Single Ionization with Dressed Projectiles: An Improved Theory for Both Long- and Short-Range Interactions

Abstract

In this work, we present a theoretical model to investigate electron emission in collisions between dressed ions with He atoms and H₂ molecules. The projectile potential is described as the sum of a long- and short-range terms. The last term includes a screening function that has its maximum at short distances. The present model is based on the Continuum Distorted Wave Eikonal Initial State (CDW-EIS) theory, but the Eikonal approximation is only made within the long-range transition amplitude. This now leads to physically correct predictions, whenever dressed projectiles are involved, in the binary-encounter peak. Indeed, double-differential cross-sections spectra is calculated and compared with existing experimental data, finding that this model is capable of reproducing some well-known phenomena depending on the projectile charge state. Namely, the dependence of the binary-encounter peak magnitude with the projectile charge state.

1. Introduction

Within the research on fundamental atomic collisions and their applications, Continuum Distorted Wave (CDW) theories have proven to be a successful approach whenever the perturbative assumption is fulfilled, generally in the high-impact-energy regime. However, as the velocity of the incoming projectile ion decreases, becoming similar to the electron orbital momentum, CDW results can significantly overestimate the experiments. In fact, Crothers and McCann [1] found that the reason for such overestimation in ionization studies is that the CDW wavefunction ansatz is not properly normalized. They also stated that this problem can be avoided by either renormalizing the initial state wave function or applying the so-called Eikonal Initial State (EIS) approximation. The latter option leads to the CDW-EIS theory, and it not only improves the intermediate-energy calculations but is also computationally less demanding than renormalising CDW.

So, being a fast and reliable theory over a wide energy range makes CDW-EIS attractive for evaluating many of the cross-sections of electronic processes involved, e.g., in the simulation of stopping power, in the prediction of the charge-state distribution in an ion beam, or in the evaluation of radiation damage. Thus, further development was concentrated on maintaining analytical expression of the transition amplitudes in its generalization to multielectronic and molecular targets, together with the study of dressed projectiles [2]. These developments, proved to make significant improvements in some implementations on radiobiology [3] and charge-state distribution [4], for example.

Part of this success relies on the fact that most of the electron yield consists of low-energy electrons known as the soft-collision emission. These electrons and those at an intermediate emission energy, which are affected by two-centre effects, are well represented in the double-differential cross-sections (DDCSs) by the EIS approximation; thus, good single differential cross-sections (SDCSs) and total cross-sections (TCSs) are generally obtained with this theory. Also the contribution of the electron loss of the projectile is adequately described by CDW-EIS models.

However, the most energetic electron emission, the so-called binary-encounter emission (BEE) peak, was found to be incorrectly described in magnitude by CDW-EIS in the case of dressed light projectiles [2]. The predicted magnitude of the BEE peak with CDW-EIS decreased with an increasing number of electrons in the projectile, which is the opposite of the well-documented BEE peak enhancement with an increasing number of electrons in the projectile ion [5,6,7,8,9,10,11,12,13].

The influence of the BEE on the total cross-sections is almost insignificant, but it can lead to a large damage on the surrounding media, due to the high energy of the emitted electrons. This is the case in both biological media [14,15,16] and solids [17,18]. In fact, it was proved that up to 15% increment in stopping power can be accounted for by an adequate description of the BEE peak enhancement in dressed projectiles simulations [18].

Theoretically, the ionic charge dependence of the BEE peak magnitude was first explained by Olson, Reinhold and Schultz [19,20] with model potentials that account for the screening of the nuclear charge of the dressed projectile. Within the framework of the Impulse Approximation, they, and also Shingal et al. [21], found that the non-Coulombic, short-range potential related to the screening of the projectile is responsible for the BEE peak enhancement. Soon later, Taulbjerg [22] further explained this phenomenon but in terms of the exchange interaction between the target and projectile electrons. Taulbjerg claimed that the electron exchange plays an essential role in this phenomenon and later on, [7,9,23], agreed with the importance of such interaction based on the good agreement between their calculations and the experimental data. However, Salin [24] questioned the validity of Taulbjerg [22] arguments, and stated that such an interaction would stand for a process where the target electron is captured by the projectile with a simultaneous projectile electron emission. Thus, its contribution to the spectra must decrease rapidly with collision energy.

