Resonant Transfer and Excitation of First-Row Ions Using Zero-Degree Auger Projectile Spectroscopy

1. Introduction

Understanding and modeling the dynamics of many-body quantum systems under intense, ultrafast perturbations remains a major challenge in physics and chemistry—whether for atoms and molecules in the gas phase or condensed matter systems [2,3]. Energetic (MeV) collisions of few-electron ions with atomic targets provide ideal laboratory testbeds for studying these fundamental challenges. Although the interaction time is much shorter than 1 femtosecond, the interplay between electron-nucleus $(e-n)$ and electron-electron $(e-e)$ interactions, together with exchange effects, creates profoundly complex dynamics. Fortunately, few-electron collision systems remain sufficiently simple to permit analyses based on individual particle interactions. Special interests and challenges exist when considering the dynamic interactions between two electrons located on different centers (also known as two-center $(e-e)$ interactions), the most celebrated case of which is the process of resonant transfer and excitation (RTE) occurring in swift ion-atom collisions.

In asymmetric collisions of heavy projectiles with light targets, two distinct peaks typically appear in the cross section as a function of impact energy [4,5]: a high-energy peak attributed to RTE [6], and a low-energy peak attributed to the process of nonresonant transfer excitation (NTE) [4,7,8]. The mechanisms for RTE and NTE differ fundamentally and are shown schematically in Figure 1. The RTE contribution is described to first order by a correlated one-step mechanism mediated by the two-center $(e-e)$ interaction [see Figure 1(a)]. It was modeled using the IA as a quasifree resonant electron scattering, analogous to the inverse Auger process [9,10,11,12]. In contrast, NTE is thought to arise from a two-step sequence of uncorrelated excitation and transfer events, each driven by independent $(e-n)$ interactions [13,14] [see Figure 1(b)]. Both transfer excitation (TE) mechanisms can occur in the same ion–atom collision, contributing coherently to the same final doubly excited projectile state. However, cross sections for RTE and NTE have typically been computed separately, allowing only incoherent addition to the total TE cross section and enabling speculations about RTE–NTE interferences [13,15,16,17,18,19,20]. Nevertheless, in collisions with light targets such as H₂ and He, NTE is greatly suppressed at the collision energies where RTE is significant and was consequently neglected in early analyses. Alternatively, a rigorous and successful TE treatment has been extremely difficult to accomplish and has only recently emerged.

Following its discovery in 1982 [4,6] RTE received considerable attention due to its unique and direct connection to dielectronic capture (DC), the analogous resonant ion-electron collision process [21]. In both cases, an impinging electron excites an ion while simultaneously being captured into a bound state. The key difference lies in the nature of the impinging electron: it is free in DC, but bound to the target atom in RTE [see Figure 1 (a) and (c)]. Both processes lead to the production of short-lived doubly excited states, but only at specific resonant collision energies. Such resonances, while common in electron-ion collisions are not generally observed in energetic ion-atom collisions since their existence is usually masked by competing $(e-n)$ interactions as in NTE [4,7,8]. These resonances are especially pronounced in collisions with ions rather than atoms, due to the greater availability of bound states in the ion-electron continuum compared to the negative ion continuum. Consequently, short-lived resonances appear across a wide range of collision energies and prove highly sensitive to both long-range Coulomb interactions and short-range correlation effects that depend strongly on the ion’s electronic structure and collision dynamics [22]. The manifestation of such resonances in collisions of few-electron ions with light targets generated considerable excitement in the ion-atom collision community. These resonances are stabilized through X-ray (RTEX) or Auger (RTEA) emission, which can be readily measured at accelerator facilities, providing much-needed DC cross sections – critical for the modeling of plasma charge-state distributions and temperatures, particularly in astrophysical and fusion environments [23,24].

In 1983, Brandt [9] successfully applied the impulse approximation (IA) to describe RTE. For fast collisions where target electrons behave as quasifree (when viewed from the projectile rest frame), RTE was shown to mimic DC. The key difference is that the impinging electron’s kinetic energy is broadened by its orbital motion around the target nucleus. RTE cross sections could thus be obtained by averaging DC cross sections over the target’s Compton profile (i.e., the electron momentum distribution along the collision axis). Conversely, DC cross sections could be indirectly extracted through the IA from RTE measurements. This elegantly simple approach yielded unexpectedly good agreement with the first RTEX measurements, attracting considerable interest by connecting the seemingly disparate fields of ion-atom and ion-electron collisions. It was quickly extended to other processes, including the electron scattering model (ESM) [25], elastic electron scattering off ions [26], resonant inelastic scattering [27], electron excitation [28], electron loss [29,30], and superelastic scattering [31] (see also Refs. [32,33] and citations therein). However, success was more modest since these processes also involve $(e-n)$ interactions—which the IA does not address—that often contribute significantly in the same collision energy regime, lacking RTE’s clear, unambiguous two-center $(e-e)$ interaction signature. The connection between the IA and Born approximation for screening/antiscreening processes was also noted [34].

From 1982 to 1992, TE was extensively studied using He and H₂ targets across highly charged ions ranging from He⁺ [16] to ${\mathrm{U}}^{90+}$ [35]. Both the emitted X-rays [36,37] and Auger electrons [1,38] from the decay of the TE-formed doubly-excited states were measured, with corresponding cross sections determined. In particular, KLL state-selective TE measurements were performed, especially for the lowest atomic number $Z_p$ projectile ions, using high-resolution Auger projectile spectroscopy [10,11,16,17,39,40,41,42,43,44], providing the most stringent tests of theory. Both X-ray [36] and Auger [1] spectroscopy measurements tested the validity of the IA and extracted DC cross sections that were unavailable from direct ion-electron collision experiments at the time [23]. The comprehensive 1992 RTEX [36] and RTEA [1] reviews summarize this remarkably productive era.

Early theoretical treatments of DC cross sections relied primarily on atomic structure calculations of Auger energies and rates using Hartree-Fock methods [23]. These evolved into more sophisticated electron-ion scattering approaches such as R-matrix [45] and convergent close-coupling methods [46], which coherently account for both long- and short-range interactions. However, experimental KLL DC cross sections remained scarce in the early 1980s due to insufficient luminosity in crossed-beam or merged-beam setups [22]. The advent of high-luminosity electron beams and highly charged ion sources in the late 1980s and early 1990s made direct DC cross section measurements accessible through merged-beam or crossed-beam experiments.

With this capability established, research interest in RTE shifted toward developing rigorous ion-atom collision theories capable of treating TE processes coherently within a uniform approach. These efforts have proven challenging, primarily due to the difficulty of incorporating multiple electrons and both TC $(e-e)$ and $(e-n$) interactions in a unified dynamical treatment [15,47]. Nevertheless, two such dynamical treatments emerged between 1988 and 1997 for two-electron collision systems: (i) the two-electron atomic orbital close-coupling (2eAOCC) method [13], and (ii) the continuum distorted wave four-body (CDW-4B) approach [48,49,50,51]. The 2eAOCC calculations targeted the benchmark He⁺(1s)+H(1s) system, but the cross sections for the dominant He$(2p^{21}D)$ RTE resonance exceeded measurements for H₂ targets (see [52] and Figure 6 in [1]). This approach paved the way for exact nonperturbative TE treatments while refining RTE modeling. The CDW-4B was applied mainly to highly asymmetric systems (e.g., S¹⁵⁺+H [48] and references therein), comparing RTE peaks from low-resolution X-ray data, which hindered quantitative interpretation. The NTE peak, occurring at too low an energy, fell outside CDW-4B’s range. Applications to He⁺+He [16,50] and ${\mathrm{He}}^{+}(1s)$+H$(1s)$ [51] disagreed strongly with experimental and 2eAOCC results [13]. Due to computational limitations and measurement challenges, no further comprehensive coherent treatments followed until recently. Alternative approaches extended the IA to double differential cross sections (DDCS), enabling uniform treatment of both resonant and nonresonant electron scattering within the ESM [25]. These were supported by state-selective RTEA measurements and R-matrix calculations [12,53].

Advances in computational speed now enable overcoming many of these barriers. In 2022, the first nonperturbative TE treatment for 0.5–18 MeV collisions of ${\mathrm{C}}^{4+}(1s^2)$ with He was reported, in excellent agreement with measured Auger single differential cross sections (SDCS) [38] of the ${\mathrm{C}}^{3+}(1s2p^{22}D)$ state. This theoretical treatment considers the dynamics of three active electrons, employs semiclassical close-coupling calculations (3eAOCC) within a full configuration interaction approach [54]. It enables a coupled, coherent description of target and projectile excitation, ionization, single-electron capture, and TE, therefore going well beyond the methods developed in the past. Collaborative efforts to extend these results to other low-$Z_p$ isoelectronic systems are under way.

Following this general introduction, more details about the ion-electron process of DC and its stabilization are first presented. This is followed by a revised IA treatment connecting ion-electron to ion-atom collisions, applied to RTE with new refinements and an exact numerical treatment using screened hydrogenic wavefunctions. The ZAPS technique and advanced capabilities of our hemispherical deflector analyzer spectrograph for high-resolution zero-degree Auger spectrometry are then briefly described. Representative RTEA results from both older parallel-plate spectrometer measurements and our new hemispherical spectrograph data are systematically compared to the revised IA calculations. Finally, all existing RTEA measurements are comprehensively tabulated, accompanied by a figure of merit quantifying the significantly improved theory-experiment agreement.

2. Dielectronic Capture (DC)

A free electron cannot be captured by a bare ion (excluding radiative recombination, which predominates for heavy ions [55]), as this violates conservation of both momentum and energy. However, capture becomes possible for non-bare ions via simultaneous excitation of a bound electron, a process termed dielectronic capture [11] (also known as radiationless capture, RC [9]).

In DC, a free electron with momentum $p=m\mathrm{v}$ and kinetic energy $\epsilon =m{\mathrm{v}}^2/2$ (where $\mathrm{v}$ is the impinging electron velocity) collides with an N-electron ion, exciting it from level i (binding energy $-{\epsilon }_i$) while being captured into doubly excited level j (binding energy $-{\epsilon }_j$) of the resulting $(N+1)$-electron ion. The DC cross section is then given by [56,57]:

$$\begin{array}{cc}\hfill {\sigma }_{DC}[i\to j](\epsilon )& ={\Omega }_{DC}[i\to j](\epsilon )L_{{\Gamma }_j}(\epsilon -E_a).\hfill \end{array}$$

(1)

Here, ${\Omega }_{DC}[i\to j](\epsilon )$ denotes the DC collision strength (in cm² eV) for the $i\to j$ transition. DC is resonant, peaking at resonance energy $E_a$, the Auger electron energy in the time-reversed $j\to i$ Auger decay. $L_{{\Gamma }_j}(\epsilon -E_a)$ is the normalized Lorentzian lineshape of level j (FWHM ${\Gamma }_j$):

$$\begin{array}{cc}\hfill L_{{\Gamma }_j}(\epsilon -E_a)& ={\displaystyle \frac{1}{\pi }}{\displaystyle \frac{\left({\Gamma }_j/2\right)}{{(\epsilon -E_a)}^2+{({\Gamma }_j/2)}^2}}.\hfill \end{array}$$

(2)

For doubly excited j-levels, ${\Gamma }_j\ll |\epsilon -E_a|$ except near resonance ($\epsilon \approx E_a$), yielding a narrow profile often approximated by a Dirac delta function [56,58]:

$$\begin{array}{cc}\hfill L_{{\Gamma }_j}(\epsilon -E_a)& \approx \delta \left(\epsilon -E_a\right),\hfill \end{array}$$

(3)

leading to:

$$\begin{array}{cc}\hfill {\sigma }_{DC}[i\to j](\epsilon )& ={\Omega }_{DC}[i\to j](\epsilon )\delta \left(\epsilon -E_a\right).\hfill \end{array}$$

(4)

2.1. The DC Collision Strength ${\Omega }_{DC}$

The collision strength ${\Omega }_{DC}[i\to j](\epsilon )$ in Equation (1) is given by [56]:

$$\begin{array}{cc}\hfill {\Omega }_{DC}(\epsilon )& \equiv {\Omega }_{DC}[i\to j](\epsilon )={\displaystyle \frac{{\pi }^2}{2}}{a}_0^2{\epsilon }_0{\displaystyle \frac{g_j}{g_i}}\left[{\displaystyle \frac{\hslash A_a[j\to i]}{\epsilon }}\right].\hfill \end{array}$$

