The number of displaced atoms inside the cascade can be simply estimated using the Kichin — Pease formula9 or the modified Kichin-Pease for­mula (Norgett-Robinson-Torrens model or NRT model).1 , 1 According to this last model, the number of displaced atoms within the cascade in the case of a 22 keV PKA and using a displacement energy of Ed = 40 eV is np = 0.4i7r/Ed ~ 220. Because of the large mean free path of fast neutrons (several
centimeters), it can be considered that only one PKA is created by the incoming neutron going through the Zr cladding used in pressurized water reactors (PWRs) (with a thickness of 0.6 mm). There­fore, if the PKA creation rate per unit volume within the cladding is known for a typical fuel assem­bly in a PWR (with typical fast neutron flux is 5 x 1017 nm-2 s-1 (E > 1 MeV)), the number of dis­placed atoms per unit volume and per second can be computed. From this value, the overall number of displacements per atom (dpa) and per second can be simply computed. This calculation can be achieved, as described by Luneville et at., by taking into account the PWR neutron spectrum as well as the neutron-atom differential cross section. It can be shown that a typical damage rate for a cladding in a PWR core is between 2 and 5 dpa year- , depending on the neutron flux history. This means that each atom of the cladding has been displaced 2-5 times per year! A more accurate correspondence between the fast fluence and the damage for a cladding in a PWR is provided by Shishov et a/.12 These authors evaluate that a fluence of 6 x 102 4 nm 2 (E > 1 MeV) corresponds to a damage of 1 dpa.

This simple approach gives a good description of the number of displaced atoms during the creation of the cascade, but does not consider intracascade elas­tic recombinations that occur during the cascade relaxation or cooling-down phase.11,13,14 In addition, this approach does not give any information on the form of the remaining damage at the end of the cascade, such as the point-defect clusters that can be created in the cascade.

In order to have a better understanding of the created damage in a-zirconium, several authors have

performed MD computations also using different types of interatomic potentials. It is shown that, at the end of the cascade creation (<2 ps), the cascade is composed of a core with a high vacancy concentra­tion, and the self interstitial atoms (SIAs) are concen­trated at the cascade periphery.14-16 The cascade creation is followed by the athermal cascade relaxa­tion that can last for a few picoseconds. During this phase, most of the displaced atoms quickly reoccupy lattice sites as a result of prompt (less than a lattice vibration period, 0.1 ps) elastic recombination if a SIA and a vacancy are present at the same time in the elastic recombination volume (with 200 ^ <V<400 ^, where V is the elastic recombination volume and ^ the atomic volume.17) Wooding eta/.16 and Gao eta/.8 have shown that at the end ofthe cascade relaxation the number of surviving point defects is very low, much lower, only 20% at 600 K, than the number of Frenkel pairs computed using the NRT model. It is also shown that all the point defects are not free to migrate but that small point-defect clusters are created within the cas­cade. This clustering is due to short-range diffusion driven by the large elastic interaction among neighbor­ing point defects and small point-defect clusters. In the case of zirconium, large point-defect clusters, up to 24 vacancies and 25 SIAs (at 600 K), can be found at the end of the cascade relaxation (Figure 1).8 According to Woo eta/.,14 the presence of these small point-defect clusters spatially separated from each other, as well as the different concentrations of single vacancies and SIAs, can have a major impact on the subsequent microstructural evolution. This effect is known as the production bias, which has to be considered when solving the rate equations in the mean-field approach of point-defect evolution.14

The form of these small clusters is also of major importance since it plays a role on the nucleation of dislocation loops. Wooding et a/.16 and Gao et a/.8 have shown that the small SIA clusters are in the form of dislocation loops with the Burgers vector 1/3(1120). The collapse of the 24-vacancy cluster to a dislocation loop on the prism plane was also found to occur.