Читать книгу Crystallography and Crystal Defects - Anthony Kelly - Страница 37

There are no special forms in either of the point groups in this system. The unit cell is a general parallelepiped and the geometry is more complicated than even for the monoclinic system. The drawing of a stereogram whose centre is taken to be the z‐axis, [001], can be carried out with the aid of Figures 2.21a and b. The two diagrams here are completely general and can be specialized to apply to any crystal system more symmetric than the triclinic by setting one or more of the axial angles to particular values and by setting two or more of the lattice parameters to be equal.

Figure 2.21 Diagrams relevant to drawing stereograms of triclinic crystals. The centre of the stereogram in (a) is the z‐axis, [001]

The choice of the centre of the stereogram as the z‐axis in Figure 2.21a means that all planes of the general form (hk0) lie on the primitive of the stereogram, just as for the monoclinic stereogram in Figure 2.19. However, for triclinic crystals there is no good reason to have the 010 pole located on the right‐hand side of the horizontal axis of the stereogram, and so here it is deliberately rotated around the primitive away from this position.

The great circle passing through 001 and 010 is the [100] zone, containing planes of the general form (0kl). Therefore, the angle between this zone and the primitive must be the angle between [100] and [001], β, as shown in Figure 2.21a. This is because the stereographic projection is a conformal projection (see Appendix 2); that is, one for which angles are faithfully reproduced. Similarly, α and γ can be specified on Figure 2.21a from the triclinic system geometry.

Angles between poles can be determined using Eq. (A3.18). Once the position of the 010 pole (or any other pole of the form hk0) has been chosen, geometry determines the positions of the remaining poles on the stereogram. Thus, for example, the position of the 001 pole is fixed knowing that it lies in the [100] zone, which has to make an angle of β with the primitive, and that 001 is a given angle from 010, determined for example using Eq. (A3.18). Likewise, the position of a pole 0kl lying in the [100] zone can be determined, as can the positions of the poles 100, h0l, hk0 and hkl.

While stereographic projections can now be produced routinely via proprietary software packages, it is still instructive to consider further aspects of the geometry of the part of the triclinic stereogram shown in Figure 2.21a in order to gain a full appreciation of the richness of information displayed on stereograms. An (hkl) plane of a triclinic crystal is shown in Figure 2.21b. The six angles φ₁–φ₆ in Figure 2.21b are the same as those marked in Figure 2.21a.

Thus, for example, φ₁ in Figure 2.21b is an angle lying in the (001) plane. It is the angle between the y‐axis and the vector [h0] common to (hkl) and (001); that is, the angle between the zone containing (001), (h0l) and (100) and the zone containing (001), (hkl) and (hk0). We can therefore mark φ₁ on the stereogram. Similarly, φ₅ in Figure 2.21b is an angle lying in the (100) plane. It is the angle between the z‐axis and the vector [0k] common to (hkl) and (100); that is, the angle between the zone containing (100), (hk0) and (010) and the zone containing (100), (hkl) and (0kl). φ₅ can therefore also be marked on the stereogram. Proceeding in this way, we can identify all of the angles φ₁–φ₆. We have, from the geometry in Figure 2.21b:

(2.10)

Furthermore, from the triangle on the (001) face in Figure 2.21b we have, using the sine rule:

(2.11)

Therefore:

(2.12a)

and similarly:

(2.12b)

and:

(2.12c)

As an aside, we note that the equations in Eq. (2.12) are also of use in finding axial ratios and axial angles from measured angles between planes on single crystal specimens of crystals belonging to the triclinic crystal system; such crystals tend to be minerals or organic materials, rather than metals, the crystal structures of which rarely tend to belong to either the monoclinic or the triclinic systems.