Partial conservation of seniority and its unexpected influence on E2 transitions in $g_{9/2}$ nuclei

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 significant mixture between the $v=2$ and $v=4$ states may be induced by the cross-orbital non-diagonal interaction matrix elements. Such strong mixture is essential to explain the observed E2 transition properties of $N=50$ isotones $^{96}$Pd and $^{94}$Ru.

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 significant mixture between the v = 2 and v = 4 states may be induced by the cross-orbital nondiagonal interaction matrix elements. Such strong mixture is essential to explain the observed E2 transition properties of N = 50 isotones 96 Pd and 94 Ru. One of the greatest challenges in nuclear physics is to understand the regular and simple patterns that emerge from the complex nuclear structure. Among those one can mention the shell structure as a consequence of the strong spin-orbit coupling, which is characterized by nucleons occupying orbitals with different lj values. While the original shell model is mostly built upon independent particle motion, the concept of seniority symmetry has been applied implicitly to account for the strong pairing correlation. The seniority quantum number refers to the minimum number of unpaired particles in a single-j shell for a given configuration |j n ; I with total angular momentum I [1]. The seniority coupling has shown remarkable success in describing the spectroscopy and electromagnetic transition properties of semi-magic nuclei restricted to a single j shell. Of particular interest are nuclei that can be well approximated by the seniority coupling in high j orbitals like 0f 7/2 . For heavier systems, we can mention the neutron-rich 70−78 Ni isotopes [2], the N = 50 and 82 [3] isotones in the 0g 9/2 proton subshell, neutron-rich isotopes 134−140 Sn with in the 1f 7/2 subshell [4] as well as 210−218 Pb in the 1g 9/2 neutron subshell [5].
Seniority remains a good quantum number within a subshell when j ≤ 7/2. All states in such systems can be uniquely specified by the total angular momentum I and seniority v. The interaction matrix elements have to satisfy a number of constraints in order to conserve seniority when j > 7/2. For a subshell with j = 9/2, where all but one two-body matrix elements conserve seniority, the condition reads [1, 6-10] where V J = j 2 ; J|V |j 2 ; J denotes a two-body matrix element and J the angular momentum of a two-particle * chongq@kth.se state |j 2 . The symmetry is broken for most effective interactions (see, e.g., Ref. [11]) in subshells with j ≥ 9/2 where the eigenstates would be admixtures of states with different seniorities. For a system with n = 4 identical fermions in a j = 9/2 shell, there are three I = 4 (and also I = 6) states, which may be constructed so that one state has seniority v = 2 and the other two have seniority v = 4. In principle, those seniority v = 4 states are not uniquely defined and any linear combination of them would result in a new set of v = 4 states. However, it was noticed that in the j = 9/2 shell two special v = 4 states with I = 4 and 6 have good seniority for any interaction [12]. They have vanishing matrix elements with the other v = 2 and v = 4 states, irrespective of two-body interactions used. In other words, those two special v = 4 states are uniquely specified and are eigenstates of any two-body interaction. In the following we those special states and the v = 4 states orthogonal to them as |α and |β , respectively. Detailed descriptions of the problem can be found in Refs. [10,[12][13][14][15][16][17][18][19]]. An analytical proof for such partial conservation of seniority is also given in Refs. [15,18]. In this letter we will show that the existence of partial conservation of seniority in j = 9/2 shells plays an essential role in our understanding of the electric quadrupole transitions of the nuclei involved. Another important objective of this paper is to explore how the unique states mentioned above, which are defined for single-j systems, are influenced by configuration mixing from other neighboring subshells. We will show that a sharp transition from pure seniority coupling to significant mixing between the v = 2 and v = 4, α states may be induced by the cross-orbital non-diagonal interaction matrix elements. Such strong mixture is essential to explain the observed E2 transition properties of N = 50 isotones 96 Pd and 94 Ru. In a similar context, Ref. [20] discussed briefly the consequences of multi-shell calculations for states that are degenerate within a single-j shell.
