D R Mason

The treatment of electronic excitations in atomistic models of radiation damage in metals

Atomistic simulations are a primary means of understanding the damage done to metallic materials by high energy particulate radiation. In many situations the electrons in a target material are known to exert a strong influence on the rate and type of damage. The dynamic exchange of energy between electrons and ions can act to damp the ionic motion, to inhibit the production of defects or to quench in damage, depending on the situation. Finding ways to incorporate these electronic effects into atomistic simulations of radiation damage is a topic of current major interest, driven by materials science challenges in diverse areas such as energy production and device manufacture. In this review, we discuss the range of approaches that have been used to tackle these challenges. We compare augmented classical models of various kinds and consider recent work applying semi-classical techniques to allow the explicit incorporation of quantum mechanical electrons within atomistic simulations of radiation damage. We also outline the body of theoretical work on stopping power and electron-phonon coupling used to inform efforts to incorporate electronic effects in atomistic simulations and to evaluate their performance.

Elastic trapping of dislocation loops in cascades in ion-irradiated tungsten foils

Using in situ transmission electron microscopy (TEM), we have observed nanometre scale dislocation loops formed when an ultra-high-purity tungsten foil is irradiated with a very low fluence of self-ions. Analysis of the TEM images has revealed the largest loops to be predominantly of prismatic 1/2〈111〉 type and of vacancy character. The formation of such dislocation loops is surprising since isolated loops are expected to be highly mobile, and should escape from the foil. In this work we show that the observed size and number density of loops can be explained by the fact that the loops are not isolated-the loops formed in close proximity in the cascades interact with each other and with vacancy clusters, also formed in cascades, through long-range elastic fields, which prevent the escape of loops from the foil. We find that experimental observations are well reproduced by object Kinetic Monte Carlo simulations of evolution of cascades only if elastic interaction between the loops is taken into account. Our analysis highlights the profound effect of elastic interaction between defects on the microstructural evolution of irradiated materials.

How good is damped molecular dynamics as a method to simulate radiation damage in metals

Classical molecular dynamics (MD) is a frequently used technique in the study of radiation damage cascades because it provides information on very small time and length scales inaccessible to experiment. In a radiation damage process, energy transfer from ions to electrons may be important, yet there is continued uncertainty over how to accurately incorporate such effects in MD. We introduce a new technique based on the quantum mechanical Ehrenfest approximation to evaluate different methods of accounting for electronic losses. Our results suggest that a damping force proportional to velocity is sufficient to model energy transfer from ions to electrons in most low energy cascades. We also find, however, that a larger rate of energy transfer is seen when the ionic kinetic energy is confined to a focused sequence of collisions. A viscous damping coefficient dependent on the local atomic environment is shown to be an excellent model for electronic energy losses in low energy cascades in metals.

Electronic damping of atomic dynamics in irradiation damage of metals

We investigate the transfer of energy from a harmonically oscillating atom in a metal to the electronic subsystem, using a direct simulation method based on time-dependent tight-binding (TDTB). We present our results in terms of a viscous damping coefficient β to enable direct comparison with previous MD and Langevin dynamics simulations, over an ionic energy range relevant for radiation damage. Analysis of our results using time-dependent perturbation theory shows that the rate of energy transfer to the electrons is independent of the frequency of the driven atom at high electronic temperatures, but at low temperature may vary by an order of magnitude. Our simulations show β also to be dependent on the electronic temperature, the position of the atom within the unit cell and even the direction of oscillation. We conclude that a TDTB simulation can give the electronic damping for an infinite metal over a limited simulation time window dependent on system size, and show how to monitor errors in dynamic simulations due to finite-size effects.

Stochastic kinetic Monte Carlo algorithms for long-range Hamiltonians

We present a higher order kinetic Monte Carlo methodology suitable to model the evolution of systems in which the transition rates are non-trivial to calculate or in which Monte Carlo moves are likely to be non-productive flicker events. The second order residence time algorithm first introduced by Athenes et al. [Phil. Mag. A 76 (1997) 565] is rederived from the n-fold way algorithm of Bortz et al. [J. Comput. Phys. 17 (1975) 10] as a fully stochastic algorithm. The second order algorithm can be dynamically called when necessary to eliminate unproductive flickering between a metastable state and its neighbours. An algorithm combining elements of the first order and second order methods is shown to be more efficient, in terms of the number of rate calculations, than the first order or second order methods alone while remaining statistically identical. This efficiency is of prime importance when dealing with computationally expensive rate functions such as those arising from long-range Hamiltonians. Our algorithm has been developed for use when considering simulations of vacancy diffusion under the influence of elastic stress fields. We demonstrate the improved efficiency of the method over that of the n-fold way in simulations of vacancy diffusion in alloys. Our algorithm is seen to be an order of magnitude more efficient than the n-fold way in these simulations. We show that when magnesium is added to an Al-2at.%Cu alloy, this has the effect of trapping vacancies. When trapping occurs, we see that our algorithm performs thousands of events for each rate calculation performed.

