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Atomic collisions involving positrons
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 treatment 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 *
Corresponding author. E-mail addresses: [email protected] (H.R.J. Walters), [email protected] (S. Sahoo), [email protected] (S. Gilmore). 0168-583X/$ - see front matter 2005 Published by Elsevier B.V. doi:10.1016/j.nimb.2005.03.089
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 correlation 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 system. It is also to be remembered that, if an electron and positron are in close proximity for a sufficiently long time, then annihilation of both particles 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
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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. Consequently, Ps bound states are classified in the same way, Ps(nlm), and have half the energy of the corresponding 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 predominantly 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 analogue of H.1 A year later Hylleraas and Ore [6] showed that two Ps atoms could combine to form the ‘‘molecule’’ Ps2, while in 1951 Ore [7] demonstrated 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 states had 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
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 remarks that positronic systems are very highly correlated; this makes it very difficult to give an adequate theoretical treatment. Since 1997 more than 50 bound states have been positively identified [12,13].
3. Positron-atom scattering When a positron scatters off an atom, A, the following processes are possible: eþ þ A !eþ þ A
Elastic scattering ð1aÞ
þ
þ
nþ
e þA e þA
þ ne
þ
PsðnlmÞ þ A
Ps þ A
2þ
Excitation
ð1bÞ
Ionization
ð1cÞ
Ps formation
ð1dÞ
Ps formation
ð1eÞ
Ps þ Aðnþ1Þþ þ ne Transfer ionization
Ps þ A
ðnþ2Þþ
ð1fÞ
þ ne Transfer ionization with Ps
þ
A þ c-rays
formation
ð1gÞ
Annihilation
ð1hÞ
This is a very much richer, and therefore much more interesting, set of possibilities than is available under electron impact where only (1a)–(1c) apply. The most powerful theoretical method presently in use to treat positron-atom scattering is the coupled-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 collisional wavefunction for the system, W, is expanded in atom states wa and Ps states /b according to X X F a ðrp Þwa ðre Þ þ Gb ðRÞ/b ðtÞ. ð2Þ W¼ a
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.
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b
1
Here, for atomic hydrogen, rp(re) is the position vector of the positron (electron) relative to the
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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 giving a discrete representation of the atom/Ps continuum. They are constructed by diagonalizing the atomic/Ps Hamiltonian (HA and HPs) in some suitable basis, e.g. a basis of Slater orbitals, hwa jH A jwa0 i ¼ ea daa0 ; h/b jH Ps j/b0 i ¼ Eb dbb0 .
ð3Þ
For a more complete discussion of pseudostates, see [14–17]. Substitution of (2) into the Schro¨dinger equation and projection with wa and /b leads to coupled equations for the Fa and Gb. Fig. 1 shows results calculated by Kernoghan et al. [15] in a 33-state approximation. It is
seen that the agreement with experiment is very good. Besides atomic hydrogen, coupled-state calculation 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 experimental developments that are challenging theory and therefore worth highlighting. The first is a pioneering coincidence experiment on positron impact 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].
Fig. 1. Positron scattering by atomic hydrogen: (a) total Ps formation; (b) ionization; (c) total cross section. Solid curve, 33-state approximation [15]; experimental data from [18,19].
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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 consists of o-Ps(1s), p-Ps(1s) is too short lived to be transportable. 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 momentum transfer cross section. A very interesting new development has been the first measurement of Ps fragmentation and the longitudinal energy distribution of the residual positrons [34,42]. From a theoretical viewpoint, one of the difficulties with Ps-atom scattering is that both partners have an internal structure whose dynamics in a collision must be described. Early calculations by McAlinden et al. [35] used the first Born approximation to study collisions in which the atom is excited or ionized and a pseudostate close coupling approximation, neglecting electron exchange 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 concentrated upon the more interesting low energy end of the impact energy scale and, once again, has profited from the powerful coupled-pseudostate approximation, now including electron exchange. To be specific, consider Ps scattering by atomic hydrogen. In the coupled-pseudostate approximation we expand the collisional wavefunction as X W¼ ½Gab ðR1 Þ/a ðt1 Þwb ðr2 Þ
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reflects the appropriate symmetrization in the spatial 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 the calculations, 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 measurements 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 representing the Ps continuum channels by using pseudostates. Fig. 3 shows the same frozen target approximation but now for the momentum transfer cross section. It is in good agreement with the measurement of Nagashima et al. [32] but in disagreement with the other experimental data. The frozen target calculations threw up a number 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
a;b S
þ ð1Þ e Gab ðR2 Þ/a ðt2 Þwb ðr1 Þ;
ð4Þ
where rp(ri) is the position of the positron (ith electron) 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 pseudostates. In a non-relativistic treatment the total electronic spin Se(=0 or 1) must be conserved. The positron spin is separately conserved. Formula (4)
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].
