Symmetry uniaxial

It was therefore with some confidence in the infra-red and Raman spectroscopic methods that a much more complex investigation was carried out on the molecular orientation in one-way drawn PET films which show uniplanar axial orientation 5). In such films the condition of fibre symmetry is removed in two ways (1). There is no longer uniaxial symmetry of the distribution of chain axes. 

Although the common nematic and smectic A phases have uniaxial symmetry, the molecules that form them are not cylindrically symmetric, although they... 

This approximation is strictly valid only for systems of uniaxial symmetry, i.e, having two equivalent magnetic easy directions. For systems with more equivalent directions (magnetite for example... 

The parameters K1/ K2/ and K3 are defined by the refractive indices of the crystal and sample and by the incidence angle [32]. If the sample has uniaxial symmetry, only two polarized spectra are necessary to characterize the orientation. If the optical axis is along the plane of the sample, such as for stretched polymer films, only the two s-polarized spectra are needed to determine kz and kx. These are then used to calculate a dichroic ratio or a P2) value with Equation (25) (replacing absorbance with absorption index). In contrast, a uniaxial sample with its optical axis perpendicular to the crystal surface requires the acquisition of spectra with both p- and s-polarizations, but the Z- and X-axes are now equivalent. This approach was used, through dichroic ratio measurements, to monitor the orientation of polymer chains at various depths during the drying of latex [33]. This type of symmetry is often encountered in non-polymeric samples, for instance, in ultrathin films of lipids or self-assembled monolayers. 

Silk fibers, a basic system with a uniaxial symmetry, have also been investigated by Raman spectromicroscopy [63] that is one of the rare techniques capable of providing molecular data on such small (3-10 pm diameter) single filaments. The amide I band of the silk proteins has been particularly studied to determine the molecular orientation using the cylindrical Raman tensor approximation. In this work, it was assumed that Co Ci, C2 and the a parameter was determined from an isotropic sample using the following expression of the depolarization ratio... 

As long as there is at least uniaxial symmetry and the fiber axis is in the detector plane, the scattering pattern can be split into four quadrants which should carry each identical information. This means that there is some harmony in the scattering pattern, from which missing data can be reconstructed13. 

Of highest practical relevance is the case in which the scattering pattern, the structural entities, and even the orientation distribution g(uniaxially oriented, F3 is most frequently fulfilled. In this case the mathematical relations are considerably simplified. Suitably the orientation distribution is normal-ized... 

Note 3 The tensorial properties of a biaxial mesophase have biaxial symmetry unlike the uniaxial symmetries of, for example, the nematic and smectic A mesophases. 

Sekimoto and Kawasaki showed that an elastic model with uniaxial symmetry can be linearly unstable if its lower surface is clamped and its upper surface is deformable [17,88]. The present author started with the free energy functional presented in Sect. 3 and found that it can be lowered in uniaxial gels by periodic folding of the surface [20,89]. Subsequent numerical simulations showed that the surface tends to touch and fold as corrugations grow [90]. 

There is some evidence that while mono adducts of the j8-diketonate shift reagents may show time-averaged uniaxial symmetry, bis adducts often do not. Thus the spectra of [M(dpm)3L2], where M = Pr, Sm, Dy, Ho, Er or Yb and L = 3-picolinc or 3,5-lutidine, have been interpreted in terms of biaxial magnetic anisotropy. One magnetic axis is defined by the molecular C2 axis present in the X-ray crystal structure and the other two axes lie in similar orientations in all the adducts. A variable-temperature study of this system shows restricted rotation of the aromatic rings with a very solvent-dependent conformational equilibrium resulting from this.558-560... 

