- Email: [email protected]
Positronium scattering from 2s and 3s excited states of hydrogen
Nuclear Instruments and Methods in Physics Research B 192 (2002) 138–145 www.elsevier.com/locate/nimb
Positronium scattering from 2s and 3s excited states of hydrogen P.K. Biswas b
a,b,*
, J.W. Darewych
a
a Department of Physics and Astronomy, York University, Toronto ON, M3J 1P3, Canada Departamento de Fısica, Instituto Tecnol ogico de Aeron autica, Pracea Marechal Eduardo Gomez, 50 S~ ao Jos e dos Campos, SP 12228-901, Brazil
Abstract We investigate ortho positronium (Ps) scattering from H(2s), H(3s) excited states of hydrogen using a model coupled-channel formalism that was previously found to yield correct binding and resonance energies for H(1s) targets [Phys. Rev. A 59 (1999) 2058]. Like Ho, and Yan and Ho, we find several resonances corresponding to PsH thresholds. We also find that the energy of some of these resonant states lie below the corresponding Ps(1s)H(ns) thresholds signifying the possible existence of stable or unstable excited PsH states. In addition, we present elastic, inelastic and quenching cross sections in the energy range 0–30 eV. Ó 2002 Published by Elsevier Science B.V. PACS: 34.90.+q; 36.10.Dr
During the past few years, positronium (Ps) beam studies have become an increasingly important subject owing to the availability of slow Ps beams in the laboratory, and because of their applicational importance both in physics and chemistry [1–6]. The neutral character of Ps and the high interaction affinity of its positron with matter, makes it attractive as a probe. However, the study of the interaction of positronium with atomic species turns out to be complicated due to the underlying many-body nature of the dynamics and associated resonant, binding and annihilation phenomena. Extensive theoretical studies have
* Corresponding author. Address: Departamento de Fısica, Instituto Tecnol ogico de Aeronautica, Pracea Marechal Eduardo Gomez, 50 S~ ao Jose dos Campos, SP 12228-901, Brazil. E-mail address: biswas@fis.ita.br (P.K. Biswas).
concentrated mostly on the PsH system [7–24], due to its relative simplicity. The PsH system has been found to exhibit resonance formation [7,8,12,14,16–20] in various partial waves (S, P, D, E, F, G, H, etc.) of the electronic spin-singlet channel, and the formation of a chemically stable positronium hydride (PsH) system [8,11,13,18–21,25] in the S-wave singlet channel with a binding energy of 1.067 eV [11,21]. We might mention that, apart from the S-, P-, D-, F-, G-, H-wave resonances, Ho [15] also found a resonance corresponding to the doubly excited unstable state H (2s2 ). Resonance formation and binding in PsH have been interpreted [8] to be a manifestation of the attachment of Rydberg states of the positron to the spin-singlet H ion. The lowest order S-wave resonance (we shall call it S(0)), which lies below the PsH threshold, corresponds to the bound state.
0168-583X/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 0 8 5 2 - 2
P.K. Biswas, J.W. Darewych / Nucl. Instr. and Meth. in Phys. Res. B 192 (2002) 138–145
Ho [12] obtained two S-wave resonances S(1) and S(2) in the PsH continuum at 4.004 and 4.944 eV, respectively. This implies that the resonant state S(1) is separated from the bound state S(0) by 4:004 þ 1:067 ¼ 5:071 eV and S(2) by 4:944 þ 1:067 ¼ 6:011 eV. We recall that the threshold for Ps(n ¼ 2) is 5.1 eV and for Ps(n ¼ 3) is 6.044 eV. Yan and Ho [14] find two low-lying P-wave resonances P(1) and P(2) at 4.283 and 5.052 eV, which are separated from the bound state by 5.35 eV (4:283 þ 1:067) and 6.119 eV (5:052 þ 1:067), respectively. Thus both the low lying S- and P-wave resonances appear near the thresholds of Ps(n). We tabulate the results of different calculations together with DE, the separation of S(1) from the bound state, in Table 1. The constancy of DE suggests that the resonance appears in the vicinity of the Ps(n ¼ 2) threshold, implying that, despite the conventional view of PsH as a positron revolving around the H ion, the identity of Ps is retained, more in keeping with a molecule-like picture of PsH. Our interest, here, is to investigate the existence of this and of other resonant states that might correspond to the thresholds of H(n) in PsH. In addition, we wish to determine if such states imply the existence of stable states or unstable excited states (with energies lower than the thresholds), which might cascade to the PsH ground state or dissociate into Ps and H. Therefore, in this work, we investigate Ps scattering from 2s and 3s metastable states of hydrogen. Recently, excited states for PsH with angular momentum ‘ ¼ 1, 2 has been reported [26] and Ps binding with alkali atoms
Table 1 S(1) resonance position and its separation from the bound state energy for the Ps(1s)H(1s) system in various models Eb (or S(0))
DE ¼ S(1) S(0)
4.004
1.067
5.071
4.02
1.05
5.07
4.55
0.634
5.184
4.45
0.672
5.122
S(1) Ps(1s)H(1s) (variational [12,21]) Ps(1s)H(1s) (5-state [20]) Ps(1s)H(1s) (22-state [18]) Ps(1s)H(1s) (stab. method [7])
139
have also been predicted [27–29]. Since the 2s and 3s metastable states of hydrogen mimic the alkali atoms, lithium and sodium, respectively, we expect to find some similarity of our results with those of Ps-Li and Ps-Na scattering. We study elastic and inelastic PsH scattering, including resonance formation, binding, and their possible correlation. In this work, we present results obtained from static exchange, 3-Ps-state [Ps(1s,2s,2p)H(ns); (n ¼ 2, 3)] and 4-state [Ps(1s, 2s, 2p)H(2s,2p)] models. Inclusion of the de-excitation channels for the excited state targets and further expansion of the coupling scheme is being undertaken, and the results will be presented in a subsequent publication. In the coupled-channel (CC) scheme, the total wave function of the PsH system is expanded in terms of the Ps(vm ) and H(/l ) eigenstates, 1 X Xn W ðr1 ; r2 ; xÞ ¼ pffiffiffi Fml ðs2 Þvm ðt2 Þ/l ðr1 Þ 2 m l o S þ ð 1Þ 12 Fml ðs1 Þvm ðt1 Þ/0 ðr2 Þ ; ð1Þ where Fml is the continuum wave function of the moving Ps with respect to the center of mass (fixed at the proton of H). S12 represents the total spin of the electrons at r1 and r2 and takes on the values 0, 1 corresponding to electronic singlet (þ) or triplet () states of the two electrons in the PsH system. In the above equation, si ¼ ðx þ ri Þ=2 and ti ¼ ðri xÞ, where x is the positron co-ordinate relative to the proton. We recast the time-independent Schr€ odinger equation into the following momentum-space Lippmann–Schwinger scattering integral equation [22] (atomic units are used throughout): fl0 m0 ;lm ðkf ;ki Þ ¼ B l0 m0 ;lm ðkf ;ki Þ Z B 00 00 ðkf ;k00 Þfl00 m00 ;lm ðk00 ;ki Þ 1 XX 00 l0 m0 ;l m dk ; 2 2 2p 00 00 k200 00 k00 þ i0 m
m
m l
ð2Þ where f represent the spin-singlet (þ) and spintriplet () amplitudes corresponding to total electronic spin S12 ¼ 0 or 1, and the two sets of equations are solved independently. The input potentials to these two sets of coupled equations are B ¼ BD BE , where BD and BE are the direct
140
P.K. Biswas, J.W. Darewych / Nucl. Instr. and Meth. in Phys. Res. B 192 (2002) 138–145
