Oblate shape

Four volumetric defects are also included a spherical cavity, a sphere of a different material, a spheroidal cavity and a cylinderical cavity (a side-drilled hole). Except for the spheroid, the scattering problems are solved exactly by separation-of-variables. The spheroid (a cigar- or oblate-shaped defect) is solved by the null field approach and this limits the radio between the two axes to be smaller than five. 

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. 

For all three types of dendrimers described above, a flattened, disk-like conformation was observed for the higher generations. However, the molecular shape at the air-water interface is also intimately associated with the polarity, and hence the type of dendrimer used. In case of the polypropylene imine) and PAMAM dendrimers the hydrophilic cores interact with the sub-phase and hence these dendrimers assume an oblate shape for all generations. The poly(benzyl ether) dendrimers, on the other hand, are hydrophobic and want to minimize contact with the water surface. This property results in a conformational shape change from ellipsoidal, for the lower generations, to oblate for the higher generations [46]. 

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. 

The deformation can be very complicated to describe in a single-particle framework, but a good understanding of the basic behavior can be obtained with an overall parameterization of the shape of the whole nucleus in terms of quadmpole distortions with cylindrical symmetries. If we start from a (solid) spherical nucleus, then there are two cylindrically symmetric quadmpole deformations to consider. The deformations are indicated schematically in Figure 6.10 and give the nuclei ellipsoidal shapes (an ellipsoid is a three-dimensional object formed by the rotation of an ellipse around one of its two major axes). The prolate deformation in which one axis is longer relative to the other two produces a shape that is similar to that of a U.S. football but more rounded on the ends. The oblate shape with one axis shorter than the other two becomes a pancake shape in the limit of very large deformations. 

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.

The nuclei show consistently large deformations of B2 = 0.35-0.45. It thus can be inferred that the predicted prolate to oblate shape transition in this mass region BEN84, NAZ85 occurs by a change of y rather than by large variations of 62. 

It is important to note that the form effect is proportional to the square of the dielectric contrast, Ae, and will always be positive for prolate particles (Ll > L2), and negative for oblate shapes (L2 > Lx). The intrinsic contribution can change sign depending on the relative magnitudes of the principal values of the polarizability tensor of the particle. 

Further studies by electron microscopy on some of the samples exhibiting the Pm3n cubic phase show the existence of grain bormdaries and stacking faults [118]. These are all consistent with the presence of quasi-spherical assemblies or more precisely to polyhedral-like micelles, and moreover suggest that the supramolecular spheres are deformable, interacting with one another through a relatively soft pair potential [119]. The majority of such quasi-spherical assemblies are thus distorted into an oblate shape. 

Figure 6.13 Prolate and oblate shapes of various particles. (Adapted from Macosko, Rheology Prinripyx, MfinKtirfmfintx, nnd Appliratinns, Copyright 1994. Reprinted by permission from John Wiley Sons.)...

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.)...

Another possibility to measure nearly isolated fulleride ions is the investigation of solid fulleride salts with bulky cations. The large cations separate the fulleride ions but in many cases they are capable of lowering their symmetry at the same time. In Ni(CsMe5)2C6o the Cgg ion was found by X-ray diffraction to be oblate shaped and to have roughly D2h symmetry [38]. This results probably from the enhanced n — n interaction between the CsMes and the fullerene units. 

Citrus peel is a natural source of limonene. However, it is also present in very small quantities in other essential plant oils. Citrus peel contains essential oils in oblate-shaped oil glands in the flavedo (0.5-1 ml essential oil/100 cm2 flavedo). 

Peterlin, P., Svetina, S., and Zeks, B. (2007) The prolate-to-oblate shape transition of phospholipid vesicles in response to frequency variation of an AC electric held can be explained by the dielectric anisotropy of a phospholipid bilayer. Journal of Physics Condensed Matter, 19 (13), 136220. 

In the Ni phase, Sa > 0, but SB < 0. The conformation of the backbone chain is discus-like, i.e., is oblate shape, in which the mean square end-to-end distance along the director, (R%) is less than the perpendicular component (R%). In the smectic phase the anisotropy of two components is greater than that in the nematic phase. At the extreme case the backbone is surpassed in a plane, which was predicted by Renz Warner (1986) to exist in smectic polymers where backbones become confined between smectic layers formed by side chains, and has been seen by Moussa et al. (1987). In the extreme case, the backbone becomes a two-dimensional random walk when SB = — and Sa = 1. 

As a test of the revised theory, further experiments were conducted [Zhao et al., 2005] on nematic solutions of a SCLCP. ER measurements indicated, via application of the Brochard hydrodynamic model, a slightly prolate conformation, R /R = 1.17 0.02, consistent with small-angle neutron scattering measurements, which indicated, that= 1.12 0.06. Observations of the shear stress transient response of a homeotropic monodomain indicated that at a concentration between 0.01 and 0.02 g/mL, the solution exhibited a transition from director-aligning to director-tumbling behavior. The latter result is inconsistent with the original Brochard model [see Eq. (1.94)], which predicts such a transition (i.e., Sas > 0) only for a polymer with an oblate shape but agrees with the modified theory [Eq. (1.96)]. 

Deviations from spherical shape can be modeled as a growth into prolate or oblate shapes. The area enclosed volume ratio constraint implies a constraint in the possible values of the two semiaxes describing the particle size. Halle [62] calculated the correlation functions for the combined particle tumbling and surface diffusion of prolate and oblate particles. His results have, for example, been applied to microemulsion systems [48,58], focusing on the ratio yVpr(0)// 5,sph(0). where the subscripts pr and sph refer to prolate and spherical particles, respectively. For a given ratio of interfacial area to enclosed volume, which specifies the radius R of the sphere, y, pr(0)//. sph(0) is a function of the prolate axial ratio, Diat, and R. Knowing Djat and R from other experiments, it is possible to determine the aggregate axial ratio from the relaxation experiment. 

Other aspects of the drop oscillation problem, such as oscillation of liquid drops immersed in another fluid [17-21], oscillations of pendant drops [22, 23], and oscillations of charged drops [24, 25], have also been considered. In particular, there are numerous works on the oscillation of acoustically levitated drops in acoustic field. In such studies, high-frequency acoustic pressnre is required to levitate the droplet and balance the buoyancy force for the experimental studies performed on the Earth. As a result of balance between buoyancy and acoustic forces, the equilibrium shape of the droplet changes from sphere to a slightly flattened oblate shape [26]. Then a modulating force with frequency close to resonant frequencies of different modes is applied to induce small to large amplitude oscillations. Figure 5.4 shows a silicon oil droplet levitated in water and driven to its first three resonant modes by an acoustic force and time evolution for each mode. 

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