**Deep Minima in the Triply Differential Cross Section for Ionization of Atomic Hydrogen by Electron and Positron Impact**

#### **C. M. DeMars 1, S. J. Ward 1,\*, J. Colgan 2, S. Amami <sup>3</sup> and D. H. Madison <sup>3</sup>**


Received: 06 May 2020; Accepted: 25 May 2020; Published: 29 May 2020

**Abstract:** We investigate ionization of atomic hydrogen by electron- and positron-impact. We apply the Coulomb–Born (CB1) approximation, various modified CB1 approximations, the three body distorted wave (3DW) approximation, and the time-dependent close-coupling (TDCC) method to electron-impact ionization of hydrogen. For electron-impact ionization of hydrogen for an incident energy of approximately 76.45 eV, we obtain a deep minimum in the CB1 triply differential cross section (TDCS). However, the TDCC for 74.45 eV and the 3DW for 74.46 eV gave a dip in the TDCS. For positron-hydrogen ionization (breakup) we apply the CB1 approximation and a modified CB1 approximation. We obtain a deep minimum in the TDCS and a zero in the CB1 transition matrix element for an incident energy of 100 eV with a gun angle of 56.13◦. Corresponding to a zero in the CB1 transition matrix element, there is a vortex in the velocity field associated with this element. For both electron- and positron-impact ionization of hydrogen the velocity field rotates in the same direction, which is anticlockwise. All calculations are performed for a doubly symmetric geometry; the electron-impact ionization is in-plane and the positron-impact ionization is out-of-plane.

**Keywords:** electron-impact ionization; hydrogen; positron-impact ionization; velocity field; vortices

#### **1. Introduction**

Studies of the angular distributions of ionized atomic electrons by charged-particle impact is a rich field that has long been studied due to its importance to other areas of physics (e.g., plasma, medical physics) and also due to the information that is available about the correlated nature of the particle interactions.

It was thought that structures in differential cross sections of atoms by the impact of fast bare charged particles could be described by using the first and second Born terms [1]. However, as discussed by Berakdar and Briggs [2] the deep minimum observed in the experimentally measured triply differential cross section (TDCS) of helium [3,4] is a different feature to those structures reported in reference [1]. Interestingly, Macek et al. [5] showed that a zero in the ionization element and the TDCS obtained using the theory of reference [2,6] corresponds to a vortex in the velocity field associated with this element.

Berakdar and Briggs [2] for electron-impact ionization of helium used the product of three Coulomb waves for the final state [6]. They expressed the transition matrix element *Tf i* as the sum of three amplitudes, *T*1, *T*<sup>2</sup> and *T*3, which represent, respectively, the initial scattering of the incoming electron with the active electron, the nucleus and the passive electron of the helium atom. In their paper [2], they explained that, in order for both the real and imaginary part of *Tf i* to be zero, the three amplitudes have to destructively interfere and that the amplitude *T*<sup>3</sup> is essential for a zero. Since for electron-hydrogen ionization the term *T*<sup>3</sup> = 0, as there is no passive electron in the target atom, Berakdar and Briggs [2] deduced that there cannot be an exact zero in *Tf i* and thus in the TDCS. Nevertheless, as pointed out by Berakdar and Briggs [2], Berakdar and Klar [7] obtained a dip in the TDCS for positron-hydrogen ionization. Also, recently, Navarrete et al. [8] and Navarrete and Barrachina [9–11], using for the final state the correlated three Coulomb wave (C3) wave function, have established that there are zeros in the transition matrix element for positron-hydrogen ionization, whereas, of course, there is also no passive electron. So, there exists the open question of whether it is possible to have zeros in the transition matrix element and the TDCS for hydrogen ionization by electron impact as well as by positron impact.

We apply the Coulomb–Born (CB1) [12–15] and modified CB1 [13,16] approximations to both positron- and electron-impact ionization of atomic hydrogen to see if we find a zero in the CB1 transition matrix *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* [17–20]. We consider direct ionization (breakup) by positron impact only and, thus, we do not consider positronium formation into the continuum [21,22]. Electron-hydrogen ionization and positron-hydrogen ionization are fundamental three-body Coulomb processes. An exact zero in the ionization transition matrix element means that there is a deep minimum in the TDCS and a corresponding vortex in the velocity field, **v**, associated with this element [5,18,19,23–27]. A significant advantage of the CB1 method over more sophisticated methods is that CB1 calculations can be performed rapidly, enabling a systematic search for zeros in *Tf i* to be done quickly and for the velocity field **v** associated with *Tf i* to be readily computed. Once a zero in *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* and in the corresponding TDCS is located, more elaborate methods can be applied for the kinematics, or approximately the kinematics, of the zero in *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* . In addition to applying the CB1 and various modified CB1 approximations to electron-hydrogen ionization, we also apply the three body distorted wave (3DW) approximation and the time-dependent close-coupling (TDCC) method to electron-hydrogen ionization at incident energies close to where we obtained a zero in *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* .

The 3DW and TDCC methods have earlier been applied for electron-impact ionization of atomic hydrogen, molecular hydrogen and helium [28]. For the kinematics considered in reference [28], minima in the TDCSs for electron-impact ionization of atomic hydrogen were obtained, but these minima are not deep or zero. However, the TDCC [28] method obtained a deep minimum in the TDCS for electron-helium ionization for an incident energy of 64.6 eV that compared very well with experimental measurements [4]. (This incident energy of 64.6 eV is about three times the binding energy of helium.) The 3DW TDCS for the incident energy of 64.6 eV shows only a dip, although it does give a strong minimum for the incident energies of 44.6 and 54.6 eV [28]. Also, in the region of the experimental data [4], the CB1 and modified CB1 TDCSs for electron-helium ionization compared reasonably well with the measurements and with the TDCC results [25,28].

Previously, the CB1 approximation has been applied to electron-impact ionization of a K-shell model carbon atom at a high energy of 1801.2 eV [24]. A deep minimum in the CB1 TDCS and a zero in *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* was obtained. Corresponding to the zero in *<sup>T</sup>*CB1 **<sup>k</sup>**,1*<sup>s</sup>* there is vortex in the velocity field associated with this element [24].

For positron-hydrogen ionization a deep minimum in the fully differential cross section has previously been explained in terms of a vortex in the generalized velocity field **u** that is associated with the transition matrix element *T*(**k**+, **k**−), where *T*(**k**+, **k**−) depends on the momentum of the scattered positron **k**<sup>+</sup> and the momentum of the ejected electron **k**<sup>−</sup> [8,9]. Navarrete and Barrachina [9–11] obtained zeros in *T*(**k**+, **k**−) and found that the velocity field rotates in opposite directions around the zeros. Recently, a deep minimum in the TDCS for positron-helium ionization was obtained using the CB1 and modified CB1 approximations [18–20,25–27].

Positron-impact ionization of atoms and molecules is of experimental interest [29–35], including studies of differential cross sections [36,37]. Structures in differential cross sections are of both theoretical and experimental interest [38].

In this paper we show that the CB1 and modified CB1 approximations give a deep minimum in the TDCS in the doubly symmetric geometry [4] for ionization of atomic hydrogen by both electron and positron impact. For electron impact the incident energy is about six times the binding energy, whereas for positron impact the incident energy is approximately seven times the binding energy. We determine the velocity field **v** associated with *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* for both projectiles and notice that there is a vortex in this field which is seen by the swirling of the field around a zero in *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* . We are not able to compare our results with Navarrete and Barrachina's results [9–11] since they used collinear geometry and we used the doubly symmetric out-of-plane geometry.

We compare for electron-impact ionization of hydrogen the CB1, various modified CB1, 3DW and the TDCC TDCSs. This comparison is valuable since the CB1 approximation, a distorted-wave approximation, is perturbative and is applicable at high energies, while the TDCC method is an ab-initio method that is typically applied at low to intermediate energies. The CB1 gives the correct asymptotic limit for the ionization amplitude for fixed scattering angle. However, the CB1 TDCS multiplied by the modulus squared of the normalization factor of the Coulomb wave function of the two outgoing electrons gives the correct behavior in the vicinity of the Wannier's threshold law [13]. This approximation that obtains the TDCS as the product of the CB1 and the modulus squared of the normalization factor is called as the improved final-state Coulomb–Born approximation (ICBA) [13], and it is one of the three modified Coulomb–Born (CB1) approximations given in reference [16]. Surprisingly, the 3DW method, while a perturbative method, can give results that are good all the way down close to threshold, depending on the final-state energy. Since the 3DW has the post collision interaction (PCI) to all orders of perturbation theory, the 3DW can work well for intermediate to low energies, depending on the strength of the PCI [39,40].

