ABSTRACT
This book is comprized of two parts. In the first part through chapters 1-10, formal theory of scattering is presented. The second part via chapters 11-19 deals with a wide class of ion-atom collisions at high energies. The common thread which tightly binds together these two parts into one coherent whole is the necessary rigor of the mathematics of scattering theory, intertwined with the first principles of physics. Although theory is the central subject of this book, experiment is also in the main focus, as indeed pursuing one without the other in physics does not bear fruit. The major theme which connects the first and second parts into a consistent
scattering formalism is the following logical chain of key physical principles supported by a firm mathematical basis, as the necessity for experimentally relevant theoretical predictions: (i) the existence of the strong limit of the Schro¨dinger time-dependent full scattering state Ψ(t) in the remote past (t→ −∞) and distant future (t → +∞), (ii) the existence of the Møller wave operators Ω±, (iii) the existence of the scattering Sˆ− as well as transition Tˆ−operators and (iv) the existence of isometry. State vectors and operators are required to converge strongly in scatter-
ing theory. This implies the corresponding weak convergence by way of the Schwartz inequality. Adjectives ‘strong’ and ‘weak’ refer to convergence in terms of the norm and absolute value, respectively. Principle (i) is the requirement of the correct asymptotic behavior of the complete scattering states via the condition that the total dynamics is experimentally indistinguishable from the unperturbed (free) dynamics in the remote past and distant future. This is formulated by the strong limits Limt→∓∞||Ψ(t)−Ψ0(t)|| = 0 by which the full states Ψ(t) = Uˆ(t)Ψ0 associated with the complete evolution operator Uˆ(t) = exp (−iHˆt) of the entire system are reduced at t → ∓∞ to the state Ψ0(t) = Uˆ0(t)Ψ0 corresponding to the unperturbed evolution operator Uˆ0(t) = exp (−iHˆ0t), where Ψ0 ≡ Ψ(0). Here, convergence in the norm is needed because the strong limit of the difference Ψ(t) − Ψ0(t) implies that both the difference Ψ(t) − Ψ0(t) itself is a zero state vector ∅ and that e.g. the spatial integral over |Ψ±(t) − Ψ0(t)|2 tends to zero at every point in the configuration space. However, had it not been the case for the strong limit, each of the two states Ψ(t) and Ψ0(t) could have independently approached ∅ at each spatial point when t → ∓∞. The most important physical significance of the scattering wave function Ψ(t) is that it represents the particle state at infinitely large negative (and/or positive) times. Therefore, principle
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(i) interprets scattering theory as the asymptotic agreement between the two descriptions, the one using the full dynamics (with H) and the other employing the unperturbed dynamics (with H0). Principle (ii) necessitates that the strong limit of the product Uˆ†(t)Uˆ0(t) ≡ Ω(t) of the two evolution operators, containing the full (Hˆ) and unperturbed (Hˆ0) Hamiltonians, respectively, as t→ ±∞ must be the stationary Møller wave operators Ω∓. Principle (iii) requires that the product of two stationary Møller operators via Ω−†Ω+ is the Sˆ−operator. Principle (iv) necessitates the relationship ||Uˆ(t)Ψ0|| = ||Ψ0|| which conserves the norm and probability. None of the principles (i)–(iv) holds for Coulomb potentials. Nevertheless,
in order to restore the usual interpretation of scattering theory for Coulomb potentials, the Hamiltonian must be appropriately modified so that the complete perturbation interaction is of short range. With such modifications, the principles (i)–(iv) could simultaneously hold, provided that the full Coulomb wave functions for electrons centered on both nuclei are used to distort the unperturbed channel states within theories that obey the correct boundary conditions, like the continuum distorted wave (CDW) method. However, when one or both of these Coulomb waves are replaced by their asymptotic forms that are logarithmic phase factors, the principle of isometry becomes invalid, since ||Uˆeik(t)Ψ0|| 6= ||Ψ0|| where Uˆeik(t)Ψ0 is the wave packet containing the Coulomb eikonal phase instead of the corresponding full Coulomb wave in Uˆ(t)Ψ0. In other words, the wave packet Uˆeik(t)Ψ0 does not represent a physical state of a system of particles. Precisely this ansatz (i.e. the replacement of a full Coulomb wave by its asymptotic behavior at large distances) which violates isometry is used in a method called the continuum distorted wave eikonal initial ‘state’ (CDW-EIS) method. As such, the CDW-EIS method is obviously an approximate CDW method, irrespective of whether the former is established by reliance upon the latter or derived independently. In the first part of this book, within formal scattering theory, a rigorous for-
