Double charge exchange in ion–atom collisions using distorted wave theories with twoelectron continuum intermediate states in one or both scattering channels
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Abstract
A general quantummechanical formalism is reviewed for double electron capture from heliumlike atomic systems by fast nuclei. The development is carried out with and without the distorted wave theory by fulfilling the correct boundary conditions. These refer to the required asymptotic behaviors of the total scattering wave functions and their appropriate connections to the perturbation interactions that produce the transitions from the initial to the final states of the system. In this general formulation any choice is allowed for the pairs of the distorting potentials and the related distorted wave functions as long as the correct boundary conditions are satisfied. This is the case with the fourbody versions of several most frequently used methods (continuum distorted wave: CDW4B, boundarycorrected continuum intermediate state: BCIS4B, Born distorted wave: BDW4B, continuum distorted wave initial/final state: CDWEIS/EFS4B, and the boundarycorrected first Born: CB14B). A comparative analysis of these methods makes in evidence both their similarities and differences. For example, the most illustrative is the juxtaposition of the post BDW4B and CDWEIS4B methods. They share the same distorting potential in the exit channel. The only difference is in the coordinates from the Coulomb logarithmic phases in the initial distorted wave functions. This difference is completely negligible in the asymptotic scattering regions. Yet, for e.g. double electron capture from helium by alpha particles, the total cross sections from these two methods differ by 1–3 orders of magnitudes. The BDW4B method is in agreement with experimental data at high impact energies. In sharp contrast, within its validity domain of impact energies, the CDWEIS4B method underestimates the measured data by orders of magnitude. This shows that what matters is not solely the correct asymptotes of distorted wave functions, but rather how they affect the contributions to the integrals over the entire regions in the Tmatrix elements for total cross sections. Such insights help understand the assessment of the overall validity and relative performance of various methods, and can provide a versatile guidance for improving the existing approximations for double charge exchange in fast ion–atom collisions.
Keywords
High energy atomic collisions Correct boundary conditions Double charge exchange Secondorder theories1 Introduction
Double charge exchange in ion–atom collisions at intermediate and high energies is prominent among many multielectron processes [1, 2, 3, 4]. These include electron transfer, excitation, ionization their combinations (transferexcitation, transferionization, ...), etc. Such processes have been studied intensively over the years both theoretically [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54] and experimentally [55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86]. When there is active participation of two or more electrons from either the projectile or the target or both, we talk about two or multiple electron transfer, excitation, ionization, transferexcitation, transfer ionization, etc. Stated equivalently, the term multiple electron atomic processes implies that more than one electron has left its initial orbital.
The socalled frozen core approximation has often been invoked in descriptions of such collisional processes. This additional approximation assumes that the electrons that do not participate to the actual transitions (the passive electrons) remain in the final state of the target and/or projectile in the same orbitals which they have occupied in the initial states. Such an approximation is expected to be reasonable at high impact energies. Nevertheless, it is pertinent here to emphasize that e.g. a singleelectron process, in the strict meaning of the term, cannot occur in collisions involving multielectron atomic systems. The explanation is that an alteration in the orbital energy of one electron (the active electron) would inevitably lead to some changes (albeit perhaps only slight) in the orbital energies of the remaining electrons (meaning that here, in fact, there could be no passive electrons). Such a strictness is often not of a particular concern in many applications that frequently rely upon the frozen core approximation, the notion of passive electrons, the effective or screened nuclear charges, etc.
In particular, for chargeexchange processes, the noncaptured electrons are viewed as playing only a static role amounting to merely screening the bare nuclear charge. Various choices of the effective nuclear charges can be made, and this should be done in a consistent manner. Physically, these effective nuclear charges should be close to the nuclear charges that reproduce the orbital energies of the initial and final bound states. One reasonable choice is guided by the fact that the nuclear charge Z and the binding energy \(\varepsilon _n<0\) of the electron in a hydrogenlike atomic system for the state with the principal quantum number n is \(Z=(2n^2\varepsilon _n)^{1/2}.\) Similarly, as suggested in Ref. [87], in the case of a multielectron atomic or ionic target, an electron to be captured from a state of the orbital energy \(\varepsilon ^\mathrm{T}_n<0\) with the orbital number n, the effective nuclear charge \(Z^\mathrm{eff}_{\mathrm{T}}\) could be chosen to satisfy the hydrogenictype relationship \(Z^\mathrm{eff}_{\mathrm{T}}=(2n^2\varepsilon ^\mathrm{T}_n)^{1/2}.\) Here, \(\varepsilon ^\mathrm{T}_n\) could be selected as the RoothanHartreeFock orbital energy for which the tabulated values can be found in Ref. [88] for many multielectron atomic systems. Also given in Ref. [88] are the variationally determined parameters (expansion coefficients, exponential damping factors) for the analytical forms of the corresponding groundstate wave functions (including some of the excited states) for neutral and ionized atoms. These latter wave functions are linear combinations of the Slatertype orbitals (STO) as the basis set functions.
This type of choice for an effective or screened nuclear charge works quite well in practice [87]. The reason is in the fact that chargeexchange is a very local process. This process occurs with nonnegligible probabilities at the places where the initial and/or final bound state wave functions are appreciable. It is at these places that the electrons to be captured experience the screened charge \(Z^\mathrm{eff}_{\mathrm{T}}\) as an average target nuclear charge. Note that due to their exponential decline with augmentation of distances, the atomic bound state wave functions take on their noticeable values only at small separations between the electrons and their parent nucleus. At high energies, the dominant contribution to charge exchange transition amplitude for complex atomic targets is predominantly determined by the electrons that are closest to their nucleus (the Kshell electrons). Small distances correspond to high momenta. Therefore, even at high impact energies, it is important to use the atomic wave functions whose momentumspace representations accurately describe high momentum components of the electronic states. Momentumspace boundstate wave functions come into play here because the charge exchange transition amplitudes are determined by the overlap integrals of the initial and final scattering states. Such overlap integrals contain the momentumspace boundstate wave functions that are initially given in the coordinate representations. This becomes most obvious from an inspection of the wellknown transition amplitude in the firstorder Oppenheimer–Brinkman–Kramers (OBK) approximation for single electron capture processes [89].
From these remarks one can infer the two main mechanisms, the velocity matching and the Thomastype double scattering, for charge exchange in the first and secondorder methods, respectively. The firstorder methods are based upon the onestep pathways, involving the direct projectiletarget interactions alone. Therein, the velocity matching mechanism operates via the circumstance that the dominant contribution to electron capture is due to the near equality between the incident speed and the orbital velocity of the active target electron. High incident velocities require high momentum components from the momentum distribution in the target momentumspace boundstate wave functions. The secondorder methods describe the two steps via target ionization followed by capture of the emitted electron. The emitted electron must have high momentum if it is to be captured by a fast projectile. This ionizationcapture mechanism is a quantummechanical version of the classical Thomas double scattering. There are two successive elastic rearranging collisions in the Thomas billiardtype twin events. In the first event, the electron is scattered elastically on the projectile through \(60^\circ \) towards the target nucleus. In the second encounter, the electron scatters again elastically through \(60^\circ \) on the target nucleus with the velocity \(\varvec{v}\) equal to the projectile large speed. This electron is finally captured by the positively charged projectile, since on top of having collinear velocities, the attractive Coulomb potential between these two particles binds them together into a newly formed atomic system.
In the present review, we will focus only upon several selected first and secondorder methods with the correct boundary conditions for double electron capture from heliumlike targets by heavy nuclei. These are the fourbody continuum distorted wave (CDW4B) [30, 31], boundarycorrected continuum intermediate state (BCIS4B) [32], Born distorted wave (BDW4B) [41, 42], continuum distorted wave initial/final state (CDWEIS/EFS4B) [47] and the boundarycorrected first Born (CB14B) [33, 34] methods. We will illuminate their similarities as well as differences and illustrate their performance in the most frequently studied example of collisions between alpha particles and helium atoms.

