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Abstract
We propose a method to build stable classical multielectron model atoms with the ionization energies optimized to experimental values. Based on the work of Kirschbaum and Wilets [Phys. Rev. A 21, 834 (1980) [CrossRef] ], which introduces auxiliary potentials to simulate quantum mechanical effects, we implement a genetic algorithm to optimize the related parameters such that the model atoms yield correct (first few) ionization energies. Ionization-energy optimized model atoms automatically show separated electron shells, consistent to normal expectations. Numerical examples are given to demonstrate the importance of correct ionization energies, as well as new perspectives to double ionization processes.
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1. Introduction
Many-particle systems have been long-standing challenges for quantum mechanics. For an $N$-particle system, the wave function spreads in a $3N$-dimensional configuration space. As $N$ increases, it is challenging computationally to predict the evolution of the wave function, or even to find the wave function itself.
The same challenge has also been encountered in strong-field atomic physics, which studies the interaction between intense laser fields and isolated (gas-phase) atoms. Such an interaction has led to novel nonlinear and nonperturbative phenomena including multiphoton and above-threshold ionization [1,2], correlated or nonsequential double ionization [3,4], high harmonic generation [5,6], attosecond pulse generation [7–10], etc. Commonly used atoms in experiments are multielectron atoms, such as rare gas atoms, alkaline earth atom vapors, etc. Numerical solutions of the time-dependent Schrödinger equation have been limited to one- or two-electron systems [11–14], whereas in experiments the lasers are strong enough to remove multiple electrons from an atom [15–26].
An alternative approach to gain insights into many-particle processes is (semi-)classical simulations. Widely known examples include molecular dynamics simulations for proteins and particle-in-cell simulations for plasmas. Classical simulations have also been used in nuclear physics [27,28] and collision physics [29–33]. Strong-field atomic physics is another area where classical simulations play an important role. Many insights about the atomic ionization process, especially double or multiple ionization processes, are obtained with classical simulations [34–55].
The very first step of classical simulations for atomic physics is to build a classical multielectron model atom, which however suffers two problems. The first problem is that the atom is not stable: One or more electrons can be emitted automatically due to electron-electron repulsions, leading to autoionization. This problem can be cured by either introducing auxiliary potentials accounting for quantum mechanical effects [33], or by softening the Coulomb potential [56,57]. The underlying idea is to limit the amount of energy sharing among the electrons.
The second (less known and more subtle) problem is incorrect ionization energies. It has been shown that even the first ionization energies deviate substantially from experimental values, either with auxiliary potentials [34,35] or with soft-core Coulomb potentials [37,38]. Whereas ionization energies certainly have measurable effects. The first (second,…) ionization energy determines the laser field strength around which the first (second,…) electron is emitted. Using elliptically polarized laser fields, the instantaneous laser electric field at the time of ionization can be mapped to the momentum distribution of the electron or of the ion. The peaks of the ion momentum distribution is closely related to the individual ionization energies [58–60]. Without solving this ionization-energy problem, classical simulations cannot reproduce the experimental momentum distributions.
The goal of the current article is to build classical model atoms with optimized (first few) ionization energies, so that classical simulations can be improved to provide better descriptions of the strong laser-atom interaction process. In this article we consider the auxiliary-potential model by Kirschbaum and Wilets [33], who introduce momentum-dependent potentials to simulate the Heisenberg uncertainty principle and the Pauli exclusion principle. A genetic algorithm is implemented to optimize the model parameters such that the first few ionization energies converge to known experimental values as close as possible. No additional parameters are introduced.
Ionization-energy optimized model atoms automatically gain an interesting and encouraging feature that separated electron shells emerge. For example, the C model atom shows four outer electrons and two inner electrons, and the (six-active-electron) Mg model atom has two outer electrons and four inner electrons. This is a direct consequence of the ionization-energy optimization. This feature allows one to revisit the traditional double ionization process. An example will be given on the following question: Are the two electrons emitted in a nonsequential double ionization process from the same electron shell?
This article is organized as follows. In Section 2 our method will be explained, including an introduction to the auxiliary potentials by Kirschbaum and Wilets and the ionization-energy optimization procedures. In Section 3 the results of optimization will be given, together with numerical examples demonstrating the usefulness of the method. A conclusion will be given in Section 4.
2. Method
2.1 Kirschbaum-Wilets (KW) potentials
Kirschbaum and Wilets introduce two momentum-dependent potentials to simulate quantum mechanical effects of the Heisenberg uncertainty principle and the Pauli exclusion principle. The Hamiltonian of an atom reads (atomic units are used unless stated otherwise)
The potentials contain three parameters $\{ \alpha, \xi _H, \xi _P \}$, controlling the hardness of the potentials and the ranges of the Heisenberg and the Pauli effects. Note that in the model atom, each electron is given a spin label $s_i$ of either up or down, and the Pauli potential only acts between electrons of the same spin label [notice the $\delta _{s_i,s_j}$ term in Eq. (1)].