malism is presented using the Abel and Cauchy strong limits. The concept of the strong limit, in general, is of critical importance to theory and experiment on scattering phenomena. The strong limits rely upon the correct boundary conditions through the requirement of the reduction of the complete dynamics under the full Hamiltonian Hˆ to the unperturbed dynamics governed by the unperturbed Hamiltonian Hˆ0. Without these limits it would be impossible to unequivocally distinguish the situations “before” and “after” the collision. And without this latter differentiation, it would be impossible to identify the real cause of the passage of the system from the initial to the final states. The wave function Ψ(t) could tend to Ψ0(t) only if the interaction (as an external perturbation exerted onto the system) becomes negligibly small as t → ±∞. On the other hand, at infinitely large times, the system has evolved to the asymptotic spatial region of scattering (r →∞). Thus the two limits t→ +∞ and r →∞ consistently require that the interaction must vanish at infinitely large time-space variables. This concept coheres with the adiabatic theorem which treats the effect of screening and adiabatic switching of the interac-
tions. These strategies via the existence of the strong limits, applications of the adiabatic theorem and other related aspects all hold in an adequate manner for short-range interactions from nuclear collisions. However, they fail for long-range Coulomb potentials from atomic collisions. For example, the rigorous work of Dollard proves that no screening whatsoever is applicable to Coulomb potentials. Yet, disregarding this fundamental objection, screened Coulomb potentials are regularly used throughout the literature on atomic collisions, as one of the inadequate attempts of avoiding to deal properly with the asymptotic convergence problem for long-range interactions. Despite constituting a mathematically-oriented exposition, chapters 1-10
of the first part are presented in the typical physics setting with the relevant connections to the main observables, as experimentally measurable quantities, that can test and eventually validate the theory. Even the most rigorous concepts of strong topology can find their critical applications to typical situations encountered in scattering experiments as well as to associated numerical computations. Rigorous mathematical treatments of scattering phenomena, accompanied naturally with the corresponding robust algorithms, are necessary to secure two key achievements in modeling from first principles with no adjustable parameters: (i) enhancement of the predictive power of ab initio physics theories, and (ii) minimization as well as control of uncertainties in theoretical formulations and the ensuing results of computations. Specifically, in this first part of the book, we begin with several funda-
mental notions and main observables in standard scattering problems. This theme is comprized of a number of interrelated topics, including: observables and elementary processes, energy as the most important physical property, classification of collisions, the role of wave packets, adiabatic switching of interaction potentials, collimation of beams of projectiles, general waves as well as quantum mechanical waves and probability character of quantum collisions. Subsequently, we address the key issue of the requirements of the theory for
the experiment. This encompasses elementary events versus multiple scatterings, average probabilities, differential and total cross sections, total probabilities, transmission phenomena, as well as quantum mechanical currents and cross sections. Next, we study continuous spectra and eigen-problems of resolvents. This subject is especially relevant to scattering where the continuum plays the dominant role. While in the corresponding bound-state spectrum, the Schro¨dinger eigen-value problem for Hamiltonians is at work, in scattering phenomena the Green resolvent operator proves to be much more appropriate. The topics analyzed here are: completeness and separability of the Hilbert spaces, the key realizations of abstract vector spaces, isomorphism of vector spaces, eigen-problems for continuous spectra, normal and Hermitean operators, strong and weak topology, compact operators for mapping of the weak to strong limits, strong differentiability and strong analyticity. Further, we treat linear and bilinear functionals. In scattering theory, func-
tionals play a very important role, as a special form of mapping between vector spaces and scalar fields. Therefore, it is necessary to present this sub-
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ject, as well, and we do so by relying upon the Ries-Freshe theorem. We also focus on the definition of a quantum scattering event. This theme is addressed through Hamiltonian operators and boundedness, evolution operators and the Møller wave operators, as well as the Cauchy strong limit in non-stationary scattering theory, ending with three criteria for a quantum collisional system. In continuation, we elaborate on the adiabatic theorem and the Abel strong limit. The concept of screening of interaction potentials at asymptotically large distances is analyzed within the adiabatic theorem. The main idea behind this theorem is that all perturbation potentials must vanish at infinitely large times when the interacting particles are at asymptotic mutual distances. Special attention is paid to the application of the adiabatic theorem to scattering states that are improper, in the sense of being non-normalizable and, as such, not belonging to the Hilbert space of physical state vectors. Further, we inter-connect the adiabatic theorem with the Møller wave operators, the Abel strong limit in stationary scattering theory, the Green resolvent operators and the Lippmann-Schwinger equations. Our next investigation is directed towards non-stationary and stationary
