The recently proposed spin-aligned neutron-proton pair coupling scheme is studied within a non-orthogonal basis in term of the multistep shell model. This allows us to identify simultaneously the roles played by other configurations such as the normal pairing term. The model is applied to four-, sixand eight-hole N = Z nuclei below the core 100 Sn.Keywords: Spin-aligned neutron-proton pair, Multistep shell model, 0g 9/2 shell Many features in nuclear structure physics can be understood in term of the seniority coupling scheme, which was first introduced in atomic physics by Racah [1]. This scheme showed to be extremely useful for the classification of nuclear states in the jj-scheme [2, 3, 4], particularly in semimagic nuclei with only one type of nucleons. The lowest-seniority pair (with v = 0) has nothing special from a coupling point of view since the nuclear state can then be constructed in a variety of equivalent ways through other pairs. In particular, the aligned like-nucleon pair coupling was proposed in Ref. [5], which may manifest itself from the energy differences of mirror nuclei [6]. The driving force behind the dominance of seniority coupling is the strong pairing interaction between like particles.The neutron-proton (np) correlation breaks the seniority symmetry in a major way. Correspondingly, the wave function is a mixture of many components with different seniority quantum numbers. It is not clear yet how
For a system with three identical nucleons in a single-j shell, the states can be written as the angular-momentum coupling of a nucleon pair and the odd nucleon. The overlaps between these nonorthonormal states form a matrix that coincides with the one derived by Rowe and Rosensteel [Phys. Rev. Lett. 87, 172501 (2001)]. The propositions they state are related to the eigenvalue problems of the matrix and dimensions of the associated subspaces. In this work, the propositions are proven from the symmetric properties of the 6j symbols. Algebraic expressions for the dimension of the states, eigenenergies, as well as conditions for conservation of seniority can be derived from the matrix.
A partial conservation of the seniority quantum number in j = 9/2 shells has been found recently in a numerical application. In this paper a complete analytic proof for this problem is derived as an extension of the work by Zamick and P. Van Isacker [Phys. Rev. C 78 (2008) 044327]. We analyze the properties of the non-diagonal matrix elements with the help of the one-particle and two-particle coefficients of fractional parentage (cfp's). It is found that all non-diagonal (and the relevant diagonal) matrix elements can be re-expressed in simple ways and are proportional to certain one-particle cfp's. This remarkable occurrence of partial dynamic symmetry is the consequence of the peculiar property of the j = 9/2 shell, where all v = 3 and 5 states are uniquely defined.The concept of seniority quantum number in many-body systems has played a very important role since its inception by Racah [1]. It refers to the minimum number of unpaired particles in a single-j shell for a given configuration |j n ; I where I is the total angular momentum. In nuclear physics it has classified the influence of the pairing force on nuclear spectra [2,3]. But this concept is nowadays been applied in a variety of fields, including Bose-Einstein condensates [4]. It is established that seniority is a good quantum number for systems with identical fermions in shells with j ≤ 7/2. All states in these systems can be uniquely specified by the total angular momentum I and seniority v. Unfortunately seniority symmetry breaks in shells with j ≥ 9/2. Efforts have been made to find cases for which that symmetry is partially fulfilled. It is thus found that the rotationally-invariant interaction has to satisfy a number of constraints in order to conserve seniority [3]. The conservation conditions are not satisfied by most general two-body interactions for which the eigenstates would be admixtures of states with different seniorities. However, it was noted that in j = 9/2 shell two special eigenstates with I = 4 and 6 have good seniority for an arbitrary interaction [5,6]. The states are eigenstates of any spherically symmetric two-body interaction. They exhibit partial dynamic symmetry and the solvability property (for details, see, e.g., Refs. [7,8]). The problem has been described in Refs. [5][6][7][8][9][10] in a variety of ways and will only be briefly presented here for completeness. For a system with n = 4 identical fermions in a j = 9/2 shell, The I = 4 (and I = 6) states may be constructed so that one state has seniority v = 2 (denoted as |j 4 , v = 2, I in the following) and the other two have seniority v = 4 (denoted as |j 4 , α 1 , v = 4, I and |j 4 , α 2 , v = 4, I where the index α symbolizes any additional quantum number needed when there are more than one state with a given seniority v and total angular momentum I. The seniority v = 4 states are not uniquely defined and any linear combination of them would result in a new sets of v = 4 states. The corresponding Hamiltonian matrix elements can be written as linear combinations of th...
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