Prolate shape

There are two Fermi seas for a given quark number with different volumes due to the exchange splitting in the energy spectrum. The appearance of the rotation symmetry breaking term, oc p U 4 in the energy spectrum (16) implies deformation of the Fermi sea so rotation symmetry is violated in the momentum space as well as the coordinate space, 0(3) —> 0(2). Accordingly the Fermi sea of majority quarks exhibits a prolate shape (F ), while that of minority quarks an oblate shape (F+) as seen Fig. 1 3. ... 

Figure 1. Modification of the Fermi sea as Ua is increased from left to right. The larger Fermi sea (F ) takes a prolate shape, while the smaller one (F+) an oblate shape for a given Ua- In the large Ua limit (completely polarized case), F+ disappears as in the right panel.

AV the Fermi sea has a prolate shape for the majority spin particles, while an oblate shape for the minority spin particles. 

Stokes-Einstein Relationship. As was pointed out in the last section, diffusion coefficients may be related to the effective radius of a spherical particle through the translational frictional coefficient in the Stokes-Einstein equation. If the molecular density is also known, then a simple calculation will yield the molecular weight. Thus this method is in effect limited to hard body systems. This method has been extended for example by the work of Perrin (63) and Herzog, Illig, and Kudar (64) to include ellipsoids of revolution of semiaxes a, b, b, for prolate shapes and a, a, b for oblate shapes, where the frictional coefficient is expressed as a ratio with the frictional coefficient observed for a sphere of the same volume. 

Figure 6.16 Schematic diagram of the splitting of the f7spherical shell model level as the potential deforms. Positive deformations correspond to prolate shapes while negative deformations correspond to oblate shapes.

Cloud shapes. While precollapse clouds often have a complicated appearance, attempts have been made to approximate their shapes with simple geometries. Triaxial spheroids seem to be required in general, though most lower-mass clouds appear to be more nearly oblate than prolate (Jones et al., 2001). On the larger scale, prolate shapes seem to give a better ht than oblate spheroids. 

Figure 6.14 Intrinsic viscosity versus Peclet numberfor dilute suspensions of spheroidal particles of (a) oblate shape and (b) prolate shape, (From Macosko 1994, adapted from Brenner 1974, with permission from Pergamon Press.)...

A prolate shape also appears to explain better the hydrodynamic properties of iron-free (142) and of iron-saturated (143) transferrin. Ferric transferrin (a/b = 3) would, however, be more elongated than the iron-free form (a/b =2) while the effective hydrodynamic volume (Ve) would be higher for the iron complex than for the apoprotein. These results not only differ from those given in Table 3 for conalbumin but are also in partial disagreement with dielectric dispersion and viscosity measurements (144) which have indicated that human transferrin assumes a more spherical shape with iron-saturation, the axial ratio decreasing from 2.5 (apo) to 2.0 (ferric). This latter investigation also indicates a slight expansion (15.4 16.9) of the hydrated volume... 

In the latter two phases backbones have the spindle-like conformation, i.e., the prolate shape with (R%) > R p), the characteristic of main chain liquid crystalline polymers. Important means of investigating the conformations of side chain liquid crystalline polymers include small angle neutron scattering from deuterium-labeled chains (Kirst Ohm, 1985), or small angle X-ray scattering on side chain liquid crystalline polymers in a small molecular mass liquid crystal solvent (Mattossi et al., 1986), deuterium nuclear resonance (Boeffel et al., 1986), the stress- or electro-optical measurements on crosslinked side chain liquid crystalline polymers (Mitchell et al., 1992), etc. Actually, the nematic (or smectic modifications) phases of the side chain liquid crystalline polymers have been substantially observed by experiments. 

