Calculations of positron scattering from atomic oxygen

Abstract

The single-center convergent close coupling (CCC) method has been applied to positron scattering from the oxygen atom. Cross sections have been calculated for total, elastic, momentum transfer, total ionization, inelastic, and excitation scattering processes from threshold to 5000 eV. Due to their relevance in transport studies, we also present stopping power and mean excitation energy results. Low-energy studies have been conducted to calculate scattering length, the hidden Ramsauer–Townsend minimum, and the energy of the positron virtual state. CCC-scaled complex optical potential calculations are utilized to calculate positronium-formation, direct ionization, and values between the positronium-formation and ionization thresholds. Good agreement is generally observed at high energies with past theory, electron-atomic oxygen experiment, and halved electron/positron O$_2$ experiment. Large discrepancies, however, have been found between current and previous calculations for positronium-formation cross section and for low-energy elastic scattering cross section.

Graphic Abstract

1 Introduction

The oxygen atom (O) is a fundamental part of life as it composes molecular oxygen (O$_2$), water (H$_2$O), and many important biomolecules, such as the bases of DNA. Consequently, atomic oxygen is the main component of the human body, comprising approximately 65% of its mass. There is great interest in positron interactions within the human body due to the application of positrons in the biomedical industry, most notably in positron emission tomography (PET) scans [1] and positherapy [2]. Quantifiable data of the scattering processes occurring in these procedures are currently an active area of research [3]. To model positron collisions with the complex molecules present in the body, theoretical methods rely upon accurate cross sections of constituent atoms [4,5,6,7]. Therefore, accurate calculations of positron scattering from the oxygen atom are a necessary step required to facilitate such modeling.

Atomic and molecular oxygen are a fundamental part of the Earth’s atmosphere. Due to the high chemical reactivity of atomic oxygen, it does not exist in substantial quantities close to the Earth’s surface. Instead, it is predominantly found in the upper parts of the atmosphere, where abundant ultraviolet radiation continuously dissociates O$_2$. Accordingly, studies have found atomic oxygen to be the main component of the atmosphere between 200 and 650 km above the Earth’s surface [8, 9], with atmospheric models predicting that, by volume, 96% of the atmosphere in low earth orbit consists of atomic oxygen [10, 11]. As a result, atomic oxygen is essential for cooling the atmosphere through collisions with other atmospheric molecules or direct radiative cooling to the ground state [9].

In the same region where atomic oxygen is dominant, positrons are readily produced through cosmic ray interactions with the atmosphere, terrestrial gamma ray flashes, or via nuclear reactions [12,13,14,15]. These positrons are either destroyed within the atmosphere, escape to space, or can become trapped and accumulated by the inner magnetosphere into a positron radiation belt [13,14,15,16]. Therefore, positron collisions with atomic oxygen are likely most common in the upper atmosphere. Accurate cross sections will, therefore, be essential for modeling positron transport through this region, or in the atmospheres of oxygen-rich bodies such as Venus [17] and Europa [18].

For positron scattering on atomic oxygen, several theoretical methods have been used to obtain a variety of important cross sections for this system. Total cross sections have been calculated by Singh et al. [19] using a spherical complex optical potential (SCOP), Reid and Wadehra [20] using a parameter-free model potential, and by Pindariya et al. [21] using a quantum mechanical formulation based upon a complex atomic spherical potential. Singh and Antony [22] and Pindariya et al. [21] have also obtained results for the total ionization cross section (TICS) through utilization of the complex scattering potential-ionization contribution (CSP-ic) technique [23]. Pindariya et al. [21] obtained results for this cross section only above the ionization threshold, but their approach could not distinguish between positronium-formation and direct ionization. Singh et al. [22], on the other hand, were able to calculate results for positronium-formation and direct ionization separately.

For the elastic cross section, calculations have been completed by Reid and Wadhera [20] and Pindariya et al. [21], using the same methods as before, and also by Dapor and Miotello [24] for incident positron energies between 500 and 4000 eV. Dapor and Miotello [24] have also calculated elastic differential cross sections (DCS) for each of these energies. Total inelastic and the total bound excitation cross sections have been presented solely by Pindariya et al. [21]. Stopping power results have been presented for this system by Gumus et al. [25] from calculations by a generalized oscillator strength (GOS) model and the penelope program [26]. No experimental studies have been conducted for positron scattering from atomic oxygen.

As cross sections for incident positron and electrons are expected to become equivalent at higher energies, direct comparisons can be made for energies above approximately 500 eV. Electron scattering from atomic oxygen has a number of experimental and theoretical results; however, there has been no recent experimental research for the $e^--$O system, even after numerous requests from researchers [27,28,29,30]. Measurements have been taken for total cross sections below the ionization threshold [31,32,33,34], ionization from threshold to 2000 eV [35,36,37], and multiple excitation and optical emission cross sections which are described in several review articles [29, 30, 38]. This system is well documented theoretically, with calculations existing for excitations, optical emission, stopping power, total ionization, and total cross sections from multiple different approaches [27, 28, 39,40,41,42,43,44,45].