We will focus on the lightest semi-magic nuclei that involve a j = 9/2 orbital. These include the Ni isotopes arXiv:1810.07747v1 [nucl-th] 17 Oct 2018 between N = 40 and 50 and N = 50 isotones between Z = 40 and 50 (see Ref. [21] for a review on the structure of nuclei in this region). Those nuclei are expected to be dominated by the coupling within the 0g 9/2 shell but the contribution from other neighboring orbitals (including 1p 1/2 , 1p 3/2 , 0f 5/2 ) may also play an important role. A microscopic description of the many-body wave function is provided by the shell model full configuration interaction approach where the superposition of a sufficiently large number of many-body basis states within a given valence model space are considered. As for the N = 50 isotones, there has been many studies within the model spaces that include the g 9/2 orbital, the 1p 1/2 0g 9/2 orbitals as well as the 0f 5/2 1p 3/2 1p 1/2 0g 9/2 orbitals. All our calculations below are done numerically within the full shell model framework with exact diagonalization.
We have done calculations for different (g 9/2 ) 4 systems within the g 9/2 orbital. The calculations are exactly the same for the spectra and E2 transition properties of the four-particle/four hole systems 94 Ru and 96 Pd (and 72 Ni and 74 Ni). In Fig. 1 a detailed calculation is given on the relative E2 transition strengths for a (9/2) 4 system calculated with a seniority-conserving (SC) interaction. Part of the results may also be found in Ref. [14]. The E2 transition matrix elements between states with the same seniority is related to each other as j n vI||E2||j n vI = (2j As a result, the E2 transitions involve v = 2 are mostly weak. On the other hand, as indicated in Fig. 1, the E2 transitions between the two special v = 4, α states and between those states are strong and are proportional to B(E2; 2 + 1 → 0 + 1 ). The transitions between those v = 4 states and the v = 2 states are also expected to be strong. However, those special states are weakly connected to the other v = 4 states.
The lowest-lying spectra for such semi-magic nuclei are usually dominated by low seniority states. The seniority coupling is also associated with the existence of long-lived isomeric states with aligned spin I = 2j − 1 and seniority v = 2 in relation to the diminishing energy gap between the isomer and the I = 2j −3 state and the suppressed E2 transition between the two. The suppression is expected to be maximum when the subshell is half-occupied. A systematic study on those E2 transitions may be found, e.g., in Ref. [22]. The situation for (9/2) 4 systems can be much more complicated since the two α states are also expected to have rather low excitation energies. Analytic expressions have been derived for their energies which depend on the strengths of the matrix elements 0g 2 9/2 |V |0g 2 9/2 J with J = 0 [14]. A schematic plot for the influence of the relative positions of low-lying states on the yrast E2 transition properties are shown in Fig. 2. The low-lying spectroscopy of 72 Ni including the 4 + 2 , 6 + 2 and 8 + 1 states was reported in Ref. [2]. The B(E2; 4 + 1 → 2 + 1 ) value for 72 Ni was measured to be 50(9) e 2 fm 4 in Ref. [23], which indicates that the 4 + 1 state may be mostly of seniority v = 4 (see, also, Fig. 4 in Ref. [10] Figure 1. The E2 transition strengths relative to B(E2; 2 + 1 → 0 + 1 ) (denoted as B20) for a system with four particles (holes) in j = 9/2 shell calculated using a seniority-conserving interaction. The 0 + states and few weakest transitions are not shown for simplicity. One has B(E2; 2 + 1 → 0 + 2 ) = 0.044B20 and B(E2; 2 + 2 → 0 + 2 ) = 0.19B20. The two v = 4, α states do not mix with others for any g 9/2 interaction. Figure 2. Illustration on E2 transitions for the yrast states of a (9/2) 4 system in different scenarios based on E2 transitions from Fig. 1: A. All lowest excited states are dominated by seniority v = 2 configurations with suppressed E2 transitions below them; B. The special v = 4, 6 + state becomes yrast with a large B(E2; 8 + 1 → 6 + 1 ) value, in which situation the 8 + 1 state may not be isomeric; C. Similar to B but with the special v = 4, 4 + state becomes yrast; D. Both special v = 4, 4 + and 6 + states become yrast where a collective-like strong inband E2 transition pattern is formed. E. One may wonder if it is possible to have a strong mixture between the v = 2 and 4 states (see text for details).
