Uniaxial simplest form

Easy axes of alignment of the magnetization, or an anisotropy, often exist within a single domain ferromagnetic particle. The simplest form of anisotropy is uniaxial anisotropy in which the magnetization favors either of two opposite directions. This may be represented in the framework of our analysis by means of an anisotropy potential of the form... 

In this section we consider the dependence of the relaxation time on the internal anisotropy of a single domain ferromagnetic particle. Thus it is applicable only to Neel relaxation. We consider the simplest form of anisotropy, namely uniaxial anisotropy described by the potential... 

Weibull developed his statistical theory of brittle fracture on the basis of the weakest link hypothesis, i.e. the specimen fails if its weakest element fails [6, 7], In its simplest form and for an uniaxial homogenous and tensile stress state, ct, and for specimens of the volume, F, the so called Weibull distribution of the probability of failure, F, is given by ... 

If there is no preferred orientation around Oxy, as is the case in the simplest form of uniaxial orientation, U31) = ( 2)- Equation (A.5) then shows that... 

A particular sample of a semicrystalline pol5mer with the simplest form of uniaxial orientation is found to have a birefringence of 0.042. Density measurements show that it has a volume crystallinity of 0.45 and X-ray measurements show that P2icosff)) for the crystalline phase is 0.91. Assuming that An ax is 0.050 for the crystalline phase and 0.045 for the amorphous phase, estimate (P2(cos 0)) for the amorphous phase. 

Simplifying assumptions a highly oriented polymer has the simplest form of uniaxial distribution and is fully oriented, there is no correction for reflection at the surface of the sample, no internal field corrections and no scattering of radiation. 

The simplest form of densification, uniaxial pressing, has been used to densify a variety of materials such as TiC -I- NiAl, TiC -i- Ni3Al [41], TiC + Ni, TiC -1- Fe [42], and others [43,44]. Two parameters play an important role in this type of postreaction densification the magnitude of the applied pressure and the time of its application relative to the completion of the combustion process. Figure 30 shows the dependence of the residual porosity in TiC-Ni/Al cermets on the applied pressure. As the figure shows, the relative density increases from about 50 to 99% as the pressure is increased from ambient (0.1 MPa) to about 28 MPa [41]. The timing of the application of the pressure relative to the synthesis reaction has also... 

Elastic Modulae. The mechanical behavior is in general terms concerned with the deformation that occurs under loading. Generalized equations that relate stress to strain are called constitutive relations. The simplest form of such a relation is Hooke s Law which relates the stress a to the strain e for uniaxial deformation of the ideal elastic isotropic solid ... 

Let us consider the simplest case of a two-sublattice antiferromagnet with uniaxial anisotropy. The relevant theoretical description is usually based on consideration of the thermodynamic potential [30], or the energy density, in the following form ... 

The sample, usually in the form of a cylinder, can be subjected to uniaxial compression (the simplest and most common test), uniaxial tension, shear, bending or torsion. In compression, the sample rests on the base-plate and is compressed by a horizontal flat plate attached to the crosshead when... 

In the simplest liquid-crystalline phase, namely the uniaxial nematic, there is at rest a special direction designated by a unit vector n called the director (see Fig. 10-2). In the plane transverse to the director, the fluid is isotropic. The most common nematics are composed of oblong molecules that tend to point in a common direction, which defines the director orientation. Oblate, or disc-like, molecules can also form uniaxial nematics for these discotic nematics, the director is defined by the average orientation of the short axis of the molecule. Lath-like molecules or micelles (shaped like rectangular slabs), in which all three dimensions of the molecule are significantly different from each other, can form biaxial nematics (Praefcke et al. 1991 Chandrasekhar 1992 Fialtkowski 1997). A biaxial... 

The discussion of the most general form of biaxial orientation involves fairly complicated mathematics. Further discussion in this section and in the sections on the characterisation of orientation is therefore largely restricted to a special simple type of distribution of orientations of the structural units. This distribution is the simplest type of uniaxial orientation, for which the following conditions apply. 

The tensor form of the order parameter should be taken into account, in the simplest case, the uniaxial one [3,4] ... 

Consider the simplest case, namely, the nematic phase consisting of uniaxial rodlike molecules. Generally, the intermolecular interaction again consists of the repulsive and attractive parts but both of them become anisotropic. The potential of pair molecular interaction can be written in the following general form ... 

Realistic intermolecular interaction potentials for mesogenic molecules can be very complex and are generally unknown. At the same time molecular theories are often based on simple model potentials. This is justified when the theory is used to describe some general properties of liquid crystal phases that are not sensitive to the details on the interaction. Model potentials are constructed in order to represent only the qualitative mathematical form of the actual interaction energy in the simplest possible way. It is interesting to note that most of the popular model potentials correspond to the first terms in various expansion series. For example, the well known Maier-Saupe potential JP2 (Sfli )) is just the first nonpolar term in the Legendre polynomial expansion of an arbitrary interaction potential between two uniaxial molecules, averaged over the intermolecular vector r,-, ... 

Consider first a uniaxial phase that is composed of chiral rod-like molecules. In the simplest situation, a helix can form in a direction perpendicular to the long axis of an object molecule. This example is analogous to the structure of the chiral nematic phase. In the direction parallel to the long axes of the molecules no twist can be effected. Now consider a similar situation, but this time the twist in the orientational order can occur in more than one direction in the plane perpen-... 

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