Introduction

Beryllium is widely used as a first wall element of the fusion reactors1 because of its unique thermo-physical properties. So, due to wall erosion, Beryllium should be one of main impurity in fusion chamber2. The radiative decay of excited impurity ions can be the source for the energy loss of the plasma and can cool the plasma. These radiative decays can be analyzed by the electron capture recombination spectroscopy (CXRS). Therefore, the exact knowledge of electron capture cross sections in collisions between Be ions and hydrogen atoms is essential3. Due to the experimental difficulties, the experimental results for electron capture cross sections in Be4+  + H collisions are entirely lacking, but those were studied intensively theoretically in the past years. The total electron capture cross sections have been studied using various models and methods such as applying the quantum–mechanical molecular orbital close-coupling (QMOCC)4, the atomic orbital close-coupling (AOCC)5, the hyper spherical close-coupling (HSCC)6 models, using the solution of the time dependent Schrödinger equation (TDSE)7, the lattice time dependence Schrödinger equation (LTDSE)8, the classical over barrier model (COBM)9 and the classical trajectory Monte Carlo method10,11. The partial electron capture cross sections in the collision between Be and hydrogen atom have been also studied using different quantum–mechanical methods such as: QMOCC4,12, AOCC5, one-electron diatomic molecule (OEDM)13, and boundary corrected continuum intermediate state (BCCIS)14 models. It is worth noting that all the results have been published for projectile energy below 100 keV/amu. The calculation of the principle quantum number, n, dependent cross sections has been studied by Jorge et al.15 by solving the time-dependent Schrödinger equation with the GridTDSE package (GTDSE) numerically in the broad energy range between 1 keV/amu and 500 keV/amu.

In this work we present the electron capture cross sections into the bound states of the projectile in Be4+ + H(1s) collisions. We treat the collision dynamics classically using a three-body classical trajectory Monte Carlo (CTMC) and a three-body quasi classical Monte Carlo (QCTMC) model when the Heisenberg correction term is added to the standard CTMC model via model potential16,17,18,19,20,21. Since there is no experimental data available, our calculated cross sections are compared with the previous theoretical results.

Results

For each collision energies, the calculation of the state selective electron capture cross sections requires to follow 107 classical trajectories. At first, we tested three calculation schemes during our simulations since the Heisenberg correlation potential may influenced the obtained results significantly. These are the following: (1) target-centered, where the correction term is taken into account between the target electron the target nucleus, (2) projectile-centered, where the correction term is taken into account between the target electron the projectile. (3) Combined one, i.e., target and projectile centered when the correction term is taken into account between target electron and both the target nucleus and projectile.

As an example Fig. 1 shows our CTMC and QCTMC results corresponding to the three calculation schema of the electron capture cross sections into the 4 s state of the projectile in Be4+  + H (1s) collision as a function of the impact energy. It can be seen that the effects of the correction term at lower energies are significant. While for the case of target-centered, the cross sections at lower incident energies are increasing compared to the standard CTMC results for the case of projectile-centered they are decreasing. The combination of the use of target- and projectile-centered corrections results increases the cross sections and we also obtained good agreement between our QCTMC results and previous full quantum mechanical results in the entire impact energy range. Therefore, in followings, for the calculation of the capture cross sections, we will use only the combination scheme.

Figure 1
figure 1

Electron capture cross sections into the 4s state of the projectile in Be4+  + H (1s) collision as a function of the impact energy. Solid red line: present CTMC results, solid black line: present target-centered QCTMC results, solid green line: present projectile-centered QCTMC results, solid blue line: present target and projectile centered QCTMC results, pink squares: AOCC results of Fritsch5, green circles: QMOCC results of Harel et al.4, black triangles: BCCIS results of Das et al.14, blue inverse triangles: OEDM results of Errea et al.13.

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Physically, due to the Heisenberg constraint, the electron cannot collapse to the target and projectile nucleus in the electron capture channel. To clarify this further, we calculated the electron capture probabilities as a function of the impact parameter.

