Twinning of low-temperature quartz (α-quartz)

Hahn, Th.; Klapper, H.

doi:10.1107/97809553602060000644

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 404-405

Section 3.3.6.3. Twinning of low-temperature quartz (α-quartz)

Th. Hahn^a ^* and H. Klapper^b

^a Institut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D-52056 Aachen, Germany, and ^bMineralogisch-Petrologisches Institut, Universität Bonn, D-53113 Bonn, Germany
Correspondence e-mail: hahn@xtal.rwth-aachen.de

3.3.6.3. Twinning of low-temperature quartz (α-quartz)

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Quartz is a mineral which is particularly rich in twinning. It has the noncentrosymmetric trigonal point group 32 with three polar twofold axes and a non-polar trigonal axis. The crystals exhibit enantiomorphism (right- and left-handed quartz), piezoelectricity and optical activity. The lattice of quartz is hexagonal with holohedral (lattice) point group $[6/m\,2/m\,2/m]$ . Many types of twin laws have been found (cf. Frondel, 1962), but only the four most important ones are discussed here:

(a) Dauphiné twins;
(b) Brazil twins;
(c) Combined-law (Leydolt, Liebisch) twins ;
(d) Japanese twins .

The first three types are merohedral (parallel-lattice ) twins and their composite symmetries belong to category (i) in Section 3.3.4.2, whereas the non-merohedral Japanese twins (twins with inclined lattices or inclined axes) belong to category (ii).

3.3.6.3.1. Dauphiné twins

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This twinning is commonly described by a twofold twin rotation around the threefold symmetry axis [001]. The two orientation states are of equal handedness but their polar axes are reversed (`electrical twins'). Dauphiné twins can be transformation or growth or mechanical (ferrobielastic) twins. The composite symmetry is $[{\cal K} = 622]$ , the point group of high-temperature quartz ( $[\beta]$ -quartz). The coset decomposition of $[{\cal K}]$ with respect to the eigensymmetry $[{\cal H} = 32]$ (index [2]) contains the operations listed in Table 3.3.6.3.

Table 3.3.6.3 | top | pdf |
Dauphiné twins of α-quartz: coset of alternative twin operations (twin law)

$[{\cal H}]$	$[2_z \times {\cal H} ]$
1
	$[6^5 (= 6^{-1})]$

$[2_{[100]}]$	$[2_z\times 2_{[100]} = 2_{[120]} ]$
$[2_{[010]}]$	$[2_z\times 2_{[010]} = 2_{[210]}]$
$[2_{[{\bar 1}{\bar 1}0]}]$	$[2_z\times 2_{[{\bar 1}{\bar 1}0]} = 2_{[1{\bar 1}0]}]$

The left coset $[2_z \times {\cal H}]$ constitutes the twin law. Note that this coset contains four twofold rotations of which the first one, [2_z] , is the standard description of Dauphiné twinning. In addition, the coset contains two sixfold rotations, [6^1] and $[6^5 = 6^{-1}]$ . The black–white symmetry symbol of the composite symmetry is $[{\cal K} =]$ [6'(3)22'] (supergroup of index [2] of the eigensymmetry group $[{\cal H} = 32]$ ).

This coset decomposition $[622 \Rightarrow 32]$ was first listed and applied to quartz by Janovec (1972, p. 993).

3.3.6.3.2. Brazil twins

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This twinning is commonly described by a twin reflection across a plane normal to a twofold symmetry axis. The two orientation states are of opposite handedness (i.e. the sense of the optical activity is reversed: optical twins) and the polar axes are reversed as well. The coset representing the twin law consists of the following six operations:

(i) three reflections across planes $[\{11{\bar 2}0\}]$ , normal to the three twofold axes;
(ii) three rotoinversions $[{\bar 3}]$ around [001]: $[{\bar 3}{^1}]$ , $[{\bar 3}{^3} = {\bar 1}]$ , $[{\bar 3}{^5} = {\bar 3}{^{-1}}]$ .

The coset shows that Brazil twins can equally well be described as reflection or inversion twins. The composite symmetry $[{\cal K} = {\bar 3}{^\prime}(3){2\over m'}1({\bar 1}{^\prime})]$ is a supergroup of index [2] of the eigensymmetry group 32.

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

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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:

(i) three twin reflections across planes $[\{10{\bar 1}0\}]$ , parallel to the three twofold axes;
(ii) three rotoinversions $[{\bar 6}]$ around [001]: $[{\bar 6}{^1}]$ , $[{\bar 6}{^3} = m_z]$ , $[{\bar 6}{^5} = {\bar 6}{^{-1}}]$ .

The composite symmetry $[{\cal K} = {\bar 6}{^\prime}(3)2 m' = {3\over m'} 2m']$ 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 [2_z] followed by the Brazil twin reflection $[m(11{\bar 2}0)]$ or, alternatively, by the inversion $[{\bar 1}]$ . The product $[2_z\times {\bar 1} = m_z]$ results in a particularly simple description of the combined law as a reflection twin on [m_z] .

