1. Introduction
The study of positron scattering processes has found significant applications in a large variety of fields, such as astrophysics, solar physics, bio-medicine, and materials science [1]. Compared to electron scatterings, the collision of the positron with atoms in the gas phase is much more challenging due to the difficulty of producing high-flux, high-energy resolution beams of positrons [2, 3]. Nonetheless, there have been many important studies of positron-atom interactions during the past decades [4], with an emphasis on positron-atom scattering cross sections [3, 5]. However, significant discrepancies in the peak position and magnitude of the cross sections exist between theory and experiment for a number of atomic targets. Reliable theoretical estimations are therefore still needed. Compared to other positron-atom collision processes, the study of inner-shell ionization cross sections are few. The inner-shell ionization has wide applications, e.g., in the production of x-rays [6, 7] and free-electron lasers [8, 9]. It also plays a significant role in analyzing materials and surfaces by different spectroscopic techniques such as Auger-electron spectroscopy, electron probe microanalysis, and electron energy-loss spectroscopy.
Extensive theoretical research has been carried out on positron-atom scattering for the ionization of valence orbitals. For the positron impact ionization of valence-shell electrons at low and intermediate energies, the channel coupling and electron correlation effects are important. As a result, various quantum mechanical methods, e.g., the convergent close-coupling method and the R-matrix method, have been developed to tackle this complexity. The convergent close-coupling method [10, 11] is considered as one of the most sophisticated methods for solving the convergence problem in coupling different reaction channels. The R-matrix method [12, 13] divides the configuration space of the physical system into inner and outer regions and solves each of the domains separately. The optical potential methods [14, 15] have been used to calculate the major discrete or continuum channels that were previously neglected in electron (positron)-atom collision. The coupled-channel optical potential method [16–19] includes an ab initio complex optical potential for the continuum channels, whereas the remaining significant discrete channels are treated by the second-order polarization potentials. For the triple differential cross sections, the Brauner–Briggs–Klar (BBK) model [20] includes the correlated double continuum wavefunction for the final state of the ionization system. The high-energy positron scattering problem can also be solved very well by the distorted-wave Born approximation (DWBA) [21–24] and even the plane-wave Born approximation (PWBA) [25]. Recently, the semi-empirical binary-encounter dipole and binary-encounter Bethe (BEB) models [26, 27] has been extended to positron-atom scatterings by Fedus and Karwasz [28]. In their work, the Wannier-type threshold law developed by Klar [29] was incorporated into the acceleration correction to produce the correct behavior of ionization cross sections at low incident energies, for a variety of atomic and molecular targets.
Until now, however, only a few measurements of atomic inner-shell positron impact ionization cross sections have been performed in the literature [30–33]. The first experiments to study inner-shell processes were carried out by Hansen et al [30]. They used a beta-ray spectrometer to velocity-select electrons and positrons emitted from radioactive sources. Using this method, Seif el Nasr et al [31] measured the positron impact ionization cross sections for the K-shell electrons of Ni, Y, and Ag atoms at 490 and 670 keV and the L-shell electrons of Yb, Ta, Au, and Pb atoms at 490 keV. Schneider et al [32] performed measurements of relative cross sections of K- and L-shell ionization of Ag and Au targets by positron impact at 30 to 70 eV using the low-energy positron source TEPOS. In a recent work, Nagashima et al [33] studied the threshold behavior of the measured ionization cross sections for the Cu K-shell and Ag L-shell using an x-ray detector with a thin crystal.
At relativistic incident energies, the ionization cross sections increase with the positron incident energy, which is known as the ‘relativistic rise' of the ionization cross sections. The previous theoretical study of Khare et al [25] using the PWBA method has obtained the K-, L-, and M-shell ionization cross sections of a number of atoms by positron impact in the energy range from the threshold to 1 GeV. Based on the DWBA method, Sergi et al [23] presented ionization cross sections for K- and L-shells of a few elements in the non-relativistic energy range, where the Dirac–Fock–Slater potential was employed as the distorting potential for the projectile positron. Later, Bote et al [24] derived analytic formulas for the ionization cross sections of the K-, L-, and M-shells of neutral atoms by positron impact with relativistic kinetic energies. In these formulas, the parameters were determined by fitting the cross sections in an extensive database that was compiled by using the distorted-wave and plane-wave Born approximations for low- and high-energy of incident electrons, respectively.
