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Redalyc.Projected Shell Model Description for Nuclear Isomers Revista Mexicana de Física ISSN: 0035-001X [email protected] Sociedad Mexicana de Física A.C. México Sun, Yang Projected shell model description for nuclear isomers Revista Mexicana de Física, vol. 54, núm. 3, diciembre, 2008, pp. 122-128 Sociedad Mexicana de Física A.C. Distrito Federal, México Available in: http://www.redalyc.org/articulo.oa?id=57016055020 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative REVISTA MEXICANA DE FISICA´ S 54 (3) 122–128 DICIEMBRE 2008 Projected shell model description for nuclear isomers Yang Sun Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, P.R. China, Joint Institute for Nuclear Astrophysics, University of Notre Dame, Notre Dame, Indiana 46545, USA. Recibido el 10 de marzo de 2008; aceptado el 7 de mayo de 2008 The study of nuclear isomer properties is a current research focus. To describe isomers, we present a method based on the Projected Shell Model. Two kinds of isomers, K-isomers and shape isomers, are discussed. For the K-isomer treatment, K-mixing is properly implemented in the model. It is found however that in order to describe the strong K-violation more efficiently, it may be necessary to further introduce triaxiality into the shell model basis. To treat shape isomers, a scheme is outlined which allows mixing those configurations belonging to different shapes. Keywords: Shell model; nuclear energy levels. Se estudian las propiedades de isomeros´ nucleares a traves´ del modelo de capas proyectadas. Se discutan isomeros´ K y de forma nuclear. Para discutir las propiedades de los isomeros´ K se tiene que incluir la mezcla de diferentes valores K en el modelo as´ı como la deformacion´ triaxial. A los isomeros´ de forma se pueden tratar en un modelo que permita la mezcla de configuraciones con formas nucleares distintas. Descriptores: Modelo de capas; niveles de energ´ıa nucleares. PACS: 21.60.Cs; 21.10.-k 1. Introduction Detailed nuclear structure studies are at the heart of un- derstanding the formation of nuclear isomers with applica- A nuclear isomer is an excited state, in which a combination tions to many aspects in nuclear physics. The study is partic- of nuclear structure effects inhibits its decay and endows the ularly interesting and important for unstable nuclei, such as isomeric state with a lifetime that can be much longer than those in neutron-rich, proton-rich, and superheavy mass re- most nuclear states. Known isomers in nuclei span the en- gions. In a quantum system, the ground state is usually more tire range of lifetimes from 1015 years for 180mTa – longer stable than the excited states. However, the lifetime of ground than the accepted age of the universe – to an informal rule state of unstable nuclei is short, which makes the labora- of thumb on the lower side of approximately 1 ns. Nuclear tory study extremely difficult. In contrast, nuclear isomers in isomers decay predominantly by electromagnetic processes those nuclei may be relatively easy to access experimentally. (γ-decay or internal conversion). There are also known in- Furthermore, the physics may be changed due to the exis- stances of the decay being initiated by the strong interac- tence of isomers in those unstable nuclei. It has been pointed tion (®-emission) or the weak interaction (¯-decay or elec- out by Xu et al. [10] that in superheavy nuclei, the isomeric tron capture). Decay by proton or neutron emission, or even states decrease the probability for both fission and ®-decay, by nuclear fission, is possible for some isomers (see recent resulting in enhanced stability for these nuclei. One expects examples [1-3]. that the isomers in very heavy nuclei could serve as stepping stones toward understanding the single-particle structure be- Often discussed in the literature are three mechanisms [4] yond the Z=82 and N=126 shell closures, which is the key leading to nuclear isomerism, although new types of isomer to locating the anticipated ‘island of stability’ [11]. may be possible in exotic nuclei [5]. It is difficult for an iso- meric state to change its shape to match the states to which Moreover, nuclear isomers may play a significant role it is decaying, or to change its spin, or to change its spin ori- in determining the abundances of the elements in the uni- entation relative to an axis of symmetry. These correspond verse [12]. In hot astrophysical environments, an isomeric to shape isomers, spin traps, and K-isomers, respectively. In