Spherical symmetrical condition

The spherical symmetrical condition is applied to the silicalite crystal particle ... 

Most theories of droplet combustion assume a spherical, symmetrical droplet surrounded by a spherical flame, for which the radii of the droplet and the flame are denoted by and respectively. The flame is supported by the fuel diffusing from the droplet surface and the oxidant from the outside. The heat produced in the combustion zone ensures evaporation of the droplet and consequently the fuel supply. Other assumptions that further restrict the model include (/) the rate of chemical reaction is much higher than the rate of diffusion and hence the reaction is completed in a flame front of infinitesimal thickness (2) the droplet is made up of pure Hquid fuel (J) the composition of the ambient atmosphere far away from the droplet is constant and does not depend on the combustion process (4) combustion occurs under steady-state conditions (5) the surface temperature of the droplet is close or equal to the boiling point of the Hquid and (6) the effects of radiation, thermodiffusion, and radial pressure changes are negligible. 

FIG. 19 Scheme of a simple fluid confined by a chemically heterogeneous model pore. Fluid modecules (grey spheres) are spherically symmetric. Each substrate consists of a sequence of crystallographic planes separated by a distance 8 along the z axis. The surface planes of the two opposite substrates are separated by a distance s,. Periodic boundary conditions are imposed in the x and y directions (see text) (from Ref. 77). 

If 5v //v /coex is not small, the simple description Eq. (14) in terms of bulk and surface terms no longer holds. But one can find AF from Eq. (5) by looking for a marginally stable non-uniform spherically symmetric solution v /(p) which leads to an extremum of Eq. (5) and satisfies the boundary condition v /(p oo) = v(/ . Near the spinodal curve i = v /sp = Vcoex /a/3 (at this stability limit of the metastable states both and S(0) diverge) one finds "... 

For swarms of spherical bubbles, the field may be expected to be approximately spherically symmetric when the origin of coordinates is fixed on the center of mass of a typical particle. Therefore, by using spherical coordinates and the initial condition ... 

Here S Nft = 0 because of the electroneutrality condition.) Equation (1.3.10) is substituted into Eq. (1.3.6) and the Laplace operator is expressed in polar coordinates (for the spherically symmetric problem) ... 

Here r and v are respectively the electron position and velocity, r = —(e2 /em)(r/r3) is the acceleration in the coulombic field of the positive ion and q = /3kBT/m. The mobility of the quasi-free electron is related to / and the relaxation time T by p = e/m/3 = et/m, so that fi = T l. In the spherically symmetrical situation, a density function n(vr, vt, t) may be defined such that n dr dvr dvt = W dr dv here, vr and vt and are respectively the radical and normal velocities. Expectation values of all dynamical variables are obtained from integration over n. Since the electron experiences only radical force (other than random interactions), it is reasonable to expect that its motion in the v space is basically a free Brownian motion only weakly coupled to r and vr by the centrifugal force. The correlations1, K(r, v,2) and fc(vr, v(2) are then neglected. Another condition, cr(r)2 (r)2, implying that the electron distribution is not too much delocalized on r, is verified a posteriori. Following Chandrasekhar (1943), the density function may now be written as an uncoupled product, n = gh, where... 

The spherical-symmetric fuel droplet burning problem is the only quiescent case that is mathematically tractable. However, the equations for mass burning may be readily solved in one-dimensional form for what may be considered the stagnant film case. If the stagnant film is of thickness <5, the free-stream conditions are thought to exist at some distance 8 from the fuel surface (Fig. 6.16). 

We demonstrated that by the selection of a representation of the Dirac Hamiltonian in the spinor space one may strongly influence the performance of the variational principle. In a vast majority of implementations the standard Pauli representation has been used. Consequently, computational algorithms developed in relativistic theory of many-electron systems have been constructed so that they are applicable in this representation only. The conditions, under which the results of these implementations are reliable, are very well understood and efficient numerical codes are available for both atomic and molecular calculations (see e.g. [16]). However, the representation of Weyl, if the external potential is non-spherical, or the representation of Biedenharn, in spherically-symmetric cases, seem to be attractive and, so far, hardly explored options. 

The first attempt to calculate realistic wave functions for electrons in metals is that of Wigner and Seitz (1933). These authors pointed out that space in a body-or face-centred cubic crystal could be divided into polyhedra surrounding each atom, that these polyhedra could be replaced without large error by spheres of radius r0, so that for the lowest state one has to find spherically symmetrical solutions of the Schrodinger equation (6) subject to the boundary condition that... 

Solve the Poisson-Boltzmann equation for a spherically symmetric double layer surrounding a particle of radius Rs to obtain Equation (38) for the potential distribution in the double layer. Note that the required boundary conditions in this case are at r = Rs, and p - 0 as r -> oo. (Hint Transform p(r) to a new function y(r) = r J/(r) before solving the LPB equation.)... 

