Читать книгу Introduction to Solid State Physics for Materials Engineers - Emil Zolotoyabko - Страница 17

As was already mentioned, twinning is very interesting and practically important phenomenon in crystal physics, which is tightly related to specific symmetry operations. The phenomenon of twinning in crystals has been extensively studied due to its emergence in crystal growth and phase transformations and its substantial effect on mechanical, electrical, and optical properties in real crystals. In fact, twinning is one of the key mechanisms of plastic deformation in metals and ceramics. Quite often it serves as a structural basis for different types of ferroelectric domains (see Chapter 12) or structural variants in shape-memory alloys.

In contrast to what we have learned until now, symmetry operations involved into twinning processes are not included into the point group of a particular crystal, in which twinning occurs. Correspondingly, twins are crystal parts (sometimes called individuals), which are transformed into each other under such symmetry operations. In fact, if a specific symmetry element, considered for twinning, belongs to the crystal point group, its application to one crystal part would produce perfect continuation of the crystal, rather than a twin. In principle, every basic symmetry element, introduced earlier in Section 1.1, may serve for twinning, if it does not belong to the point group set for a given crystal. However, most frequently twins are produced by reflection in a mirror plane or by a 180° rotation about the twofold rotation axis perpendicular to the boundary plane between the twinned parts. If a crystal has an inversion center, both operations result in identical twins.

Let us illustrate these considerations by two examples, the first being taken for monoclinic lattice, which is characterized by lattice translations a, b, and c and angle γ ≠ 90° between vectors a and b. Correspondingly, vector c is perpendicular to both the vectors a and b. In monoclinic crystals, containing mirror plane as symmetry element (classes m and 2/m, see Table 1.1), this plane is horizontal, i.e. perpendicular to the c-translation (in our setting). Consequently, the planes containing translations a and c or b and c are not mirror planes (Figure 1.19). However, if nevertheless, part of the crystal is produced according to this “forbidden” symmetry operation (as fault during growth or because of stress application), one obtains twins shown in Figure 1.19. The angle between twinned parts equals 180° – 2γ.

Figure 1.19 Illustration of twin formation in monoclinic lattice via mirror reflection in plane containing the b- and c-translations. The latter is perpendicular to the plane of drawing. The a- and b-translation vectors are indicated by red arrows. Twins are crystal parts located at right-hand and left-hand sides from the trace of the twinning plane (blue solid line).

Figure 1.20 Illustration of twin formation in orthorhombic lattice via mirror reflection in plane containing the (b + a)- and c-translations. The latter is perpendicular to the plane of drawing. The a- and b-translation vectors are indicated by red arrows. Twins are crystal parts located at right-hand and left-hand sides from the trace of the twinning plane (blue solid line).

As second example, let us consider orthorhombic lattice with lattice translations a, b, c, being mutually perpendicular to each other. In orthorhombic crystals (classes 222, mm2, and mmm, see Table 1.1), the faces of rectangular prism, which represents unit cell, are related to certain symmetry elements. For class mm2 they are mirror planes, while for class 222 the normals to these planes are the twofold rotation axes. For class mmm both assertions are valid. Therefore, if we apply these symmetry operations to 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.