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40
15 30
10
5
0 0
1 Energy (eV)
2
Fig. 3. Momentum transfer cross section for Ps(1s) + He(11S) scattering [74]. Solid curve, 22-state frozen target approximation 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, Rytso¨la¨ et al. [30]; square, Nagashima et al. [32]; circle, Skalsey et al. [33].
Cross Section (πa02)
Cross Section (πa02)
20
20
10
0
of McAlinden et al. [35] had certainly shown that real excitation/ionization was important at high impact 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 calculation which allows for (the virtual) excitation/ ionization of the He atom as well as of the Ps. This reduces the momentum transfer cross section below the frozen target value by up to 30% in the energy range shown. However, this is not a definitive result, questions concerning the use of approximate 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 spectacularly 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 Ps þ H ! H þ eþ :
ð5Þ
Fig. 4 shows a calculation of Ps(1s)–H(1s) scattering incorporating the H channel. The spectacular Rydberg resonance structure converging on
0
2
4 Energy (eV)
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].
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)–alkali 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 symmetry. Denoting by r1 and r2 (r3 and r4) the positions of the two positrons (two electrons) relative to some fixed origin O, we may write the Hamiltonian for the Ps–Ps system as
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1 1 1 1 H ¼ r21 r22 r23 r24 2 2 2 2 1 1 1 þ jr1 r2 j jr1 r3 j jr1 r4 j 1 1 1 þ : jr2 r3 j jr2 r4 j jr3 r4 j
ð6Þ
It is clear that H is unchanged by the permutations (1 2) (interchange of positrons), (3 4) (interchange of electrons), (1 2)(3 4) (simultaneous interchange of positrons and interchange of electrons), (1 3)(2 4), (1 4)(2 3), (1 3 2 4) and (1 4 2 3) (four possible interchanges of positrons with electrons, i.e. charge conjugation symmetry). These seven operations together with the identity operation form the symmetry group of Ps–Ps. Until the work of Kinghorn 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 consequently the symmetry may be classified according to the irreducible representations of D2d. These consist of 4 one-dimensional representations, labelled A1, A2, B1, B2 and a single twodimensional representation, labelled E. Bound states of Ps–Ps, i.e. Ps2, must be classified according 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 being found 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
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 emitting 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 interesting 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–Einstein condensate (BEC) of Ps, i.e. a matter–antimatter 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 conduction 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 electron 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 the first 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 antiprotons in a nested Penning trap. Two mechanisms are possible for the formation, radiative recombination and three-body recombination [59,60]. The relative contribution of these two mechanisms is presently unclear [60] but estimates suggest that the H is mainly formed in Rydberg states with
Table 1 Ps–Ps bound states S
Symmetry
Lowest accessible threshold
Binding energy (eV)
Lifetime against two c-rays annihilation (ns)
1 e
A1 B2 E B2
Ps(1s) + Ps(1s) Ps(1s) + Ps(2p) Ps(1s) + Ps(2s, 2p) Ps(1s) + Ps(2p)
0.4354 0.0541 0.4805 0.5961
0.23 0.48 0.43 0.45
Lp
S S 3 e S 1 0 P 1 e
83
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n J 48; this together with the high production rate of H is consistent with the three-body recombination mechanism [57]. A primary motivation for the production of H is that it offers the opportunity of making very high precision tests of the Weak Equivalence Principle 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 (temperature < 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 most likely candidates are H2 and He. However, these targets are difficult for theorists, particularly H2, and so theoretical studies of cooling and destruction of H have initially concentrated upon H(1s) + H(1s) collisions. Here, destruction may take place either through rearrangement into positronium (Ps) and protonium (Pn) (a bound state of the antiproton ð pÞ and the proton (p)), Hð1sÞ þ Hð1sÞ ! PsðnlmÞ þ PnðNLMÞ
ð7Þ
–p in-flight annihilation. or electron–positron, or p At the temperatures of interest ([15 K) the maximum 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 ground state 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, forming a free Ps(1s) atom, when the distance, R, be and p is less than a critical value Rc. tween the p 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 –p in-flight annihilation have which estimates of p been obtained, and, using the distorted-wave approximation, cross sections for the rearrangement 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 rearrangement channel (7). A more dynamical treatment has been given by Armour 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 dominant for the rearrangement process (7). While there are differences between results [64– 66], a typical set of cross sections in the cold collision regime would be rel ¼ 908a20 [66], rrear ¼ 0.67E1=2 a20 [66], rpp ¼ 0.14E1=2 a20 [63], where rel, rrear, and rpp are the elastic, rearrangement –p annihilation cross sections, (7), and in-flight p respectively, and E is the impact energy in the center-of-mass frame. In-flight electron–positron annihilation seems to be negligible in comparison –p [63]. Note that rel is effectively conto that of p stant at low energies while rrear and rpp diverge as E1/2. Consequently, as E reduces, i.e. as the temperature falls, the destruction processes begin to dominate the cooling from elastic scattering. Using rel and rrear from above, Armour and Chamberlain [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 pr20 cNZ eff ;