Nevertheless, as we stated before, despite the inclusion of the same type of model potentials of [18,19,20] in CDW theories [2,25], it was found that the BEE peak enhancement could not be reproduced by CDW-EIS, while it is fairly well achieved by CDW [2,26]. The reason, as mentioned in [26], is a contradiction with the asymptotic limit of the Eikonal-state approximation when dealing with these short-range collisions. If $\mathit{s}$ is the position of the electron relative to the projectile and $\mathit{v}$ is the projectile velocity in the laboratory frame, the Eikonal approximation holds for $\mathit{s}·\mathit{v}\to \infty$. Therefore, in a BEE spectra calculation with CDW-EIS, a distortion that is meant to work well at long distances is used within a region where the dominant interaction takes place at short distances, so the Eikonal factor may greatly differ from the effective Coulomb continuum it describes. Thus, in this work, we redefine the short-range transition amplitude of the CDW-EIS theory in order to account for the projectile charge-state dependence of the peak magnitude, while maintaining the analytical expressions. As the long-range term of the original CDW-EIS will remain unchanged, let us refer to the present model as CDW-lrEIS.

All quantities are referred to the laboratory frame and are expressed in atomic units unless otherwise stated.

2. Theory

Consider the single ionization of a multielectronic target due to the collision of a dressed projectile of nuclear charge $Z_{\mathrm{P}}$ and N bounded electrons. Here, we still do not include electron exchange calculations to show that the expected enhancement due to static potentials can now be achieved with a CDW-EIS-based model like the present CDW-lrEIS. Thus, we are going to proceed within an independent electron model, where the one-electron Hamiltonian is given by

$$H_{\mathrm{el}}=-{\displaystyle \frac{1}{2}}{\nabla }_r^2+V_{\mathrm{T}}\left(\mathit{x}\right)+V_{\mathrm{P}}\left(\mathit{s}\right),$$

(1)

where $\mathit{x}$ and $\mathit{s}$ are the position vectors of the so-called active electron, relative to the target and projectile nucleus, respectively. The other electrons are considered as passive, as they remain unperturbed in their initial state, bound to the target or projectile. Consequently, these electrons may be taken into account by a screening function within the interaction potentials $V_{\mathrm{T}}$ and $V_{\mathrm{P}}$, where the former corresponds to the target and the latter to the projectile. We take for $V_{\mathrm{P}}$ the Green–Sellin–Zachor model potential [27,28,29]:

$$V_{\mathrm{P}}\left(s\right)=-{\displaystyle \frac{q}{s}}-{\displaystyle \frac{1}{s}}(Z_{\mathrm{P}}-q){[\mathrm{H}({\mathrm{e}}^{\mathrm{s}/\mathrm{d}}-1)+1]}^{-1},$$

(2)

where H and d are two parameters, which are already calculated for a wide variety of ions and atoms in [28,29]. As seen, this model potential consist of the sum of a long-range interaction with net charge $q=Z_{\mathrm{P}}-N$ plus a short-range interaction that describes the screening of the projectile nuclear charge due to its bound electrons. This last term will be the one responsible for the enhancement of the DDCS with increasing number of electrons in the projectile. It originates a new term in the transition amplitude that will add, coherently, to the long-range perturbative potential, in the total transition amplitude.

The transition amplitude can be defined in its prior or post versions, whose expressions are, respectively,

$$\begin{array}{cc}\hfill a_{if}^{-}\left(\rho \right)& =-i{\int }_{-\infty }^{+\infty }〈{\chi }_f^{-}\left|\left(H_{\mathrm{el}}-i\frac{\partial }{\partial t}{|}_x\right)\right|{\chi }_i^{+}〉\mathrm{d}t,\hfill \end{array}$$

(3a)

$$\begin{array}{cc}\hfill a_{if}^{+}\left(\rho \right)& =-i{\int }_{-\infty }^{+\infty }〈{\chi }_f^{-}\left|{\left(H_{\mathrm{el}}-i\frac{\partial }{\partial t}{|}_x\right)}^{†}\right|{\chi }_i^{+}〉\mathrm{d}t.\hfill \end{array}$$

(3b)

The choice between the prior and post transition amplitudes relies on practical arguments. For example, as there are quite good atomic structure calculations, we can have a sufficiently good description of the initial electron-bound state and thus consider that all the interactions with the target nucleus and its electrons are solved by a proper orbital description whenever a prior version of the transition amplitude (3a) is used. In contrast, proceeding like in (3b) involves searching solutions in the continuum of $V_{\mathrm{T}}$, which are cumbersome to find and thus more approximations are involved in the post treatment [25].