The outline of the paper is as follows. In Section 2 we briefly describe the CB1, various modified CB1, the 3DW and the TDCC methods. We also give the expression for the velocity field **v** associated with *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* and the equation of the circulation Γ. In Section 3.1 we compare for electron-hydrogen ionization the TDCS computed with the various methods. We also show *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* and direction of the velocity field **v**ˆ = **v**/|**v**|. In Section 3.2 we present for positron-hydrogen ionization the TDCS computed with the CB1 and modified CB1 approximations and we show both *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* and the direction of the velocity field.

We use atomic units throughout the paper unless otherwise stated. We report angles in degrees and the incident energies in eV.

#### **2. Theory**

The CB1 transition matrix element *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* , derived by Botero and Macek [12–15], is the first non-zero term in a perturbative expansion of the exact transition matrix element. For electron-impact ionization from atomic hydrogen, the direct *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* is given by [12–15]

$$T\_{\mathbf{k},1s}^{\rm CB1} = \langle \psi\_{\mathbf{K}\_f}^-(\mathbf{r}) \psi\_{\mathbf{k}}^-(\mathbf{r}') \left| \frac{1}{|\mathbf{r} - \mathbf{r}'|} \right| \varphi\_i(\mathbf{r}') \psi\_{\mathbf{K}\_i}^+(\mathbf{r}) \rangle\_\prime \tag{1}$$

where **k**, **K***<sup>i</sup>* and **K***<sup>f</sup>* are the momentum of the ejected electron, the momentum of the incident particle, and momentum of the scattered particle, respectively. In Equation (1) **r** and **r** are, respectively, the position vector of the incident (or scattered) particle and of the atomic (or ejected) electron relative to the proton. Also, in this equation, *ϕ<sup>i</sup>* is the ground-state wave function of hydrogen and *ψ*<sup>±</sup> are Coulomb wave functions where the subscript − and the subscript + refer to incoming and outgoing boundary conditions, respectively [14,16].

For the doubly symmetric geometry [3,4,41] and for the effective charge *Z*eff in the Coulomb wave functions of the incident electron and the scattered electron equal to the atomic number *ZT* = 1 of hydrogen, the direct and exchange CB1 transition matrix elements are equal. Therefore, the CB1 TDCS for electron-hydrogen ionization can be expressed as:

$$\frac{d^3 \sigma^{\rm CB1}}{d \Omega\_f dE\_\mathbf{k} d\Omega\_\mathbf{k}} = (2\pi)^4 \frac{K\_f k}{K\_i} |T\_{\mathbf{k}, 1s}^{\rm CB1}|^2,\tag{2}$$

where *d*Ω**<sup>k</sup>** is the solid angle for the ejected electron, *d*Ω*<sup>f</sup>* is the solid angle for the scattered particle, and *E***<sup>k</sup>** is the energy of the ejected electron [14,24].

The modified CB1 TDCS for electron-impact ionization is the CB1 TDCS multiplied by the normalization factor given by reference [16], namely

$$|D\_3^-\left(\mathbf{r}\_{3\text{ave}}\right)|^2 = \frac{\pi \exp\left(-\frac{\pi}{k\_3}\right)}{k\_3[1 - \exp\left(-\frac{\pi}{k\_3}\right)]} |\,\_1F\_1(\frac{i}{2k\_3}, 1, -2ik\_3r\_{3\text{ave}})|^2,\tag{3}$$

where *k*<sup>3</sup> = |**K***<sup>f</sup>* − **k**|/2. Also, **r**3*ave* is the vector whose magnitude is the average spacing between the two outgoing electrons and whose direction is taken to be along **k**ˆ 3, **r**3*ave* = *r*3*ave***k**ˆ 3, which is appropriate in the Wannier region.

We follow reference [16] by considering three modified CB1 approximations which correspond to three different values of *r*3*ave*, namely *r*3*ave* = 0, *r*3*ave* = 1/*k*<sup>3</sup> and

$$r\_{3\text{ave}} = \frac{\pi^2}{16\varepsilon} \left( 1 + \frac{0.627}{\pi\sqrt{\varepsilon} \ln \varepsilon} \right)^2,\tag{4}$$

respectively. In Equation (4), *ε* = (*K*<sup>2</sup> *<sup>f</sup>* + *<sup>k</sup>*2)/2, which is the total energy of the two outgoing electrons. By taking the limit *r*3*ave* → ∞, |*D*<sup>−</sup> <sup>3</sup> (**r**3*ave*)| <sup>2</sup> → 1, the CB1 TDCS is obtained. In contrast, when *<sup>r</sup>*3*ave* = 0, |*D*<sup>−</sup> <sup>3</sup> (**r**3*ave*)| <sup>2</sup> = |*N*<sup>−</sup> *e*−*e*<sup>−</sup> | 2, where *N*<sup>−</sup> *<sup>e</sup>*−*e*<sup>−</sup> is the normalization factor of the Coulomb wave function *ψ*<sup>−</sup> **k3** of the two outgoing electrons. This modified Coulomb–Born approximation is called the ICBA in reference [13].

The modulus squared of the normalized factor of the Coulomb wave function for two outgoing particles (*e*<sup>−</sup> and *e*<sup>−</sup> for electron-impact ionization and *e*<sup>+</sup> and *e*<sup>−</sup> for positron-impact ionization) is given by [7,13,16]

$$\left|N\_{\mathfrak{c}^{-}\mathfrak{c}^{\pm}}\right|^{2} = \frac{\pm \pi \exp(\pm \frac{\pi}{k\_{3}})}{k\_{3}[\exp(\pm \frac{\pi}{k\_{3}}) - 1]}.\tag{5}$$

In an earlier paper [25], and for positron-impact ionization of hydrogen, we refer to the CB1 TDCS multiplied by |*N*<sup>−</sup> *e*−*e*<sup>±</sup> | <sup>2</sup> as the modified CB1 TDCS. Here, where we are considering three different modified CB1 TDCS for electron-impact ionization of hydrogen, we specify this particular modified CB1 TDCS as the modified CB1 TDCS with *r*3*ave* = 0 to distinguish it from the other two (where *r*3*ave* = 1/*k*<sup>3</sup> and *r*3*ave* is given by Equation (4)).

For positron-impact ionization, **K***<sup>i</sup>* and **K***<sup>f</sup>* in Equations (1) and (2) are the momentum of the incident positron and the momentum of the scattered electron, respectively, and the interaction of the positron with the proton is repulsive.

Triple differential cross sections for electron-impact ionization of hydrogen are also calculated within the 3DW approximation. The 3DW approach is described in previous publications [42–44] (and references therein), so here we only present the theoretical aspects relevant to the present paper. In this approach, the wave functions for the incident electron, scattered electron, and ejected electron are all distorted waves. The important distinction of the method is that the exact final state electron-electron interaction, normally called the post collision interaction (PCI), is included in the approximation for the final state wave function of the system. The fact that any physical effect included in the system wave function is automatically included to all orders of perturbation theory means that PCI is included to all orders of perturbation theory in this approach. In the 3DW approximation the direct and exchange amplitudes are identical for the doubly symmetric in-plane geometry for electron-hydrogen ionization.

The TDCC calculations shown here are very similar to previous calculations [28] for electron ionization of atomic hydrogen. In brief, the TDCC approach solves the time-dependent Schrödinger equation for the two electrons by propagating the corresponding differential equations in time until convergence of the probabilities for ionization has been reached. The calculations shown here included fourteen partial waves, which were found to be sufficient for the TDCS calculations shown here. We note that the small cross sections found at larger scattering angles (beyond 100◦) are quite sensitive to some of the numerical aspects of the calculations (such as mesh size, number of coupled channels). We note that these small cross sections are around three orders of magnitude lower than the peak of the cross section, which occurs at a scattering angle of approximately 39◦.