Stellar atmospheres, upper atmosphere, interstellar medium [90, 91].

Heavy ion accelerators at GSI (Darmstadt), KSU (Kanzas), GANIL (Caen), etc [92]

Storage ring accelerator such as ESR, CRYRING (at GSI), TSR (Heidelberg), ASTRID (Aarhus), etc [93, 94, 95, 96].

Ion traps (EBIT, Paul trap, Penning trap, ...), ion sources (EBIS, ECRIS, ...), etc [93, 94, 95].

Charge exchange spectroscopy in magnetically confined plasmas [97].

Hot and dense plasmas (\(T\ge 10^6\,{}^\circ \mathrm{K}, n_e\sim 10^{19}/\mathrm{cm}^310^{24}/\mathrm{cm}^3),\) hightemperature thermonuclear fusion by way of inertial confinement accomplished with heavy ion bombardment (at GSI), short highenergy laser pulses (at LMJ: Bordeaux, PHOEBUS: Limeil, NOVA, NIS: Livermole, ...) or short intense discharges (Zpinch), etc [98, 99, 100].

Hot and dilute plasmas (\(T\ge 10^6\,{}^\circ \mathrm{K}, n_e\sim 10^{14}/\mathrm{cm}^3),\) hightemperature thermonuclear fusion via magnetic confinement devices e.g. Tokamaks, Stellarators, etc [97, 101, 102]

Hadron therapy by highenergy (\(\sim 300\) MeV/amu) light ions (from proton to Carbon nuclei) for treatment of deepseated tumors in patients at either physicsbased facilities or at hospitalbuilt dedicated accelerators in several countries (USA, Germany, France, Austria, Sweden, Italy, Japan, Russia, ...) [107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124].
2 General theory with and without the distorted wave formalism
2.1 Coulombmodified initial and final scattering states without distorted waves
2.1.1 Eikonal formalism for dominant forward scatterings of heavy particles
2.1.2 The prior and post forms of the transition amplitudes
2.2 Coulombmodified initial and final scattering states with distorted waves
2.3 Determination of the initial and final distorted waves
Here we shall outline the procedure of obtaining the initial and final pairs \(\{U_{i,f},\chi ^\pm _{i,f}\}\) in the entrance and channels.
2.3.1 Entrance channel
2.3.2 Exit channel
2.4 Asymptotic behaviors of distorted waves at large interparticle distances
2.5 Different choices of the distorting potentials and distorted waves
Secondorder theories are the formalisms that include the intermediate ionization channels for electronic degrees of freedom through the Coulomb wave functions of the electrons centered on one or both nuclei. Symmetric secondorder theories, such as the CDW4B method, have the electronic Coulomb wave functions centered on both nuclei with two such functions in each channel (entrance and exit). Asymmetric secondorder theories, e.g. the BCIS4B and BDW4B methods, possess two electronic Coulomb wave functions in total, both centered on one nucleus in one channel alone (entrance or exit) for the given transition amplitude. There is also another pair of asymmetric secondorder theories, e.g. the CDWEIS4B and CDWEFS4B methods, that use four electronic distorting functions, such as two full Coulomb waves in one channel (exit/entrance) and two Coulomb logarithmic phases in the complementary channel (entrance/exit), respectively. Firstorder theories are the formalisms that do not include any intermediate ionization channels for the two captured electrons. Symmetric firstorder theories, e.g. the CB14B method, include one Coulomb wave function (or equivalently, its logarithmic phase factor) per channel (entrance or exit) for the relative motion of heavy nuclei. Both the BCIS4B and BDW4B method belong to the class of hybrid theories that treat one channel by the CDW4B method and the other by the CB14B method. The CDWEIS/EFS4B are also from the category of hybrid theories since they coincide with the CDW4B in one channel, whereas the electronic and nuclear distorting functions in the other channel are eikonalized.
2.5.1 Symmetric secondorder theories: fourbody continuum distorted wave method, CDW4B

