1 Introduction

Experimental and theoretical investigations of positron scattering have been carried out for many years. Historically, positron beams intensities were limited such that experiments were confined primarily to the measurement of total cross sections from the noble gases [8] and some diatomic and triatomic molecules [6]. Neglecting direct annihilation, at low energies, the only open channel was that for elastic scattering and consequently theoretical calculations were generally confined to this region. At higher energies, the positronium formation channel is open. The region between the positronium threshold and the first excitation threshold of the target is known as the Ore gap. There have been some recent measurements of the excitation process [5] but these cross sections are small relative to the elastic and ionization cross sections.

More intense positron beams with higher resolving power have now become available. Consequently, improved measurements of these reactions can now be made. Thus, for the most part, we have only shown more recent measurements and theoretical calculations.

A recent review of positron scattering with atoms [17] has provided recommended results convenient for comparison. However, in the intervening years, more sophisticated theoretical methods have been developed for the determination of both elastic and inelastic cross sections, including positronium formation, for the noble gases [9, 13] as well as for group II atoms [3, 16] and beyond [14]. Here, the relativistic optical potential (ROP) method [4], hereafter referred to as I, has been applied to the scattering of positrons from neon and argon from threshold to 1 keV. We have calculated elastic, positronium formation, excitation, direct and total ionization cross sections as well as the grand total cross section. The technique given in McEachran and Stauffer [10] was used to account for positronium formation in both its ground as well as excited states.

2 Theory

The elastic and inelastic scattering of positrons from neon and argon atoms are described using the relativistic optical potential (ROP) method of Chen et al. [4]. Here, we provide a brief description of this overall method, and the reader is referred to ref. I as well as McEachran and Stauffer [13], hereafter referred to as II, for the details.

The scattering of the incident positrons, with wavenumber k, by closed-shell atoms can be described by the integral equation formulation of the partial wave Dirac–Fock scattering equations. In our ROP method, these equations can be written in matrix form as

$$\begin{aligned}{} & {} \begin{pmatrix} F_\kappa (r) \\ G_\kappa (r) \\ \end{pmatrix}= \begin{pmatrix} v_1(kr) \\ v_2(kr) \\ \end{pmatrix} + {1\over k} \int _0^r \!\!\! \textrm{d}x \, G(x,r) \, \nonumber \\{} & {} \quad \biggl [ U(r) \begin{pmatrix} F_\kappa (r) \\ G_\kappa (r) \\ \end{pmatrix} - i \, U_{\textrm{opt}}(r) \begin{pmatrix} F_\kappa (r) \\ G_\kappa (r) \\ \end{pmatrix} \biggr ] \end{aligned}$$
(1)

where the local potential U(r) is given by the sum of the static and a local polarization potential, i.e.,

$$\begin{aligned} U(r) = U_{\textrm{st}}(r) + U_{\textrm{pol}}(r) \end{aligned}$$
(2)

Here, we have followed the procedure outlined by Bartschat et al. [1, 2] and have replaced the real part of the optical potential by a local polarization potential based upon the polarized orbital method of McEachran et al. [11, 12]. The static potentials were determined in the usual manner from the ground state Dirac–Fock orbitals of neon and argon, while the polarization potential \(U_{\textrm{pol}}(r)\) in Eq. (2) comprised the sum of the long-range dipole and quadrupole potentials for neon and the first four multipoles for argon. This inclusion of additional multipoles for argon is due to its much larger polarizabilities. In Eq. (1), the non-local potential \(U_{\textrm{opt}}(r)\) denotes the imaginary part of the optical potential and describes the absorption of the incident flux into the inelastic channels and thereby describes both excitation and ionization processes. This potential is given by a sum and integration over the bound and continuum states of the atom (see §2.1 below as well as equation (21b) of ref. I for details).

Finally, in Eq. (1), \(F_\kappa (r)\) and \(G_\kappa (r)\) are the large and small components of the complex scattering wavefunction, while the functions \(v_1(kr)\) and \(v_2(kr)\) are the corresponding free particle wavefunctions and are given in terms of Riccati–Bessel functions. G(r, x) is the free particle Green’s function which can be expressed in matrix form in terms of the Riccati–Bessel and Riccati–Neumann functions (see equations (23) and (24a,b) of ref. I for details). The subscript \(\kappa \) on the scattering wavefunctions is the relativistic angular momentum quantum number of the incident positron. It is related to the corresponding orbital angular momentum quantum number l according to \(\kappa ={}-l-1\) when (spin-up) and \(\kappa =l\) when (spin-down) where j is the total angular momentum quantum number of the incident positron.

