Coincidence-site lattices

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],... 

V. Randle, The Role of the Coincidence Site Lattice in Grain Boundary Engineering, The Institute of Materials, London, 1996. 

If h > k and k 0 then the super-lattice is unsymmetrically disposed with respect to the original lattice. If the lattice is turned over and superimposed on itself so that the super-lattice points coincide, then we have a coincidence site lattice (in which a fraction 1 /(h2 + hk + k2) of the original lattice points coincide). Coincidence site lattices can also be found in three dimensions, particularly for cubic lattices. 

We assume in the following discussion that the solid surface under consideration is of the same chemical identity as the bulk, that is, free of any oxide film or passivation layer. Crystallization proceeds at the interfaces between a growing crystal and the surrounding phase(s), which may be solid, liquid, or vapor. Even what we normally refer to as a crystal surface is really an interface between the crystal and its surroundings (e.g., vapor, vacuum, solution). An ideal surface is one that is the perfect termination of the bulk crystal. Ideal crystal surfaces are, of course, highly ordered since the surface and bulk atoms are in coincident positions. In a similar fashion, a coincidence site lattice (CSL), defined as the number of coincident lattice sites, is used to describe the goodness of fit for the crystal-crystal interface between grains in a polycrystal. We ll return to that topic later in this section. 

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). 

Consider a pair of adjacent crystals. We mentally expand the two neighboring crystal lattices until they interpenetrate and fill all the space. Without loss of generality, it is assumed that the two lattices possess a common origin. If we now hold one crystal fixed and rotate the other, it is found that a number of lattice sites for each crystal, in addition to the origin, coincide with certain relative orientations. The set of coinciding points form a coincidence site lattice, or CSL, which is a sublattice for both the individual crystals. 

However, the experimental results do not allow one to decide whether the rotation of the Pb UPD overlayer (cf. eq. (3.22)) can be explained by the coincident site lattice" concept (involving higher order commensurate overlayers, Fig. 3.17) or the static distortion waves" (SDW) concept (dealing with incommensurate overlayers, Fig. 3.18). The estimation of d as a function of AE from a statistical analysis of in situ STM images coincides with GKS results illustrated in Fig. 3.27 [3.176]. 

Fig. 9.41. Schematic of several representative grain boundaries with structures described by coincident site lattice model. This set of boundaries corresponds to a (001) rotation axis, and the atomic-level geometries have not been relaxed (courtesy of D. Pawaskar). The filled circles correspond to those sites (coincident sites) that are common to both lattices.

The atomic level structure of grain boundaries has been an important issue for the past several decades. In cubic materials geometrical constructs of periodic grain boundaries can be obtained for certain misorientation axis-angle combinations that are associated with coincident site lattices (CSLs). The CSLs are formed by the coincident sites of two hypothetically interpenetrating crystal lattices, where S is the reciprocal density of CSL sites. Much of the discussion of grain boundary structure and properties has revolved around the description of grain boundary structures in terms of the CSL, the displacement-shift complete (DSC) and the 0-lattice [10.10, 10.11]. 

Another possibility for obtaining CSLs exists by approximating the crystal structure by pseudocubic or tetragonal unit cells. This approach of applying the CSL to non-cubic systems has been discussed in the literature by means of the constrained coincident site lattice [10.12] which has, among others, also been applied to YBCO grain boundaries. 

Table 13 shows the features of compound tessellations (3, 6 [ 3, 6 ] to r = lOOA [Eqn. (12) assuming a = 5.3 A], each of which describes a coincidence-site lattice (CSL) (Ranganathan 1961) the multiplicity n of its mesh is termed coincidence index or Z factor and corresponds to the order of the subgroup of translation defining the two-dimensional CSL with respect to the hp lattice. As shown in Table 13, the minimal value of the E factor for the hp lattice is 7 (see also Pleasants et al. 1996). 

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