Electronic excitations and their effect on the interionic forces in simulations of radiation damage in metals

Using time-dependent tight-binding simulations of radiation damage cascades in a model metal we directly investigate the nature of the excitations of a system of quantum mechanical electrons in response to the motion of a set of classical ions. We furthermore investigate the effect of these excitations on the attractive electronic forces between the ions. We find that the electronic excitations are well described by a Fermi-Dirac distribution at some elevated temperature, even in the absence of the direct electron-electron interactions that would be required in order to thermalize a non-equilibrium distribution. We explain this result in terms of the spectrum of characteristic frequencies of the ionic motion. Decomposing the electronic force into four well-defined components within the basis of instantaneous electronic eigenstates, we find that the effect of accumulated excitations in weakening the interionic bonds is mostly (95%) accounted for by a thermal model for the electronic excitations. This result justifies the use of the simplifying assumption of a thermalized electron system in simulations of radiation damage with an electronic temperature dependence and in the development of temperature-dependent classical potentials.

Atomistic modelling of diffusional phase transformations with elastic strain

Phase transformations in 2xxx series aluminium alloys (Al–Cu–Mg) are investigated with an off-lattice atomistic kinetic Monte Carlo simulation incorporating the effects of strai na round misfitting atoms and vacancies. Atomic interactions are modelled by Finnis–Sinclair potentials constructed for these simulations. Vacancy diffusion i sm odelled by comparing the energies of trial states, where the system is partially relaxed for each trial state. No special requirements are made about the description of atomic interactions, making our approach suitable for more fundamentally based models such as tight binding if sufficient computational resources are available. Only a limited precision is required for the energy of each trial state, determined by the value of kBT . Since the change in the relaxation displacement field caused by a vacancy hop decays as 1/r 3 ,i t is suff icient to determine the next move by relaxing only those atoms in a sphere of finite radius centred on the moving vacancy. However, once the next move has been selected, the entire system is relaxed. Simulations of the early stages of phase separation in Al–Cu with elastic relaxation show an enhanced rate of clustering compared to those performed on the same system with a rigid lattice. (Some figures in this article are in colour only in the electronic version)

Faster neighbour list generation using a novel lattice vector representation

In any many-body simulation where particles are coupled using short-range potentials, a key part of the simulation is to find which particles {j } interact with particle i. The set of such particles is known as the neighbour list of particle i. A novel algorithm is developed here which efficiently returns a neighbour list. A partially occupied reference lattice may be constructed for any simulation, with the position of particles defined as being a short vector separation from a node. A lattice vector which preserves translational symmetry in a periodic supercell under addition and subtraction operations can then be constructed from a single 32-bit integer number. A novel neighbour list algorithm is then developed which uses a single set of lattice vectors to return all nodes, and therefore all particles associated with the nodes, within a fixed radius sphere of particle i .T his new algorithm preserves translational symmetry in a periodic supercell, requires a small memory overhead, and is shown to be faster than the well-known Linked-Cell method in all cases considered here.  2005 Elsevier B.V. All rights reserved.

Incorporating non-adiabatic effects in Embedded Atom potentials for radiation damage cascade simulations

In radiation damage cascade displacement spikes ions and electrons can reach very high temperatures and be out of thermal equilibrium. Correct modelling of cascades with molecular dynamics should allow for the non-adiabatic exchange of energy between ions and electrons using a consistent model for the electronic stopping, electronic temperature rise, and thermal conduction by the electrons. We present a scheme for correcting embedded atom potentials for these non-adiabatic properties at the level of the second-moment approximation, and parameterize for the bcc transition metals above the Debye temperature. We use here the Finnis-Sinclair and Derlet-Nguyen-Manh-Dudarev potentials as models for the bonding, but the corrections derived from them can be applied to any suitable empirical potential. We show with two-temperature MD simulations that computing the electronic thermal conductivity during the cascade evolution has a significant impact on the heat exchange between ions and electrons.

Resonant charging and stopping power of slow channelling atoms in a crystalline metal

Fast moving ions travel great distances along channels between low-index crystallographic planes, slowing through collisions with electrons, until finally they hit a host atom initiating a cascade of atomic displacements. Statistical penetration ranges of incident particles are reliably used in ion-implantation technologies, but a full, necessarily quantum-mechanical, description of the stopping of slow, heavy ions is challenging and the results of experimental investigations are not fully understood. Using a self-consistent model of the electronic structure of a metal, and explicit treatment of atomic structure, we find by direct simulation a resonant accumulation of charge on a channelling ion analogous to the Okorokov effect but originating in electronic excitation between delocalized and localized valence states on the channelling ion and its transient host neighbours, stimulated by the time-periodic potential experienced by the channelling ion. The charge resonance reduces the electronic stopping power on the channelling ion. These are surprising and interesting new chemical aspects of channelling, which cannot be predicted within the standard framework of ions travelling through homogeneous electron gases or by considering either ion or target in isolation.