(5)

Here, $a_0$ and ${\epsilon }_0$ are atomic units of length and energy; $g_i=(2L_i+1)(2S_i+1)$ and $g_j=(2L_j+1)(2S_j+1)$ are $LS$-coupling degeneracies; $A_a[j\to i]$ is the $(j\to i)$ Auger rate with energy $E_a={\epsilon }_i-{\epsilon }_j$. At resonance ($\epsilon =E_a$ in eV and $A_a$ in s⁻¹), ${\Omega }_{DC}$ is given in (cm² eV) by:

$$\begin{array}{cc}\hfill {\Omega }_{DC}(\epsilon =E_a[j\to i])& =2.4751\times {10}^{-30}{\displaystyle \frac{(2L_j+1)(2S_j+1)}{(2L_i+1)(2S_i+1)}}\left({\displaystyle \frac{A_a[j\to i]}{E_a[j\to i]}}\right)[{\mathrm{cm}}^2\mathrm{eV}].\hfill \end{array}$$

(6)

2.2. DC Level Relaxation by Photon or Auger Emission

The doubly excited DC state j stabilizes by photon or Auger emission. Photon decay to final state $f^{\prime }$ of the $(N+1)$-electron ion ($i\to j\to f^{\prime }$) yields DR. Auger decay returns to the N-electron final state f ($i\to j\to f$, where $f=i$ or $f\ne i$), and yields RE [10,59]:

$$\begin{array}{cc}\hfill {\sigma }_{DR}[i\to j\to f^{\prime }](\epsilon )& ={\sigma }_{DC}[i\to j](\epsilon )\omega [j\to f^{\prime }],\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\sigma }_{RE}[i\to j\to f](\epsilon )& ={\sigma }_{DC}[i\to j](\epsilon )\xi [j\to f],\hfill \end{array}$$

(8)

with the fluorescent yield defined as:

$$\begin{array}{cc}\hfill \omega [j\to f^{\prime }]& \equiv {\displaystyle \frac{A_r[j\to f^{\prime }]}{({\sum }_f{A}_a[j\to f]+{\sum }_{f^{\prime }}{A}_r[j\to f^{\prime }])}}\hfill \end{array}$$

(9)

and the Auger yield defined as:

$$\begin{array}{cc}\hfill \xi [j\to f]& \equiv {\displaystyle \frac{A_a[j\to f]}{({\sum }_f{A}_a[j\to f]+{\sum }_{f^{\prime }}{A}_r[j\to f^{\prime }])}}.\hfill \end{array}$$

(10)

DR cross sections for highly charged ions (H-like [57], He-like [60,61,62], Li-like [63,64,65]) have been accurately measured in storage ring electron coolers via merged beams, finely scanning the electron energy while detecting the recombined ions.

For Auger emission at angles $({\theta }_e,{\varphi }_e)$ relative to the electron beam, the RE collision strength and SDCS are [66]:

$$\begin{array}{cc}\hfill {\Omega }_{RE}[i\to j\to f](\epsilon )& =\xi [j\to f]{\Omega }_{DC}[i\to j](\epsilon ),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\sigma }_{RE}}{d{\Omega }_e}}[i\to j\to f](\epsilon ,{\theta }_e,{\varphi }_e)& ={\sigma }_{RE}[i\to j\to f](\epsilon ){\left|Y_{L,M_L=0}({\theta }_e,{\varphi }_e)\right|}^2.\hfill \end{array}$$

(12)

Axial symmetry populates only $M_L=0$, favoring forward/backward (${\theta }_e=0^{\circ },{180}^{\circ }$) emission along the beam (z-axis), where $|Y_{L,M_L=0}(0^{\circ }){|}^2={|Y_{L,M_L=0}({180}^{\circ })|}^2=(2L+1)/(4\pi )$. When $i=f$, RE equates to resonant elastic electron scattering.

2.3. Dielectronic Recombination Rate ${\alpha }_{DR}(T)$

The DR rate ${\alpha }_{DR}$ at plasma temperature T is [56,67]:

$$\begin{array}{cc}\hfill {\alpha }_{DR}[i\to j\to f^{\prime }](T)& \equiv {\int }_0^{\infty }{\sigma }_{DR}[i\to j\to f^{\prime }](\mathrm{v})\mathrm{v}g(\mathrm{v},T)d\mathrm{v},\hfill \end{array}$$

(13)

with Maxwell–Boltzmann distribution [56]:

$$\begin{array}{cc}\hfill \mathrm{v}g(\mathrm{v},T)d\mathrm{v}& =\sqrt{{\displaystyle \frac{2m^3}{\pi {(k_{B}T)}^3}}}{\mathrm{v}}^{2}exp\left(-{\displaystyle \frac{m{\mathrm{v}}^2}{2k_{B}T}}\right)\mathrm{v}d\mathrm{v},\hfill \end{array}$$

(14)

where $k_B$ is the Boltzmann constant. Substituting Equation (7) for ${\sigma }_{DR}$, Equation (1) for ${\sigma }_{DC}$ and converting to energy $\epsilon$ yields:

$$\begin{array}{cc}\hfill {\alpha }_{DR}[i\to j\to f^{\prime }](T)& ={\displaystyle \frac{4\omega [j\to f^{\prime }]}{{(k_{B}T)}^{3/2}\sqrt{2\pi m}}}{\int }_0^{\infty }{\Omega }_{DC}[i\to j](\epsilon )L_{{\Gamma }_j}(\epsilon -E_a[j\to i])\epsilon exp\left(-{\displaystyle \frac{\epsilon }{k_{B}T}}\right)d\epsilon .\hfill \end{array}$$

(15)

Using the delta-function approximation Equation (3) simplifies to:

$$\begin{array}{cc}\hfill {\alpha }_{DR}[i\to j\to f^{\prime }](T)& ={\displaystyle \frac{4\omega [j\to f^{\prime }]}{{(k_{B}T)}^{3/2}\sqrt{2\pi m}}}{\Omega }_{DC}(\epsilon =E_a[j\to i])E_a[j\to i]exp\left(-{\displaystyle \frac{E_a[j\to i]}{k_{B}T}}\right).\hfill \end{array}$$

(16)

Using Equation (5) at $\epsilon =E_a$ then gives ${\alpha }_{DR}$ [56]:

$$\begin{array}{cc}\hfill {\alpha }_{DR}[i\to j\to f^{\prime }](T)& ={\displaystyle \frac{1.6564\times {10}^{-22}}{{(k_{B}T)}^{3/2}}}{\displaystyle \frac{(2L_j+1)(2S_j+1)}{(2L_i+1)(2S_i+1)}}{A}_a[j\to i]\omega [j\to f^{\prime }]exp\left(-{\displaystyle \frac{E_a[j\to i]}{k_{B}T}}\right)[{\mathrm{cm}}^3/s].\hfill \end{array}$$

(17)

Thus, DC cross sections and DR rates require only atomic structure calculations for $A_a$, $E_a$, and fluorescence yield $\omega$. Although DR’s importance in corona plasmas was recognized in the 1960s, cross sections relied on theory until the first cross-beam measurement in 1983 [68]. EBIS sources [69] and storage ring electron cooling enabled high-resolution merged-beam experiments from the late 1980s [60,70]. Early DC insights came indirectly from RTE in ion–atom collisions, linked via the the IA, treating bound target electrons as quasifree to extract DC/RE/DR cross sections.

3. The Impulse Approximation (IA)

3.1. Quasi-Free Electron Scattering

A bound electron enters the collision with its normalized momentum distribution $F(\mathbf{p})$, arising from its orbital momentum $\mathbf{p}=m\mathbf{v}$ around the target. In the ion’s rest frame, the target electron moves with velocity ${\mathbf{V}}_p$ toward the ion, yielding a net impinging velocity ${\mathbf{v}}^{\prime }={\mathbf{V}}_p+\mathbf{v}$. Primes denote quantities in the projectile rest frame. For fast collisions where $\eta \equiv V_p/\mathrm{v}\gg 1$, the impinging electron’s distribution $F^{\prime }({\mathbf{p}}^{\prime })$ centers around ${\mathbf{V}}_p$.

The impulse approximation (IA) assumes that free electron–ion impact cross sections relate to their bound (or quasifree) analogues via [55]:

$$\begin{array}{cc}\hfill {\sigma }_{quasifree}(V_p)& ={\int }_{-\infty }^{+\infty }d{\mathbf{p}}^{\prime }{\sigma }_{free}({ϵ}^{\prime })F^{\prime }({\mathbf{p}}^{\prime })\delta (E_f-E_i),\hfill \end{array}$$

(18)

where the delta function $\delta (E_f-E_i)$ enforces energy conservation between the total initial energy $E_i$ and final energy $E_f$ of the interacting target-electron–ion system [55] and depend on the process at hand. The impinging momentum ${\mathbf{p}}^{\prime }$ and energy ${ϵ}^{\prime }$ of the quasifree electron are:

$$\begin{array}{cc}\hfill {\mathbf{p}}^{\prime }& =m{\mathbf{V}}_p+\mathbf{p}\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {ϵ}^{\prime }& ={\displaystyle \frac{{{\mathbf{p}}^{\prime }}^2}{2m}}={\displaystyle \frac{1}{2}}m{V}_p^2+{\displaystyle \frac{p^2}{2m}}+{\mathbf{V}}_p·\mathbf{p}.\hfill \end{array}$$

(20)

Aligning the projectile velocity along the z-axis (${\mathbf{V}}_p=V_p\widehat{\mathbf{z}}$), Equation ( becomes:

$$\begin{array}{cc}\hfill {ϵ}^{\prime }& ={ϵ}^{\prime }(V_p,p,p_z)={\displaystyle \frac{1}{2}}m{V}_p^2+{\displaystyle \frac{p_x^2+p_y^2+p_z^2}{2m}}+V_p{p}_z\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill p^{\prime }& =p^{\prime }(V_p,p,p_z)=\sqrt{2m{ϵ}^{\prime }(V_p,p,p_z)}.\hfill \end{array}$$

(22)

Both ${\mathbf{p}}^{\prime }$ and ${ϵ}^{\prime }$ depend on the orbital momentum p (or its components) and projectile velocity $V_p$. The use of the free-electron cross section ${\sigma }_{free}$ in Equation (18) assumes the projectile states remain undistorted by the target nucleus or other target electrons.

The ion rest-frame momentum distribution $F^{\prime }({\mathbf{p}}^{\prime })$ combines the fixed momentum $m{\mathbf{V}}_p$ from relative projectile-target motion with the lab-frame orbital momentum $\mathbf{p}$ around the target, distributed as ${|\Phi (\mathbf{p})|}^2$. It centers on $m{\mathbf{V}}_p$ since the impinging electron averages this momentum toward the stationary ion:

$$\begin{array}{cc}\hfill F^{\prime }({\mathbf{p}}^{\prime })d{\mathbf{p}}^{\prime }& ={\left|\Phi (\mathbf{p})\right|}^{2}d\mathbf{p}\mathrm{with}\int {\left|\Phi (\mathbf{p})\right|}^{2}d\mathbf{p}=1.\hfill \end{array}$$

(23)

For short interactions, the target electron remains frozen, so

$$\begin{array}{c}\hfill \Phi (\mathbf{p})\approx {\Phi }_i(\mathbf{p}),\end{array}$$

(24)

where ${\Phi }_i(\mathbf{p})$ is the initial target momentum wave function. Substituting in Equation (18) yields:

$$\begin{array}{cc}\hfill {\sigma }_{quasifree}(V_p)& ={\int }_{-\infty }^{+\infty }d\mathbf{p}{\sigma }_{free}({ϵ}^{\prime }){\left|{\Phi }_i(\mathbf{p})\right|}^2\delta (E_f-E_i).\hfill \end{array}$$

(25)

Equation (25), with ${ϵ}^{\prime }$ from Equation (21), forms the IA’s core equation. It assumes the initial momentum distribution $|{\Phi }_i{(\mathbf{p})|}^2$ stays undisturbed and projectile resonance states experience negligible distortion by the approaching target. This formulation first appeared in 1970 for Compton scattering of photons off target electrons [71,72]. Kleber and Jakubassa [55] applied it in 1975 to radiative electron capture (REC) using the known radiative recombination (RR) cross section. In 1982, Tanis et al. [6] observed a broad peak in the cross section for electron capture with projectile excitation (${\mathrm{S}}^{13+}(1s^{2}2s)$ + Ar), attributed to RTE. Brandt [9] explained it via the IA, linking RTE to the DC cross section and predicting the resonance’s $V_p$ dependence and peak position.