ð8Þ
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where c is the speed of light, r0 e2/mc2 is the classical electron radius, N is the number of atoms/ molecules per unit volume and Zeff is given by Z 2 Z eff Z jWðrp ; x1 ; x2 ; . . . ; xZ Þj dðrp r1 Þ drp dx1 x2 ; . . . ; 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
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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 ¼ eik0 rp wT , where k0 is the momentum of the incident positron 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 target electrons, increasing the electron density around it, and so we might expect Zeff > Z. However, experiments with trapped thermal positrons in the presence of large organic molecules have found values of Zeff as high as 5 · 104Z [69], which is something of an overcrowding of the positron. Various ideas to explain these very high values of Zeff have been put forward; amongst them is the suggestion of resonance formation. An analysis by Gribakin [70] has shown that such high values of Zeff would be possible if the positron were trapped in a vibrational Feshbach resonance. With the 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 enhancements associated with vibrational thresholds.
8. Concluding remarks
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 vibrational 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 2,2difluoropropane (C3H6F2).
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 positronic atomic collision physics. Perhaps, one of the important themes to emerge is the stimulating role 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 excitation cross sections measured, are starting to appear [68,69,71]; the first (e+, e+e) coincidence ionization experiment has been performed [23,24], with intriguing results; very difficult experiments with positronium beams [27,28] are beginning to yield interesting information on
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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 explosion in the number of predicted bound states [12,13], coupled-pseudostate methods have given a detailed insight into positron-atom and positronium-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, these include 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 resolve discrepancies at low energies [16,37,38,43], the latter to make contact with reliable high energy approximations [35]; the development of experiments to probe the mechanisms of fragmentation [23,24,34], resonance formation [71], and annihilation [68,69,71]; the production of the first matter– antimatter Bose–Einstein condensate [50,51] and the possibility of using it to make an annihilation photon laser [72]; the antihydrogen project [54– 60]. For theory, the challenge is to extend the coupled-pseudostate method to a full treatment of positron and positronium scattering by more complex targets such as the heavier noble gases, especially where conflict exists between experiments [73]; to resolve differences between theory and experiment on (e+, e+e) measurements [24]; to investigate correlation and resonance effects; to provide an ab initio description of positron-molecule scattering with particular reference to vibrational 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 recombination process in cold antihydrogen formation; to investigate how cold Rydberg antihydrogen may be efficiently de-excited to the ground state; to develop 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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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 treatment 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 *
Corresponding author. E-mail addresses: [email protected] (H.R.J. Walters), [email protected] (S. Sahoo), [email protected] (S. Gilmore). 0168-583X/$ - see front matter 2005 Published by Elsevier B.V. doi:10.1016/j.nimb.2005.03.089
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 correlation 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 system. It is also to be remembered that, if an electron and positron are in close proximity for a sufficiently long time, then annihilation of both particles 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
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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. Consequently, Ps bound states are classified in the same way, Ps(nlm), and have half the energy of the corresponding 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 predominantly 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 analogue of H.1 A year later Hylleraas and Ore [6] showed that two Ps atoms could combine to form the ‘‘molecule’’ Ps2, while in 1951 Ore [7] demonstrated 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 states had 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
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 remarks that positronic systems are very highly correlated; this makes it very difficult to give an adequate theoretical treatment. Since 1997 more than 50 bound states have been positively identified [12,13].
