Incommensurate intergrowth

Smicker. D. Van Smaalen. S.V. de Boers. J.L. Haas. C. Harris. K.D.M. Observation of the sliding mode in incommensurate intergrowth compounds Brillouin scattering from the inclusion compound of urea and heptadec-ane. Phys. Rev. Lett. 1995, 74. 734. 

Unlike crystals that are packed with identical unit cells in 3D space, aperiodic crystals lack such units. So far, aperiodic crystals include not only quasiperiodic crystals, but also crystals in which incommensurable modulations or intergrowth structures (or composites) occur [14], That is to say, quasiperiodicity is only one of the aperiodicities. So what is quasiperiodicity Simply speaking, a structure is classified to be quasiperiodic if it is aperiodic and exhibits self-similarity upon inflation and deflation by tau (x = 1.618, the golden mean). By this, one recognizes the fact that objects with perfect fivefold symmetry can exist in the 3D space however, no 3D space groups are available to build or to interpret such structures. 

Incommensurate structures have been known for a long time in minerals, whereas TTF-TCNQ is one of the very first organic material in which a incommensurate phase has been observed. There are two main types of incommensurate crystal structures. The first class is that of intergrowth or composite structures, where two (or more) mutually incommensurate substructures coexist, each with a different three-dimensional translational periodicity. As a result, the composite crystal consists of several modulated substructures, which penetrate each other and we cannot say which is the host substructure. The second class is that of a basic triperiodic structure which exhibits a periodic distortion either of the atomic positions (displa-cive modulation) and/or of the occupation probability of atoms (density modulation). When the distortion is commensurate with the translation period of the underlying lattice, the result is a superstructure otherwise, it is an incommensurately modulated structure (IMS) that has no three-dimensional lattice periodicity. 

In the years after the discovery of aperiodic crystals (incommensurate modulated, intergrowth and quasicrystals) crystallography was for me a very rich and open field of research, but not mysterious. Even the surprising combination in snow crystals of sixfold circular rotations with hyperbolic rotations [1], leading to hexagrammal scaling symmetry, fitted into the whole because the atomic positions in ice are invariant with respect to both types of crystallographic rotations [2]. 

Lefort. R. Etrillard. J. Toudic, B. Guillaume, F. Breczewski, T. Bourges, P. Incommensurate intermodulation of an organic intergrowth compound observed by neutron scattering. Phys. Rev. Lett. 1996. 77.4027. 

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