As shown in Section 1.13.5, in the framework of FP3DM, the swelling rate depends on the dislocation density and becomes small for a low dislocation den­sity, dS/df « Bdpd/k2 ! 0 at pd! 0 (see eqn [96]). Thus, it was a common belief that the swelling rate in well-annealed metals has to be low at small doses, that is, when the dislocation density increase can be neglected. Under neutron irradiation, the effect of dislocation bias on swelling is even smaller because of intracascade recombination: (dS/df)^^ =

(dS/dfED(1 — er) < (dS/df)Zf. It has been

found experimentally, however, that the void swelling rate in fully annealed pure copper irradiated with fission neutrons up to about 10—2dpa (see Singh and Foreman18) is of ~1% per dpa, which is similar to the maximum swelling rate found in materials at high enough irradiation doses. This observation was one of those that prompted the development of the PBM. The production bias term in eqn [138] allows the understanding of these observations. Indeed, at low doses of irradiation, the void size is small, and therefore, the void cross-section for the inter­action with the SIA glissile clusters is small (Krc2Nc/Lg ^ 1). As a result, the last term in the pro­duction bias term is negligible and thus the swelling rate is driven by the production bias:

 

dS df

 

[138] where f = GNRTt is the NRT irradiation dose. The first term in the brackets on the right-hand side of eqn [138] corresponds to the influence of the dislocation bias and the second one to the production bias. The factor (1 — er) describes intracascade recombination of defects, which is a function of the recoil energy and may reduce the rate of defect production by up to an order of magnitude that can be compared to the NRT value: (1 — er)! 0.1 at high PKA energy (see Section 1.13.3). As indicated by this equation, the swelling rate is a complicated function of dislocation density, dislocation bias factor, and the densities and sizes of voids and PD clusters. It also demonstrates the dependence of the swelling rate on the recoil energy, determined by eig, which increases with increasing PKA energy up to about 10-20 keV. The main predictions of the PBM are dis­cussed below.

 

1.13.6.2 Main Predictions of Production Bias Model As can be seen from eqn [138], the action and con­sequences of the two biases, the dislocation and production ones, is quite different. As shown in Section 1.13.5, the dislocation bias depends only slightly on the microstructure and predicts indefinite void growth. In contrast, the production bias can be positive or negative, depending on the microstruc­ture. The reason for this is in negative terms in eqn [138]. The first term decreases the action of the

sessile vacancy and SIA clusters, the swelling rate is given by dS/df ~ 1/2(1 — er)eg where the sink strength ratio, k//(k2 + Z/pd), is taken to be equal to 1/2, as frequently observed in experiments. Taking into account the magnitude of the cascade para­meters er and eg estimated in Golubov et a/.24 and neglecting the dislocation bias term in eqn [138], one may conclude that the maximum swelling rate under fast neutron irradiation may reach about 1% per dpa. As pointed out in Section 1.13.5, in the case of FP production, that is, in the FP3DM, the maximum swelling rate is also ~1% per dpa. This coincidence is one of the reasons why an illusion that the FP3DM model is capable of describing damage accumula­tion in structural and fuel materials in fission and future fusion reactors has survived despite the fact that nearly 20 years have passed since the PBM was introduced.

Note that the production bias provides a way to understand another experimental observation, namely, that the swelling rate in some materials decreases with increasing irradiation dose (see, e. g., Figure 5 in Golubov et a/.24). Such a decrease is predicted by eqn [138], as the negative term of the production bias, %r Nc/Lg, increases with an increase in the void size. As the first term in the 10-4

Figure 5 Experimentally measured133 and calculated24 levels of void swelling in pure copper after irradiation with 2.5 MeV electrons, 3MeV protons, and fission neutrons. The calculations were performed in the framework of the FP3DM for the electron irradiation and using the production bias model as formulated in Singh etal.22 for irradiations with protons and fission neutrons. From Golubov etal.24

production bias is proportional to the void radius and the second to the radius squared, the swelling rate may finally achieve saturation at a mean void radius equal to Rmax ~ 2яг/.19,30,35

Finally, the cascade production of the SIA clusters may strongly affect damage accumulation. As can be seen from eqn [132], the steady-state sink strength of the sessile SIA clusters is inversely proportional to the fraction of SIAs produced in cascades in the form of mobile SIA clusters, thus k2d ! 1 when eg! 0. This limiting case was considered by Singh and Foreman18 to test the validity of the original frame­work of the PBM,16,17 where all the SIA clusters produced by cascades were assumed to be immobile (hereafter this case of eg = 0 is called the Singh— Foreman catastrophe). If for some reasons this case is realized, void swelling and the damage accumulation in general would be suppressed for the density of SIA clusters, hence, their sink strength would reach a very high value by a relatively low irradiation dose, f ^ 1dpa, (see Singh and Foreman18). Thus, irradiation with high-energy particles, such as fast neutrons, provides a mechanism for suppressing damage accumulation, which may operate if the SIA clusters are immobilized. In alloys, the interaction with impurity atoms may provide such an immobili­zation. The so-called ‘incubation period’ of swelling observed in stainless steels under neutron irradia­tion for up to several tens of dpa (Garner32,33) might be due to the Singh—Foreman catastrophe. A possible scenario of this may be as follows: during the incubation period, the material is purified by RIS mainly on SIA clusters because of their high density. At high enough doses, that is, after the incu­bation period, the material becomes clean enough to provide the recovery of the mobility of small SIA clusters created in cascades that triggered on the production bias mechanism. As a result, the high number density of SIA clusters decreases via the absorption of the excess of vacancies, restoring con­ditions for damage accumulation.