and exchange parts of the potentials and k2m00 l00 ¼ ð2m= h2 ÞfE m00 !l00 g, where E is the total energy of the system. The symbols m00 and !l00 denote the energies of the intermediate state Ps and H atoms, respectively. The cross sections for the ortho Ps scattering and ortho–para conversion or quenching process are defined in terms of f as 2 drs kf 1 þ ¼ fl0 m0 ;lm ðkf ; ki Þ dXl0 m0 ;lm ki 4 2 3 ð3Þ þ fl0 m0 ;lm ðkf ; ki Þ ; 4 2 drq 1 þ ¼ fl0 m0 ;lm ðkf ; ki Þ fl0 m0 ;lm ðkf ; ki Þ ; 16 dX where rs and rq are the scattering and ortho–para conversion or quenching cross section, respectively. The direct amplitudes, BD , in Eq. (2) are the Born amplitudes, which can be expressed as [30,31] E 4D BD l0 m0 ;lm ðkf ; ki Þ ¼ 2 /l0 ðr1 Þj1 exp ðiq r1 Þj/l ðr1 Þ q
hvm0 ðt2 Þj exp ðiq t2 =2Þ exp ð iq t2 =2Þjvm ðt2 Þi; ð4Þ where q ¼ ki kf and ki , kf are the initial and final momenta of the Ps atom with respect to the target hydrogen. The above form ((4) is arrived at by using co-ordinate transformations from fx; r1 ; r2 g to fr1 ; s2 ; t2 g in the original expression for the direct Born term [30,31], which contain ninedimensional integrals, and by integrating the resulting expression over s2 [30,31]. Proper inclusion of exchange in the CC scheme through antisymmetrization, without encountering overcompleteness of the Hilbert space from non-orthogonal basis sets, is technically very cumbersome. Also, for Ps scattering, the matter of concern is the dominance of the short-range exchange interaction. Using 22-Ps [18] and 14ðPsÞ 14ðHÞ ¼ 196 channels [32], the Belfast group has found that, although their CC results converge with respect to the states of Ps and H, the crucial resonance and binding energies are yet to converge and agree with accurate variational predictions [12,21]. This underscores the need for the inclusion of additional correlation, possibly in the form of H [33] and Ps ion formation. The inclusion of
such states would increase the dimension of the calculational scheme enormously. However, we have seen that tunable model exchange potential [19,20,34,35] can be used successfully in the CC framework to yield converged results [34–39]. Therefore, in this work, we replace the nonorthogonal exchange terms by corresponding nonlocal model exchange potentials, which also take into account the required time reversal symmetry [34], BEl0 m0 ;lm ðkf ; ki Þ ¼
E 4 D /l0 ðrÞj exp ðiq rÞj/l ðr1 Þ hDi
hvm0 ðtÞj expðiq t=2Þjvm ðtÞi; ð5Þ
where hDi ¼ ðk2f þ k2i Þ=8 þ C 2 ½ða2l0 þ a2l Þ=2 þ ðb2m0 þ b2m Þ=2. The expressions a2l =2, a2l0 =2 are the binding energies of the lth and l0 th quantum states of H, and b2m , b2m0 are the binding energies of the initial and final Ps states in atomic units (au), respectively. This approximation can be regarded as a generalization of the Ochkur–Rudge model of exchange [40] to Ps scattering with additional emphasis on time reversal symmetry. The parameter C, introduced in the expression for the exchange potential, serves the purpose of treating the ionization energy parameter of the composite system as a variable, as has been done in the case of electron exchange potentials [41]. We shall present results that demonstrate the utility and sensitivity of this parameter in the CC model. After a partial-wave expansion of the coupled Lippmann–Schwinger Eq. (2), two sets of equations for the singlet (þ) and triplet ðÞ electronic states are obtained, and after discretization they are solved by the method of matrix inversion [22]. For the discretized integral equations, Gaussian mesh points ranging from 32 (for H(1s)) to 48 (for H(3s)) are used in the calculations. The maximum number of partial waves included in the calculations was 15 for the H(2s) target, and 18 for the H(3s) target. The results are found to be very slowly convergent with increasing size and hence with increasing polarizability of the target. The contribution of higher partial waves to the cross sections was included by replacing them with corresponding partial wave Born terms. The truncation of the CC approximation calculation and
P.K. Biswas, J.W. Darewych / Nucl. Instr. and Meth. in Phys. Res. B 192 (2002) 138–145
subsequent replacement by Born partial waves has been done when the Born-augmented cross sections for successive partial waves were found to converge to four significant digits. For higher energies, this convergence is limited to three significant digits. First, we present some results for Ps scattering by ground state hydrogen and compare them with the 22-Ps-coupled-pseudostate calculations [18] to illustrate the reliability of the method used in our calculation. We also demonstrate the effect of varying the parameter C, which appears in our model exchange potential. In Fig. 1, we plot our results for the S-wave singlet cross sections, obtained from a 3-Ps-state calculation of Ps(1s)H(1s) scattering (with the parameter C ¼ 1:0), together with the same from the 22-Ps-coupled-pseudostate calculation [18]. The
141
agreement is acceptable, though the 22-state results are somewhat better converged. In the 3Ps-state model, with C ¼ 1, we find that the sole S-wave resonance in the PsH continuum (which we label S(1)) lies at 4.64 eV, and that the lowest order resonance (S(0)), which lies below the PsH threshold and signifies binding, occurs at 0.165 eV. The resonance energy S(1) in the 22-state calculation is at 4.55 eV. As we gradually lower the value of C from unity, we find that 1. the whole pattern of the S-wave phase shift (including resonances) moves monotonically towards the left on the energy scale, thus simultaneously lowering the energy of the resonance state in the continuum and so increasing the binding energy [19]; 2. the relative separation between the bound and resonant states, in other words between S(0) and S(1), also increases monotonically with decreasing C [19]. Interestingly, at C ¼ 0:785, when the factor in the parenthesis of the expression for hDi becomes 0:77 au, close to the combined ionization energy (0:75 au) of Ps and H, 1 the calculation yields a resonance energy of 4.04 eV and a binding energy of 0.97 eV, that is, a relative separation of 5.01 eV in the 3-Ps-state model. In the 5-state model [20], we find a resonance energy of 4.02 eV and binding energy of 1.05 eV with the (almost unaltered) value of C ¼ 0:784 and a relative separation of 5.071 eV. The accurate variational results are 4.004 eV [12], 1.067 eV [21] and 5.07 eV (see Table 1), respectively. For the excited state hydrogen targets, we present our results for the S-wave singlet resonances in Figs. 2 and 3. For the PsH system, resonances do not appear in the static exchange approximation [20]. However, the resonances manifest
Fig. 1. S-wave spin-singlet cross section (in units of pa20 ) versus incident Ps energy (in eV) for Ps(1s) scattering from ground state hydrogen atoms. Solid curve: present results from 3-Psstate CCA with the model exchange of Eq. (5) (with C ¼ 1:0); dotted curve: results from 22-coupled-pseudostate calculation of [18].
1 This identification with the ionization energy is clear and straightforward in the Ochkur or Rudge modeling of exchange [40]. It also appears in the free electron gas model of exchange [45]. However, it has been found that better result can be obtained by considering the ionization energy as an adjustable parameter in the model potential [41].