When there is a zero in the real and imaginary parts of *T***k**,1*s*, there is a vortex in the velocity field associated with *T***k**,1*s*. Previously vortices in the velocity fields associated with atomic wave functions have been discussed by Bialynicki-Birula et al. [45]. Recently vortices in the velocity field associated with the ionization amplitude have been discussed by Macek et al. [5] and by Macek [23]. Here, we discuss vorticies in the velocity field associated with the transition matrix element which is proportional to the ionization amplitude [24]. The velocity field **v** associated with *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* for electron or positron ionization can be expressed as [24,25]

$$\mathbf{v} = \nabla\_{\mathbf{k}} \text{Im}[\ln T\_{\mathbf{k}, 1s}^{\text{CB1}}].\tag{6}$$

The circulation, Γ, for a closed contour, *c*, taken in the anticlockwise direction that encloses only one first-order zero in *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* is given by [5,23–25,45]

$$
\Gamma = \int\_{\mathcal{C}} \mathbf{v} \cdot d\boldsymbol{\ell} = \pm 2\pi. \tag{7}
$$

#### **3. Results**

#### *3.1. Deep Minimum in the TDCS for Electron-Impact Ionization of Hydrogen*

For electron-impact ionization of hydrogen we perform calculations for the doubly symmetric in-plane geometry (Figure 1a) where both outgoing particles have the same energy and same angle, *ξ*, with respect to the *z*-axis. This geometry was used by Murray and Reed in their experimental measurements for electron-helium ionization [4]. The scattered electron makes a polar angle with respect to the *z*-axis of *ξ* while the ejected electron makes an angle of –*ξ* with respect to the same axis.

**Figure 1.** Geometry figures for both in-plane and out-of-plane configurations.

A deep minimum in the TDCS occurs when the real and imaginary parts of the transition matrix element, *T***k**,1*s*, are identically zero at the same angle; this occurs at a polar angle of 87.94◦ in the CB1 approximation (Figure 2). A comparison of the TDCSs computed with the CB1, modified CB1 with *r*3*ave* = 0, modified CB1 with *r*3*ave* = 1/*k*3, and the modified CB1 with *r*3*ave* given by Equation (4) approximations can be seen in Figure 3a. The position of the minimum is the same for these four approximations.

We show the comparison between the modified CB1 with *r*3*ave* = 0, 3DW, and TDCC methods in Figure 3b. There is a dip 3DW TDCS at a polar angle of 90.2◦ and a slight dip in the TDCC TDCS at a polar angle of 90.6◦. The polar angle for the deep minimum in the CB1 TDCS is within 3 degrees of these angles.

**Figure 2.** Transition matrix element, *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* , versus the polar angle *ξ* for electron-impact ionization of hydrogen at 76.4541 eV. (**a**) Re[*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* ] (solid red line) and Im[*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* ] (dashed blue line) over the full angular range (**b**) Re[*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* ] (solid red line) and Im[*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* ] (dashed blue line) in the vicinity of the zero in *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* .

**Figure 3.** Triply differential cross section (TDCS) on a log (ln) scale versus the polar angle *ξ* for electron-impact ionization of hydrogen at 76.4541 eV. (**a**) Comparison of the CB1 TDCS (dot-dashed black line), the modified CB1 TDCS with *r*3*ave* = 0 (solid orange line), the modified CB1 TDCS with *r*3*ave* = 1/*k*<sup>3</sup> (dashed purple line) and the modified CB1 TDCS with *r*3*ave* given by Equation (4) (dotted brown line). (**b**) Comparison of the modified CB1 TDCS with *r*3*ave* = 0 (solid orange line) with an incident energy of 76.4541 eV, the 3DW TDCS (dotted blue line) for an incident energy of 76.46 eV, and the TDCC TDCS (dashed green line) for an incident energy of 76.45 eV.

A zero in the transition matrix element means that there is a vortex in the velocity field (Figure 4) associated with this element. The rotation of the velocity field is in the anticlockwise direction and the value of the circulation Γ is 2*π*. Figure 4 shows the nodal lines of Re[*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* ] and Im[*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* ] and it shows the direction of the velocity field, **v**ˆ = **v**/|**v**|, by the arrows. (The axes for this figure are *kz* and *kx*, where *kz* and *kx* are respectively the *z*- and *x*-components of the momentum **k** of the ejected electron [25].) The figure also shows a density plot of ln |*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* |.

**Figure 4.** A density plot of ln <sup>|</sup>*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* | for electron-impact ionization of hydrogen for a fixed incident energy of 76.4541 eV in-plane and a grid in the *z*- and *x*-components of the momentum (*kz*, *kx*) of the ejected electron **k**. The nodal lines of Re[*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* ] and Im[*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* ] are shown respectively, by the solid blue line and the dashed green line. The direction of the velocity field, **v**ˆ = **v**/|**v**|, is indicated by the arrows.

#### *3.2. Deep Minimum in the CB1 TDCS for Positron-Impact Ionization of Hydrogen*

For positron-impact ionization of hydrogen, we also consider a doubly symmetric geometry; however, for this projectile, we use the out-of-plane geometry that we show in Figure 1b. The gun angle *ψ* is the angle the incident particle makes with the *z*-axis [4]. We find a deep minimum in the CB1 TDCS for an incident energy of 100 eV with a gun angle of 56.13◦. We choose this incident energy because it had been previously used [9]. Using the CB1 approximation, we vary the gun angle until we obtain a zero in *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* . The zero in *<sup>T</sup>*CB1 **<sup>k</sup>**,1*<sup>s</sup>* can be seen in Figure 5.

**Figure 5.** Transition matrix element *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* versus the polar angle, *ξ*, for positron-impact ionization of hydrogen at 100 eV with a gun angle, *ψ*, of 56.13◦. (**a**) Re[*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* ] (solid red line) and Im[*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* ] (dashed blue line) over the full angular range (**b**) Re[*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* ] (solid red line) and Im[*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* ] (dashed blue line) in the vicinity of the zero in *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* .

In the positron-impact ionization of hydrogen at 100 eV with a gun angle of 56.13◦ there is a deep minimum in the TDCS at a polar angle of 46.79◦. This deep minima in the TDCS and a secondary dip around *ξ* ≈ 140◦ can be seen in Figure 6. The secondary dip in the TDCS is in the vicinity of

the intersection point of the real and imaginary parts of *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* and a second zero in Re[*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* ] (see Figure 5). We perform an out-of-plane search to see if we could find a zero in *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* around ≈ 140◦, but without success.

We note that we also obtain a deep minimum in the TDCS for a slightly lower incident energy, 95 eV, if the gun angle is reduced a little to 55.75◦. We expect that we could track the position of the deep minimum for different incident energies and obtain a locus of points that gives the angles *ψ* and *ξ* for these energies as we did previously for electron-helium ionization [25].

**Figure 6.** TDCS on a log (ln) scale versus the polar angle, *ξ*, for positron-impact ionization of hydrogen at 100 eV with a gun angle of 56.13◦. Comparison of the CB1 TDCS (dot-dashed black line) and the modified CB1 TDCS (solid orange line) results.

For positron-impact ionization of hydrogen, we find that circulation Γ is 2*π* and the velocity field that is associated with the zero in *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* rotates anticlockwise, as can be seen in Figure 7. This figure shows the direction of the velocity field, **v**ˆ = **v**/|**v**|, by the arrows and it shows the nodal lines of Re[*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* ] and Im[*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* ]. It also gives a density plot of ln |*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* |.

**Figure 7.** A density plot of ln <sup>|</sup>*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* | for positron-impact ionization of hydrogen for a fixed incident energy of 100 eV out-of-plane with a gun angle of 56.13◦ and a grid in the *z*- and *x*-components of the momentum (*kz*, *kx*) of the ejected electron **k**. The nodal lines of Re[*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* ] and Im[*T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* ] are shown respectively, by the solid blue line and the dashed green line. The direction of the velocity field, **v**ˆ = **v**/|**v**|, is indicated by the arrows.

For the specific geometries and kinematics that we consider, the direction of rotation of the velocity field is the same for positron- and electron-impact ionization of hydrogen. Interestingly, for the two ionization processes, the polar angle *ξ* of the deep minimum in the CB1 TDCS is less than 90◦, and thus, *ξ* lies in the first quadrant.

#### **4. Conclusions**

While a deep minimum in the TDCS was not expected to exist for electron-impact ionization of hydrogen, we were able to obtain one using the CB1 and modified CB1 approximations for an incident energy of 76.4541 eV [2]. The CB1 approximation is a high-energy approximation that has been applied to intermediate energies. Previously, the CB1 and modified CB1 approximations have been applied to electron-helium ionization for an incident energy of 64.6 eV [25]; and, in the region of the experimental data, the TDCS results compare reasonably well with experimental measurements [4] and with TDCC results [28]. These approximations have also been applied previously to positron-helium ionization [25].

