Ionization of Hydrogenic Systems by Positron and Electron Impacts

Abstract

The ionizations of the 1S state of hydrogenic systems with a nuclear charge of Z = 2 and 3 have been carried out using the hybrid theory. This is a continuation of the work started earlier. The present results are compared with the published cross-sections for Z = 1 [Bhatia, A.K. 2025]. The distortion of the orbit is considered irrespective of the position of the incident particle, whether it is outside or inside the orbit. Only the distortion of the target orbit in the initial state is considered, but the distortion in the final state is not considered. Cross-sections decrease as the nuclear charge increases.

1. Introduction

The hybrid theory has been applied to study several processes involving positrons and electrons, obtaining very accurate results for phase shifts [1] and photo absorption [2], agreeing with those obtained using the close-coupling [3] and R-matrix formulations [4]. Hybrid theory includes short-range and long-range interactions and is variationally correct [1]. The close-coupling and R-matrix formulations have provided very accurate results for various processes in atomic physics as well as in molecular physics. For example, excitation of 2S from 1S state of atomic hydrogen using the hybrid theory [5] cross-sections compared well with those obtained by using the standard R-matrix close-coupling method [6]. Now we will discuss ionization by positron impacts.

Dirac [7] formulated the relativistic wave equation in 1930. This wave equation helped him to predict an antiparticle of electron of spin ${\displaystyle \frac{\hslash }{2}}$. The antiparticles have been called positrons, and they are produced, among many processes, by the collision of cosmic rays in a cloud chamber. Positrons were detected experimentally by Anderson [8] in 1932. In the Sun, positron processes result in the emission of solar gamma rays at energies of nearly 100 keV to greater than 1 GeV [9]. Gamma rays have also been detected at the center of galaxy [10]. The emissions from these two sources can be distinguished by the velocities of gamma rays; the ones from the center of the galaxy have higher velocities. We describe an ionization process positron impact first. The process of ionization is indicated by

$$e^{+}+H\left(n\mathrm{S}\right)\to e^{+\hspace{1em}}+p+e^{-}$$

(1)

Electron ionization is discussed in Section 3. The ionization cross-section is given by

$$\sigma ={\displaystyle \frac{k_f}{k_i}}\int {\left|T_{fi}\right|}^2 d\mathsf{\Omega }$$

(2)

where $k_i$ and $k_i$ are the initial and final momenta of the incident particle (positron or electron), and $T_{fi}$ is the matrix describing the process. It is given by

$$T_{fi}=-{\displaystyle \frac{1}{4\pi }}<{\psi }_f|{\displaystyle \frac{2Z}{r_1}}-{\displaystyle \frac{2}{r_{12}}}|{\psi }_i>.$$

(3)

The initial and final state wave functions are indicated by ${\psi }_{i }$ and ${\psi }_f$. Z is the nuclear charge, $r_1$ and $r_{2 }$ are position coordinates of the incident positron and the target electron, and $r_{12}=|\overrightarrow{r_1}-\overrightarrow{r_2}|.$ The initial wave state wave function is given by

$${\psi }_i(r_1,r_2)=u(\overrightarrow{r_1}) {\varphi }^{pol}(r_1,r_2),$$

(4)

${\varphi }^{pol}(r_1,r_2)$ is the distorted target function depending explicitly on r₁ in addition to r₂. This represents the distortion of the target orbit [11] due to the incident positron and is given by

$${\varphi }^{pol}(r_1,r_2)=\varphi (\overrightarrow{r_2})+{\displaystyle \frac{\chi (r_1)}{r_1^2}}{e}^{-Z{r}_2}(0.5\mathrm{Z}{r}_2^2+r_2)\cos ({\theta }_{12})/\sqrt{\pi Z}$$

(5)

Reference [11] was originally devoted to the incident electrons by Temkin. However, it can be generalized to positrons by one change a sign before the second term.