3. Positron-atom scattering When a positron scatters off an atom, A, the following processes are possible: eþ þ A !eþ þ A
Elastic scattering ð1aÞ
þ
þ
nþ
e þA e þA
þ ne
þ
PsðnlmÞ þ A
Ps þ A
2þ
Excitation
ð1bÞ
Ionization
ð1cÞ
Ps formation
ð1dÞ
Ps formation
ð1eÞ
Ps þ Aðnþ1Þþ þ ne Transfer ionization
Ps þ A
ðnþ2Þþ
ð1fÞ
þ ne Transfer ionization with Ps
þ
A þ c-rays
formation
ð1gÞ
Annihilation
ð1hÞ
This is a very much richer, and therefore much more interesting, set of possibilities than is available under electron impact where only (1a)–(1c) apply. The most powerful theoretical method presently in use to treat positron-atom scattering is the coupled-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 collisional wavefunction for the system, W, is expanded in atom states wa and Ps states /b according to X X F a ðrp Þwa ðre Þ þ Gb ðRÞ/b ðtÞ. ð2Þ W¼ a
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.
79
b
1
Here, for atomic hydrogen, rp(re) is the position vector of the positron (electron) relative to the
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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 giving a discrete representation of the atom/Ps continuum. They are constructed by diagonalizing the atomic/Ps Hamiltonian (HA and HPs) in some suitable basis, e.g. a basis of Slater orbitals, hwa jH A jwa0 i ¼ ea daa0 ; h/b jH Ps j/b0 i ¼ Eb dbb0 .
ð3Þ
For a more complete discussion of pseudostates, see [14–17]. Substitution of (2) into the Schro¨dinger equation and projection with wa and /b leads to coupled equations for the Fa and Gb. Fig. 1 shows results calculated by Kernoghan et al. [15] in a 33-state approximation. It is
seen that the agreement with experiment is very good. Besides atomic hydrogen, coupled-state calculation 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 experimental developments that are challenging theory and therefore worth highlighting. The first is a pioneering coincidence experiment on positron impact 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].
Fig. 1. Positron scattering by atomic hydrogen: (a) total Ps formation; (b) ionization; (c) total cross section. Solid curve, 33-state approximation [15]; experimental data from [18,19].
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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 consists of o-Ps(1s), p-Ps(1s) is too short lived to be transportable. 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 momentum transfer cross section. A very interesting new development has been the first measurement of Ps fragmentation and the longitudinal energy distribution of the residual positrons [34,42]. From a theoretical viewpoint, one of the difficulties with Ps-atom scattering is that both partners have an internal structure whose dynamics in a collision must be described. Early calculations by McAlinden et al. [35] used the first Born approximation to study collisions in which the atom is excited or ionized and a pseudostate close coupling approximation, neglecting electron exchange 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 concentrated upon the more interesting low energy end of the impact energy scale and, once again, has profited from the powerful coupled-pseudostate approximation, now including electron exchange. To be specific, consider Ps scattering by atomic hydrogen. In the coupled-pseudostate approximation we expand the collisional wavefunction as X W¼ ½Gab ðR1 Þ/a ðt1 Þwb ðr2 Þ
81
reflects the appropriate symmetrization in the spatial 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 the calculations, 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 measurements 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 representing the Ps continuum channels by using pseudostates. Fig. 3 shows the same frozen target approximation but now for the momentum transfer cross section. It is in good agreement with the measurement of Nagashima et al. [32] but in disagreement with the other experimental data. The frozen target calculations threw up a number 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
a;b S
þ ð1Þ e Gab ðR2 Þ/a ðt2 Þwb ðr1 Þ;
ð4Þ
where rp(ri) is the position of the positron (ith electron) 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 pseudostates. In a non-relativistic treatment the total electronic spin Se(=0 or 1) must be conserved. The positron spin is separately conserved. Formula (4)
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].