Using the CB1 and a modified CB1 approximations we obtain a deep minimum in the TDCS for positron-impact ionization of hydrogen for an incident energy of 100 eV in the doubly symmetric out-of-plane geometry with a gun angle of 56.13◦. This energy has been previously considered for positron-hydrogen ionization in the collinear geometry [8,9]. A benefit of the CB1 approximation is that it is a fairly simple approximation that does not take much computing time.

While the question of whether there can be a zero in the transition matrix element and a corresponding deep minimum in the TDCS for electron-impact ionization of hydrogen has not been resolved, we have found that these exist within the CB1 approximation. Due to the fact that zeros in a transition matrix element have been found previously for positron-impact ionization of hydrogen [8,9], a zero in *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* for electron-hydrogen ionization is not that surprising even though there is no passive electron. We note that, for electron-hydrogen ionization, neither the TDCC or 3DW methods obtained a deep minimum in the TDCS; however, they did obtain a dip.

The direction of the rotation of the velocity field around a zero in *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* does not appear to be projectile dependent as for both electron- and positron-impact ionization of hydrogen the rotation is anticlockwise. This may be due to the fact that the polar angle of the zero in *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* is between 0◦ and 90◦ (first quadrant) for both cases. An earlier paper shows, for positron-helium ionization, clockwise rotation of the velocity field about the zero in *T*CB1 **<sup>k</sup>**,1*<sup>s</sup>* ; however, the polar angle for this zero is between 90◦ and 180◦ (second quadrant) [25]. For the energies that were considered in reference [25] for electron-helium ionization the direction of the velocity field is anticlockwise and the polar angle lies in the first quadrant.

The interesting finding that for both electron- and positron-impact ionization of hydrogen the velocity field rotates in the same direction is different from the previous finding for impact ionization of helium [25]. This may be due to the fact that for impact ionization of hydrogen by the two projectiles the deep minimum in the TDCS occurs at an angle less than 90◦, while for earlier results of electron and positron ionization of helium the zeros occur in different quadrants [25].

**Author Contributions:** C.M.D. and S.J.W. prepared the manuscript (with input from J.C. and D.H.M.). C.M.D., under the supervision of S.J.W., performed the CB1 and modified CB1 calculations for both electron- and positron-impact ionization of hydrogen. J.C. performed the TDCC calculations and provided a discussion of the method. The 3DW calculations are from D.H.M. and S.A. A discussion on the 3DW approximation was provided by D.H.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** S.J.W. is thankful for support from the NSF under Grant No. PHYS-1707792.

**Acknowledgments:** J. B. Kent, while an undergraduate student at UNT and under the supervision of S. J. Ward, performed CB1 and modified CB1 calculations for electron-impact ionization of hydrogen. We acknowledge that, for electron-hydrogen ionization, he was the first to locate the position of the zero in the CB1 transition matrix element and the corresponding deep minimum in the TDCS. SJW would like to thank Gaetana (Nella) Laricchia for encouraging the CB1 investigation. CMD and SJW are thankful for the CB1 Fortran codes from Javier Botero. Wolfram Research [46] and Microsoft Publisher [47] were used to generate the images in this paper. Also, Wolfram Research [46] was used in some of the calculations. JC acknowledges the support by the US Department of Energy

through the ASC PEM Program of the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

The following abbreviations are used in this manuscript:


#### **References**


c 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Resonances in Systems Involving Positrons**

#### **Anand K. Bhatia**

Heliophysics Science Division, NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA; anand.k.bhatia@nasa.gov

Received: 17 February 2020; Accepted: 5 May 2020; Published: 7 May 2020

**Abstract:** When an incident particle on a target gets attached to the target, the cross-section at that energy could be much larger compared to those at other energies. This is a short-lived state and decays by emitting an electron. Such states can also be formed by the absorption of a photon. Such states are below the higher thresholds and are called autoionization states, doubly excited states, or Feshbach resonances. There is also a possibility of such states to form above the thresholds. Then they are called shape resonances. Resonances are important in the diagnostic of solar and astrophysical plasmas. Some methods of calculating the resonance parameters are described and resonance parameters occurring in various systems are given.

**Keywords:** autoionization states; doubly excited states; Feshbach states; resonances; shape resonances

#### **1. Introduction**

In measuring scattering cross-sections, a peak or dip implies that the incident particle has formed a compound state which decays after a while. Such a state is called an autoionization state, a doubly excited state, or a resonance state, and has a very short lifetime compared to real bound states. Such states can also be formed by absorption of radiation in the target. Resonances are ubiquitous in electron-atom and electron-ion interactions. They play an important role in solar and astrophysical plasmas to infer temperatures and densities of plasmas [1]. However, they are not that common in the case of positron-target systems.

In a simple system like a positron-hydrogen, the positron and electron tend to be on the same side of the nucleus because of the attraction between the two particles, unlike in the case of the electron-hydrogen system where the two electrons tend to be on the opposite sides of the nucleus because of the repulsion between the two electrons. This shows that the correlations become very important in a positron-hydrogen system. This makes calculations of resonances difficult because many terms are required to calculate resonance parameters. Moreover, there is a positronium channel open below all the thresholds and this adds further complications.

#### **2. Methods of Calculations**

There are methods like the stabilization method, complex rotation method, Feshbach operator formalism [2], close-coupling method, and *R-*matrix method to calculate the resonance parameters. In approach [2], projection operators *P* and *Q* are defined such that *P* projects on a state and *Q* = 1 − *P* removes that state, *P<sup>2</sup>* = *P, Q2* = *Q* (idempotent)*,* and *PQ* = 0 (orthogonality). We form a wave function *Q*Ψ which is such that the lower states have been removed [3]. Therefore, using Raleigh–Ritz variational principle, we obtain eigenvalues:

$$
\varepsilon\_Q = \frac{} \tag{1}
$$

These eigenvalues give us the positions of resonances. We obtain these bound states which are embedded in the continuum and are below the higher thresholds. They correspond to resonance states within the continuum which must be calculated separately and the width for each state must be calculated. In the narrow region of the width, the scattering phase shift increases by π radians. Calculation of the shift and width requires continuum functions (cf. Equation (2.13a)) in [3]. Various approximations like the exchange approximation, method of polarized orbitals, close-coupling approximation have been used to calculate continuum functions. However, it is difficult to write projection operators *P* and *Q* when the positronium channel is open. We need to use a method which does not depend on projection operator formalism [2].

The complex rotation method, based on a theorem by Belslev and Combes [4], has been applied extensively to calculate resonance parameters with great accuracy. The advantage of this formulation is that only discrete functions are included in the wave function and the continuum function is not necessary. In this method, the radial part is rotated by an angle θ. The Hamiltonian is transformed in the same way. The angle is varied until the eigenvalues do not change. Widths of the states are also obtained in the same calculation. Since there are also eigenvalues in the continuum, the shift mentioned above is included in the resonance positions obtained in this method. This is discussed further in Section 3.

This method gives the resonance positions which include the shift due to the interaction of discrete states with the continuum and need not be calculated separately. The eigenvalues obtained in this method are complex, where the complex part gives the width of the state.

In the positron-hydrogen system, Mittleman [5] showed that the equation for the positron-hydrogen system has an attractive potential proportional to 1/r2 due to the degeneracy of the 2*s* and 2*p* states of the hydrogen atom and therefore, there should be an infinite number of resonances in this case as in the electron-hydrogen system. Mittleman [5] conclusively showed the existence of resonances without carrying out detailed calculations. The resonances in electron-hydrogen system have been observed but not in the positron-hydrogen system, at least up to now.

