Initial work of Bacon and coworkers indicated that the number of stable displacements remaining at the end of a cascade simulation, ND, exhibited a power — law dependence on cascade energy.84 For example, their analysis of iron cascade simulations between 0.5 and 10 keV at 100 K showed that the total number of surviving point defects could be expressed as

Nd = 5-67E MD79 [3]

where EMD is given in kiloelectronvolts. This rela­tionship is not followed below about 0.5 keV because true cascade-like behavior does not occur at these
low energies. Subsequent work by Stoller64-67 indi­cated that Nd also begins to deviate from this energy dependence above 20 keV when extensive subcascade formation occurs. This is illustrated in Figure 9(a) where the values of ND obtained in cascade simula­tions at 100 K is plotted as a function of cascade energy. At each energy, the data point is an average of between 7 and 26 cascades, and the error bars indicate the standard error of the mean. It appears that three well-defined regions with different energy dependencies exist. A power-law fit to the points in each energy region is also shown in Figure 9(a). The best-fit exponent in the absence of true cas­cade conditions below 0.5 keV is 0.485. From 0.5 to

(b)

ratio at low energies was first measured in 1978 and

is well known.57,85 The error bars again reflect the standard error and the dashed line through the points is only a guide to the eye. The MD/NRT ratio is greater than 1.0 at the lowest values of EMD, indicat­ing that the NRT formulation underestimates defect production in this energy range. This is consistent with the low-energy (near threshold) simulations preferentially producing displacements in the ‘easy’ directions.[8] [9] [10] [11] [12] [13] [14] The actual displacement threshold var­ies with crystallographic direction and is as low as ~19 eV in the [100] direction.20,84 Thus, using the recommended average value of 40 eV £d in eqn [2] predicts fewer defects at low energies. The aver­age value is more appropriate for the higher energy events where true cascade-like behavior occurs. In the cascade-dominated regime, the defect density within the cascade increases with energy. Although many more defects are produced, their close proxim­ity leads to a higher probability of in-cascade recom­bination and a lower defect survival fraction.

The surviving defect fraction shows a slight increase as the cascade energy increases above 20, and the indicated standard errors make it arguable that the increase is statistically significant. If signifi­cant, the increase appears to be associated with sub­cascade formation, which becomes prominent above 10-20 keV. In the channeling regions between the high-angle collisions that produce the subcascades shown in Figures 7 and 8, the moving atom loses energy in many low-angle scattering events that pro­duce low-energy recoils. These are essentially like low-energy cascades, which have higher-than-average defect survival fractions (Figure 9). These events could contribute to the incremental increase in defect sur­vival at the highest energies. The average defect sur­vival fraction of ^0.3 NRT shown for cascade energies greater than about 10 keV is consistent with values of Frenkel pair formation obtained from resistivity change measurements following low-temperature

neutron irradiation and ion irradiation.26,27,57,85

The effect of irradiation temperature is shown in Figure 10, which compares the defect survival fractions obtained from simulations at 100, 600, and 900 K. Although it is difficult to discern a consistent effect of temperature between the 600 and 900 K data points, the defect survival fraction at 100 K is always somewhat greater than at either of the two higher temperatures. A similar result for iron was reported in Bacon and coworkers.84 In addition to an interest in radiation temperature itself, the effect of temperature is relevant to the
simulations presented here because no thermostat was applied to the simulation cell to control tem­perature. As mentioned above, the energy intro­duced by the PKA will lead to some heating if the simulation cell temperature is not controlled by a thermostat. For example, in a 1 keV cascade simula­tion with 54 000 atoms, the average temperature rise will be about 140 K when all the kinetic energy of the PKA is distributed in the system. This change in temperature should be more significant at 100 K than at higher temperatures. The fact that defect survival at 600 and 900 K is lower than at 100 K suggests that the 100 K results may be

somewhat biased toward lower survival values by the PKA-induced heating. This is in agreement with the effect of temperature reported by Gao and coworkers77 in their study of 2 and 5 keV cascades with a hybrid MD model that extracted heat from the simu­lation cell. On the other hand, the difference between the 100 and 600 K results is not large, so the effect of ^200 K of cascade-induced heating may be modest.

A simple assessment of this cascade-induced heat­ing was carried out using 10 keV cascades at 100 K. Two independent sets ofsimulations were carried out, seven simulations in a cell of 128 k atoms and eight simulations in a cell of 250 k atoms. A 10 keV cascade will raise the average temperature by 604 and 309 K, respectively, for these two cell sizes. The results of these simulations are summarized in Figure 11, where the parameters plotted are the surviving defect fraction (per NRT), the fraction of interstitials in clusters (per NRT), and the fraction of interstitials in clusters (per surviving MD defect). In each case, the range ofvalues for the two populations are shown, along with their respective mean values with the standard error indicated. The mean and standard error for the combined data sets is also shown. Although the heating differed by a factor of two, it is clear that the defect survival fraction is essentially identical for both populations. There is a slight trend in the interstitial clustering results, which indicates that a higher temperature (due to a smaller number of atoms) promotes interstitial clustering. This is consis­tent with the results that will be discussed below.