(IUCr) Article Footnotes for Arithmetic classification of crystal structures

Notes

¹ $[GL(3,{\bb Z})]$ denotes the group of invertible 3 × 3 matrices with integral entries (as usual, in this paper we denote by $[{\bb Z}]$ and $[{\bb R}]$ the sets of integral and real numbers, respectively).

²Pitteri & Zanzotto (1996) discuss the relation between the arithmetic classification of simple lattices and the earlier criterion actually stated by Bravais (1850).

³These arithmetic groups arise from the analysis of the general indeterminacy in the choice of the multilattice descriptors (see Pitteri, 1985). In the case of deformable simple lattices, the descriptors are the vectors forming the lattice basis, which is determined up to a transformation in $[GL(3,{\bb Z})]$ , whence the role of the latter group in the study of simple-lattice symmetry. For multilattices, one also considers the shift vectors giving the positions of the further simple lattices constituting the multilattice (or, equivalently, of the further points in the unit cell), so that more general arithmetic groups must be considered [see for instance formulae (16) or (32) below]. The arithmetic criterion presented by Pitteri & Zanzotto (1998) can be generalized in a straightforward way also to classify `polyatomic' crystal structures constituted by atoms belonging to a finite number of distinct atomic species. Here, we confine our attention to `monoatomic' crystals, whose points are all physically indistinguishable.

⁴This happens much in the same way in which the classical arithmetic criterion for 1-lattices allows one to determine the three distinct cubic centerings P, F, I that pertain to the same cubic holohedry.

⁵Appendix B shows through an example how such computations are actually carried out.

⁶We recall that for simple lattices these two classifications are equivalent (see for instance Janssen, 1973, p. 120).

⁷Also the description of the symmetry hierarchies that exist among the 29 different types of 3D monoatomic 2-lattices, and the ensuing account of all the possibilities for symmetry breaking in these structures, constitute interesting knowledge for modeling phase transitions in crystalline materials; see Fadda & Zanzotto (2001) for detailed information on the symmetry groups of 2-lattices and their inclusion relations up to $[\Gamma_{3,1}]$ -conjugacy (partial ordering of the 29 conjugacy classes).

⁸`Regular point systems' is the term used in the literature for the special multilattices on which the space-group action is transitive, so that they are constituted by a single orbit [see Hilbert & Cohn-Vossen (1932), Engel (1986), Dolbilin et al. (1998)]. In general, however, a multilattice is not a single crystallographic orbit, but a union of finitely many orbits with the same translational invariance, and does not coincide with a single orbit even when all of its atoms belong to a single species. See §5 for more details, and for examples taken from the allotropic structures of the elements. As recalled above, general multilattices are also termed `multiregular point systems' (Dolbilin et al., 1998), or `ideal crystals' (Engel, 1986, p. 2), when it is useful to stress the fact that they are not single orbits. It is well known that, besides 3D periodicity, the characteristic feature of a regular point system is that it `looks the same' if seen from every one of its points (see Engel, 1986). This is not true for multiregular point systems, which `look different' in finitely many ways when seen from their own points.

⁹We remark that here we classify monoatomic 2-lattices independently of Fischer & Koch's principle: our procedure is not a new way of retrieving their results. Only a posteriori, through a comparison of the explicit list of the distinct types obtained from the two methods, does one find that they are equivalent for the specific case of monoatomic 2-lattices.

¹⁰To summarize, the method by Fischer & Koch classifies, by definition, structures that are regular point systems, while the arithmetic criterion classifies any multiregular point systems. It is an open question whether these two independent criteria coincide for regular point systems. This paper shows that the two classifications do coincide for regular point systems with two points in their unit cell.

¹¹For more general multilattices, with atoms in their unit cells, the relevant group is a subgroup of $[GL(n+2,{\bb Z}) ]$ ; see I or §5 below.

¹²Since the vectors $[\boldvarepsilon_\sigma]$ , [sigma] = 1, 2, 3, 4, are not linearly independent, there are infinitely many 4 × 4 matrices relating them to the vectors $[\bar{\boldvarepsilon}_\sigma ]$ . Proposition 1 states that, when $[{\boldvarepsilon}_\sigma ]$ and $[\bar{\boldvarepsilon}_\sigma]$ are essential, there is always one and only one such matrix in the group $[\Gamma_{3,1}]$ .