Figure 2 shows the present CTMC and QCTMC results with the three calculation schemes of the electron capture probabilities into the specific n = 3, 4 and nl = 3d, 4s states of the projectile at 10 keV/amu impact energy in Be4+  + H(1s) as a function of impact parameter. The impact parameter dependent electron capture probabilities, bP(b), were fitted by a Gaussian function. The peak maxima of the Gaussian fitting is also shown in Fig. 2. We note that the area under the curves is proportional to the state-selective electron capture cross sections. We found that the probability of electron capture is higher in target-centered QCTMC and lower in projectile-centered QCTMC model compared with the standard CTMC model. This behavior can be understood with the explanation of the acting forces between the interacting particles, \(F = - dU/dr\). The attractive force between an electron and both of proton and positive projectile ion, is due to the Coulomb interaction and repulsive force is due to the Heisenberg correction term as follow:

Figure 2
figure 2

Probability for electron capture into n = 3, 4 and nl = 3d, 4s states of the projectile (multiplied by impact parameter) in Be4+  + H(1s), as a function of the impact parameter, at 10 keV/amu impact energy. Dash-dotted red line: present CTMC results, dash black line: present target-centered QCTMC results, solid green line: present projectile-centered QCTMC results, solid blue line: combination of target-projectile-centered.

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$${F}_{Heisenberg}= -\left(\frac{{{\xi }_{H}}^{2}}{2{\alpha }_{H}{r}^{3}}+\frac{r{p}^{4}}{{{\xi }_{H}}^{2}}\right)exp\left\{{\alpha }_{H}\left[1-{\left(\frac{rp}{{\xi }_{H}}\right)}^{4}\right]\right\}$$
(1)

The attractive Coulomb force acts between the electron and positively charged, target and projectile, in the same way in all schemes. This force, most of the time of the collision, is much larger than \({F}_{Heisenberg}\). On the other hand, in the target-centered scheme, the repulsive force, \({F}_{Heisenberg}\), is toward the projectile, but on the contrary, this repulsive force is towards the target in projectile-centered mode. We note that this repulsive force, of course, does not show up in the standard CTMC model. According to the sum of the forces, the electron has the highest attraction to the projectile in the target-centered QCTMC and the least attraction to the projectile in the projectile-centered QCTMC. With this scenario, the case of CTMC is placed between the above two modes. Therefore, the probability of electron capture in projectile-centered QCTMC, CTMC, and target-centered QCTMC modes increases, respectively.

Another noteworthy point is that the peak maxima in CTMC and QCTMC projectile-centered cases are very close to each other and locate in lower impact parameters. This is also true in QCTMC target-centered and QCTMC combined target- and projectile-centered cases, except that the peak maxima are at higher impact parameters.

Figure 3 shows the present CTMC and QCTMC results of the electron capture cross sections into the n = 3, 4, 5 states of the projectile in Be4+  + H(1s) collision as a function of the impact energy. The present classical results are compared with Fritsch5, Harel et al.4, and Das et al.14, as well. The QCTMC results are higher than the CTMC ones at low and intermediate impact energies. This difference is more significant in n = 4 and n = 5 states. The best matching between present CTMC and QCTMC is seen at high energies. In n = 3 and n = 5 states, the present QCTMC results agree well with the available quantum–mechanical approaches such as; QMOCC4, AOCC5, and BCCIS14.

Figure 3
figure 3

Electron capture cross sections into the n = 3, 4, 5 states of the projectile in Be4+  + H(1s) collision as a function of the impact energy. Solid red line: present CTMC results, solid blue line: present QCTMC results, pink squares: AOCC results of Fritsch5, green circles: QMOCC results of Harel et al.4, black triangles: BCCIS results of Das et al.14. Also, electron capture cross sections into n = 6, 8, 10 states are recommended.