In the present work, we employ the improved relativistic binary-encounter-Bethe (RBEB) model with the Wannier-type threshold law to investigate the positron impact inner-shell ionization cross sections of general atomic targets. A thorough comparison is made with the experimental measurements and other theoretical predictions when they are available.
The rest of the paper is organized as follows. In section 2 , a brief introduction of the BEB model and its relativistic analogy are outlined. In section 3 , the present model is applied to the inner-shell positron impact ionization cross sections for different atoms. Conclusions are presented in section 4 .
2. Theory
The original BEB model developed by Kim et al [26] for electron impact ionization has the advantage that it is parameter-free and easy to use. This model depends on the binding energy ϵb and the kinetic energy $\left\langle {{\boldsymbol{p}}}^{2}/2m\right\rangle $ of the target's orbital and the given incident electron energy ϵi. Kim et al [26] combined the Mott cross section with the dipole term of the Bethe cross section, and such a combination considerably enhances the applicability and accuracy of the estimated cross section. In the original BEB model, the term ${\varepsilon }_{b}+\left\langle {{\boldsymbol{p}}}^{2}/2m\right\rangle $ in the (so-called) ‘Burgess denominator' plays an significant role in the low-energy range.
The BEB cross section for an atomic electron in the (nl)q shell, where q refers to the occupation number, and for the incident electron with energy in binding energy unit of t = ϵi/ϵb is given by
$\begin{eqnarray}\begin{array}{l}{\sigma }_{\mathrm{BEB}}=\displaystyle \frac{4\pi {a}_{0}^{2}q}{{\left({\varepsilon }_{b}/R\right)}^{2}(t+u+1)}\\ \,\times \,\left[\displaystyle \frac{\mathrm{ln}t}{2}\left(1-\displaystyle \frac{1}{{t}^{2}}\right)+1-\displaystyle \frac{1}{t}-\displaystyle \frac{\mathrm{ln}t}{t+1}\right],\end{array}\end{eqnarray}$
where R is the Rydberg energy, a0 is the Bohr radius, and $u=\left\langle {{\boldsymbol{p}}}^{2}/2m\right\rangle /{\varepsilon }_{b}$ represents the kinetic energy of the orbital in the binding energy unit.Recently, Fedus and Karwasz [28] extended the BEB model to calculate positron impact ionization of neutral atoms and molecules by removing all terms that arise from the exchange interaction among the incident electron and target electrons. The positron impact BEB ionization cross section is given by
$\begin{eqnarray}{\sigma }_{\mathrm{BEB}}=\displaystyle \frac{4\pi {a}_{0}^{2}q}{{\left({\varepsilon }_{b}/R\right)}^{2}(t+u+1)}\left[\displaystyle \frac{\mathrm{ln}t}{2}\left(1-\displaystyle \frac{1}{{t}^{2}}\right)+1-\displaystyle \frac{1}{t}\right],\end{eqnarray}$
where u + 1 is often called the acceleration energy in the binding energy unit. In addition, Fedus and Karwasz [28] derived an extra factor $\begin{eqnarray}\gamma =\displaystyle \frac{C}{{\left(t-1\right)}^{1.65}},\end{eqnarray}$