142
P.K. Biswas, J.W. Darewych / Nucl. Instr. and Meth. in Phys. Res. B 192 (2002) 138–145
Fig. 2. S-wave spin-singlet cross sections (in units of pa20 ) versus incident Ps energy (in eV) for Ps scattering from metastable H(2s) using (a) 3-Ps-state model with C ¼ 1:0, (b) 3-Ps-state model with C ¼ 0:785, (c) 4-state model with C ¼ 1:0.
Fig. 3. S-wave spin-singlet cross sections (in units of pa20 ) versus incident Ps energy (in eV) for Ps scattering from metastable H(3s) using (a) 3-Ps-state model with C ¼ 1:0, (b) 3-Ps-state model with C ¼ 0:785.
themselves with the inclusion of Ps-excitation channels in the coupling scheme. In Fig. 2(a) and (b) we plot the resonances as they appear for the case of the excited H(2s) target for the 3-Ps-state calculation with C ¼ 1:0 and 0.785. Variation of C from 1.0 to 0.785 shifts the resonance positions from 3.54 and 5.09 to 3.1, 4.98 eV. We note that the shift is lower compared to that for the ground state hydrogen case, signifying a weaker dependence on variations of the parameter C. In all cases, we find that two resonances appear in the continuum below the Ps-excitation threshold for H(2s). This feature does not change even in the 4-state model (Fig. 2(c)). As we search for the T-matrix pole, we find that a third resonant state, which we call S(0), occurs below the Ps(1s)H(2s) threshold, namely by 0.248 eV in the 3-Ps-state approximation, and 0.424 eV in the 4state approximation. These results are suggestive of the existence of unstable excited states of PsH.
We obtain these approximate negative energies for the T-matrix pole using the effective-range expansion [20,42]. The possible existence of metastable states of PsH is interesting and could be verified experimentally by applying an energy of the order of 10.8 eV (0:424 þ 1:067 þ 10:2) to 11.02 eV (0:248 þ 1:067 þ 10:2) to the ground-state PsH system. We plot the S-wave singlet cross sections versus incident Ps energy for the H(3s) target in Fig. 3(a) and (b). For H(3s) targets we find that three Swave singlet resonance states S(0), S(1), S(2) appear in the Ps(1s)H(3s) continuum. For this H(3s) case, the lowest resonance S(0) appears above the Ps(1s)H(3s) threshold. This suggests that excited states corresponding to the Ps(1s)H(3s) configuration do not arise, or at least become imperceptible. For C ¼ 1:0, the resonances for H(3s) target appear at 3.13, 4.57, and 5.14 eV (we refer to them as S(0), S(1) and S(2), respectively).
P.K. Biswas, J.W. Darewych / Nucl. Instr. and Meth. in Phys. Res. B 192 (2002) 138–145
Recently, in a calculation using the stochastic variational method [28], Ps(1s) has been reported to form bound states with lithium and sodium atoms, with binding energies 0.318 and 0.008 eV respectively. Since H(2s) and H(3s) mimic the lithium and sodium targets to a great extent, our present results are consistent with these values of the binding energies. Since the analogy of H(3s) to Na is nevertheless approximate, our failure to detect binding for the H(3s) target does not necessarily prove that Ps will not bind to Na. We now discuss the scattering cross sections for these systems. In Fig. 4, we exhibit elastic, inelastic and total cross sections for the excited H(2s) target, obtained by using the 4-state approximation [Ps(1s,2s,2p)H(2s), Ps(1s)H(2p)]. In this figure we
Fig. 4. Angle integrated cross sections (in units of pa20 ) versus incident Ps energy (in eV) for Ps(1s) scattering from metastable H(2s) targets. Results from 3-Ps-state coupling [Ps(1s,2s,2p)H(2s)] – dotted curve: elastic cross sections. Results from 4-state coupling [Ps(1s,2s,2p)H(2s) þ Ps(1s)H(2s,2p)] – long-dashed curve: elastic transition; short-dashed curve: Ps(1s ! 2s) transition; dot-dashed curve: Ps(1s ! 2p) transition; lower solid curve: H(2s ! 2p) transition; upper solid curve: total cross sections.
143
also plot the elastic scattering results of 3-Ps-state coupling [Ps(1s,2s,2p)H(2s)] in order to exhibit the effect of H(2s,2p) coupling on the elastic channel. The threshold of the H(2s) to H(2p) transition in Ps(1s)H(2s) scattering is zero as long as we assume energy degeneracy of the H(2s,2p) states. We note that the transition channel Ps(1s)H(2s) ! Ps(1s) H(2p) affects the low-energy results very significantly. However, at intermediate and medium energies, the influence of this channel is marginal. The dotted curve, representing the elastic cross sections from the 3-Ps-state model, exhibits some threshold structure near zero energy (0.15–0.4 eV) which almost disappears with the inclusion of the H(2s,2p) states in the scheme. Both approximations show some structure in the cross section in the vicinity of the resonance positions at 3.54 and 5.09 eV. This feature is also observed in the ground state hydrogen target case [18,20]. The Ps(1s ! 2s) and Ps(1s ! 2p) excitation cross sections plotted in Fig. 4 for the scattering from H(2s), are qualitatively similar to those obtained for Ps scattering from lithium [39]. The H(2s ! 2p) excitation cross section has substantial influence on the low energy elastic results, as well as on the total cross sections. The overall qualitative features of the total cross section are similar to the corresponding H(1s) target case, with the exception that the low energy cross sections are very high. However, they fall rapidly as the incident Ps energy increases. This reflects the large size of the excited target atoms. Next, in Fig. 5, we plot the elastic, inelastic and total cross sections for the H(3s) target case in the 3-Ps-state approximation. The elastic cross section for H(3s) is even higher at very low energies (360 pa20 at 0.02 eV) than it was for H(2s). It also decreases more rapidly with increasing energy. This behavior is similar to that observed for alkali atom targets [39,43]. However, the rapid decrease ceases in the intermediate energy region, where the Ps excitations have a large influence on the cross section. The elastic cross section has a similar structure in the vicinity of the resonance energy as was found for H(1s) and H(2s) targets. The Ps(1s ! 2s) and Ps(1s ! 2p) excitation cross sections are qualitatively similar to those observed in the H(2s) target case, but they are comparatively high for the H(3s) target. The reliability of the
144
P.K. Biswas, J.W. Darewych / Nucl. Instr. and Meth. in Phys. Res. B 192 (2002) 138–145
Fig. 5. Angle integrated cross sections (in units of pa20 ) versus incident Ps energy (in eV) for Ps scattering from metastable H(3s) state target using 3-Ps-state [Ps(1s,2s,2p)H(3s)] coupling. Dotted curve: elastic transition; short-dashed curve: Ps(1s ! 2s) transition; dot-dashed curve: Ps(1s ! 2p) transition; solid curve: total cross sections.
