Insights into different stages of formation of swift heavy ion tracks
Introduction
A swift heavy ion (SHI) is typically defined as an ion heavier than carbon with energy E > 1 MeV/u [1], [2]. Such fast ions may appear in nature as fragments of fission processes, in cosmic rays, and also artificially produced at accelerator facilities. Electronic energy losses by such ions constitute over 95% of the energy deposition [1], [2], whereas energy transfer to target atoms is almost negligible along the entire SHI path except near the end of the range. Depending on the target, the ion energy and mass, the electronic stopping power may reach 5–50 keV/nm along the ion trajectory.
Traversing through matter, an SHI initially excites electrons within a few angstroms around its trajectory penetrating to depths of up to hundreds of microns. In its trail the ion may leave regions with structure and phase transformations which constitute a latent ion track: structurally modified material within a diameter of ~ 1–10 nm and a length up to hundreds of microns.
Effects of SHI impacts have found a broad range of applications such as production of track-etched membranes [3], creation of nanodots [4], design of biological materials [5], hadron therapy [6], etc. Irradiation with heavy particles presents certain risks for satellites, spacecraft, and humans in space [7]. Analysis of tracks accumulated in meteorites, in particular, consisting of olivine (Mg2SiO4), potentially allows to identify a presence of superheavy elements in galactic cosmic rays [8].
Design of experiments, as well as development and implementation of technologies based on SHI irradiation, is hampered by a lack of fundamental understanding of the processes of SHI interaction with solids. Development of a reliable quantitative model of the track formation kinetics would considerably facilitate accumulation of such knowledge.
An SHI impact on a target induces a sequence of processes starting from the energy deposition into the electronic system, to the final relaxation of the excited matter. A swift heavy ion passes by an atom of the target within some attosecond (~10-18 s), exciting a number of electrons. At such times, the atoms and electrons of a material can be considered frozen in space. The energy transferred to electrons of a target around the SHI trajectory relaxes via various channels. Delta electrons fly outwards from the SHI trajectory, creating cascades of secondary electrons and transferring energy to the lattice (typically on the sub-picosecond timescales, 10-15 s to 10-12 s) [9]. Simultaneously, deep shell holes, created by an SHI impact, decay via Auger or radiative mechanisms (also during ~ 10-15 s) [10]. Atoms react to the transferred energy from the excited electrons and valence holes (during 10-13 s to 10-12 s) [11], with possible ensuing atomic disorder [12]. Partial or full recovery of the transiently damaged area around an SHI trajectory typically takes ~ 100 ps [12]. All of these points indicate not only extreme initial excitation of a target but also ultrashort spatial and temporal scales of the track formation kinetics.
Over the past decades, a number of models describing SHI irradiation effects have been developed [13]. They may be categorized into analytical or semiempirical models, and numerical ones. Among the most popular semiempirical models are the two-temperature model (TTM [14], within the SHI community also known as the inelastic thermal spike model [15]), and the Coulomb explosion [16].
The TTM considers only the exchange of the kinetic energy between electrons and atoms of the target, modeled within the thermodynamic equilibrium approximation. The model relies on the coupled thermo-diffusion equations for the electronic and the phononic systems. Fitting parameters are introduced empirically to reproduce the experimental results on track diameters [17]. The need to use fitting parameters arises due to inapplicability of the thermodynamic concepts, phononic and diffusion approximations, for description of the nanometric track formation problem at the femtosecond time scale, as will be discussed in this review.
The Coulomb explosion model is based on the assumption that the created charge non-neutrality drives the atomic repulsion thereby creating damaged regions along an SHI path. It assumes that the charge non-neutrality lasts sufficiently long, and the created holes are bound to their parenting ions. These approximations, valid for small molecules and nanoclusters [18], [19], are not justified for solids [11]. The Coulomb explosion model is rarely used nowadays for description of SHI interaction with bulk solids.
Numerical models used to describe a fast ion interaction with matter present an important alternative. Powerful and precise ab-initio approaches (such as e.g. time-dependent density functional theory) are time consuming and at present limited mostly to small systems or low particle energies [20], [21].
Electronic cascades can be described very efficiently with classical Monte Carlo (MC) simulations [22], [23]. Such methods allow simulating the initial stages of electron kinetics induced by an SHI impact [9], [24], [25], [26]. However, they do not describe the reaction of the atomic lattice to such an energy deposition.
Atomic dynamics can be modeled by means of classical Molecular Dynamics (MD) simulations [27], [28], [29]. Tracing each atom’s trajectory, MD methods provide complete information on the transient state of the atomic system, however, they do not have an access to the state of the electronic system and thus rely on an external input, which is often introduced ad hoc [27], [30], [31].