Independence of the eikonal total cross sections on the internuclear potential
Overall, the relation (2.123) displays the greatest practical usefulness of the eikonal setting because the product of the two confluent hypergeometric functions in (2.122) for the relative motion of heavy nuclei is first reduced to a single phase \((\mu v\rho )^{2i\nu _\mathrm{PT}},\) which is subsequently shown to disappear altogether from the total cross sections. It is such a complete elimination of the \(\varvec{R}\)dependent Coulomb wave functions and their logarithmic phase factors that enormously simplifies the computations of total cross sections for double charge exchange in the CDW4B method [30, 31].
2.5.2 Symmetric firstorder theories: fourbody boundarycorrected first Born method, CB14B
In Refs. [36, 37, 38], the fourbody boundarycorrected first Born method, or CB14B, has alternatively been called the fourbody CoulombBorn distorted (CBDW4B) method because therein the computations make use of the full Coulomb wave functions for the relative motion of heavy nuclei. However, the total cross sections from the CB14B method, employing the logarithmic Coulomb phase factors for the relative motion of heavy nuclei, fully agree with those from the CBDW4B method, as has explicitly been shown in computations on single capture (see section 4), and this should also be true for double capture.
2.5.3 Asymmetric secondorder theories: fourbody boundarycorrected continuum intermediate states method, BCIS4B
2.5.4 Asymmetric secondorder theories: fourbody Born distorted wave method, BDW4B
2.5.5 Asymmetric secondorder theories: fourbody continuum distorted wave eikonal initial/final state methods, CDWEIS/EFS4B
2.5.6 The link between the prior/post BDW4B and CDWEFS/EIS4B methods
2.5.7 The link between the prior/post CDW4B and CDWEFS/EIS4B methods
3 Convergence issues with the Born series for rearrangement collisions
Aaron et al. [129] pointed out that the Born series for the transition operators diverges for threebody rearrangement collisions. However, neither the transition operators nor the related total scattering wave functions are observable (physically measurable quantities). What actually matters is the status of convergence of certain observables of the main interest, e.g. scalar products that contain the transition operators and total scattering wave functions. In scalar products, such as those from cross sections, the transition operators and total scattering wave functions are embedded in integrals over all the configuration and/or momentum space. This circumstance may well wash out the pathological/divergent features of the transition operators and total scattering wave functions within the transition amplitudes. Indeed, it has been demonstrated by Corbett [130] that for a divergent Toperator series, convergence can nevertheless exist for both the series of total scattering wave function and the Tmatrix elements. This means that the conditions in the Born series for the Toperator convergence are more restrictive than those for the total wave functions. It also implies that the convergence conditions in the Born series for the total scattering wave functions are more restrictive that those for the Tmatrix elements. An appropriate illustration of this important conclusion has been reported by Dettman and Leibfried [131] for a special case of rearrangement collisions with the \(\delta \)function interactions. For this particular scattering, it has been found [131] that despite the existing operator divergence, the resulting physical transition amplitude is convergent.
In practice, besides having a divergencefree Born series, its first few successive terms should also be computed numerically to see their behaviors regarding the smoothness and convergence rate. Such an insight could help to empirically assess the possibility for convergence of the entire Born series. This has been the subject of a number of studies where the exact numerical computations were carried out in the 1st [140, 141, 142, 143, 144], 2nd [145, 146, 147, 148, 149] and 3rd [150] Born approximations to the series for the full Tmatrix elements with \(Z_\mathrm{P}=Z_\mathrm{T}=1\) in process (3.1). The outcome is that all the three Born approximations are wellbehaved, smooth functions at all impact energies and for any scattering angle. Further, these studies show that, at high impact energies (in the MeV region), the 2nd Born approximation dominates over both the 1st and the 3rd Born approximations. This steady trend, especially with the recent availability of the exact cross sections for the 3rd Born approximation [150], is an improved assessment of the convergence rate of the Born series for rearrangement collisions of the prototype (3.1).