#### **3. Calculations and Results**

The first successful calculation for the *S-*wave resonance was carried out by Doolen, Nuttal, and Wherry [6] using a sparse-matrix technique in the complex-rotation method. In this method, the radial coordinates are transformed by an angle θ:

$$r \to n\epsilon^{i0} \tag{2}$$

The Hamiltonian *H* = *T* + *V* is transformed to:

$$H = Te^{-2i0} + Ve^{-i0} \tag{3}$$

They used a wave function of the form:

$$\Phi(r\_1, r\_2, r\_{12}) = \exp(-a(r\_1 + r\_2)L\_l^0(u)L\_m^0(v)L\_n^0(w) \tag{4}$$

In the above equation, α is the nonlinear parameter, *L*<sup>0</sup> *<sup>l</sup>* is a Laguerre polynomial and:

$$u = a(r\_2 + r\_{12} - r\_1), \\ v = a(r\_1 + r\_{12} - r\_2), \text{ and } w = 2a(r\_1 + r\_2 - r\_{12}) \tag{5}$$

They found only one resonance whose complex energy is given by:

$$E(complex) = \text{Re}(E) + \text{Im}(E) = \text{Re}(E) - i\Gamma/2 \tag{6}$$

Re(E) represents the position of the resonance and the imaginary part represents the half width of the resonance. In this method, Im(E) is plotted vs. Re(E) and we look for the stationary paths as the angle θ is increased for a fixed value of the nonlinear parameter α. Their results as a function of the number of terms are given in Table 1.


**Table 1.** Position and width of the resonance below *n* = 2 threshold of the hydrogen atom.

This table shows that the resonance is at E = −0.2573741 Ry with a width of 0.0001354 Ry. It is clear from the table that a wave function with very large number of terms is needed to calculate the resonance parameters which is not so in the case of the electron-hydrogen. Prior to this, attempts by various authors failed to infer the existence of this resonance due to using a very small number of terms in their wave functions. This calculation provided incentive to look for resonances below the higher thresholds as well. Varga, Mitroy, Mezie, and Kruppa [7] carried out calculations below the n = 2, 3, and 4 thresholds. Their results are shown in Table 2. There are two resonances below *n* = 2 threshold, three below n = 3 threshold, and three below *n* = 4 threshold.

**Table 2.** Positron-hydrogen resonances [7] below higher thresholds. Units are Ry.


It is possible to find positions of higher resonances, using the relation between two resonances given by Temkin and Walker [8]:

$$
\varepsilon\_{n+1} = \varepsilon^{-2\pi/a} \varepsilon\_{n\prime} \tag{7}
$$

$$\alpha = \left(\sqrt{37} - 5/4\right)^{0.5} \tag{8}$$

This was deduced for electron-hydrogen resonances but does give reasonable values of higher resonances in positron-hydrogen system as well.

The potential between a positron and He<sup>+</sup> is repulsive. Therefore, an existence of resonances in this system seems unlikely. Using the stabilization method, Bhatia and Drachman [9] showed the existence of several resonances. This was confirmed by Ho [10] who carried out a definitive calculation using the complex rotation method described above and showed the correctness of their results [9]. Hylleraas type functions have been used in most calculations. The resonance parameters obtained by him are shown in Table 3.


**Table 3.** Positions and widths of S- and P-resonances in e+-He<sup>+</sup> system below the *n* = 2 threshold [10], Units Are Ry.

Kar and Ho [11] carried out similar calculations for e+-He system and their results agree with those obtained by Ren, Han, and Shi [12], who used hyperspherical coordinates. Their results are shown in Table 4. The resonances are very narrow compared to those in e+-He<sup>+</sup> system.


**Table 4.** Resonance parameters in the e+-He system. Units Are Ry.

#### **4. Resonances in Ps**

The Ps− system is obtained when the proton in H− is replaced by a positron. Now the nucleus has the same mass as an electron. Therefore, the mass polarization term in the Hamiltonian becomes important. The binding energy of Ps− is very close to half of the binding energy of H−. The Hamiltonian is given by:

$$H = -2\nabla\_1^2 - 2\nabla\_2^2 - 2\nabla\_1 \cdot \nabla\_2 - 2/r\_1 - 2/r\_2 + 2/r\_{12} \tag{9}$$

$$H = T + V\tag{10}$$

where r1, r2, are the coordinates of electrons with respect to the positron and *r*<sup>12</sup> = -- - → *<sup>r</sup>* <sup>1</sup> <sup>−</sup> <sup>→</sup> *r* <sup>12</sup> -- - .



**Table 5.** Singlet and triplet *S*-wave resonances in Ps−. Positions are with respect to the ground state of the positronium. Units Are Ev.

There are several doubly-excited or Feshbach-type triplet *P* even parity resonances below *n* = 2, 3, 4, 5, and 6 thresholds. These have been calculated by Ho and Bhatia [14]. They are given below each threshold in Table 6. The third resonance below *n* = 4 threshold is a shape resonance because it is above the *n* threshold.


**Table 6.** Doubly-excited <sup>3</sup>*P*<sup>e</sup> resonances in Ps−.

<sup>a</sup> Shape resonance, above the threshold.

There are also odd parity triplet and singlet states which have been calculated by Bhatia and Ho [14–16]. Their results are shown in Tables 7 and 8. The lowest odd parity state in Ps− has been observed by Michishio et al. [17] using laser beams of 2285 and 2297 angstroms and their results for the position and width agree with those given in Table 8.

**Table 7.** Odd parity triplet P-shape resonances of Ps−.


**Table 8.** Odd parity singlet P-shape resonances of Ps−.


Similar calculations have been carried out by Bhatia and Ho [18] for odd parity singlet and triplet *D* states of Ps−. Their results are given in Table 9.

**Table 9.** Triplet and singlet *D* states of Ps−. Positions are with respect to the ground state of Ps = –0.5 Ry.


<sup>a</sup> Shape resonance.

#### **5. Resonances in PsH**

As it is, it appears that the system having two neutral atoms, cannot have bound states. However, if it is viewed as a system consisting of a positron and H−, then the positron is in the field of the Coulomb field of the hydrogen ion. There should be infinite number of bound states and resonances. Drachman and Houston [19] and Ho [20] have calculated the parameters of the S-wave resonance given below in Rydberg units:

$$E\_R = -1.1726 \pm 0.0007 \text{ \AA} = (4.6 \pm 1.1) \times 10^{-6} \tag{11}$$

by Drachman and Houston [19], and:

$$E\_R = -1.205 \pm 0.001 \Gamma = (5.5 \pm 2.0) \times 10^{-3} \tag{12}$$

by Ho [20].

Similarly, there are *P*-wave and *D*-wave resonances.

A hybrid theory [21] has been developed, which considers the long-range and short-range correlations at the same time. The theory is variationally correct. In the scattering calculations, calculated phase shifts have lower bounds to the exact phase shifts. This method has been applied to calculate resonance parameters in He and Li+. Phase shifts are calculated in the resonance region [22] and then fitted to the Breit–Wigner formula (cf. Equation (17) in [22]) to infer resonance parameters (cf. Equation (17) of [22]). The results obtained agree with those obtained using the Feshbach formalism [3]. The hybrid theory can also be applied to calculate resonance parameters in positron-target systems. Such calculations have not yet been carried out.

The books mentioned in references [1,23] have several chapters describing various methods employed to calculate resonance parameters. The author has two chapters in the book mentioned in ref. [1] (written with Aaron Temkin) on methods of calculating resonance parameters, also in [3].

#### **6. Conclusions**

Resonances in electron-target systems can be calculated easily. However, in the case of positron-target systems, calculations are not easy because of the positronium channel which is present below every threshold. As indicated above, correlations are very important. Therefore, wave functions having many terms are required. There are other systems with positrons where resonance parameters have been calculated. Here, we have discussed a few such systems and have described some of the methods of calculations. A theory called hybrid theory which includes the long-range and short-range correlations and is variationally correct has given resonance parameters for He and Li+, which agree with those obtained in earlier calculations. This is achieved by calculating phase shifts in the resonance region and fitting them to the Breit–Wigner expression to infer resonance parameters. This approach could be applied to positron-target systems. There could be results in the future, using this approach.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Review* **Positron Processes in the Sun**

#### **Nat Gopalswamy**

Solar Physics Laboratory, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA; nat.gopalswamy@nasa.gov

Received: 19 March 2020; Accepted: 16 April 2020; Published: 22 April 2020

**Abstract:** Positrons play a major role in the emission of solar gamma-rays at energies from a few hundred keV to >1 GeV. Although the processes leading to positron production in the solar atmosphere are well known, the origin of the underlying energetic particles that interact with the ambient particles is poorly understood. With the aim of understanding the full gamma-ray spectrum of the Sun, I review the key emission mechanisms that contribute to the observed gamma-ray spectrum, focusing on the ones involving positrons. In particular, I review the processes involved in the 0.511 MeV positron annihilation line and the positronium continuum emissions at low energies, and the pion continuum emission at high energies in solar eruptions. It is thought that particles accelerated at the flare reconnection and at the shock driven by coronal mass ejections are responsible for the observed gamma-ray features. Based on some recent developments I suggest that energetic particles from both mechanisms may contribute to the observed gamma-ray spectrum in the impulsive phase, while the shock mechanism is responsible for the extended phase.

