Twinning with partial lattice pseudo-coincidence (lattice index )

Hahn, Th.; Klapper, H.

doi:10.1107/97809553602060000644

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 424-425

Section 3.3.9.2.4. Twinning with partial lattice pseudo-coincidence (lattice index )

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

3.3.9.2.4. Twinning with partial lattice pseudo-coincidence (lattice index )

| top | pdf |

This type can be derived from the category in Section 3.3.9.2.3 above by relaxation of the condition of exact lattice coincidence, resulting in two nearly, but not exactly, coinciding lattices (pseudo-coincidence, cf. Section 3.3.8.4). In this sense, the two Sections 3.3.9.2.3 and 3.3.9.2.4 are analogous to the two Sections 3.3.9.2.1 and 3.3.9.2.2.

The following four examples are characteristic of this group:

(1) (110) reflection twins of a pseudo-hexagonal orthorhombic crystal with a P lattice: If the axial ratio $[b/a = \sqrt{3}]$ were exact, the lattices of both twin partners would coincide exactly on a sublattice of index (due to the absence of the C centring); cf. Koch (2004), Fig. 1.3.2.2 . If deviates slightly from $[\sqrt{3}]$ , the exact coincidence lattice changes to a pseudo-coincidence lattice of lattice index . Examples are ammonium lithium sulfate, NH₄LiSO₄, many members of the K₂SO₄-type series (cf. Docherty et al., 1988) and aragonite, CaCO₃.

(2) Staurolite twinning: This topic has been extensively treated as Example 3.3.6.12. The famous 90°- and 60°-twin `crosses' are a complicated and widely discussed example for Friedel's notion of `twinning by reticular merohedry ' (Friedel, 1926, p. 461). It was followed up by an extensive analysis by Hurst et al. (1956). Both twin laws (90° and 60° crosses) can be geometrically derived from a multiple pseudo-cubic cell $[{\bf a}_c']$ , $[{\bf b}_c']$ , $[{\bf c}_c']$ (so-called `Mallard's pseudo-cube') which is derived from the structural monoclinic C-centred cell $[{\bf a}_m]$ , $[{\bf b}_m]$ , $[{\bf c}_m]$ as follows, involving a rotation of $[\sim 45^\circ]$ around [100]: $[{\bf a}_c' = {\bf b}_m + 3{\bf c}_m,\quad {\bf b}_c' = - {\bf b}_m + 3 {\bf c}_m, \quad {\bf c}_c' = 3{\bf a}_m.]$

Using Smith's (1968) lattice constants for the structural monoclinic cell with space group C2/m and a = 7.871, b = 16.620, c = 5.656 Å, β = 90° (within the limits of error), V_m = 740 Å³, the pseudo-cube has the following lattice constants: $[\matrix{a'_c = 23.753\hfill &b'_c = 23.753\hfill & c'_c = 23.613\,\,\hbox{\AA}\hfill &\cr \alpha_c = 90\hfill & \beta_c = 90\hfill & \gamma_c = 88.81^\circ\hfill & V'_c = 13323\,\,\hbox{\AA}^3.\hfill}]$

The volume ratio [V'_c/V_m] of the two cells is 18, i.e. the sublattice index is [[j] = 18] . If, however, the primitive monoclinic unit cell is used, the volume ratio doubles and the sublattice index used in the twin analysis increases to [[j] = 36] . The (metrical) eigensymmetry of the pseudo-cube is orthorhombic (due to $[\beta_c = 90^\circ]$ ), $[(2/m)_{[001]}(2/m)_{[110]}(2/m)_{[1{\bar 1}0]}]$ , referred to $[{\bf a}'_c]$ , $[{\bf b}'_c]$ , $[{\bf c}'_c]$ .

Note, however, that this pseudo-cube in reality is C-centred because the C-centring vector $[1/2({\bf a}'_c + {\bf b}'_c) = 3{\bf c}_m]$ is a lattice vector of the monoclinic lattice. This C-centring has not been considered by Friedel, Hurst and Donnay, who have based their analysis on the primitive pseudo-cube.

According to Friedel, the `symmetry elements' of the pseudo-cube are potential twin elements of staurolite, except for $[(2/m)_{[1{\bar 1}0]}]$ , which is the monoclinic symmetry direction of the structure. In Table 3.3.9.1, the twin operations of the $[90^\circ]$ and $[60^\circ]$ twins are compared with the `symmetry operations' of the pseudo-cube with respect to obliquities $[\omega]$ and lattice indices [j], referred to both sets of axes, pseudo-cubic $[{\bf a}_c']$ , $[{\bf b}'_c]$ , $[{\bf c}'_c]$ and monoclinic (but metrically orthorhombic) $[{\bf a}_m]$ , $[{\bf b}_m]$ , $[{\bf c}_m]$ . The calculations were again performed with the program OBLIQUE by Le Page (1999, 2002). In order to keep agreement with the interpretation of Friedel and Hurst et al., the pseudo-cube is treated as primitive, with [[j] = 36] .

