Energy of Vacancy Formation

In general, one may estimate as satisfactory the fit of strength data based on the first-principles simulation with experimental results. 

Vacancies in a crystal lattice act a significant part in the thermodynamic and kinetic behavior of sohds. The energy of vacancy formation Ey is a most important quantity, which determines the equilibrium vacancy concentration and contributes to processes of diffusion and strain. The activation energy of self-diffusion Qa is the sum of the energy of vacancy formation Ey and the energy of vacancy diffusion Uy. 

The first-principle calculation of point defects is much more difficult than the calculation of perfect crystals because the model looses the translational symmetry. 

The self-consistent electronic structure calculations for the energy of vacancy formation are based on the local density approximation, equation (8.17). Authors of [25] used the supercells with 27 and 32 lattice sites for bcc and fee metals, respectively. The atoms neighboring the vacancies are not allowed to relax from their perfect lattice position. It was found that the errors due to omission of the lattice relaxation around the vacancy were only of the order of one-tenth of an eV. The total energy of a supercell tot depends on the number of atoms in the supercell N, the number of vacancies v in a volume Q, Etot = E N,v, Q). In the superceU approximation, the vacancy-formation energy is given by 

The calculated values of the energy Ey are given in Table 9.3 together with the 

---

Calculate the energy of vacancy formation in aluminum, given that the equilibrium number of vacancies at 500°C is 7.57 x 10 m. State your assumptions. 

Note that in DRP-structures, the vacancies are unstable as numerical simulation shows [6.35] the property of DRP structures. Because of the difference in LO and elastic deformations, the energies of vacancy formation are different in different sites. Let Ebe the energy of vacancy formation in the jt-type site without the elastic deformations account,- UVfl the correction to the energy of vacancy formation that is due to elastic deformations present, and f Uvli) the distribution function of UVfl values normalized to unity. Then the vacancies concentration in /i-type sites is determined by the relation... 

The content of Chapter 9 sheds light on validity and application for different solids of the theory, which has been considered in previous chapters. The calculated values of cohesive energy and bulk modulus are compared with the experimental results. We present data on superconductivity, the embrittlement of metals, the electronic density of states, properties of intermetallic compounds, and the energy of vacancy formation. 

In this chapter we consider properties of solids that are related to strength of interatomic bonding. We both bear in mind such macroscopic characteristics as elastic constants, melting temperature, and microscopic parameters as amplitudes of atom vibrations, energy of vacancy formation, the Debye temperature. All these properties can be measured experimentally with a high precision. Therefore they are factors that characterize the interatomic bonding in solids. 

Table 9.3 Calculated and experimental values for energy of vacancy formation (eV/atom).

Table 11.1 Comparison of the energy of vacancy formation Ey and the surface energy y calculated from tight-binding theory (TB) with an experiment. The first-principle data obtained from the local density approximation...

It is readily seen from Figure 13.2 that energy of vacancy formation has the largest values for molybdenum, niobium, and rhodium with 5, 4, and 8 electrons in their 4d shell, respectively. 

Figure 13.2 Energy of vacancy formation Ey ofthe4d transition metals as a dependence on the number of d electrons. The experimental Ey data from [11].

The energy of vacancy formation determines generally the rate of propagation of the fatigue crack. A value of is by-tum determined by the strength of interatomic bonding in the material. 

And again, as given in Chap. 3, N is the number of lattice sites and Ep is the energy of vacancy formation. The activation energy, Q, for the jump rate, J, is given by the sum of the energy of vacancy formation and the vacancy s energy for... 

To estimate the correctness of the proposed calculations, we used the phenomenological equations for the energy of vacancy formation for a nonplanar surface... 

We now have a large number of energies of defects and disorders which have been evaluated. Table 3.9 gives the example of the energies of vacancy formation in atomic solids. 

The LMTO method was used by Zhukov and Gubanov (120) and Zhukov et al. (121) to calculate the band structures and some characteristics of the ground state of TiCo.75, VCojs (120) and TiNo.75, VNo,7s (121). These authors also estimated the lattice constants, bulk moduli and cohesive energies, and the energies of vacancy formation. The results of the APW and LMTO calculations, which will be discussed in Sec. III.D, show great similarities to each other. The assumed periodic long-range order of the vacancies leads to sharp structures in the DOSs, which are probably not realistic. 

The values of the equilibrium constant may be computed from Eq. (8). Burton has shown that vacancy formation entropies in cubic metals range from 3.6 to 5.2 e.u. For the case of copper, a value of 3.8 e.u. was computed. The energy of vacancy formation has been estimated to be about 23 kcal/mole. Thus at 1000°K, Ky for copper is found to be 6 x 10 from Eq. (8). Assuming [1 ] ci 1, this value would also correspond to the vacancy concentration according to Eq. (13). 

---