scattering in the framework of the strong limits. The strong limits of the Møller wave operators are of central importance to scattering theory. For example, the Lippmann-Schwinger equations can be derived from the Abel limit, provided that the Møller operators Ω± exist. The Cauchy and Abel limits deal with the processes t→ ∓∞ and ε→ 0± that are typical in the nonstationary and stationary scattering theory. Here, ε is an infinitesimally small positive number added via its imaginary counterpart iε to the real energy E in the Green resolvent operator to avoid singularities (poles) at the eigen-energies of Hˆ. In the non-stationary {stationary} formalism, the limiting processes t → ±∞ {ε → 0∓} imposed onto Ψ(t) {Ψ(E)} must lead to the total scattering states Ψ∓ with the incoming and outgoing boundary conditions, respectively. In the configuration space, the incoming {outgoing} asymptote is comprized of wavelets that are conceived as arriving {departing} to {from} the scattering center. It is possible to pass from the Cauchy to the Abel limit and vice versa. This passage effectively carries out the transformation from the time-dependent state vector Ψ(t) to the corresponding stationary state Ψ(E). Such a transformation is usually performed by means of the Fourier integral. The Cauchy-Abel mapping can totally replace the Fourier integral. We expand the analysis to the scattering and transition matrices, as the two
most important physical quantities that contain the entire information about any studied system. Here, we derive the Sˆ-and Tˆ-operators from the existence of the Abel strong limit. The squared absolute values of the Sˆ-and Tˆ-matrix elements are the key physical quantities that yield all the observables that could be measured experimentally, such as probabilities for transitions from the given initial to one and/or all final states, the corresponding cross sections, rate coefficients, density of states, spectral parameters, etc. To describe the characteristic features of the systems under investigation, we carry out spectral analysis of dynamical operators. As mentioned, the Cauchy and Abel
limits can entirely replace the role of the Fourier integral. Moreover, these two limits do not necessarily need to rely upon each other. This implies that e.g. stationary scattering theory could be built with the sole reliance upon the Abel strong limit and with no recourse to the time-dependent formalism. Here, we deal with such issues as the proof of the existence of the Møller wave operators in an arbitrary representation, since this is one of the three conditions that define the so-called simple scattering system. Also addressed here are the themes of vital importance to scattering theory e.g. the spectral theorem, the link of the Møller operators with the Green resolvents, the relationships of the Abel strong limit with unitary operators and the Møller wave operators. We end the first part of this book with chapter 10 by considering the existence and completeness of Møller wave operators. The specific themes encompass linearity and isometry of wave operators, boundedness of wave operators throughout the Hilbert spaces, the spectral projection operators and the completeness of the Møller operators. Overall, although the main emphasis in the first part of this book (chapters
1-10) is on the rigorous mathematical foundations of scattering theory, the basic general concepts of the related experiments are also analyzed. This is done on an intuitive level, as well as on deeper grounds that formulate a sequence of the requirements of the theory for the experiment. The second part of the book (chapters 11-19) gives a thorough and sys-
tematic overview of the current status and a critical assessment of the existing quantum mechanical four-body methods for energetic ion-atom collisions. Proper descriptions of these collisions with two active electrons, such as the ZP − (ZT; e1, e2)i and (ZP, e1)i1 − (ZT, e2)i2 scatterings require solutions of the pertinent four-body problems. We consider a number of inelastic collisions and special attention is focused on double electron capture, transfer ionization, transfer excitation, single electron detachment and single electron capture. We limit the scope of this book to intermediate and high non-relativistic impact energies. A quantum mechanical treatment is adopted to set the formal theoret-
ical framework for description of four-body rearrangement collisions. After establishing the basic notation, we present a succinct derivation of the Lippmann-Schwinger equations, the boundary-corrected Born as well as the Dodd-Greider perturbation expansions and the leading distorted-wave methods. All these theories satisfy the proper physical asymptotic conditions at large inter-particle distances in four-body collisions for which the CDW-4B method emerges as the most adequate. Subsequently, the particular one-and two-electron transitions in scatterings of completely stripped projectiles on helium-like atomic systems, as well as in collisions between two hydrogenlike atoms or ions, are analyzed in detail. Helium-like atoms or ions are the simplest many-electron systems where one can investigate the importance of electron correlation effects. The reviewed problems indicate that for helium as a target, the dynamic
electronic correlations in the given perturbation potential are much more im-
