Point groups

Janssen, T.; Janner, A.; Looijenga-Vos, A.; Wolff, P. M. de

doi:10.1107/97809553602060000624

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. C. ch. 9.8, pp. 938-939

Section 9.8.4.2. Point groups

T. Janssen,^a A. Janner,^a A. Looijenga-Vos^b and P. M. de Wolff^c ^‡

^a Institute for Theoretical Physics, University of Nijmegen, Toernooiveld, NL-6525 ED Nijmegen, The Netherlands,^bRoland Holstlaan 908, NL-2624 JK Delft, The Netherlands, and ^cMeermanstraat 126, 2614 AM, Delft, The Netherlands

9.8.4.2. Point groups

| top | pdf |

9.8.4.2.1. Laue class

| top | pdf |

Definition 1. The Laue point group [P_L] of the diffraction pattern is the point group in three dimensions that transforms every diffraction peak into a peak of the same intensity.²

Because all diffraction vectors are of the form (9.8.4.5), the action of an element R of the Laue group is given by $[R{\bf a}^*_i=\textstyle\sum\limits^{3+d}_{j=1}\Gamma^*(R)_{ji}{\bf a}^*_j, \quad i=1, \ldots,3+d. \eqno (9.8.4.15)]$ The (3 + d) × (3 + d) matrices $[\Gamma^*(R)]$ form a finite group of integral matrices $[\Gamma^*(K)]$ for K equal to [P_L] or to one of its subgroups. A well known theorem in algebra states that then there is a basis in 3 + d dimensions such that the matrices $[\Gamma^*(R)]$ on that basis are orthogonal and represent (3 + d)-dimensional orthogonal transformations [R_s] . The corresponding group is a (3 + d)-dimensional crystallographic group denoted by [K_s] . Because R is already an orthogonal transformation on V, [R_s] is reducible and can be expressed as a pair [(R,R_I)] of orthogonal transformations, in 3 and d dimensions, respectively. The basis on which acts according to $[\Gamma^*(R)]$ is denoted by $[\{({\bf a}^*_i,{\bf a}^*_{Ii})\}]$ . It spans a lattice $[\Sigma^*]$ that is the reciprocal of the lattice $[\Sigma]$ with basis elements $[({\bf a}_i,{\bf a}_{Ii})]$ . The pairs [(R, R_I)] , sometimes also noted [(R_E,R_I)] , leave $[\Sigma]$ invariant: $[(R, R_I)({\bf a}_i, {\bf a}_{Ii}) \equiv (R{\bf a}_i,R_I{\bf a}_{Ii})=\textstyle\sum\limits^{3+d}_{j=1}\Gamma(R)_{ji}({\bf a}_j,{\bf a}_{Ij}), \eqno (9.8.4.16)]$ where Γ(R) is the transpose of $[\Gamma^*(R^{-1})]$ .

In many cases, one can distinguish a lattice of main reflections, the remaining reflections being called satellites. The main reflections are generally more intense. Therefore, main reflections are transformed into main reflections by elements of the Laue group. On a standard lattice basis (9.8.4.13), the matrices Γ(R) take the special form $[\Gamma(R)=\left(\matrix{ \Gamma_E(R)&0 \cr\Gamma_M(R)&\Gamma_I(R)}\right). \eqno (9.8.4.17)]$ The transformation of main reflections and satellites is then given by $[\Gamma^*(R)]$ as in (9.8.4.15), the relation with Γ(R) being (as already said) $[\Gamma^*(R)= \tilde\Gamma(R^{-1}),]$ where the tilde indicates transposition. Accordingly, on a standard basis one has $[\Gamma^*(R)=\left(\matrix{ \Gamma^*_E(R)&\Gamma^*_M(R) \cr 0&\Gamma^*_I(R)}\right). \eqno (9.8.4.18)]$ The set of matrices $[\Gamma_E(R)]$ for R elements of K forms a crystallographic point group in three dimensions, denoted [K_E] , having elements R of O(3), and the corresponding set of matrices $[\Gamma_I(R)]$ forms one in d dimensions denoted by [K_I] with elements [R_I] of O(d).

For a modulated crystal, one can choose the $[{\bf a}^*_i]$ (i = 1, 2, 3) of a standard basis. These span the (reciprocal) lattice of the basic structure. One can then express the additional vectors $[{\bf a}^*_{3+j}]$ (which are modulation wavevectors) in terms of the basis of the lattice of main reflections: $[{\bf a}^*_{3+j}=\textstyle\sum\limits^3_{i=1}\sigma_{ji}{\bf a}^*_i, \quad j=1,2, \ldots, d. \eqno (9.8.4.19)]$ The three components of the jth row of the (d × 3)-dimensional matrix σ are just the three components of the jth modulation wavevector $[{\bf q}_j={\bf a}^*_{3+j}]$ with respect to the basis $[{\bf a}^*_1, {\bf a}^*_2, {\bf a}^*_3]$ . It is easy to show that the internal components $[{\bf a}_{Ii}]$ (i = 1, 2, 3) of the corresponding dual standard basis can be expressed as $[{\bf a}_{Ii}=-\textstyle\sum\limits^d_{j=1}\sigma_{ji}{\bf d}_j, \quad i=1,2,3. \eqno (9.8.4.20)]$ This follows directly from (9.8.4.19) and the definition of the reciprocal standard basis (9.8.4.13). From (9.8.4.16) and (9.8.4.17), a simple relation can be deduced between σ and the three constituents $[\Gamma_E(R)]$ , $[\Gamma_I(R)]$ , and $[\Gamma_M(R)]$ of the matrix Γ(R): $[-\Gamma_I(R)\sigma + \sigma\Gamma_E(R)=\Gamma_M(R). \eqno (9.8.4.21)]$ Notice that the elements of $[\Gamma_M(R)]$ are integers, whereas σ has, in general, irrational entries. This requires that the irrational part of σ gives zero when inserted in the left-hand side of equation (9.8.4.21). It is therefore possible to decompose formally σ into parts $[\sigma^i]$ and $[\sigma^r]$ as follows. $[\sigma =\sigma^i+\sigma^r,\quad\hbox{ with }\quad\sigma^i\equiv{1\over N}\sum_R\Gamma_I(R)\sigma\Gamma_E(R)^{-1}, \eqno (9.8.4.22)]$ where the sum is over all elements of the Laue group of order N. It follows from this definition that $[\Gamma_I(R)\sigma^i\Gamma_E(R)^{-1}=\sigma^i.\eqno (9.8.4.23)]$ This implies $[\Gamma_M(R)=-\Gamma_I(R)\sigma^r+\sigma^r\Gamma_E(R). \eqno (9.8.4.24)]$ The matrix $[\sigma^r]$ has rational entries and is called the rational part of σ. The part $[\sigma^i]$ is called the irrational (or invariant) part.

