1. Introduction

As a bridge between atomic properties and bulk behavior, the organization, structure, and dynamics of clusters have been a rapidly developing field in recent years [1,2]. Rare gas clusters were employed as unique model systems to investigate several intriguing phenomena, such as enhanced ionization [3–10], interatomic Coulombic decay (ICD) [11–13], Efimov effect [14–16], and so on. To study and understand all these processes, a vital first step is to image the structure of the clusters, which is routinely provided by a well-known technique coined as Laser-based Coulomb Explosion Imaging (LCEI) [17–24]. The metrology of LCEI normally relies on an assumption that the laser pulse is short enough so that the nuclear motion can be safely neglected during the ionization stage of laser-cluster interaction and the electronic excitation process of the fragmental ions is rarely taken into account [25–28]. Then, the final momentum vectors of all fragments are solely determined by the initial geometry of the cluster, i.e., the internuclear distances and bond angles. Inversely, the initial geometry of clusters can be retrieved by measuring the three-dimensional (3D) momentum vector of all ions in coincidence [29].

However, fluorescence experiment clearly indicated that the transient excited multicharged molecules produced by a strong laser field play important roles in the strong-field Coulomb explosion [30]. Matsuda and co-workers demonstrated that excitation of vibrational motion during the interaction of the oxygen trimer with an intense 40 fs laser pulse leads to a significantly different KER distribution after Coulomb explosion compared to what is observed using a 9 fs pulse with the same peak intensity [31]. More recently, it was also found that the fragmental ions can be populated into Rydberg states and the Coulomb explosion process is modified dramatically, i.e., indirect Coulomb explosion channel with Rydberg excitation [32–44]. The understanding of these non-equilibrium electron excitations and their subsequent effect on the fragmentation dynamics of molecules and polyatomic clusters is currently one of the most interesting topics in strong-field physics. In theoretical aspect, the ab initio full-dimensional simulation of the coupled dynamics of electrons and nuclei under strong-field influence, however, is limited by the complexity structure of these systems.

In this paper, we employ a feasible classical polyatomic model going beyond the classical reflection approximation [45]. It is able to take into account the influence of the electrons on the ion fragments dynamically during the laser-cluster interaction. Our model well reproduces both the KER spectra of argon and neon trimers observed in a recent experiment [17]. With the numerical simulations, we are able to reveal the close relationship between the key parameters (peak position, width, and degree of asymmetry) of the KER spectrum and the ionization, excitation, and polarization configurations of a trimer in strong laser field. For example, apart from the much-studied peak position which was commonly used to extract the bond length of a trimer, we demonstrate that the width of the KER spectrum is significantly affected by the species of the constituted atom which in a classical model is described as an ensemble of electrons having an ionization potential equal to the experimental value. Furthermore, we find that the degree of asymmetry on the KER spectrum is induced by the asymmetric ionization configuration of trimers, i.e., how the electrons are ionized before the dissociation starts. Finally, supplemented by a Dalitz plot and by tracing back the typical trajectories, we identify the pumping of different kinds of elliptical orbits as sources of the lower and higher parts of the KER spectra. Our procedure for retrieving the organization, structure, and dynamics of trimers in strong laser field has the potential to be extended to even more complex small clusters.

2. Model

When a trimer is irradiated by a strong laser pulse, the electron on one site (parent atom or ion) might be ionized prior to that on the other site (see Fig. 1). This kind of inherent asymmetry can be passed on leading to the final asymmetry on the KER spectrum. Since both the nuclear and electronic motions should be involved, it is currently computationally challenging to simulate this process by solving the fully dimensional Schrödinger equation. Thus it is desirable to develop a feasible classical or semiclassical model. Great success has been achieved by classical trajectory ensemble models in accounting for the double- [46–49] and triple-ionization [50] of atoms or even small molecules [39] in strong laser field where a soften core potential or a Heisenberg core potential is used to avoid unphysical autoionization. Realistic Coulomb potential is also adopted in the semiclassical model for double ionization of atoms [51, 52], diatomic molecules [40, 41, 53, 54] as well as two-electron triatomic molecules [42, 43, 55], where one electron escapes either by tunneling or over-the-barrier ionization in the field-lower Coulomb potential. In our model trimers we assign one active electron in the vicinity of each parent ion. Realistic Coulomb potential is used and the unphysical autoionization problem is naturally suppressed by the nature of a trimer system that all the constituted atoms are far apart from each other [7.16 a.u. for argon trimer and 6.14 a.u. for neon trimer, atomic units (a.u.) are used throughout the paper unless specified otherwise], and thus each electron is basically localized and circulates around their corresponding parent ion respectively.