In the recent years, it is becoming a new standard practice to build combined (also known as hybrid or multiscale) approaches [32], [33], [34]. Such methods combine two or more approaches describing different stages and effects of SHI track formation (e.g. excitation of electronic and atomic subsystems, nonthermal effects etc.) into one model, which then allows to treat various aspects of the problem at hand with their own efficient methods [35].
Due to a large difference between the characteristic times of the electronic and atomic systems of a solid, one of the most common combined approaches became the combination of the TTM for description of electronic evolution with MD describing the atomic dynamics [36], [37], [38]. Such methods, however, suffer from the inherent limitations of the models being combined. For example, nonequilibrium electron kinetics, ballistic transport, and processes involving valence holes cannot be properly described within the TTM [11] – thus neither can they be within a joint TTM-MD.
To alleviate the above mentioned limitations, we recently developed a combined approach that describes all the successive stages of an SHI track formation [12], [39], [40], [41]. The ion penetration and ensuing nonequilibrium kinetics of electrons, holes, and photons are traced within the MC code TREKIS [9], [42], whereas the atomic dynamics is modeled by means of an MD [39]. The model allowed us to describe effects of different electron energy spectra generated by ions of different energies, and for the first time to describe the track formation along the entire SHI trajectory.
An extensive validation of the model against the dedicated experiments was performed in our collaboration, proving a good applicability of the approach [12], [39], [40], [41]. The model does not use fitting parameters, thereby allowing us to gain qualitatively new insights into the track production process which will be described in this review.
Section snippets
Experiment
In a series of dedicated experiments, single crystalline α-Al2O3, MgO and Y3Al5O12 (YAG) specimens were irradiated with 167 MeV Xe and 710 MeV Bi ions at 300 K. The irradiations were performed at fluences ranging from 1010 to 1013 cm−2 at the cyclotron complex of FLNR JINR (Dubna, Russia) in order to investigate both the individual track regime and track overlapping regime. Average Xe ion flux was ~ 109 cm-2s−1, and Bi ion flux was ~ 3·108 cm-2s−1. During irradiation, the samples were mounted
Model
To study the entire sequence of processes occurring during an SHI track formation, we applied a hybrid simulation technique describing the coupled kinetics of the excited electronic and the atomic systems of a material irradiated with high-energy heavy ions. First, the asymptotic trajectory MC code TREKIS [9], [42] is used to determine the initial parameters characterizing an excited state of electrons and holes as well as energy transferred to lattice atoms. The calculated radial distribution
Transient electronic exitations
A penetrating SHI initially deposits its energy into the electronic system within a few angstroms around its trajectory. An example of such an energy deposition, resulting electronic excitation, and their evolution is shown in Fig. 1 for the case of 167 MeV Xe impact in Al2O3. Both graphs demonstrate propagation of several different modes of excitation from the track center.
The very front of the excitation is produced by the photons that create secondary electrons upon absorption. These photons
Conclusion
We reviewed here results of our studies of stages of the kinetics of swift heavy ion tracks formation in insulators. The developed hybrid approach combining Monte Carlo and Molecular Dynamics describes in detail each successive stage, starting from the SHI induced electronic excitation, transport of electrons, valence holes and photons, energy exchange with the target lattice, transient excitation and relaxation of the atomic system, and ultimate damage formation. The used efficient way of
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
Partial financial support from the Czech Ministry of Education, Youth and Sports, Czechia [grants numbers LTT17015, LM2015083] is acknowledged by N. Medvedev. The work was supported by the Ministry of Science and High Education of the Russian Federation, Russia in the frameworks of Project No. 16 APPA (GSI). The work of A.E. Volkov was supported by NRC Kurchatov Institute, Russia under Grant n.1603. R.A. Rymzhanov acknowledge partial financial support from AYSS JINR, Russia grant No. 20-502-06.
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2022, Nuclear Instruments and Methods in Physics Research, Section B: Beam Interactions with Materials and AtomsCitation Excerpt :Temperature evolution within the electronic and lattice systems can be understood through the inelastic Thermal Spike (i-TS) model [8,9]. Though the i-TS has been successfully used to fit experimental data from ion irradiation experiments, it should be mentioned that the i-TS is an incomplete model in the sense that it fails to account for ballistic energy transfer at short time scales [10,11] (<100 fs) as well as other non-thermal effects such as athermal melting [10,12]. In the i-TS model, an ion passes through the solid, ionizing and ejecting energetic electrons (delta electrons).