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portant than the static ones stemming from the target bound state wave function. A substantial improvement of most of the examined four-body methods over e.g. the corresponding semi-classical impact parameter model (IPM) can be attributed exclusively to the role of dynamic electron correlation effects. The main drawback of the IPM and the related independent electron model (IEM) is in effectively reducing the initial four-body problem to the associated three-body problem. In this reduction, dynamic electron-electron correlations are completely ignored from the outset. The results of the correlated CDW method for double electron capture from helium by protons was the first proof of the complete break-down of the IPM. This is also the case for other twoand many-electron transitions involving complex-structured targets. All the presently analyzed boundary-correct four-body quantum mechanical methods are seen to be able to naturally incorporate both static and dynamic correlation effects of electrons. In particular, static inter-electron correlations are shown to be very impor-
tant for ionizing collisions involving H− as a target, such as single electron detachment from H− by H+. Moreover, for these collisions, an even stronger emphasis is placed upon the proper connection between the distorted wave functions and the corresponding perturbations, as illustrated within the modified Coulomb Born (MCB) method, which is in excellent agreement with the available experimental data from the threshold through the Massey maximum to the Bethe region of high energies and beyond. By contrast, ignoring the said connection, as done in the eikonal Coulomb Born (ECB) method, leads to utterly unphysical total cross sections which overestimate the experimental data by 2-3 orders of magnitude, and tend to a peculiar constant value at high impact energies, instead of reaching the correct Bethe asymptotic limit. In the present book, particular emphasis is placed upon the critical im-
portance of preserving the proper Coulomb boundary conditions in formal four-body theory and in computational practice. This is guided by common past experience, which has shown that whenever such conditions (as one of the most basic requirements from scattering theory) are overlooked, severe and fundamental problems arose. As a consequence, models with incorrect boundary conditions are inadequate for describing experimental findings. In practice, the total scattering wave functions can satisfy the correct Coulomb boundary conditions in the initial and final asymptotic channels following the well-established procedure. For example, in addition to the long-range Coulomb distortions of the plane waves for relative motion of two charged heavy aggregates, account should be made for the intermediate ionization continua of the electrons in the entrance and exit channels for the CDW-4B method, or in either the entrance or exit channel for asymmetric distorted wave treatments, such as the boundary-corrected continuum intermediate state (BCIS-4B), Born distorted wave (BDW-4B), continuum distorted wave Born initial state (CDW-BIS) and continuum distorted wave Born final state (CDW-BFS) methods. As to the boundary-corrected first Born (CB1-4B) method, the pure electronic continuum intermediate states are not directly
taken into account. Rather the presence of the electrons is felt here through a screening of the two nuclear charges in the Coulomb wave functions of the relative motion of the heavy scattering aggregates. Double electron capture in the considered collisional systems is studied by
means of the CB1-4B, CDW-4B, BDW-4B, BCIS-4B and CDW-EIS-4B methods. Unlike the well-documented success of the CB1-3B method for singleelectron capture at intermediate and a wide range of high energies (all the way up to the outset of the Thomas double scattering), the CB1-4B method, as the prototype of four-body first-order theories, is satisfactory for double electron capture only at some intermediate energies, but flagrantly fails at higher energies. By contrast, as the prototype of four-body second-order theories, the CDW-4B method is successful for the majority of double electron capture processes, thus continuing with the excellent tradition of the corresponding three-body counterpart, which is the CDW-3B method. This is particularly true for two-electron capture from He by H+ for which it is sufficient to include only the ground-to-ground state transition, due to the absence of the excited states of the H− ion formed in the exit channel. For the same H+−He double charge exchange, the cross sections from the CB1-4B method markedly overestimate all the experimental data by 1-3 orders of magnitude at all energies (10-1000 keV). Moreover, using the CDW-4B method for double electron capture in the ZP−He collisions with ZP ≥ 3, it is found that the contribution from excited states can be important compared to that from the corresponding ground states. As such, including excited states into computations can improve the agreement between the CDW-4B method and experimental data. Specifically, within the distorted wave formalism, it is customary to refer
