Low- Small -Angle Grain Boundaries

For 0=1° and for b 2.5 x 10 , the value of D 140 A. In simple cubic materials, b = (100). The larger the angle 0 is, the closer the spacing between the dislocations will be. Thus, for an angle of 15° (when the dislocation model is no longer valid), the dislocations are approximately a few atomic spacings one from another, at 9.5 x 10 , and it is impossible to distinguish between the individual dislocations. 

The energy of a small-angle grain boundary may be calculated on the basis of the dislocation model. Schematically, the relative energy, as a function of the angle of orientation difference, is shown in Fig. 3.63c. One may express this energy as  

Note the shortcomings of this derivation (a) it is valid only for real, small angles, as assumed to be the case when using Eq. (3.81) (b) the core energy is not known. The approximation in Eq. (3.39), at best, only provides an estimation and (c) it is inherent in the derivation that A and Eq be independent of 0. 

Small-angle grain boundaries are illustrated in Fig. 3.22 by means of the etch-pits technique Relation (3.82) is rewritten as (3.89)  

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