Interatomic Coulombic decay cascades in multiply excited neon clusters

In high-intensity laser light, matter can be ionized by direct multiphoton absorption even at photon energies below the ionization threshold. However on tuning the laser to the lowest resonant transition, the system becomes multiply excited, and more efficient, indirect ionization pathways become operative. These mechanisms are known as interatomic Coulombic decay (ICD), where one of the species de-excites to its ground state, transferring its energy to ionize another excited species. Here we show that on tuning to a higher resonant transition, a previously unknown type of interatomic Coulombic decay, intra-Rydberg ICD occurs. In it, de-excitation of an atom to a close-lying Rydberg state leads to electron emission from another neighbouring Rydberg atom. Moreover, systems multiply excited to higher Rydberg states will decay by a cascade of such processes, producing even more ions. The intra-Rydberg ICD and cascades are expected to be ubiquitous in weakly-bound systems exposed to high-intensity resonant radiation.

W hen an inner-valence ionized atom or molecule is surrounded by other atoms or molecules, the energy released via the electronic relaxation of the ionized species can be transferred to a neighbouring atom or molecule, which uses the released energy to emit one of its electrons. This process is called interatomic/intermolecular Coulombic decay (ICD) 1 . Since the first prediction in 1997 (ref. 1), ICD has been extensively studied both theoretically and experimentally in a variety of weakly bound systems (for recent reviews, see refs 2,3). It is worth noting that ICD has also been observed in molecular water clusters, suggesting that ICD may also occur in waterabundant biological systems 4,5 and be an important source of genotoxic low-energy electrons and radical cations.
Intriguingly it has been shown theoretically and verified experimentally that ICD is not an isolated phenomenon, but might be part of complex cascade relaxation mechanisms, occurring for instance following an Auger decay process [6][7][8][9][10][11][12][13][14] . To understand relaxation mechanisms, it is necessary to be aware of all possible channels and determine their significance, and in the present work, we investigated this subject.
In the regime of high-intensity radiation, Kuleff et al. 15 proposed an ICD mechanism in which two neighbouring species are resonantly photo-excited to low-lying excited states, and the de-excitation of one of them leads to the ionization of the other via an ultrafast energy transfer. It was argued that this process could be an additional important ion-production mechanism in clusters exposed to intense resonant light, largely dominating the direct multi-photon ionization of clusters' constituents at moderate intensities. Recent experimental studies of large helium droplets excited to low-lying states further supported the high efficiency of this variant of the decay processes 16,17 . What happens when the cluster constituents are brought to higher excited states?
In this article, we show that in this case different relaxation dynamics take place. We report the observation of a new type of ICD, as well as of a cascade of ICD processes. For ease of analysis and interpretation, we chose to study neon clusters. Ne clusters containing about 5,000 atoms were irradiated by free electron laser (FEL) pulses of 61 nm wavelength, corresponding to the 2p 6 -2p 5 3d atomic-like transition in the cluster 18 . The FEL pulse produces a multiply excited cluster which can relax either via ICD in which one of the excited 3d electrons recombines with the 2p hole, as shown in Fig. 1a, or with a much slower rate, by fluorescence. The present experiment, however, clearly demonstrated that this 'direct' ICD channel is significantly suppressed and instead, ICD transitions that lead to the final state 2p 5 3s state (see Fig. 1b) are dominant. Hereafter, we will refer to these transitions as 'intra-Rydberg' ICD. We have also observed secondary ICD processes (ICD cascades) resulting from the intermediate excited states such as 2p 5 3s populated by the primary ICD processes (see Fig. 1c). We note that similar processes have been studied in a completely different regime and form of matter, namely Rydberg gases, and shown to be responsible for the so-called avalanche ionization phenomenon Coulombic Decay (ICD) process: Ne*(2p 5 3d) þ Ne*(2p 5 3d)-Ne þ (2p 5 ) þ Ne(2p 6 ) þ e ICD , which can be abbreviated as 3d þ 3d-2p þ e ICD . Here one excited neon atom, Ne*(2p 5 3d), returns to its ground state (Ne(2p 6 )) and the other neon atom is ionized by using the energy transferred by a virtual photon exchange. (b) Intra-Rydberg ICD: 3d þ 3d-3s þ e ICD . In this process, one Ne*(2p 5 3d) undergoes a transition to a lower excited state, Ne*(2p 5 3s), and the other one is ionized. Note that excited neon atoms remain in the neon cluster after intra-Rydberg ICD. (c) ICD cascade: after an intra-Rydberg ICD, the excited species thus produced continue to decay by secondary ICD processes either among themselves, or with other neighbouring excited species. Note that such an ICD cascade produces more ions compared with the case when the multiply-excited cluster would have decayed only by direct ICD processes. and the creation of ultracold plasma [19][20][21][22] . In Rydberg gases, however, collisions are recognized as a prerequisite for the ionizing mechanism, while in the van der Waals clusters studied by us the energy transfer between excited atoms is so efficient, due to the much smaller interatomic distances, that collisions (although not excluded) are not necessary for producing electrons and ions. Our study on clusters complements that on Rydberg gases and makes clear the importance of cascades in condensed matter.  Fig. 2 that the FEL pulse does not cause any significant electron emission from isolated atoms despite its high intensity (B5.3 Â 10 12 W cm À 2 ). The situation changes dramatically for clusters, where a strong electron signal is produced already at moderate FEL intensities (B4.5 Â 10 11 W cm À 2 ) due to the ICD processes.

