Section 3.3.6.3.3. Combined Dauphiné–Brazil (Leydolt, Liebisch) twins

Twins of this type can be described by a twin reflection across the plane (0001), normal to the threefold axis [001]. The two orientation states of this twin are of opposite handedness (i.e. the optical activity is reversed, optical twin), but the polar axes are not reversed. The coset representing the twin law consists of the following six operations:

The composite symmetry is again a supergroup of index [2] of the eigensymmetry group 32. This twin law is usually described as a combination of the Dauphiné and Brazil twin laws, i.e. as the twofold Dauphiné twin rotation followed by the Brazil twin reflection or, alternatively, by the inversion . The product results in a particularly simple description of the combined law as a reflection twin on .

Twin domains of the Leydolt type are very rarely intergrown in direct contact, i.e. with a common boundary. If, however, a quartz crystal contains inserts of Dauphiné and Brazil twins, the domains of these two types, even though not in contact, are related by the Leydolt law. In this sense, Leydolt twinning is rather common in low-temperature quartz. In contrast, GaPO₄, a quartz homeotype with the berlinite structure, frequently contains Leydolt twin domains in direct contact, i.e. with a common boundary (Engel et al., 1989).

In conclusion, the three merohedral twin laws of -quartz described above imply four domain states with different orientations of important physical properties. These relations are shown in Fig. 3.3.6.2 for electrical polarity, optical activity and the orientation of etch pits on (0001). It is noteworthy that these three twin laws are the only possible merohedral twins of quartz, and that all three are realized in nature. Combined, they lead to the composite symmetry (`complete twin': Curien & Donnay, 1959).

Figure 3.3.6.2 | top | pdf | Distinction of the four different domain states generated by the three merohedral twin laws of low-quartz and of quartz homeotypes such as GaPO4 (Dauphiné, Brazil and Leydolt twins) by means of three properties: orientation of the three electrical axes (triangle of arrows), orientation of etch pits on (001) (solid triangle) and sense of the optical rotation (circular arrow). The twin laws relating two different domain states are indicated by arrows [D (): Dauphiné law; B (): Brazil law; C (): Leydolt law]. For GaPO4, see Engel et al. (1989).

In the three twin laws (cosets) above, only odd powers of 6, and (rotations and rotoinversions) occur as twin operations, whereas the even powers are part of the eigensymmetry 32. Consequently, repetition of any odd-power twin operation restores the original orientation state, i.e. each of these operations has the nature of a `binary' twin operation and leads to a pair of transposable orientation states.