We have utilized the single-center CCC method, previously applied to positron scattering from carbon [46], to calculate e$^+$–O scattering. To facilitate calculations of positronium-formation, direct ionization, and cross sections between the positronium-formation and ionization thresholds we utilize a complex model potential calculation scaled to our single-center CCC results. We have addressed discrepancies found for low-energy positronium-formation results for positron scattering on atomic hydrogen [46] within this approach by including a scaled component that is expected to increase the accuracy at energies below the positronium-formation maximum. Therefore, we have obtained total, direct and total ionization, positronium-formation, elastic, total bound-state excitation, stopping power, and elastic differential cross sections for incident energies between threshold and 5000 eV. Quantities of mean excitation energy, scattering length, and the energy of the positron-oxygen virtual state are also obtained.

2 Method

The single-center CCC method is well-documented in the literature for positron scattering from atoms [46,47,48,49,50,51] and molecules [52,53,54]. Therefore, only a brief overview of the CCC formalism is provided.

2.1 Atomic structure calculation

We utilize a configuration interaction (CI) representation of the atomic wave function

(1)

where $\phi _i$ are antisymmetrized $N_{e}$-electron functions, $C_{i}$ are CI coefficients, N is the number of configurations, and $x_{i} = ({\textbf {r}}_{i}, \sigma _{i} )$ represents the spatial (${\textbf {r}}_{i}$) and spin ($\sigma$) coordinates of electron i.

To obtain these CI coefficients the target Hamiltonian (H) is diagonalized with a finite N number of square integrable basis functions. We obtain N target pseudostates ($\Phi ^{N}_{n}$) which, for large N, form a near-complete basis and satisfy

$$\begin{aligned} \langle \Phi ^{N}_{m}|H|\Phi ^{N}_{n} \rangle =\epsilon ^{N}_{n} \delta _{n, m}, \end{aligned}$$

(2)

where $\epsilon ^{N}_n$ is the pseudostate energy and $\delta _{n, m}$ is the Kronecker delta symbol.

2.2 Scattering calculation

The total Hamiltonian of the positron scattering from the target atom,

$$\begin{aligned} H_\textrm{T} = \sum _{i=1}^{N_\text {e}} \left( -\frac{1}{2} \nabla _{i}^{2} - \frac{Z}{r_{i}}\right) + \sum _{i>j=1}^{N_\text {e}} \frac{1}{|\varvec{r}_i-\varvec{r}_j|} - \frac{1}{2}\nabla ^{2}_{0} + V, \end{aligned}$$

(3)

is substituted into the Schrödinger equation

$$\begin{aligned} (H_\textrm{T}-E)|\Psi ^{(+)} \rangle = 0, \end{aligned}$$

(4)

where E is the energy of the collision system, V is the positron-atom interaction potential, and $\Psi ^{(+)}$ is the total scattering wave function.

To solve equation (4) the single-center CCC method relies upon expanding the total scattering wave functions with the previously obtained target pseudostates

(5)

where $F_{n}^{(+)} ({\textbf {r}}_0 )$ is a positron channel function. This expansion is substituted into the Schrödinger equation which can then be transformed through the Green’s function approach into the coupled Lippmann–Schwinger equations for the T matrix

$$\begin{aligned} \langle \varvec{k}_n^{}&\Phi ^{N}_n|T|\Phi ^{N}_i \varvec{k}_i^{} \rangle = \langle \varvec{k}_n^{} \Phi ^{N}_n|V|\Phi ^{N}_i \varvec{k}_i^{} \rangle \nonumber \\&\!+ \sum _{m=1}^{N} \!\int \textrm{d}{\varvec{k}} \frac{\langle \varvec{k}_n^{} \Phi ^{N}_n|V |\Phi ^{N}_m \varvec{k} \rangle \langle \varvec{k}\Phi ^{N}_m|T |\Phi ^{N}_i \varvec{k}_i^{} \rangle }{E +i0-\epsilon ^{N}_n - k^2/2}. \end{aligned}$$

(6)

Here $|\varvec{k}^{}_{n} \rangle$ is a plane wave with energy $k^2_{n}/2$ and V is the interaction potential between the positron and the target.

We solve coupled Lippmann–Schwinger Eq. (6) by expanding in J partial waves of total orbital angular momentum and solving each partial wave separately. Analytical Born completion is also used to increase the efficiency of calculations, as the Born approximation is applicable for high partial waves [55]. When this equation is solved, we obtain the T-matrix elements for each of the pseudostates from which relevant cross sections can be extracted. Mass stopping power and mean excitation energy can be calculated following the procedure described in [53, 54].

2.3 Calculation details

Orbitals 1s to 2p were obtained from a HF calculation of O$^{+}$, optimized on its ground state. All other orbitals were obtained from a Laguerre basis,

$$\begin{aligned} \varphi _{k \ell } (r)&= \sqrt{\frac{\alpha _{\ell }(k-1)!}{(k+\ell )(k+2\ell )!}} (2\alpha _{\ell }r)^{\ell +1} \nonumber \\ {}&\times e^{-\alpha _{\ell } r} L^{2\ell +1}_{k-1}(2\alpha _{\ell }r), \quad k=1, \ldots , N_{\ell }, \end{aligned}$$

