ABSTRACT
The correct boundary conditions are also critical for a proper description of ionizing processes of type (11.4) in the case of the wave function for the final state of the emitted electron. In fact, this is best appreciated already in the related simpler pure three-body problem with ionization in e.g. the ZP − (ZT, e) collisional process
ZP + (ZT, e)i −→ ZP + ZT + e. (16.1)
It is well-known that the correct Coulomb boundary conditions are much more difficult to fulfill for ionization (16.1) than for the corresponding electron capture (12.26). This difficulty is due to the presence of three charged particles in the exit channel of process (16.1). The main task is to use the total Schro¨dinger equation to consistently derive the final state scattering wave function with the exact asymptotic behavior when all the three free charged particles are far apart from each other. In the part of the asymptotic region where the three free charged particles are simultaneously present at infinitely large mutual separations, the exact form of the total wave function is known, and it is given by the product of three Coulomb-distorted plane waves (dressed or clothed plane waves). Each plane wave is modified by a multiplicative logarithmic Coulomb phase factor. Therefore, there will be altogether three such multiplicative distortions of the product of three plane waves in the exact asymptotic final wave function which must be matched by an adequate description of the exit channel in process (16.1). This exact Coulomb boundary condition for the final state in process (16.1) was fulfilled by Belkic´ [134] who was the first to derive the C3 function (via the product of three full separable Coulomb wave functions) from the complete Schro¨dinger equation in the distorted wave formalism1. The three Coulomb waves from this C3 wave function have the Sommerfeld parameters ZPZT/v, ZP/|κ − v| and ZT/κ that are due to the pairwise separate interactions ZP − ZT , ZP − e and ZT − e, respectively. Here, κ and p = κ − v are the
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electron momentum vectors in the target and projectile frame, respectively, whereas v is the incident velocity, as usual. In the same study, Belkic´ [134] also imposed the correct boundary condi-
tion to the entrance channel of process (16.1). This was again accomplished through the usage of the full distorted wave Schro¨dinger equation which gave Φ+i as the product of the target bound state and the so-called C2 function. Here, the C2 function is the product of two full Coulomb waves for electron e and nucleus ZT, each centered on the projectile nuclear charge ZP. Such a distorted wave treatment of the entrance channel is the same for three-body electron capture and ionization processes (12.26) and (16.1), respectively. The ensuing theory of Belkic´ [134] represents the CDW-3B method with the
correct boundary conditions in the entrance and exit channels of ionization in the general three-body process (16.1). Precisely as in the case of electron capture (12.26) treated using the CDW-3B method, the transition amplitude for ionization (16.1) in the same theory, has the product of the initial and final Coulomb waves for the relative motion of the two nuclei as the entire contribution from the inter-nuclear potential VPT = ZPZT/R. The standard eikonal result for this latter product is the well-known phase factor (µvρ)2iZPZT/v, which disappears from the corresponding total cross section Q(CDW−3B)if for process (16.1). Consequently, the inter-nuclear potential VPT gives zero contribution to the total cross section Q(CDW−3B)if for ionization in the usual mass limit MP À 1 and MT À 1 which is amply justified for processes (12.26) and (16.1). This is required for every ion-atom collision involving heavy nuclei [44]. Such a physically indispensable result of the CDW-3B method is a direct consequence of the symmetric treatment of the relative motion of the projectile and target nucleus in both the entrance and exit channels. As to the emitted electron e, the C3 wave function accounts fully for the si-
multaneous presence of the two Coulomb centers located at the nuclear charges ZP and ZT. Consequently, the ejected electron moves in the field of ZP as well as ZT, and this constitutes the so-called two-center effect. When κ ¿ v, the electron is mainly in close vicinity to its parent nucleus ZT. These small values of electron momentum κ give the main contribution to total cross sections at high energies. However, the influence of the other Coulomb center, ZP, becomes dominant for the electrons emitted nearly in the direction of the scattered projectiles (κ ≈ v). Such electrons are viewed as being ‘captured’ by ZP, albeit not in a bound state, but rather into a continuum state. This is the ECC effect, which is obviously a resonance phenomenon. In addition, the CDW-3B method predicts the forward and binary effects in the angular distributions of scattered projectiles ZP. The forward effect is manifested in the appearance of a peak near zero emission angle in the differential cross section for the ejected electrons. Similarly, the binary effect is seen as a peak when κ ≈ 2v. The forward, ECC and binary peaks have all been observed experimentally and, moreover, they were found to be in quantitative agreement with the corresponding predictions from the CDW-3B method of