A tentative search for the 6 + 2 state in 94 Ru was reported in Ref. [27]. For 94 Ru and 96 Pd, the two α states are expected to be just above the yrast I = 4 and 8 states, respectively, in most of our calculations. The 4 + 2 states in 94 Ru and 96 Pd were also predicted to be lower than 6 + 1 in the pg calculations in Refs. [28,29]. Restricted calculations with the interactions from Ref. [25,30] predict the two v = 4 states to be yrast. When extended to the full f pg space, the 6 + 2 state is calculated to be 35 keV above the 8 + 1 state with the jun45 interaction [30]. The nucleus 94 Ru has an 8 + isomer at 2.644 MeV with a half-life of 71 μs [31]. The isomeric character of this level is a consequence of the significantly suppressed E2 decay and the small energy difference with the 6 + level below it. The E2 transition probabilities in 94 Ru have been calculated in Refs. [28,[32][33][34][35][36].
The existence of the uniquely defined v = 4, α states makes it possible to understand the suppression of B(E2; 8 + 1 → 6 + 1 ) in 94 Ru from a very simple perspective. Since those two states do not mix with others, one can write the wave functions of the seniority v = 2 (here one uses v to denote states with mixed seniorities but are dominated by the configuration with seniority v), show finite values with the same sign as β I 2 for negative V SN C , which eventually lead to a full cancellation of M (E2).
As indicated in Fig. 3, the transition 8 + 2 (v = 4) → 6 + 3 (v = 4) will also be suppressed for the same reason. On the other hand, the g 9/2 matrix elements from the effective interactions for Ni isotopes (jj44Ni and jj44b) [25] also show rather large seniority non-conserving matrix element but with a different sign. In that case, as shown in Fig. 4, the predicted B(E2; 8 + v =2 → 6 + v =2 ) values for 72,74 Ni are much larger than those from other interactions and no cancellation is expected.
In order to explore the influence of the neighboring orbitals, in Fig. 4

I have done calculations with different effective interactions on the transitions
by gradually enlarging the model space. No significant influence from the mixture with those orbitals is seen for the 8 + v =2 → 6 + v =2 transition in 96 Pd (and 74 Ni). Moreover, the opening of the N/Z = 50 shell closures is not expected to influence the E2 transitions in the N = 50 isotones in a significant manner. On the other hand, if the model space is extended to include the p 1/2 orbital, the transitions for 94 Ru and 72 Ni can be influenced by the mixture between |g −6 9/2 J and |p −2 1/2 g −4 9/2 J configurations. This is related to the cancellation as induced by the four-particle and four-hole natures of the two configurations. Such kind of cancellation does not happen for 96 Pd and 74 Ni. This is partly responsible for the fact that the observed transition probability B(E2; 8 + 1 → 6 + 1 ) for 96 Pd is nearly 100 times larger than that of 94 Ru. It is also noticed that, for the same reason, the measured B(E2; 6 + 1 → 4 + 1 ) value for 96 Pd [37] is more than eight times larger than that of 94 Ru.
In Ref. [37], the B(E2; 4 + 1 → 2 + 1 ) value for 96 Pd was measured to be as small as 3.8 e 2 fm 4 , which is significantly suppressed by roughly a factor of seven in comparison with that predicted by a SC interaction. In contrast to those for → 6 + 1 and 6 + 1 → 4 + 1 , that value is expected to be significantly smaller than the that for 94 Ru where the lower limit for B(E2; 4 + 1 → 2 + 1 ) is suggested to be as large as 46 e 2 fm 4 . Such an anomalous suppression can not be reproduced by calculations within the single g 9/2 shell but should be related to the mixing with other shells. In the following I will show that such anomalous transition is related to the unexpected mixture between v = 2 and v = 4, α which is induced by cross-orbital non-diagonal matrix elements of the two-body interaction. A detailed analysis on all related transitions will be presented in a forthcoming paper. Moreover, a dramatic increase in the B(E2; 4 + v =2 → 2 + v =2 ) values of 96 Pd and 94 Ru is seen in Fig. 4 for calculations with the jun45 interaction when the model space is extended to include f 5/2 . Our detailed analysis of the corresponding wave functions shows that this calculated abrupt change is also related to the configuration mixing within g 9/2 induced by non-diagonal E2 transition strengths (divided by the square of the effective charge, in fm 4 ) for 94 Ru, 96 Pd calculated in different model spaces with effective interactions from Refs. [30] (square), [25] (diamond) and [29] (2). The transition 4 + 2 (v = 2) → 2 + 1 vanish with V J=2 p 3/2 p 3/2 g 9/2 g 9/2 ≈ 0.52 MeV. With this interaction strength, a strong mixture between the v = 2 and α configurations is still expected for 4 + 1,2 in 94 Ru.
