Grain boundaries are the lattice defects that have been known for the longest time but they are also the least understood. A grain boundary separates two regions of the same crystal structure but of diﬀerent orientation. In very coarse-grained materials it can be discerned by the naked eye if the surface is properly prepared (Fig. 2.1). Our deﬁciency of fundamental knowledge of grain boundaries is mainly due to their complex structure, which requires an extensive mathematical description for its macroscopic characterization. Already in the two-dimensional case four parameters are needed to deﬁne mathematically exactly a grain boundary (Fig. 2.2), namely an angle ϕ which describes the orientation diﬀerence between the adjacent crystals (orientation relationship), an angle Ψ which deﬁnes the spatial orientation of the grain boundary “plane” (grain boundary orientation) with respect to one crystal and the components t1, t2 of the translation vector t that characterizes the displacement of the two crystals with respect to each other (translation vector). For the three-dimensional case (the real case) one even needs eight parameters to unambiguously deﬁne a grain boundary, namely three terms for the orientation relationship, for instance the Euler angles ϕ1, Φ, ϕ2, two parameters for the spatial orientation of the grain boundary by means of the normal to the grain boundary plane n = (n1, n2, n3), with respect to one of the adjacent crystals (keeping in mind that |n| = 1) and ﬁnally the three components of the translation vector t = (t1, t2, t3). The properties, in particular energy and mobility of a grain boundary, are, in principle, a function of eight parameters. Five of these eight can be inﬂuenced externally, i.e. orientation relationship and spatial orientation of the grain boundary. The translation vector will be forced by the crystals such that the total energy will be minimal; however,

FIGURE 2.1 Microstructure of recrystallized α-brass. The grain boundaries separate areas of uniform orientation represented by a uniform shade of gray.