In our approach, the transition amplitude is obtained in a closed form. This is an important objective because it allows a subsequent efficient numerical integration for the calculation of double, single and total cross-sections.

2.1. Continuum Distorted Wave Theories

Treating the interaction between the projectile and the active electron as a perturbation, we can follow a CDW approach and propose, for the initial and final states, the following distorted wave functions:

$${\chi }_i^{+}(\mathit{r},t)={\varphi }_i(\mathit{x},t){\mathcal{L}}_i^{+}\left(\mathit{s}\right),$$

(4a)

$${\chi }_f^{-}(\mathit{r},t)={\varphi }_f(\mathit{x},t){\mathcal{L}}_f^{-}\left(\mathit{s}\right).$$

(4b)

The total electron wavefunction is thus approximated by the product of two factors, namely $\varphi$ and $\mathcal{L}$, where the first one describes the evolution of the electron relative to the target, and the latter is a distortion originated by the projectile.

With regard to target states, ${\varphi }_i(\mathit{x},t)$ is the corresponding atomic or molecular initial bound orbital of the active electron, while ${\varphi }_f(\mathit{x},t)$ is the final continuum state of the ejected electron. The bound state ${\varphi }_i$ is expanded in a base of Slater-type orbitals. For atomic targets, the base and expansion parameters are taken from the Clementi and Roetti Roothaan–Hartree–Fock solutions [30], while for the hydrogen molecule orbital, a detailed description will be given in Section 2.2. However, in both cases ${\varphi }_i(\mathit{x},t)$ is a mean-field solution of the target potential in (1). That is why, in the prior transition amplitude (3a), the target interaction is completely solved, while in the post formulation (3b), a residual term for the active electron–target electron residual interaction should be treated separately or neglected [25].

The final continuum state is described with a Coulomb function corresponding to an effective Coulomb potential. The effective nuclear charge of the latter is defined as ${\tilde{Z}}_{\mathrm{T}}=\sqrt{-2n^2{\epsilon }_i}$ [31], where ${\epsilon }_i$ is the initial orbital binding energy and n its principal quantum number. Thus,

$${\varphi }_f(\mathit{x},t)={\displaystyle \frac{N^{\star }\left(\lambda \right)}{{\left(2\pi \right)}^{3/2}}}exp\left(i\mathit{k}·\mathit{x}-{\displaystyle \frac{i}{2}}{k}^{2}t\right){}_{1}F_1[-i\lambda ,1,-i(kx+\mathit{k}·\mathit{x})]$$

(5)

where $\lambda ={\tilde{Z}}_{\mathrm{T}}/v$ and $N\left(a\right)=exp(a\pi /2)\Gamma (1+ia)$ is the normalization factor of the ${}_{1}F_1$ hypergeometric function. Here, it is worth to remark that ${\varphi }_i$ and ${\varphi }_f$ are not orthogonal, nevertheless it is an usual approximation.

The projectile distortions are taken, in principle, as

$${\mathcal{L}}_i^{\mathit{CDW},+}\left(\mathit{s}\right)=N\left(\nu \right){}_{1}F_1[i\nu ,1,i(vs+\mathit{v}·\mathit{s})],$$

(6a)

$${\mathcal{L}}_f^{\mathit{CDW},-}\left(\mathit{s}\right)=N^{\star }\left(\xi \right){}_{1}F_1[-i\xi ,1,-i(ps+\mathit{p}·\mathit{s})]$$

(6b)

where $\mathit{p}=\mathit{k}-\mathit{v}$ is the ejected electron momentum relative to the projectile, $\nu =q/v$ and $\xi =q/p$.