The cutoff function $\chi (r_1)$ can be chosen in various forms as long as $\chi (r_1)/r_1^2$ is finite for $r_1$ tending to zero. It is given by Shertzer and Temkin [12]

$$\chi \left(r_1\right)=1-e^{-2Z{r}_1}[{\displaystyle \frac{(Z{r}_1{)}^4}{3}}+{\displaystyle \frac{4(Z{r}_1{)}^3}{3}}+2(\mathrm{Z}{\mathrm{r}}_1{)}^2+2(\mathrm{Z}{\mathrm{r}}_1)+1]$$

(6)

The scattering function $u\left(\overrightarrow{r_1}\right)$ is obtained by solving the equation

$$<{\varphi }^{pol}(r_1,r_2)|\mathrm{H}\text{-}\mathrm{E}|u\left(\overrightarrow{r_1}\right){ \varphi }^{pol}\left(r_1,r_2\right)=0$$

(7)

In the above equation, H is the Hamiltonian of the system, and E is the total energy consisting of the target energy and energy of the projectile. In Rydberg units the Hamiltonian for hydrogen atoms and the incident positrons is given by

$$H=-{\nabla }_{1 }^{2 }-{\nabla }_{2 }^{2 }+{\displaystyle \frac{2Z}{r_1}}-{\displaystyle \frac{2Z}{r_2}}-{\displaystyle \frac{2}{r_{12}}}.$$

(8)

The polarization potential proportional to 1/$r_1^4$ enters Equation (7) when it is solved for the function u, which is clear because Equation (7) depends on ${\varphi }^{pol}(r_1,r_2)$, given in Equation (5).

The energy available for the ionization of the system is given by

$$E=k_i^2-Z^2 \text{-}$$

(9)

If the incident energy is equal to the binding energy, ionization is not possible. The incident must be greater than the binding energy Z². Therefore, the energy available for ionization must be greater than Z². After losing energy equal to the binding energy, the incident particle can detach the electron in the target. The resulting equation for the continuum function u(r₁) has the attractive polarization potential ${\displaystyle \frac{\alpha \left(r_1\right)}{r_1^4}}$, where $\alpha \left(r_1\right)$ is the dipole polarizability equal to 4.5 $a_0^3$ for the ground state of the hydrogen atom with Z = 1 when $r_{1 }$ is tending to infinity, $a_0$ is the Bohr otbit.

2. Calculation and Results

We assume that the detached electron and the incident particle in the final state share the remaining energy equally after ionization. There are various ways the remaining energy can be shared, but the equal sharing of energy is very likely [13], otherwise many calculations would be required. In Rydberg units, the energy available for ionization is shared by the ejected electron and the outgoing particle we have

$$k_f^2+k_e^2=k_i^2-Z^2$$

(10)

$Z^2$ is the binding energy of the 1S state of the hydrogenic system. The final state wave function is given by

$${\psi }_f\left(\overrightarrow{r_1},\overrightarrow{r_2}\right)=\mathrm{e}\mathrm{x}\mathrm{p}(i\overrightarrow{k_f}·\overrightarrow{r_1})FC(k_e,r_2)$$

(11)

As stated earlier, below Equation (2) ${\mathrm{k}}_{\mathrm{f}},\mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{k}}_{\mathrm{e}}$ are the final and the detached electron momenta, and the exponential part in Equation (11) represents the plane-wave of the outgoing particle [14], and FC represents the Coulomb functions obtained from the Handbook [15]. The exponential term is the plane-wave [14] and has an expansion given by

$$4\pi \left(i^l\right)Y_{lm}\left(\overrightarrow{k_f}\right)Y_{lm}({\mathsf{\Omega }}_1),$$

(12)

FC$(k_e,\overrightarrow{r_2})$ in Equation (11) is the Coulomb function given by

$${\displaystyle \frac{FC(r_2)}{(k_e.r_2)}}\mathrm{Y}({\mathsf{\Omega }}_2)$$

(13)

These functions at various r₂ are obtained from the Handbook [15]. The cross-sections at various energies are given in

Table 1. Ionization cross sections (units are ${10}^{-16}$ cm²) of the 1S state of the hydrogenic systems by positron impact. Energy units are Ryd, a(−b) = a × 10^−b.

