Deformed nuclei

SR. A. Kenefick and R. K. Sheline, Phys. Rev. 135. B939 (1964), have described the energy levels of Smla, with N= 90, as representing the transition between spherical and highly deformed nuclei. 

Observed properties of many nuclei have been interpreted as showing that the nuclei are not spherical but are permanently deformed (4). The principal ranges of deformation are neutron numbers 90 to 116 and 140 to 156. Most of the deformed nuclei are described as prolate ellipsoids of revolution, with major radii 20 to 40 percent larger than the minor radii. 

A simple explanation of the existence of deformed nuclei in these ranges is provided by the close-packed-spheron theory (J4) it is that the inner core (of two or five spherons) in these ranges has an elongated structure, and that this elongation is imposed by the inner core on the two surrounding layers. 

The wave functions for the two inner-core spherons can, of course, be described as the symmetric and antisymmetric combinations of l.t and Ip-functions. The Nilsson (19) treatment of neutron and proton orbitals in deformed nuclei is completely compatible with the foregoing discussion, which provides a structural interpretation of it. 

Nuclear interaction is very complicated and its explicit form is still unknown. So, in practice different approximations to nuclear interaction are used. Skyrme forces [11,12] represent one of the most successful approximations where the interaction is maximally simplified and, at the same time, allows to get accurate and universal description of both ground state properties and dynamics of atomic nuclei (see [20] a for recent review). Skyrme forces are contact, i.e. 5(fi — 2), which minimizes the computational effort. In spite of this dramatic simplification, Skyrme forcese well reproduce properties of most spherical and deformed nuclei as well as characteristics of nuclear matter and neutron stars. Additional advantage of the Skyrme interaction is that its parameters are directly related to the basic nuclear... 

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,... 

Some have taken the viewpoint that, without the special stability associated with nuclear shell structure, elements as light as Z = 106-108 would have negligibly short half-lives. The mere existence of these nuclei with millisecond half-lives is said to be a demonstration that we have already made superheavy nuclei, according to this view. The shell stabilization of these nuclei, which are deformed, is due to the special stability of the N = 162 configuration in deformed nuclei. (The traditional superheavy nuclei with Z 114, N = 184 were calculated to have spherical shapes.)... 

Since the original study of Otsuka, Arima and Iachello (OTS78), much work has been carried out on this subject. In spite of that, until now, the description of well deformed nuclei can not be considered satisfactory. We present here a new mapping technique which is directed toward the treatment of these systems. 

Mixed-Symmetry Interpretation of Some Low-Lying Bands in Deformed Nuclei and the Distribution of Collective Magnetic Multipole Strength... 

We see that many problems still need to be solved in order to obtain accurate results in Hauser-Feshbach calculations. Some examples are the energy dependence of rotational enhancement of levels in deformed nuclei, the energy and mass dependence of Ml gamma-ray transitions, the importance of E2 transitions, and better estimates of fission barriers. Work in each of these areas will benefit greatly from a better understanding of the discrete levels, particularly in nuclei away from stability. 

HOF84] R. W. Hoff and W. R. Willis, "Level-Structure Modeling of Odd-Mass Deformed Nuclei," Nuclear Chemistry Division FY 84 Annual Report, Lawrence Livermore National Laboratory, Livermore, CA, UCAR 10062-84/1 (1984). 

HOF85a] R. W. Hoff, J. Kern, R. Piepenbring, and J. B. Boisson, "Modeling Level Structures of Odd-Odd Deformed Nuclei," Proc. Conf. on Capture Gamma-Ray Spectroscopy and Related Topics, Knoxville, TN, American Institute of Physics, New York, NY, AIP Conf. Proc. 125 (1985), p. 274. 

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]. ... 

Figure 1. Schematic divisions of nuclei in three major regions (I) region near doubly closed shells (II) region of strongly deformed nuclei (Ill) region of intruder states.

A clear gap in the study of this region was a lade of information on single-particle states in deformed nuclei with N > 60. Only recently, since we began our first Y study with "r, has such information emerged. The status through the end of 1984 was recently summarized by [PETT85] ... 

Although odd-odd rotational bands are best understood in well-deformed nuclei, for quite some time they have also been known to exist, based on excited states, in nuclei very close to closed shells [SAM77] Rather intriguing examples of this occur in many of the odd-odd Sb isotopes. The high-spin level scheme of 11 Sb, resulting from the 11 In(a,3nY) reaction [BEN85], is shown in Fig. 6. It contains at least two rotational bands, the J - 7... 

An important measurement is an accurate determination of the y-ray yield threshold for exciting a particular level. Thresholds can be determined with uncertainties of only a few keV, when necessary to guarantee certain placement of a y-ray in a level scheme. Usually, thresholds determined to within 30 to 50 keV are sufficient for this purpose, even in fairly heavy nuclei. The reason for this is that, typically, excited levels will have one or more decays to low-lying levels which are spaced 50 keV or more apart, even in heavy deformed nuclei. 

Many well-deformed nuclei have been found in rare earth and actinide regions. In those nuclei the ratio of the excitation energies of the 4+ to the 2+ states E /E2 are almost equal to the rotational limit 10/3. The lowest values of the E2 are about 72 keV and 43 keV, respectively. Can we find such well-deformed nuclei in the region of the proton and neutron numbers ranging from 50 to 82 What the minimum E2 there these questions have not been answered yet because well-deformed nuclei, if any, are too proton-rich and have too poor yields to observe by conventional in-beam spectroscopic methods. The present studies on light isotopes of Sm, Nd and Ce aim at finding a clue to the questions. 

Other accessible candidates of well-deformed nuclei are Ce-126 and Ce-124. Early results revealed their level schemes as depicted in fig.5 which have been built on the p-y-y coincidences. These nuclei, together with other light Ce isotopes, form smooth systematic curves of the E2 and the E /E2. Especially the E2 of 142 keV and the E /Ez of 3.15 in Ce-124 are the... 

Finally we ask further question Can we find any more wel1-deformed nuclei in this region We may think of more proton-rich nuclei such as Sm-132, Nd=126 and Ce-122. However these nuclei are situated on the verge of the proton dripping so that their yields are a few mb or less. The observation of in-beam gamma-rays from them requires a higher quality of charged-particle multiplicity filter. 

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. 

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