state can communicate with its ground state through thermal any of these cases, decay to the ground state is strongly hin- excitations. This could alter significantly the elemental abun- dered, either by an energy barrier or by the selection rules dances produced in nucleosynthesis. The communication be- of transition. Therefore, isomer lifetimes can be remarkably tween the ground state of 26Al and the first excited isomeric long. To mention a few examples, an I¼=0+ excited state state in this nucleus has the consequence that the astrophys- in 72Kr has been found as a shape isomer [6], a 12+ state in ical half-life for 26Al can be much shorter than the labora- 98Cd has been understood as a spin trap [7], and in 178Hf, tory value [13]. One is just beginning to look at the impact there is a famous 16+, 31-year K-isomer [8], which has be- that nuclear isomers have on various other nucleosynthesis come a discussion focus because of the proposal of using this processes such as the rapid proton capture process thought isomer as energy storage [9]. to take place on the accretion disks of binary neutron stars. PROJECTED SHELL MODEL DESCRIPTION FOR NUCLEAR ISOMERS 123 There are cases in which an isomer of sufficiently long life- constructed in the multi-qp states with the following forms time (probably longer than microseconds) may change the y y y y y y y y paths of reactions taking place and lead to a different set of e ¡ e nuclei :fj0i; aº aº j0i; a¼a¼j0i; aº aº a¼a¼j0i; elemental abundances [14]. y y y y y y y y aº aº aº aº j0i; a¼a¼a¼a¼j0i;:::g With rapidly growing interest in the isomer study and in- y y y y y y y y y y creasing possibility of experimental access to isomeric states, o ¡ o nuclei :faº a¼j0i; aº aº aº a¼j0i; aº a¼a¼a¼j0i; theoretical effort is much needed. The present paper dis- ay ay ay ay ay ay j0i;:::g cusses a Projected Shell Model (PSM) description for nuclear º º º ¼ ¼ ¼ y y y y y y y isomers. As isomeric states are a special set of nuclear states, odd ¡ º nuclei :faº j0i; aº aº aº j0i; aº a¼a¼j0i; special emphasis is given when these states are treated. In y y y y y Sec. 2 of the paper, we present a description for K-isomers, aº aº aº a¼a¼j0i;:::g in which K-mixing is emphasized. We point out, however, odd ¡ ¼ nuclei :fay j0i; ay ay ay j0i; ay ay ay j0i; that an extended PSM based on triaxially-deformed basis is ¼ º º ¼ ¼ ¼ ¼ y y y y y required to describe the strong K-violation. In Sec. 3, shape aº aº a¼a¼a¼j0i;:::g isomer examples are presented and a perspective how con- figurations with different shapes can be mixed is outlined. The omitted index for each creation operator contains labels Finally, the paper is summarized in Sec. 4. for the Nilsson orbitals. In fact, this is the usual way of build- ing multi-qp states [10,17-19]. The angular-momentum-projected multi-qp states, each 2. K-mixing in the projected shell model being labeled by a K quantum number, are thus the build- ing blocks in the PSM wavefunction, which can be generally Many long-lived, highly-excited isomers in deformed nuclei written as owe their existence to the approximate conservation of the X X jÃI,σi = f I,σP^ I jÁ i = f I,σP^ I jÁ i: (1) K quantum number. The selection rule for an electromag- M · MK · · MK· · netic transition would require that the multipolarity of the ·;K·I · decay radiation, ¸, be at least as large as the change in the The index σ denotes states with the same angular momentum ^ I K-value (¸¸¢K). However, symmetry-breaking processes and · labels the basis states. PMK is the angular-momentum- I,σ make possible transitions that violate the K-selection rule. projection operator [15] and the coefficients f· are weights I,σ A microscopic description of K-violation is through the so- of the basis states. The weights f· are determined by diag- called K-mixing in the configuration space. A theoretical onalization of the Hamiltonian in the projected spaces, which model that can treat K-mixing has preferably the basis states leads to the eigenvalue align (for a given I) that are eigenstates of angular momentum I but labeled by K. X σ Diagonalization of two-body interactions mixes these states (H··0 ¡ EσN··0 ) f·0 = 0: (2) and the resulting wavefunctions contain the information on ·0 the degree of K-mixing. In this kind of approach, the mixing The Hamiltonian and the norm matrix elements in Eq. (2) are and its consequences are discussed in the laboratory frame given by rather than in a body-fixed frame in which K is originally ^ ^I ^I defined.
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