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

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∘ 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.

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_Tfias the sum of three amplitudes,_T1,_T2and_T3 , 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_Tfito be zero, the three amplitudes have to destructively interfere and that the amplitude_T3is essential for a zero. Since for electron-hydrogen ionization the term_T3=0 , as there is no passive electron in the target atom, Berakdar and Briggs [2] deduced that there cannot be an exact zero in_Tfi 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 [9,10,11], 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) [12,13,14,15] 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_Tk,1sCB1 [17,18,19,20]. 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 [5,18,19,23,24,25,26,27]. 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_Tfito be done quickly and for the velocity fieldvassociated with_Tfito be readily computed. Once a zero in_Tk,1sCB1and in the corresponding TDCS is located, more elaborate methods can be applied for the kinematics, or approximately the kinematics, of the zero in_Tk,1sCB1. 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_Tk,1sCB1.

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_Tk,1sCB1was obtained. Corresponding to the zero in_Tk,1sCB1 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 fielduthat is associated with the transition matrix elementT(_k+,_k−), whereT(_k+,_k−)depends on the momentum of the scattered positron_k+and the momentum of the ejected electron_k− [8,9]. Navarrete and Barrachina [9,10,11] obtained zeros inT(_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 [18,19,20,25,26,27].

Positron-impact ionization of atoms and molecules is of experimental interest [29,30,31,32,33,34,35], 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 fieldvassociated with_Tk,1sCB1for 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_Tk,1sCB1 . We are not able to compare our results with Navarrete and Barrachina’s results [9,10,11] 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 fieldvassociated with_Tk,1sCB1and the equation of the circulationΓ . In Section 3.1 we compare for electron-hydrogen ionization the TDCS computed with the various methods. We also show_Tk,1sCB1and direction of the velocity fieldv^=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_Tk,1sCB1and 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_Tk,1sCB1 , derived by Botero and Macek [12,13,14,15], 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_Tk,1sCB1 is given by [12,13,14,15]

_Tk,1sCB1=⟨_ψKf−(r)_ψk−(^r′)|1|r−^r′||_φi(^r′)_ψKi+(r)⟩,

wherek,_Kiand_Kf are the momentum of the ejected electron, the momentum of the incident particle, and momentum of the scattered particle, respectively. In Equation (1)rand^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,_φiis the ground-state wave function of hydrogen and^ψ± 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_Zeffin the Coulomb wave functions of the incident electron and the scattered electron equal to the atomic number_ZT=1of hydrogen, the direct and exchange CB1 transition matrix elements are equal. Therefore, the CB1 TDCS for electron-hydrogen ionization can be expressed as:

^{d3 σCB1}d_Ωfd_Ekd_Ωk=^(2π)4_Kfk_Ki^|_Tk,1sCB1|2,

whered_Ωkis the solid angle for the ejected electron,d_Ωfis the solid angle for the scattered particle, and_Ek 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

|_D3−(_r3ave)^|2=πexp(−π_k3)_k3[1−exp(−π_k3)]^{_{|1 F1}(i2_k3,1,−2i_{k3 r3ave})|2},

where_k3=|_Kf−k|/2. Also,_r3aveis the vector whose magnitude is the average spacing between the two outgoing electrons and whose direction is taken to be along_k^3,_r3ave=_{r3ave k^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_r3ave, namely_r3ave=0,_r3ave=1/_k3and

_r3ave=^π216ε^{1+0.627πεlnε2},

respectively. In Equation (4),ε=(_Kf2+^k2)/2, which is the total energy of the two outgoing electrons. By taking the limit_r3ave→∞,|_D3−(_r3ave)^|2→1, the CB1 TDCS is obtained. In contrast, when_r3ave=0,|_D3−(_r3ave)^|2=^{|_{Ne− e−−}|2}, where_{N^{e− e−}−}is the normalization factor of the Coulomb wave function_ψk3− 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−and^e−for electron-impact ionization and^e+and^e− for positron-impact ionization) is given by [7,13,16]

|_{N^{e− e±}−} ^|2=±πexp(±π_k3)_k3[exp(±π_k3)−1].

In an earlier paper [25], and for positron-impact ionization of hydrogen, we refer to the CB1 TDCS multiplied by|_{N^{e− e±}−} ^|2as 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_r3ave=0to distinguish it from the other two (where_r3ave=1/_k3and_r3ave is given by Equation (4)).

