After the cascade formation and relaxation, which last for a few picoseconds, the microstructure evolves over a longer time. The evolution of the microstruc­ture is driven by the bulk diffusion of point defects. For a better understanding of the microstructure evolution under irradiation, the elementary proper­ties of point defects, such as formation energy and migration energy, have first to be examined.

4.01.1.2.1 Vacancy formation and migration energies

Concerning the vacancy, all the atomic positions are identical in the lattice and so there is only one vacancy description leading to a unique value for the vacancy formation energy. Due to the rather low a—p phase transformation temperature, the measurement of vacancy formation and migration energy in the Zr hexagonal close-packed (hcp) phase is difficult. The temperature that can be reached is not high enough to obtain an accurately measurable concentration and mobility of vacancies.18 Nevertheless, various experi­mental techniques (Table 1), such as positron annihila­tion spectroscopy or diffusion of radioactive isotopes, have been used in order to measure the vacancy formation and migration energies or the self-diffusion

coefficient.18-26 The values obtained by the various authors are given in Table 1. It is pointed out by Hood18 that there is great discrepancy among the vari­ous results. It is particularly shown that at high tem­perature, the self-diffusion activation energy is rather low compared to the usual self-diffusion activation energy in other metals.18 However, as the temperature decreases, the self-diffusion activation energy increases strongly. According to Hood,18 this phenomenon can be explained assuming that at high temperature the vacancy mobility is enhanced by some impurity such as an ultrafast species like iron. At lower tempera­ture, the iron atoms are believed to form small pre­cipitates, explaining that at low temperatures the measured self-diffusion energy is coherent with usual intrinsic self-diffusion of hcp crystals. It is also shown that the self-diffusion anisotropy remains low for normal-purity zirconium, with a slightly higher mobil­ity in the basal plane than along the (c) axis.22,26,27 For high-purity zirconium, with a very low iron con­tent, the anisotropy is reversed, with a higher mobility along the (c) axis than in the basal plane.27

The vacancy formation and migration energies have also been computed either by MD methods, where the mean displacement distance versus time allows obtaining the diffusion coefficient, or by static computation of the energy barrier corresponding to the transition between two positions of the vacancy using either empirical interatomic potential7,28-34 or the most recent ab initio tools.35-38 Since the different sites surrounding the vacancy are not similar, due to the non-ideal c/a ratio, the migration energies are expected to depend on the crystallographic direction, that is, the migration energies in the basal plane Em and along the (c) direction E? are different. The results are given in Table 2.

The atomistic calculations are in agreement with the positron annihilation spectroscopy measurement but are in disagreement with the direct measure­ments of self-diffusion in hcp zirconium.2 As dis­cussed by Hood,18 and recently modeled by several authors,39, 0 this phenomenon is attributed to the enhanced diffusion due to coupling with the ultrafast diffusion of iron.

4.01.1.2.2 SIA formation and migration energies

In the case of SIAs, the insertion of an additional atom in the crystal lattice leads to a great distortion of the lattice. Therefore, only a limited number of configurations are possible. The geometrical descrip­tion of all the interstitial configuration sites has been

proposed for titanium byJohnson and Beeler41 and is generally adopted by the scientific community for other hcp structures (Figure 2).

• T is the simplest tetrahedral site, and O is the octahedral one, with, respectively, 4 and 6 coordi­nation numbers.

• BT and BO are similar sites projected to the basal plane with three nearest neighbors, but with dif­ferent numbers of second neighbors.

• BC is the crowdion extended defect located in the middle of a segment linking two basal atoms.

• C is the interstitial atom located between two adjacent atoms of two adjacent basal planes in the (2023) direction. This direction is not a close- packed direction, and allows easier insertion of the SIA.

• S is the split dumbbell position in the (c) direction.

The only way to have access to the SIA formation energy is from atomistic computations taking into account the different configurations of the SIA given previously. In their early work on titanium, Johnson and Beeler41 found that the most stable SIA configuration was the basal-octahedral site (BO). Several other sites were also found to be metastable, like asymmetric variants of the T and C sites. As reviewed by Willaime,35 the relative stabilities of the various SIA configurations were observed to depend strongly on the interatomic potential used (Table 3).

The mobility of SIAs can be estimated experimen­tally using electron irradiation at very low tempera­tures (4.2 K), followed by a heat treatment. During the recovery, the electrical resistivity is measured. The main recovery process was found around 100-120 K and analysis of the kinetics gives the SIA migration energy of Em ~ 0.26 eV.4

Atomistic computations have also brought results (Table 3) concerning the SIA migration energy. Sev­eral authors7,28-31,33-37 have found that the mobility of SIAs is anisotropic, with low migration activation energy for the basal plane mobility (E/J ~ 0.06 eV) and a higher migration activation energy in the (c) direction (E? ~ 0.15 eV). In the temperature range of interest for the power reactors (T~ 600 K), the diffusion coefficients obtained are the following: D1! = 8 x 10~9m2s_1 (in the basal plane) and

D? = 10~9 mm2s-1 (along the (c) direction). These authors have also shown that the anisotropy depends on the temperature. Computing the effective diffu­sion rate of SIAs in all directions, taking into account the multiplicity of the jump configurations for each type of migration, Woo and co-workers34,42 have obtained the anisotropy for self-interstitial diffusion as a function of temperature. It is shown that the SIA mobility is higher in the basal plane than along the (c) axis and that the anisotropy decreases when the temperature increases.