Table 3.3.9.1 | top | pdf |
Staurolite, 60° and 90° twins

Comparison of the twin operations with the `symmetry operations' of the primitive pseudo-cube with respect to obliquity $[\omega]$ and lattice index [[j]] , referred both to the pseudo-cubic axes, $[{\bf a}'_c]$ , $[{\bf b}'_c]$ , $[{\bf c}'_c]$ , and the monoclinic (metrically orthorhombic) axes, $[{\bf a}_m]$ , $[{\bf b}_m]$ , $[{\bf c}_m]$ . The calculations were performed with the program OBLIQUE by Le Page (1999, 2002).

(a) 90° cross (eight twin operations).

Twin operations referred to		Obliquity $[\omega]$ $[[^\circ]]$	Lattice index [j] referred to		Remarks
$[{\bf a}'_c, {\bf b}'_c, {\bf c}'_c]$	$[{\bf a}_m, {\bf b}_m, {\bf c}_m]$	Obliquity $[\omega]$ $[[^\circ]]$	$[{\bf a}'_c, {\bf b}'_c, {\bf c}'_c]$	$[{\bf a}_m, {\bf b}_m, {\bf c}_m]$	Remarks
		0	1	1	Four collinear twin operations , , $[{\bar 4}{^1}]$ , $[{\bar 4}{^3}]$
		1.19	1	6	Four `diagonal' (with respect to the monoclinic unit cell) twin operations intersecting in
		1.19	1	6
	$[2[0{\bar 1}3]_m]$	1.19	1	6
	$[m(0{\bar 3}1)_m]$	1.19	1	6

(b) 60° cross.

Twin operations referred to		Obliquity $[\omega]$ $[[^\circ]]$	Lattice index [j] referred to		Equivalent directions
$[{\bf a}'_c, {\bf b}'_c, {\bf c}'_c]$	$[{\bf a}_m, {\bf b}_m, {\bf c}_m]$	Obliquity $[\omega]$ $[[^\circ]]$	$[{\bf a}'_c, {\bf b}'_c, {\bf c}'_c]$	$[{\bf a}_m, {\bf b}_m, {\bf c}_m]$	Equivalent directions
( $[\pm120^\circ]$ )		0.87	3	3	$[[11{\bar 1}]_c = [{\bar 1}02]_m ]$
$[3[{\bar 1}11]_c]$ ( $[\pm120^\circ]$ )		0.25	3	9	$[[1{\bar 1}1]_c = [3{\bar 2}0]_m ]$
( $[\pm 90^\circ]$ )		1.19	1	6	$[[010]_c = [0{\bar 1}3]_m ]$
		1.19	1	6
		0.90	1	12	$[[011]_c = [3{\bar 1}3]_m ]$
		0.90	1	12	$[[10{\bar 1}]_c = [{\bar 3}13]_m ]$
					$[[01{\bar 1}]_c = [31{\bar 3}]_m ]$

The following interpretations can be given (cf. Fig. 13 in Hurst et al., 1956):

(a) 90° cross (Table 3.3.9.1a, Fig. 3.3.6.13a):

(i) The pseudo-tetragonal 90° cross can be explained and visualized very well with eight twin operations, a fourfold twin axis along with operations , , $[{\bar 4}{^1}]$ , $[{\bar 4}{^3}]$ and two pairs of `diagonal' twin operations 2 and m. They form the coset of the (metrically) `orthorhombic' ( $[\beta = 90^\circ]$ ) eigensymmetry $[{\cal H} = mmm]$ which results in the composite symmetry $[{\cal K} = 4'(2)/m\,2/m\,2'/m']$ .
(ii) The obliquities for all twin operations are at most 1.2°, the lattice index is for the twin axis, but for the `diagonal' twin elements it is , which is at the limit of the permissible range. Because of these facts, Friedel prefers to consider the 90° cross as a 90° rotation twin around rather than as a (diagonal) reflection twin across or $[(0{\bar 3}1)_m]$ .
(iii) Note that for the interpretation of the 90° cross the complete pseudo-cube with lattice index is not required. Because $[{\bf c}'_c = 3{\bf a}_m]$ , a pseudo-tetragonal unit cell with axes $[{\bf a}_c']$ , $[{\bf b}'_c]$ , $[(1/3){\bf c}'_c]$ and is sufficient.

