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Positronium (Ps) formation for positron impact on metastable hydrogen in 2s state has been studied by using the two-channel, two-center eikonal final state-continuum initial distorted wave (EFS-CDW) method. The differential, integrated, and total cross sections for Ps formation in different states have been calculated from each channel opening thresholds to high energy region. The results are compared with other theoretical calculations available in the literature. For Ps formation in s-state at intermediate and high energies, our results are in good agreement with the prediction of distorted wave theory. Those formed in p-states and the total Ps formation cross sections are reported for the first time. It is shown that the total Ps formation cross sections for positron scattering from H(2s) state are significantly larger at relatively low energies, while smaller at high energies, compared with those obtained from hydrogen in ground state.
Positronium (Ps) formation is one of the most fundamental charge rearrangement processes in positron scattering from atoms, molecules, and solid materials, and it plays important roles in many aspects of researches and applications.[1–4] For Ps formation from the ground state of atoms by positron impact, it has been extensively investigated by various theoretical methods as well as experimental measurements over the past few decades.[5,6] However, for target atoms in excited metastable state, it has rarely been reported in the literature. It is the same situation even for the most simplest target, hydrogen atom. In recent years, the excitation and ionization processes of metastable hydrogen in 2s state by positron impact and many other scattering processes by electron, proton, and anti-proton impact have attracted considerable interests and abundant phenomena were found.[7–10] The Ps formation process of the metastable hydrogen has not been extensively investigated so far.
For positron scattering from the 2s metastable state of H, Ps formation channel is open starting from zero collision energy (−3.4 eV) which is a great challenge for theoretical considerations. The pioneering work on the Ps formation of positron-H(2s) scattering was performed by Fojón et al.[11,12] using the Coulomb Born approximation (CBA). Only the formation of Ps in ground state was investigated and it has been shown that the integrated cross sections for H(2s) is significantly larger than those for H(1s) at low energies. However, their results increase significantly as the incident energy approaches to the threshold from higher energies. Recently, Ghoshal and Mandal[13] successfully developed a distorted-wave theory (DWT) in the momentum space and applied it together with the first Born approximation (FBA) to study Ps formation in an arbitrary s-state from an arbitrary s-state of hydrogen atom by positron impact. Both differential and integrated cross sections were reported for intermediate- and high-energy regions. The interests of Ps formation for metastable hydrogen atoms in weakly coupled and dense quantum plasmas were discussed by Nayek and Ghoshal,[14,15] and it is realized that the accuracy of the cross section in plasma environment depends on its value in pure Coulomb situation. Another interesting metastable target that is worth to be mentioned here is the helium atom in 1s2s3S state for which the Ps formation threshold is also negative (−2.06 eV). A four-body version of the continuum distorted wave-final state (CDWFS) model by Hanssen et al.[16] and the two-center convergent close-coupling (CCC) method by Utamuratov et al.[17] have been applied to such target. It is generally concluded by these authors that the cross sections for positron scattering from this metastable state of helium are significantly larger than those obtained for the ground state, a similar trend as predicted for hydrogen atom. To the best of our knowledge, no direct measurement of the Ps formation for positron impact on metastable state of either H or He is reported up to now.
In our previous work,[18–20] we have developed the two-center, two-channel eikonal final state continuum initial distorted wave (EFS-CDW) method proposed originally by Macri et al.[21,22] to study the Ps formation in arbitrary final states for neutral hydrogen atom and hydrogen-like ions by positron impact from low to high energies. All our previous work were done for targets in their ground states and the calculated results compare nicely with experimental measurements and other state-of-art theoretical methods available in the literature. Great advantages of this method are, e.g., it can be used to calculate Ps formation into arbitrary final states for both neutral and ionized targets and it is more accurate than other perturbation-based theoretical methods at low energies while keeping its simplicity and straightforwardness. In this work, we apply the EFS-CDW method to study the Ps formation process from the metastable hydrogen in 2s state by positron impact. Differential, integrated, and total Ps formation cross sections will be calculated and compared with other theoretical calculations in detail.