Nevertheless, due to normalization problems (see [1]) that arise in the low collision-energy limit of the CDW theories range of validity, the Eikonal factor

$${\mathcal{L}}_i^{\mathit{EIS},+}\left(\mathit{s}\right)=exp[-i\nu ln(\mathit{vs}+\mathit{v}·\mathit{s})]$$

(7)

was proposed for the projectile distortion in the initial channel (6a), giving place to the Continuum Distorted Wave with Eikonal Initial State (CDW-EIS) framework. CDW-EIS was proved to be a successful theory in different scenarios throughout many years. However, it should be reminded that (7) is the limit of (6a) when $\mathit{s}·{\mathit{v}}_{\mathrm{P}}\to \infty$, which can be questionable when short-range interactions emission is studied.

Denoting the screening function of the projectile potential (2) as $\Omega \left(s\right)={[\mathrm{H}({\mathrm{e}}^{\mathrm{s}/\mathrm{d}}-1)+1]}^{-1}$, we find, in the Fourier transformation of the transition amplitude, the following term:

$$F^{\mathrm{sr}}\left(\mathit{K}\right)=\left[{\displaystyle \frac{-1}{{\left(2\pi \right)}^{3/2}}}\int \mathrm{d}\mathit{x}{e}^{-i\mathit{K}·\mathit{x}}{\varphi }_f^{\star }\left(\mathit{x}\right){\varphi }_i\left(\mathit{x}\right)\right]\left[{\displaystyle \frac{(Z_{\mathrm{P}}-q)}{{\left(2\pi \right)}^{3/2}}}\int \mathrm{d}\mathit{s}{e}^{i\mathit{K}·\mathit{s}}{\mathcal{L}}_f^{-\star }\left(\mathit{s}\right){\displaystyle \frac{\Omega \left(s\right)}{s}}{\mathcal{L}}_i^{+}\left(\mathit{s}\right)\right].$$

(8)

Since the screening function should tend to zero at long distances, it is evident that the greatest contribution of this term to the total transition amplitude comes from very short distances. This is the point where we question the adequacy of the Eikonal approximation when describing short-range interactions. So, to test our inquiries, we decided to use the Coulomb continuum factor (6a) in (8) instead of its asymptotic Eikonal approximation (7). We will refer to the present model as CDW-lrEIS, which stands for long-range Eikonal Initial State. That is because the long-range term of the transition amplitude is still calculated as the usual CDW-EIS prior theory.

The DDCS is given by

$${\displaystyle \frac{{\partial }^2\sigma }{\partial E_k\partial \Omega }}={\displaystyle \frac{k}{v^2{\left(2\pi \right)}^4}}\int \mathrm{d}\mathit{\eta }{\left|F^{\mathrm{lr}}\left(\mathit{K}\right)+F^{\mathrm{sr}}\left(\mathit{K}\right)\right|}^2,$$

(9)

where $\mathit{\eta }$ is the transversal momentum transfer satisfying $\mathit{\eta }·\mathit{v}=0$ and $\mathit{K}=-\mathit{\eta }-(\Delta \epsilon /v^2)\mathit{v}$.

In (9), the long-range transition amplitude in the momentum space is

$$F^{\mathrm{lr}}\left(\mathit{K}\right)=F^{\mathrm{lr}\left(a\right)}\left(\mathit{K}\right)+F^{\mathrm{lr}\left(b\right)}\left(\mathit{K}\right)$$

(10)

where

$$F^{\mathrm{lr}\left(a\right)}\left(\mathit{K}\right)=\left[{\displaystyle \frac{-1}{{\left(2\pi \right)}^{3/2}}}\int \mathrm{d}\mathit{x}{e}^{-i\mathit{K}·\mathit{x}}{\varphi }_f^{\star }\left(\mathit{x}\right){\nabla }_x{\varphi }_i\left(\mathit{x}\right)\right]·\left[{\displaystyle \frac{1}{{\left(2\pi \right)}^{3/2}}}\int \mathrm{d}\mathit{s}{e}^{i\mathit{K}·\mathit{s}}{\mathcal{L}}_f^{\mathit{CDW},-\star }\left(\mathit{s}\right){\nabla }_s{\mathcal{L}}_i^{\mathit{EIS},+}\left(\mathit{s}\right)\right]$$

