The binding of positronium to lithium

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1 J. Phys. B: At. Mol. Opt. Phys. 31 (1998) L103 L107. Printed in the UK PII: S (98) LETTER TO THE EDITOR The binding of positronium to lithium G G Ryzhikh and J Mitroy Faculty of Science, Northern Territory University, Casuarina, NT, 0909, Australia Received 29 September 1997 Abstract. The wavefunction of the lithium positride (LiPs) ground state is calculated using an ab initio approach which uses a Gaussian basis with rij 2 correlation factors. The energy of the LiPs ground state, Hartree, has a total energy more negative than the sum of the positronium ( Hartree) and lithium ( Hartree) ground states. The ground state is therefore stable against dissociation into Li + Ps with a binding energy of Hartree. The use of positrons as a tool for spectroscopy, for example, positron annihilation spectroscopy in condensed matter physics, has gained widespread acceptance over the years [1, 2]. One of the questions that is raised by the use of positrons as a spectroscopic tool is of course the possibility that chemically stable systems containing a positron, or positronium, could be formed in the various targets under investigation [1, 2]. In spite of much work (and a lot of speculation), little hard information is known about the ability of a positron or positronium to form chemically stable systems in atoms, molecules or condensed matter. For instance, a recent review identified only five atoms and molecules (this excludes Ps and Ps 2 ) as permitting positronium binding with any degree of certainty [3]. These species are H, F, Cl, Br and OH. Most of the evidence for binding is based on calculation, and only for one species, namely H, could the result be classed as rigorous. While the most recent results for the other species come from high-quality calculations [4 6], they are based on model potentials for open-shell systems and do not have the same degree of certainty as the result for hydrogen which has been the subject of numerous high-quality calculations [7 11] and a recent experiment [12]. Recently, the e + Li species has been shown to be chemically stable with a binding energy of ɛ = Hartree [13]. This result, and the fact the PsH is chemically stable, immediately suggests that lithium positride, LiPs, would also be chemically stable. However, two earlier calculations on LiPs had failed to show that positronium could be bound to neutral lithium [14, 15]. Fortunately, we were ignorant of these results when deciding whether to investigate the LiPs species for possible binding. The condition for the LiPs ground state to be chemically stable is that the total energy of the LiPs ground state be more negative than the sum ( Hartree) of the Ps ground state energy ( Hartree) and the Li ground state energy ( Hartree) [13, 16]. The decay into Li + e + does not enter into the picture since the binding energy of the Li ground state [13, 17] is only Hartree. In this letter, a large variational calculation of the LiPs system is performed using the stochastic variational method (SVM) as implemented by Varga and Suzuki [18, 19]. The SVM was initially proposed as a method suitable for solving nuclear structure problems /98/ $19.50 c 1998 IOP Publishing Ltd L103

2 L104 involving a small number of particles [20, 21]. The SVM uses a Gaussian basis which has a number of features that make it possible to generate very accurate wavefunctions for few-body systems. First, the matrix elements of the interaction Hamiltonian can be calculated analytically, or at worst, reduced to a one-dimensional integral for any number of particles. Second, the Gaussian basis contains terms with rij 2 correlation factors. In addition, that part of the wavefunction concerned with the spatial coordinates maintains its functional form after any possible permutation of the particles. This is a very useful property for studying systems containing identical particles. One of the important aspects of any variational calculation is the optimization of the nonlinear parameters. In the SVM the exponents of the Gaussian basis are optimized using stochastic techniques. In recent years, the SVM and related methods have been used to perform high-precision variational calculations in atomic, mesoatomic, hypernuclear and multiquark systems [11, 22 24]. The program of Varga and Suzuki [18] was used for the present set of calculations. An initial series of calculations on a variety of related species were performed to validate the method. Results of our calculations for Li +, Li, Li and PsH are shown in table 1 and compared with other accurate nonrelativistic calculations. To simplify comparison with other results in table 1, all results were computed with an infinite nuclear mass. Our calculation for positronium hydride, PsH, agrees with the best previous calculation to within Hartree [11]. Results for Li underestimate the best calculation [17] by less than Hartree. Table 1. Non-relativistic energies (in Hartree) of various atomic systems compared with previous accurate results. In these calculations the nuclear mass has been assumed to be infinite. The number in parentheses refers to the total dimension of the Gaussian basis. Species Present SVM Other calculations Li (300) a PsH (400) b Li (400) c Li (600) d Li + e (800) LiPs (600) a Reference [25]. b Reference [11]. c Reference [16]. d Reference [17]. The convergence of energy of the LiPs system as a function of the dimension of the Gaussian basis is shown in table 2. In contradiction with previous works [14, 15] it is found that the ground state is chemically stable. It proved relatively easy to demonstrate that the system was stable since binding was achieved with less than 200 basis functions. The largest calculation included 600 basis functions, and resulted in a total energy of E = Hartree which is equivalent to a binding energy of ɛ = Hartree. When reference is made to the binding energy of the LiPs system it should be noted that the binding energy is relative to breakup into Li and Ps. It is clear from the convergence pattern shown in table 2 that the present energy does not represent the converged limit for the LiPs total energy. This is not surprising since there are five active particles, four electrons and one positron in the LiPs system. We suspect that the present estimate of the binding energy is accurate to about 10%.