4 Illustrations
In the preceding exposition, we illuminated the similarities and differences among various distorted wave models for the general case of the arbitrary nuclear charges \(Z_\mathrm{P}\) and \(Z_\mathrm{T}\) in process (2.1) for double electron capture. For example, in subsections 2.5.6 and 2.5.7, we highlighted the relationships among the post/prior BDW4B and CDWEFSEIS methods as well as among the prior/post CDW4B and CDWEFS/EIS methods, respectively. Moreover, throughout the analysis, we emphasized the role of the double continuum intermediate states populated by the two electrons prior to their capture by the projectiles. Such twofold electronic ionization continua are explicitly included in the secondorder methods, either in one channel (BCIS4B, BDW4B) or two channels (CDW4B, CDWEIS/EFS4B). The firstorder theories, such as the CB14B method, do not take into account the double electronic continuum intermediate states since the initial and final total scattering states are considered to be the dressed channel states \(\Phi ^\pm _{i,f}.\) The states \(\Phi ^\pm _{i,f}\) are the products of the unperturbed channel states \(\Phi _{i,f}\) and the Coulomb logarithmic phase factors \(\exp {(\pm i\nu _{i,f}\ln {(vR\mp \varvec{v}\cdot \varvec{R})})}\) generated by the asymptotically present tails of the residual Coulomb potentials, \(V^\infty _{i,f}=Z_\mathrm{P,T}(Z_\mathrm{T,P}2)/R.\) Here, \(\varvec{R}\) is the vector of the internuclear axis R. Simultaneously, \(\varvec{R}\) is also the difference between the electronic position vectors \(\{\varvec{x}_k,\varvec{s}_k\}\) relative to \(Z_\mathrm{T}\) and \(Z_\mathrm{P},\) respectively, \(\varvec{R}=\varvec{x}_k\varvec{s}_k\, (k=1,2).\) In other words, using \(\varvec{R}\) in the Coulomb logarithmic phases for the BCIS4B and BDWmethods amounts to correlating the two Coulomb centers (the nuclei of the projectile and of the target). By contrast, the Coulomb logarithmic phases in terms of \(\varvec{s}_k\) (CDWEIS4B) or \(\varvec{x}_k\) (CDWEFS4B) deal only with the projectile or the target Coulomb center at a time (hence no correlation between the two centers). This difference is immaterial only at large \(\{R,s_k\}\) (entrance channel) and large \(\{R,x_k\}\) (exit channel), according to (2.113) and (2.114), but becomes essential at finite distances when these Coulomb logarithmic phases are employed in the integrals over all distances in the matrix elements of the transition amplitudes from the BCIS4B, BDW4B and CDWEIS/EFS4B methods.
Nevertheless, it still makes sense to compare theories for (4.1) and experiments for (4.2) in Fig. 1, since it has been shown in the CDW4B method [45] that at least above 1000 keV the sum of the contributions from the singly and doubly excited final states of helium is small. However, it would be important to assess the contribution from the excited states also below 1000 keV. The theoretical results shown in Fig. 1 are all obtained using (2.225) as the oneparameter Hylleraas wave function [125] for both the initial and final ground states of helium in symmetric double charge exchange (4.1) with \(Z^\mathrm{eff}_\mathrm{K}=20.3125=1.6875\, \mathrm{(K=P, T)}.\) Using the heliumlike groundstate wave functions with one parameter [125] (Hylleraas) and four parameters [126, 127, 128] within the CDW4B method at energies 100–7000 keV, it has been verified in Refs. [32, 42], that the total cross sections are not overly sensitive to the static interelectronic correlations. It would be useful to check whether this conclusion regarding (4.1) also holds true for the BCIS4B and BDW4B methods.
The status of the firstorder theories is evident from Fig. 1 when comparing the CB14B method with the experimental data [58, 61, 69, 75, 76, 79, 80] (presently taken from the most exhaustive tabulated database Refs. [85, 86]). In Fig. 1, the cross sections from the CB14B method are seen to closely follow the measurements at impact energies from 200 to 800 keV. However, the discrepancy between the CB14B method and experiments keeps on increasing with the augmented impact energy. Thus, e.g. at 3000 keV, the CB14B method overestimates the experimental data by more than an order of magnitude. This significant and systematic overestimation is a clear indication of the importance of the double electronic continuum intermediate states that are not taken into account in the CB14B method.