(11)

and

$$F^{\mathrm{lr}\left(b\right)}\left(\mathit{K}\right)=\left[{\displaystyle \frac{-1}{{\left(2\pi \right)}^{3/2}}}\int \mathrm{d}\mathit{x}{e}^{-i\mathit{K}·\mathit{x}}{\varphi }_f^{\star }\left(\mathit{x}\right){\varphi }_i\left(\mathit{x}\right)\right]\left[{\displaystyle \frac{1}{{\left(2\pi \right)}^{3/2}}}\int \mathrm{d}\mathit{s}{e}^{i\mathit{K}·\mathit{s}}{\mathcal{L}}_f^{\mathit{CDW},-\star }\left(\mathit{s}\right){\displaystyle \frac{1}{2}}{\nabla }_s^2{\mathcal{L}}_i^{\mathit{EIS},+}\left(\mathit{s}\right)\right].$$

(12)

The difference between CDW-lrEIS and CDW-EIS is shown in Figure 1 through the effect of each model on the DDCS. There, it can be seen that the short-range transition amplitude actually leads to higher DDCSs in CDW-lrEIS, while the long-range one remains the same as CDW-EIS. For sure, both contributions must be summed coherently as in (9) since they are consequences of the same collision with the single incident dressed projectile. However, for the sake of completeness, we also show what would be the result of a non-coherent sum of such terms. From this last comparison between coherent and non-coherent sum, we can see that there is a an increment due to a constructive interference of the two complex quantities (8) and (10).

It is important to note that, as the initial state of the bound electron ${\varphi }_i\left(\mathit{x}\right)$ and the screening function $\Omega \left(s\right)$ of the projectile potential can be both described by Slater-type orbitals, then, all the integrals involved in (8), (11) and (12) can be obtained in a closed form by means of the results of Nordsieck [32,33].

Outlining, we presented the CDW-lrEIS model, which has the same short-range transition amplitude (8) as CDW. However, they differ in their long-range transition amplitude (10). In addition, CDW-lrEIS can be formulated in the prior manner while CDW must be treated in its post [25]. In our approach, prior formulations are preferred than post ones. This is because the initial bound states are better represented than the final continuum states. Nevertheless, for a detailed description of the CDW post theory, we may refer the reader to the work of Monti et al. in [25].

2.2. Hydrogen Molecule Description

The molecule is described within a one-center approximation by considering the model potential proposed by Lühr [34], namely

$$V_{\mathrm{T}}\left(x\right)=-{\displaystyle \frac{1}{x}}\left[1+exp(-{\displaystyle \frac{2x}{\sqrt{\alpha }}})\right],$$

(13)

where $\alpha =0.133081$ for the bound lenght $R_{{\mathrm{H}}_2}=1.44$. The same potential was recently used by Gulyás et al. [35] in CDW-EIS single-ionization calculations with high-energy proton projectiles. They found that the results obtained with (13) were as good as the more computationally expensive multicenter-potential calculations that they have also performed. They even found that the single-center CDW-EIS backscattering DDCSs were in better accordance with the experimental results than the multicenter ones.

Then, considering (13) as the mean-field interaction of the active electron with the molecule and ${\epsilon }_i=-0.59755$ as the orbital energy, the initial bound state wavefunction was obtained by numerical integration of the radial Schrödinger equation with the Numerov algorithm. The solution was then expanded in a sum of two 1s Slater-type orbitals, so that

$${\varphi }_i(\mathit{x},t)={\displaystyle \frac{e^{-i{\epsilon }_{i}t}}{\sqrt{\pi }}}\sum _{j=1}^2{C}_j{Z}_j^{3/2}exp(-Z_{j}x),$$

(14)

where the expansion parameters $C_j$ and $Z_j$ are given in

Table 1. Expansion coefficients and exponents for the H₂ ground state monocentric orbital described based on two 1s Slater-type orbitals.

j	${\mathit{C}}_{\mathit{j}}$	${\mathit{Z}}_{\mathit{j}}$
1	0.954069	1.129807
2	0.067364	3.297702

.

3. Results and Discussion

We now proceed to show the theoretical results on the binary-encounter emission peak obtained with the present CDW-lrEIS theory. First, we present the results with an atomic target and then with the hydrogen molecule described as in Section 2.2. A comparison with experimental data and other Continuum Distorted Wave models is performed.