total cross section would improve with the inclusion of target excitations and they should be included in any future calculations on this system. We show the results for the quenching or ortho–para conversion cross sections in Fig. 6(a) and (b). The quenching cross sections are found to increase significantly with increase of the target size. From Fig. 6(a), we see that for both targets, H(2s) and H(3s), there is a peak in the quenching cross section, when only coupling of Ps-states is included in the CC scheme. A similar peak, near 1.5 eV, was also reported [18,20] for H(1s) targets. Also, from Fig. 6(a), we find that the position of the peak advances near to the scattering threshold with increase in the size of the target. However, the employment of a 4-state model for the H(2s) target yields a quenching cross section that is high at low energies, and the peak obtained in the 3-Ps-state model disappears. Owing to the similarity between
Fig. 6. Quenching or ortho-para conversion cross section (in units of pa20 ) of ortho Ps(1s) from H(2s) and H(3s) targets as a function of energy. (a) Results for H(2s) (dashed curve) and H(3s) (dashed-dotted curve) targets using 3-Ps-state model with C ¼ 1. (b) Results for H(2s) target using 3-Ps-state (dashed curve) and 4-state (solid curve) models with C ¼ 1.
alkalis and excited state hydrogen, we expect that alkali atoms could act as good quenchers, and hence might help to increase the intensity of an experimental beam of a particular spin. To conclude, we have investigated the scattering of Ps from 2s and 3s excited states of hydrogen using a modified CC formalism with an adjustable non-local model exchange potential. As in previously published variational results [12,14,21], we find that the resonances in the PsH system appear in the vicinity of the excitation thresholds of Ps. The results presented in this work suggest the existence of unstable excited states of PsH, in which the hydrogenic component of the PsH absorbs energy and makes a transition to a (modified) excited state with n ¼ 2, 3 etc., while still remaining bound to the Ps, though with a smaller binding energy. Such behaviour is observed for H(n ¼ 2)
P.K. Biswas, J.W. Darewych / Nucl. Instr. and Meth. in Phys. Res. B 192 (2002) 138–145
targets only. We present scattering and quenching cross sections of Ps from metastable 2s and 3s states of hydrogen. The metastable states of hydrogen mimic the alkali atom targets to a great extent. Therefore, Ps(1s)H(ns) scattering, in addition to its intrinsic interest, can provide useful information and guidelines for the study of Ps-alkali scattering. Experimental studies of the latter are in progress at University College London [44]. We are continuing our study of PsH(ns) collisions, but with inclusion of de-excitation channels for the excited targets. Acknowledgements This work is supported by Natural Sciences and Engineering Research Council of Canada and the FAPESP of Brazil.
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
[29]
References [1] A.J. Garner, G. Laricchia, A. Ozen, J. Phys. B 29 (1996) 5961; A.J. Garner, G. Laricchia, Can. J. Phys. 74 (1996) 518; N. Zafar, G. Laricchia, M. Charlton, A. Garner, Phys. Rev. Lett. 76 (1996) 1595. [2] M. Charlton, G. Laricchia, Commun. At. Mol. Phys. 26 (1991) 253. [3] M. Skalsey, J.J. Engbrecht, R.K. Bithell, R.S. Vallery, D.W. Gidley, Phys. Rev. Lett. 80 (1998) 3727. [4] Y. Nagashima, T. Hyodo, K. Fujiwara, A. Ichimura, J. Phys. B 31 (1998) 329. [5] D.M. Schrader, Y.C. Jean, Positron and Positronium Chemistry, Amsterdam, Elsevier, 1988; D.M. Schrader, Nucl. Instr. and Meth. B 143 (1998) 112. [6] H. Weber et al., Phys. Rev. Lett. 61 (1988) 2542; S. Tang, C.M. Surko, Phys. Rev. A 47 (1993) R743. [7] R.J. Drachman, S.K. Houston, Phys. Rev. A 12 (1975) 885; R.J. Drachman, S.K. Houston, Phys. Rev. A 14 (1976) 894. [8] R.J. Drachman, Phys. Rev. A 19 (1979) 1900. [9] B.A.P. Page, J. Phys. B 9 (1976) 1111. [10] P.A. Fraser, Proc. Phys. Soc. 78 (1961) 329; S. Hara, P.A. Fraser, J. Phys. B 8 (1975) L472. [11] A. Ore, Phys. Rev. 83 (1951) 665. [12] Y.K. Ho, Hyperfine Interact. 73 (1992) 109; Y.K. Ho, Phys. Rev. A 17 (1978) 1675.
[30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]
[41]
[42] [43] [44] [45]
145
Y.K. Ho, Phys. Rev. A 34 (1986) 609. Z.C. Yan, Y.K. Ho, Phys. Rev. A 57 (1998) R2270. Y.K. Ho, Phys. Rev. A. 41 (1990) 68. Y.K. Ho, Z.C. Yan, J. Phys. B 31 (1998) L871. Y.K. Ho, Z.C. Yan, Phys. Rev. A 62 (2000) 052503. C.P. Campbell et al., Phys. Rev. Lett. 80 (1998) 5097. P.K. Biswas, S.K. Adhikari, Chem. Phys. Lett. 317 (1999) 129. S.K. Adhikari, P.K. Biswas, Phys. Rev. A. 59 (1999) 2058. A.M. Frolov, V.H. Smith Jr., Phys. Rev. A 55 (1997) 2662. H. Ray, A.S. Ghosh, J. Phys. B 31 (1998) 4427. P.K. Sinha, A. Basu, A.S. Ghosh, J. Phys. B 33 (2000) 2579. A. Basu, P.K. Sinha, A.S. Ghosh, Phys. Rev. A 63 (2001) 012502. D.M. Schrader et al., Phys. Rev. Lett. 69 (1992) 57. D. Bressanini, M. Mella, G. Morosi, Phys. Rev. A. 57 (1998) 1678. T. Yoshida, G. Miyako, J. Chem. Phys. 107 (1997) 3864. G.G. Ryzhikh, J. Mitroy, J. Phys. B 31 (1998) L103; G.G. Ryzhikh, J. Mitroy, K. Varga, J. Phys. B 31 (1998) 3965. D. Bressanini, M. Mella, G. Morosi, J. Chem. Phys. 108 (1998) 4756; D. Bressanini, M. Mella, G. Morosi, J. Chem. Phys. 109 (1998) 5931. P.K. Biswas, A.S. Ghosh, Phys. Lett. A 223 (1996) 173. M.T. McAlinden, F.G.R.S. MacDonald, H.R.J. Walters, Can. J. Phys. 74 (1996) 434. J.E. Blackwood, M.T. McAlinden, H.R.J. Walters, Phys. Rev. A 65 (2002) 032517. P.K. Biswas, J. Phys. B 34 (2001) 4831. P.K. Biswas, S.K. Adhikari, Phys. Rev. A. 59 (1999) 363. P.K. Biswas, Nucl. Instr. and Meth. B 171 (2000) 135. P.K. Biswas, S.K. Adhikari, J. Phys. B 31 (1998) L315, and L737. P.K. Biswas, S.K. Adhikari, J. Phys. B 33 (2000) 1575. S.K. Adhikari, P.K. Biswas, R.A. Sultanov, Phys. Rev. A 59 (1999) 4829. P.K. Biswas, Phys. Rev. A 61 (2000) 12502. M.R.H. Rudge, Proc. Phys. Soc. London 86 (1965) 763; V.I. Ochkur, Zh. Eksp. Teor. Fiz. 45 (1963) 734 [Sov. Phys. JETP 18 (1964) 503]. M.A. Morrison, L.A. Collins, Phys. Rev. A 17 (1978) 918; T.L. Gibson, M.A. Morrison, J. Phys. B 15 (1982) L221; M.A. Morrison, A.N. Feldt, D. Austin, Phys. Rev. A 29 (1984) 2518. S.S.M. Wong, Introductory Nuclear Physics, Prentice Hall, Englewood cliffs, NJ, 1990. H. Ray, J. Phys. B 32 (1999) 5681. G. Laricchia, private communication. S. Hara, J. Phys. Soc. Jpn. 22 (1967) 710.