**Keywords:** solar flares; coronal mass ejections; shocks; positrons; positronium; positron annihilation; pion decay

#### **1. Introduction**

Positrons, the antiparticle of electrons, were proposed theoretically by Dirac and were first detected by Anderson in 1933 [1]. Positrons are extensively used in the laboratory for a myriad of purposes (see review by Mills [2]). Astrophysical processes involving positrons have been found in the interstellar medium [3], and galactic bulge and disk [4]. Positrons are commonly found in the Sun [5]. The proton–proton chain, which accounts for most of the energy release inside the Sun involves the emission of a positron when two protons collide to form a deuterium nucleus. There are plenty of electrons present in the solar core, so the positrons are immediately annihilated with electrons and produce two gamma-ray photons. In one proton–proton chain, two positrons are emitted and hence contribute a total of four photons in addition to the two photons emitted during the formation of a helium-3 (3He) nucleus from the fusion of deuterium nucleus with a proton.

High-energy particles exist in the solar atmosphere, energized during solar eruptions. Solar eruptions involve flares and coronal mass ejections (CMEs). A process known as magnetic reconnection taking place in the solar corona is thought to be the process which converts energy stored in the stressed solar magnetic fields into solar eruptions (see Figure 1 for a schematic of a typical eruption). One part of the released energy heats the plasma in the eruption region, while another goes to energize electrons and ions. Electromagnetic radiation from radio waves to gamma-rays produced by the energized electrons and protons by various processes is known as a flare. Acceleration in the reconnection region is referred to as flare acceleration or stochastic acceleration. CMEs also carry the released energy as the kinetic energy of the expelled magnetized plasma with a mass as high as 1016 g and speeds exceeding 3000 km/s. Such fast CMEs drive fast-mode magnetohydrodynamic shocks that can also energize ambient electrons and ions to very high energies. Diffusive shock

acceleration and shock drift acceleration are the mechanisms by which particles are energized at the shock (see [7] for a review on acceleration mechanisms during solar eruptions). The flare and shock accelerations were referred to as first- and second-phase accelerations during an eruption by Wild et al. [8]. Accelerated particles have access to open and closed magnetic structures associated with the eruption resulting in a number of electromagnetic emissions via different emission mechanisms. Energetic particles escaping along interplanetary magnetic field lines are detected as solar energetic particle (SEP) events by particle detectors in space and on ground. These particles, originally known as solar cosmic rays, were first detected by Forbush [9] in the 1940s.

**Figure 1.** Schematic of a solar eruption and the sites of particle acceleration (e,p, ... ): one in the current sheet formed low in the corona and the other on the surface of the shock driven by the coronal mass ejection (CME) flux rope. The arrows toward the current sheet indicate the reconnection inflow while the ones diverging away indicate the outflow. The red ellipses in the photosphere represent the feet of flare loops where accelerated particles precipitate and produce flare radiation. Particles from the shock propagate away from the Sun and are detected as energetic particle events in space. The dark ellipse inside the flux rope represents a prominence that erupted along with the CME (adapted from Gopalswamy [6]).

Invoking the copious production of energetic particles from the Sun, Morrison [10] suggested that the active Sun must be a source of gamma-rays. He listed electron-positron annihilation as one of the processes expected to produce gamma-ray emission from cosmic sources [10]. Elliot [11] suggested that "positive electrons" from muon decay should lead to detectable 0.5 MeV gamma-ray line emission. Lingenfelter and Ramaty [12] performed detailed calculations of gamma-ray emission processes from the Sun. Chupp et al. [13] identified for the first time the positron annihilation radiation at 0.5 MeV along with other nuclear lines during the intense solar flares of 1972 August 4 and 7 using data from the Gamma-ray Monitor onboard NASA's Seventh Orbiting Solar Observatory (OSO-7) mission.

#### **2. Mechanisms of Positron Production**

Positrons are predominantly produced by three processes: (i) emission from radioactive nuclei, (ii) pair production by nuclear deexcitation, and (iii) decay of positively charged pions (π+) that take place when ions accelerated in the corona interact with the ions in the photosphere/chromosphere. Kozlovsky et al. [14] list 156 positron-emitting radioactive nuclei resulting from the interaction of protons and α-particles (helium nuclei) with 12 different elements and their isotopes. The most important positron-emitting radioactive nuclei that result from the interaction of protons (p) and α-particles with carbon (12C, 13C), nitrogen (14N, 15N), and oxygen (16O, 18O) are listed in Table 1. In the interaction of p and α with 16O, the oxygen nucleus, ends up in the excited state (16O\*) of 6.052 MeV;

this nucleus deexcites by emitting an electron-positron pair with a lifetime of 0.096 ns. Another such excited nucleus is 40Ca\* with a lifetime of 3.1 ns. Kozlovsky et al. [15] list another set of 23 positron emitters produced when accelerated 3He interact with targets such as 12C, 14N, 16O, 20Ne, 24Mg, 28Si, and 56Fe. The radioactive nuclei have a lifetime ranging from a tenth of a nanosecond to ~1 million years (see [14] for a list). There are 26 radioactive nuclei with a lifetime -10s.


**Table 1.** Important interactions and the resulting radioactive nuclei.

Note: \*Nucleus in excited state.

Positrons from radioactive nuclei have an energy of several hundred keV, while those from π<sup>+</sup> decay have much higher energy (up to hundreds of MeV). Almost all the positrons emitted by radioactive nuclei and a major fraction of those produced by π<sup>+</sup> decay (~80%) slow down to thermal levels (tens of eV) before directly annihilating or forming a positronium (Ps) atom by capturing an electron. The formation of Ps in this way is via radiative recombination. Ps can also be formed due to charge exchange with H and He atoms. Positronium is a hydrogen-like atom (with the proton replaced by a positron). There are two types of Ps, known as orthopositronium (O-Ps) and parapositromium (P-Ps), depending on how the spins of the positron and electron are oriented. In O-Ps, the electron and positron spins are in the same direction (triplet state); in P-Ps, the spins are oppositely directed (singlet state). O-Ps and P-Ps decay into 3 and 2 photons, respectively. O-Ps is formed preferentially: 75% of the time compared to 25% of the time for P-Ps [16]. Four key processes that determine the fate of the positrons produced in the solar atmosphere are discussed in [5]. These processes involve the interaction of positrons with the ambient hydrogen and helium in redistributing their energy, formation and quenching of Ps, and the ultimate production of gamma-rays by direct annihilation or via Ps. Direct annihilation of positrons can occur with free and bound (in H and He) electrons in the ambient medium. Positronium quenching occurs resulting in the emission of second-generation positrons when Ps collides with electrons, H, H+, and He+. Another quenching process is the conversion of O-Ps to P-Ps when O-Ps collides with electrons and H.

Pions (π<sup>0</sup> and π±) are created when accelerated protons and α-particles from the corona collide with those in the chromosphere/photosphere. A detailed list of possible interactions (p-p and p-α) are listed in Murphy et al. [17]. High-energy positrons are primarily emitted from the decay of π<sup>+</sup> into positive muons (μ+), which decay into positrons. In a similar reaction, negative pions (π−) decay into negative muons (μ−), which decay into electrons. π<sup>0</sup> decays into 2 gamma-rays most of the time (98.8%). In the remaining 1.2% of cases, π<sup>0</sup> decays into an electron positron pair and a gamma-ray. The rest energy of neutral pions is 135 MeV, while that of charged pions is 139.6 MeV. To produce these particles, the accelerated protons need to have high energies, exceeding ~300 MeV. The pions are very short-lived (π0: 10−<sup>16</sup> s; <sup>π</sup>±: 2.6 <sup>×</sup> 10−<sup>8</sup> s), while the muons live for a couple of microseconds (2.2 <sup>×</sup> <sup>10</sup>−<sup>6</sup> s). The >300 MeV protons needed for pion production seem to be accelerated both in the flare reconnection and CME-driven shocks (see Figure 1).

