Rotational ellipsoid

With a decrease of sg the ellipsoid approaches a disk on the contrary, when 8q oo, it tends to a sphere. In order to find the gravitational field of the rotating ellipsoid, it is convenient to represent Equation (2.149) in a slightly different form. For this purpose introduce notations rj — sin and, therefore. 

The clear linearity of the data demonstrates the affine character of the deformation. From the intercept, b2, the semiminor axis of the rotation ellipsoid is determined. After transformation to units of reciprocal-space, the meridional long period follows from Eq. (10.3). 

As for the shapes of the Fermi surfaces for the a, Ji, y and 8 branches, it is useful to check the 1 IF2 vs. cos2 0 relation. Almost linear dependencies of the 1/F2 vs. cos2 0 curves (Figure 6.7) indicate that the Fermi surface shapes are fairly ellipsoidal in the a, Ji, y and 8 branches and that the y and 8 branches are better fitted to the ellipsoids than the a and Ji branches. Taking consideration of this behavior and symmetric angular variations of the frequencies in the (1010) and (1120) planes, it can be assumed that these four Fermi surfaces are rotational ellipsoids about the [0001] axis. Based on the above ideas, the hole Fermi surfaces corresponding to the a, Ji, y and 8 branches can be constructed. 

An important detail of the theory and its experimental verification is the fact that our setup uses isotropic detection (no polarizers) together with polarized excitation. With 2n or 47t detection a correct description of orientational correlation with three eigen- values of the diffusion tensor is obtained. They are sums of the rotational diffusion constants D, Dy, around the individual axes of the rotational ellipsoid [5,7] 2Dz + 4 Dx,y, 5Dz + D y. 

Fig. 1 Accessories for diffuse reflectance spectroscopy (A) Integrating sphere with hemispherical radiation collection (B) Accessory based on ellipsoidal mirrors, used within the sample compartment of the spectrometer (C) Rotational ellipsoidal mirror device with dedicated detector and (D) Bifurcated fiber optic-based accessory (also shown is the random mixture of fibers for illumination and detection compared with devices based on reflection optics the acceptance cone for radiation delivery and collection is limited and depends on the refractive indices of the core and cladding material).

The stepwidth vector s is defined prior to the optimization and kept constant during the optimization run. Due to the normalization of d (step 2), all generated configurations are placed on a rotation ellipsoid of dimensionality corresponding to s. This makes it difficult for the algorithm to locate the exact extreme of an optimization function. 

The effective diameter of a rapidly rotating ellipsoidal molecule can be taken as the length of the major axis, about 15 A for the androstane molecule. The surface density, 1/2, corresponding to tPetf/S = 1, is 4 X 1013 molecules/cm2. 

These and others experimental facts have been theoretically analyzed with the use of the methods of the self-consistent field and scaling [7i-17], The results of an analysis can be lined in the following simplified model in the flocculent adsorption layer (a distance between the centers of the adsorptive molecules I > 2Rj, the polymeric chain is in practically the same conformational state as in the solution in dripless adsorptive layer (/ < 27 an interaction between the adsorbed chains compresses the pol5mieric bdls in the adsorption plate and stretches them in a form of the chain by blobs [76], cylinders [72] or rotation ellipsoids [75] along the normal to the surface. 

In the real solution the Flory ball is deformed into the rotation ellipsoid, compressed or elongated along the axe connecting the beginning and the end of a chain. The semi-axes X. of the conformational ellipsoid can be expressed via the and vie the multiplicities / of the linear deformation ... 

Thus, if the particle has no permanent dipole moment, the rise and decay signals of the birefringence are symmetrical [cf Eqs. (13) and (14)]. The rotational diffusion coefficient 9 is usually obtained directly from the decay curve, Eq. (14) and Eq. (11), together with the apparent hydrodynamic radius Vf, determined in light scattering experiments, can be used to obtain information about the size and eccentricity alb of the rotating ellipsoid. 

Table 10.1 Proportional Coefficient for Calculation of the Magnetic Moment of Rotation Ellipsoids with Different Ratio of Semiaxes Values...

The molecules either have a generic infinite rotation axis (cones, rods, rotational ellipsoids, spherocylinders or discs) or acquire this average uniaxial form due to free rotation around the longitudinal molecular axis Then /(fl) becomes 4 -independent [13] / = /(i )/4ti with/(i ) =f(n — d), see Fig. 3.15. This figure shows that angle 0 and ti are equally and the most populated by molecules and these two angles correspond to the condition n = —n. The angles close to ti/2 are the less populated. Now our task is to find the form offi d) and relate it to experimentally measured parameters. 

Globular proteins soluble in water or dilute salt solutions rotational ellipsoids with frictional coefficients f/f=l.l-1.5. Representatives are Albumins (see), Globulins (see), Protamines (see). Histones (see). Prolamines (see) and Glutelins (see). 

Thus, the propagation of the electromagnetic waves into a crystal will have three main directions of refraction or the main optical axes (for each of them being characteristic one refraction index as n, n, n ), in a quadratic relation with the Cartesian coordinates associated to the crystalline system (x, y, z), through the form of a rotational ellipsoid which generates the so-called surface of the refraction index (indicatrix), with the equation ... 

Figure 12.9. Modeling of intermediate chiral surfaces (a) flattened rotational ellipsoids are not equidistant, so they are substituted by (b) equidistant disk-like surfaces, referred to as the oblate chiral surfaces. The length <5 is a measure of how much they are flattened, i.e., how much they deviate from spherical shape. In (c) their three-dimensional layout is presented (from [14]).

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