For simplicity, we restrict our consideration to nematic polymers of prolate shape. This includes (i) main chain nematic polymers and (ii) side chain nematic polymers, but in such a way that they preferentially align with the backbone, thus giving the features of (i). This has been called the Nni phase. The other prolate possibility in the Nn phase is that the backbone takes a prolate shape while the side chains are forced to have oblate symmetry. We shall only discuss the nematic polymers of (i), but the method and the conclusions are quite general. 

In the following we will limit the discussion to the case where only one quadrupole nucleus is present in the molecule. Ethylenimine will be used as an example in all numerical calculations. As mentioned above, the nucleus has a small positive prolate shaped quadrupole moment. If the potential well at the equilibrium position of the nucleus lacks spherical symmetry, the nucleus will tend to align itself with respect to the molecular frame as depicted in Fig. III. 17. 

Both the liquid-drop model and the single-particle model assume that the mass and charge of the nucleus are spherically symmetric. This is true only for nuclei close to the magic numbers other nuclei have distorted shapes. The most common assumption about the distortion of the nuclide shape is that it is ellipsoidal, i.e. a cross-section of the nucleus is an ellipse. Figure 11.6 shows the oblate (flying-saucer-like) and prolate (egg-shaped) ellipsoidally distorted nuclei the prolate shape is the more common. Deviation from the spherical shape is given by... 

Thus 6obs 0 for / 1/2. of,s usually givra in area (m ). Most commonly 10 used as unit and referred to as one bam, is > 0 for the more common prolate shape, and < 0 for oblate. Some measured values are given in Table 11.3. 

In a liquid, as a result of inter-molecular interactions, molecules are continuously rotating and translating. Thus, if we could tag a molecule and study its detailed motion, we would find it executing a random Brownian motion not only in the three-dimensional positional spaee, but also a similar motion in the three-dimensional orientational space. For simplicity, let us first consider the motion of a tagged prolate-shaped moleeule in a solvent of spherical molecules, as shown in Figure 3.A.1 below. 

Figure 3.A.I. Motion of a prolate-shaped molecule in a solvent of spherical molecules.

When a droplet undergoes BCD actuation between two electrified planar electrodes, the droplet deformed into a prolate shape not by the translational motion but by an electric stress on the droplet surface under a strrMig enough electric field [8]. Based on this fact, the interfacial tension in fluid-fluid systems can be measured easily. The deformation (D), defined as the difference between the length of the major axis (a) and the minor axis (b) of the deformed droplet divided by... 

In a 2005 review article, Cwiok, Heenan, and Nazarewicz (2005) present new theoretical results for properties of even-even heavy and SHE element nuclei with 94 < Z < 28 and with 134 < N < 188. They use self-consistent formalism and a modern nuclear energy density functional to formulate the following major conclusions concerning SHEs (1) SHE nuclei around Z= 116 and N= 176 are expected to exhibit coexistence of oblate and prolate shapes,... 

Atomic nuclei can be stretched like cigars (prolate shape) or compressed like discs (oblate shape). The deformation is described by the electric quadrupole moment Q (prolate Q>0 oblate Q<0). The principal interaction is, of course, the normal electrostatic (Coulomb) force on the charged nucleus (monopole interaction). The differential interaction, which depends... 

Atomic nuclei can be stretched like cigars (prolate shape) or compressed like discs (oblate shape). The deformation is described by the electric quadrii-pole moment Q (prolate Q > 0 oblate Q < 0). The principal interaction is, of course, the normal electrostatic (Coulomb) force on the charged nucleus monopole interaction). The differential interaction, which depends on the structure of the nucleus and on the valuation of the field across its finite extension, is of course very much smaller quadrupole interaction). It gives rise to an electric hyperfine structure. The energy contribution depends on the direction of the nuclear spin in relation to the electric field gradient. For the electric hyperfine interaction one obtains... 

A positive value of the C aj u d2 u) component of the quadrupole corresponds to a distribution of the 4f electrons elongated along the z axis (prolate shape). On the contrary a negative value of C aj u jO I ) corresponds to an pancake-like distribution (oblate shape). 

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