The models used for comparison are the CDW of [25], CDW-EIS of [2] and CDW-EIS with dynamic effective charge of [36], noted as “${\xi }_{\mathrm{K}}$”. Briefly, the latter considers an effective charge which depends on the momentum transfer in the projectile distortion (4b). This accounts for different degrees of screening of the projectile nuclear charge. At low momentum transfers, the dynamic charge equals q, the ion net charge, while at high momentum transfers, which correspond to close encounter collisions, the dynamic charge tends to the projectile nuclear charge $Z_{\mathrm{P}}$.

It is important to note that the two CDW-EIS theories for dressed projectiles, namely CDW-lrEIS or CDW-EIS ${\xi }_{\mathrm{K}}$, will give the same results as CDW-EIS whenever bare projectiles are considered.

3.1. Helium Target

In Figure 2 we show the results obtained with CDW-lrEIS for single-ionization DDCSs of helium by O^q+ ions impacts at 1875 keV/u in the BEE energy range. Also, the measurements of [13] and previous calculations with CDW, CDW-EIS and CDW-EIS ${\xi }_{\mathrm{K}}$ from [26] are included for comparison. The peaks appearing at lower energy corresponds, as stated in [13], to projectile KLL-Auger electrons. This is a relaxation process not accounted with our direct ionization theories, so our calculations will not reproduce them.

It is evident that, apart from the bare O⁸⁺ + He system, the CDW-EIS theories underestimate the experimental BEE peak magnitude. A better result is achieved with the dynamic charge correction in CDW-EIS ${\xi }_{\mathrm{K}}$ for the O⁷⁺ projectile. However, this is no longer the case for lower ionic charge states, as we can see that the discrepancy with the experiments increases. As the charge state of the projectile decreases, only CDW and CDW-lrEIS describe well the BEE peak magnitude. Also, the shape of the peak is well represented by these theories. It is known that the shape depends on the momentum distribution of the electron in its initial state (Compton profile), and thus it relies on the description of the initial state.

The similarity between results from CDW and CDW-lrEIS is due to the equality in the formulation of the short-range transition amplitude. Nevertheless, we can expect some differences due to the long-range terms, at least in the bare projectile case, but they do not seem significant at this collision energy. However, in Figure 3, where we show the peak magnitude ratios relative to the bare ion impact, we do see a bigger difference between CDW and CDW-lrEIS, which demonstrates the great sensitivity of this kind of plot. Even though some overestimation of the growth rate of the peak with decreasing q is seen, CDW-lrEIS does give an enhacement of the BEE peak, showing an improvement over CDW-EIS and CDW-EIS ${\xi }_{\mathrm{K}}$.

In average, with CDW and CDW-lrEIS, the difference between the experimental ratios is about 13% and 14%, respectively. Meanwhile, the other EIS approximations yield a difference of 28% in the case of CDW-EIS ${\xi }_{\mathrm{K}}$, and 50% with CDW-EIS. The improvement achieved with CDW-lrEIS is again significant.

3.2. Hydrogen Molecule Target

In Figure 4a,b, we show the theoretical and experimental results of binary-encounter electron yield of the hydrogen molecule by impact of B^q+ ions, with $q=5,4,3$ and 2. The measurements, performed by Benis and colleagues, were taken from [26] and the theoretical calculations consider the molecular description presented in Section 2.2. For comparison, we have also calculated, using CDW-EIS, the corresponding DDCS for a hydrogen target because multiplying the hydrogen target DDCS by a factor os two is a common approximation in high-energy collisions. This was performed in Figure 4a,b for the bare projectile case and noted as “2H”.

For the dressed B^q+ projectiles at 4.9 MeV, some other structures appear in the experiments superimposed in the left side of the BEE peak. As stated in [26], these correspond to projectile KLn (n ≥ 2) Auger emission. Therefore, they will not be described by our direct ionization theories.

From the results with the B⁵⁺ projectile, it can be seen that approximating the BEE of the H₂ molecule by taking twice the corresponding DDCS of an H atom might not be an inadequate approximation for calculating the peak magnitude. However, the shape of the peak was better represented by the calculated H₂ ground state monocentric orbital. The cause of this is might be that the initial electron momentum distribution of the H₂ orbital, the so-called Compton profile, is better described in this way.