(7)

where $L^{2\ell +1}_{k-1}$ are the associated Laguerre polynomials, $\alpha _{\ell }$ are exponential fall-off parameters, and $N_{\ell }$ is the number of functions for each $\ell$. For this calculation, $N_{\ell } = 18-\ell$ with $\alpha _{\ell }=0.4$ for $0 \le \ell \le 4$ and $\alpha _{\ell }=1.0$ for $\ell \le 8$. The present close-coupling expansion includes $2s^2 np^4$, $2s np^5$, and $np^6$. The $2s^22p^3n\ell$ continuum is included for $\ell \le 8$. We also include the $2s2p^4n\ell$ continuum, but with $\ell \le 4$, and $2s2p^3 n\ell ^2$ configurations for orbitals $n\ell$ between 3s and 5s. To limit the size of the calculation the only $2s2p^3 n\ell n' \ell '$ configurations included were $2s2p^3 4\,s 4p$, $2s2p^3 4s4d$, $2s2p^3 4d4f$, $2s2p^3 4p4d$, and $2s2p^3 4d5s$. The restriction of these configurations is expected to slightly decrease the accuracy of the current structure model. States were included with energies up to 75 eV above ionization, resulting in a total of 1543 states.

For energies above 500 eV, calculations were completed with $\ell _\textrm{max}=4$, which is sufficient to obtain convergent results at these energies. Below the positronium-formation threshold, a 1409 state model with $\alpha _{\ell }=5.0$ and $N_{\ell }$=$25-\ell$ containing all states with energies up to 300 eV was used. For energies above 100 eV, TICS and stopping power were extrapolated using a Born model with the same configuration structure as the CCC model but with $N=25-\ell$, the $2s2p^4n\ell$ continuum extended to $\ell \le 8$, and all 3863 generated states included. Calculations are completed to $J=10$ for all energies, except for presented DCS results which were obtained with $J \le 50$ and for low energies where convergence can be achieved for $J \le 5$.

A convergence study for the TICS is presented in Fig. 1. Calculations are fully converged for $\mathrm \ell _{max}= 7$ for energies $\ge 20$ eV, $\mathrm \ell _{max}= 6$ for energies $\ge 100$ eV, and $\mathrm \ell _{max}= 4$ for energies $\ge 300$ eV.

Fig. 1

Convergence study for the total ionization cross section of O where $\ell _\textrm{max}$ represents the maximum $\ell$ in which calculations are completed

2.4 CCC-scaled complex model potential

A full description of the complex potential we utilize for these calculations is provided in [46]. Therefore, only a summary is provided here. We calculate the optical potential ($V_\textrm{opt}$) through,

$$\begin{aligned} V_\textrm{opt}(r,E_i) = V_\textrm{st}(r) + V_\textrm{pol}(r) + iV_\textrm{abs}(r,E_{i}), \end{aligned}$$

(8)

where $V_\textrm{st}$ is the static potential, $V_\textrm{pol}$ the polarization potential, and $V_\textrm{abs}$ the absorption potential. For $V_\textrm{abs}$, we utilize the absorption potential of Staszeweska et al. [56],

$$\begin{aligned} V_\textrm{abs} (r, E_{i})= & {} -\rho (r) \left[ \sqrt{\frac{T_{loc}}{2}} \left( \frac{8\pi }{10k_{F}^{3}(r) E_i} \right) \right. \nonumber \\{} & {} \left. \times \theta (k_{i}^{2} - k_{F}^{2}(r) - 2\Delta ) (A_{1}+A_{2}+A_{3}) \right] . \nonumber \\ \end{aligned}$$

(9)

Here, $\Delta$ is the absorption threshold, $k_{\text {F}}(r)$ is the magnitude of the Fermi wave vector, $T_\text {loc}$ is the local kinetic energy of the positron, and $\theta (x)$ is the Heaviside step function. The dynamic functions $A_{1},A_{2}$, and $A_{3}$ are provided by [56] and are unmodified in the current formulation.

As positronium-formation cannot be explicitly included in the absorption potential, we rely upon the delta variational technique to calculate positronium-formation cross sections. This entails modifying the absorption threshold with the form given by Chiari et al. [57],

$$\begin{aligned} \Delta (E) = \Delta _\text {e} - (\Delta _\text {e} - \Delta _\text {p})e^{-(E_i - \Delta _\text {p})/E_{m}}. \end{aligned}$$

(10)

here $\Delta _\text {e}$ and $\Delta _\text {p}$ are the electronic-excitation and positronium-formation threshold energies, respectively. For the adjustable parameter $E_m$, we have used the energy at which the single-center CCC TCS has a maximum.

To solve the scattering equations for the complex model potential, we solve the Lippmann–Schwinger equations for a potential scattering system to obtain the T-matrix elements. These on-shell partial-wave T-matrix elements are then utilized to obtain the TCS, elastic, and inelastic cross sections. The positronium-formation cross section ($\sigma _\text {Ps}$) is given by the difference between the inelastic cross sections obtained from calculations with $\Delta (E)=\Delta _\text {e}$ and $\Delta (E)$ given by Eq. (10). For energies below the maximum of the positronium-formation, we have modified this approach as described in Sect. 2.5.

