Positron binding to atomic zinc

Transcription

1 J. Phys. B: At. Mol. Opt. Phys. 32 (1999) Printed in the UK PII: S (99) Positron binding to atomic zinc J Mitroy and G Ryzhikh Atomic and Molecular Physics Laboratories, Research School of Physical Sciences, Australian National University, Canberra, ACT, 0200, Australia Received 17 November 1998 Abstract. The fixed-core stochastic variational method has been used to predict the existence of a Zne + bound state with a binding energy of Hartree and a 2γ annihilation rate of s 1. The underlying validity of the model Hamiltonian was substantiated by predictions of the binding energies of Zn and Zn +. The convergence of the binding energy was very slow and further minimization of the energy was halted once a formal demonstration of binding had been achieved. The results of a single-positron model calculation of Zne + taken in conjunction with an examination of the convergence pattern suggest that the true binding energy of the underlying model Hamiltonian is Hartree larger than the quoted value. 1. Introduction The question of whether it is possible for positrons to bind to atoms has recently been settled in the affirmative [1 3]. In the first instance, the stochastic variational method (SVM) and related methods [4 8] were used to demonstrate that the energy of the Lie + ground state was lower than the sum of the energies for Li + and positronium ground states [1, 2]. Explicit variational demonstrations of the electronic stability of the ground state of positronic beryllium, i.e. Bee + [8], and metastable positronic helium, i.e. He (3 S e) e + [9], have also been reported. Additional calculations using a modified SVM method which treats the behaviour of the valence particles outside a closed-shell core have predicted binding to heavier atoms such as Na, Mg, Cu and Ag [8 13]. In this paper, the fixed-core stochastic variation method (FCSVM) is used to extend the list of atoms that will bind a positron to include neutral zinc. Zinc is naturally a promising candidate to bind a positron since positron binding is predicted to occur for both beryllium and magnesium. Furthermore, positron binding with a binding energy of Hartree was indicated by the many-body perturbation theory (MBPT) calculations of Dzuba et al [14]. It must be remarked that the MBPT calculation yielded an estimate of the Mge + binding energy which was three times larger than the value seen in the present series of calculations [8, 14, 15]. Positron binding was also indicated by a relativistic polarized orbital calculation (retaining only dipole terms) of positron zinc scattering which gave a positive scattering length of 1144 a 0 [16]. This would imply a binding energy of au. A non-relativistic polarized orbital calculation of positron zinc scattering retaining higher-order multipoles in the potential gave an estimated binding energy of Hartree [17]. Normal address: Faculty of Science, Northern Territory University, Casuarina, NT, 0909, Australia. On leave from: Institute for Nuclear Research, Russian Academy of Science, Moscow , Russia /99/ $ IOP Publishing Ltd 1375

2 1376 J Mitroy and G Ryzhikh 2. Details of the model The calculations for Zne + were performed with the fixed-core stochastic variational method. This method has been used in a number of previous calculations of positron atom and positronium atom complexes. Since the details of this method have been described elsewhere [8], only a minimal description of the fixed-core SVM is given here. The SVM (and FCSVM) diagonalize the working Hamiltonian in a basis of explicitly correlated Gaussian (ECG) functions. Of crucial importance to the success of the SVM is the fact that the Hamiltonian matrix elements for the ECGs are relatively simple to compute. This makes it possible to optimize the exponents of the large-dimension ECG basis using a trial and error procedure. The Hamiltonian for the positron and valence electrons was H = V dir(r 0 ) V dir (r 1 ) V dir (r 2 ) V exc (r 1 ) V exc (r 2 ) 1/r 01 1/r 02 +1/r 12 + V 1pol (r 0 ) + V 1pol (r 1 ) + V 1pol (r 2 ) +V 2pol (r 0, r 1 ) + V 2pol (r 0, r 2 ) + V 2pol (r 1, r 2 ) + λp. (1) In this expression, r 0 is the positron coordinate, while r 1 and r 2 are the coordinates of the valence electrons. (Later in this paper we use r e, r p and r ep to denote the mean e nucleus distance, the mean e + nucleus distance and the mean e e + distance, respectively). The direct interaction between the core and the active electrons and positron was computed from the Zn 2+ 1s 3d orbitals, which were taken from an HF calculation of the 3d 10 4s 2 1 S e ground state. The exchange interaction between the valence electron and the HF core was computed exactly with the only approximations being those inherent in using a basis set expansion. The operator λp was constructed by summing over the orbitals occupied by the core electrons with λp = λ φ i φ i. (2) i This acts as a projection operator to the core orbitals provided the positive constant λ is large enough [8]. The coefficient λ was set to for the present FCSVM and FCSVM pol calculations. The dipole polarizability for the Zn 2+ core was taken as a0 3 [18]. Both one- and two-body polarization potentials were included in the calculation. The polarization potentials are V 1pol (r) = α d (r) (3) 2r 4g2 and V 2pol (r i, r j ) =± α dr i r j ri 3 g(r i )g(r j ). (4) r3 j The sign of V pol is positive for electron positron interactions and negative for electron electron interactions. The cut-off function is g 2 (r) = [ 1 exp ( r 6/ ρ 6)]. (5) The cut-off parameter ρ was set to 1.70 a 0 by fitting fixed-core Hartree Fock calculations of the 3d 10 nl states to the experimental binding energies [19]. The theoretical energy levels are compared with experiment in table 1. Inclusion of the polarization potentials resulted in binding energies that were much closer to experiment. Fixed-core SVM calculations with and without polarization potentials were performed and are referred to as the FCSVM pol and FCSVM models, respectively. Two minor modifications

3 Positron binding to atomic zinc 1377 Table 1. Energies (in Hartree) of the low-lying states of Zn + relative to the energy of the 3d 10 core (E = Hartree). The experimental binding energies (averaged for spin orbit splitting) are taken from [19]. Fixed core State Fixed core +V pol Experiment 4s 2 S e s 2 S e p 2 P o p 2 P o d 2 D e d 2 D e f 2 F o f 2 F o Table 2. Energies (in Hartree) of the Zn + ground state, the Zn ground state and the Zne + ground state relative to the energy of the 3d 10 core. Atom FCSVM FCSVM pol Experiment Zn a Zn a Zne a Reference [19]. were made to the effective Hamiltonian in order to facilitate the SVM calculations [8]. The core orbitals used for the core-exchange and λp interactions were taken from an HF calculation of the 3d 10 4s 2 ground state using an existing Gaussian-type orbital (GTO) basis [20]. Also, the polarization potential was constructed by approximating g(r)/r 3 with a linear combination of Gaussian functions [8]. Because of the slightly different Hamiltonians (e.g. the use of a Gaussian expansion for the polarization potential), the binding energies reported for the Zn + ground state in table 2 are slightly different from those in table 1. The basis set used for neutral zinc included 320 ECGs and the FCSVM and FCSVM pol energies for this basis are given in table 2. The FCSVM pol model energy was Hartree. This energy has converged to an accuracy of 10 5 Hartree. The present FCSVM pol ionization potential of Hartree underestimates the experimental ionization potential of Hartree [19] by about 4%. The non-polarized FCSVM model predictions are clearly inferior to the FCSVM pol predictions. 3. The Zne + ground state The FCSVM and FCSVM pol energies for Zne + relative to the Zn 2+ core are listed in table 2. Binding energies, 2γ annihilation rates (Ɣ 2γ ), and radial expectations are listed in table 3. A total of 1042 ECGs were used in the expansion of the Zne + wavefunction. The binding energies of the Zne + system in the FCSVM and FCSVM pol models were and Hartree, respectively. These energies are not well converged with respect to the inclusion of additional terms in the ECG basis or the further optimization of the existing basis. The calculations on Zne + were extremely time consuming as a result of two factors. First, a total of 41 Gaussians were used in the expansion of the Zn 1s 3d core orbitals. Therefore the computation of the λp and core-exchange matrix elements was very time