The crystal field does, however, have a dramatic effect on the magnetic anisotropy of lanthanide complexes. For complexes of less than cubic symmetry the three principal values of the susceptibility tensor are unequal. For uniaxial symmetry, Xx — Xy Xz and for biaxial symmetry Xx Xy Xz- Very extensive studies632-640 have been carried out on the single crystal susceptibilities of the D3d lanthanide hexakis(antipyrene) triiodides over the temperature range 80-300 K, and crystal field parameters were obtained. This crystal field-induced anisotropy is responsible for the effectiveness of lanthanide complexes as NMR shift reagents, and single crystal anisotropies of lanthanide complexes have been determined in this connection also.563... 

In order to evaluate quantitatively the orientation of vibrational modes from the dichroic ratio in molecular films, we assume a uniaxial distribution of transition dipole moments in respect to the surface normal, (z-axis in Figure 1). This assumption is reasonable for a crystalline-like, regularly ordered monolayer assembly. An alternative, although more complex model is to assume uniaxial symmetry of transition dipole moments about the molecular axis, which itself is tilted (and uniaxially symmetric) with respect to the z-axis. As monolayers become more liquid-like, this may become a progressively more valid model (8,9). We define < > as the angle between the transition dipole moment M and the surface normal (note that 0° electric field of the evenescent wave (2,10), in the ATR experiment are given by equations 3-5 (8). 

Taking into account a uniaxial symmetry of the distribution function Wo, on averaging of Eq. (4.79), one gets a generic formula... 

Assuming uniaxial symmetry of the time-dependent solution and separating the variables in Eq. (4.125) in the form... 

Formula (4.370) leads to a simple expression for the macroscopic optical anisotropy. If the Oz axis of the coordinate framework coincides with the applied held direction h, then the uniaxial symmetry condition yields (n2) = (n2) and vx = vy. Hence for the refractive index anisotropy (birefringence) we have... 

Nematic phases are characterised by a uniaxial symmetry of the molecular orientation distribution function f(6), describing the probability density of finding a rod with its orientation between 6 and 6 + d0 around a preferred direction, called the director n (see Fig. 15.49). An important characteristic of the nematic phase is the order parameter (P2), also called the Hermans orientation function (see also the discussion of oriented fibres in Sect. 13.6) ... 

Figure 1.11 (a) Single crystal type (selected orientations of 0 = 0°, 55° and 90°) and (b) first derivative powder type EPR lineshapes for a randomly oriented S = 1 /2, / = 1/2 spin system with uniaxial symmetry. The angular dependence curve (0 vs field) for m, = 1/2 is shown in (c). 

The term director is usually only defined in the limit of vanishing shear rate where the order parameter tensor has uniaxial symmetry. Even at finite shear rates, however, the order parameter tensor S remains well-defined it can be represented pictorially by an ellipsoidal shape, whose major axis is the largest eigenvalue of S. If, for example, the major axis of S lies in the x y plane, then its angle 9 measured clockwise with respect to the x direction is given by... 

Mssbauer spectroscopy The inclusion of anisotropic fluctuations, modeled as T+ T in uniaxial symmetry, in the presence of applied magnetic fields, exchange anisotropy, or inter-particle interactions, must be used as a starting assumption unless the more restricted assumption that all relevant fluctuations are much faster than the measurement frequency (x+, t Xm, in uniaxial symmetry) is justified independently. All other relevant realistic features (distributions of characteristics and properties ) must also be included, by applying as many justified theoretical constraints as possible. 

In systems with uniaxial symmetry the field may be applied along any one of the two possible directions, namely, along the c-axis and along a direction lying in the basal plane. Under these conditions the above equation may be expressed as... 

This can be also illustrated as follows a fatty man can be surrounded by 4 - 5 closely-packed fat men. But the same man can be surrounded even by 10 slim men. The latter can parallel the increasing dimensionality of space for the fat man. Consequently, the elongation of fluctuations due to the action of the strong electric field in the homogeneous phase of binary mixtures of limited miscibility is equivalent to increased dimensionality d = A. The uniaxial symmetry is natural also for the isotropic phase nematic liquid crystals. It may be considered for supercooled nitrobenzene due to intermolecular interactions (Fig. 5). ... 

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