff is given by

Zeff � ZZ

jWðrp; x1; x2; . . . ; xZÞj2dðrp � r1Þ

� drp dx1x2; . . . ; dxZ . ð9Þ

In (9) Z is the number of electrons in the atom/molecule, rp is the positron coordinate, xi � (ri, si)

stands for the space and spin coordinates of the

ith electron, and W is the collisional wave function

for the system. Because of the delta function, Zeff

gives ‘‘pin-point’’ information on correlation in

Fig. 5. Zeff for (a) butane, (b) propane and (c) ethane, as a

function of positron energy (taken from [71]). Vertical bars

along the abscissae indicate the strongest infrared-active vibra-

tional modes. Arrows on the ordinate indicate Zeff for a

Maxwellian distribution of positrons at 300 K. The dashed

curve in (a) is for d-butane (C4D10), that in (b) for 2,2-

difluoropropane (C3H6F2).

the system; it measures the ‘‘effective’’ number of

electrons seen by the positron in the target. In

the first Born approximation where the positron

does not disturb the target electrons, W ¼eik0�rpwT, where k0 is the momentum of the incidentpositron and wT is the undistorted wave function

for the target, it is trivial to show that Zeff has

the sensible value Zeff = Z.

More realistically, the positron attracts the tar-

get electrons, increasing the electron density

around it, and so we might expect Zeff > Z. How-

ever, experiments with trapped thermal positrons

in the presence of large organic molecules havefound values of Zeff as high as �5 · 104Z [69],

which is something of an overcrowding of the pos-

itron. Various ideas to explain these very high val-

ues of Zeff have been put forward; amongst them is

the suggestion of resonance formation. An analy-

sis by Gribakin [70] has shown that such high val-

ues of Zeff would be possible if the positron were

trapped in a vibrational Feshbach resonance. Withthe refinement of experimental techniques, it has

now become possible to produce positron beams

of a sufficiently narrow energy width to test this

explanation. Fig. 5 shows such measurements on

ethane, propane, and butane [71]; the high thermal

values of Zeff are seen to come from large enhance-

ments associated with vibrational thresholds.

8. Concluding remarks

In this brief review, we have had, of necessity,

to be selective; nevertheless we hope that we have

been able to give a fair flavour of activities in pos-

itronic atomic collision physics. Perhaps, one of

the important themes to emerge is the stimulatingrole of experiment. Despite the handicap of low

intensities, compared with electron scattering for

example, much progress has been made. Positron

beams with a higher energy resolution, such that

resonances begin to be seen and vibrational excita-

tion cross sections measured, are starting to ap-

pear [68,69,71]; the first (e+,e+e�) coincidence

ionization experiment has been performed[23,24], with intriguing results; very difficult ex-

periments with positronium beams [27,28] are

beginning to yield interesting information on

86 H.R.J. Walters et al. / Nucl. Instr. and Meth. in Phys. Res. B 233 (2005) 78–87

fragmentation, including fragment distributions

[34]; and cold antihydrogen has been produced

[54,56,57], albeit in quite highly excited states.

On the theoretical side, there has been an explo-

sion in the number of predicted bound states[12,13], coupled-pseudostate methods have given

a detailed insight into positron-atom and positro-

nium-atom collisions [14–17,20–22,35–40,43], a

theoretical understanding of the very high positron

annihilation rates in molecules has been developed

[70], and the cooling and survivability of ground

state antihydrogen has been studied [62–67].

But new challenges await. For experiment, theseinclude the identification of the predicted bound

states, recoil ion momentum spectroscopy and

observation of the radiative decay of the 1P0 state

of Ps2 perhaps provide opportunities [12,48]; the

development of more intense and better resolved

positron and positronium beams; the extension

of positronium beam experiments to both lower

and higher impact energies, the former to resolvediscrepancies at low energies [16,37,38,43], the lat-

ter to make contact with reliable high energy

approximations [35]; the development of experi-

ments to probe the mechanisms of fragmentation

[23,24,34], resonance formation [71], and annihila-

tion [68,69,71]; the production of the first matter–

antimatter Bose–Einstein condensate [50,51] and

the possibility of using it to make an annihilationphoton laser [72]; the antihydrogen project [54–

60]. For theory, the challenge is to extend the cou-

pled-pseudostate method to a full treatment of

positron and positronium scattering by more com-

plex targets such as the heavier noble gases, espe-

cially where conflict exists between experiments

[73]; to resolve differences between theory and

experiment on (e+, e+e�) measurements [24]; toinvestigate correlation and resonance effects; to

provide an ab initio description of positron-mole-

cule scattering with particular reference to vibra-

tional Feshbach resonances [70,71]; to develop a

fuller understanding of the Ps–Ps system and its

relationship to exciton processes in solids; to

understand the positron–antiproton recombina-

tion process in cold antihydrogen formation; toinvestigate how cold Rydberg antihydrogen may

be efficiently de-excited to the ground state; to de-

velop a more dynamical understanding of cold

antihydrogen collisions divorced from the Born–

Oppenheimer approximation.

Acknowledgements

This work was supported by EPSRC grants

GR/N07424, GR/R83118/01, and GR/R62557/01.

We are grateful to C.M. Surko and Physical

Review A for permission to use Fig. 5.

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