For positron-impact ionization,_Kiand_Kf 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 [42,43,44] (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_Tk,1s, there is a vortex in the velocity field associated with_Tk,1s . 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 fieldvassociated with_Tk,1sCB1 for electron or positron ionization can be expressed as [24,25]

v=_∇kIm[ln_Tk,1sCB1].

The circulation,Γ, for a closed contour, c, taken in the anticlockwise direction that encloses only one first-order zero in_Tk,1sCB1 is given by [5,23,24,25,45]

Γ=_∫cv·dℓ=±2π.

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 ofξwhile the ejected electron makes an angle of –ξwith respect to the same axis.

A deep minimum in the TDCS occurs when the real and imaginary parts of the transition matrix element,_Tk,1s, 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_r3ave=0, modified CB1 with_r3ave=1/_k3, and the modified CB1 with_r3ave 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_r3ave=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.

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 circulationΓis2π . Figure 4 shows the nodal lines ofRe[_Tk,1sCB1]andIm[_Tk,1sCB1]and it shows the direction of the velocity field,v^=v/|v|, by the arrows. (The axes for this figure are_kzand_kx, where_kzand_kxare respectively the z- and x-components of the momentumk of the ejected electron [25].) The figure also shows a density plot ofln|_Tk,1sCB1|.

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 angleψ 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_Tk,1sCB1. The zero in_Tk,1sCB1 can be seen in Figure 5.

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 aroundξ≈ 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_Tk,1sCB1and a second zero inRe[_Tk,1sCB1] (see Figure 5). We perform an out-of-plane search to see if we could find a zero in_Tk,1sCB1around ≈ 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 anglesψandξ for these energies as we did previously for electron-helium ionization [25].

For positron-impact ionization of hydrogen, we find that circulationΓis2πand the velocity field that is associated with the zero in_Tk,1sCB1 rotates anticlockwise, as can be seen in Figure 7. This figure shows the direction of the velocity field,v^=v/|v|, by the arrows and it shows the nodal lines ofRe[_Tk,1sCB1]andIm[_Tk,1sCB1]. It also gives a density plot ofln|_Tk,1sCB1|.

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 angleξof the deep minimum in the CB1 TDCS is less than^90°, and thus,ξlies 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_Tk,1sCB1for 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_Tk,1sCB1does 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_Tk,1sCB1is 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_Tk,1sCB1; 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].

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

Figure 2. Transition matrix element,Tk,1sCB1, versus the polar angleξfor electron-impact ionization of hydrogen at 76.4541 eV. (a)Re[Tk,1sCB1](solid red line) andIm[Tk,1sCB1](dashed blue line) over the full angular range (b)Re[Tk,1sCB1](solid red line) andIm[Tk,1sCB1](dashed blue line) in the vicinity of the zero inTk,1sCB1.

Figure 3. Triply differential cross section (TDCS) on a log (ln) scale versus the polar angleξfor electron-impact ionization of hydrogen at 76.4541 eV. (a) Comparison of the CB1 TDCS (dot-dashed black line), the modified CB1 TDCS withr3ave=0(solid orange line), the modified CB1 TDCS withr3ave=1/k3(dashed purple line) and the modified CB1 TDCS withr3ave given by Equation (4) (dotted brown line). (b) Comparison of the modified CB1 TDCS withr3ave=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.

Figure 4. A density plot ofln|Tk,1sCB1| 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 electronk. The nodal lines ofRe[Tk,1sCB1]andIm[Tk,1sCB1]are shown respectively, by the solid blue line and the dashed green line. The direction of the velocity field,v^=v/|v|, is indicated by the arrows.

Figure 5. Transition matrix elementTk,1sCB1versus the polar angle,ξ, for positron-impact ionization of hydrogen at 100 eV with a gun angle,ψ, of 56.13°. (a)Re[Tk,1sCB1](solid red line) andIm[Tk,1sCB1](dashed blue line) over the full angular range (b)Re[Tk,1sCB1](solid red line) andIm[Tk,1sCB1](dashed blue line) in the vicinity of the zero inTk,1sCB1.

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.

Figure 7. A density plot ofln|Tk,1sCB1|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 electronk. The nodal lines ofRe[Tk,1sCB1]andIm[Tk,1sCB1]are shown respectively, by the solid blue line and the dashed green line. The direction of the velocity field,v^=v/|v|, is indicated by the arrows.

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:

MDPI	Multidisciplinary Digital Publishing Institute
DOAJ	Directory of open access journals
TDCS	Triply differential cross section
CB1	Coulomb-Born
TDCC	Time-dependent close-coupling
3DW	Three body distorted wave

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