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Projected Shell Model Description for Nuclear Isomers

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Projected Shell Model Description for Nuclear Isomers Projected shell model description for nuclear isomers Yang Sun1,2 1Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, P. R. China 2Joint Institute for Nuclear Astrophysics, University of Notre Dame, Notre Dame, Indiana 46545, USA Nuclear isomer 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. PACS numbers: 21.60.Cs, 21.20.-k INTRODUCTION in those unstable nuclei. It has been pointed out by Xu et al. [10] that in superheavy nuclei, the isomeric states decrease the probability for both fission and α-decay, resulting in en- A nuclear isomer is an excited state, in which a combina- hanced stability for these nuclei. One expects that the isomers tion of nuclear structure effects inhibits its decay and endows in very heavy nuclei could serve as stepping stones toward the isomeric state with a lifetime that can be much longer than understanding the single-particle structure beyond the Z = 82 most nuclear states. Known isomers in nuclei span the entire and N = 126 shell closures, which is the key to locating the range of lifetimes from 1015 years for 180mTa – longer than the anticipated ‘island of stability’ [11]. accepted age of the universe – to an informal rule of thumb on Moreover,nuclear isomers may play a significant role in de- the lower side of approximately 1 ns. Nuclear isomers de- termining the abundances of the elements in the universe [12]. cay predominantly by electromagnetic processes (γ-decay or In hot astrophysical environments, an isomeric state can com- internal conversion). There are also known instances of the municate with its ground state through thermal excitations. decay being initiated by the strong interaction (α-emission) This could alter significantly the elemental abundances pro- or the weak interaction (β-decay or electron capture). Decay duced in nucleosynthesis. The communication between the by proton or neutron emission, or even by nuclear fission, is ground state of 26Al and the first excited isomeric state in this possible for some isomers (see recent examples [1, 2, 3]). nucleus has the consequence that the astrophysical half-life Often discussed in the literature are three mechanisms [4] for 26Al can be much shorter than the laboratory value [13]. leading to nuclear isomerism, although new types of isomer One is just beginning to look at the impact that nuclear iso- may be possible in exotic nuclei [5]. It is difficult for an iso- mers have on various other nucleosynthesis processes such as meric state to change its shape to match the states to which the rapid proton capture process thought to take place on the it is decaying, or to change its spin, or to change its spin ori- accretion disks of binary neutron stars. There are cases in entation relative to an axis of symmetry. These correspond to which an isomer of sufficiently long lifetime (probably longer shape isomers, spin traps, and K-isomers, respectively. In any than microseconds) may change the paths of reactions taking of these cases, decay to the ground state is strongly hindered, place and lead to a different set of elemental abundances [14]. either by an energy barrier or by the selection rules of transi- With rapidly growing interest in the isomer study and in- tion. Therefore, isomer lifetimes can be remarkably long. To creasing possibility of experimental access to isomeric states, π + 72 mention a few examples, an I = 0 excited state in Kr has theoretical effort is much needed. The present paper discusses been foundas a shape isomer [6], a 12+ state in 98Cd has been arXiv:0803.1700v1 [nucl-th] 12 Mar 2008 a Projected Shell Model (PSM) description for nuclear iso- 178 understood as a spin trap [7], and in Hf, there is a famous mers. As isomeric states are a special set of nuclear states, + 16 , 31-year K-isomer [8], which has become a discussion special emphasis is given when these states are treated. In sec- focus because of the proposal of using this isomer as energy tion II of the paper, we present a description for K-isomers, storage [9]. in which K-mixing is emphasized. We point out, however, Detailed nuclear structure studies are at the heart of under- that an extended PSM based on triaxially-deformed basis is standing the formation of nuclear isomers with applications required to describe the strong K-violation. In section III, to many aspects in nuclear physics. The study is particularly shape isomer examples are