Introduction to Solid State Physics for Materials Engineers. Emil Zolotoyabko

one part of the respective crystal, we will obtain its perfect continuation. Situation is changed, if we consider mirror plane, which contains one of the face diagonals of the prism as well as translation vector situated normally to the face chosen. This geometry is shown in Figure 1.20, for mirror plane, which contains vectors a + b (as face diagonal) and vector c. Application of such mirror plane to a part of the crystal produces twin as is clearly seen in Figure 1.20. If the angle between vector b and the trace of the twinning plane equals α, then the angle between twinned parts is 2α.

Despite all twins can be considered as being produced by certain symmetry operations, historically twins are also classified with respect to physical processes, through which they appear. In this classification, twins are sub-divided by three categories: growth twins, transformation twins, and deformation twins. Note that these classes not always have lucid boundaries, since crystals may experience deformations also during growth and especially during phase transformations. Note that adjacent ferroelectric domains in perovskite structure (e.g. in BaTiO₃ considered in Chapter 12) are good example of transformation twins, if the latter are produced during paraelectric/ferroelectric phase transformation.

      Twin boundaries are planar structural defects, which increase free energy of the system, as compared with perfect crystal structure. Correspondingly, twinning planes that require less energy for twin formation are more favorable from energy point of view. For this reason, quite often, twin boundary is a compactly packed atomic plane that prevents re-arrangement of short (and therefore strong) bonds during atomic movements, which accompany the twin formation.