We then scale the cross section for inelastic scattering minus positronium-formation to reproduce the total cross section $\sigma _\text {tot}$ obtained from the single-center CCC calculations at high energies. Following this, we scale the positronium-formation cross section $\sigma _\text {Ps}$ to reproduce the single-center $\sigma _\text {tot}$ at the maximum cross section between the ionization threshold and 10 eV above this threshold. The direct ionization is obtained by subtracting the calculated positronium-formation values from our CCC TICS results. To calculate direct ionization for energies below 10 eV above the ionization threshold, we have followed the CSP-ic method [23, 46]. Excitation cross sections are then found by subtracting the direct ionization cross section from the total direct inelastic cross section. Calculations that have utilized this approach to obtain a range, or all, of their values are referred to as CCC-pot.

2.5 Low-energy positronium-formation scaling

Through analysis of calculations for atomic hydrogen, available in appendix of  [46], we found that the CCC-pot approach significantly underestimates two-center calculations for positronium-formation at energies below its maximum. As atomic oxygen has a positronium-formation threshold almost equal to that of atomic hydrogen, we expect this same deficit to occur for this calculation. To accommodate for this, we have extended the CCC-pot calculation to include a scaling factor which increases the positronium-formation component of $V_\textrm{abs}$ for energies between the positronium-formation threshold and the maximum positronium-formation cross section calculated in our CCC-pot method. The values for this factor were determined to produce positronium-formation equal to that of the two-center calculation for atomic hydrogen for energies below the positronium-formation maximum. The unscaled and scaled results for atomic hydrogen are shown in Fig. 2 alongside the two-center CCC calculation [58].

Fig. 2

Impact of scaling factor on CCC-pot positronium-formation cross section compared against two-center CCC results [58]

To allow for calculation at any energy below the maximum of positronium-formation, this factor was then fitted with a function of the form

$$\begin{aligned} F_{s}(E) = \frac{p_{1}E_{f} ^{5} +p_{2}E^{4}_{f} + p_{3}E^{3}_{f}+ p_{4}E^{2}_{f}+p_{5}E_{f} + p_{6}}{E_{f}^{5} + q_{1}E^{4}_{f} + q_{2}E^{3}_{f} + q_{3}E^{2}_{f}+q_{4}E_{f} + q_{5} },\nonumber \\ \end{aligned}$$

(11)

where

$$\begin{aligned} E_{f}(E) = E - E_\textrm{Ps}^\textrm{H}. \end{aligned}$$

(12)

Here, E is the incident energy and $E_\textrm{Ps}^\textrm{H}$ is the positronium-formation threshold of atomic hydrogen, with both units in eV. The parameters for Eq. 11 are shown in Table 1.

Table 1 Fit parameters

Table 2 Excitation energies (eV) for atomic oxygen triplet target bound states

To apply this to atoms beyond atomic hydrogen, we have formulated an approach based upon the difference in energy between the positronium-formation threshold and maximum for each atom. This replaces Eq. 12 with,

$$\begin{aligned} E_{f}(E) = \frac{E -E_\textrm{Ps}^\textrm{A}}{U_{s}} \end{aligned}$$

(13)

where

$$\begin{aligned} U_{s} = \frac{E^{A}_\textrm{max} - E_\textrm{Ps}^\textrm{A} }{ E^{H}_\textrm{max} - E_\textrm{Ps}^\textrm{H}}. \end{aligned}$$

(14)

Here, $E_\textrm{Ps}^\textrm{A}$ is the positronium-formation threshold for atom A and $E^{A}_\textrm{max}$ refers to the energy that the maximum positronium-formation is found for this atom, with $E^{H}_\textrm{max}$ representing this for the hydrogen atom. For Eq. 13 units are in eV, whereas Eq. 14 is dimensionless. It is expected that this approach will decrease errors at lower incident energies and better approximate two-center CCC calculations.

3 Results

3.1 O structure

Excitation energies for triplet target states of the atomic oxygen atom are presented in Table 2. Agreement between the current calculation and past theory and experiment is satisfactory, with differences between NIST [59] and CCC ranging from 0.002 to 0.278 eV. Oscillator strengths for several transitions between triplet bound states are shown in Table 3. Good agreement is found with NIST and past theory for all presented transitions except for the $2p^33s$ $^3D^{0}$ transition, though this is a weak transition with a small oscillator strength. This discrepancy is expected to mainly result from the restrictions imposed on the included configurations.

Table 3 Oscillator strengths for bound triplet states of O

An important quantity for scattering, mostly at low incident energies, is the dipole polarizability ($\alpha _{D}$). For O, this is predicted by experiment to be $5.2 \pm 0.4$ a$_{0}^{3}$ [60]. Our largest 3863 state Born model had $\alpha _{D}=5.50$ a$_{0}^3$, which is within experimental error and slightly higher than other theory [40]. Our 1543 state model had a dipole polarizability of 6.00 a$_{0}^3$ and our small-energy 1490 state model of 5.31 a$_{0}^3$, which is within 3% of the largest model and within experimental uncertainty.

