Phase space filling

The regions of phase space, filled with trajectories of similar asymptotic behaviour, are separated by the sets (for n = 2 these are straight lines) called separatrices. The directions along which separatrices reach a stationary point are called the main directions (axes). The method of determination of separatrices is given in Appendix A2.6. 

The phase space filling (PSF) model was developed to describe non-resonant NLO properties in semiconductors [112, 113], especially in quantum well structures [114]. Greene et al. adapted this model to one-dimensional polymer chains [115, 116]. The model is only applicable to systems were the low energy absorption band is excitonic, as is the case with PDAs [117], Formation of excitons is limited by the number of available electron states that are necessary to form the exciton. With an increasing number of excitons, the dipole momentum for forming a new exciton is reduced. The exciton band bleaches. 

The new phases were discovered by the combination of exploratory synthesis and a phase compatibility study. As commonly practised, the new studies were initially made through the chemical modification of a known phase. Inclusion of salt in some cases is incidental, and the formation of mixed-framework structures can be considered a result of phase segregation (for the lack of a better term) between chemically dissimilar covalent oxide lattices and space-filling, charge-compensating salts. Limited-phase compatibility studies were performed around the region where thermodynamically stable phases were discovered. Thus far, we have enjoyed much success in isolating new salt-inclusion solids via exploratory synthesis. 

At the heart of the AIM theory is the definition of an atom as it exists in a molecule. An atom is defined as the union of a nucleus and the atomic basin that the nucleus dominates as an attractor of gradient paths. An atom in a molecule is thus a portion of space bounded by its interatomic surfaces but extending to infinity on its open side. As we have seen, it is convenient to take the 0.001 au envelope of constant density as a practical representation of the surface of the atom on its open or nonbonded side because this surface corresponds approximately to the surface defined by the van der Waals radius of a gas phase molecule. Figure 6.15 shows the sulfur atom in SC12. This atom is bounded by two interatomic surfaces (IAS) and the p = 0.001 au envelope. It is clear that atoms in molecules are not spherical. The well-known space-filling models are an approximation to the shape of an atom as defined by AIM. Unlike the space-filling models, however, the interatomic surfaces are generally not flat and the outer surface is not necessarily a part of a spherical surface. 

A partial differential equation is then developed for the number density of particles in the phase space (analogous to the classical Liouville equation that expresses the conservation of probability in the phase space of a mechanical system) (32>. In other words, if the particle states (i.e. points in the particle phase space) are regarded at any moment as a continuum filling a suitable portion of the phase space, flowing with a velocity field specified by the function u , then one may ask for the density of this fluid streaming through the phase space, i.e. the number density function n(z,t) of particles in the phase space defined as the number of particles in the system at time t with phase coordinates in the range z (dz/2). 

Fig. 18 Phase space of PI-fc-PS-fc-PEO in vicinity of ODT. Filled and open circles-. ordered and disordered states, respectively, within experimental temperature range 100 < T/° C< 225. Outlined areas compositions with two- and three-domain lamellae (identified by sketches) shaded regions three network phases, core-shell double gyroid (Q230), orthorhombic (O70), and alternating gyroid (Q214). Overlap of latter two phase boundaries indicates high- and low-temperature occurrence, respectively, of each phase. Dashed line condition tfin = 0peo associated with symmetric PI-fc-PS-fc-PEO molecules. From [75]. Copyright 2004 American Chemical Society...

Figure 8.20 Structure and phase sequence of prototypical bent-core mesogen NOBOW (8) are given, along with space-filling model showing one of many conformational minima obtained using MOPAC with AMI force field. With observation by Tokyo Tech group of polar EO switching for B2 smectic phases formed by mesogens of this type, banana LC field was bom. Achiral, polar C2v layer structure, with formation of macroscopic spontaneous helix in polarization field (and concomitant chiral symmetry breaking), was proposed to account for observed EO behavior.

Free-phase liquid filling the pore spaces of the aquifer matrix and thus free to migrate ... 

Reduced dimensional parameters (strain parameters and near-neighbours diagrams) By comparing the space-filling theoretical curves and the actual values of intermetallic phases it has been observed that an incompressible sphere model of the atom gives only a rough description when discussing metallic structures. 

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