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40
15 30
10
5
0 0
1 Energy (eV)
2
Fig. 3. Momentum transfer cross section for Ps(1s) + He(11S) scattering [74]. Solid curve, 22-state frozen target approximation 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, Rytso¨la¨ et al. [30]; square, Nagashima et al. [32]; circle, Skalsey et al. [33].
Cross Section (πa02)
Cross Section (πa02)
20
20
10
0
of McAlinden et al. [35] had certainly shown that real excitation/ionization was important at high impact 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 calculation which allows for (the virtual) excitation/ ionization of the He atom as well as of the Ps. This reduces the momentum transfer cross section below the frozen target value by up to 30% in the energy range shown. However, this is not a definitive result, questions concerning the use of approximate 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 spectacularly 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 Ps þ H ! H þ eþ :
ð5Þ
Fig. 4 shows a calculation of Ps(1s)–H(1s) scattering incorporating the H channel. The spectacular Rydberg resonance structure converging on
0
2
4 Energy (eV)
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].
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)–alkali 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 symmetry. Denoting by r1 and r2 (r3 and r4) the positions of the two positrons (two electrons) relative to some fixed origin O, we may write the Hamiltonian for the Ps–Ps system as
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1 1 1 1 H ¼ r21 r22 r23 r24 2 2 2 2 1 1 1 þ jr1 r2 j jr1 r3 j jr1 r4 j 1 1 1 þ : jr2 r3 j jr2 r4 j jr3 r4 j
ð6Þ
It is clear that H is unchanged by the permutations (1 2) (interchange of positrons), (3 4) (interchange of electrons), (1 2)(3 4) (simultaneous interchange of positrons and interchange of electrons), (1 3)(2 4), (1 4)(2 3), (1 3 2 4) and (1 4 2 3) (four possible interchanges of positrons with electrons, i.e. charge conjugation symmetry). These seven operations together with the identity operation form the symmetry group of Ps–Ps. Until the work of Kinghorn 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 consequently the symmetry may be classified according to the irreducible representations of D2d. These consist of 4 one-dimensional representations, labelled A1, A2, B1, B2 and a single twodimensional representation, labelled E. Bound states of Ps–Ps, i.e. Ps2, must be classified according 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 being found 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
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 emitting 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 interesting 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–Einstein condensate (BEC) of Ps, i.e. a matter–antimatter 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 conduction 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 electron 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 the first 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 antiprotons in a nested Penning trap. Two mechanisms are possible for the formation, radiative recombination and three-body recombination [59,60]. The relative contribution of these two mechanisms is presently unclear [60] but estimates suggest that the H is mainly formed in Rydberg states with
Table 1 Ps–Ps bound states S
Symmetry
Lowest accessible threshold
Binding energy (eV)
Lifetime against two c-rays annihilation (ns)
1 e
A1 B2 E B2
Ps(1s) + Ps(1s) Ps(1s) + Ps(2p) Ps(1s) + Ps(2s, 2p) Ps(1s) + Ps(2p)
0.4354 0.0541 0.4805 0.5961
0.23 0.48 0.43 0.45
Lp
S S 3 e S 1 0 P 1 e
83
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n J 48; this together with the high production rate of H is consistent with the three-body recombination mechanism [57]. A primary motivation for the production of H is that it offers the opportunity of making very high precision tests of the Weak Equivalence Principle 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 (temperature < 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 most likely candidates are H2 and He. However, these targets are difficult for theorists, particularly H2, and so theoretical studies of cooling and destruction of H have initially concentrated upon H(1s) + H(1s) collisions. Here, destruction may take place either through rearrangement into positronium (Ps) and protonium (Pn) (a bound state of the antiproton ð pÞ and the proton (p)), Hð1sÞ þ Hð1sÞ ! PsðnlmÞ þ PnðNLMÞ
ð7Þ