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The standard statistical error [see Eq. (10)] at 1000 keV/amu impact energy is around 4% in CTMC and QCTMC, respectively. Figure 3 also shows the cross sections for higher states where no previous data are available.

Figure 4 shows our CTMC and QCTMC results of the electron capture cross sections into 3s, 3p and 3d states of the projectile in Be4+ + H(1s) as a function of the impact energy. The comparison is made with Fritsch5, Harel et al.3, Das et al.14, and Errea et al.13. The QCTMC model significantly improve the cross sections compared to the CTMC at low and intermediate impact energies. Moreover, the unique agreement is obtained between the present QCTMC results and (1) QMOCC results4 in the 3s state, (2) OEDM13, and BCCIS results14 in 3p state (3) QMOCC4, AOCC5, OEDM13, and BCCIS14 results in 3d state of the Be3+ at energies lower than 100 keV/amu, respectively.

Figure 4
figure 4

Electron capture cross sections into 3s, 3p, and 3d states of the projectile in Be4+  + H(1s) collision as a function of the impact energy. Solid red line: present CTMC results, solid blue line: present QCTMC results, pink squares: AOCC results of Fritsch5, green circles: QMOCC results of Harel et al.4, black triangles: BCCIS results of Das et al.14, blue inverse triangles: OEDM results of Errea et al.13.

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Figure 5 represents our CTMC and QCTMC results of the electron capture cross sections into 4s, 4p, 4d, and 4f states of the projectile in Be4+ + H(1s) as a function of the impact energy. We have compared the present classical results with the quantum–mechanical approaches such as QMOCC4, AOCC5, OEDM13, and BCCIS14. The QCTMC model remarkably increases the cross sections compared with the CTMC at low and intermediate energies. The difference between the present CTMC and QCTMC results at low energies gradually increases from 4s to 4f states. It can be seen that our CTMC results have the best agreement with the AOCC results of Fritsch5 in 4s and 4p states. Moreover, the QCTMC cross sections in 4s and 4p states have better agreement with the QMOCC results of Harel et al.4 and the OEDM results of Errea et al.13. However, both present classical results are in excellent agreement with the BCCIS results of Das et al.14 at intermediate energies. The standard statistical error for 4s, 4p, 4d, and 4f states is around 1.6% in the range of 1–500 keV/amu impact energies. At the same time, for the projectile energy range of 500–1000 keV/amu, the estimated uncertainties around 4%.

Figure 5
figure 5

Electron capture cross sections into 4s, 4p, 4d, and 4f states of the projectile in Be4+  + H(1s) collision, as a function of the impact energy. Solid red line: present CTMC results, solid blue line: present QCTMC results, pink squares: AOCC results of Fritsch5, green circles: QMOCC results of Harel et al.4, black triangles: BCCIS results of Das et al.14, blue inverse triangles: OEDM results of Errea et al.13.

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Figure 6 shows the present CTMC and QCTMC results of the electron capture cross sections into 5s, 5p, 5d, and 5f states of the projectile in Be4+ + H(1s) as a function of the impact energy. The obtained results are compared with QMOCC4, BCCIS14, and OEDM13 methods, as well. According to Fig. 6, the QCTMC method outstandingly enhances the cross sections compare to the CTMC results at impact energies lower than about 60 keV/amu. Good agreements are obtained between the present QCTMC results with the OEDM results of Errea et al.13 and the QMOCC results of Harel et al.4 in 5s, 5d, and 5f states of the projectile. The present CTMC and QCTMC results in all 5l-states agree well with the BCCIS results of Das et al.14 at intermediate energies. The QCTMC and CTMC cross sections are approximately matched at the impact energies greater than 100 keV/amu.

Figure 6
figure 6

Electron capture cross sections into 5s, 5p, 5d, and 5f states of the projectile in Be4+  + H(1s) collision, as a function of the impact energy. Solid red line: present CTMC results, solid blue line: present QCTMC results, green circles: QMOCC results of Harel et al.4, black triangles: BCCIS results of Das et al.14, blue inverse triangles: OEDM results of Errea et al.13.