from the Wannier-type threshold law [29, 34] in order to correct the cross section for low incidence energy, and where C is a positive constant that can be adjusted for different inner shells. With these modifications, the BEB ionization cross section with the correction of Wannier-type threshold law is given by $\begin{eqnarray}\begin{array}{l}{\sigma }_{\mathrm{BEB}-{\rm{W}}}=\displaystyle \frac{4\pi {a}_{0}^{2}q}{{\left({\varepsilon }_{b}/R\right)}^{2}(t+u+1+\gamma )}\\ \,\times \,\left[\displaystyle \frac{\mathrm{ln}t}{2}\left(1-\displaystyle \frac{1}{{t}^{2}}\right)+1-\displaystyle \frac{1}{t}\right].\end{array}\end{eqnarray}$
In order to calculate the inner-shell ionization cross sections, Kim et al [27] extend the BEB model to relativistic incident electron energies:
$\begin{eqnarray}\begin{array}{l}{\sigma }_{\mathrm{RBEB}}=\displaystyle \frac{4\pi {a}_{0}^{2}{\alpha }^{4}q}{2b^{\prime} ({\beta }_{t}^{2}+{\beta }_{u}^{2}+{\beta }_{b}^{2})}\\ \ \ \ \ \ \ \times \,\left\{\displaystyle \frac{1}{2}\left[\mathrm{ln}\left(\displaystyle \frac{{\beta }_{t}^{2}}{1-{\beta }_{t}^{2}}\right)-{\beta }_{t}^{2}-\mathrm{ln}(2b^{\prime} )\right]\left(1-\displaystyle \frac{1}{{t}^{2}}\right)\right.\\ \qquad \quad \left.+1-\displaystyle \frac{1}{t}+\displaystyle \frac{b{{\prime} }^{2}}{{\left(1+t^{\prime} /2\right)}^{2}}\displaystyle \frac{t-1}{2}\right\},\end{array}\end{eqnarray}$
where we removed the exchange term. In the above formula, α is the fine-structure constant and all energies are expressed in terms of the ratio of the electron speed to the speed of light c, $\begin{eqnarray}t^{\prime} ={\varepsilon }_{i}/{{mc}}^{2},\ \ \ \ \ \ \ \ \ \ \ \ \ {\beta }_{t}^{2}=1-\displaystyle \frac{1}{{\left(1+t^{\prime} \right)}^{2}},\end{eqnarray}$
$\begin{eqnarray}u^{\prime} =\left\langle {{\boldsymbol{p}}}^{2}/2m\right\rangle /{{mc}}^{2},\ \ \ {\beta }_{u}^{2}=1-\displaystyle \frac{1}{{\left(1+u^{\prime} \right)}^{2}},\end{eqnarray}$
$\begin{eqnarray}b^{\prime} ={\varepsilon }_{b}/{{mc}}^{2},\ \ \ \ \ \ \ \ \ \ \ \ \ {\beta }_{b}^{2}=1-\displaystyle \frac{1}{{\left(1+b^{\prime} \right)}^{2}}.\end{eqnarray}$
In order to ensure the behavior of cross sections to follow the Wannier-type threshold law, the factor γ is incorporated in the RBEB partial ionization cross section, which is given by
$\begin{eqnarray}\begin{array}{l}{\sigma }_{\mathrm{RBEB}-{\rm{W}}}=\displaystyle \frac{4\pi {a}_{0}^{2}{\alpha }^{4}q}{2b^{\prime} ({\beta }_{t}^{2}+{\beta }_{u}^{2}+{\beta }_{b}^{2}+\gamma )}\\ \qquad \quad \times \,\left\{\displaystyle \frac{1}{2}\left[\mathrm{ln}\left(\displaystyle \frac{{\beta }_{t}^{2}}{1-{\beta }_{t}^{2}}\right)-{\beta }_{t}^{2}-\mathrm{ln}(2b^{\prime} )\right]\left(1-\displaystyle \frac{1}{{t}^{2}}\right)\right.\\ \qquad \qquad \left.+1-\displaystyle \frac{1}{t}+\displaystyle \frac{b{{\prime} }^{2}}{{\left(1+t^{\prime} /2\right)}^{2}}\displaystyle \frac{t-1}{2}\right\}.\end{array}\end{eqnarray}$
3. Results and discussions
The RBEB model depends on the binding energy ϵb of each atomic shell. In the following calculations, the orbital binding energies extracted from the X-RAY DATA BOOKLET (https://xdb.lbl.gov/) [35] for some selected neutral targets are shown in tables 1 and 2 for the K- and L- shells, respectively. The average orbital kinetic energies of the atoms are calculated by using the Jena Atomic Calculator (JAC) toolbox [36] in the framework of Dirac–Hartree–Fock theory.