The following section presents a careful exposition of the RTE IA treatment, building on the original by Brandt [9] and the further refinements by Lee et al [26]. A small refinement is pointed out that primarily affects low-$Z_p$ ions ($Z_p\lesssim 10$). In addition, an exact numerical treatment of the IA never considered previously is also shown to be possible further illuminating some of the approximate IA results.

3.2. IA RTE

3.2.1. Total IA RTE Cross Section

Brandt [9] extended the IA from radiative capture [55] to RTE by applying Equation (25) with the corresponding DC cross section:

$$\begin{array}{cc}\hfill {\sigma }_{RTE}[i\to j](V_p)& ={\int }_{-\infty }^{+\infty }d\mathbf{p}{\sigma }_{DC}[i\to j]({ϵ}^{\prime }){\left|{\Phi }_i(\mathbf{p})\right|}^2\delta (E_f-E_i).\hfill \end{array}$$

(26)

Here the total initial and final energies are:

$$\begin{array}{cc}\hfill E_i& =-{\epsilon }_i^P-{\epsilon }_i^T+{ϵ}^{\prime }\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill E_f& =-{\epsilon }_f^P-{\epsilon }_f^T.\hfill \end{array}$$

(28)

Here, ${ϵ}^{\prime }$ is the impinging target electron kinetic energy given by Equation (21), while ${\epsilon }_i^T,{\epsilon }_f^T$ denote initial and final target and ${\epsilon }_i^P,{\epsilon }_f^P$ projectile binding energies, respectively (all negative quantities). Defining $E_I\equiv {\epsilon }_i^T-{\epsilon }_f^T$ (target ionization energy) and $E_a={\epsilon }_i^P-{\epsilon }_f^P$ (Auger energy for $j\to i$ decay) yields:

$$\begin{array}{cc}\hfill E_f-E_i& =(E_a+E_I)-{ϵ}^{\prime }.\hfill \end{array}$$

(29)

At resonance, $E_f=E_i$, so the impinging resonance electron energy is:

$$\begin{array}{cc}\hfill {ϵ}_R^{\prime }& =E_a+E_I,\hfill \end{array}$$

(30)

with $E_I=15.5$ eV (H₂) or 24.59 eV (He). Thus Equation (26) simplifies to:

$$\begin{array}{cc}\hfill {\sigma }_{RTE}(V_p)& ={\int }_{-\infty }^{+\infty }d\mathbf{p}{\sigma }_{DC}({ϵ}^{\prime }){\left|{\Phi }_i(\mathbf{p})\right|}^2\delta ({ϵ}^{\prime }-{ϵ}_R^{\prime }),\hfill \end{array}$$

(31)

omitting $[i\to j]$ labels for brevity.

Quadratic IA Treatment

The three-dimensional integral over $d\mathbf{p}=d{p}_{x}d{p}_{y}d{p}_z$ reduces by assuming only $p_z$ contributes significantly [Equation (21)]:

$$\begin{array}{c}\hfill {ϵ}^{\prime }={ϵ}^{\prime }(V_p,p_z)\approx {\displaystyle \frac{1}{2}}m{V}_p^2+{\displaystyle \frac{p_z^2}{2m}}+V_p{p}_z(\mathrm{quadratic}\mathrm{IA}).\end{array}$$

(32)

This retains both linear and quadratic $p_z$ terms, improving on Brandt’s original linear approximation. The Compton profile then is:

$$\begin{array}{c}\hfill J(p_z)\equiv {\int }_{-\infty }^{\infty }d{p}_x{\int }_{-\infty }^{\infty }d{p}_y{\left|{\Phi }_i(\mathbf{p})\right|}^2,\end{array}$$

(33)

normalized such that ${\int }_{-\infty }^{\infty }J(p_z)d{p}_z=2$ for two-equivalent-electron targets (He, H₂). Equation (31) becomes:

$$\begin{array}{cc}\hfill {\sigma }_{RTE}(V_p)& ={\int }_{-\infty }^{\infty }d{p}_z{\sigma }_{DC}({ϵ}^{\prime }(V_p,p_z))J(p_z)\delta ({ϵ}^{\prime }(V_p,p_z)-{ϵ}_R^{\prime }).\hfill \end{array}$$

(34)

Changing variables in the delta function gives:

$$\begin{array}{cc}\hfill {\sigma }_{RTE}(V_p)& ={\int }_{-\infty }^{+\infty }d{p}_z{\sigma }_{DC}({ϵ}^{\prime }(V_p,p_z))J(p_z){\displaystyle \frac{\delta (p_z-p_z^R)}{{\left|\frac{d{ϵ}^{\prime }(V_p,p_z)}{d{p}_z}\right|}_{p_z=p_z^R}}},\hfill \end{array}$$

(35)

where $p_z$ at resonance, $p_z^R$, is a function of $V_p$ satisfying the resonance condition ${ϵ}^{\prime }(V_p,p_z^R)={ϵ}_R^{\prime }=E_a+E_I$:

$$\begin{array}{cc}\hfill p_z^R(V_p)& =-m{V}_p+\sqrt{2m(E_a+E_I)}(\mathrm{quadratic}\mathrm{IA}).\hfill \end{array}$$

(36)

The derivative evaluates to:

$$\begin{array}{c}\hfill {\left.{\displaystyle \frac{d{ϵ}^{\prime }(V_p,p_z)}{d{p}_z}}\right|}_{p_z=p_z^R}=\sqrt{{\displaystyle \frac{2(E_a+E_I)}{m}}}(\mathrm{quadratic}\mathrm{IA}).\end{array}$$

(37)

Thus:

$$\begin{array}{cc}\hfill {\sigma }_{RTE}(V_p)& ={\Omega }_{DC}({ϵ}^{\prime }(V_p,p_z^R)){\displaystyle \frac{J(p_z^R(V_p))}{\sqrt{\frac{2(E_a+E_I)}{m}}}}(\mathrm{quadratic}\mathrm{IA}),\hfill \end{array}$$

(38)

with ${\Omega }_{DC}[i\to j](\epsilon )$ from Equation (5) evaluated for $\epsilon ={ϵ}_R^{\prime }=E_a+E_I$:

$$\begin{array}{cc}\hfill {\Omega }_{DC}({ϵ}_R^{\prime })& ={\displaystyle \frac{{\pi }^2}{2}}{a}_0^2{\epsilon }_0{\displaystyle \frac{g_j}{g_i}}{\displaystyle \frac{\hslash A_a}{E_a+E_I}}={\Omega }_{DC}(\epsilon =E_a)\left({\displaystyle \frac{E_a}{E_a+E_I}}\right).\hfill \end{array}$$

(39)

Substituting in Equation (38) yields:

$$\begin{array}{cc}\hfill {\sigma }_{RTE}(V_p)& ={\Omega }_{DC}(E_a){\displaystyle \frac{J(p_z^R(V_p))}{\sqrt{\frac{2(E_a+E_I)}{m}}}}\left({\displaystyle \frac{E_a}{E_a+E_I}}\right)(\mathrm{revised}\mathrm{quadratic}\mathrm{IA}),\hfill \end{array}$$

(40)

where $p_z^R(V_p)$ is from Equation (36) and ${\Omega }_{DC}(E_a)$ in cm² eV from Equation (6).

This result introduces the new factor ${\displaystyle \frac{E_a}{E_a+E_I}}$ absent in prior treatments [1,11,66]. All older IA treatments evaluated DC at ${ϵ}^{\prime }=E_a$ rather than $E_a+E_I$ and is therefore called revised quadratic. For heavy ions ($Z_p>10$), $E_I\ll E_a$ justifies neglecting $E_I$ [9]. For lighter ions, however, this correction can be substantial. For example in He⁺ + He ($2p^{21}D$, $E_a=35.29$ eV) reduces ${\sigma }_{RTE}$ in Equation (40) by 0.589 or 41%. For ${\mathrm{C}}^{4+}$+He ($1s2p^{22}D$, $E_a=242.15$ eV) reduces by 0.908 or 10%. This correction improves agreement with experimental data (which were mostly smaller than the older IA ${\sigma }_{RTE}$ for $Z_p\le 9$) [1] without altering the $V_p$ dependence, governed by $J(p_z^R(V_p))$. The values of this correction are listed in Table 1, Table 2 and Table 3 together with ${\Omega }_{RE}(E_a)$.

The older quadratic treatment (without the new correction factor) is then given by:

$$\begin{array}{cc}\hfill {\sigma }_{RTE}(V_p)& ={\Omega }_{DC}(E_a){\displaystyle \frac{J(p_z^R(V_p))}{\sqrt{\frac{2(E_a+E_I)}{m}}}}(\mathrm{quadratic}\mathrm{IA}).\hfill \end{array}$$

(41)

Linear IA Treatment

Neglecting the $p_z^2/2m$ term in Equation (32) yields:

$$\begin{array}{cc}\hfill {ϵ}^{\prime }& \approx {\displaystyle \frac{1}{2}}m{V}_p^2+V_p{p}_z(\mathrm{linear}\mathrm{IA}),\hfill \end{array}$$

(42)

and therefore

$$\begin{array}{cc}\hfill p_z^R& ={\displaystyle \frac{E_a+E_I}{V_p}}-{\displaystyle \frac{1}{2}}m{V}_p(\mathrm{linear}\mathrm{IA}),\hfill \end{array}$$

(43)

with the derivative $d{ϵ}^{\prime }/d{p}_z$ now giving:

$$\begin{array}{cc}\hfill {\left.{\displaystyle \frac{d{ϵ}^{\prime }(V_p,p_z)}{d{p}_z}}\right|}_{p_z=p_z^R}& =V_p(\mathrm{linear}\mathrm{IA}),\hfill \end{array}$$

(44)

similarly leading to the slightly different cross section:

$$\begin{array}{cc}\hfill {\sigma }_{RTE}(V_p)& ={\Omega }_{DC}(E_a){\displaystyle \frac{J(p_z^R(V_p))}{V_p}}\left({\displaystyle \frac{E_a}{E_a+E_I}}\right)(\mathrm{revised}\mathrm{linear}\mathrm{IA}),\hfill \end{array}$$

(45)

with the older linear IA without the correction given as previously by:

$$\begin{array}{cc}\hfill {\sigma }_{RTE}(V_p)& ={\Omega }_{DC}(E_a){\displaystyle \frac{J(p_z^R(V_p))}{V_p}}(\mathrm{linear}\mathrm{IA}).\hfill \end{array}$$

(46)

The linear treatment was found to be inferior to the quadratic [11,26,39] and is only shown here for completeness. It was the one first derived by Brandt [9] who even ignored the binding energy $E_I$ completely (i.e., set $E_I=0$) as he was mostly interested in RTEX measurements of that time involving ions with $Z_p>10$ for which $E_a\gg E_I$.

In both quadratic and linear IA treatments ${\sigma }_{RTE}$ maximizes at $p_z^R=0$ (where $J(0)$ peaks), at identical $V_p^{\max }$ or $E_p^{\max }$ given by:

$$\begin{array}{cc}\hfill V_p^{\max }& =\sqrt{{\displaystyle \frac{2(E_a+E_I)}{m}}},\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill E_p^{\max }& ={\displaystyle \frac{1}{2}}{M}_p{\left(V_p^{\max }\right)}^2={\displaystyle \frac{M_p}{m}}(E_a+E_I),\hfill \end{array}$$

(48)

and with the same maximum RTE cross section, ${\sigma }_{RTE}^{\max }\equiv {\sigma }_{RTE}(V_p^{\max })$ given by:

$$\begin{array}{cc}\hfill {\sigma }_{RTE}^{\max }& ={\Omega }_{DC}(E_a)\left({\displaystyle \frac{E_a}{E_a+E_I}}\right){\displaystyle \frac{J(0)}{V_p^{\max }}}(\mathrm{revised}\mathrm{IA}),\hfill \end{array}$$

(49)

or in cm²:

$$\begin{array}{cc}\hfill {\sigma }_{RTE}^{\max }[{\mathrm{cm}}^2]& ={\displaystyle \frac{{\Omega }_{DC}(E_a)[{\mathrm{cm}}^2\mathrm{eV}]}{{\epsilon }_0}}\left({\displaystyle \frac{E_a}{E_a+E_I}}\right){\displaystyle \frac{J(0)[\mathrm{a}.\mathrm{u}.]}{\sqrt{2(E_a+E_I)[\mathrm{a}.\mathrm{u}.]}}}(\mathrm{revised}\mathrm{IA}),\hfill \end{array}$$

(50)

with $J(0)[a.u.]=1.5395$ (H₂), 1.0641 (He) [73].

