Recent studies show that for systems with four identical fermions in the $j=9/2$ shell two special states, which have seniority $v=4$ and total spins I=4 and 6, are eigenstates of any two-body interaction. These states have good seniority for an arbitrary interaction. In this work an analytic proof is given to this peculiar occurrence of partial conservation of seniority which is the consequence of the special property of certain coefficients of fractional parentage. Further calculations did not reveal its existence in systems with other $n$ and/or $I$ for shells with $j\leq15/2$.
Generalized seniority provides a truncation scheme for the nuclear shell model, based on pairing correlations, which offers the possibility of dramatically reducing the dimensionality of the nuclear shell-model problem. Systematic comparisons against results obtained in the full shell-model space are required to assess the viability of this scheme. Here, we extend recent generalized seniority calculations for semimagic nuclei, the Ca isotopes, to open-shell nuclei, with both valence protons and valence neutrons. The even-mass Ti and Cr isotopes are treated in a full major shell and with realis…
In the present work available experimental data up to high-spin states of $^{119-126}$Sn isotopes with different seniority ($v$), including $v$ = 4, 5, 6, and 7 have been interpreted with shell model, by performing full-fledged shell model calculations in the 50-82 valence shell composed of $1g_{7/2}$, $2d_{5/2}$, $1h_{11/2}$, $3s_{1/2}$, and $2d_{3/2}$ orbitals. The results have been compared with the available experimental data. These states are described in terms of broken neutron pairs occupying the $h_{11/2}$ orbital. Possible configurations of seniority isomers in these nuclei are discus…
In this paper, we investigate the generalized seniority scheme and the validity of Generalized Seniority Schmidt Model in and around the $Z=82$ semi-magic region. A consistently same multi-j configuration is used to explain all the nuclear spectroscopic properties such as $g-$factors, $Q-$moments and $B(E2)$ trends for the ${13/2}^+$, ${12}^+$ and ${33/2}^+$ isomers in all the three Hg, Pb and Po isotopic chains. The inverted parabolic $B(E2)$ trends for the first $2^+$ states in Hg, Pb and Po isotopes are also explained using the generalized seniority scheme. A comparison with the experimenta…
Analytic proof of partial conservation of seniority in $j=9/2$ shells
最近在一个数值应用中发现了$j=9/2$壳的量子数的部分守恒性。在本文中，作为Zamick和P. Van Isacker [Phys. Rev. C 78（2008）044327]工作的扩展，导出了这个问题的完整解析证明。我们利用分数双亲的单粒子系数和二粒子系数分析了非对角矩阵元素的性质。它被发现，所有的非对角（和相关的对角）矩阵元素可以用简单的方式重新表示，并与某些单粒子cfp的成比例。这种部分动力学对称性的显著发生是$j=9/2$壳层的特殊性质的结果，其中所有$v=3$和5个状态都是唯一定义的。
There exist two uniquely defined $v=4$ states in systems within a $j=9/2$ subshell, which automatically conserve seniority and do not mix with other states. Here I show that the partial conservation of seniority plays an essential role in our understanding of the electric quadrupole transitions of the semimagic nuclei involving $j=9/2$ subshells, including the long-lived $8^{+}$ isomer in $^{94}$Ru. The effects of configuration mixing from neighboring subshells on the structure of those unique states are analysed. It is shown that a sharp transition from pure seniority coupling to a significan…
Even tin isotopes of mass number $A = 108 \sim 124$ are calculated with realistic interactions in the generalized-seniority approximation of the nuclear shell model. For each nucleus, we compute the lowest ten thousand states ($5000$ of each parity) up to around $8$ MeV in excitation energy, by allowing as many as four broken pairs. The lowest fifty eigen energies of each parity are compared with the exact results of the large-scale shell-model calculation. The wavefunctions of the mid-shell nuclei show a clear pattern of the stepwise breakup of condensed coherent pairs with increasing excitat…
We study the quadrupole moments and the B(E2; $2^+ \rightarrow 0^+$) values for the ${11/2}^-$ states and the first $2^+$ states, respectively, by using a multi-j generalized seniority approach in the Cd (Z = 48), Sn (Z = 50) and Te (Z = 52) isotopic chains. The g-factor trends have also been discussed. Although, Cd and Te isotopes represent two-proton hole and two-proton particle systems, thus involving both kind of particles (protons and neutrons) in contrast to Sn (Z = 50) where only neutrons play a role, we find that a similar model based on neutron valence space alone is able to explain n…
The neutron-rich nuclei 130Cd and 206Hg, so important in the astrophysical processes, may also be useful in tracking the evolution of nuclear shell gaps as one traverses the neutron-rich region. The high spin 8+ isomer in 130Cd and the 10+ isomer in 206Hg turn out to be the lampposts, which may shed light on the shell gaps and validity of the seniority scheme in the neutron-rich systems. We explore the robustness of the N=82 and N=126 magic numbers in the neutron-rich 130Cd and 206Hg nuclides, respectively. A parallel between the two nuclides in terms of the high-spin isomers allows us to inve…
We present a comparative study of the high-spin ${11/2}^-$, ${10}^+$ and ${27/2}^-$ nuclear isomers, observed commonly in the full semi-magic $Z=50$ isotopic and $N=82$ isotonic chains, which are found to display nearly identical systematics in excitation energy and half-lives. A constant and particle number independent energy gap of $\sim$ 4 MeV between the $0^+$ ground states and ${10}^+$ isomers and, the ${11/2}^-$ and ${27/2}^-$ isomers exists before the mid-shell, which declines near the mid-shell, and then becomes constant at $\sim$ 3 MeV after the mid-shell. Large scale shell model calc…