Table 1. Binding energy ϵb (eV), kinetic energy $\left\langle {{\boldsymbol{p}}}^{2}/2m\right\rangle $ (eV), and electron occupation number q for K-shell of the Ni, Cu, Y, and Ag atoms. |
| Element | Subshell | ϵba | $\left\langle {{\boldsymbol{p}}}^{2}/2m\right\rangle $ | q |
|---|---|---|---|---|
| Ni | K | 8333 | 10726 | 2 |
| Cu | K | 8979 | 11562 | 2 |
| Y | K | 17038 | 21930 | 2 |
| Ag | K | 25514 | 33385 | 2 |
aBearden et al [35]. |
Table 2. Binding energy ϵb (eV), kinetic energy $\left\langle {{\boldsymbol{p}}}^{2}/2m\right\rangle $ (eV), and electron occupation number q for L-shell of the Ag, Yb, Ta, Au, and Pb atoms. |
| Element | Subshell | ϵba | $\left\langle {{\boldsymbol{p}}}^{2}/2m\right\rangle $ | q |
|---|---|---|---|---|
| Ag | L1 | 3806 | 6269 | 2 |
| L2 | 3524 | 6949 | 2 | |
| L3 | 3351 | 7010 | 4 | |
| Yb | L1 | 10 486 | 20484 | 2 |
| L2 | 9978 | 20071 | 2 | |
| L3 | 8944 | 15481 | 4 | |
| Ta | L1 | 11 682 | 23323 | 2 |
| L2 | 11 136 | 22788 | 2 | |
| L3 | 9881 | 17055 | 4 | |
| Au | L1 | 14 353 | 30159 | 2 |
| L2 | 13 734 | 29322 | 2 | |
| L3 | 11 919 | 20483 | 4 | |
| Pb | L1 | 15 861 | 34329 | 2 |
| L2 | 15 200 | 33294 | 2 | |
| L3 | 13 035 | 22347 | 4 |
aBearden et al [35]. |
3.1. K-shell ionization
Figure 1 compares the present RBEB-W K-shell (1s1/2) positron impact ionization cross sections of the Ni, Cu, Y, and Ag atoms with the existing measurements by Seif el Nasr et al [31], Schneider et al [32], and Nagashima et al [33]. We also compare the present results with other theoretical predictions, such as the DWBA calculations by Bote et al [24] and the PWBA calculations by Khare et al [25]. For these K-shell cross sections, the adjusting parameter C is set to 0.2 to give a better agreement with the experimental results for all targets. For comparison, we also display in figure 1 the predictions of the original RBEB model to show how the γ factor affects the positron impact cross section over the entire range of impact energies. As can be seen from the differences between the RBEB and RBEB-W models in the low-energy region, the original RBEB model produces a sharp rise at corresponding thresholds, while the RBEB-W model agree very well with the PWBA and DWBA estimations in the low-energy region for all these four atoms. For incident energies larger than 100 keV, the RBEB-W and RBEB results tend to overlap. The peak positions of the present RBEB-W calculations are nearly the same as those of PWBA and DWBA for these atoms, while the peaks of the present calculations are generally the smallest among all theoretical predictions.