The conventional viewpoint, which assumes that the ionic atmosphere is spherically symmetric, does not take account of the inevitable effects of ionic polarization. From an analysis of the general solution (19), however, it is evident that the ionic atmosphere must be spherically symmetric for nonpolarizable ions, and the DH model is therefore adequate. (Moreover, in very dilute solution polarization effects are negligibly small, and it does not matter whether we choose a polarizable or unpolarizable sphere for our model.) But once we have made the realistic step of conferring a real size on an ion, the ion becomes to some extent polarizable, and the ionic cloud is expected to be nonspherical in any solution of appreciable concentration. Accordingly, we base our treatment on this central hypothesis, that the time-average picture of the ionic solution is best represented with a polarizable ion surrounded by a nonspherical atmosphere. In order to obtain a value for the potential from the general solution of the LPBE we must first consider the boundary conditions at the surface of the central ion. 

Diffusion into a sphere represents a three-dimensional situation thus we have to use the three-dimensional version of Fick s second law (Box 18.3, Eq. 1). However, as mentioned before, by replacing the Cartesian coordinates x,y,z by spherical coordinates the situation becomes one-dimensional again. Eq. 3 of Box 18.3 represents one special solution to a spherically symmetric diffusion provided that the diffusion coefficient is constant and does not depend on the direction along which diffusion takes place (isotropic diffusion). Note that diffusion into solids is not always isotropic, chiefly due to layering within the solid medium. The boundary conditions of the problem posed in Fig. 18.6 requires that C is held constant on the surface of the sphere defined by the radius ra. 

If the potential energy, U, and the boundary and initial conditions are spherically symmetrical, then the survival and recombination probabilities are, respectively... 

On taking the Laplace transform, and noting that the steady-state limit (f->°° where dp/dt 0) is equivalent to the limit s -+ 0, and using the initial condition (131), this reduces to the spherically symmetric... 

The spherically symmetric component of this diffusion equation is of prime concern because the boundary and initial conditions are themselves spherically symmetric, for instance the Smoluchowski conditions (eqns. (3)—(5)]. Selecting the spherical equation for a coulomb potential U = rc/r [see eqn. 39)]... 

Since the boundary and initial conditions are spherical symmetric, we may assume that only the l = 0 component in the expresion of h,(r, s r0) contributes a non-zero term to the radial current of particles together. It may be proved by taking the angular integral of pj. This simplifies the equation for ftj(r, ro) further. The form given above differs from Hong and Noolandi [72] by a factor of 4tt and the term D in the delta... 

In the mean-field approximation, each particle develops a spherically symmetric diffusion field with the same far-field boundary condition fixed by the mean concentration, (c). This mean concentration is lower than the smallest particles ... 

In this addendum, we will derive the spectral function from Weyl s theory and in particular demonstrate the relationship between the imaginary part of the Weyl-Titchmarsh m-function, mi, and the concept of spectral concentration. For simplicity we will restrict the discussion to the spherical symmetric case with the radial coordinate defined on the real half-line. Remember that m could be defined via the Sturm-Liouville problem on the radial interval [0,b] (if zero is a singular point, the interval [a,b], b > a > 0), and the boundary condition at the left boundary is given by [commensurate with Eq. (5)]... 

This solution, formally expressed as Equation (1.131), is essentially nonlocal in space, although the problem is originally formulated in terms of local Equations (1.119). The spatial nonlocality arises from boundary conditions on S. Simple solutions are available only for spherically symmetrical cases (Born ion or Onsager point dipole). The equilibrium solvation energy is expressed as... 

The hope of understanding the concept of molecular structure quantum-mechanically would obviously be at its most realistic for the smallest of molecules at the absolute zero of temperature. However, under these conditions completely different pictures emerge for the molecule in, either total isolation, or in a macroscopic sample. In the latter case the molecule appears embedded in a crystal, which is quantum-mechanically described by a crystal hamiltonian with the symmetry of the crystal lattice. The isolated molecule has a spherically symmetrical hamiltonian. The two models can obviously not define the same quantum molecule. 

The electron density in an atom, molecule or crystal is described by a wave function which is subject to strict characteristic boundary conditions. As shown before (eqn. 5.3) an electron on a spherically symmetrical atom obeys the one-dimensional radial Helmholtz equation... 

Problems associated with the quantum-mechanical definition of molecular shape do not diminish the importance of molecular conformation as a chemically meaningful concept. To find the balanced perspective it is necessary to know that the same wave function that describes an isolated molecule, also describes the chemically equivalent molecule, closely confined. The distinction arises from different sets of boundary conditions. The spherically symmetrical solutions of the free molecule are no longer physically acceptable solutions for the confined molecule. 

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