We present freshly evaluated B$(E2\uparrow;0^+\rightarrow2^+)$ values across the even-even Sn-isotopes which confirm the presence of an asymmetric behavior as well as a dip in the middle of the full valence space. We explain these features by using the concept of generalized seniority. The dip in the B$(E2)$ values near $^{116}$Sn is understood in terms of a change in the dominant orbits before and after the mid shell, which also explains the presence of asymmetric peaks in the B$(E2)$ values. This approach helps in deciding the most active valence spaces for a given set of isotopes, and singl…
The particle-hole symmetry (equivalence) of the full shell-model Hilbert space is straightforward and routinely used in practical calculations. In this work we show that this symmetry is preserved in the subspace truncated at a certain generalized seniority, and give the explicit transformation between the states in the two types (particle and hole) of representations. Based on the results, we study the particle-hole symmetry in popular theories that could be regarded as further truncations on top of the generalized seniority, including the microscopic interacting boson (fermion) model, the nu…
The low-lying structures of the midshell $νg_{9/2}$ Ni isotopes $^{72}$Ni and $^{74}$Ni have been investigated at the RIBF facility in RIKEN within the EURICA collaboration. Previously unobserved low-lying states were accessed for the first time following $β$ decay of the mother nuclei $^{72}$Co and $^{74}$Co. As a result, we provide a complete picture in terms of the seniority scheme up to the first $(8^+)$ levels for both nuclei. The experimental results are compared to shell-model calculations in order to define to what extent the seniority quantum number is preserved in the first neutron $…
We study mode-entanglement in the seniority model, derive analytic formulas for the one-body reduced density matrix of states with seniority $ν= 0,\;1,\;2$, and $ν=3$, and also determine the particle number dependence of the one-body reduced density matrix for arbitrary seniority. We carry out numerical calculations for the lightest calcium isotopes and for $^{94}{\rm Ru}$ nucleus, and investigate the structure of their ground and low energy yrast states. We investigate the fulfillment of several predictions of the seniority model regarding the behavior of one-mode entropies, which we compare …
Isomeric studies in neutron-rich nuclei present a powerful tool to explore the structure at the nuclear extremes. We recently used the shell model calculations with Renormalized Charge Depen- dent Bonn (RCDB) effective interaction to calculate the properties of the $6^+$ seniority isomers in $^{134-138}$Sn in an attempt to resolve the anomalous B(E2) behavior of the $6^+$ isomer in $^{136}$Sn [Phys. Rev. C 91, 024321 (2015)]. We further explore these isomers by using the generalized seniority scheme for multi-j orbitals recently presented by us [Phys. Lett. B 753, 122 (2016)]; the B(E2) values…
There has been a great deal of attention to the low lying energy spectum in a nucleus becuse of the abundance of experimetal data. Likewise ,perhaps to a lesser extent but still significant the high end for a given configuratiion has been examined. Here ,using single j shell calculations as a guide we examine the middle part of the spectum resulting form single j shell calculations.Seniority arguments are used to partially explain the midshell behaviours even though in general seniority is not a good quantum number for mixed systems of neutrons and protons.
Symmetries are known to play an important role in the low lying level structure of Sn isotopes, mostly in terms of the seniority and generalized seniority schemes. In this paper, we revisit the multi-j generalized seniority approach for the first excited $2^+$ and $3^-$ states in the Cd ($Z=48$), Sn ($Z=50$) and Te ($Z=52$) isotopes, where the Cd and Te isotopes represent two-proton hole and two-proton particle nuclei, thus involving both kind of particles (protons and neutrons) in contrast to Sn isotopes. Interestingly, the approach based on neutron valence space alone is able to explain the …
Effective interactions and shell model studies of heavy tin isotopes
本文给出了在1f ~（7/2），2 p ~（3/2），0 h ~（9/2），2 p ~（1/2），1f ~（5/2），0 i ~（13/2）单粒子轨道定义的壳模型空间下，对134 ~ 142}Sn的偶、奇锡同位素的大尺度壳模型计算结果.采用基于现代核子-核子相互作用的有效两体相互作用。壳模型的结果反过来分析其配对的内容，使用广义的资历方法。我们的研究结果表明，一个配对模型的图片捕捉了大量的结构和最低躺态的偶，奇同位素的相关性。
In nuclei with valence nucleons are say identical nucleons and say these nucleons occupy several-$j$ orbits, then it is possible to consider pair creation operator $S_+$ to be a sum of the single-$j$ shell pair creation operators $S_+(j)$ with arbitrary phases, $S_+=\sum_j α_j S_+(j); α_j=\pm 1$. In this situation, it is possible to define multi-orbit or generalized seniority that corresponds to the quasi-spin $SU(2)$ algebra generated by $S_+$, $S_-=(S_+)^\dagger$ and $S_0=(\hat{n} -Ω)/2$ operators; $\hat{n}$ is number operator and $Ω=[\sum_j (2j+1)]/2$. There are now multiple pairing quasi-s…
Conservation laws in the $1f_{7/2}$ shell model of $^{48}$Cr found in numeric studies by Escuderos, Zamick and Bayman [A. Escuderos, L. Zamick, and B. F. Bayman, arXiv:0506050 (2005)] and me [K. Neergård, Phys. Rev. C \textbf{90}, 014318 (2014)] are explained by symmetry under particle-hole conjugation and the structure of the irreps of the symplectic group Sp(4). A generalization is discussed.