–p in-flight annihilation. or electron–positron, or p At the temperatures of interest ([15 K) the maximum 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 ground state 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, forming a free Ps(1s) atom, when the distance, R, be and p is less than a critical value Rc. tween the p 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 –p in-flight annihilation have which estimates of p been obtained, and, using the distorted-wave approximation, cross sections for the rearrangement 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 rearrangement channel (7). A more dynamical treatment has been given by Armour 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 dominant for the rearrangement process (7). While there are differences between results [64– 66], a typical set of cross sections in the cold collision regime would be rel ¼ 908a20 [66], rrear ¼ 0.67E1=2 a20 [66], rpp ¼ 0.14E1=2 a20 [63], where rel, rrear, and rpp are the elastic, rearrangement –p annihilation cross sections, (7), and in-flight p respectively, and E is the impact energy in the center-of-mass frame. In-flight electron–positron annihilation seems to be negligible in comparison –p [63]. Note that rel is effectively conto that of p stant at low energies while rrear and rpp diverge as E1/2. Consequently, as E reduces, i.e. as the temperature falls, the destruction processes begin to dominate the cooling from elastic scattering. Using rel and rrear from above, Armour and Chamberlain [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 pr20 cNZ eff ;
ð8Þ
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where c is the speed of light, r0 e2/mc2 is the classical electron radius, N is the number of atoms/ molecules per unit volume and Zeff is given by Z 2 Z eff Z jWðrp ; x1 ; x2 ; . . . ; xZ Þj dðrp r1 Þ drp dx1 x2 ; . . . ; 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
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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 ¼ eik0 rp wT , where k0 is the momentum of the incident positron 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 target electrons, increasing the electron density around it, and so we might expect Zeff > Z. However, experiments with trapped thermal positrons in the presence of large organic molecules have found values of Zeff as high as 5 · 104Z [69], which is something of an overcrowding of the positron. Various ideas to explain these very high values of Zeff have been put forward; amongst them is the suggestion of resonance formation. An analysis by Gribakin [70] has shown that such high values of Zeff would be possible if the positron were trapped in a vibrational Feshbach resonance. With the 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 enhancements associated with vibrational thresholds.
8. Concluding remarks
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 vibrational 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 2,2difluoropropane (C3H6F2).
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 positronic atomic collision physics. Perhaps, one of the important themes to emerge is the stimulating role 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 excitation cross sections measured, are starting to appear [68,69,71]; the first (e+, e+e) coincidence ionization experiment has been performed [23,24], with intriguing results; very difficult experiments with positronium beams [27,28] are beginning to yield interesting information on
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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 explosion in the number of predicted bound states [12,13], coupled-pseudostate methods have given a detailed insight into positron-atom and positronium-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, these include 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 resolve discrepancies at low energies [16,37,38,43], the latter to make contact with reliable high energy approximations [35]; the development of experiments to probe the mechanisms of fragmentation [23,24,34], resonance formation [71], and annihilation [68,69,71]; the production of the first matter– antimatter Bose–Einstein condensate [50,51] and the possibility of using it to make an annihilation photon laser [72]; the antihydrogen project [54– 60]. For theory, the challenge is to extend the coupled-pseudostate method to a full treatment of positron and positronium scattering by more complex targets such as the heavier noble gases, especially where conflict exists between experiments [73]; to resolve differences between theory and experiment on (e+, e+e) measurements [24]; to investigate correlation and resonance effects; to provide an ab initio description of positron-molecule scattering with particular reference to vibrational 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 recombination process in cold antihydrogen formation; to investigate how cold Rydberg antihydrogen may be efficiently de-excited to the ground state; to develop 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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