The complex phase shifts are determined from the asymptotic form of the scattering wavefunction \(F_\kappa (r)\). These phase shifts are then used to determine the various elastic and inelastic cross sections (see equations (5) to (9) of II).

2.1 The optical potential

For neon, the following 15 bound excited states, in intermediate coupling notation, \(3\mathrm s[3/2]_1\), \(3\mathrm s[1/2]_1\), \(3\mathrm p[1/2]_0\), \(3{\bar{\textrm{p}}}[1/2]_0\), \(3\mathrm p[5/2]_2\), \(3\mathrm p[3/2]_2\), \(3{\bar{\textrm{p}}}[3/2]_2\), \(4\mathrm s[3/2]_1\), \(4\mathrm s[1/2]_1\), \(3{\bar{\textrm{d}}}[1/2]_1\), \(3\mathrm d[3/2]_1\), \(3{\bar{\textrm{d}}}[3/2]_1\), \(3\mathrm d[7/2]_3\), \(3\mathrm d[5/2]_3\), and \(3{\bar{\textrm{d}}}[5/2]_3\), were included in \(U_{\textrm{opt}}(r)\), in order to simulate excitation processes. Similarly, for argon, the following 17 bound excited states, \(3{\bar{\textrm{d}}}[1/2]_1\), \(3\mathrm d[3/2]_1\), \(3{\bar{\textrm{d}}}[3/2]_1\), \(3\mathrm d[7/2]_3\), \(3{\bar{\textrm{d}}}[5/2]_3\), \(3\mathrm d[5/2]_3\), \(4\mathrm s[3/2]_1\), \(4\mathrm s[1/2]_1\), \(4\mathrm p[1/2]_0\), \(4{\bar{\textrm{p}}}[1/2]_0\), \(4\mathrm p[5/2]_2\), \(4{\bar{\textrm{p}}}[3/2]_2\), \(4\mathrm p[3/2]_2\), \(5\mathrm s[3/2]_1\), \(5\mathrm s[1/2]_1\), \(5\mathrm p[1/2]_0\) and \(5{\bar{\textrm{p}}}[1/2]_0\), were included in \(U_{\textrm{opt}}(r)\), in order to simulate excitation processes.

Also included in \(U_{\textrm{opt}}(r)\) were all continuum states with orbital angular momentum given by \(l_\textrm{c}=0,1,2,3\) and 4 in order to simulate ionization processes (including Ps formation). The integration over the continuum states in the absorption potential was approximated by using Gauss–Legendre integration usually with 16 points. In a relativistic close coupling expansion, it is necessary to couple the total angular momentum of the electron in the excited state (bound or continuum) to the total angular momentum of the incident positron in order to obtain the total angular momentum J of the positron-atom system. This total angular momentum J is then conserved during the collision process. Under the above circumstances, this gave rise to a maximum of 53 excitation channels and 34 ionization channels in \(U_{\textrm{opt}}(r)\) for neon and 34 and 55 similar channels for argon. In general, the excitation cross section is obtained by only including the bound states in the optical potential. Similarly, the ionization cross section is obtained by only including the continuum states in the optical potential. The total cross section (excluding Ps formation) is then obtained by including both the bound and continuum states in the optical potential.

2.2 Positronium formation

Positronium formation was simulated using the method given in McEachran and Stauffer [10]). In that paper, the imaginary part of the optical potential was treated as a perturbation within the Hulthén–Kato formulation [4]. This approximation is not used here. Here, the Ps formation cross section in the ground state is determined by first calculating the direct ionization cross section and then the comparable cross section when the ionization threshold is reduced by 6.8 eV, the binding energy of positronium. The Ps formation cross section is then taken to be the difference between these two results. Similarly, Ps formation in an excited state with principle quantum number n is simulated by reducing the ionization thresholds by \(6.8/n^2\) eV. This procedure is a modification of the method originally suggested by Reid and Wadehra [18, 19]. Any method for simulating rather than directly calculating Ps formation will contain one or more adjustable parameters. In the work of Blanco et al. [3], their adjustable parameter involved both the Ps formation and the first excited-state thresholds, while in the method of McEachran and Stauffer [10], there is one adjustable parameter which is chosen according to where the Ps formation effectively vanishes. For the noble gases, for which there are many experimental measurements, this parameter was chosen to be \(120+E_\textrm{ion}\) eV. It should be noted that the method of McEachran and Stauffer influences the asymptotic behavior of the Ps formation cross section but has very little influence on its peak value.

