The Shell Model and Interatomic Potentials for Ceramics
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MRS BULLETIN/FEBRUARY 1996
metal-ceramic bonding, stress relief, and grain growth. The course of oxidation can be spectacularly affected by impurities and alloying, and this can only be understood by considering the atomicscale processes involved. What then is a ceramic? Those we consider here are materials that, to a first approximation, may be treated as an assembly of non-overlapping spherical ions, held together by Coulomb attraction. We thus exclude covalent materials such as SiC and also materials such as intermetallics, which may take up crystal structures normally regarded as characteristic of ionic bonding. Electrondensity maps, showing near-spherical charge densities, are often cited as evidence in favor of the ionic picture, but they can be misleading as they are dominated by the high electron densities near the nuclei, and there is no unique way of
Electric Field
Electron Density Distorted by Field
Shell Moved Relative to Core
Figure 1. The shell model of ionic polarizability compared with the distortion of electron density.
drawing the outlines of the ionic spheres. For the same reason, electron-density maps do not offer a resolution of the confused matter of ionic charge or ionicity. Although a direct definition of ionic charge could in principle be made in terms of the integrated electron density within an ionic volume, the choice of volume would always involve some arbitrary decisions. In general, one should use full formal charges (+1 for sodium, - 1 for chlorine, in units of the electron charge lei). An ion such as oxygen presents a problem. The O2~ ion is not bound in free space, and so some would argue that it is unreasonable to give it a formal charge of —2. In the crystal, however, the ion is stabilized by the electrostatic potential of the neighboring positive ions. We shall return to the question of ionic charges, and to the oxide ion, later. To use this basic model for simulations, it is necessary to describe the interactions between the ions. In principle, it may be shown that the total energy of a crystal made up of N ions at lattice spacing R may be written as a sum of onebody, two-body, and higher order interactions:
+ 2
(1)
However, this formal expression gives no guidance on the order to which the sum must be taken or on the functional forms of the potentials. This is the point at which some physical insight is required.
The Shell Model The dominant term in the crystal energy is the Coulomb term arising from the interaction of the formal charges, which accounts for more than 80% of the cohesive energy. This attraction is countered by a short-range repulsion, arising from the exclusion principle, as ioniccharge densities overlap. Small additional contributions come from dispersion (van der Waals) interactions. Such a model, however, is incomplete. This is clear when one considers the dielectric response of the crystal, and dielectric polarization is a major term in the energies of defects. At low frequencies, one can explain the dielectric properties in terms of opposite moveme
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