In order to have a more quantitative insight into the role of the intermediate double ionization continua of the two electrons, it is necessary to pass onto comparisons of measurements with the secondorder methods. To achieve this goal, we examine the overall performance of the CDW4B and CDWEIS4B methods. Thus, it is seen in Fig. 1 that the cross sections from the CDWEIS4B method are strongly suppressed relative to those of the CDW4B method. For example, at the impact energy of 100 keV, the cross sections of the CDWEIS4B method underestimate the results of the CDW4B method by nearly four orders of magnitude. Such a gigantic and unprecedented discrepancy is still huge at larger energies. For instance, even at 1000 keV, the predictions by the CDWEIS4B method are an order of magnitude lower than those due to the CDW4B method. The magnitudes of the cross sections from the CDW4B method are seen in Fig. 1 to significantly underestimate the experimental data at impact energies 200–3000 keV. Also the lineshapes (the behaviors of the cross sections as a function of the impact energy) are different in the CDW4B method and measurements. At still higher energies (4000 and 7000 keV), the CDW4B method is in excellent agreement with the measurement from Ref. [79]. However, particularly at 4000 keV, the situation is inconclusive since the cross sections from the two independent measurements [79, 80] differ by more than an order of magnitude (see the pertinent remark on p. 3837 in Ref. [32]).
Further, it is observed in Fig. 1 that the cross sections from the BCIS4B and the BDW4B methods agree quite well with each other concerning both their magnitudes and lineshapes. The cross sections from these two methods would be identical if the helium bound state wave function were exact. For example, the same cross sections are obtained from the BCIS3B and BDW3B methods in the case of electron capture from hydrogenlike atomic systems by nuclei in process (3.1) for any initial and final state. In Fig. 1, at intermediatetohigh energies 100–3000 keV, the curvatures of the cross sections from the BCIS3B and BDW3B methods are similar to the common lineshape that could be drawn through the depicted experimental data to guide the eye. As to the magnitudes of the cross sections, the BCIS4B and BDW4B methods underestimate the experimental data below 1000 keV, albeit by a much smaller factor than in the CDW4B method. On the other hand, the BCIS4B and BDW4B methods are in excellent agreement with the experimental data at impact energies 1000–3000 keV. However, no similar validity assessment is possible at still higher energies (4000, 7000 keV) because of the mentioned more than an order of magnitude discrepancy between the two measurements reported in Refs. [79, 80]. In fact, at 4000 keV, the cross sections from the BCIS4B and BDW4B methods are near the midpoint between the two symbols for experimental data from Refs. [79, 80]. Needless to say, it would be highly desirable and important to perform some new measurements on (4.2) so as to clarify this unusual disagreement between the two independent measurements of Schuch et al. [79] and Afrosimov et al. [80].
Evidently, there is no postprior discrepancy for (4.1) and (4.2) because these are symmetric double capture events in collisions between alpha particles and helium. Moreover, in the case of general heavy particle collisions (\(m_\mathrm{P,T}\gg 1\)), and not just single or double capture processes, the numerical values of total cross sections are expected to be the same for computations with either the logarithmic Coulomb phase factors or the associated full Coulomb wave functions for the relative motion of the two heavy scattering aggregates. This has explicitly been confirmed in the CB14B as well as in the BCIS3B and BCIS4B methods for single electron capture in heavy particle collisions [151, 152]. The same conclusion is anticipated to apply also to double capture. Surprisingly, however, the total cross sections in e.g. the fourbody boundarycorrected continuum intermediate state method for double electron capture from heliumlike atomic systems by heavy nuclei are significantly different (especially at intermediatetolower impact energies) when computed with the full Coulomb wave function [49, 50, 51, 52, 53] and with its logarithmic Coulomb phase [1, 3, 32]. To clarify the matter, especially regarding an earlier observation [3] made on this discrepancy, we performed a new thorough computation the results of which will be published shortly.
5 Discussion and conclusions
In this work, a parallel structure presentation is expounded by juxtaposing the conventional and distorted wave formalisms of the general quantummechanical scattering theory for double electron capture from heliumlike atomic systems by heavy nuclei. The former deals directly with the original, analytically unsolvable collisional problem. The latter solves (by analytical means) a flexible model problem with certain judicious choices of the distorting potentials and the corresponding distorted wave functions.