Results
Electron emission spectra of neon clusters and assignment. Electron spectra from Ne clusters are shown in Fig. 3. On top of the well-known exponential thermal distribution of the electrons 25 , stemming from the increase of the cluster potential by charge accumulation and nanoplasma formation, we also observe distinct structures at about 1.8 and 11 eV that become more pronounced with increasing FEL intensity. Before assigning the observed structures, let us first consider the possible ICD transitions in Ne clusters under the present conditions, using the atomic energy level data for Ne (ref. 26). Table 1 summarizes all energetically allowed transitions. Apart from the 3d to 2p transition, the 3d to 3s transition in one Ne atom is able to eject a 3d electron from a neighbouring excited atom. This intra-Rydberg ICD process can be viewed as a 'super Coster-Kronig' transition since all participating electrons occupy n ¼ 3 shells and, as we will see, dominates the other ICD   Table 1. processes. Moreover, the Ne*(2p 5 3s) produced by the intra-Rydberg ICD processes can continue to decay by ionizing neighbouring Ne*(2p 5 3d) or Ne*(2p 5 3s) (see Fig. 1c).
As we see from Table 1, an interacting pair of Ne*(2p 5 3d) atoms within a cluster can undergo 4 different ICD transitions (2 primary and 2 secondary) emitting electrons with different energies. These energies, obtained using atomic data, are shown with triangles in Fig. 3. The comparison of these ICD electron energies with the electron spectra suggests that the 1.8 eV peak could be attributed to the 3d þ 3d-3s þ e ICD decay route, while the structure around 11 eV could stem from the secondary ICD process: 3s þ 3s-2p þ e ICD . The transition 3s þ 3d-2p þ e ICD emitting electrons with B15 eV, and the 'direct' 3d þ 3d-2p þ e ICD transition (B18.6 eV) are not visible in the experimental spectra, as a consequence of their much lower decay rates (see below).
We note a small discrepancy between the estimated energies of the ICD electrons and the observed peak positions. One reason for the discrepancy could be the interatomic interaction potentials within the cluster that are not included in the estimation using atomic energy levels. Also, the energy of the emitted electrons may be shifted due to the increase of the cluster Coulomb field, as we discuss later.
Theoretical modelling of the ICD transition rates. To validate our interpretation of the measured electron spectra, we performed numerical simulations based on a system of rate equations. The system of equations describes the time evolution of the population of the excited atoms, which can decay pair-wise via an energetically allowed ICD transition with the corresponding rate. To obtain the rate of each possible ICD transition, we made use of the asymptotic formula for the ICD width G (derived in ref. 15) which involves the oscillator strength f of the de-excitation transition in one of the atoms in the pair, and the cross-section s to ionize the other excited electron in the other atom with the energy o released by the de-excitation transition: In the above expression, given in atomic units, c denotes the speed of light and R is the distance between the excited atoms. For all possible ICD transitions, the required atomic quantities (oscillator strengths and ionization cross-sections) were computed at the Hartree-Fock level. The rate equations were solved for the number of electrons emitted in all possible ICD channels along the cascade (see Methods). The results are shown as a histogram in Fig. 4 together with the experimental data. The model cannot predict absolute intensities, but only relative ones, assuming all excited pairs have decayed and all emitted electrons have reached the detector with their initial energies. The model is also not sensitive to the initial concentration of Ne*(2p 5 3d) and, therefore, to the FEL pulse intensity. That is why, to compare with the experimental spectrum, an exponential function accounting for the thermal distribution of the electron emission has been extracted from the experimental curve and added to the theoretical results, and the resulting data have been normalized to the intensity of the measured peak at B11.6 eV. We see that despite the simplicity of the model, it correctly describes the experimental observations: two strong peaks at B2 and B11.6 eV are predicted with an intensity ratio of approximately 2:1, while the transitions at B15.1 and B18.6 eV have very low rates and are thus not observed experimentally. The high-intensity of the line at B2 eV is due to the high decay rate of the 3d þ 3d-3s þ e ICD transition which nearly completely suppresses the 'direct' 3d þ 3d-2p þ e ICD process (see column 6 in Table 1). According to our calculations, the 3s þ 3s-2p þ e ICD transition is also much more efficient compared with the other secondary ICD transitions and, therefore, is the dominant decay mode in the second stage of the decay giving rise to the peak at B11.6 eV. We note that relaxing via a two-step ICD cascade produces up to 1.5 times more ions compared with the direct ICD transition, the upper limit corresponding to the case when all atoms that have relaxed to their 3s excited state in the first ICD step decay pairwise in a second ICD transition.