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According to Figs. 3, 4, 5, and 6, the present CTMC and QCTMC results of the electron capture cross sections into specific states of the projectile in Be4+ + H(1s) are given for several typical impact energies in Table 1. As we already mentioned, the QCTMC cross sections are larger compared to CTMC ones at lower incident energies. However, as the energy increases, this difference gradually decreases so that at very high energies, this difference is negligible. To explain this behavior physically, we focus on the force between the electron and the hydrogen nucleus. Typically, the net Coulomb force is applied between two bodies, which is inversely related to the square of the distance between them. Heisenberg correction term [see Eq. (6)] generates a repulsive force in the opposite direction to the Coulomb force. In this case, the attraction force between the electron and the target's nucleus decreases, increasing the electron's reactivity with the projectile's ion in the electron capture channel. Also, the long- distance of the projectile to the electron practically reduces this repulsive force's effect on the calculations (see Fig. 1).

Table 1 The present CTMC and QCTMC results of the electron capture cross sections into specific states of the projectile in Be4+ + H(1s).
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On the other hand, the passing projectile ion at low energies causes the extension of the interaction time. Therefore, the effect of these factors increases the cross section at low energies in the QCTMC model. Also, the interaction time is shorter at high energies. Furthermore, due to the small Heisenberg repulsive force, the correction term gradually loses its effects; therefore, the CTMC and QCTMC results are approximately the same.

Discussions

The electron capture cross sections into n = 3, 4, 5, 6, 8, 10 and nl = 3l, 4l, 5l states of the projectile have been presented in Be4+  + H(1s) in the framework of CTMC and QCTMC methods. For the determination of the cross sections 107 trajectories were calculated for each impact energies. We found that the QCTMC cross sections are higher than the CTMC ones at low energies. We have used the previous AOCC, QMOCC, BCCIS, and OEDM quantum–mechanical approaches for comparison with our present data. Including the potential correction term to mimic the Heisenberg uncertainty principle in the classical Hamiltonian, we have shown that our QCTMC capture cross sections into the projectile states, n = 3, 5 and nl = 3s, 3p, 3d, 4s, 4p, 5s, 5d, 5f are in excellent agreement with quantum–mechanical results. We believe that our model, with its simplicity, can be an alternative way to calculate accurate cross sections and maybe can replace the results of the quantum–mechanical models, where the quantum mechanical calculations become complicated.

Methods

The QCTMC model, in principle, takes into account the Heisenberg and Pauli constraints in adding a correction term into the standard original Hamiltonian16. For hydrogen atom, which has only one electron, the Pauli correction can automatically neglect. Therefore, the quasi classical Hamiltonian consists of correction potential, VH, inspired by the Heisenberg principles can be written as:

$${H}_{QCTMC}={H}_{0}+{V}_{H}$$
(2)

where,

$${V}_{H}=\sum_{n=a.b}\sum_{i=1}^{N}f\left({r}_{ni} .{p}_{ni};{\xi }_{H} . {\alpha }_{H}\right)$$
(3)

a and b denote the nuclei, and the i index the electrons. r and p are the distance and momentum of an electron with respect to a nucleus, which is defined as follows:

$${r}_{ab}={r}_{b}-{r}_{a}$$
(4)
$${p}_{ab}=\frac{{m}_{a}{p}_{b}-{m}_{b}{m}_{a}}{{m}_{a}+{m}_{b}}$$
(5)