A	Z = 1	Z = 2	Z = 3
0.20	0.4160	0.1095(−1)	0.1037(−2)
0.25	0.6716	0.1044(−1)	0.8550(−3)
0.30	0.3224	0.7192(−2)	0.7220(−3)
0.40	0.2539	0.6744(−2)	0.6101(−3)
0.50	0.2021	0.8813(−2)	0.7405(−3)
0.60	0.1630	0.8027(−2)	0.7530(−3)
1.00	0.7557(−1)	0.3669(−2)	0.8754(−3)
2.00	0.1629(−1)	0.3509(−2)	0.8550(−3)
3.00	0.5601(−2)	0.2944(−2)	0.6232(−3)
4.00	0.2301(−2)	0.2702(−2)	0.6080(−3)

, where $k_i^2=Z^2$ + A, indicating that it is easier to indicate the incident energy in this form rather than indicating energy for each Z. Since the incident energy is in Rydberg units, A is also in Rydberg units: or example, A = 1 Rydberg represents an incident energy equal to 2 Rydberg for Z = 1; for Z = 2, the incident energy = 5 Rydberg, and for Z = 3, the incident energy = 10 Rydberg. The Z = 1 results are taken from a previous publication [16] In this publication, we have also given cross sections of the ionization of the excited states.

3. Electron Impact Ionization

In this section, we discuss electron impact ionization. Electrons were discovered by J. J. Thompson in 1897. This led to the development of physics beyond classical physics. Positrons and electrons helped us to understand various processes, like elastic scattering, excitations, ionizations, resonances, annihilation, photon absorption, recombination, etc. The electron impact ionization process is indicated by the equation

$$e^{-\hspace{1em}}+H\left(nS\right)\to e^{-}+\left(p+e^{-}\right)$$

(14)

The distortion of the target is now given by Equation (5), with a negative sign instead of the positive sign before the second term. Also in Equation (3), there is a negative sign before Z. The Hamiltonian in Equation (8) for electron–hydrogen systems requires a change the sign of the third and fifth terms. The initial state wave function is given by

$${\mathit{\psi }}_{\mathit{i}}({\mathit{r}}_{\mathbf{1}},{\mathit{r}}_{\mathbf{2}})\mathbf{=}\mathbf{[}\mathit{u}(\overrightarrow{{\mathit{r}}_{\mathbf{1}}}){\mathit{\varphi }}^{\mathit{p}\mathit{o}\mathit{l}}({\mathit{r}}_{\mathbf{1}},{\mathit{r}}_{\mathbf{2}})\pm \mathit{u}(\overrightarrow{{\mathit{r}}_2}){\mathit{\varphi }}^{\mathit{p}\mathit{o}\mathit{l}}({\mathit{r}}_2,{\mathit{r}}_{\mathbf{1}})\mathbf{]}\mathbf{/}\sqrt{\mathbf{2}}$$

(15)

In the above equation, the upper sign refers to the singlet states, and the lower sign refers to the triplet states. The total cross-section is the sum of the singlet and triplet cross-sections.

The rest of the equations remain the same. In

Table 2. Ionization cross-sections (units are ${10}^{-16}$ cm²) of the 1S state nuclear charge Z of the hydrogenic systems by electron impact.