The above equations simplify for the case d = 1. The elements $[\sigma_{1i}=\sigma_i]$ are the three components of the wavevector q, the row matrix $[\sigma\Gamma_E(R)]$ has the components of $[R^{-1}{\bf q}]$ and $[\Gamma_I(R)]$ = ɛ = ±1 since, for d = 1, q can only be transformed into ±q. One has the corresponding relations $[{\bf q} = {\bf q}^i+{\bf q}^r,\quad\hbox{ with }\quad{\bf q}^i\equiv{1\over N}\sum_R\varepsilon R{\bf q}, \eqno (9.8.4.25)]$ and $[R{\bf q}\equiv\varepsilon{\bf q}\hbox{ (modulo reciprocal lattice} \Lambda^*)\semi \quad R{\bf q}^i=\varepsilon{\bf q}^i. \eqno (9.8.4.26)]$ The reciprocal-lattice vector that gives the difference between $[R{\bf q}]$ and $[\varepsilon{\bf q}]$ has as components the elements of the row matrix $[\Gamma_M(R)]$ .

9.8.4.2.2. Geometric and arithmetic crystal classes

| top | pdf |

According to the previous section, in the case of modulated structures a standard basis can be chosen (for M* and correspondingly for $[\Sigma^*]$ ). According to equation (9.8.4.15), for each three-dimensional point-group operation R that leaves the diffraction pattern invariant, there is a point-group transformation [R_E] in the external space (the physical one, so that [R_E=R] ) and a point-group transformation [R_I] in the internal space, such that the pair [(R,R_I)] is a (3 + d)-dimensional orthogonal transformation [R_s] leaving a (3 + d)-dimensional lattice $[\Sigma]$ invariant. For incommensurate crystals, this internal transformation is unique and follows from the transformation by R of the modulation wavevectors [see equations (9.8.4.15) and (9.8.4.18) for the $[{\bf a}^*_{3+j}]$ basis vectors]: there is exactly one [R_I] for each R. This is so because in the incommensurate case the correspondence between M* and $[\Sigma^*]$ is uniquely fixed by the embedding rule (9.8.4.10) (see Subsection 9.8.4.1). Because the matrices Γ(R) and the corresponding transformations in the (3 + d)-dimensional space form a group, this implies that there is a mapping from the group [K_E] of elements [R_E] to the group [K_I] of elements [R_I] that transforms products into products, i.e. is a group homomorphism. A point group [K_s] of the (3 + d)-dimensional lattice constructed for an incommensurate crystal, therefore, consists of a three-dimensional crystallographic point group [K_E] , a d-dimensional crystallographic point group [K_I] , and a homomorphism from [K_E] to [K_I] .

Definition 2. Two (3 + d)-dimensional point groups [K_s] and [K'_s] are geometrically equivalent if they are connected by a pair of orthogonal transformations [(T_E,T_I)] in [V_E] and [V_I] , respectively, such that for every [R_s] from the first group there is an element [R'_s] of the second group such that [R_ET_E=T_ER'_E] and [R_IT_I=T_IR'_I] .

A point group determines a set of groups of matrices, one for each standard basis of each lattice left invariant.

Definition 3. Two groups of matrices are arithmetically equivalent if they are obtained from each other by a transformation from one standard basis to another standard basis.

The arithmetic equivalence class of a (3 + d)-dimensional point group is fully determined by a three-dimensional point group and a standard basis for the vector module M* because of relation (9.8.4.15).

In three dimensions, there are 32 geometrically non-equivalent point groups and 73 arithmetically non-equivalent point groups. In one dimension, these numbers are both equal to two. Therefore, one finds all (3 + 1)-dimensional point groups of incommensurately modulated structures by considering all triples of one of the 32 (or 73) point groups, for each one of the two one-dimensional point groups and all homomorphisms from the first to the second.

Analogously, in (3 + d) dimensions, one takes one of the 32 (73) groups, one of the d-dimensional groups, and all homomorphisms from the first to the second. If one takes all triples of a three-dimensional group, a d-dimensional group, and a homomorphism from the first to the second, one finds, in general, groups that are equivalent. The equivalent ones still have to be eliminated in order to arrive at a list of non-equivalent groups.

References

International Tables for Crystallography (2006). Vol. C. ch. 9.8, pp. 938-939