### 7.3.3 The Significance of the O-Lattice

 We now must address the essential question: What is the signifcance of the O-lattice for grain- and phase boundaries? What is the physical meaning? There is an easy answer and a difficult implementation: First of all, the O-lattice in itself has no physical meaning whatsoever - in this it exactly like the CSL. However, since it is defined in both crystals, if you design a boundary between two crystals of given orientation (and thus with a given O-lattice) to intersect as many O-lattice points as possible, you will obtain the best physical fit along the boundary, i.e. the lowest grain boundary energies. The crystals then can be expected to increase the area of best fit between O-lattice points and to concentrate the misfit in the regions between O-lattice points - this will be a dislocation with Burgers vector=lattice vector. This is a direct consequence of the basic equation (I- A)r0=T(I), because we can replace T(I) by b(I), the set of possible Burgers vectors. This is were it becomes important what kind of unit cell we pick for the O-lattice. The intersection lines of the actual plane of the boundary with the cell walls of the O-lattice are the disloations in the grain boundary exactly as in the case of the small angle grain boundary treated before As long as the spacing of the O-lattice is large compared to the crystal lattices, this makes sense. This condition is always met for small deformation, i.e. for small angle boundaries. For O-lattices with latice constants in the same order of magnitude as the crystals, the spacing between the dislcoation would be too small as to be physically meaningful - exactly as before. The O-lattice theory then is simply the generalized version of Franks formula, but now applicable to "small deformation" boundaries of any kind. This is already a remarkable achievement; but, as we will see, the O-lattice theory also incorporates arbitrary ("large angle") boundaries.