Group–Theoretical Representation of Holohedral Forms of Crystals By

Crystallography Reports, Vol. 45, No. 4, 2000, pp. 529–536. Translated from Kristallografiya, Vol. 45, No. 4, 2000, pp. 583–590. Original Russian Text Copyright © 2000 by Nizamutdinov, Vinokurov, Khakimov, Bulka, Galeev, Khasanova. CRYSTALLOGRAPHIC SYMMETRY Group–Theoretical Representation of Holohedral Forms of Crystals by Conjugated Simple Forms. Intergrowth of Crystals N. M. Nizamutdinov*, V. M. Vinokurov*, E. M. Khakimov**, G. R. Bulka*, A. A. Galeev*, and N. M. Khasanova* * Kazan State University, ul. Lenina 18, Kazan, 420008 Russia1 ** Kazan State Pedagogical University, Kazan, Russia Received June 8, 1998; in final form, April 12, 1999 Abstract—The schemes for division of holohedral simple forms of crystals into conjugated simple forms are derived by decomposition of the symmetry group of the primitive sublattice into double cosets. The number of equivalently oriented simple forms in the intergrowth of crystals, whose primitive space sublattices are parallel to one another, is equal to the number of holohedral permutational conjugated simple forms, which has the value 992 for all the 32 symmetry classes. © 2000 MAIK “Nauka/Interperiodica”. 1 INTRODUCTION disymmetrization [7]. If the orientations of the space lattices of individual crystals of the intergrowth coin- If the point group GK of the crystal does not coincide cide, the detection of such crystals by the diffraction [6] with the space group Gsp of the lattice, the faces of the and electron paramagnetic resonance (EPR) methods crystal equivalently oriented with respect to the three- becomes rather difficult because their diffraction pat- dimensional lattice are nonequivalently oriented with terns and EPR spectra become indistinguishable. The respect to the crystal structure. As a results of this non- present article is aimed at studying the intergrowth of equivalence, the holohedral form decomposes into con- individual crystals with the parallel primitive sublat- jugated simple forms allowed by the group GK [1, 2]. tices by the methods of the abstract theory of groups The division of holohedral forms into conjugated sim- and representations. We also consider their possible ple ones was derived geometrically rigorously in [1–4] detection by the EPR method. The study is performed on the basis of the theory of crystal symmetry. The in two main stages: (i) decomposition of the space present study is aimed to derive the schemes of division group of the primitive space sublattice Gssp into the of holohedral forms caused by lowering of the symme- adjacent classes with respect to the subgroup G and try of the point group G with respect to the group of K K double cosets by modulus (GK, GH), where GH is the the primitive sublattice Gssp (a holohedral group) by the group of a face of the holohedral simple form, and (ii) group-theoretical method. derivation of simple forms for cubic and rhombohedral ⊂ Under the condition that GK Gssp, the crystal struc- crystals from their holohedral forms. The mathematical apparatus of the abstract group theory [8] used in our ture can have nK equivalent orientations with respect to the primitive space sublattice [1]: study corresponds to the merohedry method of the theory of crystal form [1, 2]. ≡[] nK Gssp : G K = gG ()/ ssp gG (), K (1) where [Gssp : GK] is the index of the group GK in Gssp, GROUP–THEORETICAL METHOD and g(Gssp) and g(GK) are the orders of the groups Gssp OF DERIVATION OF CONJUGATED SIMPLE FORMS OF CRYSTALS and GK, respectively. The number nK is the number of individual crystals with parallel primitive sublattices in FROM THEIR HOLOHEDRAL FORMS the intergrowth, called hereafter holohedral inter- There are seven system space groups Gsp for crystals growth. Various individual crystals in the intergrowth of various systems—Ci, C2h, D2h, D3d, D4h, D6h, and Oh have different orientations and differently interact with [9, 10]. Only for the trigonal system (Gsp = D3d), the applied magnetic fields. The possible formation of primitive sublattice is characterized by the group differ- intergrown crystals of a substance is used in the inter- ent from Gsp (Gssp = D6h), which reduces the number of pretation of the nature of modulated crystal structures holohedral groups to six [1, 2, 11, 12]. [5, 6] and is also taken into account in the studies of Consider the holohedral class of crystals, GK = Gssp. 1 e-mail: [email protected] Let GH be the symmetry group of the face of the holo- 1063-7745/00/4504-0529 $20.00 © 2000 MAIK “Nauka/Interperiodica” 530 NIZAMUTDINOV et al. hedral