4 1378 J Mitroy and G Ryzhikh Table 3. Binding energies for Zne + (in Hartree), the total spin-averaged 2γ annihilation rates (in s 1 ) and the spin-averaged 2γ annihilation rates with the core electrons (in s 1 ). The mean positron nucleus distance r p, the mean electron nucleus distance r e and the mean electron positron distance r ep (all in a 0 ) do not take into consideration the contribution from the core electrons. Model FCSVM FCSVM pol Binding energy Ɣ 2γ (total) Ɣ 2γ (core) r p r e r ep consuming. Secondly, the orthogonality constraints with the six core orbitals resulted in electron wavefunctions with a more complicated nodal structure. The proper representation of these wavefunctions required a larger ECG basis which, in turn, slowed down the calculation. Therefore, it was decided to halt the calculation once a formal demonstration of binding had been achieved. Examination of the convergence pattern, and model potential calculations reported in the next section, suggests that the true binding energies for the present model Hamiltonians could be larger by an amount of the order of Hartree. Although, the Zne + energy has not converged, the FCSVM and FCSVM pol energies for neutral zinc have been computed to an accuracy of about 10 5 Hartree. Therefore, the present calculations provide rigorous evidence of positron zinc binding with respect to the underlying model Hamiltonians. As was the case for Bee + and Mge +, the non-polarized FCSVM predicts a larger positron binding energy than the FCSVM pol model. Even though the FCSVM pol model has an additional polarization potential acting on the positron, the inclusion of core polarization also decreases the effective polarizability of the two valence electrons. The overall polarizability of the core plus valence electrons will therefore decrease and lower the positron binding energy. A similar situation exists in calculations of the electron affinity of calcium where the inclusion of core valence correlations has been shown to decrease the electron affinity [24 27]. The small value of r e (2.85 a 0 ), and the large value of r p (12.38 a 0 ) suggest that the best heuristic description of the Zne + ground state would be as a positron orbiting a polarized Zn atom. The small 2γ annihilation rate (Ɣ 2γ )of s 1 and the large electron positron expectation value, r ep =12.36 a 0, also support a picture of Zne + with a small degree of positronium (Ps) clustering. The value of Ɣ 2γ is the smallest we have computed so far for any positron atom bound state. The contribution of the closed-shell core to Ɣ 2γ was s 1. An examination of the successive estimates of Ɣ 2γ had shown a tendency for Ɣ 2γ to increase as the energy decreased. Therefore, the true value of the FCSVM pol Ɣ 2γ is probably larger than the present value due to the incomplete convergence of the Zne + wavefunction. The size of the 2γ annihilation rate (Ɣ 2γ ) is consistent with the trend for Ɣ 2γ to decrease as the atomic ionization potential (IP) of the parent atom increased [13]. The parent Zn atom has a relatively large IP and therefore should have a small Ɣ 2γ. The relationship between Ɣ 2γ and IP can be seen in figure 1. The explanation advanced for the observed trend was that an electron with a large atomic IP was most likely to be strongly attracted to the core and was therefore less likely to be found in a Ps cluster [13].

