Ellipsoidal Nuclei

It is evident that, since concentration gradient suppresses nuclei growth along the longitudinal direction, nature will find the possibility to increase the nucleus s volume (and reduce Gibbs free energy) by transversal growth. Hiis statement means that the nuclei formed in the diffusion zone must be nonspherical. Because of this, it is necessary to take into account the shape optimization for each fixed volume of a nucleus. 

The first attempt in this direction was made in 1991 [15]. Nuclei (embryos) were taken as spheroidal (ellipsoids of rotation) with a symmetry axis being directed along Vc, and the parameters J n (jj x) and Rj. (J x). In this case, AG is a function of two arguments, the volume V and the shape parameter, y = R /R at a fixed concentration gradient, Vc=l/L. We get 

Let interdifiusion in the binary compound A- B lead to the formation of a metastable parent phase (soHd solution or amorphous phase) with a concentration gradient inversely proportional to -v/Dt, where D is the diflfusion coefficient in the parent phase. Let us study the possibility for the nuclei of the stable intermediate phase to appear under the mentioned gradienL We approximate the concentration dependence of Gibbs potential for both phases by a parabola with a minimum at cj = Cq = 1/2 (this approximation is essential only for comparison with the analytical solutions and is optional at MC simulations). The concentration profile of the parent phase near the forming nucleus wiU be approximated by a linear dependence. 

Consider the simulation procedure. Each cell may be found in either one of the two states, the old or the new one. The change of phase state leads to the change of bulk and surface energy. For example, if the cell changes the phase from the old to the new one, the change of the system s Gibbs potential equals 

1) At first, all cells belong to the old (parent) phase. Randomly we choose a cell as that where the nucleation (new phase) takes place and try to transform it according to the Metropolis algorithm (AN = 6). 

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The microscopic study of the formation and growth of nuclei in solid ammonium perchlorate was studied by a number of authors (Vol. II. p. 481). Raevskii and Manclis [79a found that the decomposition centres of orthorhombic fonn consists of a large number of ellipsoid nuclei of 1-2 /i/m. They are not stationary but moving at the speed of the order 7 -10 m/min at 230 C. Their activation cncrg> is 31 and 33 kcal/mol depending on the direction of tlie movement. 

The dominant mode of spin-lattice relaxation of nuclei with spins greater than results from the quadrupolar nature of such nuclei. These nuclei are considered to have an ellipsoidal rather than a spherical shape. When I — 1, as in or there are three stable orientations in the magnetic field parallel, orthogonal, and antiparallel, as shown in Figure 5-3. When these ellipsoidal nuclei tumble in solution within an unsymmetrical electron cloud of the molecule, they produce a fluctuating electric field that can bring about relaxation. 

An ellipsoidal nucleus with two spherons in the inner core has major radius greater than the minor radii by the radius of a spheron. about 1.5 f, which is about 25 percent of the mean radius. The amount of deformation given by this model is accordingly in rough agreement with that observed (18). In a detailed treatment it would be necessary to take into account the effect of electrostatic repulsion in causing the helions to tend to occupy the poles of the prolate mantle, with tritons tending to the equator. 

Figure 19.10 Free energy to form ellipsoidal nucleus, AS( ), as a function of the aspect...

Two-Component System with Isotropic Interfaces and Strain Energy Present. An example of this case is the solid-state precipitation of a 5-rich (i phase in an A-rich a-phase matrix. For steady-state nucleation, Eq. 19.16 again applies. However, for a generalized ellipsoidal nucleus, the expression for AQ will have the form of Eq. 19.28. Also, /3 must be replaced by an effective frequency, as discussed in Section 19.1.2. 

The average potential of the ellipsoidal nucleus is essentially a harmonic oscillator potential O Fig, 22),... 

The strength of the quadrupolar interaction is proportional to the quadrupole moment Q of a nucleus and the electric field gradient (EFG) [21-23]. The size of Q depends on the effective shape of the ellipsoid of nuclear charge distribution, and a non-zero value indicates that it is not spherically symmetric (Fig. 1). 

The nucleocapsid of the C. sonorensis polydnavirus is prolate ellipsoid in shape and has two envelopes (37) one envelope is obtained in the nucleus and the other as the virus buds through the calyx cell membrane... 

In considering the physical forces acting in fission, use may be made of the Bohr liquid drop model of the nucleus. Here it is assumed that in its uonual energy state, a nucleus is spherical and lias a homogeneously distributed electrical charge. Under the influence of the activation eneigy furnished by the incident nentron, however, oscillations are set up which tend to deform the nucleus. In the ellipsoid form, the distribution of the protons is such that they are concentrated in the areas of the two foci. The electrostatic forces of repulsion between the protons at the opposite ends of the ellipse may then further deform the nucleus into a dumbbell shape. Rrom this condition, there can be no recovery, and fission results. 

Chapter 7. THE STRUCTURE OF THE NUCLEUS OF THE ATOM What exclaimed Roger, as Karen rolled over on the bed and rested her warm body against his. I know some nuclei are spherical and some are ellipsoidal, but where did you find out that some fluctuate in between ... 

