Order parameters defined

Above the transition temperature cf> = 0 and below the critical temperature 4> > 0. At T = 0, = 1. Any order parameter defined to have a magnitude of unity at zero temperature is said to be normalized. Criteria for testing the theory include a number of critical exponents, which are accurately known from experimental measurement. 

In common with most who have addressed the problem posed by this kind of configurational integral ratio, we shall exploit the statistics of an appropriate order-parameter defined on the common configuration space and of the general form... 

In the case of hard spheres the energy function does not provide a usefully graded measure of how far we are from a wormhole The order parameter defined in Eq. (53) will generally be infinite because of hard-sphere overlap in the configurations created by the switch. But it is easy to find an alternative All... 

The next refinement of the model takes into account that the shape of most molecular species differs from being rod-like typical nematogenic molecules are given in Table 4.6-1. The resulting behaviour of such a bi-axial molecule is often associated with hindered rotation, however it can also be understood from a rigid-body model where different moments of inertia lead to oscillations of different angular amplitudes in spite of identical (thermal) excitation and identical repulsive forces (Korte, 1983). This can be summarized by order parameters defined as above but referring to one of the two shorter. 

Fig. 20 The absolute values of the first-order parameters, defined in Eq. 98, for...

In principle, a free energy expansion of this type should be valid for nematic liquid crystals, with s denoting the usual orientational order parameter defined by (2.3.1). The term of order is not precluded by symmetry, for the states s and —s represent two entirely different kinds of molecular arrangement which are not symmetry related and do not have equal free energies. In the former case, the molecules are more nearly parallel to the unique axis, while in the latter they are more nearly perpendicular to it. However, in the nematic phase s is usually quite large (greater than about 0.4) so that very many more terms have to be included in the expansion in order to draw any valid conclusions. Consequently, the... 

FIGURE 2.4 Same as Figure 2.3 but for the order parameter defined by Equation 2.22. The black line is the center of the polaron obtained from Figure 2.3. 

FIGURE 2.8 Time dependence of y (the optical order parameter defined in Equation 2.21 of a polaron moving in a two-chain system with different electric field strengths Eo). 

In contrast to conventional liquid crystals, A m changes depending on the temperature and composition, and should be decided by the equilibrium condition. The symbol jj expresses the nematic order parameter defined by... 

Figure 7. Plot of P2) as a function of time defined for a system of 200 chains of semi-rigid units of 10 beads with additional orienting potential the ftill line represents the order parameter calculated with respect to a fixed axis, while the dashed line represents the order parameter defined with respect to an averaged sixis.

In the original solution of this model by SK the low-temperature phase is characterized by a single order parameter, defined by... 

Structural properties of planar atomic layer are usually represented by parameters which are sensitive to the deformation of the ideal triangular plane structure. The calculation of such parameter in the case of cylindrical surface requires xmfolding of cylindrical layers into a plane geometry [6,7]. Then, one can apply the order parameter defined by Mermin [15] ... 

With reference to Figure 2.1, 0 is the angle made by the molecular axis with the director axis. The average < > is taken over the whole ensemble this kind of order is usually termed long-range order. It is called microscopic because it describes the average response of a molecule. The scalar order parameter defined previously is sufficient to describe liquid crystalline systems composed of molecules that possess cylindrical or rotational symmetry around the long axis L... 

The order parameters defined previously in terms of the directional averages can be translated into expressions in terms of the anisotropies in the physical parameters such as magnetic, electric, and optical susceptibilities. For example, in terms of the optical dielectric anisotropies Ae = one can define a so-called macroscopic order parameter which characterizes the bulk response... 

---