Lattices and metric

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

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International Tables for Crystallography (2006). Vol. C. ch. 9.8, pp. 937-938

Section 9.8.4.1. Lattices and metric

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

9.8.4.1. Lattices and metric

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A periodic crystal structure is defined in a three-dimensional Euclidean space V and is invariant with respect to translations n which are integral linear combinations of three fundamental ones $[{\bf a}_1,{\bf a}_2,{\bf a}_3]$ : $[{\bf n}=\textstyle\sum\limits^3_{i=1}n_i{\bf a}_i, \quad n_i\hbox{ integers}. \eqno (9.8.4.1)]$ These translations are linearly independent and span a lattice Λ. The dimension of Λ is the dimension of the space spanned by $[{\bf a}_1,{\bf a}_2,{\bf a}_3]$ and the rank is the (smallest) number of free generators of those integral linear combinations. In the present case, both are equal to three. Accordingly, $[\{\Lambda\}=V\quad\hbox{ and }\quad \Lambda \approx Z^3. \eqno (9.8.4.2)]$ The elements of [Z^3] are triples of integers that correspond to the coordinates of the lattice points. The Bragg reflection peaks of such a crystal structure are at the positions of a reciprocal lattice Λ*, also of dimension and rank equal to three. Furthermore, the Fourier wavevectors H belong to Λ* (after identification of lattice vectors with lattice points): $[{\bf H}=\textstyle\sum\limits^3_{i=1}h_i{\bf a}^*_i, \quad h_i\hbox{ integers} \eqno (9.8.4.3)]$ where $[\{{\bf a}^*_i\}]$ is the reciprocal basis $[{\bf a}_i\cdot {\bf a}^*_k=\delta_{ik}.]$ The two corresponding metric tensors g and [g^*] , $[g_{ik}={\bf a}_i\cdot{\bf a}_k\quad\hbox{ and }\quad g^*_{ik}={\bf a}^*_i\cdot{\bf a}^*_k, \eqno (9.8.4.4)]$ are positive definite and dual: $[\textstyle\sum\limits^3_{k=1}g_{ik} g^*_{kj}=\delta_{ij}.]$ We now consider crystal structures defined in the same three-dimensional Euclidean space V with Fourier wavevectors that are integral linear combinations of n = (3 + d) fundamental ones $[{\bf a}^*_1]$ , $[\ldots]$ , $[{\bf a}^*_n]$ : $[{\bf H}=\textstyle\sum\limits^n_{i=1}h_i {\bf a}^*_i, \quad h_i\hbox{ integers}. \eqno (9.8.4.5)]$ The components $[(h_1,\ldots,h_n)]$ are the indices labelling the corresponding Bragg reflection peaks.

A crystal is incommensurate when d > 0 and the vectors $[{\bf a}^*_i]$ linearly independent over the rational numbers. In that case, the crystal does not have lattice periodicity and is said to be aperiodic. The above description can still be convenient, even in the case that the vectors $[{\bf a}^*_i]$ are not independent over the rationals: one or more of them is then expressed as rational linear combinations of the others. A typical example is that of a superstructure arising from the (commensurate) modulation of a basic structure with lattice periodicity.

Let us denote by M* the set of all integral linear combinations of the vectors $[{\bf a}^*_1, \ldots,{\bf a}^*_n]$ . These are said to form a basis. It is a set of free Abelian generators, therefore the rank of M* is n. The dimension of M* is the dimension of the Euclidean space spanned by M* $[\{M^*\}=V \quad\hbox{ and }\quad M^*\approx Z^n. \eqno (9.8.4.6)]$ The elements of [Z^n] are precisely the set of indices introduced above. Mathematically speaking, M* has the structure of a (free Abelian) module. Its elements are vectors. So we call M* a vector module. This nomenclature is intended as a generic characterization. When a series of structures is considered with different values of the components of the last d vectors with respect to the first three, the generic values of these components are irrational, but accidentally they may become rational as well. This situation typically arises when considering crystal structures under continuous variation of parameters like temperature, pressure or chemical composition. In the case of an ordinary crystal, rank and dimension are equal, the crystal structure is periodic, and the vector module becomes a (reciprocal) lattice.

