Order parameter dimensionality

Key Words Dipolar glasses, Ferroelectric relaxors, Conducting polymers, NMR line shape, Disorder, Local polarization related to the line shape, Symmetric/asymmetric quadrupole-perturbed NMR, H-bonded systems, Spin-lattice relaxation, Edwards-Anderson order parameter, Dimensionality of conduction, Proton, Deuteron tunnelling. 

The existence of large-distance fluctuations of the order parameter in the critical region leads also to the universality hypothesis, according to which the properties of thermodynamic functions of different physical systems are given by the space dimensionality d and the order parameter dimensionality n. The systems with equal d and n form a class of universality with the same critical indices. 

The critical exponents a, j8, y, 5, are expressed in the perturbation expansion in terms of the parameter e=d -d, and the expansion coefficients depend upon an order parameter dimensionality. In the diluted systems d = 6, in tricritical points and in three-dimensional dipolar-coupled Ising ferromagnets d = 3. Among lanthanide compounds uniaxial and... 

McMillan s model [71] for transitions to and from tlie SmA phase (section C2.2.3.2) has been extended to columnar liquid crystal phases fonned by discotic molecules [36, 103]. An order parameter tliat couples translational order to orientational order is again added into a modified Maier-Saupe tlieory, tliat provides tlie orientational order parameter. The coupling order parameter allows for tlie two-dimensional symmetry of tlie columnar phase. This tlieory is able to account for stable isotropic, discotic nematic and hexagonal columnar phases. 

Figure 2 Two-dimensional X-Y order-parameter[upper panel] and spinpower panel] maps for temperatures 1000 K, 1400 K and 1550 K pattice sizes 24x60x60 in unit of fee cubes].

Figure 5 Two-dimensional X-Y order-parameter and spin maps for temperature 1687 K.

For smectic phases the defining characteristic is their layer structure with its one dimensional translational order parallel to the layer normal. At the single molecule level this order is completely defined by the singlet translational distribution function, p(z), which gives the probability of finding a molecule with its centre of mass at a distance, z, from the centre of one of the layers irrespective of its orientation [19]. Just as we have seen for the orientational order it is more convenient to characterise the translational order in terms of translational order parameters t which are the averages of the Chebychev polynomials, T (cos 2nzld)-, for example... 

The critical radius at Tg is a multiple of Droplets of size N > N are thermodynamically unstable and will break up into smaller droplets, in contrast to that prescribed by F N), if used naively beyond size N. This is because N = 0 and N = N represent thermodynamically equivalent states of the liquid in which every packing typical of the temperature T is accessible to the liquid on the experimental time scale, as already mentioned. In view of this symmetry between points N = 0 and N, it may seem somewhat odd that the F N) profile is not symmetric about. Droplet size N, as a one-dimensional order parameter, is not a complete description. The profile F N) is a projection onto a single coordinate of a transition that must be described by order parameters—the... 

Figure 36 is a three dimensional representation of the order parameter P at 350 K after 19.2 ns of simulation, where about 25% of the system has transformed into the crystalline state. The black regions near both side surfaces correspond to the crystalline domains with higher P values, while the white regions are in a completely isotropic state of P = 0. Detailed inspection of these data has shown that no appreciable order is present in the melt. A simple interface model between the crystal and the isotropic melt seems to be more plausible at least in this case of short chain Cioo-... 

A switch to double-angle vectors given by Eq. (2.3.12) not only significantly simplifies the treatment of orientation phase transitions in planar systems of nonpolar molecules but also leads to a number of substantial inferences on the transition nature. First of all note that the long-range-order parameter t] (vanishing in a disordered phase and equal to unity at T = 0) in a -dimensional space (specified by the orientations of long molecular axes) can be defined as ... 

We review Monte Carlo calculations of phase transitions and ordering behavior in lattice gas models of adsorbed layers on surfaces. The technical aspects of Monte Carlo methods are briefly summarized and results for a wide variety of models are described. Included are calculations of internal energies and order parameters for these models as a function of temperature and coverage along with adsorption isotherms and dynamic quantities such as self-diffusion constants. We also show results which are applicable to the interpretation of experimental data on physical systems such as H on Pd(lOO) and H on Fe(110). Other studies which are presented address fundamental theoretical questions about the nature of phase transitions in a two-dimensional geometry such as the existence of Kosterlitz-Thouless transitions or the nature of dynamic critical exponents. Lastly, we briefly mention multilayer adsorption and wetting phenomena and touch on the kinetics of domain growth at surfaces. 

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