text: It is convenient to categorize grain boundaries according to the extent of misorientation between the two grains. Low-angle grain boundaries (LAGB) or subgrain boundaries are those with a misorientation less than about 15 degrees. Generally speaking they are composed of an array of dislocations and their properties and structure are a function of the misorientation. In contrast the properties of high-angle grain boundaries, whose misorientation is greater than about 15 degrees (the transition angle varies from 10 to 15 degrees depending on the material), are normally found to be independent of the misorientation. However, there are special boundaries at particular orientations whose interfacial energies are markedly lower than those of general high-angle grain boundaries.
  The simplest boundary is that of a tilt boundary where the rotation axis is parallel to the boundary plane. This boundary can be conceived as forming from a single, contiguous crystallite or grain which is gradually bent by some external force. The energy associated with the elastic bending of the lattice can be reduced by inserting a dislocation, which is essentially a half-plane of atoms that act like a wedge, that creates a permanent misorientation between the two sides. As the grain is bent further, more and more dislocations must be introduced to accommodate the deformation resulting in a growing wall of dislocations – a low-angle boundary. The grain can now be considered to have split into two sub-grains of related crystallography but notably different orientations.
  An alternative is a twist boundary where the misorientation occurs around an axis that is perpendicular to the boundary plane. This type of boundary incorporates two sets of screw dislocations. If the Burgers vectors of the dislocations are orthogonal, then the dislocations do not strongly interact and form a square network. In other cases, the dislocations may interact to form a more complex hexagonal structure.
  These concepts of tilt and twist boundaries represent somewhat idealized cases. The majority of boundaries are of a mixed type, containing dislocations of different types and Burgers vectors, in order to create the best fit between the neighboring grains.
  If the dislocations in the boundary remain isolated and distinct, the boundary can be considered to be low-angle. If deformation continues, the density of dislocations will increase and so reduce the spacing between neighboring dislocations. Eventually, the cores of the dislocations will begin to overlap and the ordered nature of the boundary will begin to break down. At this point the boundary can be considered to be high-angle and the original grain to have separated into two entirely separate grains.
  In comparison to low-angle grain boundaries, high-angle boundaries are considerably more disordered, with large areas of poor fit and a comparatively open structure. Indeed, they were originally thought to be some form of amorphous or even liquid layer between the grains. However, this model could not explain the observed strength of grain boundaries and, after the invention of electron microscopy, direct evidence of the grain structure meant the hypothesis had to be discarded. It is now accepted that a boundary consists of structural units which depend on both the misorientation of the two grains and the plane of the interface. The types of structural unit that exist can be related to the concept of the coincidence site lattice, in which repeated units are formed from points where the two misoriented \
  In coincident site lattice (CSL) theory, the degree of fit (Σ) between the structures of the two grains is described by the reciprocal of the ratio of coincidence sites to the total number of sites.
  In this framework, it is possible to draw the lattice for the two grains and count the number of atoms that are shared (coincidence sites), and the total number of atoms on the boundary (total number of site). For example, when Σ=3 there will be one atom of each three that will be shared between the two lattices. Thus a boundary with high Σ might be expected to have a higher energy than one with low Σ. Low-angle boundaries, where the distortion is entirely accommodated by dislocations, are Σ1. Some other low-Σ boundaries have special properties, especially when the boundary plane is one that contains a high density of coincident sites. Examples include coherent twin boundaries (e.g., Σ3) and high-mobility boundaries in FCC materials (e.g., Σ7). Deviations from the ideal CSL orientation may be accommodated by local atomic relaxation or the inclusion of dislocations at the boundary.