Finally, at the RTEA maximum the ratio of projectile velocity to target velocity $\eta$ is given by:

$$\begin{array}{c}\hfill {\eta }^{\max }\equiv {\displaystyle \frac{V_p^{\max }}{v}}=\sqrt{{\displaystyle \frac{(E_a+E_I)}{K_I}}},\end{array}$$

(51)

where $K_I=m{\mathrm{v}}^2/2$ is the kinetic energy of the target electron with $K_I(H_2)=31.93$ eV and $K_I(He)=39.5$ eV for H₂ and He, respectively [74]. Both ${\eta }^{\max }$ and $E_p^{\max }$ as well as ${\sigma }_{RTEA}^{\max }={\sigma }_{RTE}^{\max }\xi$ are listed in Table 1, Table 2 and Table 3. It is reminded that the validity criterion of the IA is $\eta \gg 1$.

Table 1. Results for $2p^{21}D$ state production by RTEA in H-like [$X^{Z_p-1}(1s)$] ion collisions with He and H₂ targets (Refs. in last column). Theoretical RE collision strengths ${\Omega }_{RE}\equiv {\Omega }_{RE}(E_a)$ are computed using tabulated parameters $A_a$ (Auger rate), $E_a$ (Auger energy), and $\xi$ (Auger yield). Shown are zero-degree IA RTEA maximum SDCS $d{\sigma }_{RTEA}^{\max }(0^{\circ })/d{\Omega }_e^{\prime }$, projectile energy $E_p^{\max }$, and validity criterion ${\eta }^{\max }=V_p^{\max }/\mathrm{v}$. Experimental maximum SDCS $d{\sigma }_{\mathrm{Exp}}^{\max }(0^{\circ })/d{\Omega }_e^{\prime }$ yield r ratio and extracted ${\Omega }_{RE}^{\mathrm{Exp}}=r{\Omega }_{RE}$. Where SDCS unavailable, ${\Omega }_{RE}^{\mathrm{Exp}}$ appear in parentheses (from Refs.) and used to compute r in parentheses. Blank entries repeat values above; − indicates SDCS were not available. Boron results in Figure 5 (left); helium in Figure 6.

		Theory								Experiment
Collision	$Z_p$	$A_a$a	$E_a$a	$\xi$a	${\displaystyle \frac{E_a}{E_a+E_I}}$	${\eta }^{\max }$	$E_p^{\max }$	${\Omega }_{RE}$	${\displaystyle \frac{d{\sigma }_{RTEA}^{\max }}{d{\Omega }_e^{\prime }}}$	${\displaystyle \frac{d{\sigma }_{\mathrm{Exp}}^{\max }}{d{\Omega }_e^{\prime }}}$b	rc	$r{\Omega }_{RE}$d
System		($s^{-1}$)	(eV)				(MeV/u)	(cm2eV)	(cm2/sr)	(cm2/sr)		(cm2eV)
		$(\times {10}^{14})$						$(\times {10}^{-19})$	$(\times {10}^{-20})$	$(\times {10}^{-20})$		$(\times {10}^{-19})$
						Equation (51)	Equation (48)	Equations 11 and 6)	Equation (65)		Equation (66)	Equation (67)	Refs
	$X^{Z_p-1}(1s)+T\to X^{Z_p-2}(2p^{21}D)\to X^{Z_p-1}(1s)+e_A^{-}$
3He++He	2	1.09	35.33e	1.000	0.590	1.23	0.109	191.6	8.38	0.48	0.057	11	[75]
3He++H2					0.695	1.26	0.093		15.51	3.57	0.230	44.7	[52]
7Li2++He	3	1.60f	74.32g	0.999	0.751	1.58	0.180	133.4	5.79	-	-	-	[76]
11B4++H2	5	2.86	193.26h	0.996	0.926	2.56	0.381	91.17	4.86	5.30	1.10	99.4	[77]
								(90.41)		-	(0.845)	(77)i	[12]
								(84)		-	(0.879)	(80.1)	[78]
12C5+He	6	2.52	273.30h	0.994	0.917	2.75	0.543	56.73	1.73	1.80	1.04	59.9	[43]
12C5+H2					0.946	3.01	0.526		2.62	2.50	0.95	54.0	[43]
								(66.5i)		-	(1.16)	(66)i	[53]
14N6++H2	7	2.68	366.9j	0.991	0.960	3.46	0.697	44.75 (51.0)i	1.82	-	(1.16)	(52)i	[53]
16O7++H2	8	3.12	474.2j	0.986	0.968	3.91	0.893	40.14 (40.1)i	1.46	-	(1.05)	(42)i	[53]
19F8++H2	9	2.90	595.0j	0.977	0.975	4.37	1.113	29.46	0.965	0.88	0.907	26.7	[40]
								(32.31)		-	(1.12)	(33)	[12]
								(35.00)		-	(1.19)	(36)i	[53]
								(32.3)		-	(1.09)	(32.2)	[78]
24Mg11++H2	12	3.08	1038.61	0.927	0.985	5.74	1.922	17.00 (19.1)	0.428	-	(1.31)	(22.2)	[78]

^a $A_a$, $E_a$ and $\xi$ from various sources as indicated. $A_a$ are mostly from the MZ code based on Z-expansion method with relativistic corrections [79] except for helium [80] and boron [12]. For carbon the truncated-diagonalization method [81] gives the same results, ^b Absolute experimental error reported ~20-30% in most Refs, ^c Values of r are also shown in Figure 8 as a function of projectile atomic number Z_p with an overall uncertainty of ≤ 30%, ^d Values in parentheses as reported in the corresponding reference unless otherwise indicated, ^e From Ref. [82], ^f Complex rotation method [83], ^g Measured by Rodbro et al. [76], ^h Reported in Ref. [12], ⁱ Reported in Ref. [78], ^j Reported in Ref. [78], originally from Ref. [84], ^k Reported in Ref. [78] as T.W. Gorczyca (private communication).

Table 1. Results for $2p^{21}D$ state production by RTEA in H-like [$X^{Z_p-1}(1s)$] ion collisions with He and H₂ targets (Refs. in last column). Theoretical RE collision strengths ${\Omega }_{RE}\equiv {\Omega }_{RE}(E_a)$ are computed using tabulated parameters $A_a$ (Auger rate), $E_a$ (Auger energy), and $\xi$ (Auger yield). Shown are zero-degree IA RTEA maximum SDCS $d{\sigma }_{RTEA}^{\max }(0^{\circ })/d{\Omega }_e^{\prime }$, projectile energy $E_p^{\max }$, and validity criterion ${\eta }^{\max }=V_p^{\max }/\mathrm{v}$. Experimental maximum SDCS $d{\sigma }_{\mathrm{Exp}}^{\max }(0^{\circ })/d{\Omega }_e^{\prime }$ yield r ratio and extracted ${\Omega }_{RE}^{\mathrm{Exp}}=r{\Omega }_{RE}$. Where SDCS unavailable, ${\Omega }_{RE}^{\mathrm{Exp}}$ appear in parentheses (from Refs.) and used to compute r in parentheses. Blank entries repeat values above; − indicates SDCS were not available. Boron results in Figure 5 (left); helium in Figure 6.

		Theory								Experiment
Collision	$Z_p$	$A_a$a	$E_a$a	$\xi$a	${\displaystyle \frac{E_a}{E_a+E_I}}$	${\eta }^{\max }$	$E_p^{\max }$	${\Omega }_{RE}$	${\displaystyle \frac{d{\sigma }_{RTEA}^{\max }}{d{\Omega }_e^{\prime }}}$	${\displaystyle \frac{d{\sigma }_{\mathrm{Exp}}^{\max }}{d{\Omega }_e^{\prime }}}$b	rc	$r{\Omega }_{RE}$d
System		($s^{-1}$)	(eV)				(MeV/u)	(cm2eV)	(cm2/sr)	(cm2/sr)		(cm2eV)
		$(\times {10}^{14})$						$(\times {10}^{-19})$	$(\times {10}^{-20})$	$(\times {10}^{-20})$		$(\times {10}^{-19})$
						Equation (51)	Equation (48)	Equations 11 and 6)	Equation (65)		Equation (66)	Equation (67)	Refs
	$X^{Z_p-1}(1s)+T\to X^{Z_p-2}(2p^{21}D)\to X^{Z_p-1}(1s)+e_A^{-}$
3He++He	2	1.09	35.33e	1.000	0.590	1.23	0.109	191.6	8.38	0.48	0.057	11	[75]
3He++H2					0.695	1.26	0.093		15.51	3.57	0.230	44.7	[52]
7Li2++He	3	1.60f	74.32g	0.999	0.751	1.58	0.180	133.4	5.79	-	-	-	[76]
11B4++H2	5	2.86	193.26h	0.996	0.926	2.56	0.381	91.17	4.86	5.30	1.10	99.4	[77]
								(90.41)		-	(0.845)	(77)i	[12]
								(84)		-	(0.879)	(80.1)	[78]
12C5+He	6	2.52	273.30h	0.994	0.917	2.75	0.543	56.73	1.73	1.80	1.04	59.9	[43]
12C5+H2					0.946	3.01	0.526		2.62	2.50	0.95	54.0	[43]
								(66.5i)		-	(1.16)	(66)i	[53]
14N6++H2	7	2.68	366.9j	0.991	0.960	3.46	0.697	44.75 (51.0)i	1.82	-	(1.16)	(52)i	[53]
16O7++H2	8	3.12	474.2j	0.986	0.968	3.91	0.893	40.14 (40.1)i	1.46	-	(1.05)	(42)i	[53]
19F8++H2	9	2.90	595.0j	0.977	0.975	4.37	1.113	29.46	0.965	0.88	0.907	26.7	[40]
								(32.31)		-	(1.12)	(33)	[12]
								(35.00)		-	(1.19)	(36)i	[53]
								(32.3)		-	(1.09)	(32.2)	[78]
24Mg11++H2	12	3.08	1038.61	0.927	0.985	5.74	1.922	17.00 (19.1)	0.428	-	(1.31)	(22.2)	[78]

^a $A_a$, $E_a$ and $\xi$ from various sources as indicated. $A_a$ are mostly from the MZ code based on Z-expansion method with relativistic corrections [79] except for helium [80] and boron [12]. For carbon the truncated-diagonalization method [81] gives the same results, ^b Absolute experimental error reported ~20-30% in most Refs, ^c Values of r are also shown in Figure 8 as a function of projectile atomic number Z_p with an overall uncertainty of ≤ 30%, ^d Values in parentheses as reported in the corresponding reference unless otherwise indicated, ^e From Ref. [82], ^f Complex rotation method [83], ^g Measured by Rodbro et al. [76], ^h Reported in Ref. [12], ⁱ Reported in Ref. [78], ^j Reported in Ref. [78], originally from Ref. [84], ^k Reported in Ref. [78] as T.W. Gorczyca (private communication).

Table 2. As in Table 1 (including footnotes unless otherwise indicated), but for the production of the $1s2p^{22}D$ state in collisions of ground state He-like [$X^{(Z_p-2)+}(1s^2)$] ions. Results for carbon and fluorine are shown in Figure 4 and for boron in Figure 5 (right).

