Defect Calculations

These limitations are largely eliminated in sophisticated defect calculations described in the following section. This approach can also include more sophisticated site exclusion rules, which allow defects to either cluster or keep apart from each other. Nevertheless, the formulas quoted are a very good starting point for an exploration of the role of defects in solids and do apply well when defect concentrations are small and at temperatures that are not too high. 

From these early beginnings, computer studies have developed into sophisticated tools for the understanding of defects in solids. There are two principal methods used in routine investigations atomistic simulation and quantum mechanics. In simulation, the properties of a solid are calculated using theories such as classical electrostatics, which are applied to arrays of atoms. On the other hand, the calculation of the properties of a solid via quantum mechanics essentially involves solving the Schrodinger equation for the electrons in the material. 

Quantum mechanical methods follow a similar path, except that the starting point is the solution of the Schrodinger equation for the system under investigation. The most successful and widely used method is that of Density Functional Theory. Once again, a key point is the development of a realistic model that can serve as the input to the computer investigation. Energy minimization, molecular dynamics, and Monte Carlo methods can all be employed in this process. 

There are two other methods in which computers can be used to give information about defects in solids, often setting out from atomistic simulations or quantum mechanical foundations. Statistical methods, which can be applied to the generation of random walks, of relevance to diffusion of defects in solids or over surfaces, are well suited to a small computer. Similarly, the generation of patterns, such as the aggregation of atoms by diffusion, or superlattice arrays of defects, or defects formed by radiation damage, can be depicted visually, which leads to a better understanding of atomic processes. 

Calculations are now carried out routinely using a wide variety of programs, many of which are freely available. In particular, the charge on a defect can be included so that the formation energies, interactions, and relative importance of two defects such as a charged interstitial as against a neutral interstitial are now accessible. Similarly, computation is not restricted to intrinsic defects, and the energy of formation of 

Sometimes, the system of interest is not the inhnite crystal, but an anomaly in the crystal, such as an extra atom adsorbed in the crystal. In this case, the inhnite symmetry of the crystal is not rigorously correct. The most widely used means for modeling defects is the Mott-Littleton defect method. It is a means for performing an energy minimization in a localized region of the lattice. The method incorporates a continuum description of the polarization for the remainder of the crystal. 

Cardone, Fundamentals of Semicondutors Springer-Verlag, Berlin (1996). 

Hoffmann, Solids and Surfaces A Chemist s View of Bonding in Extended Structures VCH, New York (1988). 

Tsidilkovski, Band Structure of Semiconductors Pergamon, Oxford (1982). 

Harrison, Solid State Theory Dover, New York (1979). 

---

The chemistry of interest is often not merely the inhnite crystal, but rather how some other species will interact with that crystal. As such, it is necessary to model a system that is an inhnite crystal except for a particular site where something is diherent. The same techniques for doing this can be used, regardless of whether it refers to a defect within the crystal or something binding to the surface. The most common technique is a Mott-Littleton defect calculation. This technique embeds a defect in an inhnite crystal, which can be considered a local perturbation to the band structure. 

Local-density potentials greatly simplify the computational problems associated with defect calculations. In practice, however, such calculations still are very computer-intensive, especially when repeated cycles for different atomic positions are treated. In most cases the cores are eliminated from the calculation by the use of pseudopotentials, and considerable effort has gone into the development of suitable pseudopotentials for atoms of interest (see Hamann et al., 1979). 

It is important to realize that each of the electronic-structure methods discussed above displays certain shortcomings in reproducing the correct band structure of the host crystal and consequently the positions of defect levels. Hartree-Fock methods severely overestimate the semiconductor band gap, sometimes by several electron volts (Estreicher, 1988). In semi-empirical methods, the situation is usually even worse, and the band structure may not be reliably represented (Deak and Snyder, 1987 Besson et al., 1988). Density-functional theory, on the other hand, provides a quite accurate description of the band structure, except for an underestimation of the band gap (by up to 50%). Indeed, density-functional theory predicts conduction bands and hence conduction-band-derived energy levels to be too low. This problem has been studied in great detail, and its origins are well understood (see, e.g., Hybertsen and Louie, 1986). To solve it, however, requires techniques of many-body theory and carrying out a quasi-particle calculation. Such calculational schemes are presently prohibitively complex and too computationally demanding to apply to defect calculations. 