(I) the proper asymptotic behaviors of the total scattering wave functions in the initial and final states of the entire system,

(II) the consistent connections between the perturbation potentials and the total scattering wave functions in the entrance and exit channels, and

(III) the presence of shortrange perturbation potentials in the transition amplitudes.

(i) the unperturbed channel states have to be multiplied by the corresponding logarithmic Coulomb phase factors for the relative motions of the scattering aggregates,

(ii) the asymptotically present Coulombic interactions must be subtracted from the original perturbation interactions, and

(iii) the modified perturbation potentials (that cause the transitions of the whole system from the initial to the final states) must be shortrange, squareintegrable interactions (i.e. falling off faster than 1 / r with the augmentation of the interparticle separation r).
To appreciate this particular feature of Coulomb scattering theory, it is instructive to see the repercussions of obedience and disobedience of the correct boundary conditions. Thus, when the steps (I)–(III) and (i)–(iii) are accomplished in their entirety, the pertinent Møller wave operators exist and they permit the definition of the transition or Toperators and the scattering or Smatrices. In contradistinction, a disrespect of either the whole procedure (I)–(III) and (i)–(iii), or implementing these steps incompletely (e.g. in one channel, but not in the other) would mean that the Møller wave operators do not exist, with the consequence of being unable to introduce the Toperators and Smatrices [153]. Under such circumstances, attempts to deal with any particular approximate methods, in spite of the nonexistent Toperators and Smatrices, are not justified. We presently illustrate how to practically implement the steps (I)–(III) and (i)–(iii) as the gestalt, regarding both the full transition amplitudes and its several approximate methods.
The purpose of this general, unified and exact setting is to lay the ground for a systematic and consistent derivation of all the existing approximate methods by making different choices of the distorting potentials and distorted wave functions. In principle, any choice of the distorting potentials and distorted wave functions is permitted as long as these latter two objects respect the correct boundary conditions through the steps (I)–(III) and (i)–(iii). This is shown here for the fourbody formulations of the continuum distorted wave (CDW4B) [30, 31], the boundarycorrected intermediate state (BCIS4B) [32], the Born distorted wave (BDW4B) [41, 42], the continuum distorted wave – initial/final state (CDWEIS/EFS4B) [47] and the boundarycorrected first Born (CB14B) [33, 34] methods.