Figure 1. K-shell positron impact ionization cross sections of the Ni, Cu, Y, and Ag atoms are shown in (a)-(d), respectively. The results of the present work RBEB-W (black solid curve) and RBEB (green dashed dotted dotted curve) are compared with the DWBA (blue short dashed curve) results by Bote et al [24], and the PWBA (red dashed curve) results by Khare et al [25]. Comparison of the theoretical results with the experimental data by Seif el Nasr et al [31], Schneider et al [32] and Nagashima et al [33] are also shown. |
The DWBA cross sections are obtained by means of two analytical formulas together with a number of parameters. These parameters were obtained by fitting the distorted-wave and plane-wave Born cross sections from the calculation of Bote et al [24]. As a result, the cross sections by DWBA are not smooth in the connection region. It is obvious that the inclusion of the gamma factor of equation (9 ) into the RBEB model is indispensable to reproduce the Wannier-type threshold law for positron scattering. For the Ni atom shown in figure 1(a), the present result is within the error bar of the experimental data by Seif el Nasr et al [31] at the incident energy of 490keV, while the DWBA calculations underestimate the cross sections at 490 and 670 keV. Figure 1(b) shows that all theoretical estimates for the Cu atom are in good agreement with the measurements by Nagashima et al [33], except for the original RBEB results. In figure 1(c), all theoretical results agree well with the experimental data by Seif el Nasr et al [31]. Apparently, the RBEB-W estimations are in better agreement with the DWBA results than the original RBEB predictions at low incident energies. In figure 1(d), the present results are even in better agreement with the experimental data by Schneider et al [32] and Seif el Nasr et al [31] than the DWBA predictions.
3.2. L-shell ionization
Figure 2 compares the present total L-shell (L1, L2, and L3) positron impact ionization cross sections of the Ag, Yb, Ta, Au, and Pb atoms with existing experimental measurements by Seif el Nasr et al [31], Nagashima et al [33], and Schneider et al [32]. We also compare the present results with other theoretical results, such as the DWBA results by Bote et al [24] and the PWBA results by Khare et al [25]. For this shell, the adjusting parameter C is set to 0.5 for all subshells.
Figure 2. L-shell positron impact ionization cross sections of the (a) Ag, (b) Yb, (c) Ta, (d) Au, and (e) Pb atoms, respectively. The sub-figure (f) is for L3-shell ionization of Au. The results of the present work RBEB-W (black solid curve) and RBEB (green dashed dotted dotted curve) are compared with the DWBA (blue short dashed curve) results by Bote et al [24], and the PWBA (red dashed curve) results by Khare et al [25]. Comparison of the theoretical results with the experimental data by Seif el Nasr et al [31], Nagashima et al [33], and Schneider et al [32] are also shown. |
In figure 2(a), the present RBEB cross sections have the lowest peak among the three theoretical estimations, and is the closest to the experimental data by Nagashima et al [33] for the Ag atom. In the intermediate energy range where the cross section has a minimum, the present results are in good agreement with the PWBA predictions, while the DWBA calculations of Bote et al [24] show an unsmooth ‘dip' structure in the cross sections, similar to the situation in the K-shell ionization. Figures 2(b)-(e) show the total L-shell ionization cross sections of the Yb, Ta, Au, and Pb atoms, respectively. The peak values of the present estimations and PWBA results are close to each other. When compared with the experimental data by Seif el Nasr et al [31], both the present result and the existing PWBA predictions are within the error bars, while DWBA underestimate the measurement for the Pb atom. For high incident energies, the cross sections of DWBA are obtained from PWBA calculations. Therefore, the results of DWBA and PWBA are close to each other in the high-energy region. For L3-subshell of the Au atom shown in figure 2(f), all the theoretical estimations agree well with the measurements by Schneider et al [32] at the low incident energies, while the PWBA and DWBA results are closer to each other in the high-energy region. Both the RBEB and RBEB-W models generally predict larger cross sections at high impact energies of the incident positrons.
4. Conclusions
The relativistic BEB model with the Wannier-type threshold law is applied to the partial ionization cross sections of K- and L-shells for a number of atoms by positron impact. The present inner-shell ionization results show good agreement with the experimental data and other theoretical calculations. The constant in the acceleration term derived from the Wannier law is found to be 0.2 and 0.5 for the K- and L-shells, respectively. In our future work, we will consider how to extend the present model to the molecules which contain molecular orbitals arising from the linear combination of atomic core orbitals.