1 Institut de Physique de Rennes, CNRS UMR 6251, Univ. Rennes 1, 35042 Rennes, France
2 CNRS, UMR 6251, IPR, 263 Avenue du Général Leclerc, 35042 Rennes Cedex, France.
3 Laboratoire Léon Brillouin, CEA-CNRS, CEA Saclay, 91191 Gif-sur-Yvette, France
4 Institut Laue-Langevin, 38042 Grenoble Cedex 9, France.
5 Department of Chemistry, Kansas State University, Manhattan, KS 66506, USA.
6 Facultad de Ciencias, Universidad del Pais Vasco, Apdo 644, Bilbao, Spain.

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Symmetry and order are essential properties in the description of the various states of matter and their physical properties. These have long been synonymous with periodicity for solids, also used to define the notion of crystal, paradigm for many topics of condensed matter during the XX^th century.

One way to probe the structure of matter at the atomic scale is to use radiation diffraction (such as X-rays, or the wave associated with particles: neutrons, electrons ...). The presence of a long-distance order is then characterized by the presence of diffraction peaks, forming an image reflecting the symmetry of the object. This led to the discovery in the recent decades of non-periodic materials, but still with long range order as evidenced by discrete peaks in their diffraction spectrum. Physicists represent this type of crystals as periodic crystals but in a super-space (with dimension 3 + d, which represent the 3 usual dimensions of space and the dimension d of the internal space).

The systems understudy are aperiodic supramolecular model systems consisting of an urea single crystal (host structure) and alkane molecules (invited molecules), whose length is determined by the number n_C of carbon atoms in the molecule (n_C>7). Crystalline plane of urea have a structure of hexagonal symmetry. Along the C perpendicular axis, the structure exhibits a double helix forming channels in which alkane molecules may be inserted. At room temperature, these molecules are ordered along the C axis of the urea lattice, but according to an aperiodic order.

 

 

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Neutron diffraction spectrum along the C* axis. Crossing the first critical temperature Tc₁ superstructure rods (h = 1/2) along C* appear that indicates a change in the symmetry of the system. Along the rods, incommensurate discrete diffraction peaks are characteristic of the new aperiodic ordered structure. This mesh doubling is completely original because the observed phase transition between the two incommensurate phases can be only properly described by means of the 4-dimensional space. A second structural phase appears below Tc₂.

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