Fig. 3

Total cross section for positron scattering on oxygen. Theoretical CCC results are shown alongside optical potential calculations by Singh et al. [19], Reid and Wadehra [20], and Pindariya et al. [21]. Experimental O$_2$ results for positron scattering are from Charlton et al. [62], Chiari et al. [57], and Dababneh et al. [63]. Electron O$_2$ results are by García et al. [64]. The vertical dotted lines represent the positronium-formation and ionization thresholds

3.2 O scattering

3.2.1 Total cross section

The total cross section arising in the CCC calculation of atomic oxygen is presented in Fig. 3 alongside the theoretical calculations of Singh et al. [19], Reid and Wadehra [20], and Pindariya et al. [21]. In this case, the CCC-pot is equivalent to the 1490 state CCC calculation below the positronium-formation threshold and the 1543 state model above 23.5 eV. The calculation is equivalent to the CCC-scaled complex model potential between these two energies. Due to the lack of existing experimental data for electron and positron scattering from atomic oxygen, we compare against experimental results for positron and electron results for O$_2$, which are divided by 2, in accordance with the additivity rule. This rule is expected only to be relevant for higher energies. Therefore, only qualitative comparisons can be made for lower energies.

CCC results are higher than the halved positron O$_2$ experimental results of Charlton et al. [62] and Dababneh et al. [63] for energies above 40 eV. However, comparison with the more recent measurements of Chiari et al. [57], shows near-perfect agreement. For high incident energies, positron and electron results are expected to become equal to the Born approximation, which is independent of projectile charge. We observe this for energies above 1000 eV, with excellent agreement found with the halved electron O$_2$ measurements of García et al. [64].

The peak cross section of the CCC TCS is of similar magnitude to the calculation of Pindariya et al. [21] and the halved positron O$_2$ experiment of [57]. However, the results of Singh et al. [19] are much larger than our CCC result at the peak TCS and also depicts significantly different behavior for lower energies. At low energies, our results behave similarly to O$_2$ experiment, with results flat from 1 eV to the positronium-formation threshold. On the other hand, Singh et al. [19] predict the cross section to decrease significantly from 1 eV to the positronium threshold and the experiment of Dababneh et al. [63] predicts the cross section to increase over this energy range. Consequently, this calculation predicts a value at 1 eV over five times higher than the CCC result. We expect that CCC results converge to exact results at energies below the positronium-formation threshold, given in Fig. 3. Therefore, the discrepancy is likely due to the polarization potential being too high in the model potential or from the exclusion of virtual positronium-formation in the SCOP approach. A similar discrepancy was noted between the CCC and results of Singh et al. [19] for the carbon atom [46], but not as pronounced.

3.2.2 Total ionization cross section

The total ionization (electron-loss) cross section (TICS) for positron scattering on oxygen, which is equivalent to the sum of direct ionization and positronium-formation, is shown in Fig. 4. As with the total cross section, for energies below 23.5 eV the CCC-pot results are from the CCC-scaled complex model potential, whereas, for energies above this limit they are from the single-center CCC calculation. The results of [21] are in close perfect agreement with those of CCC from near the direct ionization threshold through to high energies. The results of [22] are considerably higher for energies above 10 eV.

Fig. 4

Total ionization (electron-loss) cross section for positron scattering on oxygen. Current CCC results are presented alongside the calculations of [19] and [21]. The CCC-pot calculations incorporate an estimate of the positronium-formation cross section contribution from its threshold, see text

Fig. 5

Direct ionization cross section for positron scattering on oxygen. Theoretical results include CCC-pot and Singh and Antony [22] calculations. Experimental measurements for electron scattering from Zipf [36], Brook et al. [35], and Thompson et al. [37] are expected to converge to positron scattering ones at high energies

3.2.3 Direct ionization and positronium-formation cross section

The $e^{+}-\textrm{O}$ direct ionization cross section is presented in Fig. 5. This includes the current CCC-pot results alongside the theoretical calculation of Singh and Antony [22] and $e^{-}-\textrm{O}$ experiment. At high energies, electron and positron results are expected to become equal. We find this to occur by 1000 eV, with excellent agreement between the CCC and electron experiment above this energy. Similar to the TICS, the calculation of Singh and Antony [22] is significantly larger than the CCC calculation for incident energies above 35 eV. Cross sections for positronium-formation in $e^{+}-\textrm{O}$ scattering are presented in Fig. 6. Current CCC-pot results are presented together with the only other theoretical calculation, by Singh and Antony [22]. The experimental results are for the O$_2$, and have therefore been divided by two. The positronium-formation threshold energy of O$_2$ is lower than that of O, which is why near-threshold energies, there is a discrepancy in the behavior of the O$_2$ experiment and theoretical values for O. Compared to the halved O$_{2}$ experiment, we find agreement with the measurements of Marler and Surko [65] for energies above 40 eV, and Griffith [67] above 100 eV. Apart from the lower energies the results of Singh and Antony [22] are generally considerably above those of CCC-pot. A similar disagreement was found comparing these approaches for atomic carbon [46]. The absence of existing experiment for atomic oxygen makes it difficult to judge the true accuracy of the CCC-pot calculations, particularly at the lower energies. However, as our calculation is scaled directly to the ab-initio CCC calculation and is in agreement with halved O$_2$ experiment at the higher energies, we believe lower energy CCC-pot cross section uncertainty is no more than 20%.