Our paper is organized as follows. In Section
A detailed description of the EFS-CDW model to study the Ps formation in arbitrary states by positron scattering from atoms or ions can be found in our previous work.[18–20] Here, we briefly review the essential features of the method. Unlike the process in other distorted-wave formalisms where a distortion potential is constructed initially and from which the distorted wave can be obtained, we firstly define the initial and final states of the scattering system by continuum distorted and eikonal wave functions, respectively, to approximately represent the distortion effects of the Coulomb potentials in both the entry and exit channels,
![]() | (1) |
![]() | (2) |
![](cpb_26_3_033401/cpb_26_3_033401_sec2_ieqn2.gif)
Once the wave functions in both channels are determined, the use of either post or prior form makes no differences in calculation. Here we use the prior form to construct the transition matrix. The distortion potential Wi for the initial wave function
![]() | (3) |
![](cpb_26_3_033401/cpb_26_3_033401_sec2_ieqn4.gif)
The prior form of the exact transition matrix element is given by the two-potential formula[25]
![]() | (4) |
![](cpb_26_3_033401/cpb_26_3_033401_sec2_ieqn5.gif)
![]() | (5) |
The EFS-CDW method has close relationship with the Born approximation due to the fact that it includes the entire first- and second- and partial three- and higher-orders of Born expansion series. It is useful to rewrite the transition matrix element Eq. (
![]() | (6) |
The first amplitude corresponds to the B-CDW model introduced by Bransden, et al.[26] and the second term contains all remaining corrections due to the final state. Reminding the multiple scattering theory of Dewangan and Khadkikar,[27] it can be derived that the first term can be constructed as[28]
![]() | (7) |
![]() | (8) |
![]() | (9) |
![]() | (10) |
![]() | (11) |
![](cpb_26_3_033401/cpb_26_3_033401_sec2_ieqn6.gif)
The differential cross sections provide more details of the scattering mechanism than the integrated ones in investigating the Ps formation process. In Fig.
![]() | Fig. 1. (color online) Differential (a) Ps(1s), (b) Ps(2s), (c) Ps(3s), and (d) Ps(4s) formation cross sections for positron impact on H(2s) at 50 eV. The results of DWT method by Ghoshal and Mandal[13] are shown for comparison. |
It is well known that the minimum in DWT method is formed due to the destructive interference of the higher angular momentum states of the scattered waves, if a partial-wave expansion of the transition matrix is performed.[13] Such structure exists extensively in all kinds of distorted wave methods as well as the first-order Born approximation. In the present EFS-CDW model, the complicated structure at about 45° corresponds entirely on the multiple-scattering mechanism including the Thomas double-scattering, which is a natural result of the second- and higher-orders of Born expansion series. The relationship between the present EFS-CDW method and the Born expansions has been discussed above. Details of the scattering mechanism of Thomas double-scattering in either quantum or classical pictures are available everywhere[19,29,30] and will not be discussed here. Another quite interesting phenomenon is the increasing of oscillation characters of the Thomas peak when Ps is formed in higher s-states. It should be attributed to the radial distributions of the Ps atom in different s-states where multi-shell structures exist. Very similar structures as those in Fig.
It should be mentioned here that although our results differ significantly from the calculations of DWT at intermediate angles, the integrated cross sections are generally the same due to the forward character of the Ps formation process. The differential cross sections are highly peaked at 0° and fall rapidly at large angles. Although the calculation of Ps differential cross sections makes no further difficulties in various theoretical methods, there are enormous difficulties in experimental measurements of such quantity. It is just recently available that the first absolute differential Ps formation cross sections are measured for several noble gas atoms and small molecules.[31] It is highly expected that the differential Ps formation cross sections for H might be available in the near future.
The integrated cross sections are readily obtained by integrating the differential cross sections. In Fig.
![]() | Fig. 2. (color online) Integrated (a) Ps(1s), (b) Ps(2s), (c) Ps(3s), and (d) Ps(4s) formation cross sections for positron impact on H(2s). The results of CBA model by Fojò et al.,[11,12] FBA and DWT methods by Ghoshal and Mandal[13] and Nayek and Ghoshal[14,15] are shown for comparison. |
We show in Fig.
Making summations over all s- and p-state Ps formations, the Ps(n) formation cross sections are obtained approximately and they are displayed in Fig.
We sum all the Ps(
![]() | Fig. 5. (color online) Total Ps formation cross sections for positron impact on H(2s) and H(1s). Also shown in the inset are the comparison of the Ps formation cross sections of H(1s) with other theoretical calculations and experimental measurements. For details of the references, see Fig. 7 of Ref. [18]. |
From Fig.
In this work, we have applied the EFS-CDW model to investigate the differential, integrated, and total Ps formation cross sections for positron scattering from metastable H in 2s state. Our results are compared with other theoretical calculations available in the literature such as CBA, FBA and DWT methods. The calculated differential cross sections agree well with the DWT predictions, however, with a multi-scattering structure appearing at about 45° which is an intrinsic feature of the present model. The integrated cross sections for Ps(1s) formation generally have a similar trend as other calculations at relatively high energies except that, at low energies, our results decrease rapidly while CBA increase continuously when the positron energy approaches to the threshold. More calculations are warranted to shed light on this energy range. In our calculations, Ps(n = 2) formations dominate the total Ps formation cross sections for energies below 10 eV and, in higher-energy ranges, the n−3 scaling law is well kept. Comparison with the Ps formation process for H in ground state shows that the total Ps formation cross sections for positron scattering from H(2s) state are significantly larger at relatively low energies, while smaller at high energies. We expect our work could provide useful informations for further studies.
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