¹³As usual, $[\mu^T]$ denotes the transpose of any matrix $[\mu]$ .

¹⁴The classical 14 Bravais types in three dimensions, recalled in the Introduction, are indeed obtained by considering the conjugacy classes of lattice groups of simple lattices (`arithmetic holohedries') within $[GL(3,{\bb Z})]$ .

¹⁵Unlike with 1-lattices, multilattices can realize non-holohedral lattice groups and point groups, that is, it can be $[C_\Lambda\not\leq L_\Lambda ]$ (see for instance type number 25 in Table 1).

¹⁶In International Tables for Crystallography (1996), this definition is given directly for the space groups; as our lattice group $[\Lambda]$ determines the space group, our definition is equivalent.

¹⁷These submanifolds of $[I(\Lambda)]$ are not fixed sets in $[{\cal Q}_{3,1}^{\rm ess}]$ but their projections on the subspace $[\langle K_{11}, K_{12},\ldots, K_{33}\rangle\simeq{\cal C}^+({\cal Q}_3) ]$ are fixed sets for the simple-lattice action (21). As recalled in II, there is the prejudice that, physically, excess skeletal symmetry should not be a stable feature of a multilattice (see for instance Landau & Lifshitz, 1959, §130).

¹⁸We notice that if an operation $[{\bf Q}]$ of the point group $[P(\boldvarepsilon_\sigma)]$ corresponds, through equation (25) or (28), to a matrix $[\mu\in\Lambda^+(\boldvarepsilon_\sigma) ]$ , then, with the choice made in (1) for the origin O, the operation $[({\bf Q}|{\bf 0})]$ belongs to the space group $[{\cal S}[{\cal M}(\boldvarepsilon_\sigma)]]$ of the 2-lattice $[{\cal M}(\boldvarepsilon_\sigma)]$ . However, the operation $[-{\bf Q}]$ , corresponding to the matrix $[-\mu\in-\Lambda^+(\boldvarepsilon_\sigma)]$ , gives the affine operation $[(-{\bf Q}|{\bf p})\in{\cal S}[{\cal M}(\boldvarepsilon_\sigma)] ]$ , which involves also a translation $[{\bf p}]$ . For instance, the fact that 2-lattices are always centrosymmetric means that in their space groups there is always the operation $[(-{\bf 1}|{\bf p}) ]$ .

¹⁹For the centered types of skeletal lattices, International Tables for Crystallography (1996) utilize a conventional non-unit cell that is not suitable for the computations related to equation (28), which need an actual skeletal basis.

²⁰This is the space group of the affine 1-lattice $[O+{\cal L}({\bf e}_a) ]$ , whose point group is isomorphic to the holohedry $[P({\bf e}_a) ]$ (see §2.4). Of course there are 14 such groups (one for each Bravais type) with their corresponding asymmetric units, indicated explicitly in International Tables for Crystallography (1996). That the domain $[{\cal A}({\bf e}_a)]$ in this lemma is bounded comes from the fact that $[{\cal A}({\bf e}_a) ]$ is a special case of a `fundamental domain' for a space-group action on $[{\bb R}^3]$ . See Appendix B for an example.

²¹Indeed, only `a few' of the positions $[{\bf p}]$ in $[A({\bf e}_a)]$ give the correct Bravais type (see Corollary 1 below). The other positions produce smaller lattice groups in $[\Gamma_{3,1}]$ , for which $[{\bf e}_a]$ is a cell having excess symmetry; such positions are consequently not relevant. An example is given in Appendix B4.

²²Recall the comment below Lemma 4.

²³Hence, in the cubic case the site symmetry must contain four threefold operations; in the tetragonal case, it must contain a fourfold operation; in the hexagonal case it must either contain a sixfold operation or a threefold operation; in the rhombohedral case, it must contain a threefold operation. Alternative (a) in Corollary 1 gives the holohedral 2-lattices; alternative (b) gives the non-holohedral 2-lattices, and only occurs once; see point (2) in §4.

²⁴Out of the roughly one hundred Strukturberichte usually considered (see Strukturberichte, 1913-1940), 27 are monoatomic and, among the latter, six are 1-lattices and five, which are all included in our list of 29, are 2-lattices. The latter comprise for instance the well known structures of $[\alpha]$ -U, of diamond, of the h.c.p. metals etc.; see Table 1.

