(International Tables for Crystallography) A general introduction to space groups

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

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International Tables for Crystallography (2016). Vol. A. ch. 1.3, pp. 22-23

Section 1.3.2.1. Basic properties of lattices

B. Souvignier^a ^*

^aRadboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands
Correspondence e-mail: souvi@math.ru.nl

1.3.2.1. Basic properties of lattices

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The two-dimensional vector space $[{\bb V}^2]$ is the space of columns $[\pmatrix{ x \cr y }]$ with two real components $[x,y \in {\bb R}]$ and the three-dimensional vector space $[{\bb V}^3 ]$ is the space of columns $[\pmatrix{ x \cr y \cr z }]$ with three real components $[x,y,z \in {\bb R}]$ . Analogously, the [n] -dimensional vector space $[{\bb V}^n]$ is the space of columns $[{\bf v} = \pmatrix{v_1 \cr \vdots \cr v_n }]$ with n real components.

For the sake of clarity we will restrict our discussions to three-dimensional (and occasionally two-dimensional) space. The generalization to n-dimensional space is straightforward and only requires dealing with columns of n instead of three components and with bases consisting of n instead of three basis vectors.

Definition

For vectors $[{\bf a}, {\bf b}, {\bf c}]$ forming a basis of the three-dimensional vector space $[{\bb V}^3]$ , the set $[ {\bf L}: = \{ l {\bf a} + m {\bf b} + n {\bf c} \mid l,m,n \in {\bb Z} \} ]$ of all integral linear combinations of $[{\bf a}, {\bf b}, {\bf c} ]$ is called a lattice in $[{\bb V}^3]$ and the vectors $[{\bf a}, {\bf b}, {\bf c}]$ are called a lattice basis of $[{\bf L}]$ .

It is inherent in the definition of a crystal pattern that the translation vectors of the translations leaving the pattern invariant are closed under taking integral linear combinations. Since the crystal pattern is assumed to be discrete, it follows that all translation vectors can be written as integral linear combinations of a finite generating set. The fundamental theorem on finitely generated abelian groups (see e.g. Chapter 21 in Armstrong, 1997) asserts that in this situation a set of three translation vectors $[{\bf a}, {\bf b}, {\bf c} ]$ can be found such that all translation vectors are integral linear combinations of these three vectors. This shows that the translation vectors of a crystal pattern form a lattice with lattice basis $[{\bf a}, {\bf b}, {\bf c}]$ in the sense of the definition above.

By definition, a lattice is determined by a lattice basis. Note, however, that every two- or three-dimensional lattice has infinitely many bases.

Example

The square lattice $[{\bf L} = {\bb Z}^2 = \left\{ \pmatrix{ m \cr n } \mid m, n \in {\bb Z}\right\} ]$ in $[{\bb V}^2]$ has the vectors $[{\bf a} = \pmatrix{ 1 \cr 0 },\quad {\bf b} = \pmatrix{ 0 \cr 1 } ]$ as its standard lattice basis. But $[{\bf a}' = \pmatrix{ 1 \cr -2 },\quad {\bf b}' = \pmatrix{ -2 \cr 3 } ]$ is also a lattice basis of $[{\bf L}]$ : on the one hand $[{\bf a}' ]$ and $[{\bf b}']$ are integral linear combinations of $[{\bf a}, {\bf b} ]$ and are thus contained in $[{\bf L}]$ . On the other hand $[-3 {\bf a}' - 2 {\bf b}' = \pmatrix{ -3 \cr 6 } + \pmatrix{ 4 \cr -6 } = \pmatrix{ 1 \cr 0 } = {\bf a} ]$ and $[-2 {\bf a}' - {\bf b}' = \pmatrix{ -2 \cr 4 } + \pmatrix{ 2 \cr -3 } = \pmatrix{ 0 \cr 1 } = {\bf b}, ]$ hence $[{\bf a}]$ and $[{\bf b}]$ are also integral linear combinations of $[{\bf a}', {\bf b}']$ and thus the two bases $[{\bf a}, {\bf b} ]$ and $[{\bf a}', {\bf b}']$ both span the same lattice (see Fig. 1.3.2.1).

Figure 1.3.2.1 | top | pdf |

Conventional basis $[{\bf a}, {\bf b}]$ and a non-conventional basis $[{\bf a}', {\bf b}']$ for the square lattice.

The example indicates how the different lattice bases of a lattice $[{\bf L}]$ can be described. Recall that for a vector $[{\bf v}]$ = $[x {\bf a} + y {\bf b} + z {\bf c} ]$ the coefficients [x,y,z] are called the coordinates and the vector $[\pmatrix{ x \cr y \cr z }]$ is called the coordinate column of $[{\bf v}]$ with respect to the basis $[{\bf a}, {\bf b}, {\bf c}]$ . The coordinate columns of the vectors in $[{\bf L}]$ with respect to a lattice basis are therefore simply columns with three integral components. In particular, if we take a second lattice basis $[{\bf a}', {\bf b}', {\bf c}']$ of $[{\bf L}]$ , then the coordinate columns of $[{\bf a}']$ , $[{\bf b}']$ , $[{\bf c}']$ with respect to the first basis are columns of integers and thus the basis transformation $[{\bi P}]$ such that $[({\bf a}', {\bf b}', {\bf c}') = ({\bf a}, {\bf b}, {\bf c}) {\bi P} ]$ is an integral 3 × 3 matrix. But if we interchange the roles of the two bases, they are related by the inverse transformation $[{\bi P}^{-1} ]$ , i.e. $[({\bf a}, {\bf b}, {\bf c}) = ({\bf a}', {\bf b}', {\bf c}') {\bi P}^{-1} ]$ , and the argument given above asserts that $[{\bi P}^{-1}]$ is also an integral matrix. Now, on the one hand $[\det {\bi P}]$ and $[\det {\bi P}^{-1} ]$ are both integers (being determinants of integral matrices), on the other hand $[\det {\bi P}^{-1} = 1 / \det {\bi P}]$ . This is only possible if $[\det {\bi P} = \pm 1]$ .

Summarizing, the different lattice bases of a lattice $[{\bf L}]$ are obtained by transforming a single lattice basis $[{\bf a}, {\bf b}, {\bf c} ]$ with integral transformation matrices $[{\bi P}]$ such that $[\det {\bi P} = \pm 1]$ .

References

Armstrong, M. A. (1997). Groups and Symmetry. New York: Springer.Google Scholar

International Tables for Crystallography (2016). Vol. A. ch. 1.3, pp. 22-23