Positronium scattering from 2s and 3s excited states of hydrogen P.K. Biswas b
a,b,*
, J.W. Darewych
a
a Department of Physics and Astronomy, York University, Toronto ON, M3J 1P3, Canada Departamento de Fısica, Instituto Tecnol ogico de Aeron autica, Pracea Marechal Eduardo Gomez, 50 S~ ao Jos e dos Campos, SP 12228-901, Brazil
Abstract We investigate ortho positronium (Ps) scattering from H(2s), H(3s) excited states of hydrogen using a model coupled-channel formalism that was previously found to yield correct binding and resonance energies for H(1s) targets [Phys. Rev. A 59 (1999) 2058]. Like Ho, and Yan and Ho, we find several resonances corresponding to PsH thresholds. We also find that the energy of some of these resonant states lie below the corresponding Ps(1s)H(ns) thresholds signifying the possible existence of stable or unstable excited PsH states. In addition, we present elastic, inelastic and quenching cross sections in the energy range 0–30 eV. Ó 2002 Published by Elsevier Science B.V. PACS: 34.90.+q; 36.10.Dr
During the past few years, positronium (Ps) beam studies have become an increasingly important subject owing to the availability of slow Ps beams in the laboratory, and because of their applicational importance both in physics and chemistry [1–6]. The neutral character of Ps and the high interaction affinity of its positron with matter, makes it attractive as a probe. However, the study of the interaction of positronium with atomic species turns out to be complicated due to the underlying many-body nature of the dynamics and associated resonant, binding and annihilation phenomena. Extensive theoretical studies have
* Corresponding author. Address: Departamento de Fısica, Instituto Tecnol ogico de Aeronautica, Pracea Marechal Eduardo Gomez, 50 S~ ao Jose dos Campos, SP 12228-901, Brazil. E-mail address: biswas@fis.ita.br (P.K. Biswas).
concentrated mostly on the PsH system [7–24], due to its relative simplicity. The PsH system has been found to exhibit resonance formation [7,8,12,14,16–20] in various partial waves (S, P, D, E, F, G, H, etc.) of the electronic spin-singlet channel, and the formation of a chemically stable positronium hydride (PsH) system [8,11,13,18–21,25] in the S-wave singlet channel with a binding energy of 1.067 eV [11,21]. We might mention that, apart from the S-, P-, D-, F-, G-, H-wave resonances, Ho [15] also found a resonance corresponding to the doubly excited unstable state H (2s2 ). Resonance formation and binding in PsH have been interpreted [8] to be a manifestation of the attachment of Rydberg states of the positron to the spin-singlet H ion. The lowest order S-wave resonance (we shall call it S(0)), which lies below the PsH threshold, corresponds to the bound state.
0168-583X/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 0 8 5 2 - 2
P.K. Biswas, J.W. Darewych / Nucl. Instr. and Meth. in Phys. Res. B 192 (2002) 138–145
Ho [12] obtained two S-wave resonances S(1) and S(2) in the PsH continuum at 4.004 and 4.944 eV, respectively. This implies that the resonant state S(1) is separated from the bound state S(0) by 4:004 þ 1:067 ¼ 5:071 eV and S(2) by 4:944 þ 1:067 ¼ 6:011 eV. We recall that the threshold for Ps(n ¼ 2) is 5.1 eV and for Ps(n ¼ 3) is 6.044 eV. Yan and Ho [14] find two low-lying P-wave resonances P(1) and P(2) at 4.283 and 5.052 eV, which are separated from the bound state by 5.35 eV (4:283 þ 1:067) and 6.119 eV (5:052 þ 1:067), respectively. Thus both the low lying S- and P-wave resonances appear near the thresholds of Ps(n). We tabulate the results of different calculations together with DE, the separation of S(1) from the bound state, in Table 1. The constancy of DE suggests that the resonance appears in the vicinity of the Ps(n ¼ 2) threshold, implying that, despite the conventional view of PsH as a positron revolving around the H ion, the identity of Ps is retained, more in keeping with a molecule-like picture of PsH. Our interest, here, is to investigate the existence of this and of other resonant states that might correspond to the thresholds of H(n) in PsH. In addition, we wish to determine if such states imply the existence of stable states or unstable excited states (with energies lower than the thresholds), which might cascade to the PsH ground state or dissociate into Ps and H. Therefore, in this work, we investigate Ps scattering from 2s and 3s metastable states of hydrogen. Recently, excited states for PsH with angular momentum ‘ ¼ 1, 2 has been reported [26] and Ps binding with alkali atoms
Table 1 S(1) resonance position and its separation from the bound state energy for the Ps(1s)H(1s) system in various models Eb (or S(0))
DE ¼ S(1) S(0)
4.004
1.067
5.071
4.02
1.05
5.07
4.55
0.634
5.184
4.45
0.672
5.122
S(1) Ps(1s)H(1s) (variational [12,21]) Ps(1s)H(1s) (5-state [20]) Ps(1s)H(1s) (22-state [18]) Ps(1s)H(1s) (stab. method [7])
139
have also been predicted [27–29]. Since the 2s and 3s metastable states of hydrogen mimic the alkali atoms, lithium and sodium, respectively, we expect to find some similarity of our results with those of Ps-Li and Ps-Na scattering. We study elastic and inelastic PsH scattering, including resonance formation, binding, and their possible correlation. In this work, we present results obtained from static exchange, 3-Ps-state [Ps(1s,2s,2p)H(ns); (n ¼ 2, 3)] and 4-state [Ps(1s, 2s, 2p)H(2s,2p)] models. Inclusion of the de-excitation channels for the excited state targets and further expansion of the coupling scheme is being undertaken, and the results will be presented in a subsequent publication. In the coupled-channel (CC) scheme, the total wave function of the PsH system is expanded in terms of the Ps(vm ) and H(/l ) eigenstates, 1 X Xn W ðr1 ; r2 ; xÞ ¼ pffiffiffi Fml ðs2 Þvm ðt2 Þ/l ðr1 Þ 2 m l o S þ ð 1Þ 12 Fml ðs1 Þvm ðt1 Þ/0 ðr2 Þ ; ð1Þ where Fml is the continuum wave function of the moving Ps with respect to the center of mass (fixed at the proton of H). S12 represents the total spin of the electrons at r1 and r2 and takes on the values 0, 1 corresponding to electronic singlet (þ) or triplet () states of the two electrons in the PsH system. In the above equation, si ¼ ðx þ ri Þ=2 and ti ¼ ðri xÞ, where x is the positron co-ordinate relative to the proton. We recast the time-independent Schr€ odinger equation into the following momentum-space Lippmann–Schwinger scattering integral equation [22] (atomic units are used throughout): fl0 m0 ;lm ðkf ;ki Þ ¼ B l0 m0 ;lm ðkf ;ki Þ Z B 00 00 ðkf ;k00 Þfl00 m00 ;lm ðk00 ;ki Þ 1 XX 00 l0 m0 ;l m dk ; 2 2 2p 00 00 k200 00 k00 þ i0 m
m
m l
ð2Þ where f represent the spin-singlet (þ) and spintriplet () amplitudes corresponding to total electronic spin S12 ¼ 0 or 1, and the two sets of equations are solved independently. The input potentials to these two sets of coupled equations are B ¼ BD BE , where BD and BE are the direct
140
P.K. Biswas, J.W. Darewych / Nucl. Instr. and Meth. in Phys. Res. B 192 (2002) 138–145
