Periodicity incommensurate

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. 

Even in the thermodynamic equilibrium, 0 z) can form a periodic soliton-like structure with a period incommensurate with the helical pitch qo = 2itfPo [41]. When the field is applied the conical distortion can appear, especially for certain values of the twist elastic constants, K22 > Kzz [29, 42]. Such conical distortion can be accompanied by a compression of the hehcal pitch. The results of some earlier controversial observations were discussed in [6] and [1]. Probably, the conical deformation has been observed in a ternary cholesteric mixture by measuring the field-induced changes in the... 

These structures were firstly observed for terminally polar mesogens [11, 12]. However, recent experiments give clear evidence of the presence of smectic A layering [37, 38], re-entrant nematic behaviour [39], two-dimensional lattices [40, 41] and smectic layering with incommensurate periodicities [42] for non-polar sterically asymmetric LCs. 

Fig. 19—Shear stress and chain angle as a function of sliding distance, from simulations of alkanethiolates on Au(111) at temperature 1 K (a) results from commensurate sliding show a stick-slip motion with a period of 2.5 A, (b) in incommensurate case both shear stress and chain angle exhibit random fluctuations with a much smaller average friction [45],...

Figure 20 provides important information on the activities of individual molecules, considering the fact that the performance of tail groups is a manifestation of molecule motion. The periodic jumps observed in Fig. 20(a) indicate that the alkane chains are plucked by the opposite mono-layer at the moment of slip. Looking at Fig. 20(b), however, it is difficult to tell whether or not the plucking mechanism also involves in an incommensurate sliding. 

The system is regarded as commensurate if there is a rational value for the ratio a=al2iT=L/N, where a denotes the mean space of the particles, 27t is the potential period, and L indicates the number of potential periods within the calculation window. By using different combinations of L and N, one may be able to alter the commensurability of the system, which will become incommensurate when a approaches an irrational value. The golden mean, a=( /5- l)/2 144/233, for example, is usually chosen to characterize a typical incommensurate case. 

At finite velocity kinetic friction behaves quite differently in the sense that the commensurability plays a less significant role. Besides, the system shows rich dynamic properties since Eq (16) may lead to periodic, quasi-periodic, or chaotic solutions, depending on damping coefficient y and interaction strength h. Based on numerical results of an incommensurate case [18,19], we outline a force curve of F in Fig. 23 asafunction ofv, in hopes of gaining a better understanding of dynamic behavior in the F-K model. 

Solid contacts are incommensurate in most cases, except for two crystals with the same lattice constant in perfect alignment. That is to say, a commensurate contact will become incommensurate if one of the objects is turned by a certain angle. This is illustrated in Fig. 30, where open and solid circles represent the top-layer atoms at the upper and lower solids, respectively. The left sector shows two surfaces in commensurate contact while the right one shows the same solids in contact but with the upper surface turned by 90 degrees. Since the lattice period on the two surfaces, when measured in the x direction, are 5 3 A and 5 A, respectively, which gives a ratio of irrational value, the contact becomes incommensurate. 

A Bloch function for a crystal has two characteristics. It is labeled by a wave vector k in the first Brillouin zone, and it can be written as a product of a plane wave with that particular wave vector and a function with the "little" period of the direct lattice. Its counterpart in momentum space vanishes except when the argument p equals k plus a reciprocal lattice vector. For quasicrystals and incommensurately modulated crystals the reciprocal lattice is in a certain sense replaced by the D-dimensional lattice L spanned by the vectors It is conceivable that what corresponds to Bloch functions in momentum space will be non vanishing only when the momentum p equals k plus a vector of the lattice L. 

Figure 8.10 Idealized structures of Bi2Ca2Sr2Cu3O10+s (a) atomic structure projected down [100] (b) structure as Bi-O and perovskite lamellae (c) incommensurate modulation (exaggerated) along the b axis with a period b 5.8 b (orthorhombic) and (d) incommensurate modulation (exaggerated) along the b axis with a period b w 5.8 h (monochnic).

Unlike the bulk morphology, block copolymer thin films are often characterized by thickness-dependent highly oriented domains, as a result of surface and interfacial energy minimization [115,116]. For example, in the simplest composition-symmetric (ID lamellae) coil-coil thin films, the overall trend when t>Lo is for the lamellae to be oriented parallel to the plane of the film [115]. Under symmetric boundary conditions, frustration cannot be avoided if t is not commensurate with L0 in a confined film and the lamellar period deviates from the bulk value by compressing the chain conformation [117]. Under asymmetric boundary conditions, an incomplete top layer composed of islands and holes of height Lo forms as in the incommensurate case [118]. However, it has also been observed that microdomains can reorient such that they are perpendicular to the surface [ 119], or they can take mixed orientations to relieve the constraint [66]. 

The two surfaces that comprise a contact can be oriented in any number of specific ways however, for crystalline surfaces, interfacial symmetries correspond to either of two broad classifications. The first type of orientation is called the commensurate case and is found when two identical surfaces are perfectly aligned. The term incommensurate corresponds to the case in which two crystalline surfaces are misoriented or have different periodicities. An example of a commensurate systems is given as structure A in Figure 7, whereas structures B through D are incommensurate. Interestingly, the orientation of the surfaces within a contact has a tremendous influence on... 

Therefore, whenever we introduce symmetries into our systems, we risk observing behavior that is inconsistent with that observed when these symmetries are absent. Because opposing surfaces are almost always incommensurate unless they are prepared specifically, it will be important to avoid symmetries in simulations as much as possible. Unfortunately, it can be difficult to make two surfaces incommensurate in simulations, particularly when the interface is composed of two identical crystalline surfaces. These difficulties arise from the fact that only a limited number of geometries conform to the periodic boundary conditions in the lateral direction. Each geometry needs to be analyzed separately... 

A group of crystals show diffraction patterns in which two or more 3D lattices having periods commensurate or incommensurate to each other may be recognized. In other words, the crystal consists of two or more interpenetrating substructures (two or more different atom sets) with different periods at least along one direction (see Fig. 3.42). Names such as composite crystals, vernier structures, misfit-layer structures, and chimney-ladder structures have been used for this group of structures. 

The resulting (A + T) sequence is perfectly ordered (its diffraction pattern would be built by peaks ) but it is not periodic (the two periods a and ar/2 are incommensurate) (See Fig. 3.45). Approximations to this composite structure may be built considering the representation of r through a continuous infinite fraction ... 

The essential properties of incommensurate modulated structures can be studied within a simple one-dimensional model, the well-known Frenkel-Kontorova model . The competing interactions between the substrate potential and the lateral adatom interactions are modeled by a chain of adatoms, coupled with harmonic springs of force constant K, placed in a cosine substrate potential of amplitude V and periodicity b (see Fig. 27). The microscopic energy of this model is ... 

Chen et ai, 2002), which exhibits eight-fold periodicity along the [110] azimuth and incommensurate order along [001] and HBC/Au(lll) (Sellam et ai, 2001) -3 6 ... 

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