Spheroidal cavities

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. 

One of the aspects that has been of interest is the incorporation of an external atom in the spheroidal cavity. A variety of metal atoms can, in principle, be trapped in this cavity. Some of the studies have claimed that it is possible to push atoms such as lanthanum, iron and helium inside the spheroidal cavity of CgQ and other fullerenes. Substitution of the carbon in CgQ by boron and nitrogen has been attempted. Interestingly, nitrogen not only substitutes for carbon in the cage but also adds on to Cgo and C-iq. 

Whereas macrocycles define a two-dimensional, circular hole, macrobicycles define a three-dimensional, spheroidal cavity, particularly well suited for binding the spherical alkali cations (AC) and alkaline-earth cations (AEC). 

Thus, it is the polyazamacrocyclic ligands developed by Lehn which have proved to be the most successful and versatile cryptands, rather than those carbon-bridgehead cryptands sought initially by Stoddart (25-27) which define a spheroidal cavity (e.g., 6). On the other hand, the bridged macrocyclic polyethers developed by Parsons (28), such as 7, show high complexing ability with alkali metal cations (28-30)... 

Many catalysts are porous, a feature which greatly increases their surface area [48]. Pores above 50 nm in width are termed macropores, those with widths below 2 nm are called micropores, and those of intermediate size are mesopores. Not only the size but also the shape of pores can vary widely and common descriptions refer to open and closed cylinders, slits, cones, spheroidal cavities, and ink-bottle shapes. The type of pore can frequently be identified from the shape of the hysteresis loop in physical gas adsorption experiments [32, 49], The absence of hysteresis indicates that the pores are closed perfect cylinders, or that the solid is microporous or, indeed, non-... 

Probably the most precious occurrence of anion recognition is that related to size exclusion selectivity. This takes place when the receptor, providing for instance a spheroidal cavity, includes only spherical anions of radius less than or equal to a definite value. In this context, the smallest anion, fluoride, has offered vast opportunities. 

Let us consider a many-electron atom of nuclear charge Z confined by a hard prolate spheroidal cavity. In this study the nuclear position will correspond to one of the foci as shown in Figure 4. In terms of prolate spheroidal coordinates, the nuclear position then corresponds to one of the foci for a family of confocal orthogonal prolate spheroids and hyperboloids defined, respectively, by the variables f and rj as [73] ... 

M. M. Yovanovich, General Conduction Resistance for Spheroids, Cavities, Disks, Spheroidal and Cylindrical Shells, AIAA 77-742, AIAA 12th Thermophysics Conference, Albuquerque, New Mexico, June 27-29,1977. 

Ramshaw has pointed out that in the y- 0 limit, one can regard the core parameter 0 as a geometric parameter describing the degree of ellipticity of a spheroidal cavity used in defining a local field in a phenomenological mean field treatment. Such a treatment, described by Ramshaw, is the analog of the Debye-Hiickel treatment of a simple ionic fluid. 

T, the host and guest molecules occupy Wyckoff positions 4(c) of point symmetry 2 and 2(a) of point symmetry 4, respectively. Eight host molecules cluster around a spheroidal cavity centered at ( /4, t), which accommodates the guestmolecule. The... 

Clathrates of tetraphenylene 6 form a single isostruc-tural series (P42/n) with H G ratio 2 1, in which guests range in size from dichloromethane to cyclohexane. Guests occupy common spheroidal cavities generated by the isostructural host framework. Elaboration of the host 6 to the more buUqf tetrabenzo[ ,/z, ,t]tetraphenylene 4 produce, no fewer than three distinct isostructural series of clathrates (two triclinic, one monoclinic). 

For a spherical cavity the problem was solved for the first time by Kirkwood, in 1934 [38] and next generalised by several authors to multipoles of different order and to polarisable solutes [e.g. 36, 37, 40, 46, 68]. Solutions for more or less general multipolar developments in spheroidal cavities have been reported by several authors [45, 47, 48, 79, 95] and this is also the case for general ellipsoidal cavities starting from the work of Westheimer and Kirkwood, in 1938 [41] and proceeding in several steps [96-99] up to the general algorithm for arbitrary... 

Appendix B Spheroidal Cavity Embedded By One or More Concentric Dielectric ShelE Surrounded By the Bulk... 

B.l Spheroidal Cavity Embedded by One Concentric Dielectric Shell Surrounded by the Bulk... 

We can investigate first the charge distribution inside a prolate spheroidal cavity, defined by the condition X = Xc = add = constant and dielectric constant c, with a first shell of polarisable dielectric, extending from X = Xc to X = X = a /d = constant > Xc, characterised by dielectric constant Beyond this shell, there is the bulk region, from X = Xi to infinity, with dielectric constant b- The solution of Laplace s Equation (3.10) is the same of the single spheroidal cavity case, but it is needed to introduce coefficients so that the expressions for the potential in the three regions (viz. cavity, shell, and bulk) are ... 

The expressions for the oblate spheroidal cavity can be obtained with the substitutions (4.47). 

This work have been carried out by the method of numerical adsorption experiments on model porous netwoilc, a mesoporous fragment of wlch is shown in Pig. 1. This model porous space consists of spheroidal cavities ( voids ) and cylindrical necks between these voids. 

Spheroidal cavities were considered to be filled provided that... 

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