and exchange parts of the potentials and k2m00 l00 ¼ ð2m= h2 ÞfE m00 !l00 g, where E is the total energy of the system. The symbols m00 and !l00 denote the energies of the intermediate state Ps and H atoms, respectively. The cross sections for the ortho Ps scattering and ortho–para conversion or quenching process are defined in terms of f as 2 drs kf 1 þ ¼ fl0 m0 ;lm ðkf ; ki Þ dXl0 m0 ;lm ki 4 2 3 ð3Þ þ fl0 m0 ;lm ðkf ; ki Þ ; 4 2 drq 1 þ ¼ fl0 m0 ;lm ðkf ; ki Þ fl0 m0 ;lm ðkf ; ki Þ ; 16 dX where rs and rq are the scattering and ortho–para conversion or quenching cross section, respectively. The direct amplitudes, BD , in Eq. (2) are the Born amplitudes, which can be expressed as [30,31] E 4D BD l0 m0 ;lm ðkf ; ki Þ ¼ 2 /l0 ðr1 Þj1 exp ðiq r1 Þj/l ðr1 Þ q
hvm0 ðt2 Þj exp ðiq t2 =2Þ exp ð iq t2 =2Þjvm ðt2 Þi; ð4Þ where q ¼ ki kf and ki , kf are the initial and final momenta of the Ps atom with respect to the target hydrogen. The above form ((4) is arrived at by using co-ordinate transformations from fx; r1 ; r2 g to fr1 ; s2 ; t2 g in the original expression for the direct Born term [30,31], which contain ninedimensional integrals, and by integrating the resulting expression over s2 [30,31]. Proper inclusion of exchange in the CC scheme through antisymmetrization, without encountering overcompleteness of the Hilbert space from non-orthogonal basis sets, is technically very cumbersome. Also, for Ps scattering, the matter of concern is the dominance of the short-range exchange interaction. Using 22-Ps [18] and 14ðPsÞ 14ðHÞ ¼ 196 channels [32], the Belfast group has found that, although their CC results converge with respect to the states of Ps and H, the crucial resonance and binding energies are yet to converge and agree with accurate variational predictions [12,21]. This underscores the need for the inclusion of additional correlation, possibly in the form of H [33] and Ps ion formation. The inclusion of
such states would increase the dimension of the calculational scheme enormously. However, we have seen that tunable model exchange potential [19,20,34,35] can be used successfully in the CC framework to yield converged results [34–39]. Therefore, in this work, we replace the nonorthogonal exchange terms by corresponding nonlocal model exchange potentials, which also take into account the required time reversal symmetry [34], BEl0 m0 ;lm ðkf ; ki Þ ¼
E 4 D /l0 ðrÞj exp ðiq rÞj/l ðr1 Þ hDi
hvm0 ðtÞj expðiq t=2Þjvm ðtÞi; ð5Þ
where hDi ¼ ðk2f þ k2i Þ=8 þ C 2 ½ða2l0 þ a2l Þ=2 þ ðb2m0 þ b2m Þ=2. The expressions a2l =2, a2l0 =2 are the binding energies of the lth and l0 th quantum states of H, and b2m , b2m0 are the binding energies of the initial and final Ps states in atomic units (au), respectively. This approximation can be regarded as a generalization of the Ochkur–Rudge model of exchange [40] to Ps scattering with additional emphasis on time reversal symmetry. The parameter C, introduced in the expression for the exchange potential, serves the purpose of treating the ionization energy parameter of the composite system as a variable, as has been done in the case of electron exchange potentials [41]. We shall present results that demonstrate the utility and sensitivity of this parameter in the CC model. After a partial-wave expansion of the coupled Lippmann–Schwinger Eq. (2), two sets of equations for the singlet (þ) and triplet ðÞ electronic states are obtained, and after discretization they are solved by the method of matrix inversion [22]. For the discretized integral equations, Gaussian mesh points ranging from 32 (for H(1s)) to 48 (for H(3s)) are used in the calculations. The maximum number of partial waves included in the calculations was 15 for the H(2s) target, and 18 for the H(3s) target. The results are found to be very slowly convergent with increasing size and hence with increasing polarizability of the target. The contribution of higher partial waves to the cross sections was included by replacing them with corresponding partial wave Born terms. The truncation of the CC approximation calculation and
P.K. Biswas, J.W. Darewych / Nucl. Instr. and Meth. in Phys. Res. B 192 (2002) 138–145
subsequent replacement by Born partial waves has been done when the Born-augmented cross sections for successive partial waves were found to converge to four significant digits. For higher energies, this convergence is limited to three significant digits. First, we present some results for Ps scattering by ground state hydrogen and compare them with the 22-Ps-coupled-pseudostate calculations [18] to illustrate the reliability of the method used in our calculation. We also demonstrate the effect of varying the parameter C, which appears in our model exchange potential. In Fig. 1, we plot our results for the S-wave singlet cross sections, obtained from a 3-Ps-state calculation of Ps(1s)H(1s) scattering (with the parameter C ¼ 1:0), together with the same from the 22-Ps-coupled-pseudostate calculation [18]. The
141
agreement is acceptable, though the 22-state results are somewhat better converged. In the 3Ps-state model, with C ¼ 1, we find that the sole S-wave resonance in the PsH continuum (which we label S(1)) lies at 4.64 eV, and that the lowest order resonance (S(0)), which lies below the PsH threshold and signifies binding, occurs at 0.165 eV. The resonance energy S(1) in the 22-state calculation is at 4.55 eV. As we gradually lower the value of C from unity, we find that 1. the whole pattern of the S-wave phase shift (including resonances) moves monotonically towards the left on the energy scale, thus simultaneously lowering the energy of the resonance state in the continuum and so increasing the binding energy [19]; 2. the relative separation between the bound and resonant states, in other words between S(0) and S(1), also increases monotonically with decreasing C [19]. Interestingly, at C ¼ 0:785, when the factor in the parenthesis of the expression for hDi becomes 0:77 au, close to the combined ionization energy (0:75 au) of Ps and H, 1 the calculation yields a resonance energy of 4.04 eV and a binding energy of 0.97 eV, that is, a relative separation of 5.01 eV in the 3-Ps-state model. In the 5-state model [20], we find a resonance energy of 4.02 eV and binding energy of 1.05 eV with the (almost unaltered) value of C ¼ 0:784 and a relative separation of 5.071 eV. The accurate variational results are 4.004 eV [12], 1.067 eV [21] and 5.07 eV (see Table 1), respectively. For the excited state hydrogen targets, we present our results for the S-wave singlet resonances in Figs. 2 and 3. For the PsH system, resonances do not appear in the static exchange approximation [20]. However, the resonances manifest
Fig. 1. S-wave spin-singlet cross section (in units of pa20 ) versus incident Ps energy (in eV) for Ps(1s) scattering from ground state hydrogen atoms. Solid curve: present results from 3-Psstate CCA with the model exchange of Eq. (5) (with C ¼ 1:0); dotted curve: results from 22-coupled-pseudostate calculation of [18].
1 This identification with the ionization energy is clear and straightforward in the Ochkur or Rudge modeling of exchange [40]. It also appears in the free electron gas model of exchange [45]. However, it has been found that better result can be obtained by considering the ionization energy as an adjustable parameter in the model potential [41].
