Isotropy spherical

IlyperChem avoids th e discon tin nily an d, in isotropy problem of th e implied cutoff by iin posing a sin oothed spherical cn toff within the implied cutoff. When a system is placed in a periodic box, a switched cnLoITis aiitoinatically added. The default outer radius, where the interaction is completely turned off, is the smallest of 1/2 R., 1/2 R.. and 1/2 R, so that the cutoff avoids discontinuities and is isotropic, fh is cutoff may be turned off or modified in the. Molecular Mechanics Options dialog box after solvation and before calcii lation. ... 

The only difference is that a(0) is now an operator acting in jm) space of angular momentum eigenfunctions. This space consists of an infinite number of states, unlike those discussed above which had only four. This complication may be partly avoided if one takes into account that the scalar product in Eq. (4.55) does not depend on the projection index m. From spherical isotropy of space, Eq. (4.55) may be expressed via reduced matrix elements (/ a(0 /) as follows... 

Under the assumption of isotropy the element in momentum space may be replaced by a spherical shell, in polar coordinates 4np2dp. Substituting p2 = 2mH this factor becomes... 

We are interested in final states of stellar evolution. Therefore we can restrict ourselves to static configurations. Also, fluid-like behavior seems appropriate in the microscopical dimensions. Therefore we are looking for static configurations. Also, some fluid-like behavior is expected in the sense that stresses in the macroscopical directions freely equilibrate. Then in 3 spatial directions isotropy is expected and thence spherical symmetry. Finally, in the lack of any information so far, we may assume symmetry in the extra dimension. Then in... 

The isotropy properties assumed for the model universe imply that it is statistically spherically symmetric about the chosen origin. If, for the sake of simplicity, it is assumed that the characteristic sampling times over which the assumed statistical isotropies become exact are infinitesimal, then the idea of statistical spherical symmetry, gives way to the idea of exact spherical symmetry thereby allowing the idea of some kind of rotationally invariant radial coordinate to exist. As a first step toward defining such an idea, suppose only that the means exists to define a succession of nested spheres, Sj C S2C ClSp, about the chosen origin since the model universe with infinitesimal characteristic sampling times is stationary, then the flux of particles across the spheres is such that these spheres will always contain fixed numbers of particles, say N, N2,. . . , NP, respectively. 

Formally, alignment and orientation follow from the general symmetry properties of statistical tensors pkK introduced in Section 8.4. Spherical symmetry leads to k = k = 0, axial symmetry to k — 0, and alignment requires k = even, and orientation k = odd. Since the dipole approximation in the photoionization process restricts k to k < 2, a photoionized axially symmetric state can only have Poo, p10, and p20 Poo describes isotropy, and the alignment is given by [BKa77]... 

For most particulate composites the mismatch between the particles and the matrix is more important than the anisotropy of either component (though alumina/aluminium titanate composites provide a notable exception and are described below). The main features of the stresses can therefore be understood in terms of a simple elastic model assuming thermoelastic isotropy and consisting of a spherical particle in a concentric spherical shell of matrix with dimensions chosen to give the appropriate volume fractions. The particles are predicted to be under a uniform hydrostatic stress, ap after cooling. If the particles have a larger thermal expansion coefficient than the matrix, this stress is tensile, and vice versa. For small particle volume fractions the stress... 

The conservation of angular momentum is a consequence of isotropy or spherical rotational symmetry of space (1.3.1). An alternative statement of a conservation law is in terms of a nonobservable, which in this case is an absolute direction in space. Whenever an absolute direction is observed, conservation no longer holds, and vice versa. The alignment of spin, that allows of no intermediate orientations, defines such a direction with respect to conservation of angular momentum. One infers that space is not rotationally symmetrical at the quantum level. 

It is not necessary to invoke a complete molecular isotropy if V depends only on 0. When dV/dS = fiE sin 3, we obtain the well-known Debye equation. It is very interesting to remark that at this perturbation order there is no sign of whether the rotator is a symmetric or a spherical top. 

Provided that G > Gp (for liquid foams x of solutions > x of air) we obtain Eq. (39) from Eq. (41) by substitution of k instead of G. In contrast to Wagner s formula, Odelevsky s formula holds for all concentrations of the disperse phase (gas) and for all types of gas-filled systems gaseous emulsions (d < 0.74), spherical (0.74 < d< 0.9) and polyhedral ( > 0.9) foams. It requires isotropy of the matrix structures and equal diameters of the disperse phase inclusions. Therefore, the dependence of the ratio of the foam to the solution electroconductivity on the degree of foaming in the general form is given by equation... 