#### **3. Gamma-Rays Due to Positrons**

Nonthermal electromagnetic emission produced by charged particles is one of the key evidences for particle acceleration in the Sun (see, e.g., [18] for a review). Energetic electrons are readily inferred from nonthermal emission they produce at wavelengths ranging from millimeters to kilometers. Theories of radio emission have helped us understand the acceleration mechanism for the electrons and the radio emission mechanism. Energetic electrons also produce hard X-rays and gamma-rays. On the other hand, the electromagnetic indicators of energetic ions are limited to gamma-rays, ranging from a few hundred keV to >1 GeV. Positrons, which are produced by various processes noted in Section 2, contribute to the gamma-ray emission from the Sun at various energies via different processes.

Figure 2 shows the total gamma-ray spectrum from the Sun, exhibiting both line emission and continuum as predicted [10]. This spectrum is constructed from all possible processes that emit gamma-rays during solar eruptions [19]. At low energies (<10 MeV), there are many lines superposed on the electron bremsstrahlung continuum, the lowest being the 0.511 MeV line (positron annihilation, marked e+). The next narrow line is the neutron capture line at 2.223 MeV. When energetic protons spallate ambient nuclei, neutrons are produced and emitted over a broad angular distribution; the downward neutrons slow down and are captured by a proton in the ambient medium forming a deuterium nucleus and releasing the binding energy as the 2.223 MeV line. The other lines are due to nuclear deexcitation of varying widths due to various combinations of incumbent and insurgent ions. The continuum emission has several components. Below the 0.511 MeV line, there is a weak Ps continuum that merges with the strong primary electron bremsstrahlung continuum. Primary electrons are those arriving from the acceleration site in the corona, as opposed to secondary electrons, which are produced in the chromosphere/photosphere due to the impact of accelerated ions from the corona (e.g., via π– decay). The quasi-continuum between 0.7 MeV and 10 MeV (on which the discrete lines are superposed) is due to a multitude of broad nuclear lines caused by insurgent heavy ions interacting with ambient H and He nuclei. If the primary electron bremsstrahlung continuum is hard, it can be detected above background even at energies exceeding 10 MeV. When there is significant pion production both a broad line like feature centered near 70 MeV from neutral pions and a hard, secondary positron bremsstrahlung continuum can also be detected above 10 MeV (see below).

**Figure 2.** Overall theoretical spectrum of gamma-ray emission from the Sun from 0.1 MeV to 2 GeV. The blue dashed line labeled "brem" represents the contribution from the bremsstrahlung of energetic electrons accelerated during solar eruptions. The red line represents computed spectrum taking into account of all possible processes that contribute to gamma-ray emission. At energies below ~10 MeV there is a quasi-continuum with several lines superposed. e+ and n represent the 0.511 MeV positron annihilation line and the 2.223 MeV neutron capture line. 12C and 16O mark the next intense lines produced by nuclear deexcitation. The pion continuum at high energies is denoted by π, which involves contribution from both neutral and charged pions. The energy for these emission components is from energetic particles (electrons and ions) accelerated in the solar corona during solar eruptions (adapted from Ramaty and Mandzhavidze [19]).

#### *3.1. The 0.511 MeV Gamma-Ray Line*

The width of the solar annihilation line can range from ~1 keV to 10 keV full width at half maximum (FWHM) depending on the ambient conditions of the medium in which the annihilation takes place: e.g., temperature, density, and the ionization state [20]. Detailed calculations and comparison with observations have confirmed that the 0.511 MeV line is produced over a range of these parameters in the chromosphere/photosphere region [5]. One of the major contributors to the line width is the temperature in the region of annihilation because the ambient electrons have a distribution of speeds at a given temperature. The environmental conditions also determine the formation and destruction of Ps. Parapositronium annihilates emitting two 0.511 MeV photons (2γ decay), with lifetime ∼0.125 ns. On the other hand, orthopositronium annihilates in three 341 keV photons ((3γ decay) with a lifetime ∼142 ns [21]. Depending on the initial energy of the positron capturing an electron, the 3γ decay results in a gamma-ray continuum at energies below the annihilation line (see Figure 2). The flux ratio of the 3γ continuum (from O-Ps) to the 2γ line (from direct annihilation with free and bound electrons, and the decay of P-Ps) is an important parameter that can be used to infer the properties of the ambient medium (density, temperature, and ionization state). Murphy et al. [5] found that the flux ratio at a given temperature in an ionized medium remains constant up to an ambient hydrogen density of ~10<sup>13</sup> cm−<sup>3</sup> and then rolls over to values lower by 2–3 orders of magnitude at densities ~1017 cm−3. The constant value depends on the ambient temperature starting from ~3 for 2000 K and decreasing to ~0.001 at 10 MK. For a neutral atmosphere, the temperature dependence is weak: the constant value of the flux ratio is ~5 at densities below ~10<sup>14</sup> cm−<sup>3</sup> and rolling over to ~0.008 at an ambient density of ~1017 cm<sup>−</sup>3.

The 0.511 MeV emission can originate from different environments at different times during an event. Figure 3 shows the profile of the 0.511 MeV line during the 2003 October 28 eruption, considered to be an extreme event. In this event, the profile was quite wide during the first 2 min of the event compared to the last 16 min. Detailed calculations by Murphy et al. [5] revealed that the early part of the gamma-ray emission (broad line) might have occurred in an environment with a temperature in the range (3–4) <sup>×</sup> 105 K and densities 1015 cm−3. This implies that temperature in the chromosphere has transition-region values. Even though a 6000-K Vernazza et al. [22] model could marginally fit the observations, it was ruled out based on other considerations such as the low atmospheric density inferred (~2 <sup>×</sup> 1013 cm−3). On the other hand, the narrow line late in the event is consistent with an environment in which the temperature is very low (~5000 K), the density is same as before (~10<sup>15</sup> cm<sup>−</sup>3), but the ionization fraction in the gas is ~20%. These results point to the inhomogeneous and dynamic nature of the chromosphere inferred from other considerations [23]. The derived conditions also depend on the atmospheric model, which itself has been revised [24].

#### *3.2. Pion Continuum*

The pion continuum described briefly above is shown in Figure 4 with different components: (i) the π<sup>0</sup> decay continuum, which has a characteristic peak around 68 MeV, (ii) the bremsstrahlung continuum due to positrons emitted by μ<sup>+</sup> resulting from π<sup>+</sup> decay (π<sup>+</sup> brm), (iii) the positron annihilation continuum due to in-flight annihilation of positrons from π<sup>+</sup> decay (π<sup>+</sup> ann), and (iv) the bremsstrahlung continuum due to electrons emitted by μ− resulting from π− decay (π− brm). The sum of the four components (Total) represents the spectrum of gamma-rays resulting from pion decay assuming that the accelerated particle angular distribution is isotropic. The π<sup>0</sup> continuum dominates at energies >100 MeV and determines the spectrum at these energies but its contribution is very tiny at 10 MeV. For example, the π− bremsstrahlung has four times larger contribution than from π<sup>0</sup> decay, while π<sup>+</sup> bremsstrahlung and annihilation contributions are larger by factors of 20 and 75, respectively. Below 10 MeV, the gamma-ray spectrum is mostly determined by π<sup>+</sup> bremsstrahlung. Thus, below ~30 MeV, the combined contribution from positrons dominate the spectrum. The π<sup>0</sup> continuum exceeds the π<sup>+</sup> bremsstrahlung around 30 MeV, producing the characteristic "giraffe" shoulder around this energy. Such a spectrum was first derived from the observations of pion

continuum in the 1982 June 3 event by Forrest et al. [25], who identified a gamma-ray emission component that lasted for ~20 min beyond the impulsive phase of the flare. The spectrum in Figure 4 was calculated in great detail by Murphy et al. [17] to explain the 1982 June 3 event assuming that the primary protons have a shock spectrum [26]. They also performed a similar calculation assuming a stochastic particle spectrum thought to be produced in the reconnection site. While the early part of the 1982 June 3 gamma-rays can be explained by the steep stochastic spectrum, the part extending beyond the impulsive phase (late part) needs to be explained by the shock spectrum, which is much harder than the stochastic spectrum. Furthermore, these authors found that the shock spectrum is similar to the SEP spectrum observed in space.