²⁵This is an interesting point because, as is well known, many crystalline materials undergo phase transitions from 1- to 2-lattice structures and vice versa, such as the h.c.p. $[\leftrightarrow ]$ b.c.c. or h.c.p. $[\leftrightarrow]$ f.c.c. transitions (see for instance Nishiyama et al., 1978).

²⁶The $[\nu]$ × $[\nu]$ permutation matrix $[\alpha]$ of a permutationf of $[\{ 1,\ldots, \nu \} ]$ is defined as usual by $[\alpha^{\,j}_i r_j = r_{f(i)}]$ for any numbers $[r_1, \ldots, r_\nu]$ ; so the entries of the matrix $[\alpha^{\,j}_i]$ are all 0, except in the f(i)th row of the ith column where they are 1.

²⁷In the literature, it is not customary to describe the structures (a) and (b) with the origin on a lattice point as we do. The coordinates of the four atoms in their conventional cell are usually given as follows: $[(\,{{1}\over{3}}{{2}\over{3}}z)]$ $[(\,{{2}\over{3}}{{1}\over{3}}z+{{1}\over{2}}\,)]$ $[ (\,{{1}\over{3}}{{2}\over{3}}\bar z) ]$ $[(\,{{1}\over{3}}{{2}\over{3}}\bar z+{{1}\over{2}}\,)]$ for structure (a); $[(00{{1}\over{4}}\,)]$ $[(00{{3}\over{4}}\,)]$ $[ (\,{{2}\over{3}}{{1}\over{3}}{{1}\over{4}}\,) ]$ $[(\,{{1}\over{3}}{{2}\over{3}}{{3}\over{4}}\,)]$ for structure (b).

²⁸The orbits 2(a) and 2(b) of P6₃/mmc actually give the same primitive hexagonal 1-lattice, while 2(c) and 2(d) give the same 2-lattice (the h.c.p. structure). We notice, however, that 2(a) and 2(b) do not belong to the same Wyckoff set of P6₃/mmc; they only are equivalent when considered in the same lattice complex of P6/mmm.

²⁹This happens in the same way in which, for instance, the primitive face-centered and body-centered cubic structures are said to belong to distinct 1-lattice (Bravais) types, because their symmetry groups determine distinct `arithmetic' conjugacy classes in $[GL(3,{\bb Z})]$ . Exactly as the arithmetic criterion produces the 14 Bravais types of 1-lattices (and the 29 types of 2-lattices in Table 1), it can also produce all the distinct types of 4-lattices that are possible, and not only the three considered here.

³⁰Other examples are possible, for instance with 3-lattices; see Proposition 5 in I.

³¹This means that the two criteria distinguish in the same manner the 2-lattice structures that are `essentially' different from a physical point a view, that is, which have different nearest-neighbor relations etc., grouping together only the 2-lattices whose atomic landscape around each of their lattice points is essentially the same. This is what happens, for instance, with the A-, B- or C-centerings in base-centered orthorhombic 1-lattices; such centerings are indeed `essentially' the same and are equivalent for both classification principles when applied to 1-lattice structures.

³²Fischer & Koch (Koch & Fischer, 1975; Fischer & Koch, 1996) determine 402 types of regular point systems (lattice complexes). As single orbits have a maximal number of atoms in their unit translational cells, there is a finite number of distinct types of regular point systems, also in the arithmetic sense. This is not the case for multiregular point systems (multilattices), because, at least mathematically, the latter can have any given number of points in their unit cell.

³³As remarked at the end of §3, for centered lattices the multiplicity of a special Wyckoff position given in International Tables for Crystallography (1996) is a multiple of the index of the site-symmetry group; in this base-centered case, the multiplicative factor is 2.

³⁴Notice that, as explained in §3, we are using the information contained in the list of Wyckoff positions of a space group (in this case C2/m2/m2/m) only to find rapidly the solutions $[{\bf p}]$ to equation (28)₂, given $[{\bf e}_a]$ in (46). In general, the subdivision into Wyckoff sets of these Wyckoff positions is not related to the final subdivision into arithmetic types of the 2-lattices resulting from these solutions.