 

Fig. 1 Electronic configuration for different charge states of a trimer: (a) X₃, (b) ${\text{X}}_3^{+}$, (c) ${\text{X}}_3^{2+}$, and (d) ${\text{X}}_3^{3+}$. The unknown atom X is chosen as Ar and Ne for comparison in our following calculations. The grey contours represent the equipotential curves of electron in the combined Coulomb field of the ions.

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Specifically, four charge states of the trimer as shown in Figs. 1(a)–1(d), i.e., X₃, ${\text{X}}_3^{+}$, ${\text{X}}_3^{2+}$, and ${\text{X}}_3^{3+}$ (X is Ar or Ne), are considered in our following simulations as the initial states of our trimers. They are formed during the rising edge of the laser field, and can lead to the same ion fragments by ionizing 0–3 more electrons via the Coulomb explosion pathway ${\text{X}}_3^{n+}\to {\text{X}}^{+}+{\text{X}}^{+}+{\text{X}}^{+}+(3-n)\text{e}$ (n=0, 1, 2, and 3). For X₃ in Fig. 1(a), there are three X⁺ ions and three active electrons, respectively. Each electron is sampled from the microcanonical distribution around its corresponding parent X⁺ ion with a binding energy equal to the first ionization potential of X atom. For ${\text{X}}_3^{+}$ [Fig. 1(b)], ${\text{X}}_3^{2+}$ [Fig. 1(c)], and ${\text{X}}_3^{3+}$ [Fig. 1(d)], we replace one, two, and three X⁺ ions with X²⁺, respectively, and the corresponding electron has a binding energy equal to the first ionization potential of X⁺ ion. We assume that the initial predissociations do not significantly modify the ionization dynamics [41] and simplify our model by initializing the nuclei at rest.

Once the ensemble of electrons and ions are prepared, we trace the trajectories individually by solving the Newton’s equations taking full account of the electron-electron, electron-nucleus, and laser-particle (electron and nucleus) interactions as in [37]. Here, the Hamiltonian of the trimer system can be written as

3. Results and discussion

3.1. KER spectrum and Dalitz plot: comparison between experiment and theory

The calculated KER spectra of argon trimer starting from four different charge states are illustrated in Fig. 2(a). It can be seen that all the initial charge states (ICS) result in nearly the same peak position. This is because for any ICS the final dominant breakup geometry is always found to be equilateral triangle. However, the widthes of the distributions are substantially different. While the ICS of Ar₃ gives rise to a very narrow distribution as shown by the black curve in Fig. 2(a), the KER spectrum produced by the ICS of ${\text{Ar}}_3^{+}$ [red curve in Fig. 2(a)] is much wider and obviously asymmetric. The KER spectra associated with the ICSs of ${\text{Ar}}_3^{2+}$ and ${\text{Ar}}_3^{3+}$ are even wider [see the green and blue curves in Fig. 2(a), respectively], and become symmetric again for ${\text{Ar}}_3^{3+}$. Similar features are also found for the neon trimer as shown in Fig. 2(c). By comparing the curves with the same color in Figs. 2(a) and 2(c), one finds that the KER spectra of the neon trimer are obviously narrower than that of the argon trimer for the same charge state. This can be understood because the orbital radius of the outer electron of a neon atom is smaller than that of an argon atom and thus the corresponding dynamics is less affected by the neighboring ions.

 

Fig. 2 The calculated KER spectra of argon (a) and neon trimers (c) starting from different initial charge states (ICS) at the intensities of 3 × 10¹⁴ W/cm² and 2.4 × 10¹⁵ W/cm², respectively. (b)(d) Comparison between the measured and the calculated KER spectra for the Coulomb explosion of argon and neon trimers. Here, the experimental data (scatters) are taken from [17], while the red curve in (b) and the green curve in (d) are simulations on argon and neon trimers with ${\text{Ar}}_3^{+}$ and ${\text{Ne}}_3^{2+}$ as the initial electronic configurations, respectively.