presented and a perspective how interesting and important for unstable nuclei, such as those in configurations with different shapes can be mixed is outlined. neutron-rich, proton-rich, and superheavy mass regions. Ina Finally, the paper is summarized in section IV. quantum system, the ground state is usually more stable than the excited states. However, the lifetime of ground state of unstable nuclei is short, which makes the laboratory study ex- K-MIXING IN THE PROJECTED SHELL MODEL tremely difficult. In contrast, nuclear isomers in those nuclei may be relatively easy to access experimentally. Furthermore, Many long-lived, highly-excited isomers in deformed nu- the physics may be changed due to the existence of isomers clei owe their existence to the approximate conservation of 2 the K quantum number. The selection rule for an electro- The index σ denotes states with same angular momentum κ ˆ I magnetic transition would require that the multipolarity of and labels the basis states. PMK is the angular-momentum- I,σ the decay radiation, λ, be at least as large as the change in projection operator [15] and the coefficients fκ are weights I,σ the K-value (λ ≥ ∆K). However, symmetry-breaking pro- of the basis states. The weights fκ are determined by diag- cesses make possible transitions that violate the K-selection onalization of the Hamiltonian in the projected spaces, which rule. A microscopic description of K-violation is through the leads to the eigenvalue equation (for a given I) so-called K-mixing in the configuration space. A theoretical σ model that can treat K-mixing has preferably the basis states ∑(Hκκ′ − Eσ Nκκ′ ) fκ′ = 0. (2) that are eigenstates of angular momentum I but labeled by K. κ′ Diagonalization of two-body interactions mixes these states The Hamiltonian and the norm matrix elements in Eq. (2) are and the resulting wavefunctionscontain the information on the given by degree of K-mixing. In this kind of approach, the mixing and I I Hκκ′ = hφκ |Hˆ Pˆ ′ |φκ′ i, Nκκ′ = hφκ |Pˆ ′ |φκ′ i. (3) its consequences are discussed in the laboratory frame rather Kκ Kκ′ Kκ Kκ′ than in a body-fixed frame in which K is originally defined. Angular-momentum-projectionon a multi-qp state |φκ i with a sequence of I generates a band. One may define the rotational energy of a band (band energy) using the expectation values The model of the Hamiltonian with respect to the projected |φκ i φκ ˆ ˆI φκ The projected shell model (PSM) [15, 16] seems to fulfil I Hκκ h |HPKκ Kκ | i Eκ = = . (4) κκ φ ˆI φ the requirement. It is a shell model that starts from a deformed N h κ |PKκ Kκ | κ i basis. In the PSM, the shell-modelbasis is constructedby con- sidering a few quasiparticle (qp) orbitals near the Fermi sur- In a usual approximation with independent quasiparticle faces and performing angular momentum projection (if nec- motion, the energy for a multi-qp state is simply taken as the essary, also particle-number projection) on the chosen con- sum of those of single quasiparticles. This is the dominant figurations. With projected multi-qp states as the basis states term. The present theory modifies this quantity in the fol- of the model, the PSM is designed to describe the rotational lowing two steps. First, the band energy defined in Eq. (4) bands built upon qp excitations. introduces the correction brought by angular momentum pro- jection and the two-body interactions, which accounts for the Suppose that a PSM calculation begins with axially de- couplings between the rotating body and the quasiparticles in formed Nilsson single-particle states, with pairing correla- a quantum-mechanical way. Second, the corresponding rota- tions incorporated into these states by a BCS calculation. This † † tional states (labeled by K) are mixed in the subsequent pro- defines a set of deformed qp states (with aν and aπ being the cedure of solving the eigenvalue equation (2). The energies creation operator for neutrons and protons, respectively) with are thus further modified by configuration mixing. respect to the qp vacuum |0i. The PSM basis is then con- For deformed states with axial symmetry, each of the basis structed in the multi-qp states with the following forms states in (1), i.e. the projected |φκ i,isa K-state. For example, n−1 † † † † † † † † an n-qp configuration gives rise to a multiplet of 2 states, e − e nuclei : {|0i,aν aν |0i,aπ aπ |0i,aν aν aπ aπ |0i, † † † † † † † † with the total K expressed by K = |K1 ± K2 ±···± Kn|, where aν aν aν aν |0i,aπ aπ aπ aπ |0i,...} Ki is for an individual neutron or proton.
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