The dependence of in-cascade interstitial clustering on cascade energy is shown in Figure 12 for simula­tion temperatures of 100, 600, and 900 K, where the average number of interstitials in clusters of size two or larger at each energy has been divided by the total number of surviving interstitials in part (a), and by the number of displaced atoms predicted by the NRT model for that energy in part (b). The data points and error bars in Figure 12 indicate the mean and stan­dard error at each energy. The error bars can be used to make two significant comments. First, the relative scatter is much higher at lower energies, which is similar to the case of defect survival shown in Figure 10. Second, comparing again with Figure 10, the standard errors about the mean for interstitial clustering are greater at each energy than they are for defect survival.

The fact that the interstitial clustering fraction exhibits greater variability between cascades at a given energy than does defect survival is essentially related to the variety of defect configurations that are possible. A given amount of kinetic energy tends to produce a given number of stable point defects; this simple observation is embedded in the NRT model, that is, the number of predicted defects is linear in the ratio of the energy available to the energy per defect. However, any specific number of point defects can be arranged in many different ways.

At the lowest energies, where relatively few defects are created, some cascades produce no interstitial clusters and this is primarily responsible for the larger error bars at these energies. The average frac­tion of interstitials in clusters is about 20% of the NRT displacements above 5 keV, which corresponds to about 60% of the total surviving interstitials. Although it is not possible to discern a systematic effect of temperature below 10 keV, there is a trend toward greater clustering with increasing tempera­ture at higher energies. This can be more clearly seen in Figure 12(a) where the ratio of clustered inter­stitials to surviving interstitials is shown, and in the high-energy values in Table 2. This effect of tem­perature on interstitial clustering in these adiabatic simulations is consistent with the observations of Gao and coworkers77 mentioned above, that is, they found that the interstitial clustering fraction increases with temperature.

The interstitial cluster size distributions exhibit a consistent dependence on cascade energy and tem­perature as shown in Figure 13 (where a size of 1 denotes the single interstitial). The cascade energy dependence at 100K is shown in Figure 13(a), where the size distributions from 10 and 50 keV are included. The influence of cascade temperature is shown for 10 keV cascades in Figure 13(b), and for 20 keV cascades in Figure 13(c). All interstitial clus­ters larger than size 10 are combined into a single class in the histograms in Figure 13. The interstitial cluster size distribution shifts to larger sizes as either the cascade energy or temperature increases. An increase in the clustering fraction at the higher temperatures is most clearly seen as a decrease in the number of mono-interstitials. Comparing Figures 13(b) and 13(c) demonstrates that the temperature dependence increases as the cascade energy increases. The largest interstitial cluster observed in these simulations was contained in a 20 keV cascade at 600 K as shown in Figure 14. This large cluster was composed of 33 interstitials (<111> crowdions), and exhibited con­siderable mobility via what appeared to be a 1D glide in a < 111 > direction.64,66

Although the number of point defects produced and the fraction of interstitials in clusters was shown to be relatively independent of neutron energy spec­trum,82 the increase in the number of large clusters at higher energies suggested that the in-cascade clus­ter size distributions may exhibit more sensitivity to neutron energy spectrum than did these other para­meters. At 100 K, there are no interstitial clusters larger than 8 for cascade energies of 10 keV or

less. Therefore, the fraction of interstitials in clusters of 10 or more was chosen as an initial parameter for evaluation of the size distributions. This partial interstitial clustering fraction is shown in Figure 15. As the large clusters are relatively uncommon, the fraction of interstitials contained in them is corre­spondingly small. This leads to the relatively large standard errors shown in the figure. However, it is clear that the energy dependence of the formation of these large clusters is much stronger than simply the
total fraction of interstitials in clusters. Infrequent large clusters such as the 33-interstitial cluster shown in Figure 14 play a significant role in the sharp increase in this clustering fraction observed between 100 and 600 K for the 20 keV cascades.

One unusual observation reported by Wooding and coworkers60 and Gao and coworkers86 was that some of the interstitial clusters exhibited a complex 3D morphology rather than collapsing into planar dislocation loops which are expected to have lower

Average interstitial cluster distributions: 10keV cascades at 100 and 900K

Average interstitial cluster distributions: cascades at 100K

‘со Ф to о _co .CO Ф _g О c q

(b)

Iron cascade simulations Mean and standard error: 100K 600K 900K

Y

Vacancy Interstitial

X

z

Figure 14 Residual defects at ~30 ps from a 20 keV cascade at 600 K containing a 33-interstitial cluster.