		Theory								Experiment
Collision	$Z_p$	$A_a$a	$E_a$a	$\xi$a	${\displaystyle \frac{E_a}{E_a+E_I}}$	${\eta }^{\max }$	$E_p^{\max }$	${\Omega }_{RE}$	${\displaystyle \frac{d{\sigma }_{RTEA}^{\max }}{d{\Omega }_e^{\prime }}}$	${\displaystyle \frac{d{\sigma }_{\mathrm{Exp}}^{\max }}{d{\Omega }_e^{\prime }}}$b	rc	$r{\Omega }_{RE}$d
System		($s^{-1}$)	(eV)				(MeV/u)	(cm2eV)	(cm2/sr)			(cm2eV)
		$(\times {10}^{14})$						$(\times {10}^{-19})$	$(\times {10}^{-20})$	$(\times {10}^{-20})$		$(\times {10}^{-19})$
						Equation (51)	Equation (48)	(Eqs. 11,6)	Equation (65)		Equation (66)	Equation (67)	Refs
	$X^{Z_p-2}(1s^2)+T\to X^{Z_p-3}(1s2p^{22}D)\to X^{Z_p-2}(1s^2)+e_A^{-}$
7Li++He	3	0.169e	55.67f	1.000	0.694	1.43	0.146	74.9	3.33	-	-	-	[85]
11B3++H2	5	0.633e	166.50f	1.000	0.915	2.39	0.332	94.1	5.30	6.32	1.19	89.1	[12]
12C4++He	6	0.756	242.15g	0.999	0.908	2.59	0.485	77.48	2.475	2.45	1.00	77.51	[38]
16O6++He	8	0.988	434.31	0.987	0.946	3.41	0.837	56.30	1.428	1.80 h	1.28	72.1	-
19F7++He	9	1.07	551.20	0.979	0.957	3.82	1.050	47.01	1.076	0.96	0.95	44.7	[86]
19F7++H2					0.973	4.21	1.033		1.595	1.34	0.84	39.5	[86]

^e B-spline method [87], ^f Measured by Rodbro et al [76], ^g Measured by Rodbro et al [76] and recalibrated by Bruch et al. [88], ^h to be published.

Table 2. As in Table 1 (including footnotes unless otherwise indicated), but for the production of the $1s2p^{22}D$ state in collisions of ground state He-like [$X^{(Z_p-2)+}(1s^2)$] ions. Results for carbon and fluorine are shown in Figure 4 and for boron in Figure 5 (right).

		Theory								Experiment
Collision	$Z_p$	$A_a$a	$E_a$a	$\xi$a	${\displaystyle \frac{E_a}{E_a+E_I}}$	${\eta }^{\max }$	$E_p^{\max }$	${\Omega }_{RE}$	${\displaystyle \frac{d{\sigma }_{RTEA}^{\max }}{d{\Omega }_e^{\prime }}}$	${\displaystyle \frac{d{\sigma }_{\mathrm{Exp}}^{\max }}{d{\Omega }_e^{\prime }}}$b	rc	$r{\Omega }_{RE}$d
System		($s^{-1}$)	(eV)				(MeV/u)	(cm2eV)	(cm2/sr)			(cm2eV)
		$(\times {10}^{14})$						$(\times {10}^{-19})$	$(\times {10}^{-20})$	$(\times {10}^{-20})$		$(\times {10}^{-19})$
						Equation (51)	Equation (48)	(Eqs. 11,6)	Equation (65)		Equation (66)	Equation (67)	Refs
	$X^{Z_p-2}(1s^2)+T\to X^{Z_p-3}(1s2p^{22}D)\to X^{Z_p-2}(1s^2)+e_A^{-}$
7Li++He	3	0.169e	55.67f	1.000	0.694	1.43	0.146	74.9	3.33	-	-	-	[85]
11B3++H2	5	0.633e	166.50f	1.000	0.915	2.39	0.332	94.1	5.30	6.32	1.19	89.1	[12]
12C4++He	6	0.756	242.15g	0.999	0.908	2.59	0.485	77.48	2.475	2.45	1.00	77.51	[38]
16O6++He	8	0.988	434.31	0.987	0.946	3.41	0.837	56.30	1.428	1.80 h	1.28	72.1	-
19F7++He	9	1.07	551.20	0.979	0.957	3.82	1.050	47.01	1.076	0.96	0.95	44.7	[86]
19F7++H2					0.973	4.21	1.033		1.595	1.34	0.84	39.5	[86]

^e B-spline method [87], ^f Measured by Rodbro et al [76], ^g Measured by Rodbro et al [76] and recalibrated by Bruch et al. [88], ^h to be published.

Table 3. As in Table 1 (including footnotes unless otherwise indicated), but for the production of the $1s2s2p^{23}D$ and ¹$D$ states in collisions of Li-like [$X^{(Z_p-3)+}(1s^{2}2s)$] ions.

		Theory								Experiment
Collision	$Z_p$	$A_a$a	$E_a$a	$\xi$a	${\displaystyle \frac{E_a}{E_a+E_I}}$	${\eta }^{\max }$	$E_p^{\max }$	${\Omega }_{RE}$	${\displaystyle \frac{d{\sigma }_{RTEA}^{\max }}{d{\Omega }_e^{\prime }}}$	${\displaystyle \frac{d{\sigma }_{\mathrm{Exp}}^{\max }}{d{\Omega }_e^{\prime }}}$b	rc	$r{\Omega }_{RE}$d
System		($s^{-1}$)	(eV)				(MeV/u)	(cm2eV)	(cm2/sr)	(cm2/sr)		(cm2eV)
		$(\times {10}^{14})$						$(\times {10}^{-19})$	$(\times {10}^{-20})$	$(\times {10}^{-20})$		$(\times {10}^{-19})$
						Equation (51)	Equation (48)	(Eqs. 11,6)	Equation (65)		Equation (66)	Equation (67)	Refs
	$X^{Z_p-3}(1s^{2}2s)+T\to X^{Z_p-4}(1s2s2p^{23}D)\to X^{Z_p-3}(1s^{2}2s)+e_A^{-}$
11B2++H2	5	0.490	173.60	0.891	0.918	2.43	0.345	46.6	2.58	2.35	0.91	42.4	[77]
16O5++He	8	1.05	448.68	0.879	0.948	3.46	0.861	38.2	0.955	0.260e	0.272	10.4	[10]
16O5++H2					0.967	3.81	0.845		1.422	0.875	0.615	23.5	[39]
19F6++He	9	1.11	567.68	0.878	0.958	3.87	1.080	31.8	0.718	0.34e	0.474	15.1	[11]
19F6++H2					0.973	4.27	1.063			0.52f	0.489	15.6	[11]
19F6++H2+H2										0.78g	0.733	23.3	[39]
	$X^{Z_p-3}(1s^{2}2s)+T\to X^{Z_p-4}(1s2s2p^{21}D)\to X^{Z_p-3}(1s^{2}2s)+e_A^{-}$
11B2++H2	5	0.387	176.83	0.398	0.919	2.45	0.351	5.39	0.297	0.60	2.0	10.9	[77]
16O5++He	8	0.89	455.35	0.460	0.949	3.49	0.870	5.55	0.138	0.056e	0.406	2.25	[10]
16O5++H2					0.967	3.84	0.854		0.205	0.20	0.976	5.42	[39]
19F6++He	9	1.10	575.55	0.469	0.959	3.90	1.095	5.53	0.124	0.088e	0.710	3.92	[11]
19F6++H2					0.974	4.30	1.078		0.184	0.20g	1.02	6.01	[39]

^a $A_a,E_a,\xi$ for the ³$D$ state from Ref. [89], while for the ¹$D$ state from Ref. [11] for oxygen and fluorine and from [90] for boron and carbon, ^b Absolute experimental error reported around 20-25% and 30% for Refs. [10,11], ^e After subtraction of NTEA contributions. Collisions with H2 targets have minimal NTEA not subtracted, ^f Absolute electron detector efficiency based on normalization to target Ne K-Auger cross sections from 3 MeV collisions with protons, ^g Revised absolute electron detector efficiency based on normalization to BEe DDCS by bare ions.

Table 3. As in Table 1 (including footnotes unless otherwise indicated), but for the production of the $1s2s2p^{23}D$ and ¹$D$ states in collisions of Li-like [$X^{(Z_p-3)+}(1s^{2}2s)$] ions.

		Theory								Experiment
Collision	$Z_p$	$A_a$a	$E_a$a	$\xi$a	${\displaystyle \frac{E_a}{E_a+E_I}}$	${\eta }^{\max }$	$E_p^{\max }$	${\Omega }_{RE}$	${\displaystyle \frac{d{\sigma }_{RTEA}^{\max }}{d{\Omega }_e^{\prime }}}$	${\displaystyle \frac{d{\sigma }_{\mathrm{Exp}}^{\max }}{d{\Omega }_e^{\prime }}}$b	rc	$r{\Omega }_{RE}$d
System		($s^{-1}$)	(eV)				(MeV/u)	(cm2eV)	(cm2/sr)	(cm2/sr)		(cm2eV)
		$(\times {10}^{14})$						$(\times {10}^{-19})$	$(\times {10}^{-20})$	$(\times {10}^{-20})$		$(\times {10}^{-19})$
						Equation (51)	Equation (48)	(Eqs. 11,6)	Equation (65)		Equation (66)	Equation (67)	Refs
	$X^{Z_p-3}(1s^{2}2s)+T\to X^{Z_p-4}(1s2s2p^{23}D)\to X^{Z_p-3}(1s^{2}2s)+e_A^{-}$
11B2++H2	5	0.490	173.60	0.891	0.918	2.43	0.345	46.6	2.58	2.35	0.91	42.4	[77]
16O5++He	8	1.05	448.68	0.879	0.948	3.46	0.861	38.2	0.955	0.260e	0.272	10.4	[10]
16O5++H2					0.967	3.81	0.845		1.422	0.875	0.615	23.5	[39]
19F6++He	9	1.11	567.68	0.878	0.958	3.87	1.080	31.8	0.718	0.34e	0.474	15.1	[11]
19F6++H2					0.973	4.27	1.063			0.52f	0.489	15.6	[11]
19F6++H2+H2										0.78g	0.733	23.3	[39]
	$X^{Z_p-3}(1s^{2}2s)+T\to X^{Z_p-4}(1s2s2p^{21}D)\to X^{Z_p-3}(1s^{2}2s)+e_A^{-}$
11B2++H2	5	0.387	176.83	0.398	0.919	2.45	0.351	5.39	0.297	0.60	2.0	10.9	[77]
16O5++He	8	0.89	455.35	0.460	0.949	3.49	0.870	5.55	0.138	0.056e	0.406	2.25	[10]
16O5++H2					0.967	3.84	0.854		0.205	0.20	0.976	5.42	[39]
19F6++He	9	1.10	575.55	0.469	0.959	3.90	1.095	5.53	0.124	0.088e	0.710	3.92	[11]
19F6++H2					0.974	4.30	1.078		0.184	0.20g	1.02	6.01	[39]

^a $A_a,E_a,\xi$ for the ³$D$ state from Ref. [89], while for the ¹$D$ state from Ref. [11] for oxygen and fluorine and from [90] for boron and carbon, ^b Absolute experimental error reported around 20-25% and 30% for Refs. [10,11], ^e After subtraction of NTEA contributions. Collisions with H2 targets have minimal NTEA not subtracted, ^f Absolute electron detector efficiency based on normalization to target Ne K-Auger cross sections from 3 MeV collisions with protons, ^g Revised absolute electron detector efficiency based on normalization to BEe DDCS by bare ions.

Exact IA Treatment

Equation (31) can also be evaluated numerically using explicit ${|\Phi (\mathbf{p})|}^2$ (e.g., hydrogenic) and ${\sigma }_{DC}$ from Equation (1, termed here exact IA. This does not seem to have been noticed previously and is presented here for the first time.