142
P.K. Biswas, J.W. Darewych / Nucl. Instr. and Meth. in Phys. Res. B 192 (2002) 138–145
Fig. 2. S-wave spin-singlet cross sections (in units of pa20 ) versus incident Ps energy (in eV) for Ps scattering from metastable H(2s) using (a) 3-Ps-state model with C ¼ 1:0, (b) 3-Ps-state model with C ¼ 0:785, (c) 4-state model with C ¼ 1:0.
Fig. 3. S-wave spin-singlet cross sections (in units of pa20 ) versus incident Ps energy (in eV) for Ps scattering from metastable H(3s) using (a) 3-Ps-state model with C ¼ 1:0, (b) 3-Ps-state model with C ¼ 0:785.
themselves with the inclusion of Ps-excitation channels in the coupling scheme. In Fig. 2(a) and (b) we plot the resonances as they appear for the case of the excited H(2s) target for the 3-Ps-state calculation with C ¼ 1:0 and 0.785. Variation of C from 1.0 to 0.785 shifts the resonance positions from 3.54 and 5.09 to 3.1, 4.98 eV. We note that the shift is lower compared to that for the ground state hydrogen case, signifying a weaker dependence on variations of the parameter C. In all cases, we find that two resonances appear in the continuum below the Ps-excitation threshold for H(2s). This feature does not change even in the 4-state model (Fig. 2(c)). As we search for the T-matrix pole, we find that a third resonant state, which we call S(0), occurs below the Ps(1s)H(2s) threshold, namely by 0.248 eV in the 3-Ps-state approximation, and 0.424 eV in the 4state approximation. These results are suggestive of the existence of unstable excited states of PsH.
We obtain these approximate negative energies for the T-matrix pole using the effective-range expansion [20,42]. The possible existence of metastable states of PsH is interesting and could be verified experimentally by applying an energy of the order of 10.8 eV (0:424 þ 1:067 þ 10:2) to 11.02 eV (0:248 þ 1:067 þ 10:2) to the ground-state PsH system. We plot the S-wave singlet cross sections versus incident Ps energy for the H(3s) target in Fig. 3(a) and (b). For H(3s) targets we find that three Swave singlet resonance states S(0), S(1), S(2) appear in the Ps(1s)H(3s) continuum. For this H(3s) case, the lowest resonance S(0) appears above the Ps(1s)H(3s) threshold. This suggests that excited states corresponding to the Ps(1s)H(3s) configuration do not arise, or at least become imperceptible. For C ¼ 1:0, the resonances for H(3s) target appear at 3.13, 4.57, and 5.14 eV (we refer to them as S(0), S(1) and S(2), respectively).
P.K. Biswas, J.W. Darewych / Nucl. Instr. and Meth. in Phys. Res. B 192 (2002) 138–145
Recently, in a calculation using the stochastic variational method [28], Ps(1s) has been reported to form bound states with lithium and sodium atoms, with binding energies 0.318 and 0.008 eV respectively. Since H(2s) and H(3s) mimic the lithium and sodium targets to a great extent, our present results are consistent with these values of the binding energies. Since the analogy of H(3s) to Na is nevertheless approximate, our failure to detect binding for the H(3s) target does not necessarily prove that Ps will not bind to Na. We now discuss the scattering cross sections for these systems. In Fig. 4, we exhibit elastic, inelastic and total cross sections for the excited H(2s) target, obtained by using the 4-state approximation [Ps(1s,2s,2p)H(2s), Ps(1s)H(2p)]. In this figure we
Fig. 4. Angle integrated cross sections (in units of pa20 ) versus incident Ps energy (in eV) for Ps(1s) scattering from metastable H(2s) targets. Results from 3-Ps-state coupling [Ps(1s,2s,2p)H(2s)] – dotted curve: elastic cross sections. Results from 4-state coupling [Ps(1s,2s,2p)H(2s) þ Ps(1s)H(2s,2p)] – long-dashed curve: elastic transition; short-dashed curve: Ps(1s ! 2s) transition; dot-dashed curve: Ps(1s ! 2p) transition; lower solid curve: H(2s ! 2p) transition; upper solid curve: total cross sections.
143
also plot the elastic scattering results of 3-Ps-state coupling [Ps(1s,2s,2p)H(2s)] in order to exhibit the effect of H(2s,2p) coupling on the elastic channel. The threshold of the H(2s) to H(2p) transition in Ps(1s)H(2s) scattering is zero as long as we assume energy degeneracy of the H(2s,2p) states. We note that the transition channel Ps(1s)H(2s) ! Ps(1s) H(2p) affects the low-energy results very significantly. However, at intermediate and medium energies, the influence of this channel is marginal. The dotted curve, representing the elastic cross sections from the 3-Ps-state model, exhibits some threshold structure near zero energy (0.15–0.4 eV) which almost disappears with the inclusion of the H(2s,2p) states in the scheme. Both approximations show some structure in the cross section in the vicinity of the resonance positions at 3.54 and 5.09 eV. This feature is also observed in the ground state hydrogen target case [18,20]. The Ps(1s ! 2s) and Ps(1s ! 2p) excitation cross sections plotted in Fig. 4 for the scattering from H(2s), are qualitatively similar to those obtained for Ps scattering from lithium [39]. The H(2s ! 2p) excitation cross section has substantial influence on the low energy elastic results, as well as on the total cross sections. The overall qualitative features of the total cross section are similar to the corresponding H(1s) target case, with the exception that the low energy cross sections are very high. However, they fall rapidly as the incident Ps energy increases. This reflects the large size of the excited target atoms. Next, in Fig. 5, we plot the elastic, inelastic and total cross sections for the H(3s) target case in the 3-Ps-state approximation. The elastic cross section for H(3s) is even higher at very low energies (360 pa20 at 0.02 eV) than it was for H(2s). It also decreases more rapidly with increasing energy. This behavior is similar to that observed for alkali atom targets [39,43]. However, the rapid decrease ceases in the intermediate energy region, where the Ps excitations have a large influence on the cross section. The elastic cross section has a similar structure in the vicinity of the resonance energy as was found for H(1s) and H(2s) targets. The Ps(1s ! 2s) and Ps(1s ! 2p) excitation cross sections are qualitatively similar to those observed in the H(2s) target case, but they are comparatively high for the H(3s) target. The reliability of the
144
P.K. Biswas, J.W. Darewych / Nucl. Instr. and Meth. in Phys. Res. B 192 (2002) 138–145
Fig. 5. Angle integrated cross sections (in units of pa20 ) versus incident Ps energy (in eV) for Ps scattering from metastable H(3s) state target using 3-Ps-state [Ps(1s,2s,2p)H(3s)] coupling. Dotted curve: elastic transition; short-dashed curve: Ps(1s ! 2s) transition; dot-dashed curve: Ps(1s ! 2p) transition; solid curve: total cross sections.
total cross section would improve with the inclusion of target excitations and they should be included in any future calculations on this system. We show the results for the quenching or ortho–para conversion cross sections in Fig. 6(a) and (b). The quenching cross sections are found to increase significantly with increase of the target size. From Fig. 6(a), we see that for both targets, H(2s) and H(3s), there is a peak in the quenching cross section, when only coupling of Ps-states is included in the CC scheme. A similar peak, near 1.5 eV, was also reported [18,20] for H(1s) targets. Also, from Fig. 6(a), we find that the position of the peak advances near to the scattering threshold with increase in the size of the target. However, the employment of a 4-state model for the H(2s) target yields a quenching cross section that is high at low energies, and the peak obtained in the 3-Ps-state model disappears. Owing to the similarity between
Fig. 6. Quenching or ortho-para conversion cross section (in units of pa20 ) of ortho Ps(1s) from H(2s) and H(3s) targets as a function of energy. (a) Results for H(2s) (dashed curve) and H(3s) (dashed-dotted curve) targets using 3-Ps-state model with C ¼ 1. (b) Results for H(2s) target using 3-Ps-state (dashed curve) and 4-state (solid curve) models with C ¼ 1.
