Hexagonal 6mm

Isotropy of piezoelectric ceramics is destroyed during poling process but remains in the direction perpendicular to the poling field direction. Stractuie of tensor material coefficients of oomm cylindrical polar symmetiy is the same as for the hexagonal 6mm symmetry for dielectric, piezoelectric and elastic tensors. 

Not all the tensor components are independent. Between Eqs (6.29a) and (6.29b) there are 45 independent tensor components, 21 for the elastic compliance sE, six for the permittivity sx and 18 for the piezoelectric coefficient d. Fortunately crystal symmetry and the choice of reference axes reduces the number even further. Here the discussion is restricted to poled polycrystalline ceramics, which have oo-fold symmetry in a plane normal to the poling direction. The symmetry of a poled ceramic is therefore described as oomm, which is equivalent to 6mm in the hexagonal symmetry system. 

FIGURE 6.12 In (a) is the hkO diffraction plane of a P63 canavalin crystal having sixfold symmetry, and in (b) the view along the unique axis of a rhombohedral canavalin crystal that exhibits 6mm symmetry. In neither case would one choose orthogonal axes on which to index the reflections. In (a), the natural choice would be the a and b axes indicated. In (b), two choices of hexagonal axes are reasonable, but those indicated are chosen as a and b because they correspond with the real unit cell of smallest volume. 

FIGURE 6.21 In (a), the MO diffraction plane of R3 canavalin exhibits 6mm symmetry, but because of Friedel s law it could arise as a consequence of either a true sixfold axis or a threefold axis plus the Friedel center of symmetry. In (b), the M2 image, which is along the same direction but does not contain Friedel related reflections, exhibits only threefold symmetry. This demonstrates that the crystal does in fact belong to the trigonal system and not the hexagonal system. 

Figure 3.5 Symmetry of the plane lattices (a, b) oblique primitive, mp, 2 (c, d) rectangular primitive, op, 2mm (e, f) rectangular centred, oc, 2mm (g, h) square, tp, 4mm (i, j) hexagonal primitive, hp, 6mm...

Finally, the hexagonal primitive (hp) lattice, (Figure 3.5i), has a hexad rotation axis at each lattice point. This generates diads and triads as shown. In addition, there are six mirror lines through each lattice point. In other parts of the unit cell, two mirror lines intersect at diads and three mirror lines intersect at triads, (Figure 3.5j). The lattice point symmetry is described by the symbol 6mm. 

Figure 4.10 Symmetry elements present in a hexagonal crystal (a) directions in a hexagonal lattice (b) the point group 6mm has a rotation hexad along the c-axis which generates a set of mirrors, each at an angle of 30° to its neighbours (c) the point group 6/m 2/m 2/m has a hexad along the c-axis, a mirror plane normal parallel to the c-axis, diads along [100] and [120], and mirror plane normals parallel to these directions...

Thus, a symmetric tensor of rank 2 is characterized by an ellipsoid of revolution for threefold, fourfold and sixfold symmetries. With respect to piezoelectricity, the hexagonal groups impose the same conditions on the tensor as the corresponding tetragonal groups (422 <- 622,4mm 6mm, etc.). Because a piezoelectric polarization can only develop along a polar direction, the effect is zero in any direction perpendicular to a fourfold or sixfold axis. Consequently,... 

Table 4.4-8 Hexagonal, point group 6mm (Cg ) materials, cont.

Wurtzite (sytL, radial blende, HT ZnS) [Named after the Frendi diemist, Ch.A. Wurtzej (ICSD 67453 and PDF 36-1450) ZnS Ms 97.456 67.10 wt.% Zn 32.90 wt.% S Traces Fe (Sulfides and sulfosalts) Coordinence Zn(4) Hexagonal a = 382.30 pm c = 625.65 pm B4,hP4(Z=2) S.G. P6,mc P.G. 6mm Wurtzite type Uniaxial (-b) e= 2.356 2.378 S= 0.022 R = 17.4% 3.5-4 4030 Habit pyramidal, radial, tabular, coUoform. Color orange red, light brown or dark brown. Luster resinous. Luster adamantine, resinous. Streak brown yellow. Geav e 1010, 0001. Fracture unevea CondioidaL Chemical attacked by strong mineral acids, such as HG, or HNO, with evolution of H S and yeUow precipitate of sulfur. Infusible. 

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