Discussion
Let us discuss the intensity and the shape of the peak at around 11.6 eV. We see that the peak becomes more prominent at higher FEL intensities, has an asymmetric shape, and a developing  plateau on the low-energy side. Bostedt et al. 27 reported a similar feature in the photoelectron spectra of small argon clusters recorded at 38.7 eV. They suggested a multi-step ionization model, where sequential 1-photon-1-electron photoionization is assumed, and the plateau is attributed to the deceleration of the electrons by the developing attractive Coulomb potential of the cluster ion. In the present experiment, the 3s þ 3s-2p þ e ICD process can take place after at least two primary ICD transitions have produced two positive charges in the cluster. Therefore, in the multi-step ICD process one can expect a deceleration of the ICD electrons by the accumulation of positive ions along the 3d-3s-2p cascade, shifting the energies of the emitted ICD-electrons and thus forming a plateau in the electron spectrum. In addition, the increase of the cluster potential by charge accumulation would result in trapping the low-energy ICD electrons and thus nanoplasma formation. The intensity ratio of thermal electrons to ICD electrons increases with the increase in the FEL intensity up to 2.0 Â 10 11 W cm À 2 and becomes approximately constant in the FEL intensity range of 2.0 Â 10 11 -4.5 Â 10 11 W cm À 2, , as depicted in the inset of Fig. 3, suggesting that the source of thermal electrons could be trapping of the low-energy electrons created by the intra-Rydberg ICD by the charged cluster created by the ICD cascade. Before concluding, a brief account will be given of the comparison between the present work and previously reported ones 16,17,24 . LaForge et al. 16 and Ovcharenko et al. 17 reported results on helium droplets at the resonance derived from the atomic He 1s2p state. The energy of de-excitation from 1s2p to 1s 2 s is, however, not sufficient to ionize another 1s2p excited atom and thus is transferred only to the nuclear motion. In the present work, a novel phenomenon is observed for multiple excitation to higher Rydberg states, where the de-excitation energy of the intra-Rydberg transition can be transferred to another Rydberg atom and ionize it. Ovcharenko et al. clearly identified inelastic scattering of the energetic direct ICD electrons by the excited atoms and called it collective autoionization since more than two excited atoms are involved. In the present experiment, however, we could not identify such processes because of much lower density of excited atoms in the cluster, due to lower FEL intensity by more than one order of magnitude and lower inelastic scattering cross sections. The work of Yase et al. on the same excitation of neon clusters 24 may be more closely related to the present work. However, that work focused on high power (10 13 -10 14 W cm À 2 ) and observed different physics. There, the excited 3d electrons were instantaneously delocalized and a nanoplasma was formed. As a result ICD was suppressed. In the present work, nanoplasma formation was significantly reduced by reducing FEL intensity and thus ICD could be observed.
In conclusion, in the electron emission spectra of Ne clusters exposed to intense EUV-FEL pulses producing multiple excitations to 3d levels, we have identified a novel intra-Rydberg ICD mechanism occurring in the same electronic shell, which is much more efficient than the ICD to the ground state. Moreover, multiple population of higher Rydberg states in a cluster also triggers a very efficient cascade of ICDs in a step-wise de-excitation to the ground state, producing in that way more ions. A two-step cascade may produce up to 50% more ions compared with a single step relaxation. We note that, like all previously discovered ICD processes, these intra-Rydberg ICD and ICD cascades are expected to be general phenomena occurring in any weakly bound media (notably: water) exposed to intense light that is resonant with a higher-lying excited state of its constituents.

