displacement energies [233], which the joined ZBL+EAM potential form reproduces well [72, 234]. The details on the ZBL form are presented in Sec. 4.1.4.3.
5.2.4 Assumptions of the two-temperature model revisited
The electronic stopping process occurs when the ionic projectile travels ballistically, whereas the e-p coupling applies to small displacement motion of ions. However, the exact boundary between these processes is not clearly defined from the modelling perspective. In the two- temperature theory, the e-p coupling should be effective assuming both electronic and ionic systems are thermalised, i.e. have achieved equipartition of energy and have a defined, roughly uniform, temperature. The electron-electron scattering time is very short, of the order of τe−e = 10 fs, and so an electronic temperature can be assumed at all times. However, the
thermalisation time of the ionic system in a cascade scenario is not known a priori. We have looked at the convergence of the kinetic and potential energies of the ionic system for two exemplar simulations (with friction) at 200 keV (see Fig. 5.2). We note that the ionic system thermalises at around 0.3 ps, and it is past that time that the e-p should presumably become effective. There are several other assumptions of the 2T model, as applied to cascade simulations, which can be challenged, however handling the non-thermalised ions remains the most significant one. Others include the usual 2T model criticism, such as the applicability of the diffusion processes on a nanometre space-scale and to such high electronic temperatures.
The ionic system thermalisation time of 0.3 ps for a 200 keV pka cascade remains an estimate and it is still not clear which of the following modelling options describes the physical process correctly:
1. Apply e-s above an energy cutoff and e-p when both systems locally (i.e. in a temperature voxel) thermalise. This is tricky to implement, as it is hard to obtain an exact condition for thermalisation for several tens of atoms in a temperature voxel. Also, as different criteria are used for the e-s and e-p both could be active at the same - it is debatable if this is a physical scenario. A simpler alternative to this could be a global time cutoff thermalisation time, which can be tuneable parameter,
2. Apply e-s above an energy cutoff and e-p below it. This is computationally easy to implement and excludes e-s and e-p processes overlap.
In this chapter, we will be exploring the effect of e-p and e-s using both methods on the residual radiation damage in α-Fe.
5.3
Simulation considerations
5.3.1 The simulation setup
All simulations were performed with NPT pre-equilibrated (Nose-Hoover NPT ensemble at 300 K, 1 atm for a minimum of 30 ps) bulk iron with periodic boundary conditions. A variable time-step, ∆tM D, with a maximum of ∆tM D = 1 fs was used to simulate the atomic part of
10−3 10−2 10−1 100 101 102 −3 −2 −1 0 1 2 3x 10 5 Time [ps] Energy [eV] Etot E pot Ekin STATIS dir1 10−3 10−2 10−1 100 101 102 −3 −2 −1 0 1 2 3x 10 5 Time [ps] Energy [eV] Etot Epot Ekin STATIS dir2
Figure 5.2: Kinetic and potential energies evolution in two exemplar 200 keV pka events. The ‘global’ thermalisation time for both can be assumed to be τi−i∼ 0.3 ps.
the cascade, to correctly account for the initial large velocities in the system. The electronic system was evolved with a variable time-stepping algorithm too, which depended on the local Te gradients. The time-step of the finite difference solver (∆tF D) was typically a few times
smaller than for the ionic one. All simulations were run for 55k steps, which corresponded approximately to a simulation time of t = 45 ps.
The following sets of simulations were performed, each of which consisted of 20 or 16 runs for 50 and 100 keV pka conditions, respectively.
1. 2M atoms, 50 keV pka, 6× friction MD variations (3× reduced impact energy, 3× friction Eccutoffs),
2. 4M atoms, 50 keV pka, 9× different 2T-MD conditions (various friction Ec and therm-
alisation τi−i cutoffs),
3. 8M atoms, 50 keV pka, 3× different types of boundary conditions for NVE runs (periodic, 2× variation on the thickness of the boundary thermostat),
4. 16M atoms, 100 keV pka, 5× simulations types (1× NVE, 2× friction MD runs, 2× 2T-MD runs).