The deformation can be very complicated to describe in a single-particle framework, but a good understanding of the basic behavior can be obtained with an overall parameterization of the shape of the whole nucleus in terms of quadmpole distortions with cylindrical symmetries. If we start from a (solid) spherical nucleus, then there are two cylindrically symmetric quadmpole deformations to consider. The deformations are indicated schematically in Figure 6.10 and give the nuclei ellipsoidal shapes (an ellipsoid is a three-dimensional object formed by the rotation of an ellipse around one of its two major axes). The prolate deformation in which one axis is longer relative to the other two produces a shape that is similar to that of a U.S. football but more rounded on the ends. The oblate shape with one axis shorter than the other two becomes a pancake shape in the limit of very large deformations. 

In three dimensions a spatial wave group moves around an harmonic ellipsoid and remains compact, in contrast to the dispersive wave packets of classical optics. The distinction is ascribed to the fact that the quantum wave packet is built up from discrete harmonic components, rather than a continuum of waves. The wave mechanics of a hydrogen electron is conjectured to produce wave packets of the same kind. At small quantum numbers the wave spreads around the nucleus and becomes more particle-like, at high quantum numbers, as it approaches the ionization limit where the electron is ejected from the atom. 

When not indicated otherwise, our observations refer to cells in the sub apical area between 300 and 600 pm from the root tip. In the actively growing root, this area is the site of active cell division along with the first stages of cell differentiation, depending on the tissue. Root cells from 2 h-imbibed seeds contained numerous protein bodies19,24 of spheroidal shape, about 1.5-3 pm in diameter and nearly completely filled with highly omiophilic protein material they also contained abundant lipid reserves in the form of minute droplets, mainly concentrated at the cell periphery. The nucleus had spheroid or ellipsoidal shape and showed a distinct nucleolus. The cytoplasm contained numerous mitochondria with a dense matrix as well as relatively small and scarcely differentiated plastids with no or very little starch (Fig. 15.3a,b). 

In some cases, the nucleus may not be at the centre of the confining potential. If there is rotational symmetry about the line of nuclear displacement, the elliptic coordinates may lead to separable solutions. For example, in the case of a hydrogen atom with the nucleus at the focus (0,0, —R/2) of a confining ellipsoid, one can take the elliptic coordinates of the electron as... 

The study of the electronic structure of diatomic species, which can nowadays be done most accurately with two-dimensional numerical finite difference techniques, both in the non-relativistic [90,91] and the relativistic framework [92-94], is still almost completely restricted to point-like representations of the atomic nuclei. An extension to allow the use of finite nucleus models (Gauss-type and Fermi-type model) in Hartree-Fock calculations has been made only very recently [95]. This extension faces the problem that different coordinate systems must be combined, the spherical one used to describe the charge density distribution p r) and the electrostatic potential V(r) of each of the two nuclei, and the prolate ellipsoidal one used to solve the electronic structure problem. 

A non-zero nuclear quadrupole moment implies that the nuclear charge is not distributed spherically the shape of the nucleus resembles rather an ellipsoid. The nuclear quadrupole moment is better expressed through... 

Both the liquid-drop model and the single-particle model assume that the mass and charge of the nucleus are spherically symmetric. This is true only for nuclei close to the magic numbers other nuclei have distorted shapes. The most common assumption about the distortion of the nuclide shape is that it is ellipsoidal, i.e. a cross-section of the nucleus is an ellipse. Figure 11.6 shows the oblate (flying-saucer-like) and prolate (egg-shaped) ellipsoidally distorted nuclei the prolate shape is the more common. Deviation from the spherical shape is given by... 

When we studied the radio-frequency spectrum of D2 we hit another surprise [5]. The separation of the spectral lines in D2 were greater than in H2 even though the nuclear spin-spin interaction and the nuclear spin molecular rotation interaction should be much less. We found a similar anomaly for HD. We finally interpreted this as due the deuterium nucleus having a quadrupole moment (being ellipsoidal in shape) which gave rise to a spin dependent electrical interaction. The existence of the quadrupole moment, in turn, implied the existence of a new elementary particle force called a tensor force. In this way, magnetic resonance made a fundamental contribution to particle physics. 

The PCM Method. Accurate ab initio calculation of solvent effects requires use of a molecular shape more realistic than spherical or ellipsoidal shapes. In the polarizable-continuum model (PCM) of Miertus, Scrocco, and Tomasi, each atomic nucleus in the solute molecule M is surrounded by a sphere of radius 1.2 times the van der Waals radius of that atom. The cavity region is taken as the volume occupied by these overlapping atomic spheres. (Recall the van der Waals molecular surface— Section 15.8.)... 

The simplest shape for the hole is a sphere or an ellipsoid. This has the advantage that the electrostatic interaction between M and the dielectric medium may be calculated analytically. More realistic models employ molecular shaped holes, generated for example by interlocking spheres located on each nucleus. Taking the atomic radius as a suitable factor (a typical value is 1.2) times a van der Waals radius defines a van der Waals surface. Such as surface may have small pockets where no solvent molecules can enter and a more appropriate descriptor may be defined as the surface traced out by a spherical particle of a given radius (a typical radius of 1.4 A to model a water molecule) rolling on the van der Waals surface. This is denoted the Solvent Accessible Surface (SAS) and is illustrated in Rgure 14.9. 

Also it should be pointed out that the minima and maxima are observed to be more pronounced in lead than in a nucleus like Ta with a large quadrupole moment, i.e. in nuclei which are known to have an ellipsoidal shape. More data are needed on nuclei which are believed to have a spherical shape unfortunately these generally have many isotopes. 

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