InternationalReciprocal spaceTables for Crystallography Volume B Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 4.6, pp. 593-594
## Section 4.6.2.3. 1D composite structures |

In the simplest case, a *composite structure* (CS) consists of two intergrown periodic structures with mutually incommensurate lattices. Owing to mutual interactions, each subsystem may be modulated with the period of the other. Consequently, CSs can be considered as coherent intergrowths of two or more incommensurately modulated substructures. The substructures have at least the origin of their reciprocal lattices in common. However, in all known cases, at least one common reciprocal-lattice plane exists. This means that at least one particular projection of the composite structure exhibits full lattice periodicity.

The unmodulated (basic) 1D subsystems of a 1D incommensurate intergrowth structure can be related to each other in a 2D parameter space (Fig. 4.6.2.4). The actual atoms result from the intersection of the physical space with the hypercrystal. The hyperatoms correspond to a convolution of the real atoms with infinite lines parallel to the basis vectors and of the 2D hyperlattice .

An appropriate basis is given by where and are the lattice parameters of the two substructures and *c* is an arbitrary constant. Taking into account the interactions between the subsystems, each one becomes modulated with the period of the other. Consequently, in the 2D description, the shape of the hyperatoms is determined by their modulation functions (Fig. 4.6.2.5).

A basis of the reciprocal lattice can be obtained from the condition : The metric tensors for the reciprocal and the direct 2D lattices for are The Fourier transforms of the hypercrystals depicted in Figs. 4.6.2.4 and 4.6.2.5 correspond to the weighted reciprocal lattices illustrated in Figs. 4.6.2.6 and 4.6.2.7. The 1D diffraction patterns in physical space are obtained by a projection of the weighted 2D reciprocal lattices upon . All Bragg reflections can be indexed with integer numbers in both the 2D description and in the 1D physical-space description with two parallel basis vectors .

The reciprocal-lattice points and , , , on the main axes and are the main reflections of the two substructures. All other reflections are referred to as satellite reflections. Their intensities differ from zero only in the case of modulated substructures. Each reflection of one subsystem coincides with exactly one reflection of the other subsystem.