Methods
Experimental methods. The experiments were performed at the SPring-8 Compact SASE Source (SCSS) test accelerator in Japan 23 . Intense EUV-FEL pulses were focused on a Ne cluster beam, and the kinetic energy distribution of emitted electrons was measured with a velocity map imaging (VMI) spectrometer 28 . The Ne clusters were prepared by adiabatic expansion of Ne gas through a 0.25 mm nozzle 29 . The Ne stagnation pressure was 10.8 bar, and the nozzle was cooled to 80 K using liquid nitrogen. The average cluster sizes hNi were estimated to be about 5,000 atoms according to the well-known scaling law 30,31 . The FEL light source provided EUV pulses with a photon energy of 20.3 eV (bandwidth was 0.3 eV FWHM) and a duration of 30 fs at a repetition rate of 30 Hz. The FEL beam was steered by two upstream SiC plane mirrors, passed a gas attenuator, a 4-jaw slit, and a gas monitor detector before entering the pre-focusing system of the beam line. The FEL intensity was reduced by the 4-jaw slit down to 5% of full power and was adjusted by the gas attenuator. The skimmed FEL beam with a mean spot size of 14 mm (FWHM) was focused onto the cluster beam at the center of the VMI spectrometer. The emitted electrons were accelerated toward a position-sensitive detector consisting of a set of micro channel plates (MCPs) followed by a phosphor screen, and recorded using a CCD camera synchronized to the arrival of the FEL pulse in the interaction chamber. The measured two-dimensional (2D) projection allows for the reconstruction of the three-dimensional (3D) momentum distribution of the ejected electrons using the pBASEX algorithm 32 . The energy scale of the electron spectrum was calibrated using the 3p photo-line of atomic argon (IP ¼ 15.76 eV) with an energy resolution of 5%.
Theoretical modelling. The present calculations were performed by solving a system of coupled rate equations that describes the population of the excited atoms decaying pair-wise by a cascade of ICD processes. For extracting the rate constant for each type of ICD transition, the Ne cluster was represented by a face-centered cubic system with a lattice constant of 4.4 Å (corresponding to a nearest-neighbor distance of 3.1 Å) filling a sphere with a radius of 29.6 Å, resulting in a cluster of 5,017 neon atoms. Assuming that the FEL pulse can excite about 10% of the atoms in the cluster, the total decay rate (width) of each transition was obtained as the sum of the partial ICD widths, G, of all possible pairs of 500 randomly-distributed excited neon atoms estimated via equation (1). The latter was derived by expanding the Coulomb transition matrix element between the ICD initial and final electronic states of the dimer in a multipole series, and keeping only the dipole-dipole term 15,33 . In this asymptotic approximation, the ICD width is the product of the transition probabilities for the radiative de-excitation in one of the atoms and for the photoionization of the other one. Those transitions are characterized by the excitation oscillator strength f and by the photoionization cross-section s which were computed at the Hartree-Fock level. We note that although within this approximation the radiative transition 3d-3s is dipole-forbidden, the ICD process 3d þ 3d-3s þ e ICD is allowed, since the outgoing electron carries the angular momentum required for the conservation of the total momentum in the Coulomb matrix element. Moreover, in a large cluster the orbital momentum is no longer a good quantum number as the atomic orbitals overlap locally to form mixed states. Thus, a strict distinction between p, d, and so on. Rydberg state is nearly impossible. The important difference is between intra-and inter-shell transitions. That is why, in order to estimate the ICD width for 3d þ 3d-3s þ e ICD transition on the same level of theory, the oscillator strength of the atomic dipole-allowed 3d-3p transition was utilized in equation (1). This is a rough assumption, but the model is not very sensitive to the exact value of the oscillator strength. It can be a few orders of magnitude smaller and will still quench the 3d þ 3d-2p þ e ICD process. This is a result of the other quantities entering in equation (1), the ionization cross-section s and the virtual photon energy o, which also largely favour the 3d þ 3d-3s þ e ICD transition. The ionization cross-section for removing a 3d electron with a 2 eV photon (corresponding to the 3d-3s transition) is about 3 orders of magnitude larger than the cross-section for removing the same electron with a 19 eV photon (corresponding to the 3d-2p transition). In addition, the decay rate depends as 1/o 2 on the virtual-photon energy.
We note also that since the model does not take the cluster dynamics into account and assumes that all excited pairs have decayed by ICD to produce electrons with the respective energies, it is not sensitive to the initial population of Ne*(2p 5 3d) in the cluster and, therefore, to the FEL intensity.
Data availability. The data that support the findings of this study are available from the corresponding author on reasonable request.