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In Figs. 2(b) and 2(d), we compare our calculated KER spectra with the experimental data taken from [17]. Good agreement is found between theory (with the ICSs of ${\text{Ar}}_3^{+}$ and ${\text{Ne}}_3^{2+}$) and experiment both for argon trimer and neon trimer, even though their measured KER spectra seem substantially different, i.e., the spectrum of argon trimer [open squares in Fig. 2(b)] has a lower peak position, narrower distribution, and looks apparently more asymmetric, as compared with that of neon trimer [open circles in Fig. 2(d)]. The differences in peak position and width of the KER curves are, as demonstrated in [17] and further supported in [28], dominated by the ground state geometries of the trimers. However, the difference in degree of asymmetry is somewhat unexpected. Our simulations well reproduce all the above differences, and thus provide a possible dynamical explanation for the experimental observations.

To give more direct access to the breakup geometry of the three-particle fragmentation, we now map the momentum vectors, event by event, into a Dalitz plot [56], i.e., by rescaling the kinetic energy of each fragment by ε_i = E_ki/(E_k1 + E_k2 + E_k3) and then plotting the data on the $\left[{\epsilon }_3-1/3,({\epsilon }_2-{\epsilon }_1)/\sqrt{3}\right]$ coordinate system, where E_ki(i = 1, 2, 3) are the final kinetic energies of three fragments. In this way, one is able to obtain the geometrical configuration of the exploded trimers intuitively. In a Dalitz plot, the coordinate center corresponds to a equilateral triangle, and the outer circle with radius 1/3 corresponds to linear configurations, as shown in the middle of the first row of Fig. 3. Our simulation data presented in Dalitz plots are shown in Figs. 3(a) and 3(b), for argon (from ICS ${\text{Ar}}_3^{+}$) and neon (from ICS ${\text{Ne}}_3^{2+}$) trimers, respectively. In both plots, the concentration of the distribution to the coordinate origin indicates the dominance of equilateral triangular configuration of fragmentation. Upon closer inspection, however, one finds that the Dalitz plot of neon trimer exhibits a much broader distribution in comparison with that of argon trimer, again, in consistency with the experimental observations shown in Fig. 2 of [17]. Moreover, the structure of the center of the Dalitz plot (colored from red to yellow) is roughly a O-shape for argon trimer [Fig. 3(a)], in contrast to the ∇-shape for neon trimer [Fig. 3(b)], indicating isosceles triangular configuration of the fragment momentum vectors is more pronounced in neon trimer. To better understand these differences, we further split the Dalitz plot into subplots according to the value of the KER of trimers. As can be seen in Figs. 3(c)–(e), for argon trimer, the pattern of the distribution evolves from a ∇-shape in the low-KER region R_L to a O-shape in the medium-KER region R_M, and finally to a Δ-shape in the high-KER region R_H, indicating the dominant break-up geometry changes from obtuse isosceles triangles to equilateral triangle, and finally to acute isosceles triangles. In contrast, the Dalitz plots of neon trimer always peak around the center of the coordinate system in all three different KER regions. The above interesting predictions on the differential Dalitz plots as a function of the KER has not been done in experiments so far and is waiting for future examination.

 

Fig. 3 Dalitz plots for Coulomb explosion of argon (a) and neon (b) trimers at the same laser intensities as in Fig. 2. (c)–(e) Dalitz plots for the argon trimer corresponding to the KER regions of R_L, R_M, and R_H indicated in Fig. 2(b). (f)–(h) Same as (c)–(e) but for neon trimer.

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3.2. Valence-shell electron excitation and polarization

In order to unveil the underlying mechanisms responsible for the dramatic differences in the KER spectra and Dalitz plots between the argon and neon trimers, we trace back the evolution history of the Coulomb explosion for both trimers over a large ensemble of trajectories. Typical trajectories then emerge as shown in Figs. 4(a) and 4(b) for argon trimer ended up with low- and high-KER, respectively, and similarly in Figs. 4(c) and 4(d) for neon trimer.

 

Fig. 4 Typical trajectories of the argon/neon trimers in the low (a)/(c) and high (b)/(d) KER regions, respectively. The evolution time is confined in between 13 o.c. and 13.2 o.c. for clear visualization. The inset figures are schematic drawings of the corresponding elliptical orbits.