A	Z = 1	Z = 2	Z = 3
0.20	1.3479	0.1593	0.2005(−2)
0.25	2.9976	0.6477(−2)	0.2965(−4)
0.30	2.6911	0.3887(−1)	0.1532(−3)
0.40	0.5122	0.9925(−1)	0.9551(−2)
0.50	0.6307	0.2084(−3)	0.6370(−3)
0.60	0.3712	0.1237(−1)	0.3639(−1)
1.00	0.1396	0.4444(−1)	0.3859(−2)
2.00	0.1339	0.2031(−1)	0.4637(−3)
3.00	0.1439	0.9645(−2)	0.3933(−2)
4.00	0.1675(−1)	0.1464(−1)	0.4476(−2)

we the results. The Z = 1 results are taken from a previous publication [16]

The ionization cross sections (units are ${10}^{-16}$ cm²) of the 1S state of the hydrogenic systems by electron impact are given in Table 2. Energy units are Ryd, a(−b) = a × 10^−b. The incident energy is given by E = $Z^2$ + A, where A is in Ryd units. For Z = 1 and A = 1, the incident energy is 2 Ryd, for Z = 2 it is 5 Ryd, and for Z = 3 it is 10 Ryd. Angular momentum up to 7 could be required to obtain converged results. Some details of the present calculations are given in reference [16] and they are not repeated here.

An experiment was carried out by Spicher et al. [17] to determine the cross sections of the ionization of hydrogen by positron and electron impacts. It was pointed out in reference [16] that the calculated cross sections agreed with the experimental cross-sections. The maximum of the calculated cross-sections is close to the experimentally determined position. Another experiment was carried out by Jones et al. [18]. They obtained lower cross sections than those obtained in reference [17]. However, Jones et al. [18] present their results for cross-sections in curves only, and it is difficult to infer accurate results from their curves [18]. Furthermore, most of the publications have other electrons present in the targets.

4. Related Publications

Over the years, there have been various publications on this subject. We mention a few: Roy et al. [19] using Coulomb functions calculated accurate cross sections by electron and positron impacts in the threshold region. For electrons, the excess energy is E = 0.1 eV, while for positrons the excess energy is E = 0.14 eV. The threshold law of Wannier [20] for electrons is that the cross-section is proportional to $E^{1.127}$, while the derivation by Klar [21] for positrons shows that the cross-section is proportional to $E^{2.650}$. Temkin [22] has given a Coulomb–dipole theory of ionization in which cross-sections for electrons and positrons are proportional to ${\displaystyle \frac{E}{(\mathrm{l}\mathrm{n}(E){)}^2}}$. However, up to now, there are no experimental confirmations of threshold laws. Kadyrov et al. [23], using exterior complex scaling, have calculated fully differential ionization cross-sections. Younger [24] has calculated ionization cross-sections using the Born approximation. Bote and Salvant [25] carried out calculations of inner-shell ionization of argon, copper, and gold atoms by electrons and positrons using various approximations, like plane-wave, distorted-wave, etc. Khare et al. [26] have also carried out calculations for inner-shell ionization for C, N, O, Na, Al, Ar, Ag, and Au using the plane-wave Born approximation. However, it is not possible to infer cross-sections from these publications for comparison with the present results for hydrogenic systems with a Z equal to one and with a Z > 1; also there are interactions of electrons in various shells with the electron in the 1S shell. The present calculation is not based on a plane-wave Born approximation.

5. Conclusions

We have used the hybrid theory to calculate positron impact and electron impact ionization cross-sections of the 1S state of hydrogenic systems with Z = 1 [16] and Z = 2 and 3. The distortion of the orbit of the target is considered only in the initial state. In Table 1 and Table 2, we present cross sections. The present formulation can be extended to calculate electron or positron impact cross sections for the excited states. It is indicated in [16]. that the sum of the cross sections for positron and electron impacts is close to the experimental results for Z = 1 [17]. The presents results would be useful if there were similar experiments for a Z greater than one. Calculations were carried out using quadruple precision and using the noniterative method of Omidvar [27], which is briefly described in Appendix A in Reference [5], and using the gFortran to compile Fortran 77 on the computer the Heliophysics Science Division. We can assume that cross sections are accurate to 4 decimal places because of the use of quadrupole precision. It is hard to compare with available results when the calculated cross sections involve interactions from other electrons. Jones et al. [18] have given their results in the form of curves and not in a numerical form. It is very difficult to accurate results from their curves for comparison.

Some details of the calculations are given in [28].