Harrowell and Oxtoby have shown how the density functional theory for the solid-liquid interface outlined in Section III D can be generalized to study the nucleation of a crystal. If the critical nucleus is assumed spherical (a reasonable approximation for the alkali metals considered, given the near isotropy of the calculated surface free energy) then the inhomogeneous density of Eq. (3.13) can reasonably be generalized to... 

The classical ideas about the isotropy of electrical properties of spherical-top molecules are usually extrapolated to the magnetic properties. This leads to the conclusion about the isotropy of the magnetic susceptibility in high-symmetry molecules and hence about the disappearance of the orientational contribution to the birefringence in magnetic fields (the Cotton-Mouton effect). In the case of degenerate electronic terms or in the pseudodegeneracy situation, these conclusions are incorrect and have to be reconsidered. 

Similarly, the magnetic hyperfine interaction [Eq. (1)] can be written as AI S if the A tensor is isotropic although anisotropy can arise from non-zero orbital and dipole contributions due to admixtures with higher electronic states. Only small departures from isotropy have been found in A for one of iron transport compounds (28) with the spherically symmetric 6S state. 

Such bodies are termed helicoidally isotropic. Their resistance is characterized by the two scalars K, K, and the pseudoscalar K,. Examples of isotropic helicoids are described by Brenner (B22). Spherical isotropy is clearly a special case of this more general type. On account of the inequalities set forth in Eq. (50), the various scalars satisfy the relations... 

The usual spherical isotropy is a special case of this.) This requires that we set... 

In contrast, in sodium chloride the molecular orbitals that assure the cohesion of the lattice are practically reduced to s and orbitals of chlorine atoms. The corresponding density is practically spherical. The displacement of a sodium atom is facilitated by the isotropy. NaCil is soft H = 2.5. 

Well, this is true to some extent. For example, the Universe does not show an exact isotropy because the matter there does not show spherical symmetry. Moreover, even if only one object were in the Universe, this very object would itself destroy the anisotropy of the Universe. We should rather think of this as a kind of idealization (approximation of reality). 

In this chapter we will present the approach adopted by Ward and co-workers. Similar treatments have also been given independently by Yamagata and Hirota and by Slonim and Urman. Due to their appearance in the Japanese and Russian literature only, these latter previous treatments did not achieve prominence in the western literature. Furthermore, although it is perfectly possible to develop the theory in a very elementary manner, using Euler angle transformations, and this was the method of the earlier work, we choose to work here in terms of spherical harmonic analysis. The compactness of this representation has many advantages, particularly if the treatment is to be extended beyond transverse isotropy. 

We consider first the situation of transverse isotropy. The transverse isotropy is assumed to arise as follows. First the units of structure are transversely isotropic. Secondly, there is no preferential orientation of the units of structure in a plane perpendicular to the draw direction. We rewrite the Van Vleck equation in terms of spherical harmonics ... 

After elucidating conditions in liquids on the assumption of exactly spherical particles, attempts have been made to treat cases in which accurate spherical form and isotropy of the molecular force field are lacking but with deviations from the spherical form not very great and the lattice of the crystalline phase well known. A particularly important substance, whose behavior in the liquid state has long aroused great interest, is water. 

The assumption that there exists an appropriate self-consistent field implies break-down of the space isotropy in some way. Such anisotropy is introduced either hy a statement that the initial point preserves its position r1[0) s 0 or by additional fixing of the second end of the chain at f(L) = h. In the first case, the field is spherically symmetrical about the origin of coordinate. In the second case, (f(0) = 0 and r[L) = h are fixed), the field has the symmetry Dooh with respect to these two points (focuses). 

We could attempt to introduce an SCF approximation directly into (6.16). Such a discussion would be instructive, but only heuristic. The formal derivation is presented and generalized in Section VID. The assumption of the existence of a suitable self-consistcnt field implies that somehow we destroy the isotropy of space. The anisotropy associated with the introduction of an SCF is introduced either by specifying that the initial segment is at some fixed point in space (conveniently chosen as the origin) or by specifying the end-to-end vector R in addition. In the first case, the assumption that r(0) = 0 leads to a polymer distribution which is spherically symmetric about the origin. The field representing the excluded volume then of course has the same symmetry. We want to introduce some approximation that will permit us to calculate both the distribution and the field in a completely self-consistent manner. In the second approach, the specification of r(0) = 0 and r L) = R leads to a field of T>oo7 symmetry about these two end (focal) points. 

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