In spherical coordinates Equation (31) can be written as:

$$\begin{array}{cc}\hfill {\sigma }_{RTE}(V_p)& ={\int }_0^{\infty }{p}^{2}dp{\int }_0^{\pi }sin\theta d\theta {\int }_0^{2\pi }d\varphi {\sigma }_{DC}({ϵ}^{\prime }){|{\Phi }_i(\mathbf{p})|}^2\delta ({ϵ}^{\prime }-(E_a+E_I)),\hfill \end{array}$$

(52)

with exact ${ϵ}^{\prime }$ given by:

$$\begin{array}{cc}\hfill {ϵ}^{\prime }={ϵ}^{\prime }(V_p,z,p)& ={\displaystyle \frac{1}{2}}m{V}_p^2+{\displaystyle \frac{p^2}{2m}}+V_{p}zp(\mathrm{exact}\mathrm{IA}),\hfill \end{array}$$

(53)

where $z=cos\theta$. Setting $|{\Phi }_i{(\mathbf{p})|}^2={|{\Phi }_i(p)|}^2/(4\pi )$, the $\varphi$ integral yields:

$$\begin{array}{cc}\hfill {\sigma }_{RTE}(V_p)& ={\displaystyle \frac{1}{2}}{\int }_0^{\infty }{p}^{2}dp{\int }_{-1}^{1}dz{\sigma }_{DC}({ϵ}^{\prime }(V_p,z,p)){|{\Phi }_i(p)|}^2\delta ({ϵ}^{\prime }(V_p,z,p)-(E_a+E_I)).\hfill \end{array}$$

(54)

Using the Lorentzian form of ${\sigma }_{DC}$ [Equation (1)] we have:

$$\begin{array}{cc}\hfill {\sigma }_{RTE}(V_p)& ={\displaystyle \frac{1}{2}}{\Omega }_{DC}(E_a){\int }_0^{\infty }{p}^{2}dp{\int }_{-1}^{1}dz\left[{\displaystyle \frac{E_a}{{ϵ}^{\prime }(V_p,z,p)}}\right]\left[{\displaystyle \frac{{\Gamma }_j/(2\pi )}{{({ϵ}^{\prime }(V_p,z,p)-(E_a+E_I))}^2+{({\Gamma }_j/2)}^2}}\right]{|{\Phi }_i(p)|}^2(\mathrm{exact}\mathrm{IA}).\hfill \end{array}$$

(55)

For screened hydrogenic ground states [91] we have $|{\Phi }_i{(p)|}^2={|F_{10}(p)|}^2$ with:

$$\begin{array}{cc}\hfill |F_{10}{(p)|}^2& ={\displaystyle \frac{2^5}{\pi }}{\displaystyle \frac{{(Z_t^{\star }{p}_0)}^5}{{[p^2+{(Z_t^{\star }{p}_0)}^2]}^4}},\hfill \end{array}$$

(56)

and $Z_t^{\star }=1.5954$ (He), 1.10273 (H₂), matching nicely the measured He and H₂ Compton profiles [73] normally used with the IA, when multiplied by 2 to account for both electrons. The exact calculation is shown for comparison with the quadratic and revised quadratic in Figure 4, Figure 5 and Figure 6.

3.2.2. Single Differential Cross Sections (SDCS)

Most RTE measurements detect emitted X-rays or Auger electrons at angles $(\theta ,\varphi )$ relative to the ion beam direction, yielding SDCS. For Auger measurements, the RTEA rest frame SDCS for the $i\to j\to f$ transition in LS-coupling is [66] (assuming $S_i=S_f=0$):

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\sigma }_{RTEA}}{d{\Omega }_e^{\prime }}}({\theta }_e^{\prime },{\varphi }_e^{\prime })& =\xi (L_j{S}_j\to L_f{S}_f){\sigma }_{RTE}(L_j{S}_j){\left|Y_{L_j,M_{L_j=0}}({\theta }_e^{\prime },{\varphi }_e^{\prime })\right|}^2,\hfill \end{array}$$

(57)

where $\xi (L_j{S}_j\to L_f{S}_f)$ is the Auger yield—the probability of decay from state j to final state $L_f{S}_f$.

The IA RTEA SDCS then takes the final compact form:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\sigma }_{RTEA[i\to j\to f]}}{d{\Omega }_e^{\prime }}}(E_p,{\theta }_e^{\prime },{\varphi }_e^{\prime })& ={\displaystyle \frac{d{\sigma }_{RTEA}^{\max }}{d{\Omega }_e^{\prime }}}({\theta }_e^{\prime },{\varphi }_e^{\prime }){\displaystyle \frac{J(p_z^R(E_p))}{J(0)}},\hfill \end{array}$$

(58)

where

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\sigma }_{RTEA}^{\max }}{d{\Omega }_e^{\prime }}}({\theta }_e^{\prime },{\varphi }_e^{\prime })& \equiv {\sigma }_{RTEA}^{\max }{\left|Y_{L{M}_L=0}({\theta }_e^{\prime },{\varphi }_e^{\prime })\right|}^2=\xi {\sigma }_{RTE}^{\max }{\displaystyle \frac{(2L+1)}{4\pi }}{[P_L(cos{\theta }_e^{\prime })]}^2,\hfill \end{array}$$

(59)

where $P_L(x)$ is the Legendre polynomial of order L and $Y_{LM}$ the spherical harmonic. At $\theta =0^{\circ }$ observation in the laboratory the rest frame angle ${\theta }_e^{\prime }=0^{\circ }$ or ${180}^{\circ }$. Details of the kinematic transformations of angles and cross sections between laboratory and rest frame are given in Ref. [92]. Thus:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\sigma }_{RTEA[i\to j\to f]}}{d{\Omega }_e^{\prime }}}(E_p,\theta =0^{\circ })& ={\displaystyle \frac{d{\sigma }_{RTEA}^{\max }}{d{\Omega }_e^{\prime }}}(\theta =0^{\circ }){\displaystyle \frac{J(p_z^R(E_p))}{J(0)}},\hfill \end{array}$$

(60)

with

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\sigma }_{RTEA}^{\max }}{d{\Omega }_e^{\prime }}}(\theta =0^{\circ })& =\xi [j\to f]{\displaystyle \frac{{\sigma }_{RTE}^{\max }[i\to j]}{4\pi }}(2L_j+1).\hfill \end{array}$$

(61)

Here ${\sigma }_{RTE}^{\max }[i\to j]$ follows from Equation (49) using DC collision strength ${\Omega }_{DC}$, remaining constant across IA variants and independent of $E_p$. The RTEA SDCS are seen to be proportional to the Compton profile $J(p_z^R(E_p))$ and its $E_p$ dependence, with the constant of proportionality most sensitive to the product $A_a\xi$, i.e., the parameters with the largest uncertainty. This proportionality to the target Compton profile is amply confirmed by experiment, showing that fast collisions preserve the initial target electron momentum distribution, a key IA validation.

The projectile velocity in a.u. is:

$$\begin{array}{cc}\hfill V_p[\mathrm{a}.\mathrm{u}.]& =\sqrt{2(m[\mathrm{u}]/M_p[\mathrm{u}])·{10}^6{E}_p[\mathrm{MeV}]/{\epsilon }_0}=6.3498\sqrt{E_p[\mathrm{MeV}/\mathrm{u}]}.\hfill \end{array}$$

(62)

Quadratic $p_z^R$ [Equation (36)] becomes:

$$\begin{array}{cc}\hfill p_z^R[\mathrm{a}.\mathrm{u}.]& =-6.3498\sqrt{E_p[\mathrm{MeV}/\mathrm{u}]}+0.2711\sqrt{(E_a+E_I)[\mathrm{eV}]}(\mathrm{quadratic}\mathrm{IA}).\hfill \end{array}$$

(63)

Linear IA [Equation (43)]:

$$\begin{array}{cc}\hfill p_z^R[\mathrm{a}.\mathrm{u}.]& =5.7875\times {10}^{-3}{\displaystyle \frac{(E_a+E_I)[\mathrm{eV}]}{\sqrt{E_p[\mathrm{MeV}/\mathrm{u}]}}}-{\displaystyle \frac{1}{2}}·6.3498\sqrt{E_p[\mathrm{MeV}/\mathrm{u}]}(\mathrm{linear}\mathrm{IA}).\hfill \end{array}$$

(64)

The maximum zero-degree SDCS follows from Eqs. 61 and 50:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\sigma }_{RTEA}^{\max }}{d{\Omega }_e^{\prime }}}(0^{\circ })[{\mathrm{cm}}^2/\mathrm{sr}]& ={\displaystyle \frac{{\Omega }_{RE}[{\mathrm{cm}}^2\mathrm{eV}]}{{\epsilon }_0}}{\displaystyle \frac{(2L+1)}{4\pi }}\left({\displaystyle \frac{E_a}{E_a+E_I}}\right){\displaystyle \frac{J(0)[\mathrm{a}.\mathrm{u}.]}{\sqrt{2(E_a+E_I)[\mathrm{a}.\mathrm{u}.]}}}(\mathrm{revised}\mathrm{IA}),\hfill \end{array}$$

(65)

where ${\Omega }_{RE}$ uses atomic parameters from Eqs. 11, 6. Note that when also considering the $(2L+1)$ factor in ${\Omega }_{RE}$ (see Equation (6), ${\displaystyle \frac{d{\sigma }_{RTEA}^{\max }(0^{\circ })}{d{\Omega }_e^{\prime }}}\propto {(2L+1)}^2$, favoring high-L states. This is why D KLL-states ($L=2$) dominate RTE measurements and are the focus of this report as well as most of the measurements.

Matching the maximum theoretical RTEA SDCS to experiment at the RTE peak yields the overall scaling factor r defined by:

$$\begin{array}{cc}\hfill r& \equiv {\displaystyle \frac{\frac{d{\sigma }_{\mathrm{Exp}}^{\max }}{d{\Omega }_e^{\prime }}(0^{\circ })}{\frac{d{\sigma }_{RTEA}^{\max }}{d{\Omega }_e^{\prime }}(0^{\circ })}}\hfill \end{array}$$

(66)

which can then be used to extract an “experimental" ${\Omega }_{RE}^{\mathrm{Exp}}$ collision strength

$$\begin{array}{cc}\hfill {\Omega }_{RE}^{\mathrm{Exp}}& =r{\Omega }_{RE}.\hfill \end{array}$$

(67)

All available RTEA SDCS measurements were reviewed and ${\Omega }_{RE}^{\mathrm{Exp}}$ was extracted according to Equation (67). Theoretical ${\Omega }_{RE}$ and ${\displaystyle \frac{d{\sigma }_{RTEA}^{\max }(0^{\circ })}{d{\Omega }_e^{\prime }}}$ were computed from Equation (65) using best literature values of $E_a$, $A_a$, $\xi$. Ratios r and ${\Omega }_{RE}^{\mathrm{Exp}}$ appear in Table 1, Table 2 and Table 3 for D states from H-like, He-like, Li-like ions in collisions with He/H₂. Figure 8 plots r versus $Z_p$, quantifying theory-experiment agreement.

3.2.3. Double Differential Cross Sections (DDCS) and the Electron Scattering Model (ESM)

The ESM [25,26] has shown that the IA can also be applied directly to DDCS [12] with great success. According to the ESM the quasifree DDCS can be related to the free electron SDCS by:

$$\begin{array}{cc}\hfill {\left.{\displaystyle \frac{d^2\sigma ({\epsilon }^{\prime },{\theta }_e^{\prime },{\varphi }_e^{\prime })}{d{\Omega }_e^{\prime }d{\epsilon }^{\prime }}}\right|}_{\mathrm{quasifree}}& ={\left.{\displaystyle \frac{d\sigma ({\epsilon }^{\prime },{\theta }_e^{\prime },{\varphi }_e^{\prime })}{d{\Omega }_e^{\prime }}}\right|}_{\mathrm{free}}{\displaystyle \frac{J(p_z)}{\sqrt{\frac{2({\epsilon }^{\prime }+E_I)}{m}}}},\hfill \end{array}$$

(68)

where in the quadratic IA instead of Eqs. 32 and 36 we have:

$$\begin{array}{cc}\hfill {\epsilon }^{\prime }& \approx {\displaystyle \frac{1}{2}}m{V}_p^2+{\displaystyle \frac{p_z^2}{2m}}+V_p{p}_z\hfill \end{array}$$

(69)

and therefore

$$\begin{array}{cc}\hfill p_z& =-m{V}_p+\sqrt{2m{\epsilon }^{\prime }}.\hfill \end{array}$$

(70)

Substituting the RE SDCS [Equation (12)] for the free electron SDCS and integrating over ${\epsilon }^{\prime }$ recovers the RTEA SDCS [Equation (57)]:

$$\begin{array}{cc}\hfill {\left.{\displaystyle \frac{d\sigma ({\epsilon }^{\prime },{\theta }_e^{\prime },{\varphi }_e^{\prime })}{d{\Omega }_e^{\prime }}}\right|}_{\mathrm{quasifree}}& =\int {\left.{\displaystyle \frac{d^2\sigma ({\epsilon }^{\prime },{\theta }_e^{\prime },{\varphi }_e^{\prime })}{d{\Omega }_e^{\prime }d{\epsilon }^{\prime }}}\right|}_{\mathrm{quasifree}}d{\epsilon }^{\prime }\hfill \\ \hfill & ={\Omega }_{RE}(E_a){\left|Y_{L{M}_L=0}({\theta }_e^{\prime },{\varphi }_e^{\prime })\right|}^2\left({\displaystyle \frac{E_a}{E_a+E_I}}\right){\displaystyle \frac{J(p_z^R)}{\sqrt{\frac{2(E_a+E_I)}{m}}}}(\mathrm{revised}\mathrm{quadratic}\mathrm{IA}),\hfill \end{array}$$

(71)

with $p_z^R$ from Equation (36). Energy conservation shifts the free-electron $\delta (\epsilon -E_a)$ [Equation (4)] to quasifree $\delta ({\epsilon }^{\prime }-(E_a+E_I))$, introducing again the factor $E_a/E_a+E_I$.