Fig. 6

Positronium-formation cross section for positron scattering on oxygen. Theoretical results include CCC-pot and Singh and Antony [22] calculations. Halved experimental O$_{2}$ measurements are from Marler and Surko [65], Archer et al. [66], and Griffith [67]

3.2.4 Inelastic cross section

Figure 7 presents the total inelastic cross section for positron scattering on oxygen from its threshold to 5000 eV. Above 500 eV, close agreement is found between the CCC and the calculation of Pindariya et al. [21]. The CCC is slightly above the results of Pindariya et al. [21] between 100 and 500 eV. Below this, Pindariya et al. [21] are notably larger than the CCC results, except for below 20 eV, where they are smaller. The calculation of Reid and Wadehra [20] overestimates other theory over their calculated energy range of 100–5000 eV.

Fig. 7

Total inelastic cross section for positron scattering on oxygen. CCC results are shown alongside those of Pindariya et al. [20, 21]

3.2.5 Elastic and momentum transfer cross section

The integrated elastic cross section is shown in Fig. 8 alongside the theoretical calculations of Reid and Wadehra [20], Pindariya et al. [21], Dapor and Miotello [24], and, for the electron case, NIST. Above 500 eV, we find near-perfect agreement with the calculations of Dapor and Miotello [24] and above 3000 eV with the NIST electron results. Comparisons are also made with the halved electron experimental results of Dapor and Miotello [68] for O$_2$. At 1000 eV, the electron and positron theory lies within the experimental uncertainty.

In Fig. 9 we present the momentum transfer cross section for this system. Current results are compared with the calculations of Dapor and Miotello [24], the electron calculations of NIST, and halved electron-O$_2$ experimental data of Iga et al. [68]. As with the elastic cross section, for energies above 500 eV excellent agreement is observed between the CCC and the calculations of  Dapor and Miotello [24]. The NIST results for the electron case are larger than the positron theoretical results, with no agreement viewed by 5000 eV. The experimental data are found to lie between the positron and electron calculations.

Fig. 8

Total elastic cross section for positron scattering on oxygen. Theoretical results include the CCC, Reid and Wadehra [20], Pindariya et al. [21], and Dapor and Miotello [24]. Experimental results for positrons for electrons are from Iga et al. [68]. The vertical dotted lines represent the positronium-formation and ionization thresholds

Fig. 9

Momentum transfer cross section for positron scattering on oxygen. CCC results are presented alongside the positron calculation of [24] and the NIST electron results [59]. Halved O$_2$ experimental results for electrons are from Iga et al. [68]. The vertical dotted lines represent the positronium-formation and ionization thresholds

The DCS for various energies between 500 and 3500 eV are shown in Fig. 10. As with the integrated cross section, excellent agreement is viewed with the calculations of Dapor and Miotello [24] for each of these energies. In Fig. 11 we present a selection of elastic DCS for incident energies between 1 and 100 eV. A technique similar to Green et al. [69] is used to ensure convergence in the partial wave expansion, resulting in a cusp at low scattering angles for incident energies above 1 eV. As with carbon [46], this feature is not present at lower energies due to the triplet ground state resulting in p-wave scattering being dominant at low energies.

Fig. 10

High-energy elastic differential cross sections for positron scattering on oxygen. CCC results are presented alongside the theoretical calculations of Dapor and Miotello [24]

Fig. 11

Elastic differential cross sections for positron scattering on oxygen for energies between 1 and 100 eV

3.2.6 Low-energy study

Elastic and momentum transfer cross sections for energies below 1 eV are shown in Fig. 12. Although the IAM approximation does not hold for low energies, we find good agreement with the halved O$_2$ experiment from Chiari et al. [57]; this experiment does not have forward-scattering corrections so likely underestimates the actual result for O$_2$. However, similar behavior exhibited provides some evidence of the accuracy of the current results. At lower energies, as scattering becomes isotropic, the elastic and momentum transfer cross sections are expected to become equal. We observe this for energies below $10^{-3}$ eV.

Using,

$$\begin{aligned} \sigma _\textrm{el} = 4 \pi A^2, \end{aligned}$$

(15)

where A is the scattering length, and the asymptotic value of our elastic cross section at $10^{-5}$ eV, we determine $A=-1.862$ for this system, where the sign is derived from our phase shift at low energies [46]. As |A| is larger than the mean radius of O, the scattering cross section is greater than the geometric size of the atom at low energies. This results from the elastic cross section being enhanced due to a virtual level of the positron projectile [70]. The energy of this can be calculated through

$$\begin{aligned} \epsilon = \frac{1}{2A^2}, \end{aligned}$$

(16)

which for this system is $\epsilon =$ 3.92 eV.