The shared parameter $\alpha$ controls the hardness of the potentials and its value is assigned somewhat arbitrarily by different authors. In [33] it is assigned to be 5 a.u., in [48] it is 2 a.u., and in [52] it is 4 a.u. Then the parameters $\xi _H$ and $\xi _P$ are assigned with approximating physical arguments. Room for improvement or optimization of these parameters certainly exists. For example, it has been shown that the ionization energies are obviously not correct [61,62]. Even the first ionization energies can deviate substantially from experimental values.
2.2 Ionization energies
From Koopmans’ theorem [63], the first ionization energy of an atom $A$ is defined as
where $H_{min}(A)$ denotes the minimum energy of $A$, and $H(A^{+})$ means the energy of the ion $A^{+}$ with (a) the outermost electron removed from $A$, and (b) the remaining electrons of the same position and momentum configuration as atom $A$.Similar definition can be given to the $i$th ionization energy
for $i=1, 2, 3,\ldots$2.3 Optimization using a genetic algorithm
A genetic algorithm (GA) is an optimization heuristic that mimics the natural selection process and has been widely used in optimization or search problems [64]. Bio-inspired operators such as mutation, crossover and selection are exploited, and the evolution is guided by a user-supplied fitness function.
In our method we use GA to search for the parameter combination $\{ \alpha, \xi _H, \xi _P \}$ such that the multielectron model atom yields the correct first few ionization energies, as defined above, with respect to known experimental values. The procedure is listed as follows:
(i) Given a combination $\{ \alpha, \xi _H, \xi _P \}$, the potentials $V_H$ and $V_P$ are fixed, so are the ionization energies of the model atom.
(ii) The ionization energies are calculated according to Eq. (5). This calculation involves nested optimization processes, which are done also with the help of GA. We need to optimize the positions and momenta of the electrons of $A^{i+}$ ($i=0,1,2,3,\ldots$) to find the minimum energy configuration of this ion (or atom). This step is the most time-consuming one.
(iii) A fitness function is defined as
3. Results and discussions
3.1 Optimized parameters for selected model atoms
From the optimization procedure described above, it is to be expected that the computational load increases quickly with the number of electrons. This is indeed the case in practice. To limit the load of numerical calculations, in this paper we limit the number of electrons to $N=6$. That is, we build classical model atoms with six active electrons. For the C atom, all the electrons are included. For rare gas atoms such as Ne, Ar, Kr, Xe, the electrons in the outmost $p$-shell are included. For alkaline earth atoms such as Mg, two outermost electrons and four inner electrons are included. Model atoms with six active electrons are adequate in describing a substantial portion of strong laser-atom interaction processes. The optimization procedure, however, can be extended to include more active electrons if necessary, provided more computational power is investigated.
Also we fix $\alpha =1$ a.u. for all model atoms. Then the Heisenberg and Pauli potentials felt by different model atoms are of the same character (hardness). This consistency might be preferred. The practical reason from our optimization experience was that the fitness function given in Eq. (6) depends rather weakly on $\alpha$. That is, changing $\alpha$ does not help much in the optimization of the ionization energies. Therefore we fix $\alpha =1$ a.u. for all model atoms. Other values could also be used (as used by other authors), and the model atoms will not be altered substantially. The parameters $\xi _H$ and $\xi _P$ are then optimized for each model atom with the goal that the ionization energies of the model atoms are as close as possible to the experimental values. The major computational load is to find the ionization energies according to Eq. (5), which involves multiple energy-minimization calculations for each parameter combination.
The optimized model parameters and the first six ionization energies of Ne, Ar, Kr, Xe, Mg, and C atoms are shown in Table 1, together with the experimental values of ionization energies for comparison. After optimization, the majority of the tabulated ionization energies agree fairly with the experimental values, especially for the first two ionization energies, which all agree to the experimental values within a few percent. Some relatively large discrepancies are also noticed, such as the fifth and the sixth ionization energies of the C atom.
Table 1. Optimized model parameters, and the first six optimized ionization energies of Ne, Ar, Kr, Xe, Mg, and C atoms, in comparison to the corresponding experimental values. All quantities are in atomic units.
Earlier works on the KW model show that without optimization, even the first ionization energies disagree substantially with the experimental values [36,52,61,62]. From this respect, the optimization results shown in Table 1 are obvious progresses for the KW model. On the other hand, it is to be pointed out that with the existing form of the KW potentials, it is impossible to optimize all the ionization energies of an atom very precisely to the known experimental values.
3.2 Stability of the model atoms
After the model potentials being fixed and the ground state electron configurations being found, we need to check that the model atom is stable. Stability is a well-known problem for classical model atoms with two or more electrons: One or more electrons may “evaporate" from the atom automatically without any external laser field.
The stability of the model atom can be checked by letting the electrons evolve according to the laws of classical mechanics without applying any laser field. The Hamiltonian equations of motion are
Fig. 1. Distances of each electron from the ion core in a weak perturbing laser field, for (a) Ne (b) Ar (c) Kr (d) Xe (e) Mg and (f) C model atoms. The wavelength of the laser is 800 nm and the intensity is 10$^{12}$ W/cm$^{2}$.