to the first/second-order methods as those theories that exclude/include the electronic continuum intermediate states, respectively. On the other hand, any such second-order distorted wave method, may simultaneously be the lowestorder term in a consistent perturbation series expansion of the full transition amplitude. Then by a parallel nomenclature, this lowest-order term of a perturbation development (with or without distorted wave formalism) would be called a first-order approximation to the full transition amplitude. Thus, for example, the CDW-4B method is a second-order method (when viewed from the distorted wave perspective), as it includes the electronic continuum intermediate states. However, the same CDW-4B method is simultaneously the rigorous first-order term in the Dodd-Greider perturbation series. As such, the CDW-4B method is also the first-order approximation to the exact DoddGreider expansion. This latter fact should explicitly be indicated (whenever there is a chance for confusion) with a more specific acronym such as CDW4B1. This is especially helpful whenever a reference should be made to the second term in the Dodd-Greider expansion, in which case the acronym CDW4B2 is definitely needed. Regarding double electron capture in the He2+−He(1s2) collisions, excited
states are expected to play a minor role due to the dominance of the resonant 1s2 − 1s2 transition. The CDW-4B method confirms this anticipation, but
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does not quantitatively reproduce a part of the available experimental data at impact energies 200-3000 keV that are within the domain of the validity of this theory for the He2+ − He(1s2) collisions. Interestingly, for this symmetric scattering, the CB1-4B method substantially outperforms the CDW-4B method at energies 200-1500 keV. Such a surprising situation might seem to have been ameliorated in the past by using a crude approximation to the Green function from the second-order propagator of a perturbation expansion which, however, is not of the Dodd-Greider type. It is well-known that this latter circumstance could lead to certain serious difficulties. All ordinary distorted wave perturbation expansions (non-Dodd-Greider),
similar to the undistorted Born series, contain disconnected or dangerous diagrams that cause the transition operator to diverge for rearranging collisions. A seemingly improved agreement of the mentioned ‘augmented’ CDW-4B method for the two-electron transfer in the He2+ − He(1s2) collisions should therefore be taken with considerable caution, since the approximate Green function is merely off-shell. Moreover, only two hydrogen-like ground states centered on the projectile and target nucleus were taken into account from the sum over the discrete and continuous parts of the whole spectrum. More systematic work is needed for this particular colliding system, first by treating the on-and off-shell contributions on the same footing, and second by assessing the convergence rate in the spectral representation of the Green function from the second-order term of a chosen perturbation series. Needless to say, it would be important to use the second term in the Dodd-Greider perturbation series to obtain a relatively reasonable estimate of the proper CDW-4B2 method for double charge exchange. Comparing such an estimate to the contribution from the associated CDW-4B1 method would give an invaluable indication about convergence of the Dodd-Greider perturbation series which does not contain any disconnected diagrams. As to the BCIS-4B and BDW-4B methods, they have been applied to dou-
ble electron capture in the He2+−He collisions. At moderately high energies (1-3 MeV), good agreement with experiments is found using the BCIS-4B and BDW-4B methods. However, at still higher energies (4 and 6 MeV), the cross sections from the BCIS-4B and BDW-4B methods overestimate the experimental findings (that are the only two measured data points above 3 MeV). Below 1 MeV all the way up to 100 keV, the BCIS-4B and BDW-4B methods underestimate the available experimental data by a factor ranging from 2 to 10. Otherwise, throughout the range 100-7000 keV, the BCIS-4B and BDW4B methods agree closely with each other. They both exhibit a broad Massey maximum near 175 keV. Such a behaviour is opposed to the CDW-4B method which gives the cross sections that continue to rise with decreasing impact energy, as usual, without any sign of the resonance peak. This latter pattern occurs because of an enhanced contribution from the discrete-continuum coupling, which is mediated by the typical∇·∇ potential operator (for each of the two electrons) in the perturbation interaction from the transition amplitude of the CDW-4B method. Here, we have an example of discrete-continuum
interference, because one gradient in the perturbation acts on e.g. the initial bound state centered on the target nucleus, whereas the other gradient is applied to the electronic full Coulomb wave centered on the projectile. The gradient-gradient perturbation describes the same electron being simultaneously bound to the target nucleus, and unbound in the field of the projectile nucleus. Therefore, this coupling of discrete and continuum states via∇ ·∇ is a typical two-center effect. Once the underlining scalar product is carried out, at least two complex-valued terms are obtained in the transition amplitude T−if . The ensuing terms can have constructive or destructive interference in |T−if |2, depending on the value of the impact energy. Specifically, when the impact energy decreases, constructive interference prevails between the two mentioned parts in |T−if |2 and this causes the cross sections to increase in the CDW-4B method. Also, the Coulomb normalization constant for the full electronic continuum intermediate states increases with decreasing energy. These features are common to both the CDW-3B and CDW-4B methods. However, in the CDW-4B method, this constructive interference is further enhanced, since there are two gradient-gradient operators for each of the two actively participating electrons.
