Model spherical

There are many large molecules whose mteractions we have little hope of detemiining in detail. In these cases we turn to models based on simple mathematical representations of the interaction potential with empirically detemiined parameters. Even for smaller molecules where a detailed interaction potential has been obtained by an ab initio calculation or by a numerical inversion of experimental data, it is usefid to fit the calculated points to a functional fomi which then serves as a computationally inexpensive interpolation and extrapolation tool for use in fiirtlier work such as molecular simulation studies or predictive scattering computations. There are a very large number of such models in use, and only a small sample is considered here. The most frequently used simple spherical models are described in section Al.5.5.1 and some of the more common elaborate models are discussed in section A 1.5.5.2. section Al.5.5.3 and section Al.5.5.4. 

Douketis C, Socles G, Marchetti S, Zen M and Thakkar A J 1982 Intermolecular forces via hybrid Hartree-Fock SCF plus damped dispersion (HFD) energy calculations. An improved spherical model J. Chem. Phys. 76 3057... 

Waisman E and Lebowitz J K 1972 Mean spherical model integral equation for charged hard spheres... 

Blum L 1980 Primitive electrolytes in the mean spherical model Theoretical Chemistry Advances and Perspectives vol 5 (New York Academic)... 

Another variant that may mrn out to be the method of choice performs the alchemical free energy simulation with a spherical model surrounded by continuum solvent, neglecting portions of the macromolecule that lie outside the spherical region. The reaction field due to the outer continuum is easily included, because the model is spherical. Additional steps are used to change the dielectric constant of that portion of the macromolecule that lies in the outer region from its usual low value to the bulk solvent value (before the alchemical simulation) and back to its usual low value (after the alchemical simulation) the free energy for these steps can be obtained from continuum electrostatics [58]. 

B. Bergersen, Z. Racz. Dynamical generation of long-range interactions Random Levy flights in the kinetic Ising and spherical models. Phys Rev Lett 67 3047-3050, 1991. 

Suppose a copper atom is thought of as occupying a sphere 2.6 X 10-8 cm in diameter. If a spherical model of the copper atom is made with a 3.2 cm diameter, how much of an enlargement is this ... 

For ail samples, both a.p. and s.o., irrespective of the preparation method, the experimental intensity ratios, V2p/Zr3d, increased proportionally to the V-content up to 3 atoms nm 2 (pjg 2). The ratio approaches those calculated with the spherical model proposed recently by Cimino et al. [27] (full line in Fig. 2). For ZV samples with V-content < 3 atoms nm 2, this finding shows that vanadium species are uniformly spread on the Zr02 surface. On ZV catalysts with a larger V content (not shown in Fig. 2), the intensity ratios were markedly larger than the corresponding values yielded by the spherical model. The results obtained on samples with V-content > 3 atoms nm 2 point therefore to a V surface enrichment. 

Several different types of semi-variograms are useful. The spherical model with nugget, the random model, and the spherical model with no nugget are discussed below. 

The reorganization energy of the slow polarization for the reactions at metal electrodes can be calculated with the use of Eqs. (34.11). For a spherical model of the reacting ion, it is equal approximately to... 

Ferromagnetic ordering of two-dimensional systems with dipole-dipole and exchange interactions in the approximation of a spherical model was examined in Ref. 73. The main simplifying assumption of the spherical model is the replacement of the condition er = 1 on the orientation vectors with the weaker condition... 

Fig. 4.7. 3He/H in simple Galactic H n regions, i.e. those thought to be reasonably well represented by homogeneous spherical models (Balser et al. 1999), and one planetary nebula, as a function of their oxygen abundance. 3He/H is plotted on a logarithmic scale relative to the proto-solar value of 1.5 x 10-5. After Bania, Rood and Balser (2002). Reprinted by permission from Macmillan Publishers Ltd. Courtesy Tom Bania.

An effective spherical model is sometimes adopted (11, 17, 18) in which... 

Figure lb. Left the non-spherical ionic atmosphere. The excess charge dq has an absolute maximum in one volume element. Right the reduced ionic cloud non-spherical model (n = 1). 

The use of a wetted spherical model affords the opportunity of studying combustion under steady-state conditions. Forced convection of the ambient gas may be employed without distortion of the object. Sufficiently large models may be employed when it is desired to probe the gas zones surrounding the burning sphere. It is apparent that the method is restricted to conditions where polymerization products and carbonaceous residues are not formed. In the application of such models, the conditions of internal circulation, radiant heat transmission, and thermal conductivity of the interior are somewhat altered from those encountered in a liquid droplet. Thus the problem of breakup of the droplet as a result of internal temperature rise cannot be investigated by this method. 

Exotic atomic nuclei may be described as structures than do not occur in nature, but are produced in collisions. These nuclei have abundances of neurons and protons that are quite different from the natural nuclei. In 1949, M.G, Mayer (Argonne National Laboratory) and J.H.D. Jensen (University of Heidelberg) introduced a sphencal-shell model of die nucleus. The model, however, did not meet the requirements and restrains imposed by quantum mechanics and the Pauli exclusion principle, Hamilton (Vanderbilt University) and Maruhn (University of Frankfurt) reported on additional research of exotic atomic nuclei in a paper published in mid-1986 (see reference listedi. In addition to the aforementioned spherical model, there are several other fundamental shapes, including other geometric shapes with three mutually peipendicular axes—prolate spheroid (football shape), oblate spheroid (discus shape), and triaxial nucleus (all axes unequal). 

Returning to 3D lattice models, one may note that sine-Gordon field theory of the Coulomb gas should enable an RG (e — 4 — D) expansion [15], but this path has obviously not yet followed up. An attempt to establish the universality class of the RPM by a sine-Gordon-based field theory was made by Khodolenko and Beyerlein [105]. However, these authors did not present a scheme for calculating the critical exponents. Rather they argued that the grand partition function can be mapped onto that of the spherical model of Kac and Berlin [106, 297] which predicts a parabolic coexistence curve, i.e. fi — 1/2. This analysis was severely criticized by Fisher [298]. Actually, the spherical model has some unpleasant thermodynamic features, never observed in real fluids. In particular, it is associated with a divergence of the compressibility KTas the coexistence curve (rather than the spinodal line) is approached. By a determination of the exponent y, this possibility could also be ruled out experimentally [95, 97]. 

A spherical model was used in Ref. [15] in order to obtain the shape of the domains, reversed under the fdb conditions. This model was widely applied for studies of different processes that take place in the field of afm tip (see Ref. [65]), including ferroelectric polarization reversal [66-69], In this model the field of the tip apex is supposed to coincide with a field of a metallic sphere, the radius of which is equal to the radius of curvature of the tip apex. Using a simple approximation it may be supposed that the tip charge is concentrated in the center of the sphere [15,64-69], We will take into account a more general model and check the accuracy of the simple spherical model application to the ferroelectric domain breakdown condition. 

Let us consider the real charge distribution in the vicinity of the tip, in contrast to the simple spherical model, where it was supposed that the overall charge is concentrated in the center of the spherical curvature of the tip apex. The charge qo = RU is generated in the center of the spherical curvature of the tip apex under an applied voltage U. Moreover, an infinite series of image charges qn is located at a distance r from the center of the sphere [70,71] ... 

When domain dimensions are much larger than the tip radius R, Equations (11), (12) and (13) become identical to the corresponding equations obtained in the framework of the theory [15,64] developed using the simple spherical model. In this model the energy of the long domain W r,l), created within the condition of fdb, equals to the summation of the energies from Equation (10.2) and Equation (10.11)... 

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