form. This face can be divided into gH equivalent is equal to the index [GK : GKiH] of the group GKiH in GK: elementary (nonsymmetric) parts with respect to the ≡ [ ] ≤ ≤ nKiH = g(GK) : g ( G KiH) GK : G K iH , 1 i l. (6) symmetry elements of the group GH, where gH is the order of the group GH. If one of these parts, f, is multi- The groups GK of the crystals of the system under con- plied by the symmetry elements of the group GH, we sideration are invariant subgroups of the group G ∈ ssp arrive at the face GH f. Using the elements hi Gssp for and, therefore, expansion (3) acquires the form GH f, we arrive at all the other faces hiGH f of the simple l ( ) form, namely, ∑hi GH f. Applying the elements of the GKGssp = ∑ nKiHhi GKGH . (7) group Gssp to f, we can also represent this simple form i = 1 as Gssp f. Equating the holohedral forms obtained by The product (GKGH) in (7), which acts onto the elemen- these two methods, we obtain tary part of f, yields one conjugated form ≤ ≤ [ ] ( ) ( ) Gssp = ∑hiGH, 1 i nH, nH = Gssp: GH , (2) GKGH f = GK GH f . (8) i It follows from (7) and (8) that all the conjugated forms are of the same type. Therefore, we restrict our consid- where hiGH is the right coset with respect to GH [8]. To each face of the simple form in (2) there uniquely cor- eration to only one conjugated form GK(GH f). The group of the face G of conjugated form (8) is deter- responds one coset. The index nH of the subgroup GH in F mined as the intersection G = G ∩ G (5). Form (8) Gssp is equal to the number of faces of the holohedral F H K simple form. Hereafter, for the sake of brevity, the sim- can be obtained by multiplying the face (GF f) with the ple forms are referred to as forms. aid of the coset representatives in the expansion of GK with respect to the subgroup GF: Let a crystal of the system Gssp have the point group ⊂ GK Gssp. In order to determine the number of conju- ( ∩ ) gated forms l and their faces nKiH, apply the group oper- GK = ∑ h j GH GK = ∑ h jGF, (9) ations of GK to the face hiGH f of the holohedral form. j = 1 j = 1 With this aim, multiply hiGH in (2) by GK on the left- ≤ ≤ [ ] 1 j GK : G F = nKH. hand side. The product GKhiGH is the double coset by modulus (GK, GH) [8]. The product GKhiGH contains all The number l of the conjugated forms is calculated by the cosets corresponding to the faces equivalent with the formula respect to GK. The GKhiGH class uniquely corresponds [ ] [ ] l = gssp : gH : gK : gF to one simple form of the crystal with GK. The number (10) ≡ [ ] [ ] nKiH equals the number of the right cosets with respect Gssp : GH : GK : GF . to GH in the product GKhiGH. Multiplying the left-hand × × ≠ If GK' = GK G and GF' = GF G at G C1, then the side of (2) by GK, we arrive at the expansion substitution of GK by GK' and GF by GF ' in (9) does not l l change the system of the coset representatives {hi}. In G G = ∑ n G h G , n = ∑ n . (3) this case, the same form can be used for crystals with K ssp KiH K i H H KiH different symmetries, and the condition for the exist- i = 1 i = 1 ence of the same simple form can be written as Expansion (3) is, in fact, the scheme for making the faces of the holohedral form nonequivalent. Using the GK = ∑hiGF, GK' = ∑hi'GF', (11) –1 i i conjugated groups hiGH hi , we can represent expan- ≤ ≤ sion (2) in the form h i = h i' , 1 i n K H . –1 If G = G , then the condition for the admission of the G f = ∑(h G h )(h f ), 1 ≤ i ≤ n , (4) F F ' ssp i H i i H same simple form reduces to the condition of “equal i power” of the systems of representatives (11). In this –1 case, the groups GK and GK' are of the same order. where the group hiGHhi is the symmetry group of the ith face of the holohedral form, and hi f is one of the elementary parts of this face. The symmetry group of the SIMPLE FORMS AND INTERGROWN face of the ith conjugated form (3) is determined as the CUBIC CRYSTALS –1 intersection of the groups GK and hiGH hi , For the cubic system, Gssp = Gsp = Oh. If GK = Oh, all the simple forms—a cube, a rhombododecahedron, tet- –1 ∩ ≤ ≤ GKiH = hiGHhi GK, 1 i l. (5) rahexahedron, octahedron, tetragon-trioctahedron, trigon-trioctahedron, and hexakis octahedron—are The number nKiH of the faces of the ith conjugated form holohedral. The following holohedral forms possess CRYSTALLOGRAPHY REPORTS Vol. 45 No. 4 2000 GROUP–THEORETICAL REPRESENTATION OF HOLOHEDRAL FORMS 531 the maximum number of faces having the highest pos- has the form sible symmetry. tetrahexahedra pentagon-dodecahedra A cube has the face group GH = G4V.

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