Deformed-shell nuclei

The effects of a rather distinct deformed shell at = 152 were clearly seen as early as 1954 in the alpha-decay energies of isotopes of californium, einsteinium, and fermium. In fact, a number of authors have suggested that the entire transuranium region is stabilized by shell effects with an influence that increases markedly with atomic number. Thus the effects of shell substmcture lead to an increase in spontaneous fission half-Hves of up to about 15 orders of magnitude for the heavy transuranium elements, the heaviest of which would otherwise have half-Hves of the order of those for a compound nucleus (lO " s or less) and not of milliseconds or longer, as found experimentally. This gives hope for the synthesis and identification of several elements beyond the present heaviest (element 109) and suggest that the peninsula of nuclei with measurable half-Hves may extend up to the island of stabiHty at Z = 114 andA = 184. 

Taking into account the success of the spherical and deformed shell models, it is tempting to calculate the total energy of nucleus by summation of single-particle proton and neutron energies up to the Fermi level. Then... 

As we have seen, the nucleons reside in well-defined orbitals in the nucleus that can be understood in a relatively simple quantum mechanical model, the shell model. In this model, the properties of the nucleus are dominated by the wave functions of the one or two unpaired nucleons. Notice that the bulk of the nucleons, which may even number in the hundreds, only contribute to the overall central potential. These core nucleons cannot be ignored in reality and they give rise to large-scale, macroscopic behavior of the nucleus that is very different from the behavior of single particles. There are two important collective motions of the nucleus that we have already mentioned that we should address collective or overall rotation of deformed nuclei and vibrations of the nuclear shape about a spherical ground-state shape. 

As we learned in Chapter 2, it is necessary to include shell effects in the liquid drop model if we want to get reasonable values for nuclear masses. Similarly, we must devise a way to include these same shell effects into the liquid drop model description of the effect of deforming nuclei. Strutinsky (1967) proposed such a method to calculate these shell corrections (and also corrections for nuclear pairing) to the liquid drop model. In this method, the total energy of the nucleus is taken as the sum of a liquid drop model (LDM) energy, LDM and the shell (8S) and pairing (8P) corrections to this energy,... 

A special situation occurs at Zr (Z=40), where the neutron subshell closure (N=56) gives this nucleus a double subshell closure. Thus, particle-hole pair excitations across both subshell gaps are possible. This could produce situations much like those in doubly closed shell 0, in which the lowest-lying intruder deformed band has been shown to arise due to such excitations [BR064]. 

The deformation is related to the nuclear shell structure. Nuclei with magic numbers are spherical and have sharp boundary surfaces (they are "hard"). As the values of N and Z depart from the magic numbers the nucleus increases its deformation. 

The ground state energy, E, of a nucleus can be regarded as a sum of the liquid drop model energy (including deformation), pairing correction, 6p, and the shell... 

Strutinsky developed an extension of the liquid drop model which satisfactorily explains the fission isomers and asymmetric fission. For such short half-lives the barrier must be only 2-3 MeV. Noting the manner in which the shell model levels vary with deformation ( 11.5, the "Nilsson levels"), Strutinsky added shell corrections to the basic liquid-drop model and obtained the "double-well" potential energy curve in Figure 14.14b. In the first well the nucleus is a spheroid with the major axis about 25 % larger than the minor. In the second well, the deformation is much larger, the axis ratio being about 1.8. A nucleus in the second well is metastable (i.e. in isomeric state) as it is unstable to y-decay to the first well or to fission. Fission from the second well is hindered by a 2 - 3 MeV barrier, while from the first well the barrier is 5 - 6 MeV, accounting for the difference in half-lives. 

Figure 16.5 shows the variation in nuclear deformation calculated for the fission barrier of 298114 Qf particular interest are the small local fluctuations at small deformation. The minimum of 8 MeV at zero deformation constrains the nucleus to a spherical shape. Spontaneous fission is a very slow process in this situation since it involves tunneling through the 8 MeV barrier. These local fluctuations in the potential energy curve in Figure 16.S result from adding corrections for shell effects to a liquid drop model. The resistance to deformation associated with closed shell nuclei produces much longer half-lives to spontaneous fission than would be expected from calculations based on a standard liquid drop model.

Positive and negative ions will be stable in one another s presence only if the attractive forces are not too great. There will always be a deformation of the electron cloud of the negative ion due to the attractive force of the nucleus of the positive ion. If this deformation is large enough, electrons from the outer shell of the anion will be drawn into the orbital system of the cation, and a covalent bond will be set up. The process may be crudely represented as shown in Fig. 7. 

With increasing angular momentum (see region B in O Fig. 2.18), the strong centrifugal force disturbs the shell structure and drives the prolate nucleus into a triaxial deformed shape. 

The splitting of levels leads to an increased level density. Energetic gaps ( closed shells ) found in a spherical nucleus may disappear in a deformed nucleus. New gaps may be formed at different nuclide numbers at specific deformations. The sum of the energetic effects of all the occupied quantum states in a nucleus shows an oscillating behavior, i.e., the stability of a nucleus (as far as shells are concerned) as a function of deformation shows maxima and minima. 

A model, based on the construction of a potential energy surface from a combination of liquid drop terms for protons and neutrons as a function of deformation and shell corrections, has been published by Wilkins, Steinberg, and Chasman (Wilkins et al. 1976). In the model, a specific excitation energy (nuclear temperature) is assumed at scission and the probability to reach this specific state is calculated. The general trends of mass-yield curves in the fission of very different nuclides from "Po to Fm and for different excitation energies of the fissioning nucleus from 0 MeV (spontaneous fission) to highly excited fission reactions are reproduced correctly. In particular, the transition from the symmetric fission of (the compound nuclei) Po to a triple-humped mass-yield curve for Ra to double-humped yield curves for and Cf and, finally, a partial return to symmetry for Fm is... 

Nilsson (1955) extended the single-particle shell model to deformed potentials. The solutions of the Schrodinger equation then depend on deformation also. In the independent-partide model (Wagemans 1991) the sum of the single-particle energies of an even-even nucleus is given by... 

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