MD simulations have been employed to investigate displacement cascade evolution in a wide range of materials. The literature is sufficiently broad that any list of references will be necessarily incomplete; Malerba,41 Stoller,43 and others52-70 provide only a representative sample. Additional references will be given below as specific topics are discussed. The recent review by Malerba41 provides a good summary of the research that has been done on iron. These MD investigations of displacement cascades have estab­lished several consistent trends in primary damage formation in a number of materials. These trends include (1) the total number of stable point defects produced follows a power-law dependence on the cas­cade energy over a broad energy range, (2) the ratio of MD stable displacements divided by the number obtained from the NRT model decreases with energy until subcascade formation becomes prominent, (3) the in-cascade clustering fraction of the surviving defects increases with cascade energy, and (4) the effect of lattice temperature on the MD results is rather weak. Two additional observations have been made regarding in-cascade clustering in iron, although the fidelity of these statements depends on the interatomic potential employed. First, the interstitial clusters have a complex, three-dimensional (3D) morphology, with both sessile and glissile configura­tions. Mobile interstitial clusters appear to glide with a low activation energy similar to that of the mono­interstitial (~0.1—0.2 eV).71 Second, the fraction of the vacancies contained in clusters is much lower than the interstitial clustering fraction. Each of these points will be discussed further below.

The influence of the interatomic potential on cascade damage production has been investigated by several researchers.42,72-74 Such comparisons generally show only minor quantitative differences between results obtained with interatomic potentials of the same general type, although the differences in cluster­ing behavior are more significant with some potentials. Variants of embedded atom or Finnis-Sinclair type potential functions (see Chapter 1.10, Interatomic Potential Development) have most often been used.

However, more substantial differences are sometimes observed that are difficult to correlate with any known aspect ofthe potentials. The analysis recently reported by Malerba41 is one example. In this case, it appears that the formation of replacement collision sequences (RCS) (discussed in Section 1.11.4.1) was very sensi­tive to the range over which the equilibrium part of the potential was joined to the more repulsive pair potential that controls short-range interactions. This changed the effective cascade energy density and thereby the number of stable defects produced.

Therefore, in order to provide a self-consistent database for illustrating cascade damage production over a range of temperatures and energies and to provide examples of secondary variables that can influence this production, the results presented in this chapter will focus on MD simulations in iron using a single interatomic potential. This

potential was originally developed by Finnis and Sinclair21 and later modified for cascade simulations by Calder and Bacon.58 The calculations were carried out using a modified version of the MOLDY code written by Finnis.75 The computing time with this code is almost linearly proportional to the number of atoms in the simulation. Simulations were carried out using periodic, Parrinello—Rahman boundary condi­tions at constant pressure.76 As no thermostat was applied to the boundaries, the average temperature of the simulation cell was increased as the kinetic energy of the PKA was dissipated. The impact of this heating appears to be modest based on the observed effects of irradiation temperature discussed below, and on the results observed in the work of Gao and coworkers.77 A brief comparison of the iron cas­cade results with those obtained in other metals will be presented in Section 1.11.5.

The primary variables studied in these cascade simulations is the cascade energy, EMD, and the irra­diation temperature. The database of iron cascades includes cascade energies from near the displacement threshold (~100 eV) to a 200 keV, and temperatures in the range of 100—900 K. In all cases, the evolution of the cascade has been followed to completion and the final defect state determined. Typically this is reached after a few picoseconds for the low-energy cascades and up to ~15 ps for the highest energy cascades. Because of the variability in final defect production for similar initial conditions, several simu­lations were conducted at each energy to produce statistically meaningful average values. The para­meters of most interest from these studies are the number of surviving point defects, the fraction of these defects that are found in clusters, and the size distribution of the point defect clusters. The total number of point defects is a direct measure of the residual radiation damage and the potential for long — range mass transport and microstructural evolution. In-cascade defect clustering is important because it can promote microstructural evolution by eliminat­ing the cluster nucleation phase.

The parameters used in the following discussion to describe results of MD cascade simulations are the total number of surviving point defects and the fraction of the surviving defects contained in clus­ters. The number of surviving defects will be expressed as a fraction of the NRT displacements listed in Table 1, whereas the number of defects in clusters will be expressed as either a fraction of the NRT displacements or a fraction of the total surviv­ing MD defects. Alternate criteria were used to define a point defect cluster in this study. In the case of interstitial clusters, it was usually deter­mined by direct visualization of the defect struc­tures. The coordinated movement ofinterstitials in a given cluster can be clearly observed. Interstitials bound in a given cluster were typically within a second nearest-neighbor (NN) distance, although some were bound at third NN. The situation for vacancy clusters will be discussed further below, but vacancy clustering was assessed using first, sec­ond, third, and fourth NN distances as the criteria. The vacancy clusters observed in iron tend to not exhibit a compact structure according to these defi­nitions. In order to analyze the statistical variation in the primary damage parameters, the mean value (M), the standard deviation about the mean (s), and the standard error of the mean (e) have been calculated for each set of cascades conducted at a given energy and temperature. The standard error of the mean is calculated as e = ff/я0 5, where n is the number of cascade simulations completed.78 The standard error of the mean provides a measure of how well the sample mean represents the actual mean. For example, a 90% confidence limit on the mean is obtained from 1.86e for a sample size of nine.79 These statistical quantities are summarized in Table 2 for a representative subset of the iron cascade database.

Table 2 Statistical analysis of primary damage parameters derived from MD cascade simulations

Energy (keV) Temperature (K) Number of cascades Surviving MD Clustered interstitials (mean /

displacements standard deviation / standard error)

(mean / standard deviation / standard error)

Table 2 Continued

Energy (keV) Temperature (K) Number of cascades Surviving MD Clustered interstitials (mean /

displacements standard deviation / standard error)

(mean / standard deviation / standard error)