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 $[\alpha]$ -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 $[{\cal K} = 6/m\,2/m\,2/m]$ (`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 ( [2_z] ): Dauphiné law; B ( $[{\bar 1}]$ ): Brazil law; C ( [m_z] ): Leydolt law]. For GaPO₄, see Engel et al. (1989).

In the three twin laws (cosets) above, only odd powers of 6, $[{\bar 3}]$ and $[{\bar 6}]$ (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.

3.3.6.3.4. Japanese twins (or La Gardette twins)

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Among the quartz twins with `inclined axes ' (`inclined lattices'), the Japanese twins are the most frequent and important ones. They are contact twins of two individuals with composition plane $[(11{\bar 2}2)]$ . This results in an angle of $[84^\circ 33^\prime]$ between the two threefold axes. One pair of prism faces is parallel (coplanar) in both partners.

There exist four orientation relations, depending on

(i) the handedness of the two twin partners (equal or different);
(ii) the azimuthal difference (0 or $[180^\circ]$ ) around the threefold axis of the two partners.

These four variants are illustrated in Fig. 3.3.6.3 and listed in Table 3.3.6.4. The twin interface for all four twin laws is the same, $[(11{\bar 2}2)]$ , but only in type III do twin mirror plane and composition plane coincide.

Table 3.3.6.4 | top | pdf |
The four different variants of Japanese twins according to Frondel (1962)

Handedness of twin partners	Azimuthal difference (°)	Twin element = twin law	Label in Fig. 65 of Frondel (1962)
L–L or R–R	0	Irrational twofold twin axis normal to plane $[(11{\bar 2}2)]$	I(R), I(L)
L–L or R–R	180	Rational twofold twin axis $[[11{\bar 1}]\equiv[11{\bar 2}{\bar 3}]]$ ^† parallel to plane $[(11{\bar 2}2)]$	II(R), II(L)
L–R or R–L	0	Rational twin mirror plane $[(11{\bar 2}2)]$	III
L–R or R–L	180	Irrational twin mirror plane normal to direction $[[11{\bar 1}]\equiv[11{\bar 2}{\bar 3}]]$ ^†	IV

^† The line $[[11{\bar 1}]\equiv[11{\bar 2}{\bar 3}]]$ is the edge between the faces $[z(01{\bar 1}1)]$ and $[r(10{\bar 1}1)]$ and is parallel to the composition plane $[(11{\bar 2}2)]$ . It is parallel or normal to the four twin elements. Transformation formulae between the three-index and the four-index direction symbols, UVW and uvtw, are given by Barrett & Massalski (1966

, p. 13).

Figure 3.3.6.3 | top | pdf |

The four variants of Japanese twins of quartz (after Frondel, 1962; cf. Heide, 1928). The twin elements 2 and m and their orientations are shown. In actual twins, only the upper part of each figure is realized. The lower part has been added for better understanding of the orientation relation. R, L: right-, left-handed quartz. The polarity of the twofold axis parallel to the plane of the drawing is indicated by an arrow. In addition to the cases I(R) and II(R) , I(L) and II(L) also exist, but are not included in the figure. Note that a vertical line in the plane of the figure is the zone axis $[[11{\bar 1}]]$ for the two rhombohedral faces r and z, and is parallel to the twin and composition plane ( $[11{\bar 2}2]$ ) and the twin axis in variant II.

In all four types of Japanese twins, the intersection symmetry (reduced eigensymmetry) $[{\cal H}^\ast]$ of a pair of twin partners is 1. Consequently, the twin laws (cosets) consist of only one twin operation and the reduced composite symmetry $[{\cal K}^\ast]$ is a group of order 2, represented by the twin element listed in Table 3.3.6.4. If one were to use the full eigensymmetry $[{\cal H} = 32]$ , the infinite sphere group would result as composite symmetry $[{\cal K}]$ .

Many further quartz twins with inclined axes are described by Frondel (1962). A detailed study of these inclined-axis twins in terms of coincidence-site lattices (CSLs) is provided by McLaren (1986).

References

Curien, H. & Donnay, J. D. H. (1959). The symmetry of the complete twin. Am. Mineral. 44, 1067–1071.Google Scholar

Engel, G., Klapper, H., Krempl, P. & Mang, H. (1989). Growth twinning in quartz-homeotypic gallium orthophosphate crystals. J. Cryst. Growth, 94, 597–606.Google Scholar

Frondel, C. (1962). The system of mineralogy, 7th edition, Vol. III. Silica minerals, especially pp. 75–99. New York: Wiley.Google Scholar

Janovec, V. (1972). Group analysis of domains and domain pairs. Czech. J. Phys. B, 22, 974–994.Google Scholar

McLaren, A. C. (1986). Some speculations on the nature of high-angle grain boundaries in quartz rocks. In Mineral and rock deformation: laboratory studies, edited by B. E. Hobbs & H. C. Heard. Geophys. Monogr. 36, 233–245.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 404-405