(b) 60° cross (Table 3.3.9.1b, Fig. 3.3.6.13b):

(iv) The widespread 60° cross is much more difficult to interpret and visualize. The four threefold twin axes around $[\langle 111\rangle]$ of the pseudo-cube split into two pairs, both with very small obliquities $[ \,\lt\, 1^\circ]$ . One pair, and $[[{\bar 1}02]_m]$ , has a favourable index ; however, the other one, and $[[3{\bar 2}0]_m]$ is with unacceptably high. According to Friedel's theory, this makes the best choice as threefold twin axis.
(v) There is a further $[\pm 90^\circ]$ twin rotation around or with small obliquity, $[\omega = 1.2^\circ]$ , but very high lattice index, . Note that this is the same axis that has been used already for the 90° twin, but with a 180° rotation.
(vi) The greatest deviation from the `permissibility' criterion is exhibited by the twin axes and $[= 2[3{\bar 1}3]_m]$ and the twin planes, pseudo-normal to them, and $[(2{\bar 3}1)_m]$ . The obliquity $[\omega = 0.9^\circ]$ is very good but the twin index is , a value far outside Friedel's `limite prohibitive'. These operations, however, are the `standard' twin operations that are always quoted for the 60° twins. Following Friedel (1926, p. 462), the best definition of the 60° twin is the $[\pm 120^\circ]$ rotations around with $[\omega = 0.87^\circ]$ and .
(vii) If the (true) C-centring of the pseudo-cube is taken into account, however, no $[ \langle 111\rangle]$ pseudo-threefold axes remain; hence, the 60° cross cannot be explained by the lattice construction of the pseudo-cube.

(3) Growth twins of monoclinic PrS₂ and of tetragonal SmS_1.9: These two rather complicated examples belong to the structural family of MeX₂ dichalcogenides which is rich in structural relationships and different kinds of twins. The `basic structure' and `aristotype' of this family is the tetragonal ZrSSi structure with axes $[a_b = b_b \approx 3.8]$ , $[c_b \approx 7.9\ \hbox{\AA}]$ , $[V_b \approx 114\ \hbox{\AA}^3]$ , space group [P4/nmm] (b stands for basic). The crystal chemistry of this structural family is discussed by Böttcher et al. (2000).

(a) PrS₂ (Tamazyan et al., 2000a)

PrS₂ is a monoclinic member of this series with space group (unique axis a!) and axes $[a \approx 4.1]$ , $[b \approx 8.1]$ , $[c \approx 8.1\ \hbox{\AA}]$ , $[\alpha \approx 90.08^\circ]$ , $[V \approx 269\ \hbox{\AA}^3]$ . The structure is strongly pseudo-tetragonal along [001] (with cell a, b/2, c) and is a `derivative structure' of ZrSSi. Hence pseudo-merohedral twinning that makes use of this structural tetragonal pseudosymmetry would be expected, with twin elements 4[001] or or 2[120] etc. and because $[b \approx 2a]$ , but, surprisingly, this twinning has not been observed so far. It may occur in other PrS₂ samples or in other isostructural crystals of this series.

Instead, the monoclinic crystal uses another structural pseudosymmetry, the approximate orthorhombic symmetry along [100] with $[\alpha \approx 90^\circ]$ , to twin on , , or (coset of ) with composite symmetry $[{\cal K} = 2/m\,2'/m'\,2'/m']$ , and (cf. Fig. 4 of the paper).

The monoclinic PrS₂ cell has a third kind of pseudosymmetry that is not structural, only metrical. The cell is pseudo-tetragonal along [100] due to $[ b \approx c]$ and $[\alpha \approx 90^\circ]$ . This pure lattice pseudosymmetry, not surprisingly, is not used for twinning, e.g. via 4[100] or or $[m(0{\bar 1}1)]$ or 2[011] or $[2[0{\bar 1}1]]$ .
(b) SmS_1.9 (Tamazyan et al., 2000b)

This structure is (strictly) tetragonal with axes $[a = b \approx 8.8]$ , $[c\approx 15.9\ \hbox{\AA}]$ , $[V \approx 1238\ \hbox{\AA}^3]$ and space group . It is a tenfold superstructure of ZrSSi with the following basis-vector relations: $[ {\bf a} = 2{\bf a}_b + {\bf b}_b,\quad {\bf b} = -{\bf a}_b + 2{\bf b}_b,\quad {\bf c} = 2{\bf c}_b,]$ leading to lattice constants $[a \approx \sqrt{5}a_b]$ , $[b \approx \sqrt{5}b_b]$ , $[c \approx 2c_b]$ . This well ordered tetragonal supercell now twins on or 2[210] or or 2[130] (which is equivalent to a rotation around [001] of 36.87°) to form a $[\Sigma 5]$ twin by `reticular merohedry ' ( ) with lattice constants $[a' = a \sqrt{5} = 19.72]$ , $[b \sqrt{5} = 19.72]$ , c′ = c = 15.93 Å, V = 6192 Å³. This is illustrated in Fig. 3.3.8.1.