Cooperation of anti-aligning and aligning shell-model forces for N=Z
对于单个J壳层中的两个中子和两个质子或两个中子空穴和两个质子空穴，|具有同位旋和资历为零和最低角动量为零的状态|由具有角动量2 j的准核子对的吸引相互作用产生的chi>分别对所有相关j和具有角动量2 j和0的准核子对的大含量具有大的重叠。在1f 7/2和1g 9/2壳层中，这两个通道中有效相互作用的大的负矩阵元相对于其余的大部分，因此_合作_产生基本上是以下的线性组合的基态：|phi>和|具有可比系数。其他通道中的相互作用矩阵元素对这些系数的比值有显著影响。国|占计算的基态的约80%。后者与|chi> are_less_ in 1g9/2 shell than in 1f7/2 shell.
The neutron-rich $^{213}$Pb isotope was produced in the fragmentation of a primary 1 GeV $A$ $^{238}$U beam, separated in FRS in mass and atomic number, and then implanted for isomer decay $γ$-ray spectroscopy with the RISING setup at GSI. A newly observed isomer and its measured decay properties indicate that states in $^{213}$Pb are characterized by the seniority quantum number that counts the nucleons not in pairs coupled to angular momentum $J=0$. The conservation of seniority is a consequence of the Berry phase associated with particle-hole conjugation, which becomes gauge invariant and t…
Recent observation of the 6$^+$ seniority isomers and measurements of the B${(E2)}$ values in the $^{134-138}$Sn isotopes lying close to the neutron drip line have raised some questions about the validity of the currently used effective interactions in the neutron-rich region. Simpson $\it{et}$ $\it{al.}$ [Phys. Rev. Lett. $\textbf{113}$, 132502 (2014)] had to modify the diagonal and non-diagonal $νf_{7/2}^2$ two body matrix elements of the "${Vlk}$" interaction by ${\sim150}$ keV in their shell model calculations in order to explain the data of $^{136}$Sn. In contrast, we are able to explain …
We report the discovery of a partial conservation law obeyed by a schematic Hamiltonian of two protons and two neutrons in a j shell. In our Hamiltonian the interaction matrix element of two nucleons with combined angular momentum J is linear in J for even J and constant for odd J. It turns out that in some stationary states the sum J_p + J_n of the angular momenta J_p and J_n of the proton and neutron pairs is conserved. The energies of these states are given by a linear function of J_p + J_n. The systematics of their occurrence is described and explained.
The nucleon-pair approximation (NPA) can be a compact alternative to full configuration-interaction (FCI) diagonalization in nuclear shell-model spaces, but selecting good pairs is a long-standing problem. While seniority-based pairs work well for near-spherical nuclides, they do not work well for deformed nuclides with strong rotational bands. We propose an alternate approach. We show how one can write any Slater determinant for an even number of particles as a general pair condensate, from which one can project out pairs of good angular momentum. We implement this by generating unconstrained…
In an old paper [ 1] B.H. Flowers discussed calculations for odd A nuclei in the g9/2 shell. He finds when he varies the parameters of a certain interaction that the lowest energy state is always a seniority v=1 state with angular momentum J=9/2+. However experiments show about half the states have J=9/2+ lower and the other ones have J=7/2+ lower. More recently we considered selected nuclei between Z=40 and Z=50. For lower odd N Zr isotopes we find J=7/2+ comes below J=9/2+ while for odd Z isotones near Z=50 the opposite is true i.e. J=9/2+ comes below J= 7/2+.We justify using different inter…
The variational reduced density matrix theory has been recently applied with great success to models within the truncated doubly-occupied configuration interaction space, which corresponds to the seniority zero subspace. Conservation of the seniority quantum number restricts the Hamiltonians to be based on the SU(2) algebra. Among them there is a whole family of exactly solvable Richardson-Gaudin pairing Hamiltonians. We benchmark the variational theory against two different exactly solvable models, the Richardson-Gaudin-Kitaev and the reduced BCS Hamiltonians. We obtain exact numerical result…
