International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 43-44   | 1 | 2 |

## Section 1.3.2.6.5. The case of nonstandard period lattices

G. Bricognea

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#### 1.3.2.6.5. The case of nonstandard period lattices

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Let Λ denote the nonstandard lattice consisting of all vectors of the form , where the are rational integers and are n linearly independent vectors in . Let R be the corresponding lattice distribution: .

Let A be the nonsingular matrix whose successive columns are the coordinates of vectors in the standard basis of ; A will be called the period matrix of Λ, and the mapping will be denoted by A. According to Section 1.3.2.3.9.5 we have for any , and hence . By Fourier transformation, according to Section 1.3.2.5.5 , which we write: with  is a lattice distribution: associated with the reciprocal lattice whose basis vectors are the columns of . Since the latter matrix is equal to the adjoint matrix (i.e. the matrix of co-factors) of A divided by det A, the components of the reciprocal basis vectors can be written down explicitly (see Section 1.3.4.2.1.1 for the crystallographic case ).

A distribution T will be called Λ-periodic if for all ; as previously, T may be written for some motif distribution with compact support. By Fourier transformation, so that is a weighted reciprocal-lattice distribution, the weight attached to node being times the value of the Fourier transform of the motif .

This result may be further simplified if T and its motif are referred to the standard period lattice by defining t and so that , , . Then hence so that in nonstandard coordinates, while in standard coordinates.

The reciprocity theorem may then be written: in nonstandard coordinates, or equivalently: in standard coordinates. It gives an n-dimensional Fourier series representation for any periodic distribution over . The convergence of such series in will be examined in Section 1.3.2.6.10 .