(A) with the full Coulomb wave functions (no eikonal mass limit), and

(B) with their Coulomb phase factors (due to the eikonal mass limit).

(a) the relative performance of the analyzed distorted wave methods,

(b) the role of the double continuum intermediate states of two electrons in the secondorder methods (CDW4B, BCIS4B, BDW4B, CDWEIS/EFS4B) relative to the firstorder methods (e.g. CB14B) in which such effects are absent from the onset,

(c) the extent of the influence of double electronic full Coulomb wave functions according to their inclusions in two channels, entrance and exit (CDW4B, CDWEIS/EFS4B) and one channel, entrance or exit (BCIS4B, BDW4B),

(d) the consequences of replacing the double electronic full Coulomb wave functions by their asymptotes (CDWEIS/EFS4B) given by the Coulomb logarithmic phases, especially when compared to the CDW4B method with no such additional approximations, and

(e) the effect of using the two asymptotically equivalent logarithmic Coulomb phases for the electronic motions in one channel (CDWEIS/EFS4B, BDW4B).

(a\(^{\prime }\)) At high impact energies, the CDW4B, BCIS4B and BDW4B methods perform well relative to the experimental data. In sharp contrast, the CDWEIS/EFS4B methods completely fail at all energies. On the other hand, the CB14B method is satisfactory at intermediate, but becomes totally inadequate at high energies.

(b\(^{\prime }\)) Twofold continuum states of the two electrons in the intermediate stage of collisions are of decisive importance in double charge exchange. It is precisely the neglect of these ionizing states that invalidates the firstorder methods (CB14B) at high energies. In the secondorder methods, large discrepancies at all energies exist among the methods that take these electronic continuum states either in a single channel (BCIS4B, BDW4B) or in both channels (CDW4B, CDWEIS/EFS4B).

(c\(^{\prime }\)) At lower impact energies, the double electronic continua in two channels from the CDW4B method for double capture make the total cross sections too large. This is the same effect encountered in the CDW3B method for single electron capture processes. However, for e.g. double capture from helium by alpha particles, at intermediate and higher energies (250–3000 keV), the total cross sections of the CDW4B method (with or without the otherwise small contribution from the final excited heliumlike states) are too low with respect to the experimental data. For the same process, the BCIS4B and BDW4B methods underestimate the measurements below 1000 keV, but above this energy are in a good agreement with the experimental data at high energies.

(d\(^{\prime }\)) The CDW4B and CDWEIS/EFS4B methods differ in that the latter method replaces the two electronic full Coulomb waves in only one channel by their logarithmic phases. Such a replacement has a detrimental effect on the resulting total cross sections in the CDWEIS/EFS4B methods that underestimate the experimental data by orders of magnitudes. This completely eliminates the CDWEIS/EFS4B methods from any useful application to double charge exchange processes.

(e\(^{\prime }\)) The only difference between the BDW4B and CDWEIS/EFS4B methods is in one channel and it is in the forms of the Coulomb logarithmic phase factors. The CDWEIS/EFS4B methods employ the two Coulomb phases for the two electrons in terms of their distances from the same nucleus in the given channel. Both such phases have the same joint limit at infinitely large electronnucleus distances. It is this latter Coulomb logarithmic phase factor (in terms of the asymptotic electronnucleus distances) which is used in the BDW4B method. The ensuing total cross sections from the BDW4B and CDWEIS/EFS4B methods differ by more than two orders of magnitudes at lower energies, and the discrepancy between these two theories persists to within a factor of ten at higher energies. The CDWEIS/EFS4B methods flagrantly underestimate the experimental data that agree reasonably well with the BDW4B method.
Notes
Acknowledgements
This work is supported by the research grants from Radiumhemmet at the Karolinska University Hospital and the City Council of Stockholm (FoUU) to which the author is grateful.
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