Density modulation

In the smectic Aj (SmA ) phase, tlie molecules point up or down at random. Thus, tire density modulation can be described as a Fourier series of cosines ... 

The Maier-Saupe tlieory was developed to account for ordering in tlie smectic A phase by McMillan [71]. He allowed for tlie coupling of orientational order to tlie translational order, by introducing a translational order parameter which depends on an ensemble average of tlie first haniionic of tlie density modulation noniial to tlie layers as well as / i. This model can account for botli first- and second-order nematic-smectic A phase transitions, as observed experimentally. 

A phase itself, the amplitude of the density modulation is constant and twist and splay distortions are forbidden, thus the expression for the free energy density simplifies to equation (C2.2.10). 

Smectic A and C phases are characterized by a translational order in one dimension and a liquid-like positional order in two others. In the smectic A phase the molecules are oriented on average in the direction perpendicular to the layers, whereas in the smectic C phase the director is tilted with respect to the layer normal. A simple model of the smectic A phase has been proposed by McMillan [8] and Kobayashi [9] by extending the Maier-Saupe approach for the case of one-dimensional density modulation. The corresponding mean field, single particle potential can be expanded in a Fourier series retaining only the leading term ... 

Fig. 10.10 SEM micrograph of a tunable IR Bragg filter (1.1 xl. 85 mm) of density-modulated PS, suspended at two microactuator arms. Its rest position is more than 180°...

At the same moment, microsegregation of the pinned chains may result in a density modulation along the cylinder axis [82]. 

The situation discussed here is equivalent to a periodic distortion of the lattice with a period 2a, as developed above. When the perturbation //per is given by lattice vibrations, that is mediated by electron-phonon interactions, the electronic density modulation is expressed in terms of a charge-density wave (CDW), while when electron-electron repulsions dominate the modulation is induced by SDWs (Canadell Whangbo, 1991). 

In Landau-Brazovskii theory, the density modulation (or composition for block copolymers) is written as... 

Polycatenars form a broken-layer type 2D density modulated phases because of the mismatch between the cross-section areas required for the terminal chains and for the mesogenic core. For such compounds layers become locally bent to provide more space for tails and, since the bending of layers is finite, this leads to... 

So far we have discussed 2D density modulated phases that are formed by deformation or breaking of the layers. However, there are also 2D phases with more subtle electron density modulations. In some cases additional peaks observed in the XRD pattern (Fig. 10) are related to a double layer periodicity in the structure. As double layer periodicity was observed in the bent-core liquid crystals formed by the asymmetric as well as symmetric molecules [22-25] it should be assumed that the mechanism leading to bilayers must be different from that of the pairing of longitudinal dipole moments of molecules from the neighboring layers, which is valid for smectic antiphases made by asymmetric rod-like molecules. 

Bent-core liquid crystals are especially interesting materials for basic research as in these systems the polar and tilt order are decoupled and polarization splay seems to be an inherent property of the system. Both effects lead to a variety of structures with unusual properties, e.g., formation of the 2D density modulated phases built of the smectic layers fragments. We have presented the current knowledge... 

Pociecha D, Vaupotic N, Gorecka E, Mieczkowski J, Gomola K (2008) 2-D density-modulated structures in asymmetric bent-core liquid crystals. J Mater Chem 18 881-885... 

Gorecka E, Vaupotic N, Pociecha D (2007) Electron density modulations in columnar banana phases. Chem Mater 19 3027-3031... 

The first algorithm to use memory matrices was presented at Brookhaven s first international workshop on reconstruction techniques with the name of density modulation (Barbieri, 1974a). This method recognizes the vortices with equations 3.7 and 3.8, and then subtracts them from the list of the unknowns. By indicating with N k and the number of negative and positive vortices that fall in the ray /, k), the values of the reconstructed matrix at iteration q +1 are obtained with the following instructions ... 

The results obtained with density modulation depend, as we have seen, upon the choice of a parameter T that represents how many times an illegal value must appear in a cell in order to be considered a vortex. If T= 10, for example, it is reasonable to conclude that the point in question is a true vortex, but in this case the procedure is lengthy and the number of unknowns decreases very slowly. The choice of T= 5, on the other hand, increases the speed of the algorithm but also increases the probability of making mistakes in vortex recognition. 

The first reconstructions performed with density modulation were made with the choice T=5, and the results (Figure 3.7D) clearly showed that some points had been erroneously classified as vortices. Despite these mistakes, however, the reconstructions obtained with density modulation were greatly superior to those of the other algorithms (Figure 3.7B and 3.7C), and the memory method therefore is effective even when the choice of its parameters is not ideal. The most important result, however, is another one. The original pictures (Figure 3.7A)... 

The hypotheses that were made about density modulation, therefore, are valid a memory matrix does allow us to obtain new information about the structure that we are reconstructing, and we can progressively move towards the point where a complete reconstruction becomes possible. 

Figure3.8 A grey, or chiaroscuro, picture (A) reconstructed from 12 projections with Convolution (B), ART (C) and density modulation (D). As in the previous case, a complete reconstruction would have required 120 projections equally spaced in the 180° range.

Barbieri, M. 1974a. Density Modulation reconstruction technique. In R.B. Marr (ed.), Techniques of Three-Dimensional Reconstruction, Proceedings of an International Workshop held at Brookhaven National Laboratory, Upton, New York, BNL 20425, pp. 139-141. 

AC current density generator or induced current density modulation... 

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. 

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