5.3. Simulation considerations
The projectile was initialised to travel in random directions (randomly generated set of vectors, which were identical across the simulation sets). The projectile was selected to be the middle atom of the simulation cell for 50 keV runs, and around the 2/3 point on the diagonal of the simulation cell in the case of 100 keV runs (in which case the projectile was fired in a random direction into one of the octants). It needs to be noted that only a few of the collision cascades would occasionally wrap around the simulations cell (typically less than 1 in 10 runs), which shows that the simulation cell was of a suitable size to contain the events.
The 200 keV and 500 keV simulation runs (both friction and standard NVE ones) were composed of 100M and 500M atoms, respectively. Stochastic boundary conditions of 10 ˚A thickness and set to 300 K were used. Only four crystallographic directions were simulated: <1.2,1.2,1.8>, <1.3,1.2,1.8>, <1.6,1.7,1.8> and <1.6,1.9,1.0>. These were manually selected to avoiding channelling, unlike in the previous case.
In the two-temperature runs, the electronic system was three times bigger than the ionic one in all directions, so that the ions are effectively embedded in the sea of electrons, and thus the electrons can effectively carry away the energy outside of the MD cell. For instance, in the 16M atom simulation (554 ˚A box size), we used 55 Ti and 165 Te voxels across each
dimension. Robin’s boundary conditions (see Sec. 4.3.3) at 300 K were applied on all sides of the continuum Te solver. Such a configuration is schematically presented in Fig. 4.5.
5.3.2 Measuring defects, displacements and defect clusters
To analyse the damage produced in a collision cascade we keep track of the time evolution of defects and displacements. Displacements (Ndisp) are atoms that move a certain distance
(d) from their initial position at time t = 0, in this case taken to be d = 0.75 ˚A. This method quantifies the overall (cumulative) damage introduced; however some of it can recover on the picosecond and longer timescales due to recombination effects. Hence, we also consider ‘defects’ (N ), which are either interstitial or vacancies. Note that in this notation N = 2NF P,
where NF P is the number of Frenkel pairs. In the model, interstitials are defined as atoms
that are closer to any of the crystalline positions, which are already occupied, by a distance d. Analogously, a vacancy is a crystalline position, for which no atoms exists closer to it than d = 0.75 ˚A. Keeping track of both displacements and defects allows us to quantify the maximum damage, defect recombination dynamics and the resultant primary radiation damage formation.
While taking d = 0.75 ˚A as a ‘sphere’ cutoff value is arbitrary, we note that the dynamics recovery is not very sensitive to d, provided that it is in a sensible range. Too small (d ∼ 0.2˚A) would be affected by thermal fluctuations, while too large (d > 1.0 ˚A) would not identify the defect atoms correctly, as this value would be too close to the nearest neighbour distance. However, two complications arise when using a method with an arbitrary and non-standardised cutoff for defect analysis. Firstly, the value of d can vary across different materials, as it is related to the crystal structure and the lattice parameter. It also differs across literature even for the same materials, making a meaningful comparison with previous modelling results next to impossible [72]. Secondly, while a cutoff works fine for point defects, it identifies crowdions (a
long chain of atoms along a lattice direction containing an additional atom - a single net point defect) incorrectly, as a line of interstitials, whilst in reality it is just one ‘net’ defect inserted into a particular crystal direction. Furthermore, the gross number of defects (i.e. the sum of both vacancies and interstitials in a cluster) reported for a crowdion is highly dependent on the cutoff d chosen. More sophisticated methods, such as Wigner-Seitz (W-S) decomposition, exist (see Fig. 5.3). However, while W-S correctly computes ‘net’ defects, there is significantly more computational expense associated with it.
Clusters are composed of defects that are closer than two nearest-neighbour (2nn) distance (plus a 0.2 perturbation), d = 3.44 ˚A in α-Fe. This criterion (2nn) is very common across liter- ature [71, 235], however other criteria, particularly for vacancy clustering, have been reported (from 1nn to 4nn) [72]. Defect clustering and related analysis was performed with an in-house built code.
Figure 5.3: Comparison of Wigner-Seitz and sphere defects analysis methods on an exemplar defect structure (a set of crowdions) at two projections: (a) (-111) and (b) (-1-11). Intersti- tials (vacancies) identified by the sphere method are indicated in red (blue), while the W-S interstitials are in pink.