Substituting in Equation (68) the non-resonant elastic ion-electron scattering SDCS yields the binary encounter electron (BEe) peak. Its excellent agreement with experiment for bare ions [26] has established it as the preferred standard for the in situ absolute calibration of electron spectrometer efficiencies [92]. Bhalla [25] unified the resonant (RTEA) and non-resonant elastic electron scattering (BEe) for H-like ions + He, thus also including BEe-RTEA interference. R-matrix-enhanced free-electron SDCS further improved results for low-$Z_p$ H-like [12,53,93], boron He-like [12], resonant inelastic [27,94], and even superelastic scattering [31]. An example of the IA ESM–R-matrix agreement at the DDCS level is shown in Figure 7 for 21 MeV ${\mathrm{F}}^{8+}$ + H₂ collisions shown here for the first time.

A detailed discussion of the ESM is beyond the scope of this review, as it primarily involves electron-ion scattering theory and computational methods (e.g., R-matrix). Interested readers are referred to the previously cited literature for further details.

Figure 2. Schematic geometries of (a) tandem parallel-plate analyzer (2PPA) spectrometer and (b) hemispherical deflector analyzer (HDA) spectrograph. Red trajectories show ${\theta }_e=0^{\circ }$ electrons; green indicates equipotentials. The 2PPA (a) suffers transmission losses from inter-stage electron retardation for enhanced resolution, while the HDA (b) uses an injection lens for efficient pre-retardation and a 2D PSD for optimal efficiency. Images generated with the SIMION ion optics software [95].

Figure 2. Schematic geometries of (a) tandem parallel-plate analyzer (2PPA) spectrometer and (b) hemispherical deflector analyzer (HDA) spectrograph. Red trajectories show ${\theta }_e=0^{\circ }$ electrons; green indicates equipotentials. The 2PPA (a) suffers transmission losses from inter-stage electron retardation for enhanced resolution, while the HDA (b) uses an injection lens for efficient pre-retardation and a 2D PSD for optimal efficiency. Images generated with the SIMION ion optics software [95].

Figure 3. [left] ZAPS setup at NCSR “Demokritos”: ${\theta }_e=0^{\circ }$ Auger electrons from projectile–H₂ collisions enter HDA via injection lens for energy analysis and 2D PSD detection. The ion beam traverses the analyzer exiting undisturbed from a small hole at the back plate. It is collected in a Faraday cup (FC) used for normalization. [right] Background-subtracted boron KLL spectrum (3.7 MeV ${\mathrm{B}}^{2+}(1s^{2}2s)+$H₂, projectile frame) near $E_p^{\max }=3.8$ MeV for D states, resolving excitation ($1s2s^{22}\mathrm{S}$, $1s2s2p^4\mathrm{P}$, ²${\mathrm{P}}_{\pm }$, $1s2p^{22}D$) and TE ($1s2s2p^{23}D$, ¹$D$) lines. DDCS peak integration yields state-selective SDCS.

Figure 3. [left] ZAPS setup at NCSR “Demokritos”: ${\theta }_e=0^{\circ }$ Auger electrons from projectile–H₂ collisions enter HDA via injection lens for energy analysis and 2D PSD detection. The ion beam traverses the analyzer exiting undisturbed from a small hole at the back plate. It is collected in a Faraday cup (FC) used for normalization. [right] Background-subtracted boron KLL spectrum (3.7 MeV ${\mathrm{B}}^{2+}(1s^{2}2s)+$H₂, projectile frame) near $E_p^{\max }=3.8$ MeV for D states, resolving excitation ($1s2s^{22}\mathrm{S}$, $1s2s2p^4\mathrm{P}$, ²${\mathrm{P}}_{\pm }$, $1s2p^{22}D$) and TE ($1s2s2p^{23}D$, ¹$D$) lines. DDCS peak integration yields state-selective SDCS.

4. Zero-Degree Auger Projectile Spectroscopy (ZAPS)

ZAPS detects electrons emitted from fast ionic projectiles at ${\theta }_e=0^{\circ }$ relative to the beam direction, where optimal kinematic conditions maximize energy resolution. This enables LS-resolved (state-selective) Auger spectra, providing a powerful probe of fast ion–atom collision dynamics and stringent tests of theory. ZAPS favors low-$Z_p$ projectiles ($Z_p<30$) due to high Auger yields ($>90$% for $Z_p<10$), surpassing X-ray fluorescence. The technique, introduced in the early 1980s [96], is reviewed in Refs. [92,97].

Early ZAPS used two 45° parallel-plate analyzers (2PPA) in series (tandem) [Figure 2(a)], but since the 2000s, a single-stage hemispherical deflector analyzer (HDA) with injection lens and 2D position sensitive detector (2D PSD) has been used in our group [Figure 2(b)]. Such an HDA spectrograph can record a $\sim 20\%$ energy window simultaneously, avoiding 2PPA voltage scanning, thus boosting detection efficiency by over two orders of magnitude [98].

Our ZAPS setup at the NCSR “Demokritos” 5.5 MV Tandem accelerator [99] appears in Figure 3. The ions in the beam interact in the gas cell containing the target atoms, producing doubly-excited states that Auger-decay. The injection lens pre-retards/focuses the forward emitted electrons into the HDA for energy analysis, with 2D PSD imaging along the dispersion axis [100,101]. Spectral projection yields Auger lines, as shown in Figure 3.

We note that ZAPS has predominantly been used to measure KLL Auger lines since these can be well-resolved so as to provide state-selective information. First-row ions have mostly been used in RTEA investigations since the Auger lines in the laboratory frame increase fast with $E_p^{\max }$ eventually going beyond the range of electrostatic analyzers. Even for Mg (see Table 1) with a $2p^{21}D$ Auger energy $E_a=1038.6$ eV, the corresponding laboratory energy for a forward emitted electron at $E_p^{\max }=1.922$ MeV/u is about $4E_a$ or close to 4100 eV, requiring a special high voltage spectrometer [78]. For higher energies a large two-stage magnetic spectrograph was proposed for use at the new experimental storage ring (NESR) at GSI [102], but so far not implemented. Such spectrometers have been used successfully in the past in nuclear physics [103].

In the next section, some typical Auger RTEA cross section measurements using both the older 2PPA spectrometer and the HDA spectrograph are presented and results are compared to the IA RTEA predictions.

Figure 4. Zero-degree Auger SDCS for the production of the $(1s2p^{22}D)$ state in $C^{4+}$ (left - adapted from Ref. [38]) and $F^{7+}$ (right - adapted from Ref. [42]) ion collisions with He. Squares: measurements corrected for metastable fraction. Blue lines: revised quadratic IA RTEA (this work - see Equation (60)) with the binding energy correction factor $E_a/(E_a+E_I)$ in Equation (49. Black lines: older quadratic IA RTEA (see Equation (18) Ref. [1]) same as Equations (60) and (49), but without this correction factor. Red line: 3eAOCC calculations [38]. $E_p^{\max }$ is the IA $E_p$ prediction at which the RTEA SDCS peaks. As long as ${\eta }^{\max }\gg 1$, the energy range of the RTEA peak can be expected to be well described by the IA. Values of ${\eta }^{\max }$ and $E_p^{\max }$ are listed in the tables. We note the improved agreement of the revised IA. The exact IA (green line) is seen to be in even better agreement. Similarly for the fluorine RTEA.

Figure 4. Zero-degree Auger SDCS for the production of the $(1s2p^{22}D)$ state in $C^{4+}$ (left - adapted from Ref. [38]) and $F^{7+}$ (right - adapted from Ref. [42]) ion collisions with He. Squares: measurements corrected for metastable fraction. Blue lines: revised quadratic IA RTEA (this work - see Equation (60)) with the binding energy correction factor $E_a/(E_a+E_I)$ in Equation (49. Black lines: older quadratic IA RTEA (see Equation (18) Ref. [1]) same as Equations (60) and (49), but without this correction factor. Red line: 3eAOCC calculations [38]. $E_p^{\max }$ is the IA $E_p$ prediction at which the RTEA SDCS peaks. As long as ${\eta }^{\max }\gg 1$, the energy range of the RTEA peak can be expected to be well described by the IA. Values of ${\eta }^{\max }$ and $E_p^{\max }$ are listed in the tables. We note the improved agreement of the revised IA. The exact IA (green line) is seen to be in even better agreement. Similarly for the fluorine RTEA.

Figure 5. Same as Figure 4, but for the production of ${\mathrm{B}}^{3+}(2p^{21}D)$ state in ${\mathrm{B}}^{4+}(1s)+$ H₂ collisions (left) and the production of ${\mathrm{B}}^{2+}(1s2p^{22}D)$ state in ${\mathrm{B}}^{3+}(1s^2)+$ H₂ collisions (right). Black squares - measurements: Filled (this work), Open (older work [12]). Again the exact IA (green line) is seen to give a slightly better overall agreement.

Figure 5. Same as Figure 4, but for the production of ${\mathrm{B}}^{3+}(2p^{21}D)$ state in ${\mathrm{B}}^{4+}(1s)+$ H₂ collisions (left) and the production of ${\mathrm{B}}^{2+}(1s2p^{22}D)$ state in ${\mathrm{B}}^{3+}(1s^2)+$ H₂ collisions (right). Black squares - measurements: Filled (this work), Open (older work [12]). Again the exact IA (green line) is seen to give a slightly better overall agreement.

5. Results and Discussion

Figure 4 and Figure 5 demonstrate the agreement between the three zero-degree IA RTEA SDCS calculations and experimental data for $1s2p^{22}D$ and $2p^{21}D$ state production in He-like and H-like C, F, and B ions colliding with He and H₂. Boron results are presented here for the first time. At the lowest collision energies the increase of the experimental SDCS is due to NTE, not handled by the IA.

He-like ion beams contain a mixture of ground ($1s^2$) and metastable ($1s2s$) states, complicating absolute SDCS determinations. However, we recently developed an in situ technique exploiting gas stripping at the tandem accelerator terminal, which preferentially produces ground-state ions [104,105] and allows for the fractional determination of the mixture [106]. In the reported energy range, $1s2p^{22}D$ production occurs almost exclusively from ground-state ions. Experimental SDCS were thus corrected by dividing by the ground-state fraction $f_g$.

All three IA calculations shown in the figures are multiplied by the overall scaling factor r which is computed by normalizing the revised quadratic IA (blue line) to the peak of the experimentally measured SDCS. r is given in parentheses in the figure legends and also listed in the tables. The exact IA (green) exhibits a marginally narrower profile in better overall agreement with the data. It is also shifted slightly to lower collision energies relative to $E_p^{\max }$ the quadratic IA prediction, reflecting the neglected $p_x^2$, $p_y^2$ terms in Equation (21).

The revised quadratic IA SDCS (blue) are systematically smaller than the original quadratic treatment (black) due to the $E_a/(E_a+E_I)$ target binding energy correction factor (also present in the exact IA), yielding improved absolute agreement with experiment—most pronounced for lower-$Z_p$ ions where $E_a$ is smaller. This effect is particularly evident in Figure 6 for He⁺+He/H₂, though here the validity of the IA is clearly questionable ($V_p\sim v$, ${\eta }^{\max }=1.23$) as evidenced by the much smaller r values and the disagreement in the RTEA profiles. The ${\eta }^{\max }$ value (shown per figure) quantifies IA applicability, requiring ${\eta }^{\max }\gg 1$—clearly violated for He⁺+He collisions in the displayed energy range where the IA nevertheless is seen to predict the RTE peak maximum. Improved agreement with experiment is also due to the use of more accurate Auger rates. For example in Figure 4 for carbon we used the Auger rate $A_a=0.756\times {10}^{14}$ s⁻¹ from the 2017 MZ code based on Z-expansion method with relativistic corrections [79] (see Table 2) rather than the one used previously of $A_a=0.932\times {10}^{14}$ s⁻¹ [38] taken from an 1986 multi-configuration Dirac-Fock calculation [107].