The Heisenberg correction function is expressed as16

$$f\left({r}_{ab} .{p}_{ab};{\xi }_{H} . {\alpha }_{H}\right)=\frac{{\xi }_{H}}{4{\alpha }_{H}{r}_{ab}^{2}{\mu }_{ab}}exp\left\{{\alpha }_{H}\left[1-{\left(\frac{{r}_{ab}{p}_{ab}}{{\xi }_{H}}\right)}^{4}\right]\right\}$$
(6)

where subscripts a and b indicate pairs of particles with reduced mass \({\mu }_{ab}\). The parameter \({\xi }_{H}\) reflects the size of the core while \({\alpha }_{H}\) is a hardness parameter. These parameters universally are used where \({\alpha }_{H}\) = 4 and \({\xi }_{H}\) = 0.9428, respectively19,20,21. The Heisenberg potential between the target electron and both target core and projectile (p; projectile, e; electron, T; target) are defined as follows:

$$f\left({\overrightarrow{r}}_{pe} .{\overrightarrow{P}}_{pe};{\varepsilon }_{H} .{\alpha }_{H}\right)=\frac{{{\xi }_{H}}^{2}}{4{\alpha }_{H}{\overrightarrow{r}}_{pe}^{2}{\mu }_{pe}}exp\left\{{\alpha }_{H}\left[1-{\left(\frac{{\overrightarrow{r}}_{pe}{\overrightarrow{P}}_{pe}}{{\xi }_{H}}\right)}^{4}\right]\right\}$$
(7)
$$f\left({\overrightarrow{r}}_{Te} .{\overrightarrow{P}}_{Te};{\varepsilon }_{H} .{\alpha }_{H}\right)=\frac{{{\xi }_{H}}^{2}}{4{\alpha }_{H}{\overrightarrow{r}}_{Te}^{2}{\mu }_{Te}}exp\left\{{\alpha }_{H}\left[1-{\left(\frac{{\overrightarrow{r}}_{Te}{\overrightarrow{P}}_{Te}}{{\xi }_{H}}\right)}^{4}\right]\right\}$$
(8)

The total cross sections are computed with the following formula:

$$\sigma =\frac{2\pi {b}_{max}}{{T}_{N}}\sum_{j}{b}_{j}^{(i)}$$
(9)

and the statistical uncertainty of the cross sections is given by:

$$ \Delta \sigma = \sigma \left( {\frac{{T_{N} - T_{N}^{{\left( i \right)}} }}{{T_{N} T_{N}^{{\left( i \right)}} }}} \right)^{{1/2}} $$
(10)

where TN is the total number of trajectories calculated for impact parameters less than bmax, \(T_{N}^{(i)}\) is the number of trajectories that satisfy the criteria for the corresponding final channels (electron capture), and bj(i) is the actual impact parameter for the trajectory corresponding electron capture processes.

In the classical approaches, the classical principal (nc) and the orbital angular momentum (lc) quantum numbers are defined by

$$ n_{c} = Z_{T} Z_{e} \left( {\frac{{\mu_{Te} }}{2U}} \right)^{1/2} $$
(11)
$${l}_{c}=\sqrt{{m}_{e}\left[{\left(x\dot{y}-y\dot{x}\right)}^{2}+{\left(x\dot{z}-z\dot{x}\right)}^{2}+{\left(y\dot{z}-z\dot{y}\right)}^{2}\right]}$$
(12)

where \({\mu }_{Te}\) is the reduced mass of the target nucleus and the target electron. x, y, and z are the Cartesian coordinates of the electron relative to the nucleus and \(\dot{x}\), \(\dot{y}\), and \(\dot{z}\) are the corresponding velocities. The classical values of nc are “quantized” to a specific level n22 if they satisfy the relation:

$${\left[\left(n-1\right)\left(n-1/2\right)n\right]}^{1/3}\le {n}_{c}\le {\left[\left(n+1\right)\left(n+1/2\right)n\right]}^{1/3}$$
(13)

Since lc is uniformly distributed for a given n level23, the quantal statistical weights are reproduced by choosing bin sizes such that

$$l\le \frac{n}{{n}_{c}}{l}_{c}\le l+1$$
(14)

where l is the quantum–mechanical orbital angular momentum.