Spherical clusters

FIGURE 6.1 Different shapes of two-dimensional clusters spherical, ellipsoidal, linear, crescent, ring, and spiral. 

Pt substrates. All the results reported here for the Pt/Sn system were obtained using a multichannel hemispherical electron analyzer and a conventional, non monochromatized. Mg Ka or A1 Ka photon source. Unless otherwise specified, the experimental data were analyzed by means of the single scattering cluster -spherical wave (SSC-SW) model. 

Figure B3.3.10. Contour plots of the free energy landscape associated with crystal niicleation for spherical particles with short-range attractions. The axes represent the number of atoms identifiable as belonging to a high-density cluster, and as being in a crystalline environment, respectively, (a) State point significantly below the metastable critical temperature. The niicleation pathway involves simple growth of a crystalline nucleus, (b) State point at the metastable critical temperature. The niicleation pathway is significantly curved, and the initial nucleus is liqiiidlike rather than crystalline. Thanks are due to D Frenkel and P R ten Wolde for this figure. For fiirther details see [189].

Figure Cl. 1.2. (a) Mass spectmm of sodium clusters (Na ), N= 4-75. The inset corresponds to A = 75-100. Note tire more abundant clusters at A = 8, 20, 40, 58, and 92. (b) Calculated relative electronic stability, A(A + 1) - A(A0 versus N using tire spherical electron shell model. The closed shell orbitals are labelled, which correspond to tire more abundant clusters observed in tire mass spectmm. Knight W D, Clemenger K, de Heer W A, Saunders W A, Chou M Y and Cohen ML 1984 Phys. Rev. Lett. 52 2141, figure 1.

The spherical shell model can only account for tire major shell closings. For open shell clusters, ellipsoidal distortions occur [47], leading to subshell closings which account for the fine stmctures in figure C1.1.2(a ). The electron shell model is one of tire most successful models emerging from cluster physics. The electron shell effects are observed in many physical properties of tire simple metal clusters, including tlieir ionization potentials, electron affinities, polarizabilities and collective excitations [34]. 

When micelles are formed just above the cmc, they are spherical aggregates in which surfactant molecules are clustered, tails together, to form a spherical particle. At higher concentrations the amount of excess surfactant is such that the micelles acquire a rod shape or, eventually, even a layer structure. 

Figure 3 Periodic boundary conditions realized as the limit of finite clusters of replicated simulation cells. The limit depends in general on the asymptotic shape of the clusters here it is spherical. Cations are presented as shaded circles anions as open circles.

The metal cluster will be modeled as an infinitely deep spherical potential well with the represented by an infinitely high spherical barrier. Let us place this barrier in the center of the spherical cluster to simplify the calculations. The simple Schrodinger equation, containing only the interaction of the electrons with the static potential and the kinetic energy term and neglecting any electron-electron interaction, can then be solved analytically, the solutions for the radial wave functions being linear combinations of spherical Bessel and Neumann functions. 

When soaps are dispersed in water, the long hydrocarbon tails cluster together on the inside of a tangled, hydrophobic ball, while the ionic heads on the surface of the cluster stick out into the water layer. These spherical dusters, called micelles, are shown schematically in Figure 27.1. Grease and oil droplets... 

Micelle (Section 27.2) A spherical cluster of soaplike molecules that aggregate in aqueous solution. The ionic heads of the molecules lie on the outside, where they are solvated by water, and the organic tails bunch together on the inside of the micelle. 

Figure 6-12. Model for Ihe Calculation of the van der Waals potential experienced by a single T6 molecule on a Tfi ordered surface. Each molecule is modeled as a chain of 6 polarizable spherical units, and the surface as 8-laycr slab, each layer containing 266 molecules (only pan of the cluster is shown). Tire model is based on X-ray diffraction and dielectric constant experimental data. The two configurations used for evaluating the corrugation of the surface potential are shown. Adapted with permission front Ref. [48].

High nuclearity clusters [Au(AuPR3) ]JC+ with central gold atoms adopt two types of structure those with 12n + 18 electrons have essentially spherical structures, whereas those with 12n + 16 electrons have the peripheral atoms on the surface of a toroid. 

Sodium carboxymethyl chitin and phosphoryl chitin had most evident influences on the crystallization of calcium phosphate from supersaturated solutions. They potently inhibited the growth of hydroxyapatite and retarded the rate of spontaneous calcium phosphate precipitation. These chitin derivatives were incorporated into the precipitate and influenced both the phase and morphology of the calcium phosphate formed (flaky precipitate resembling octacalcium phosphate instead of spherical clusters in the absence of polysaccharide) [175]. 

A more realistic model of this solution was developed in 1950 by Rory and Krigbaum, and assumes that the polymer consists of approximately spherical clusters of segments. These clusters have a maximum density of segments at their centre and this density decreases with distance from the centre in an approximately Gaussian distribution. 

Mildvan AS, Grisham CM (1974) The Role of Divalent Cations in the Mechanism of Enzyme Catalyzed Phosphoryl and Nucleotidyl. 20 1-21 Mingos DMP, Hawes JC (1985) Complementary Spherical Electron Density Model. 63 1-63 Mingos DMP, Johnston RL (1987) Theoretical Models of Cluster Bonding. 68 29-87 Mingos DMP, McGrady JE, Rohl AL (1992) Moments of Inertia in Cluster and Coordination Compounds. 79 1-54... 

Large clusters of carbon atoms have been studied with ab initio calculations using basis sets of double-C quality. Planar, single-sheet graphite fragments with 6 - 54 atoms were investigated, as well as the spherical "Buckminsterfullerene ... 

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