Another interesting feature is that the electrons separate into different shells as would be expected. This can be seen in Mg and C. For Mg, the two outer electrons are separated in distance from the other four electrons. As shown in Fig. 1(e), the two outer electrons almost enfold the inner electrons and a shell-like structure can be seen. And for C, the two inner electrons are separated from the outer four electrons. No such separation is seen in the rare gas atom models, as would also be expected. This separation of shells is the result of relatively large gaps in the ionization energies, as shown in Table 1. For Mg, there is a relatively large gap between the second and the third ionization energies. And for C, there is a gap between the fourth and the fifth ionization energies. As will be discussed later, this feature of the model may be used to rethink double ionization from a new perspective.
3.3 Momentum distribution of doubly charged ions with elliptically polarized laser fields
When a model atom is exposed to an external laser field, the first (second,…) ionization energy determines the laser electric field strength around which the first (second,…) electron is pulled out. In fact, it has been understood and demonstrated that the instantaneous laser electric field at the time of electron emission can be mapped to the final momentum of the electron (or of the ion) registered at the detector, if elliptically polarized laser fields are used [59,60]. For sequential double ionization, the momentum distribution of the doubly charged ion shows a four-peak structure along the minor polarization direction. It has been realized that a classical model atom must be constructed with the ionization energies correct in order to reproduce the experimentally observed peak positions. Therefore double ionization with elliptical polarization is a perfect example to test our method.
Figure 2 shows the momentum distribution of Ar$^{2+}$ along the minor polarization direction using the same laser parameters as used in the experiment of Pfeiffer et al. [59]. The laser wavelength is 788 nm, (Gaussian) pulse duration is 33 fs, ellipticity is 0.78, and peak intensity is 4.0 PW/cm$^{2}$. Only with the ionization-energy optimized model parameters as shown in Table 1 for Ar can we obtain the signature four-peak structure, as shown by the solid curve in Fig. 2. This calculation results agree very well with the experimental results shown in Fig. 1 of [59]. Instead, if the unoptimized parameter combinations $\{ \alpha = 5 \text { a.u.}, \xi _H = 0.954 \text { a.u.}, \xi _P = 2.767 \text { a.u.}\}$ are used as given in [52], then the obtained momentum distribution is shown by the dashed line in Fig. 2, which is qualitatively wrong without showing the four-peak structure.
Fig. 2. Momentum distribution along the minor polarization direction for Ar$^{2+}$, generated by elliptically polarized laser pulses. The laser parameters are given in the text. The solid curve is obtained with ionization-energy optimized model atoms, whereas the dashed curve is obtained with un-optimized model atoms.
3.4 Nonsequential double ionization from different electron shells
Our model allows us to revisit some old topics of strong-field ionization from a new perspective. For example, let us ask the following question: Are the two electrons emitted in nonsequential double ionization from the same electron shell? Because our model atoms naturally incorporate features of electron shells, such as in Mg and C, they are useful to attempt the above question.
We use the Mg model atom, which is one of the favorite atomic targets to study double ionization [65–70], to answer the question whether the two electrons emitted in nonsequential double ionization are the two outermost electrons. In our Mg model atom, two electrons are labeled as outer electrons, four are labeled as inner electrons, and they show a clear spatial separation, as seen from Fig. 1(e). When this model atom interacts with a laser field and results in double ionization, we can track the origin of the two emitted electrons and classify all double ionization events into three channels: (1) The outer-outer channel, in which both emitted electrons were outer electrons; (2) The inner-outer channel, in which one electron was an inner electron and the other was an outer electron; (3) The inner-inner channel, in which both electrons were inner electrons.
Figure 3 shows the probability of double ionization, as well as of each channel, as a function of laser intensity for (a) linearly polarized laser field and (b) circularly polarized laser field. For both polarizations, the laser has a wavelength of 800 nm, and a 10-cycle trapezoidal pulse envelope (2 cycles turning on, 6 cycles plateau, and 2 cycles turning off). One sees that for linear polarization, quite surprisingly, most double ionization events are not from the outer-outer channel. The inner-outer channel is the dominant one for the intensity range shown. For circular polarization, the outer-outer channel dominates, as would normally be expected.
Fig. 3. Probability of double ionization for the Mg model atom, as well as of the three double ionization channels (as labeled), as a function of laser intensity, for (a) linearly polarized laser field, and (b) circularly polarized laser field.
The purpose of this example is to give a demonstration of the usefulness of our method. Further discussions about double ionization from different electron shells will be reported in another paper.
4. Conclusion
We propose a method to build stable (autoionization free) classical multielectron model atoms with the first-few ionization energies optimized to experimental values. Our method is based on the auxiliary potential method of Kirschbaum and Wilets. We implement a genetic algorithm to optimize the model parameters. We demonstrate, using numerical examples, the necessity of this optimization in order to agree with experimental results that depend critically on the ionization energies. Also, ionization-energy optimized model atoms automatically show an interesting and encouraging feature that separated electron shells emerge, and this feature allows us to revisit some traditional double ionization processes from a new perspective.
Funding
Joint Fund of National Natural Science Foundation of China and Chinese Academy of Engineering Physics (NSAF) (U1930403); Science Challenge Project (TZ2018005); National Natural Science Foundation of China (11774323).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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