Figure 6. Same as Figure 4, but for the production of He$(2p^{21}D)$ state in ${\mathrm{He}}^{+}(1s)$ collisions with H₂ (right) and He (left). Large scaling factors of $r=0.057$ and $r=0.23$ are needed to bring the IA results closer to the data (see Table 1). The predicted $E_p^{\max }$ though is seen to be close to the observed experimental maxima. The exact IA is also shown and is seen to be slightly shifted towards lower energies bringing it into better agreement with the H₂, but not the He data. In any case, the IA is not expected to hold since ${\eta }^{\max }\sim 1$ and not much larger than 1. Experimental results for He from Ref. [16] and for H₂ targets from Ref. [52].

Figure 6. Same as Figure 4, but for the production of He$(2p^{21}D)$ state in ${\mathrm{He}}^{+}(1s)$ collisions with H₂ (right) and He (left). Large scaling factors of $r=0.057$ and $r=0.23$ are needed to bring the IA results closer to the data (see Table 1). The predicted $E_p^{\max }$ though is seen to be close to the observed experimental maxima. The exact IA is also shown and is seen to be slightly shifted towards lower energies bringing it into better agreement with the H₂, but not the He data. In any case, the IA is not expected to hold since ${\eta }^{\max }\sim 1$ and not much larger than 1. Experimental results for He from Ref. [16] and for H₂ targets from Ref. [52].

The scaling factor r for each collision system is reported in the tables and plotted in Figure 8, with up to $\sim 30\%$ absolute error encompassing the $r=1$ (grey zone) across most $Z_p$—unlike the systematic disagreement with decreasing $Z_p$ reported previously (see Figure 8 in [1]). This clear improvement stems primarily from more accurate Auger rates $A_a$ available since the early 1990s and used in the calculation of ${\Omega }_{RE}$ (see tables). For $Z_p=2$ (He⁺), the large discrepancy $r=0.2$ for collisions with H₂ reflects the IA breakdown ($V_p\sim v$ and not $V_p\gg v$), yet astonishingly the scaled IA RTEA SDCS profiles still closely match the data [Figure 6 (right)]. Remarkably, a simple overall r-scaling seems to suffice for most systems (including boron H-/He-like ions), suggesting robust target Compton profile integrity during collisions ($\Phi (p)\approx {\Phi }_i(p)$, Equation (24) as already pointed out, a key feature of the IA. Any disagreement trends therefore likely reflect increasing projectile disturbance by the target as their electron binding weakens. H-/He-like ions have the tightest bound K-shell electrons and show best agreement; Li-like systems – with the more loosely bound $2s$ spectator electron – deviate more [10]. This is particularly evident for the $1s2s2p^{21}D$ state where $\xi$ is also uniquely much less than 1 (see Table 3), potentially signaling less accurate multi-electron $A_a$ and $\xi$ values. However, this conjecture seems to be in conflict with the concept of “ion surgery" for fast light targets in collision with multi-electron ions [108]. Accordingly, low $Z_t$ targets (e.g., He/H₂) act as a “needle", selectively ionizing the projectile inner shell without substantially disturbing the outer shells which are thus preserved as evidenced by the limited number of charge states produced in the collision [75,97] in contrast to collisions with heavier targets. Clearly, RTEA investigations of Be- and B-like ions with even more spectator electrons might be able to shed more light on this. In addition, the $1s2s2p^{23,1}D$ states are more complicated since they can Auger decay either to the $1s^{2}2s$ or the $1s^{2}2p$ final states. Clearly, these complication are absent for the $2p^{21}D$ and $1s2p^{22}D$ states and might also be responsible for their borderline behavior.

Thus, further investigation on how and why the IA breaks down would be of interest. In particular, RTEA measurements of Li⁺ and Li²⁺+He ions, never undertaken to date, should lie near the limit of the IA validity (${\eta }^{\max }\sim 1.4$) and would therefore also be interesting. In Table 1 and Table 2 we have listed expected values of lithium ${\Omega }_{RE}$ and $d{\sigma }_{RTEA}^{\max }(0^{\circ })/d{\Omega }_e^{\prime }$ for future reference. Overall, the systematic analysis reported here for the D states indicates that the IA seems to be valid even when $V_p$ is only a bit larger than $\mathrm{v}$, i.e., for $\eta \gtrsim 2.3$ as shown in Figure 8 (right).

Finally, in Figure 4 for carbon (left) the first rigorous ion-atom collision calculations of RTE is shown. This treatment focused on the production of the ${\mathrm{C}}^{3+}(1s2p^{22}D)$ state via the process

$$\begin{array}{cc}\hfill C^{4+}(1s^2)+\mathrm{He}& \to C^{3+}(1s2p^{22}D)+{\mathrm{He}}^{+}(\mathrm{transfer}\mathrm{excitation})\hfill \end{array}$$

(72)

$$\begin{array}{cc}\hfill & |\to C^{4+}(1s^2)+e_A^{-}({\theta }_e=0^{\circ }),(0^{\circ }\mathrm{Auger}\mathrm{stabilization})\hfill \end{array}$$

(73)

with the Auger SDCS determined via ZAPS [92] by detecting the ²$D$ Auger electron at ${\theta }_e=0^{\circ }$ relative to the ion beam direction. This full configuration interaction, semiclassical close-coupling approach [54] considers the dynamics of three active electrons (known as 3eAOCC) - two on the projectile and one on the target - with the second He target electron considered to be frozen. This approach allows for a coupled and coherent treatment of all processes such as target and projectile excitation, ionization, single electron capture, as well as all TE processes (RTE, NTE etc.) and therefore goes well beyond the methods developed in the past. Two peaks are observed in the computed SDCS (red line). The high-energy peak is seen to be in very good agreement with experiment at/above resonance (0.5 MeV/u), confirming RTE dominance via two-center $(e-e)$ interactions at large impact parameters b [38]. The low-energy peak (outside the range of the accelerator), remarkably, is found to arise from a one-step$(e-n)$ interaction (head-on collisions, small b) and not the conventional two-step NTE process. To this effect, it was named non-correlated transfer excitation (NCTE) [38]. This surprising discovery shows that even after almost 45 years since the introduction of the RTE and NTE mechanisms to explain transfer excitation, the presence of a new one-step mechanism was revealed by this fully quantum mechanical treatment, providing further insights into bielectronic processes in many-body quantum systems. We are in the process of further testing the accuracy of the 3eAOCC calculations for other low-$Z_p$ isoelectronic systems. A 4eAOCC treatment is also underway to test the validity of the independent electron approximation used for the He target.

Figure 7. Projectile rest frame electron DDCS for 21.32 MeV F⁸⁺ collisions with H₂ in the electron energy range of the F${\mathrm{F}}^{7+}(2p^{21}D)$ resonance (adapted from Ref. [12]). Excellent agreement with ESM-R-matrix calculations is seen. Upon integrating the areas under the $2p^{21}D$ peak the SDCS were obtained for both experiment and theory from which the RE collision strengths, ${\Omega }_{RE}$, were computed when this method was used. Values of extracted ${\Omega }_{RE}$ are listed in Table 1 and compared to our present results.

Figure 7. Projectile rest frame electron DDCS for 21.32 MeV F⁸⁺ collisions with H₂ in the electron energy range of the F${\mathrm{F}}^{7+}(2p^{21}D)$ resonance (adapted from Ref. [12]). Excellent agreement with ESM-R-matrix calculations is seen. Upon integrating the areas under the $2p^{21}D$ peak the SDCS were obtained for both experiment and theory from which the RE collision strengths, ${\Omega }_{RE}$, were computed when this method was used. Values of extracted ${\Omega }_{RE}$ are listed in Table 1 and compared to our present results.

Figure 8. (Left) Ratio r (from Table 1, Table 2 and Table 3) as a function of projectile atomic number $Z_p$ for the projectile ion D states indicated. Except for He⁺ ($Z_p=2$) where the IA breaks down, and a few borderline Li-like ion cases, agreement is seen to be excellent within the up to 30% expected uncertainty (grey zone). (Right) Plot of $\eta$, the ratio of the projectile velocity $V_p$ over the target electron velocity v versus collision energy $E_p$ in the range covering all the results listed in the three tables. The IA validity criterion requires $\eta \gg 1$, hardly the case here, where clear breakdown is only observed for He⁺ collisions for $\eta \lesssim 2.3$ (grey area).

Figure 8. (Left) Ratio r (from Table 1, Table 2 and Table 3) as a function of projectile atomic number $Z_p$ for the projectile ion D states indicated. Except for He⁺ ($Z_p=2$) where the IA breaks down, and a few borderline Li-like ion cases, agreement is seen to be excellent within the up to 30% expected uncertainty (grey zone). (Right) Plot of $\eta$, the ratio of the projectile velocity $V_p$ over the target electron velocity v versus collision energy $E_p$ in the range covering all the results listed in the three tables. The IA validity criterion requires $\eta \gg 1$, hardly the case here, where clear breakdown is only observed for He⁺ collisions for $\eta \lesssim 2.3$ (grey area).

6. Summary and Conclusions

We have reviewed the progress of resonant transfer excitation followed by Auger stabilization (RTEA) investigations since the comprehensive review of 1992 [1]. We continue to focus on state-resolved single differential cross section (SDCS) measurements of the most strongly populated KLL D states obtained through zero-degree Auger projectile spectroscopy (ZAPS).

These state-selective SDCS measurements enable simple, direct comparisons with the impulse approximation (IA) predictions while providing its most stringent tests. The IA predictions rely primarily upon well-established atomic structure parameters available in the literature. Prior to 1992, 14 such measurements were reported—some employing disparate absolute calibration methods and treatment of the Auger angular distributions. The 1992 review revealed a systematic discrepancy where IA predictions were consistently larger than the experimental SDCS ($r<1$) with r dropping with decreasing $Z_p$, casting some doubts on the impulse approximation’s validity, particularly for low-$Z_p$ ions.

Since that time, an additional 16 RTEA measurements have been reported, including several new collision systems presented here for the first time. These later experiments benefit from standardized absolute detection efficiency calibration using the IA binary encounter electron peak (nonresonant elastic scattering). For He-like ion beams, which contain mixtures of ground and metastable states, a more accurate in situ determination of the ground-state fraction has also been implemented. Moreover, the original two-stage parallel-plate spectrometer has been superseded by the single-stage hemispherical deflector spectrograph that offers two orders of magnitude greater efficiency. Taken together, these experimental advances have yielded significantly more reliable and precise experimental SDCS.

We have revisited the fundamental connection between the electron-ion process of dielectronic capture (DC) and its ion-atom counterpart, RTE, provided by the IA. Our systematic comparison of RTEA SDCS—now computed using modern, more accurate Auger rates while uniformly accounting for the highly anisotropic Auger emission at zero-degree observation and novel target binding energy corrections—demonstrates excellent agreement with practically all available measurements spanning both pre- and post-1992 eras. In addition, we have also presented a new exact IA treatment which is shown to be in even better agreement than the revised quadratic IA results. Most important, no systematic discrepancy remains. The IA’s validity is now firmly established down to boron for H-like and He-like ions, with He⁺ representing the sole clear exception where IA breakdown occurs as expected. Remarkably, the IA is found to be valid down to $\eta \sim 2.3$, clearly much lower than what would be expected by the generally assumed IA validity criterion of $\eta \gg 1$. However, some Li-like ion cases seem to be on the borderline for reasons not clearly understood. Overall, these results confirm that state-selective RTEA measurements in combination with the IA can serve as a uniquely reliable source of Auger rates and yields, essential for computing RE collision strengths used in plasma modeling.

Over the last four decades, ZAPS has delivered the most accurate state-selective SDCS measurements for low-$Z_p$ ion-atom collision processes leading the way in the investigations of RTE. Very recently, the first nonperturbative 3eAOCC transfer excitation calculations have emerged. Excellent agreement with carbon $1s2p^{22}D$ ZAPS data around the RTE peak was found, while further revealing an novel mechanism at lower energies quite different from the conventional NTE. Ongoing experimental/theoretical isoelectronic studies promise new insights into many-body quantum dynamics under intense, ultra-fast perturbations. In addition, RTEA studies of Li⁺ and Li²⁺ collisions with He/H₂ would provide valuable tests of IA validity. Moreover, investigating KLL RTEA for Be-like and B-like ions would further examine IA applicability to projectiles with multiple spectator electrons possibly shedding more light on the role of spectator electrons during the collision as already seen for Li-like ions. Finally, Coster-Kronig (CK) transitions in much heavier ions could also be explored using electrostatic analyzers. For example, the $L_1{L}_3{M}_5$ CK transition yields electron energies of about 340 eV for lead and 1039 eV for uranium with extremely high Auger rates, have never been explored and could well be of RTE interest.