energy. Similar clusters have been seen in materials such as copper, although they appear to be less fre­quent in copper.54 The existence of such clusters has been confirmed with interatomic potentials that were developed more recently and with ab initio
calculations.87 Representative examples of these clusters are shown in Figure 16, where a ring-like four-interstitial cluster is shown in (a) and a five — interstitial cluster is shown in (b). Unlike the mobile clusters that are composed of [111] crowdions such

as the one shown in Figure 14, the SIA clusters in Figure 16 are not mobile. As such, they have the potential for long lifetimes in the microstructure and may act as nucleation sites for larger interstitial-type defects. Figure 17 shows a somewhat larger sessile cluster containing eight SIA. This particular cluster was examined in detail by searching a large number of low-order crystallographic projections in an attempt to find a projection in which it would appear as a loop. Such a projection could not be found. Rather, the cluster was clearly 3D with a single di-, tri-, and di-interstitial on adjacent, close-packed (110) planes as shown in the figure. The eighth inter­stitial is a [110] dumbbell that lies perpendicular to the others and on the left side in Figure 17(a). Figure 16(b-d) are [101] projections through the three center (101) planes in Figure 17(a).

It is possible that the typical 10-15 ps MD simu­lation was not sufficient for the cluster to reorient and collapse. To examine this possibility, the simulation time of a 10keV cascade at 100 K that contained a similar eight SIA cluster was continued up to 100 ps. Very little cluster restructuring was seen over the time from 10 to 100 ps. In fact, the cluster had coa­lesced into nearly its final configuration by 10 ps. Gao and coworkers86 carried out a more systematic inves­tigation of sessile cluster configurations with extended simulations at 300 and 500 K. They found that many sessile clusters had converted to glissile within a few hundred picoseconds, but at least one eight SIA cluster remained sessile for ^500 ps even after aging at temperatures up to 1500 K. Given the impact that stable sessile clusters would have on the longer timescale microstructural evolution as
discussed in Chapter 1.13, Radiation Damage The­ory, further research is needed to characterize the long-term evolution of cascade-created point defect clusters. It is significant to point out that the conver­sion of glissile SIA clusters into sessile clusters has also been observed. For example, in a 20 keV cascade at 100 K, a glissile eight SIA cluster was trapped and converted into a sessile nine SIA cluster when it reacted with a single [110] dumbbell. The simulation was continued for more than 200 ps and the cluster remained sessile.

The mechanism responsible for interstitial clus­tering has not been fully understood. For example, it has not been possible to determine whether the motion and agglomeration of individual interstitials and small interstitial clusters during the cascade event contributes to the formation of the larger clus­ters that are observed at the end of the event. Alter­nate clustering mechanisms in the literature include the suggestion by Diaz de la Rubia and Guinan88 that large clusters could be produced by a loop punching mechanism. Nordlund and coworkers62 proposed a ‘liquid isolation’ model in which solidification of a melt zone isolates a region with excess atoms.

However, a new mechanism has recently been elucidated by Calder and coworkers,80 which seems to explain how both vacancy and interstitial clusters are formed, particularly the less frequent large clus­ters. Their analysis of cluster formation followed an investigation of the effects of PKA mass and energy, which demonstrated that the probability of produc­ing large vacancy and SIA clusters increases as these parameters increase.89 The conditions of this study produced a unique dataset that motivated the effort to unravel how the clusters were produced. They developed a detailed visualization technique that enabled them to connect the individual displace­ments of atoms that resulted in defect formation by comparing the start and end positions of atoms in the simulation cell. This defined a continuous series of links between each vacancy and interstitial that were ultimately produced by a chain of displacements. These chains could be displayed in what are called lines of ‘spaghetti.’80 Regions of tangled spaghetti define a volume in which atoms are highly agitated and a certain fraction of which are displaced. Stable interstitials and interstitial clusters are observed on the surface in this volume.

From their analysis of cascade development and the final damage state, Calder and coworkers were able to demonstrate a correlation between the production of large SIA clusters and a process taking place very early in the development of a cascade. Specifically, they established a direct connection between such clusters and the formation of a hypersonic recoil atom that passed through the supersonic pressure wave created by the initiation of the cascade. This highly energetic recoil may create a subcascade and a secondary supersonic shockwave at an appropriate distance from the primary shockwave. In this case, SIA clusters tend to be formed at the point where the primary and secondary shockwaves interfere with one another. This process is illustrated in Figure 18.80 Atoms may be transferred from the primary shock­wave volume into the secondary shockwave volume, creating an interstitial supersaturation in the latter and a vacancy supersaturation in the former. In this case, the mechanism of creating large SIA clusters early in the cascade process correspondingly leads to the formation of large vacancy clusters by the end of the thermal spike phase, that is, after several picose­conds. It is notable that the location of the SIA cluster is determined well before the onset of the thermal spike phase, by about 0.1 ps. Calder’s spaghetti analy­sis provides the opportunity for improved definition of parameters such as cascade volume and energy density; the interested reader is directed to Calder and coworkers80 for more details.