alkalis and excited state hydrogen, we expect that alkali atoms could act as good quenchers, and hence might help to increase the intensity of an experimental beam of a particular spin. To conclude, we have investigated the scattering of Ps from 2s and 3s excited states of hydrogen using a modified CC formalism with an adjustable non-local model exchange potential. As in previously published variational results [12,14,21], we find that the resonances in the PsH system appear in the vicinity of the excitation thresholds of Ps. The results presented in this work suggest the existence of unstable excited states of PsH, in which the hydrogenic component of the PsH absorbs energy and makes a transition to a (modified) excited state with n ¼ 2, 3 etc., while still remaining bound to the Ps, though with a smaller binding energy. Such behaviour is observed for H(n ¼ 2)
P.K. Biswas, J.W. Darewych / Nucl. Instr. and Meth. in Phys. Res. B 192 (2002) 138–145
targets only. We present scattering and quenching cross sections of Ps from metastable 2s and 3s states of hydrogen. The metastable states of hydrogen mimic the alkali atom targets to a great extent. Therefore, Ps(1s)H(ns) scattering, in addition to its intrinsic interest, can provide useful information and guidelines for the study of Ps-alkali scattering. Experimental studies of the latter are in progress at University College London [44]. We are continuing our study of PsH(ns) collisions, but with inclusion of de-excitation channels for the excited targets. Acknowledgements This work is supported by Natural Sciences and Engineering Research Council of Canada and the FAPESP of Brazil.
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
[29]
References [1] A.J. Garner, G. Laricchia, A. Ozen, J. Phys. B 29 (1996) 5961; A.J. Garner, G. Laricchia, Can. J. Phys. 74 (1996) 518; N. Zafar, G. Laricchia, M. Charlton, A. Garner, Phys. Rev. Lett. 76 (1996) 1595. [2] M. Charlton, G. Laricchia, Commun. At. Mol. Phys. 26 (1991) 253. [3] M. Skalsey, J.J. Engbrecht, R.K. Bithell, R.S. Vallery, D.W. Gidley, Phys. Rev. Lett. 80 (1998) 3727. [4] Y. Nagashima, T. Hyodo, K. Fujiwara, A. Ichimura, J. Phys. B 31 (1998) 329. [5] D.M. Schrader, Y.C. Jean, Positron and Positronium Chemistry, Amsterdam, Elsevier, 1988; D.M. Schrader, Nucl. Instr. and Meth. B 143 (1998) 112. [6] H. Weber et al., Phys. Rev. Lett. 61 (1988) 2542; S. Tang, C.M. Surko, Phys. Rev. A 47 (1993) R743. [7] R.J. Drachman, S.K. Houston, Phys. Rev. A 12 (1975) 885; R.J. Drachman, S.K. Houston, Phys. Rev. A 14 (1976) 894. [8] R.J. Drachman, Phys. Rev. A 19 (1979) 1900. [9] B.A.P. Page, J. Phys. B 9 (1976) 1111. [10] P.A. Fraser, Proc. Phys. Soc. 78 (1961) 329; S. Hara, P.A. Fraser, J. Phys. B 8 (1975) L472. [11] A. Ore, Phys. Rev. 83 (1951) 665. [12] Y.K. Ho, Hyperfine Interact. 73 (1992) 109; Y.K. Ho, Phys. Rev. A 17 (1978) 1675.
[30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]
[41]
[42] [43] [44] [45]
145
Y.K. Ho, Phys. Rev. A 34 (1986) 609. Z.C. Yan, Y.K. Ho, Phys. Rev. A 57 (1998) R2270. Y.K. Ho, Phys. Rev. A. 41 (1990) 68. Y.K. Ho, Z.C. Yan, J. Phys. B 31 (1998) L871. Y.K. Ho, Z.C. Yan, Phys. Rev. A 62 (2000) 052503. C.P. Campbell et al., Phys. Rev. Lett. 80 (1998) 5097. P.K. Biswas, S.K. Adhikari, Chem. Phys. Lett. 317 (1999) 129. S.K. Adhikari, P.K. Biswas, Phys. Rev. A. 59 (1999) 2058. A.M. Frolov, V.H. Smith Jr., Phys. Rev. A 55 (1997) 2662. H. Ray, A.S. Ghosh, J. Phys. B 31 (1998) 4427. P.K. Sinha, A. Basu, A.S. Ghosh, J. Phys. B 33 (2000) 2579. A. Basu, P.K. Sinha, A.S. Ghosh, Phys. Rev. A 63 (2001) 012502. D.M. Schrader et al., Phys. Rev. Lett. 69 (1992) 57. D. Bressanini, M. Mella, G. Morosi, Phys. Rev. A. 57 (1998) 1678. T. Yoshida, G. Miyako, J. Chem. Phys. 107 (1997) 3864. G.G. Ryzhikh, J. Mitroy, J. Phys. B 31 (1998) L103; G.G. Ryzhikh, J. Mitroy, K. Varga, J. Phys. B 31 (1998) 3965. D. Bressanini, M. Mella, G. Morosi, J. Chem. Phys. 108 (1998) 4756; D. Bressanini, M. Mella, G. Morosi, J. Chem. Phys. 109 (1998) 5931. P.K. Biswas, A.S. Ghosh, Phys. Lett. A 223 (1996) 173. M.T. McAlinden, F.G.R.S. MacDonald, H.R.J. Walters, Can. J. Phys. 74 (1996) 434. J.E. Blackwood, M.T. McAlinden, H.R.J. Walters, Phys. Rev. A 65 (2002) 032517. P.K. Biswas, J. Phys. B 34 (2001) 4831. P.K. Biswas, S.K. Adhikari, Phys. Rev. A. 59 (1999) 363. P.K. Biswas, Nucl. Instr. and Meth. B 171 (2000) 135. P.K. Biswas, S.K. Adhikari, J. Phys. B 31 (1998) L315, and L737. P.K. Biswas, S.K. Adhikari, J. Phys. B 33 (2000) 1575. S.K. Adhikari, P.K. Biswas, R.A. Sultanov, Phys. Rev. A 59 (1999) 4829. P.K. Biswas, Phys. Rev. A 61 (2000) 12502. M.R.H. Rudge, Proc. Phys. Soc. London 86 (1965) 763; V.I. Ochkur, Zh. Eksp. Teor. Fiz. 45 (1963) 734 [Sov. Phys. JETP 18 (1964) 503]. M.A. Morrison, L.A. Collins, Phys. Rev. A 17 (1978) 918; T.L. Gibson, M.A. Morrison, J. Phys. B 15 (1982) L221; M.A. Morrison, A.N. Feldt, D. Austin, Phys. Rev. A 29 (1984) 2518. S.S.M. Wong, Introductory Nuclear Physics, Prentice Hall, Englewood cliffs, NJ, 1990. H. Ray, J. Phys. B 32 (1999) 5681. G. Laricchia, private communication. S. Hara, J. Phys. Soc. Jpn. 22 (1967) 710.