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As a reference system to understand the role of electron excitation and polarization, we start our discussion with the ICS of X₃, for which the active electrons of three atoms are all stripped off during the laser-cluster interaction leaving three singly charged ions X⁺ that repel each other symmetrically like point charges [Fig. 1(a)]. Without electron excitation and polarization, the final KER is a constant according to the relationship KER=3/R, which explains the very narrow distribution on the KER spectrum [see black curves in Figs. 2(a) and 2(c)]. By comparison, the broadening and the asymmetry of the KER spectrum are then attributed to the pumping of different elliptical orbits that breaks the symmetry of three fragments.

For the argon trimer (ICS ${\text{Ar}}_3^{+}$), the two electrons around the parent ion Ar⁺ are ionized, while the third electron is still bounded around its parent ion Ar²⁺ with some excitation and polarization. The value of KER is determined by the trajectory configuration of the excited electron, which can be pumped to different elliptical orbits: the orbit leaning towards or shifting away from the two Ar⁺ ions, as shown in Figs. 4(a) and 4(b) [see also the insets for schematic draws], respectively. The former, termed as inward polarization in the following, has a lower Coulomb potential energy of the remained active electron in two Ar⁺ ions, and thus leads to lower final KER value; while the latter, coined as outward polarization, has a higher value of potential energy when the laser field is turned off and finally gives rise to higher KER value.

Due to the Coulomb attraction of the two Ar⁺ ions, the remained active electron has a higher probability to be pumped to the configuration with inward polarization, which explains the relatively longer tail on the low-KER spectrum of the argon trimer as seen in Fig. 2(b). In addition, the shielding effect of the electron cloud make the effective charge of the Ar²⁺ ion experienced by the two Ar⁺ ions to be less than an unit charge in the inward configuration and more than an unit charge in the outward configuration, respectively. Thus, in the former case, the Ar²⁺ ion acquires a lower kinetic energy as compared with that of the two Ar⁺ ions, which in turn leads to the vertex angle associated with the Ar²⁺ to be larger than 60° [Fig. 5(a)]. This explains the ∇-shape in the Dalitz plot for the low-energy region of the KER spectrum of Ar₃ as shown in Fig. 3(c). The explanation for the change from ∇-shape to the Δ-shape [Fig. 3(e)] as the KER increases is in the same line and thus we are not going to the detail.

 

Fig. 5 Long-term evolution of the corresponding trajectories in Fig. 4 from 13.2 o.c. to 2000 o.c. for clear visualization of the motion of the ions. The red and green lines shows the trajectories of the Ar⁺/Ne⁺ and Ar²⁺/Ne²⁺, and the blue lines represents the excited electron bounded around the Ar²⁺/Ne²⁺ ions. The vertex angle θ included by the legs (dashed lines) of the isosceles triangle is also illustrated.

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Likewise, for the neon trimer (ICS ${\text{Ne}}_3^{2+}$), there are also two kinds of typical trajectories as shown in Figs. 4(c) for the inward polarization and 4(d) for the outward polarization, which contribute mainly to the low- and high-energy regions in the KER spectra of neon trimer, respectively. By comparison with the ICS of ${\text{Ar}}_3^{+}$, there remain two active electrons around two Ne²⁺ ions. Whereas the Coulomb attraction of the Ne⁺ ion tends to increase the probability of the inward configuration, the repulsion between two active electrons tends to decrease its probability. The competition between these two mechanisms washes out the asymmetry in the KER spectrum of neon trimer in Fig. 2(d). It is interesting to note that both the inward polarization and the outward polarization give rise to a similar final configuration, i.e., the vertex angle associated with the Ne⁺ ion is larger than 60° [Figs. 5(c) and 5(d)]. This is the reason for the similarity of the Dalitz plots over all different KER regions as shown in Figs. 3(f)–3(h).

4. Conclusion

In conclusion, with classical polyatomic models we have studied the Coulomb explosion of argon and neon trimers in strong laser field and well reproduced the KER spectra of both trimers. Our results have provided physical explanations for the substantial differences in the experimentally observed KER spectra of argon and neon trimers, which is related to the femtosecond-to-attosecond time-scale electron dynamics, i.e., valence-shell electron excitation and polarization via pumping different elliptical obits. The feasibility of classical polyatomic models to loosely bound systems allows us to study theoretically in the future the real-space imaging and related dynamical processes in even more complex clusters.