Belkic´ [134]. Finally, it should be emphasized that the CDW-3B method is also computationally appealing, since the transition amplitude T (CDW−3B)if for process (16.1) has been obtained in Ref. [134] in a purely analytical, closed form. Similarly to charge exchange, the CDW-3B method for ionization overestimates the experimentally measured total cross sections at lower incident energies below and close to the Massey peak [377]. This is primarily due to the presence of the normalization constant N+(νP) ≡ N+(ZP/v) of the electronic full Coulomb wave function in the entrance channel. Namely, the factor |N+(ZP/v)|2 = (2piZP/v)/[1 − exp (−2piZP/v)] is enhanced with decreasing impact velocity v. In other words, the modified (scaled) cross sections Q(CDW−3B)if /|N+(ZP/v)|2 will be significantly reduced below the Massey peak [377] relative to Q(CDW−3B)if . The simplest way of eliminating the normalization N+(ZP/v) is to approximate the mentioned full Coulomb wave from the CDW-3B method by its asymptotic form. This simplification of the CDW-3B method [134] gives the CDW-EIS-3B method [151]. All told, the CDW-3B and CDW-EIS-3B methods share the same final scattering state and the ∇ ·∇ perturbation in the transition amplitude. They differ only in the description of the entrance channel, where the CDW-EIS-3B method simplifies the CDW-3B method through the replacement of the full Coulomb wave function by its asymptotic form given by the corresponding logarithmic phase factor. As expected from the discussed behavior of the auxiliary quantities Q (CDW−3B) if /|N+(ZP/v)|2, explicit computations show that indeed below the
Massey peak [377], the cross sections Q(CDW−EIS−3B)if are always substantially
smaller than the corresponding values Q(CDW−3B)if . This, in turn, considerably improves the agreement between theory and measurement at intermediate and lower energies. In a subsequent study of the same process (16.1), Garibotti and Miraglia
[135] rederived the C3 wave function of Belkic´ [134] in the exit channel. However, they failed to satisfy the correct boundary condition in the entrance channel for which they used the undistorted wave function Φi. Therefore, their so-named multiple scattering (MS) method2 as a whole is inadequate, since it disregards the proper boundary conditions that must be simultaneously fulfilled in the entrance and exit channels [44, 134]. There is another severe drawback of the MS method [135], and that is a non-zero contribution to the total cross section Q from the inter-nuclear potential VPT = ZPZT/R, which yields the mentioned Coulomb wave function for the relative motion of ZP and ZT. Otherwise, this latter Coulomb wave is one of the three Coulomb waves from the C3 wave function of Belkic´ [134] in the exit channel of process (16.1). By contrast to the MS method [135], the inter-nuclear potential VPT
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disappears altogether from the total cross section in the CDW-3B method [134], as stated. This is due to multiplication of two Coulomb waves for the relative motions of nuclei ZP and ZT from the entrance and exit channel, such that merely the well-known phase factor (µvρ)2iZPZT/v survives as the only remainder of the influence of VPT on the transition amplitude. Finally, the phase (µvρ)2iZPZT/v disappears from the total cross section in the CDW3B method and this is the signature of zero contribution of the inter-nuclear potential VPT to Q
(CDW−3B) if . Ironically, in addition to its basic faults, the
MS method [135] is computationally demanding, since its transition amplitude cannot be calculated analytically, as opposed to the CDW-3B method [134]. To alleviate this difficulty, Garibotti and Miraglia [135] approximated their T−matrix element by an expression derived from the additional peaking approximation3. Needless to say, all these deficiencies of the MS method of Garibotti and Miraglia [135] should have been obvious by mere reference to the earlier study by Belkic´ [134] within the CDW-3B method. More recently, the C3 wave function has been reinvented by Brauner, Briggs
and Klar [138] and others [139]–[144]. Concretely, Brauner et al. [138] adapted the derivation of the C3 wave function of Belkic´ to ionization of atoms by electron impact without due citation of the original work [134]. However, such an adaptation from Ref. [138] is a matter of trivial specification of the required masses and this could have been done directly in the already known C3 wave function [134]. Otherwise, the C3 wave function [134] adapted to ionization by impact of electrons [138, 139, 164, 165] and photons [141, 142] has been shown to be accurate in comparison with experimental data [139]– [144], as is the case with the CDW-3B method [134] and its simplification, which is the CDW-EIS-3B method [151]. Extensive literature exists on many important applications of the CDW-
3B method [134] to a variety of ionizing collisions [145]–[150]. Over the last three decades, regarding both differential and total cross sections for single electron emission from atoms by multiply charged ions, the CDW-3B method [134] has been firmly established as the most successful high-energy theory of ionization valid above 100 keV/amu in accordance with the associated limit (14.69), which has been found empirically for electron capture [44]. Naturally, the CDW-3B theory is not adequate below its lower limit of validity, but here the CDW-EIS-3B method [151] comes to rescue the situation by providing total cross sections that are in excellent agreement with experimental data even at energies smaller than 100 keV/amu. The CDW-3B method can be extended to the CDW-4B method for single
ionization of a helium-like atomic system by a bare nucleus in process (11.4), as done by Belkic´ [368]. The prototype of this latter process is single electron
detachment from H− by H+
H+ +H−(1s2) −→ H+ +H+ e. (16.2) This is an important example of ionizing four-body collisions, since it offers the possibility to analyze the dependence of cross sections on inter-electron correlations that are known to be strong whenever H− is involved.