Regarding the BEE peak enhancement, as might be expected from the previous section, it is found that CDW-EIS results with and without dynamic charge ${\xi }_{\mathrm{K}}$, once again do not describe well the BEE peak enhancement. More interesting to note is the difference between CDW and CDW-lrEIS in Figure 4a. From the B⁵⁺ results, it is evident that the source of this difference is due to the CDW long-range transition amplitude. Then, with the B⁴⁺ projectile, the difference is smaller and for B³⁺ and B²⁺ it disappears. This is because the screening potential is greater with an increasing number of electrons, see the factor $(Z_{\mathrm{P}}-q)$ in (8), which equals N. Therefore, the short-range transition amplitude outweighs the long-range one. In addition, from the fact that both theories hold the same short-range transition amplitude, it can then be explained why they lead to the same results as we consider more electrons in the projectile ion.

Moreover, the difference between CDW and CDW-lrEIS in the B⁵⁺ case also disappears with increasing collision energy as seen from the comparison between Figure 4a,b. This can be attributed to the normalization problem of CDW, the one pointed out by Crothers and McCann [1] for bare projectiles, and analysed in the Appendix A. Then, as the projectile ions have more electrons, the short-range interactions gain importance, and this difference is washed out. The growth in the DDCS with an increasing number of electrons is indeed due to the contribution of the short-range transition amplitude.

In Figure 5, we show the BEE peak enhancement ratios of Figure 4a,b relative to the bare projectile results. At both impact energies of 445 keV/u and 1182 keV/u, it can be seen that CDW-lrEIS adjusts well to the experimental results. For the at 445 keV/u results, the CDW ratios are affected by the normalization problem: Higher values of the DDCS for the bare projectile leads to lower ratios. These ratios differ from the experimental ones by around 15%. At 1182 keV/u, CDW and CDW-lrEIS results converge with each other with an average difference of 5%. A similar behaviour was seen before in Figure 3, at 1875 keV/u.

In terms of the Sommerfeld parameter $S=Z_{\mathrm{P}}/v$, for the 1182 keV/u B^q+ and the 1875 keV/u O^q+ projectiles, we have $S=0.7$. Meanwhile, for the 445 keV/u B^q+, $S=1.2$. Therefore, we can state that for high collision energies, where $S<1$, both CDW and CDW-lrEIS could be suitable, providing no divergences appear in CDW cases. However, at low collision energies, for which $S\simeq 1$, CDW-lrEIS holds better results than CDW, so CDW-lrEIS has a wider impact-energy domain in which it can be applied. This is a known fact with bare projectiles [1], but here we are seeing it with dressed projectiles.

4. Conclusions

In this work, we have revisited the CDW-EIS theory for dressed projectiles. Throughout the years, CDW-EIS has demonstrated to be a reliable theory reproducing fairly well the whole electron emission spectra in collision systems of ions with atoms or molecules. However, the binary-encounter emission enhancement with increasing number of electrons in the projectile is a physical effect that CDW-EIS cannot reproduce. Thus, we reformulated the short-range transition amplitude without the Eikonal state approximation. This model, referred to as CDW-lrEIS, does describe the corresponding BEE peak enhancement, as we showed in the present work.

From the above result, it is evident that the projectile distortion plays a fundamental role in the short-range transition amplitude. When dressed projectiles are involved, the Eikonal approximation could lead to unrealistic results at the BEE peak emission energies.

What is more, we show that CDW-lrEIS prior yields similar results that CDW post from [25]. This is a great achievement, due to the simpler calculations and better-behaved mathematical expressions involved in CDW-lrEIS in comparison with CDW. In addition, CDW-lrEIS is equivalent to CDW-EIS in the case of bare projectiles, and so, it can be more reliable at low energies than CDW. This is because the long-range Eikonal Initial State is properly normalized at all times [1].

Summarising, CDW-lrEIS prior has the advantages of both CDW-EIS prior and CDW post. It works well in a wider collision energy domain than CDW, while it describes the same short-range contribution as CDW, but with simpler expressions. For example, it does not have to deal with the target residual potential as in (3b) [25].

Finally, by means of the monocentric model potential for the H₂ molecule from [34], we calculated the initial bound state of the electron. Then, we found that this orbital leads to a better agreement with measurements of the BEE peak shape than considering the H₂ single-ionization DDCS as twice the corresponding DDCS of the hydrogen atom. Evidently, the present molecular description provides a better description of the initial electron momentum distribution.