5 Positron binding to atomic zinc 1379 Figure 1. Scatter diagram showing Ɣ 2γ (in 10 9 s 1 ) versus atomic IP (Hartree) for a number of positronic ions: He( 3 S e )e +, Lie +, Bee +, Nae +, Mge +, Cue +, Zne + and Age +. The ionization potentials and Ɣ 2γ rates of the various species were derived from FCSVM and FCSVM pol calculations. Results for systems with one valence electron are denoted by circles ( ), while crosses ( ) are used for the systems with two valence electrons. 4. Simple model of Zne + Although the annihilation rate is the smallest rate so far computed in the present series of calculations [8 13], most of the annihilation rate can be attributed to the formation of a positronium cluster. This was shown by constructing a simple single-positron model of Zne + and computing the annihilation rate. The e + potential in the Zne + atom was modelled as V(r)=V d (r) + V pol (r) (6) where the static potential for V d (r) was computed from the HF wavefunction for the Zn 4s 21 S e ground state. The polarization potential V pol (r) has the same functional form as equation (3). Although there is some uncertainty in the dipole polarizability for zinc [22, 23], a value of 45 a 3 0 was adopted in the expectation that the effects of an error in α d would be compensated by the adjustable short-range parameter. Since the positron binding energy is computed by subtracting two energies of comparable size it was not used to tune the potential since the FCSVM and FCSVM pol Zne + energies were not computed with sufficient precision. However, an expectation value like r p is not subject to cancellation errors and during the long optimization runs it had been noted that r p remained relatively stable (e.g. 20% variations). In addition, previous investigations with simple model potentials to represent the interaction between the positron and the atom had shown that tuning the potential to reproduce reasonably precise binding energies resulted in positron wavefunctions that gave good estimates of r p [12, 21]. Therefore, it seems sensible to tune the potential to r p. Tuning r p to 12.4 a 0, resulted in ρ = a 0 and a 2γ annihilation rate of s 1 (and a binding energy of Hartree). This is a factor of five smaller

6 1380 J Mitroy and G Ryzhikh than the FCSVM pol Ɣ 2γ. This reduction is even more marked when the annihilation with the valence 4s 2 electrons is considered in isolation. The value Ɣ 2γ was s 1, an order of magnitude smaller than the FCSVM pol rate. These decreases in the annihilation rate were preserved when different values of r p where used to tune the model potential. A value of ρ = a 0 resulted in r p =11.4 a 0, an annihilation rate of s 1, and a binding energy of Hartree. A value of ρ = a 0 resulted in r p =13.4 a 0,an annihilation rate of s 1, and a binding energy of Hartree. A similar tendency for simple model potential calculations to underestimate the annihilation rate also occurred for Bee +, Mge + and Cue + [12, 21]. The inability of the simple potential model to correctly estimate the annihilation rate reinforces the point that some mechanism for incorporating a Ps cluster in the wavefunction must be included if accurate annihilation rates are to be obtained. The Zne + wavefunction can be written schematically as a linear combination of two terms, = α (A)φ(r p )+β ( A +) ω Ps (R). (7) The first term, (A)φ(r p ), represents a positron attached to the neutral atom core, while the second term, ( A +) ω Ps (R), represents a positronium atom orbiting a residual positively charged core. The simple model potential calculation shows that the contribution to Ɣ 2γ from the first term is small. Under these circumstances the annihilation rate can be approximated as Ɣ 2γ β 2 s 1. (8) The Zne + annihilation rate of s 1 implies that β 0.3 in equation (7). Even though the positron is located at large distances from the nucleus, the Ps cluster still constitutes a significant fraction of the wavefunction. The explicit inclusion of electron positron correlations into the wavefunction is necessary even in systems where the positron and electron are found at quite different distances from the nucleus. Besides giving evidence for the existence of a Ps cluster, the simple model potential can also provide some information about the convergence limit of the binding energies. Given that previous single-positron model calculations have shown that is possible to obtain reasonable estimates of r p by tuning ρ to the positron binding energy, it seems reasonable to invert the procedure and tune ρ to r p to obtain a better estimate of the binding energy. The three different values of ρ gave binding energies of , and Hartree. These estimates should be regarded as giving a rough indication of the energy range that will probably contain the actual FCSVM pol binding energy. 