O-lattice theory

The grain boundary energy 7gb should be proportional to . For small values of high coincidence occurs and the number of broken bonds can be minimized. = 1 corresponds to complete coincidence of the ideal crystal. Experimentally it was found that the correlation between 7Gb and is not that simple due to volume expansions or translations at the grain boundaries. A principal problem of the coincident site lattice model is that, even arbitrarily small variations of the lattice orientation lead mathematically to a complete loss of coincidence. This is physically not reasonable because an arbitrarily small deviation should have a small effect. This problem was solved by the O-lattice theory [343], For a comprehensive treatment of solid-solid interfaces and grain boundaries, see Refs. [344,345],... 

The simple CSL model is directly applicable to the cubic crystal class. The lower symmetry of the other crystal classes necessitates the more sophisticated formalism known as the constrained coincidence site lattice, or CCSL (Chen and King, 1988). In this book we treat only cubic systems. Interestingly, whenever an even value is obtained for E in a cubic system, it will always be found that an additional lattice point lies in the center of the CSL unit cell. The true area ratio is then half the apparent value. This operation can always be applied in succession until an odd value is obtained thus, E is always odd in the cubic system. A rigorous mathematical proof of this would require that we invoke what is known as O-lattice theory (Bollman, 1967). The O-lattice takes into account all equivalence points between two neighboring crystal lattices. It includes as a subset not only coinciding lattice points (the CSL) but also all nonlattice sites of identical internal coordinates. However, expanding on that topic would take us well beyond the scope of this book. The interested reader is referred to Bhadeshia (1987) or Bollman (1970). 

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