Fig. 12

Elastic and momentum transfer cross sections for positron scattering on atomic oxygen at energies below 1 eV. Halved O$_2$ experimental results are from Chiari et al. [57]

As the scattering length is negative this implies a Ramsauer–Townsend minimum that occurs at incident energy [71]

$$\begin{aligned} E_\text {min} = \left( \frac{e^2}{2a_{0}} \right) \left( \frac{3 |A| a_{0}^{2}}{\pi \alpha _{D}} \right) ^2. \end{aligned}$$

(17)

Using this equation, we obtain an energy of $E_\textrm{min}=1.53$ eV, which (due to neglected higher order terms in Eq. 17) is 0.5 eV lower than the value of 2.05 eV observed in the s-wave scattering component of the elastic cross section in Fig. 13. In this figure, the s-, p-, d-, and higher-wave cross sections refer to the angular momentum of the incident positron L for which their contributions to the elastic cross section are extracted from the relevant partial-waves J. The Ramsauer–Townsend minimum is not observed in the total elastic cross section as the magnitude of the s-wave component becomes small relative to other partial waves as the incident energy increases. This process can be observed in Fig. 13, with the peak of the p-wave and increasing values of the d- and higher-wave cross sections supplementing the decreased s-wave contribution at its minimum. As we observed in carbon [46], although the effect of the Ramsauer–Townsend minimum is hidden for the elastic cross section, its impact can be observed for the momentum transfer cross section, which has a minimum at 1 eV due to this effect.

Fig. 13

Elastic cross sections for positron scattering on atomic oxygen and its s-, p-, d-, and higher $\ell$-wave contributions. Here, L refers to the angular momentum of the incident positron

3.2.7 Bound-state excitation cross section

There is no other existing positron theory or experiment for specific bound-state excitations. Therefore, we compare against existing electron theory and experiment, which are expected to be equivalent to positron results for high incident energies. To obtain stable results for energies below 20 eV, values were obtained for calculations with $\ell _\textrm{max}=2$ and only bound states included.

Figure 14 presents the excitation of the ground state to $2p^3 3s$ $^3 S^o$. Excellent agreement is found for energies above 500 eV with the recommended electron theoretical results of Johnson et al. [72]. At lower energies, our positron calculation lies within the experimental error of existing electron measurements and is close to the BSR calculation of [39].

Fig. 14

The $2p^4$ $^3P$ $\rightarrow$ $2p^3 3s$ $^3 S^o$ excitation cross section for positron scattering on oxygen. Theoretical results for incident electrons are from Tayal and Zatsarinny [39] and the recommended values from Johnson et al. [72]. Experimental electron measurements are from Johnson et al. [61], Kanik et al. [73], and Vaughan and Doering [74]

Figure 15 presents the $2p^4$ $^3P$ $\rightarrow$ $2p^3 3\,s$ $^3 S^o$ excitation cross section. Comparing with the BSR results, we find that differences decrease with increasing energy and that they become close by 100 eV. The CCC results lie within the uncertainty of the experiment of Gulcicek et al. [75] below 20 eV.

Fig. 15

The $2p^4$ $^3P$ $\rightarrow$ $2p^3 3p$ $^3 P$ excitation cross section for positron scattering on oxygen. Theoretical results for incident electrons are from Tayal and Zatsarinny [39]. Experimental electron measurements are from Gulcicek et al. [75]

The $2p^3 3d$ $^3 D^o$ excitation cross section is shown in Fig. 16. Due to the differences observed with the oscillator strength for this transition and those of NIST, we have applied OOS scaling, which is the scaling of the CCC cross section by the ratio between these two oscillator strengths. Agreement is observed after this scaling with the recommended electron results of Johnson et al. [72] for energies above 500 eV. Closer agreement is observed with the BSR calculation for the unscaled result of this excitation.

Fig. 16

The $2p^4$ $^3P$ $\rightarrow$ $2p^3 3d$ $^3 D^o$ excitation cross section for positron scattering on oxygen. Theoretical results for incident electrons are from Tayal and Zatsarinny [39] and recommended from Johnson et al. [72]. Experimental electron measurements are from Kanik et al. [73]

Figure 17 shows the $2p^3 3s$ $^3 D^o$ excitation cross section. After OOS scaling we find excellent agreement with the recommended electron results of Johnson et al. [72] at high energies. At lower energies our positron calculation is found to be lower than the electron theory, but within the uncertainty of electron experiment.

Fig. 17

The $2p^4$ $^3P$ $\rightarrow$ $2p^3 3s$ $^3 D^o$ excitation cross section for positron scattering on oxygen. Theoretical results for incident electrons are from Tayal and Zatsarinny [39] and recommended results from Johnson et al. [72]. Experimental electron measurements are from Vaughan and Doering [74] and Kanik et al. [73]

Fig. 18

Total bound-state excitation cross section for positron scattering on oxygen. Theoretical results are from the CCC, Pindariya et al. [21], and, for the electron case, Joshipura and Patel [41] and Johnson et al. [72]

The total bound-state excitation cross section is presented in Fig. 18. The CCC-pot calculation is equivalent to the CCC-scaled complex model potential calculation for energies below 23.5 eV and the CCC for energies above this. Little agreement is observed with the only existing positron calculation of Pindariya et al. [21], which is significantly larger than the CCC result for energies below 300 eV. However, both calculations find a maximum at $\approx$ 20 eV. For energies above 500 eV, the electron results of Joshipura and Patel [41] agree with the positron results of Pindariya et al. [21] above 500 eV, but both are lower than the CCC results. The presented calculation of Johnson et al. [72] is a sum of only the three previously presented excitations. Therefore, it is expected to underestimate the total as it does not contain enough states to approximate the full excitation spectrum. Consequently, the summed recommended values follow the shape of the CCC calculation, which contains 28 bound states, for energies above 100 eV but is much lower even at 1000 eV. As the summed Johnson et al. [72] results are higher than the other presented electron and positron calculation at this energy, this suggests they underestimate the total bound-state excitation cross section at higher energies.

