Spherical Nuclei

simple algebraic transformations of Equation 4.2 give 

First of all, it is necessary to find the optimal place for nucleation, proceeding from the condition 

These relations correspond to a minimum if a Therefore, the optimal place for nucleation shifts with the change of the concentration gradient, but the 

Hereinafter, we consider only nuclei appearing in the optimal place. In this case, the dependence of AG on the nucleus size is simple and is given by 

Simple transformations give the following expressions for. which corre- 

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A theoretical treatment, similar to that given above for spherical nuclei, may be provided for disc-shaped nuclei of only one or two molecules thickness (5jc) on reactant surfaces. For these... 

The form of this expression differs from that for spherical nuclei [ 28] (see above). Here, the equilibrium condition... 

This is the general form in the case where we have spherical nuclei. If we do not have spherical nuclei, then the equation must be modified to ... 

In the nucleation-and-growth transitions, nuclei of the new phase possessing a critical size have to be first formed in the parent phase. The change in free energy, AG, due to the formation of spherical nuclei is given by... 

On the basis of this equation the authors draw the conclusion that after completion of linear decomposition, i.e. at the point p0, t0, spherical nuclei of decomposition are formed. 

The first term in the square bracket in this equation is the electric monopole moment, which is equal to the nuclear charge, Ze. The second term in the square bracket is the electric dipole moment while the third term in the square bracket is the electric quadmpole moment. For a quantum mechanical system in a well-defined quantum state, the charge density p is an even function, and because the dipole moment involves the product of an even and an odd function, the corresponding integral is identically zero. Therefore, there should be no electric dipole moment or any other odd electric moment for nuclei. For spherical nuclei, the charge density p does not depend on 0, and thus the quadmpole moment Q is given by... 

Another interesting case of nuclear rotation occurs in the spherical nuclei. The observation of equally spaced 7-ray transitions implies collective rotation, but such bands have been observed in near spherical 199Pb. It has been suggested that these bands arise by a new type of nuclear rotation, called the shears mechanism. A few... 

In single-configurational approximation yj(0) f 0 only for s electrons. That is why in (22.44) the symbol xp has subscript s. The whole dependence of the shift under investigation on the pecularities of the electronic shells of an atom is contained only in the multiplier y s(0). Unfortunately, formula (22.44) does not account for the deviations of the shape of the nucleus from spherical symmetry. Therefore it is unfit for non-spherical nuclei. The accuracy of the determination of all these quantities may be improved by accounting for both the correlation and relativistic effects [157, 158]. A universal program to compute isotope shifts in atomic spectra is described in [159]. 

More systematic calculations with the present method will certainly hel to clarify our understanding of the 0-decay properties of spherical nuclei fa off the line of stability, which are needed in r-process studies. In particular, a study of the effects of the 0-decay of low-lying states thermally populated in the high temperature r-process environment is due. Sue effects have not been included in any r-process model yet attempted. Finally we mention that a different approach (i.e. RPA) is probably called for in order to deal with deformed nuclei effectively [BRA85]. ... 

The shell-correction energy is plotted in Figure 5a using data from Reference [55]. Two equally deep minima are obtained, one at Z = 108 and N = 162 for deformed nuclei with deformation parameters p2 0.22, p4 -0.07 and the other one at Z = 114 and N = 184 for spherical SHEs. Different results are obtained from self-consistent Hartree-Fock-Bogoliubov, HFB, calculations and relativistic mean-field models [56,57], They predict for spherical nuclei shells at Z = 114, 120 or 126 (dashed lines in Figure 5a) and N = 184 or 172. 

The term "superheavy elements" was first coined for elements on a remote "island of stability" around atomic number 114 (Chapter 8). At that time this island of stability was believed to be surrounded by a "sea of instability". By now, as shown in Chapter 1, this sea has drained off and sandbanks and rocky footpaths, paved with cobblestones of shell-stabilized deformed nuclei, are connecting the region of shell-stabilized spherical nuclei around element 114 to our known world. 

The main features of cold fusion reactions with the spherical nuclei of Pb or ° Bi as targets are low excitation energies of the compound nuclei (Ex w 15 to 20 MeV) with the consequence of emission of only one or two neutrons, low probability of fission, and relatively high fusion cross sections otus- On the other hand, the reaction products have relatively small neutron numbers and short half-fives. Suitable projectiles are neutron-iich stable nuclei, such as " Ca, Ti, " Cr, Fe, Ni, Zn, and Kr. 

The islands of relative stability are shown in Fig. 14.9. The stability gap around mass number 4 = 216 is evident from this figure nuclides with half-lives > 1 s do not exist for = 216. The search for superheavy elements concentrates on the islands around Z = 108 and Z= 114. At neutron numbers A = 162 the nuclei should exhibit a high degree of deformation, whereas spheric nuclei are expected for N = 184. 

TENSIONS, AND CRITICAL SIZES FOR SPHERICAL NUCLEI IN HOMOGENEOUS NUCLEATIONf... 

When spins have values greater than there are more than two available spin states. For I = 1 nuclei such as H and " N, the magnetic moments may precess about three directions relative to Bq parallel (/, = -fl), perpendicular (0), and opposite (-1). In general, there are (21 + 1) spin states—for example, six for I = 5/2 ( O has this spin). The values of L extend from +/ to — / in increments of 1 ( + /,+/ — 1, +/ — 2,... —/). Hence, the energy-state picture is more complex for quadrupolar than for spherical nuclei. 

M. Kleban, B. Nerlo-Pomorska, K. Pomorski, J. F. Berger, J. Decharge, The ground state properties of spherical nuclei calculated by Hartree-Fock-Bogolyubov procedure with Gogny DIS force, Acta Phys. Pol., B 33 (2002) 383-388. 

The filling of the pores by such spherical nuclei are a consequence of the large overpotential and the wetting interface Ti02-x(0H)zNi0y. 

If one assumes spherical nuclei with a solid-liquid interfacial energy between the growing nucleus and the melt 7 1 and radius r, and if one ignores... 

The free energy changes and their sum are shown qualitatively for a spherical nuclei in Figure 2.20. It is clear from the figure that... 

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