Lattices and vector modules are, mathematically speaking, free Z modules. For such a module, there exists a dual one that is also free and of the same rank. In the periodic crystal case, that duality can be expressed by a scalar product, but for an aperiodic crystal this is no longer possible. It is possible to keep the metrical duality by enlarging the space and considering the vector module M* as the projection of an n-dimensional (reciprocal) lattice $[\Sigma^*]$ in an n-dimensional Euclidean space [V_s] . $[M^*\rightarrow\Sigma^*, \quad\{\Sigma^*\}=V_s\quad \hbox{ and }\quad\Sigma^*\approx Z^n, \eqno (9.8.4.7)]$ with the orthogonal projection $[\pi_E]$ of [V_s] onto V defined by $[M^*=\pi_E\Sigma^*. \eqno (9.8.4.8)]$ This corresponds to attaching to the diffraction peak with indices $[(h_1,\ldots,h_n)]$ the point of an n-dimensional reciprocal lattice having the same set of coordinates. The orthocomplement of V in [V_s] is called internal space and denoted by [V_I] . The embedding is uniquely defined by the relations $[a^*_{si}=({\bf a}^*_i,{\bf a}^*_{Ii}), \quad i=1,\ldots, n, \eqno (9.8.4.9)]$ where $[\{a^*_{si}\}]$ is a basis of $[\Sigma^*]$ and $[\{{\bf a}^*_i\}]$ a basis of M*. The vectors $[{\bf a}^*_{Ii}]$ span [V_I] .

The crystal density ρ in V can also be embedded as $[\rho_s]$ in [V_s] by identifying the Fourier coefficients $[\hat\rho]$ at points of M* and of $[\Sigma^*]$ having correspondingly the same components. $[\hat\rho_s(h_1,\ldots,h_n)\equiv\hat\rho(h_1,\ldots,h_n). \eqno (9.8.4.10)]$ Then $[\rho_s]$ is invariant with respect to translations of the lattice $[\Sigma]$ with basis $[a_{si}=({\bf a}_i, {\bf a}_{Ii}) \eqno (9.8.4.11)]$ dual to (9.8.4.9). In the commensurate case, this correspondence requires that the given superstructure be considered as the limit of an incommensurate crystal [for which the embedding (9.8.4.10) is a one-to-one relation].

As discussed below, point-group symmetries R of the diffraction pattern, when expressed in terms of transformation of the set of indices, define n-dimensional integral matrices that can be considered as being n-dimensional orthogonal transformations [R_s] in [V_s] , leaving invariant the Euclidean metric tensors: $[g_{sik}=a_{si}\cdot a_{sk}\quad \hbox{ and }\quad g^*_{sik}=a^*_{si}\cdot a^*_{sk}. \eqno (9.8.4.12)]$ The crystal classes considered in the tables suppose the existence of main reflections defining a three-dimensional reciprocal lattice. For that case, the embedding can be specialized by making the choice $[\eqalign{ a^*_{si} &=({\bf a}^*_i,0) \cr a^*_{s(3+j)}&=({\bf a}^*_{3+j},{\bf d}^*_j)}\ \quad \eqalign{ i&=1,2,3, \cr j&=1,2,\ldots, d=n-3,} \eqno (9.8.4.13)]$ and, correspondingly, $[\eqalign{ a_{si}&=({\bf a}_i,{\bf a}_{Ii}) \cr a_{s(3+j)} &=(0,{\bf d}_j)}\ \quad\eqalign{ i&=1,2,3, \cr j&=1,2,\ldots, d,} \eqno (9.8.4.14)]$ with $[{\bf d}^*_i\cdot{\bf d}_k=\delta_{ik}]$ and $[{\bf a}^*_i\cdot{\bf a}_k=\delta_{ik}]$ . These are called standard lattice bases.

References

International Tables for Crystallography (2006). Vol. C. ch. 9.8, pp. 937-938