No Arabic abstract
Point defects in body-centred cubic Fe, Cr and concentrated random magnetic Fe-Cr are investigated using density functional theory and theory of elasticity. The volume of a substitutional Cr atom in ferromagnetic bcc Fe is approximately 18% larger than the volume of a host Fe atom, whereas the volume of a substitutional Fe atom in antiferromagnetic bcc Cr is 5% smaller than the volume of a host Cr atom. Elastic dipole $boldsymbol{P}$ and relaxation volume $boldsymbol{Omega}$ tensors of vacancies and self-interstitial atom (SIA) defects exhibit large fluctuations, with vacancies having negative and SIA large positive relaxation volumes. Dipole tensors of vacancies are nearly isotropic across the entire alloy composition range, with diagonal elements $P_{ii}$ decreasing as a function of Cr content. Fe-Fe and Fe-Cr SIA dumbbells are more anisotropic than Cr-Cr dumbbells. Fluctuations of elastic dipole tensors of SIA defects are primarily associated with the variable crystallographic orientations of the dumbbells. Statistical properties of tensors $boldsymbol{P}$ and $boldsymbol{Omega}$ are analysed using their principal invariants, suggesting that point defects differ significantly in alloys containing below and above 10% at. Cr. The relaxation volume of a vacancy depends sensitively on whether it occupies a Fe or a Cr lattice site. A correlation between elastic relaxation volumes and magnetic moments of defects found in this study suggests that magnetism is a significant factor influencing elastic fields of defects in Fe-Cr alloys.
The low energy structures of irradiation-induced defects have been studied in detail, as these determine the available modes by which a defect can diffuse or relax. As a result, there are many studies concerning the relative energies of possible defect structures, and empirical potentials are commonly fitted to or evaluated with respect to these energies. But recently [Dudarev et al Nuclear Fusion 2018], we have shown how to determine the stresses, strains and swelling of reactor components under irradiation from the elastic properties of ensembles of irradiation-induced defects. These elastic properties have received comparatively little attention. Here we evaluate relaxation volumes of irradiation-induced defects in tungsten computed with empirical potentials, and compare to density functional theory results where available. Different empirical potentials give different results, but some potential-independent trends in relaxation volumes can be identified. We show that the relaxation volume of small defects can be predicted to within 10% from their point-defect count. For larger defects we provide empirical fits for the relaxation volume of as a function of size. We demonstrate that the relaxation volume associated with a single primary-damage cascade can be estimated from the primary knock-on atom (PKA) energy. We conclude that while annihilation of vacancy- and interstitial- character defects will invariably reduce the total relaxation volume of the cascade debris, empirical potentials disagree whether coalescence of defects will reduce or increase the total relaxation volume.
The defect relaxation volumes obtained from density-functional theory (DFT) calculations of charged vacancies and interstitials are much larger than their neutral counterparts, seemingly unphysically large. In this work, we investigate the possible reasons for this and revisit the methods that address the calculation of charged defect structures in periodic DFT. We probe the dependence of the proposed energy corrections to charged defect formation energies on relaxation volumes and find that corrections such as the image charge correction and the alignment correction, which can lead to sizable changes in defect formation energies, have an almost negligible effect on the charged defect relaxation volume. We also investigate the volume for the net neutral defect reactions comprised of individual charged defects, and find that the aggregate formation volumes have reasonable magnitudes. This work highlights an important issue that, as for defect formation energies, the defect formation volumes depend on the choice of reservoir. We show that considering the change in volume of the electron reservoir in the formation reaction of the charged defects, analogous to how volumes of atoms are accounted for in defect formation volumes, can renormalize the formation volumes of charged defects such that they are comparable to neutral defects. This approach enables the description of the elastic properties of isolated charged defects within the overall neutral material, beyond the context of the overall defect reactions that produce the charged defect.
Extensive first-principle calculations on embedded clusters containing few O, Y, Ti, and Cr atoms as well as vacancies are performed to obtain interaction parameters to be applied in Metropolis Monte Carlo simulations, within the framework of a rigid lattice model. A novel description using both pair and triple parameters is shown to be more precise than the commonly used pair parameterization. Simulated annealing provides comprehensive data on the energetics, structure and stoichiometry of nm-size clusters at T=0. The results are fully consistent with the experimental finding of negligible coarsening and a high dispersion of the clusters, with the observation that the presence of Ti reduces the cluster size, and with the reported radiation tolerance of the clusters. In alloys without vacancies clusters show a planar structure, whereas the presence of vacancies leads to three-dimensional configurations. Additionally, Metropolis Monte Carlo simulations are carried out at high temperature in order to investigate the dependence of nanocluster composition on temperature. A good agreement between the existing experimental data on the ratios (Y+Ti):O, Y:Ti, (Y+Cr):O, and Y:Cr, and the simulation results is found. In some cases it is even possible to draw the conclusion that the respective alloys contained a certain amount of vacancies, and that the clusters analyzed were frozen-in high-temperature configurations. The comparison of experimental data with those obtained by simulations demonstrates that the assumption of nanoclusters consisting of nonstoichiometric oxides which are essentially coherent with the bcc lattice of the Fe-Cr matrix leads to reasonable results.
Anomalies in the temperature dependences of the recoil-free factor, f, and the average center shift, <CS>, measured by 57-Fe Mossbauer Spectroscopy, were observed for the first time in the archetype of the sigma-phase alloys system, Fe-Cr. In both cases the anomaly started at the temperature close to the magnetic ordering temperature, and in both cases it was indicative of lattice vibrations hardening. As no magnetostrictive effects were found, the anomalies seem to be entirely due to a spin-phonon coupling. The observed changes in f and in <CS> were expressed in terms of the underlying changes in the potential, Delta E_p, and the kinetic energy, Delta E_k, respectively. The former, with the maximum value larger by a factor of six than the latter, decreases, while the latter increases with T. The total mechanical energy change, Delta E, was, in general, not constant, as expected for the Debye-like vibrations, but it resembled that of Delta E_p. Only in the range of 4-15 K, Delta E was hardly dependent on T.
The adiabatic elastic modulus is often useful in the high frequency response of materials. Unfortunately, it can be much more difficult to directly measure the adiabatic elastic modulus of material than the isothermal elastic modulus. We derive the relationship between the adiabatic and isothermal elastic tensors from the first law of thermodynamics.