Nuclear prolate

The close-packed-spheron theory of nuclear structure may be described as a refinement of the shell model and the liquid-drop model in which the geometric consequences of the effectively constant volumes of nucleons (aggregated into spherons) are taken into consideration. The spherons are assigned to concentric layers (mantle, outer core, inner core, innermost core) with use of a packing equation (Eq. I), and the assignment is related to the principal quantum number of the shell model. The theory has been applied in the discussion of the sequence of subsubshells, magic numbers, the proton-neutron ratio, prolate deformation of nuclei, and symmetric and asymmetric fission. 

In quadrupole splitting, the existence of a nonspherical nuclear charge distribution produces an electric quadrupole moment, Q, which indicates that the charge distribution in the nucleus is prolate, when Q > 0, or oblate, if Q < 0 [137-140],... 

Fig. 1. Nuclear shapes and nuclear quadrupole moments (a) prolate nucleus, I> 1, Q> 0 (b) spherical nucleus, I = 1/2. eQ — 0 (c) oblate nucleus, I> 1, eQ <0...

The integration over the coordinates of the electron number 2 will be done over a prolate spheiical coordinate system rB2,rv2,rBi), and the analytic expression of the nuclear attraction integral shown in equation (7.5) obtained. This integration method ensures that the variable rjj will not produce any mathematical problem as it has been observed in section 6.5. Note that the constant parameter, which is the usual intercenter distance, when dealing with this kind of coordinate system, will act here as a variable and it has to be integrated when considering the electron (1) coordinates. 

Figure 4 Schematic diagram showing an atom of nuclear charge Z located at one of the foci (z = D) of a prolate spheroidal box (bold line) characterized by = o> which forms part of a family of confocal prolate spheroids ( ), each one orthogonal to a family of confocal hyperboloids (ij ) (all shown with thin dotted lines). d and denote an electron position relative to the nucleus and to the other focal point. The x-y plane corresponds to 17 = 0.

Let us consider a many-electron atom of nuclear charge Z confined by a hard prolate spheroidal cavity. In this study the nuclear position will correspond to one of the foci as shown in Figure 4. In terms of prolate spheroidal coordinates, the nuclear position then corresponds to one of the foci for a family of confocal orthogonal prolate spheroids and hyperboloids defined, respectively, by the variables f and rj as [73] ... 

The family of confocal ellipsoids and hyberboloids represented by the prolate spheroidal coordinates allows us now to treat the case of a many-electron atom spatially limited by an open surface in half-space. A special case of the family of hyperboloids corresponds to an infinite plane defined by jj = 0 according to Equations (35) and (36). We now treat the specific case of an atom whose nuclear position is located at the focus a distance D from the plane as shown in Figure 4. 

Extensions to the spherical jellium model have been made to incorporate deviations from sphericality. Clemenger [15] replaced the Woods-Saxon potential with a perturbed harmonic oscillator model, which enables the spherical potential well to undergo prolate and oblate distortions. The expansion of a potential field in terms of spherical harmonics has been used in crystal field theory, and these ideas have been extended to the nuclear configuration in a cluster in the structural jellium model [16]. 

In the latter two phases backbones have the spindle-like conformation, i.e., the prolate shape with (R%) > R p), the characteristic of main chain liquid crystalline polymers. Important means of investigating the conformations of side chain liquid crystalline polymers include small angle neutron scattering from deuterium-labeled chains (Kirst Ohm, 1985), or small angle X-ray scattering on side chain liquid crystalline polymers in a small molecular mass liquid crystal solvent (Mattossi et al., 1986), deuterium nuclear resonance (Boeffel et al., 1986), the stress- or electro-optical measurements on crosslinked side chain liquid crystalline polymers (Mitchell et al., 1992), etc. Actually, the nematic (or smectic modifications) phases of the side chain liquid crystalline polymers have been substantially observed by experiments. 

In a 2005 review article, Cwiok, Heenan, and Nazarewicz (2005) present new theoretical results for properties of even-even heavy and SHE element nuclei with 94 < Z < 28 and with 134 < N < 188. They use self-consistent formalism and a modern nuclear energy density functional to formulate the following major conclusions concerning SHEs (1) SHE nuclei around Z= 116 and N= 176 are expected to exhibit coexistence of oblate and prolate shapes,... 

Fig. 1.1. Classical picture of the origin of nuclear electric quadrupole moments through deformation of a (rotating) charged sphere. (A) prolate ellipsoid with eQ positive, (B) oblate ellipsoid with eQ negative...

Nuclear electric quadrupole A parameter which describes the effective shape of the ellipsoid of nuclear charge distribution. A nonzero quadrupole moment Q indicates that the charge distribution is not spherically symmetric. By convention, the value of Q is taken to be positive if the ellipsoid is prolate and negative if it is oblate. 

A table of the microwave spectra of triatomic molecules [1] contains 139 lines for OF2 with frequencies ranging from 8299.51 to 59846.20 MHz, which were taken from the literature, and, in an appendix, 26 additional lines from 84401.810 to 95998.940 MHz and one line at 7803.49 MHz (transitions between rotational levels Jk-,k, with J = total angular momentum quantum number excluding nuclear spin and K", K" = projections of J on the symmetry axis of the limiting prolate or oblate symmetric top). Eight of these known rotational transitions and seven new transitions, belonging to the Vi =1 and Vg = 2 vibrationally excited states, were more recently observed by IR-microwave (MW) double resonance or IR-MW-MW triple resonance [12]. 

---