5. Comparison with other calculations There have been three previous calculations [14, 16, 17] indicating positron binding to zinc. The predictive value of these calculations has been uncertain since they were based on perturbation theory and the error in the energy due to theoretical limitations is comparable in size to the predicted binding energies. For example, the polarized orbital calculation of McEachran and Stauffer [17] gives an estimated binding energy of Hartree. However, this calculation yields a dipole polarizability of 54 a0 3, a number that exceeds the known polarizability [22] due to the omission of relativistic effects and 4s 2 +4p 2 configuration mixing in the zero-order wavefunction. The situation is further confused since the latest tabulation [22] lists two different polarizabilities, 47.9 and 37.8 a0 3, each with a quoted uncertainty of 25%. The model potential of equation (6) was used to reproduce the quoted binding energy of Hartree with a polarizability of 54 a0 3. The resulting polarization potential was then

7 Positron binding to atomic zinc 1381 rescaled to find the critical polarizability which would continue to support binding. When α d was reduced to 42 a0 3 the potential could no longer support binding. It is evident that the prediction of McEachran and Stauffer depends crucially on the magnitude of an imperfectly known dipole polarizability. Because of the similarities in the underlying Hamiltonians, the binding energy of McEachran and Stauffer [17] is best compared with the non-polarized FCSVM energy. Although, the two energies, and Hartree, are apparently close together, it must be recalled that the present FCSVM energy could underestimate the binding energy by at least Hartree (the exponents of the ECG basis were optimized for the FCSVM pol Hamiltonian). The calculation of Szmytkowski [16] was a polarized orbital calculation similar to that of McEachran and Stauffer [17]. However, Szmytkowski incorporated relativistic effects into the calculation and truncated the polarization potential after the dipole terms. Although the estimated binding energies of Szmytkowski are smaller (relativistic, Hartree; nonrelativistic, Hartree) this is expected due to the omission of higher-order multipoles in the polarization potential. The comparison of these two polarized orbital calculations demonstrates that higher multipole terms must be included in any calculation of the positron atom interaction that does not explicitly include the Ps A + -type channels. It is difficult to reconcile the MBPT calculation of Dzuba et al [14] with the present calculation. The MBPT calculation does not predict binding when diagrams analogous to the polarized orbital calculations are included in the perturbation expansion. The MBPT calculation only predicts binding (ε = Hartree) when a diagram presenting a Ps cluster is included in the MBPT expansion. The MBPT binding energy is about five times larger than the present binding energy. This is reminiscent of the situation for Mge +, where the MBPT binding energy overestimated the FCSVM pol binding energy by a factor of three. Although the present FCSVM energies have not converged, the FCSVM method does have one very important advantage over the perturbative approaches. The method can treat electron positron correlations explicitly and is potentially capable of generating a wavefunction that is close to an exact solution of the model Hamiltonian. This property is very important for these positronic ion systems which have very strong correlations between the valence electrons and the positron. We would hope that the present and previous [8 13] binding energies can be used to fine tune the details of the perturbative approaches so they can then be applied with more confidence to systems currently not accessible to SVM and FCSVM calculations. 