matrix elements involving f 5/2 . The overlaps between the two special I = 4 and 6, α states with the states constructed from the coupling of two J = 2 pairs |j 2 J=2 ⊗ j 2 J=2 I=4 and two J = 2 and J = 4 pairs |j 2 J=2 ⊗ j 2 J=4 I=6 are as large as α I = 10 √ 255/ √ 25591 ≈ 0.9982 and 2 √ 6783/ √ 27257 ≈ 0.9977, respectively. It means that the cross-orbital configurations of the form |(j 1 j 2 ) ⊗ (g 9/2 ) 2 I=4,6 may overlap largely with the v = 4, α states through the nondiagonal matrix elements V J=2,4 j1j2g 9/2 g 9/2 . Those configurations also show non-zero non-diagonal matrix elements with the v = 2 states. These matrix elements lead to a co-existence of the two v = 2 and 4 configurations which does not happen in calculations within the g 9/2 .
As for I = 4, it is found that the non-diagonal matrix elements with j 1 j 2 = p 2 3/2 , p 1/2 p 3/2 , p 1/2 f 5/2 , p 3/2 f 5/2 coupled to J = 2 can indeed induce significant mixture between the v = 2 and v = 4, α states. But it happens only in a relatively small window of strengths for the two-body matrix elements. As for calculations in Fig. 4, only those from the jun45 interaction (more exactly, the V J=2 p 3/2 f 5/2 g 9/2 g 9/2 element) fall in that window. That is why there is no abrupt change seen in other calculations. It should also be mentioned that those non-diagonal matrix elements V J=2,4 j1j2g 9/2 g 9/2 have very limited influence on the energies of the states of concern. In relation to that, it has always been a challenging task to pin down the sign and the strengths of the non-diagonal interaction matrix elements for the shell-model Hamiltonian which may be approximated from realistic nucleon-nucleon potentials.
In Fig. 5 I evaluated the overlaps between the calculated wave functions and the v = 2 and v = 4, α for the first two 4 + states in 96 Pd in a model space containing orbitals p 1/2, p 3/2 and g 9/2 . That is the minimal space that can induce significant mixture between the two v = 2 and 4 configurations. As indicated in Figs. 4 and 5, no significant mixture between the two components is seen in the calculation with the original jun45 in-teraction since the V J=2 p 3/2 p 3/2 g 9/2 g 9/2 interaction is slightly outside the strength window. But a strong mixture between the two v = 2 and 4 configurations is expected for both 4 + 1,2 if the interaction got more repulsive. The transition pattern shown in Fig. 5 gives us an unique opportunity to understand the 4 + → 2 + E2 transitions of 96 Pd and 94 Ru as measured in Ref. [37,38]: The E2 transition in 96 Pd corresponds to a vanishing 4 + v =2 → 2 + v =2 transition seen in right-hand side of Fig.  5 while the large E2 transition in 94 Ru indicates that the nucleus is indeed located in the transitional region where the transition strength is very sensitive to the mixture of the two configurations.
To summarize, in this work I present a novel analysis on the electric quadrupole transition properties of semimagic nuclei with four particles or four holes in the g 9/2 orbital from a partial seniority conservation perspective. This is related to the existence of uniquely defined v = 4 states which, for systems within a j = 9/2 subshell, do not mix with other states. It is shown that the diminishing B(E2; 8 + 1 → 6 + 1 ) in 94 Ru can be mostly understood as the cancellation between few terms induced by the seniority-non-conserving interaction. Moroever, I stud-ied the influence of the neighboring 1p 1/2 , 1p 3/2 , 0f 5/2 orbitals. It is seen that the cross-orbital interaction matrix elements can induce significant mixture between the v = 2 and the unique αstates. The limited experimental information available do indicate that such a sharp phase transition can be seen in nuclei like 96 Pd and 94 Ru. In the future, besides the measurement on the predicted states and E2 transitions mentioned in the present work, it can also be of great interest to explore other j = 9/2 nuclei, including the N = 82 isotones and neutron-rich Pb isotopes, with different two-body interaction strengths and different neighboring orbitals to get a better understanding of such phase transitions.