When 1 mol of U-238 decays to Th-234, 5 X 10 kg of matter is converted to energy (the mass defect). Calculate the amount of energy released. 

Co by contrast is seen in Figures 3 and 4 to have high-energy barriers at both delithiated and partially lithiated compositions along either type of pathway Oh Td Oh or Oh Oh) into the Li/vacancy layer. Results of TM ion defect calculations at full lithiation, i.e., Xu = 1, which are not shown, indicate that both Co and Mn are prevented from entering the Li layer by the lack of octahedral lithium vacancies. 

For Co, the energy of /JS-(Li jCo)tet(LijX2o3)oct08 by contrast never drops below that of l-l iJZoOz- The results of Figure 8 for crystalline structures, like those in Table 1, mimic the results of the tetrahedral defect calculations (Figure 4). In each case tetrahedral Co is found to be unfavorable at all lithium concentrations and oxidation states considered, while tetrahedral Mn is found to be favorable at the Lii/2Mn02 composition when it has a -1-2 valence. 

The energy of all the 3d metals entering tetrahedral coordination from the /-Li rM02 structure decreases as Xu goes from 0 to 1/2. This is similar to the defect calculations on Co and Mn in section 4 that found tetrahedral defect energies in the layered structure to decrease for both as Li content increases from 0 to 1/2. 

Lasaga A. C. (1980). Defects calculations in silicates Olivine. Amer. Mineral, 65 1237-1248. 

Jaoul O., Bertran-Alvarez Y., Eiebermann R.C., and Price G.D. (1995) Ee-Mg interdiffusion in olivine up to 9 GPa at T = 600-900°C experimental data and comparison with defect calculations. Phys. Earth Planet In. 89, 199-218. 

Bennett, . H., Exact Defect Calculations in Model Substances, In Diffusion in Solids Recent Developments Eds. Nowick, A. S. Burton, J. J. Academic Press, New York, 1975 p. 73. 

C.H. Bennett, Exact defect calculations in model substances, in Algorithms for Chemical Computation, edited by A.S. Nowick and J.J. Burton, ACS Symposium Series No. 46 63 (1977). 

Theoretical point defect calculations for solid solutions are difficult because the defect surroundings are not isotropic. This is particularly true for metals considering... 

In concluding this section, we note that defect calculations may be used to study defect mobilities as well as defect formation and interaction processes. Assuming the validity of the hopping model of defect transport the frequency of defect jumps can then be written as ... 

A few key points emerge from this study first, the interpretation of the observed conductivity data is confirmed (as shown in Fig. 8.2.). Second, the results, both theoretical and experimental, show that ionic size may have a large effect on ionic conductivity and this factor should clearly be born in mind in designing solid electrolytes. The third point is that the results show the quantitative success of this class of defect calculation in treating a subtle effect. 

Configurations 14 and 110 were examined with sites A, A, N and B1 and compared to the defect calculations. Both the periodic empirical potential calculations and the periodic DFT calculations gave mainly the same sites as the Mott-Littleton study. With the surprising exception that the DFT calculation found no minimum at site N for 14 or 1 10. 

Transition state calculations were also run with the periodic force field model for moving ions out of site N. Interestingly the barrier increases for the periodic model over the defect calculation, this contradicts the failure by the DFT to find a site N which would implies a negligible barrier to migration. 

The mass of any atom is less than the combined masses of its separated parts. This difference in mass is known as the mass defect, also called mass loss. Electrons have masses so small that they can be left out of mass defect calculations. For helium, He, the mass of the nucleus is about 99.25% of the total mass of two protons and two neutrons. According to the equation E = mc, energy can be converted into mass, and mass can be converted into energy. So, a small quantity of mass is converted into an enormous quantity of energy when a nucleus forms. 

Force constant and inertia defect calculations have been performed for COCIF [604,1312,1531,1549,1550,1575-1578]. CNDO calculations have been used to calculate (> C=0) for COCIF [1680], and the fundamental frequencies for COCIF have been calculated from the force constants calculated for COXj (X = F, Cl or Br) [1578]. 

---