Fig. 19

Stopping power for positron scattering on oxygen. Theoretical calculations for this system are from the current CCC-pot calculation, the penelope code [26], and Gumus et al. [25]. Calculations for $e^-$–O by Gupta et al. [45] and semi-empirical results for the $e^-$–O$_2$ system by Williart et al. [77] are also presented. The positronium-formation component of the stopping power (PosF) is also shown for the CCC-pot calculation

3.2.8 Mean excitation energy and stopping power

In Fig. 19, we present the stopping power for a positron incident on atomic oxygen. The calculation of stopping power requires the cross section and excitation energy for each reaction channel. The stopping power for the positronium-formation channel was obtained from the total positronium-formation cross sections, as discussed in Ref. [53]. Reaction channels corresponding to bound-state excitation and ionization are treated in the following way. Below 23.5 eV, for bound-state excitation, we have used the present estimates for the cross sections. For ionization, we have used cross sections for positive-energy pseudostates from the single-center CCC model scaled to obtain the present direct ionization cross section. For energies above 23.5 eV, the stopping power is obtained directly from the single-center CCC calculation.

For energies above 200 eV, CCC, the GOS model of Gumus et al. [25] and the penelope code [26] predicts similar behavior, with the CCC results slightly below these other models. Below 200 eV, the CCC-pot calculation is in significant disagreement with these other approaches and predicts a substantially lower stopping power. As the models used for these positron calculations apply simple approximations, do not account for processes such as positronium-formation, and do not directly model the atom’s target structure, the current CCC calculation is expected to be more accurate for the presented energies.

According to Bragg’s additivity rule [76], which is accurate for high energies, the stopping power of O$_{2}$ is expected to be equal to that of O. CCC calculations of stopping power for H and H$_2$ [53] targets found this rule accurate for energies above 100 eV. As electron and positron results are equivalent for high energies, we also present the $e^-$–O calculation of Gupta et al. [45] and the $e^-$–O$_2$ semi-empirical calculations of Williart et al. [77]. For energies above 1000 eV, we find excellent agreement between the current CCC results and those of Williart et al. [77]. The results of Gupta et al. [45] are in close agreement with those of the CCC and Williart et al. [77] for energies above 3000 eV.

Fig. 20

Mean excitation energy for positron scattering on oxygen. CCC-pot theoretical results are presented alongside experimental results for the $e^-$-O$_2$ system from Williart et al. [77]

The mean excitation energy, directly obtained from the stopping power and the inelastic cross section, is shown in Fig. 20. There are no existing calculations of mean excitation energies for positron scattering on atomic oxygen or O$_2$. Williart et al. [77] only provide experimental measurements and their uncertainty for incident energies of 300 and 600 eV for the e$^-$–O$_2$ system. As their stopping power result was obtained with their experimental measurements of mean excitation energy, we have extracted the experimental values for other energies using their presented stopping power and inelastic cross section. There is no discussion provided of the experimental uncertainty for these other incident energies.

The mean excitation energy between atomic oxygen and O$_2$ is expected to be similar at higher energies. For CCC calculations of e$^-$ and e$^+$ scattering on H$_2$ [53], the e$^+$ mean excitation energy was found to follow similar behavior as the e$^-$ result but at a magnitude $\approx$ 20% higher. A similar trend is found here but with a smaller difference between the e$^+$ and e$^-$ result. At 600 eV, the CCC results lies just within the electron experimental uncertainty. Above this energy, as our results decrease, the difference between them and this experiment decreases to be within 10% at 5000 eV. This experiment and the CCC results exhibit similar behavior, with the mean excitation energy increasing to 600 eV. Above this energy, the mean excitation energy becomes nearly constant in the experiment and slowly decreases in the CCC-pot calculation. The sharp rise observed in the CCC-pot at threshold is due to positronium-formation.

4 Conclusion

A comprehensive set of cross sections and other quantities have been calculated for positron scattering from atomic oxygen within the single-center CCC approach. The target structure model for atomic oxygen adopted in the present calculations is in good agreement with the energy levels and oscillator strengths of NIST and previous theory, reflecting its accuracy. The CCC-scaled complex model potential results was used to obtain cross sections between the positronium-formation and ionization thresholds, positronium-formation, and direct ionization cross sections. A modification is introduced to this approach that is expected to result in more accurate positronium-formation cross sections between its threshold and maximum.

The total, elastic, momentum transfer, direct ionization, positronium-formation, total ionization, inelastic, excitation, and stopping power cross sections were calculated between threshold and 5000 eV. Quantities such as the scattering length, mean excitation energy, hidden Ramsauer–Townsend minimum, and the energy of the virtual positron-oxygen state were also calculated. Good agreement is observed between current results and high-energy electron atomic oxygen experiment and theory for total ionization, total, elastic, and bound-state excitations. Halved O$_2$ experiments for both positron and electrons are also in good agreement with the present calculations. For past positron theoretical calculations, agreement is observed at high energies with some available approaches. However, large differences are observed at lower energies and for the positronium-formation cross section. Future work, particularly experimental, is recommended on this system to address the discrepancies found.