**Figure 3.** Profiles of the 0.511 MeV annihilation line during the 2003 October 28 event observed by the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI). (Top) During the early phase (first 2 min of observation) the data points are fitted with a Vernazza atmosphere [22] at a temperature of 6000 K and a Gaussian with a width of ~6.7 keV corresponding to a temperature of (3–4) <sup>×</sup> 10<sup>5</sup> K. (Bottom) During the late phase (last 16 min of observation), the data are fitted with a Gaussian (width ~1 keV) and a 5000-K atmosphere with 20% ionization (adapted from Murphy et al. [5]).

≳

**Figure 4.** Contributions (in units of photons per unit energy interval) to the solar gamma-ray spectrum from π<sup>0</sup> decay, bremsstrahlung due to electrons from π– decay (π– brm), bremsstrahlung due to positrons from π<sup>+</sup> decay (π<sup>+</sup> brm), annihilation radiation of positrons from π<sup>+</sup> decay (π<sup>+</sup> ann). The energetic particles responsible for the production of pions were taken to be accelerated from a shock. The top two curves (solid and dashed) represent the total "giraffe" spectrum that combines these four components. The solid curve is for the ambient atmosphere while the dashed curve is for the abundance of the 1982 June 3 event derived from the associated solar energetic particle (SEP) event. The nuclear deexcitation line spectrum for the two abundances are also superposed. (Adapted from Murphy et al. [17]).

The time-extended gamma-ray emission first detected by Forrest et al. [25] using the Gamma-Ray Spectrometer on board the Solar Maximum Mission has been observed by many different missions, but such events were rare [27,28]. Two events had durations exceeding ~2 h [29,30]. The Large Area Telescope (LAT) on the Fermi satellite has detected dozens of such time-extended gamma-ray events from the Sun at energies >100 MeV, thanks to the detector's high sensitivity [31]. The average duration of these gamma-ray events is about 9.7 h and with six events lasting for more than 12 h [32,33], including an event that lasted for almost a day. The time-extended events are known by different names "long duration gamma-ray flare (LDGRF)" [34,35], "sustained gamma-ray emission (SGRE)" [33,36], and "late-phase gamma-ray emission (LPGRE)" [32]. The Fermi/LAT observations have revived the interest in the origin of the high-energy particles in these events because the accelerator needs to inject >300 MeV ions toward the chromosphere/photosphere to produce the pions required for the gamma-ray events.

Works focusing on the time-extended nature of these gamma-ray events explore ways to extend the life of the >300 MeV protons from stochastic (impulsive-phase) acceleration, e.g., by particle trapping in flare loops (e.g., [37]). In this scenario, the largest spatial extent of the gamma-ray source is the size of the post-eruption arcade (or flare area) discerned from coronal images taken in extreme ultraviolet wavelengths. In the shock scenario, the gamma-ray source is spatially extended because the angular extent of the shock is much larger than that of the flare structure [38,39]; shock acceleration is naturally time-extended evidenced by type II radio bursts and SEP events [33]

Gopalswamy et al. [40] demonstrated the spatially extended nature of the gamma-ray source during the 2014 September 1 event (see also [36,39,41]). They used multiview data from the Solar and Heliospheric Observatory (SOHO) and the Solar Terrestrial Relations Observatory (STEREO) missions to obtain a detailed picture of the eruption, including magnetic structures that extend beyond the flare structure (post-eruption arcade, PEA) as described in Figure 5. It must be noted that both the flux rope and PEA are products of the eruption process (magnetic reconnection). The PEA remains anchored to the solar surface, while the flux rope is ejected into the heliosphere with speeds exceeding 2000 km/s. The flux rope is a large structure rapidly expanding into the heliosphere compared to the compact flare structure. The flux rope drives a shock because of its high speed and the shock accelerates the required >300 MeV protons. The protons travel down to the chromosphere/photosphere along the field lines located between the flux rope and shock and produce the gamma-rays. Particles traveling away from the shock into the heliosphere are detected as SEP events. Shocks are known to accelerate particles as they propagate into the heliosphere beyond Earth's orbit, but the high-energy particles required for pion production may be accelerated only to certain distance from the Sun; this distance determines the duration of an SGRE event. In the case of the 2014 September 1 event, the SGRE lasted for about 4 h. With a shock speed >2300 km/s obtained from coronagraph observation, one can infer that the shock stopped accelerating >300 MeV protons to sufficient numbers by the time it reached a distance of about 50 solar radii from the Sun. Evidence for the shock shown in Figure 5 is the interplanetary type II radio burst that lasted until the end of the SGRE event and a bit beyond. The estimated duration of the type II burst (about 7.5 h) is in agreement with the linear relation found between the two durations [33].

**Figure 5.** (**a**) An extreme ultraviolet image obtained by the STEREO mission showing the spatial structure of the eruption region consisting of dimming regions (D1, D2) and the post-eruption arcade (PEA). (**b**) A flux rope (green) and shock (red) structures superposed on a SOHO white-light image showing the CME. The blue dot (at heliographic coordinates N14E90) represents the centroid of the Fermi/LAT gamma-ray source located between the flux rope (FR) leg and the shock front. The flux rope legs are rooted in the dimming regions D1 and D2. (**c**) A schematic showing the FR and the surrounding shock. Particles accelerated near the shock nose travel along magnetic field lines in the space between the FR and shock, precipitate in the chromosphere/photosphere to produce gamma-rays via the pion-decay mechanisms discussed in the text (adapted from Gopalswamy et al. [40]).

In a given eruption, both flare and shock populations are expected to be present, the flare particles being the earliest. Murphy et al. [17] concluded that the nuclear deexcitation line flux is primarily due to the flare population, while the 0.511-line flux has roughly equal contributions from flare and shock populations. On the other hand, the extended phase emission is solely due to the shock population. The conclusion on the extended phase emission initially derived from studying the 1982 June 3 event, seems to be applicable to all events with time-extended emission [33]. Recently, Minasyants et al. [42]

found that in certain gamma-ray events with high fluxes of >100 MeV photons, the development of the flare and CME are simultaneous. The CME started during the impulsive phase of the flare. They found that the >100 MeV flux is highly correlated with the CME speed, although the sample is small. Figure 6 shows the relation, replotted with a second-order polynomial fit instead of their linear fit. Normally, one would have thought the impulsive phase gamma-ray flux should be related to the impulsive-phase proton population, and not to the shock population. On further examination, it is found that a type II radio burst started in the impulsive phase of the events, indicating early shock formation. This is typical of eruptions that produce ground level enhancement (GLE) in SEP events [43–45], implying particle acceleration by shocks to GeV energies within the impulsive phase. This result suggests that the shock population may also have contribution to >100 MeV photons in the impulsive phase.

**Figure 6.** Scatter plot between >100 MeV gamma-ray flux (F) from Fermi/LAT against the CME speed (V) for events in which the CME onset was during the impulsive phase. The speeds of three CMEs are different from those in Minasyants et al. [42]. The polynomial fit to the data points and the square of the correlation coefficient (R) are shown on the plot.

STEREO observations have revealed that shocks can form very close to the Sun, as close as 1.2 solar radii [46]. Shocks typically take several minutes to accelerate particles to GeV energies after their formation. This means, shocks are present within the closed field regions of the corona early on, sending particles toward the Sun and augmenting the impulsive phase particles. The precipitation sites are expected to be different from the PEA as discussed in Figure 5. Once the shock propagates beyond ~2.5 solar radii, accelerated particles can move both ways, away and toward the Sun.

#### **4. Conclusions**

Positrons are important particles both in the laboratory and in astrophysics. They are extremely useful in understanding high-energy phenomena on the Sun. They provide information on various processes starting from particle acceleration, transport, and interaction with the dense part of the solar atmosphere. Positrons provide information on the physical conditions in the chromosphere/photosphere where they are produced and destroyed by different processes, leaving tell-tale signatures in the gamma-ray spectrum. In addition, gamma-rays and positrons provide information on the magnetic structures involved in solar eruptions that support the acceleration and transport of the highest-energy particles in the inner heliosphere. Spatially resolved gamma-ray observations beyond what is currently available (e.g., [47]) are needed to resolve the issue of the relative contributions from stochastic and shock accelerations in solar eruptive events.

**Funding:** This research was funded by NASA's Living with a Star Program.

**Acknowledgments:** I thank Gerald H. Share and Pertti A. Mäkelä for reading versions of this manuscript and providing useful comments.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


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