3 Neon results

In this section, we show our neon results and compare with the recent recommended cross section composed of the most current calculations and measurements.

3.1 Elastic scattering

For neon, the elastic energy regime is from zero to 14.76 eV (the first ionization threshold minus the binding energy of positronium). Since the positron is distinguishable from the constituents of the target, all the phase shifts must be zero at zero energy although the scattering length is nonzero giving rise to a finite nonzero cross section at zero energy. In Fig. 1, we show our present results along with the momentum transfer and viscosity cross sections. The latter two cross sections are useful in the determination of transport properties in gases. The current ROP results supersede the polarized orbital results given in Jones et al. [7].

3.2 Positronium formation

In the Ore gap (14.76–21.56 eV), positronium formation is the only inelastic channel open but at higher energies, excitation and eventually direct ionization of neon can occur. We present two sets of cross sections for Ps formation, one where the threshold for ionization is lowered to coincide with the threshold for Ps formation in the ground state and the second which is the sum of cross sections calculated when the threshold for ionization is successively lowered to coincide with Ps formation into the first four states. Our calculations for the total positronium formation cross section are compared with the recommended cross section from Ratnavelu et al. [17] in Fig. 2. In our first calculation, our cross sections do not rise as quickly from the threshold as the experimental data for the steeply rising part of the cross section and our results peak at about \(0.373\times 10^{-20}\) m\(^2\) at 35 eV which is below all the experiments. However, our second set of cross sections rises slightly more steeply and peaks at about \(0.495\times 10^{-20}\) m\(^2\) also at 35 eV which is within the typical experimental uncertainties. Furthermore, this agreement extends to 60 eV, the limit of the experimental measurements. Our calculations for the excited-state positronium formation for the first three excited states are shown in Fig. 3. More theoretical and experimental work is required here.

Fig. 1
figure 1

Elastic scattering (blue curve), momentum transfer (red curve), and viscosity (black curve) cross section for positron scattering from neon

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Fig. 2
figure 2

Total positronium formation cross section for positron scattering from neon: red curve, present work; blue curve, recommended cross section from review [17]

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Fig. 3
figure 3

Excited-state positronium formation cross section for positron scattering from neon: blue curve, present work for Ps(\(n=2\)); black curve, present work for Ps(\(n=3\)); red curve, present work for Ps(\(n=4\))

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Fig. 4
figure 4

Excitation cross section for positron scattering from neon: red curve, present work for the total excitation cross section; blue circles, experimental results for 3 s excitation [5]; green circles, experimental results for 3p excitation [5]; black circles, experimental results for 4 s excitation [5]

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3.3 Electronic excitation

Here, the excitation cross section refers to the sum of the individual excitation cross sections of the 15 excited states, listed above, from the ground state. Thus, either a \(2\mathrm p\) or \(2{\bar{\textrm{p}}}\) electron from the outermost subshells of the neon ground state is excited to one of these excited states. The first three excited states were recently experimentally determined by Cheong et al. [5] within about 10 eV of their threshold and are compared to the present work in Fig. 4.

3.4 Direct ionization

The direct ionization cross section refers to the process where both the incident positron and ejected electron are free. Figure 5 shows our results for the direct ionization cross sections over the energy range from threshold to 1 keV. This ionization cross section also included the ionization of the 2 s electron after 48.5 eV. Our direct ionization cross section is compared to the recommended cross section of Ratnavelu et al. [17] and are shown to be consistent.

3.5 Grand total cross section

This term refers to the total scattering cross section and theoretically is the sum of cross sections for all allowed processes, elastic and inelastic. Figure 6 compares our results with the recommended grand total of Ratnavelu et al. [17] with good agreement above 25 eV and below 15 eV.

