Incommensurate crystal

M. McMahon, R. Nelmes, Incommensurate crystal structures in the elements at high pressures. Z. Kristallogr. 219 (2004) 742. 

V. L Pokhrovsky, A. L. Talapov, Theory of Incommensurate Crystals, Soviet Scientific Reviews Suppl. Sec. A, Vol. 1, (Ed. I.M. Khalatnikov), Harwood Academics, Switzerland, 1984. 

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. 

Measurements have been made on a wide variety of molecules adsorbed on Au, Ag, or Pb surfaces [3,4,131,132]. The phase of the adsorbed layer changes from fluid to crystal as the density is increased. As expected, motion of fluid layers produces viscous dissipation that is, the friction vanishes linearly with the sliding velocity. The only surprise is that the ratio between friction and velocity, called the drag coefficient, is orders of magnitude smaller than would be implied by the conventional no-slip boundary condition. When the layer enters an incommensurate phase, the friction retains the viscous form. Not only does the incommensurate crystal shde without measurable static friction, the drag coefficient is as much as an order of magnitude smaller than for the liquid phase ... 

These are crystal structures which have perfect long-range order but which are only approximately periodic, incommensurate crystals on the one hand, and quasi-crystals on the other. 

Quasi-crystals have macroscopic symmetries which are incompatible with a crystal lattice (Section 2.4.1). The first example was discovered in 1984 when the alloy AlMn is rapidly quenched, it forms quasi-crystals of icosahedral symmetry (Section 2.5.6). It is generally accepted that the structure of quasicrystals is derived from aperiodic space filling by several types of unit cell rather than one unique cell. In two-dimensional space, the best-known example is that of Penrose tiling. It is made up of two types of rhombus and has fivefold symmetry. We assume that the icosahedral structure of AlMn is derived from a three-dimensional stacking analogous to Penrose tiling. As is the case for incommensurate crystals, quasi-crystals can be described by perfectly periodic lattices in spaces of dimension higher than three in the case of AlMn, we require six-dimensional space. 

Janner. A. Janssen, T. Symmetry of incommensurate crystal phases. 1. Commensurate basic structures. Acta Crystallogr. 1980. A36, 399. 

The structural study of aperiodic material is a field that is still evolving. Their detailed study at the atomic level will possibly deliver the origin of the mechanism leading to aperiodicity rather than periodicity. We can also expect to use the structural properties of incoinmensurabilities as a probe to improve our knowledge of the chemical and physical aspects of atomic interactions. In addition, we can also expect to discover and take advantage of some specific properties associated with the aperiodicities of crystalline material. A series of studies on the NMR and NQR properties of incommensurate crystals already revealed some specific characteristics of these systems. 

In the same NMR spectrum the pure polymer resonates at a separate chemical shift value. Host-guest cross peaks are diagnostic of the nanoscale topological relationship, giving an insight into the incommensurate crystal structure. This kind of spectroscopy was extended from polymethylene chains to a number of polymer nanocomposites, including rubbery polymers. The most interesting examples, those are formed with elastomers, where the crystalline adducts act as reinforcement for the elastomeric material [55]. 

Berker, A.N. and Indekeu, J.O. (1987) in Incommensurate Crystals, Liquid Crystals, and Quasi-Crystals (eds J.F. Scott and N.A. Clark), Plenum Press, New York, p. 205. 

Elemental antimony under normal conditions has a structure of A7 and at 8.5-12 GPa transforms into a modification with a complex atomic lattice (Sb-II) [20], which at 28 GPa converts to a bcc structure. The heavier analogue Bi has the same phase transitions as Sb, at generally lower pressures Bi-I (A7 structure) transforms at 2.5 GPa to a strongly distorted sc structure (Bi-II). Already at 2.8 GPa there occurs a structural reorganization into the Bi-III phase, which has an incommensurate crystal structure with Nc 9. This arrangement is very similar to that found for high-pressure allotropes of As and Sb. Upon further compression, up to P > 8 GPa, bismuth transforms to a bcc solid [20]. 

---