SmS_1.9 represents the first thoroughly investigated and documented tetragonal ( $[\Sigma 5]$ ) twin known to us. The sublattice of this twin is the tetragonal coincidence lattice with smallest lattice index , i.e. the `least-diluted' systematic tetragonal sublattice.

(4) Growth twins of micas: A rich selection of different twin types, both merohedral and pseudo-merohedral , with and 3, is provided by the mineral family of micas, which includes several polytypes. A review of these complicated and interesting twinning phenomena is presented by Nespolo et al. (1997). Detailed theoretical derivations of mica twins and allotwins, both in direct and reciprocal space, are published by Nespolo et al. (2000).

In conclusion, it is pointed out that the above four categories of twins, described in Sections 3.3.9.2.1 to 3.3.9.2.4, refer only to cases with exact or approximate three-dimensional lattice coincidence (triperiodic twins). Twins with only two- or one-dimensional lattice coincidence (diperiodic or monoperiodic twins) [e.g. the (100) reflection twins of gypsum and the (101) rutile twins] belong to other categories, cf. Section 3.3.8.2. The examples above have shown that for triperiodic twins structural pseudosymmetries are an essential feature, whereas purely metrical (lattice) pseudosymmetries are not a sufficient tool in explaining and predicting twinning, as is evidenced by the case of staurolite, discussed above in detail.

References

Böttcher, P., Doert, Th., Arnold, H. & Tamazyan, R. (2000). Contributions to the crystal chemistry of rare-earth chalcogenides. I. The compounds with layer structures LnX₂. Z. Kristallogr. 215, 246–253.Google Scholar

Docherty, R., El-Korashy, A., Jennissen, H.-D., Klapper, H., Roberts, K. J. & Scheffen-Lauenroth, T. (1988). Synchrotron Laue topography studies of pseudo-hexagonal twinning. J. Appl. Cryst. 21, 406–415.Google Scholar

Friedel, G. (1926). Lecons de cristallographie, ch. 15. Nancy, Paris, Strasbourg: Berger-Levrault. [Reprinted (1964). Paris: Blanchard].Google Scholar

Hurst, V. J., Donnay, J. D. H. & Donnay, G. (1956). Staurolite twinning. Mineral. Mag. 31, 145–163.Google Scholar

Koch, E. (2004). Twinning. In International tables for crystallography, Vol. C. Mathematical, physical and chemical tables, edited by E. Prince, 3rd ed., ch. 1.3. Dordrecht: Kluwer Academic Publishers.Google Scholar

Le Page, Y. (1999). Low obliquity in pseudo-symmetry of lattices and structures, and in twinning by pseudo-merohedry. Acta Cryst. A55, Supplement. Abstract M12.CC001.Google Scholar

Le Page, Y. (2002). Mallard's law recast as a Diophantine system: fast and complete enumeration of possible twin laws by [reticular] [pseudo] merohedry. J. Appl. Cryst. 35, 175–181.Google Scholar

Nespolo, M., Ferraris, G. & Takeda, H. (2000). Twins and allotwins of basic mica polytypes: theoretical derivation and identification in the reciprocal space. Acta Cryst. A56, 132–148.Google Scholar

Nespolo, M., Takeda, H. & Ferraris, G. (1997). Crystallography of mica polytypes. In EMU notes in mineralogy, edited by St. Merlino, Vol. 1, ch. 2, pp. 81–118. Budapest: Eötvos University Press.Google Scholar

Smith, J. V. (1968). The crystal structure of staurolite. Am. Mineral. 53, 1139–1155.Google Scholar

Tamazyan, R., Arnold, H., Molchanov, V. N., Kuzmicheva, G. M. & Vasileva, I. G. (2000a). Contribution to the crystal chemistry of rare-earth chalcogenides. II. The crystal structure and twinning of rare-earth disulfide PrS₂. Z. Kristallogr. 215, 272–277.Google Scholar

Tamazyan, R., Arnold, H., Molchanov, V. N., Kuzmicheva, G. M. & Vasileva, I. G. (2000b). Contribution to the crystal chemistry of rare-earth chalcogenides. III. The crystal structure and twinning of SmS_1.9. Z. Kristallogr. 215, 346–351.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 424-425