3 L105 Table 2. Convergence of the LiPs energy (in Hartree) as a function of basis size. The third column shows the binding energy (in Hartree) relative to the Li + Ps threshold at Hartree. Basis size Total energy Binding energy Energy expectation values were also computed with the present optimized wavefunctions for a finite mass. The 7 Li nucleus has a mass of M = m e and for this species we obtained E = Hartree for Li, and E = Hartree for LiPs giving a binding energy of ɛ = Hartree. Finite-mass considerations have almost no effect on whether the system will be bound or not. Relativistic effects will also not affect whether binding is possible or not since one estimate of the relativistic energy correction for neutral Li is Hartree [26]. Although the best ab initio estimate of the LiPs binding energy, Hartree, is subject to uncertainties due to incomplete convergence of the LiPs energy, and less importantly subject to relativistic and finite mass corrections, the statement that the system is chemically stable will remain valid under any possible refinements of the model. While the LiPs ground state is chemically stable, it is not stable against electron positron annihilation. The dominant decay process for electron positron annihilation is into two γ - rays. The two-photon annihilation rate Ɣ 2γ was computed using the general formula [11], namely Ɣ 2γ = πα 4 c δ /a 0 = δ s 1. (1) In the above expression, δ is the expectation value of electron positron Dirac δ-function, / δ = δ(r i r 0 ) (2) i where r 0 is the positron coordinate and the r i (i = 1, 2, 3, 4) are the electron coordinates. The value obtained for the LiPs annihilation rate, Ɣ 2γ, was s 1. The annihilation rate for PsH was also computed for validation reasons and the value we obtained, Ɣ 2γ = s 1, was consistent with the best previous estimate [11], namely Ɣ 2γ = s 1. The probability density for finding an electron or positron as a function of the distance from the nucleus is depicted in figure 1. P (r) is the electron probability density and P + (r) is the positron probability density. These functions are normalized so that and 0 0 P (r) dr = 4 (3) P + (r) dr = 1. (4) The electron density exhibits the shell structure of the atom with the probability densities for the core and valence electrons having two distinct maxima. The peak of the probability

4 L106 Figure 1. The electron and positron probability densities, P ± (r), plotted as a function of r (in units of a 0 ). The electron density is represented by the full curve while the positron density is shown by the broken curve. density for the valence electrons occurs near 3.6 a 0. The peak of the probability density for the positron occurs near 5.2 a 0. That the positron orbits at a larger distance than the electrons is expected since the electrons are attracted to the nucleus, while the positron is repelled from the nucleus. The binding of positronium to lithium immediately raises the question as to whether it is possible to bind positronium to the heavier alkali atoms such as sodium and potassium. This question cannot be directly answered by SVM calculations since a full n-body calculation on a system containing 12 electrons and one positron (for sodium) is out of the question. The complexity of the problem needs to be reduced by using the frozen-core approximation before progress can be made. The authors would like to thank Dr D M Schrader for useful correspondence and Dr K Varga for the use of the SVM program. This work was supported by a research grant from the Australian Research Council. References [1] Schrader D M and Y C Jean (eds) 1988 Positron and Positronium Chemistry (Amsterdam: Elsevier) [2] Schrader D M and Wedlich D M 1989 From Atoms to Polymers: Isoelectronic Analogies ed J F Liebman and A Greenburg (New York: Wiley) [3] Schrader D M Bound states of positrons with atoms and molecules: theory Nucl. Instrum. Methods under review [4] Schrader D M, Yoshida T and Iguchi K 1993 J. Chem. Phys [5] Schrader D M, Yoshida T and Iguchi K 1992 Phys. Rev. Lett [6] Yoshida T, Miyako G, Jiang N and Schrader D M 1996 Phys. Rev. A [7] Ore A 1951 Phys. Rev [8] Lebeda C F and Schrader D M 1969 Phys. Rev [9] Clary D C 1976 J. Phys. B: At. Mol. Phys [10] Ho Y K 1986 Phys. Rev. A [11] Frolov A M and Smith VHJr1997 Phys. Rev. A [12] Schrader D M, Jacobsen F M, Frandsen N-P and Mikkelsen U 1992 Phys. Rev. Lett [13] Ryzhikh G G and Mitroy J 1997 Phys. Rev. Lett

5 L107 [14] Harju A, Barbiellini B and Nieminen R M 1996 Phys. Rev. A [15] Saito S L 1995 Chem. Phys. Lett [16] McKenzie D K and Drake GWF1991 Phys. Rev. A 44 R6973 [17] Fischer C F 1993 J. Phys. B: At. Mol. Opt. Phys [18] Varga K and Suzuki Y 1998 Comput. Phys. Commun [19] Varga K and Suzuki Y 1995 Phys. Rev. C [20] Kukulin V I 1975 Izv. Acad. Nauk [21] Kukulin V I and Krasnopolsky V M 1977 J. Phys. G: Nucl. Phys [22] Kopylov V A, Frolov A M and Kolesnikov N N 1984 Izv. Vuzov [23] Kolesnikov N N and Tarasov V I 1982 Sov. J. Nucl. Phys [24] Barbour I M and Ponting D K 1979 Nucl. Phys [25] Pekeris C L 1958 Phys. Rev [26] Davidson E R, Hagstrom S A, Chakaravorty S J, Umar V M and Froese Fischer C 1991 Phys. Rev. A

Positron binding to atomic zinc

J. Phys. B: At. Mol. Opt. Phys. 32 (1999) 1375 1383. Printed in the UK PII: S0953-4075(99)99402-6 Positron binding to atomic zinc J Mitroy and G Ryzhikh Atomic and Molecular Physics Laboratories, Research