Fig. 5
figure 5

Direct ionization cross section for positron scattering from neon: red curve, present work; blue curve, recommended cross section from review [17]

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Fig. 6
figure 6

Grand total cross section for positron scattering from neon: solid black curve, present work; blue curve, recommended grand total cross section [17]; dashed curve, present work for the elastic scattering cross section

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Fig. 7
figure 7

Elastic scattering (blue curve), momentum transfer (red curve), and viscosity (black curve) cross section for positron scattering from argon

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4 Argon results

In this section, we show our argon results and compare with the recent recommended cross section or most current calculations and measurements.

4.1 Elastic scattering

For argon, the elastic energy regime is from zero to 8.96 eV. In Fig. 7, we show our present results for the elastic scattering, momentum transfer, and viscosity cross sections. The current ROP results supersede the polarized orbital results given in Jones et al. [7].

4.2 Positronium formation

Here, the Ore gap for positronium formation is from 8.96 to 15.76 eV. Once again, we present two sets of cross sections for Ps formation, one corresponding to Ps formation in the ground state (\(n=1\)) only and the other corresponding to Ps formation in the first four states (\(n=1\) to 4). Our calculations for the total positronium formation cross section are compared with the recommended cross section of Ratnavelu et al. [17] in Fig. 8. The agreement is good other than the slower rise in the cross section. Our calculations for the excited-state positronium formation for the first three excited states are shown in Fig. 9. These results are compared to the experimental measurements and theoretical results of Ps(\(n=2\)) formation presented in Murtagh et al. [15]. The previous near-threshold results are in disagreement by a factor of 2. At higher energies where the previous results disagree, the experimental results of Murtagh et al. [15] are consistent with the present calculation. Again more work is required to resolve discrepancies.

Fig. 8
figure 8

Total positronium formation cross section for positron scattering from argon: red curve, present work; blue curve, recommended cross section from review [17]

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Fig. 9
figure 9

Excited-state positronium formation cross section for positron scattering from argon: blue curve, present work for Ps(\(n=2\)); blue circles, experimental results from Murtagh et al.; blue dashed curve, theoretical results presented in Murtagh et al.; black curve, present work for Ps(\(n=3\)); red curve, present work for Ps(\(n=4\))

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4.3 Electronic excitation

Here, the excitation cross section refers to the sum of the individual excitation cross sections of the 17 excited states of argon, listed above, from the ground state. Thus, either a \(3\mathrm p\) or \(3{\bar{\textrm{p}}}\) electron from the outermost subshells of the argon ground state is excited to one of these excited states. Once again, the direct ionization cross section refers to the process where both the incident positron and ejected electron are free. Figure 10 shows our results for excitation from threshold to 40 eV. We include the first two excitation partial cross sections to illustrate the marked difference between the present work and the recommended cross section.

4.4 Direct ionization

Our direct ionization cross section is compared to the recommended cross section of Ratnavelu et al. [17] in Fig. 11. This ionization cross section also included the ionization of a 3 s electron after 29.3 eV. There is generally good agreement at intermediate energies with discrepancies near threshold and at high energies.

4.5 Grand total cross section

Once again, this term refers to the total scattering cross section and theoretically is the sum of cross sections for all allowed processes, elastic and inelastic. Figure 12 compares our results with the recent recommended cross section of Ratnavelu et al. [17]. The present work differs from the recommended cross section but appears to converge at both high and low energies.

Fig. 10
figure 10

Excitation cross section for positron scattering from argon: red curve, present work for the total excitation cross section; light blue curve, recommended 4 s excitation cross section [17]; blue curve, present work for the 4 s \(^2P_{1/2}\); blue dashed curve, present work for the 4 s \(^2P_{3/2}\)

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Fig. 11
figure 11

Direct ionization cross section for positron scattering from argon: red curve, present work; blue curve, recommended cross section from review [17]

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Fig. 12
figure 12

Grand total cross section for positron scattering from argon: black curve, present work; red curve, present work for the elastic scattering cross section; blue curve, recommended cross section from review [17]

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5 Conclusions

We have presented positron scattering cross sections for collisions with neon and argon. Overall, there is reasonable agreement between the present calculations and the corresponding recommended cross sections found in the literature. The discrepancies present are typically found at, or near, the opening of a new scattering or production channel. In the case of excited-state positronium formation, the large experimental uncertainties limit a detailed discussion. This work highlights the need for more experimental determinations of state-specific cross sections.