6. Discussion and conclusions Two calculations have been performed that give evidence for the existence of an electronically stable ground state of positronic zinc. The structure and properties of the Zne + system show obvious similarities with the previous analysis for Bee + [8]. Although the predictions of binding are rigorous within the framework of the underlying model, the use of a model Hamiltonian prevents the overall prediction of binding from being rigorous. However, model Hamiltonians similar to the present Hamiltonian have been used to compute accurate energy levels and oscillator strengths for neutral Zn and Zn + [28 31]. Since zinc has a nuclear charge of 30 some consideration might be given to the inclusion of relativistic effects and the influence this might have on the prediction of binding. First, any direct relativistic effect on the positron is unlikely to be important since the nucleus positron interaction is repulsive and therefore the positron does not achieve high speeds in the vicinity of the nucleus. This is substantiated by investigations of positron atom scattering [32, 33]. The main relativistic effect is undoubtedly the indirect effect that would result from the zinc

8 1382 J Mitroy and G Ryzhikh atom having a different charge distribution. Even here, the effects would be minimized as the repulsive positron nucleus interaction would act to keep the positron away from the interior of the atom and thereby make it less sensitive to any relativistic modifications of the core orbitals. The dominant relativistic effects would be those associated with the contraction and tighter binding of the Zn 4s orbital. However, these effects are included to a certain extent in the FCSVM pol calculation since the polarization potential was tuned to the empirical Zn + binding energies. The FCSVM pol model predicts that Zne + is dominated by a configuration best described as a positron bound to the polarized core. However, the computed annihilation rate of s 1 does suggest that a significant fraction of the wavefunction (e.g. 10%) consists of a positronium cluster outside a Zn + 3d 10 4s core. The present calculations have implications for positron binding to other group IIB atoms. Neutral cadmium is naturally a very strong candidate for positron binding since it has a polarizability that exceeds that of zinc [22]. Given the existence of positron binding to zinc, one might even regard the existence of a Cde + ground state as a sure thing. However, the polarizability of mercury is smaller than the polarizability of zinc. In this case it is not possible to make a strong prediction about positron binding, although positron binding is a definite possibility. Acknowledgments The authors would like to thank Mr A Cowie and Mr S Caple of NTU Information Technology Support for providing access to additional computer facilities. This work was supported by a research grant from the Australian Research Council. References [1] Ryzhikh G G and Mitroy J 1997 Phys. Rev. Lett [2] Strasburger K and Chojnacki H 1998 J. Chem. Phys [3] Yuan J, Esry B D, Morishita T and Lin C D 1998 Phys. Rev. A 58 R4 [4] Kukulin V I 1975 Bull. Acad. Sci. USSR [5] Kukulin V I and Krasnopolsky V M 1977 J. Phys. G: Nucl. Phys [6] Varga K and Suzuki Y 1995 Phys. Rev. C [7] Varga K and Suzuki Y 1997 Comput. Phys. Commun [8] Ryzhikh G G, Mitroy J and Varga K 1998 J. Phys. B: At. Mol. Opt. Phys [9] Ryzhikh G G and Mitroy J 1998 J. Phys. B: At. Mol. Opt. Phys [10] Ryzhikh G G, Mitroy J and Varga K 1998 J. Phys. B: At. Mol. Opt. Phys. 31 L265 [11] Ryzhikh G G and Mitroy J 1998 J. Phys. B: At. Mol. Opt. Phys. 31 L401 [12] Ryzhikh G G and Mitroy J 1998 J. Phys. B: At. Mol. Opt. Phys [13] Ryzhikh G G and Mitroy J 1998 J. Phys. B: At. Mol. Opt. Phys [14] Dzuba V A, Flambaum V V, Gribakin G F and King W A 1996 J. Phys. B: At. Mol. Opt. Phys [15] Gribakin G F and King W A 1996 Can. J. Phys [16] Szmytkowski R 1993 Acta Phys. Polon. A [17] McEachran R and Stauffer A D 1998 Nucl. Instrum. Methods B [18] Johnson W R, Kolb D and Huang K-N 1983 At. Mol. Nucl. Data Tables [19] Moore C E 1971 Atomic Energy Levels vol III (Washington, DC: US Govt Printing Office) [20] Partridge H 1989 J. Chem. Phys [21] Bromley MJW,Mitroy J and Ryzhikh G G 1998 J. Phys. B: At. Mol. Opt. Phys [22] Miller T M 1993 CRC Handbook of Chemistry and Physics ed D R Lide and HPRFrederikse (Boca Raton, FL: Chemical Rubber Company) pp [23] Stiehler J and Hinze J 1995 J. Phys. B: At. Mol. Opt. Phys [24] Sundholm D and Olsen J 1994 Chem. Phys [25] van der Hart H W, Laughlin C and Hansen J E 1996 Phys. Rev. Lett

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