1 Introduction

Nuclear low-lying spectra and electromagnetic properties are highly regulated with simple patterns. For example, doubly even nuclei always have Iπ=0⁺ ground states and Iπ=2⁺ first excited states, with few exceptions. Another example is that $2_1^{+}$ quadrupole moments of doubly even nuclei are almost proportional to the $2_1^{+}$ ones with a unified ratio of -1 across the whole nuclide chart [1]. These regularities are also predominant in an ensemble of the nuclear structural model, where all the two-body interaction matrix elements are random numbers [2, 3]. Such predominant patterns in random-interaction ensembles demonstrate how simple regularity emerges out of complex nuclei, even with interactions mostly deviating from reality [4-8]. It is also feasible to summarize the interaction property of certain robust nuclear patterns via an interaction sampling for such patterns within random-interaction ensembles.

As recently reported, the Q and μ values of the Iπ=11/2^- isomers of the neutron-rich cadmium isotopes are also regulated by simple linear systematics [9], where the Q linear systematics is attributed to the seniority scheme beyond the neutron h_11/2 orbit. Several theoretical efforts to understand such linear systematics have been made, such as the BCS with quadrupole-quadrupole force [10], the density functional theory [11], and the schematic shell-model description [12]. In particular, Ref. [11] further attributed the simpler Q linear systematics to pairing. However, it is still a challenge to explain how such simplicity exists in complex Cd isotopes [13], and whether there are other possible explanations besides the seniority scheme or pairing. As mentioned above, a random-interaction study may help to answer such questions. Therefore, our work aims to probe the potential linear Q and μ systematics in random-interaction ensembles, and clarify its interaction properties, if any. Previous random-interaction studies normally focused on the simple regularities of an individual doubly even nucleus. Therefore, random-interaction calculation is an innovative approach for the study on the systematics of odd-mass isotopes.

2 Calculation Framework

Our random-interaction calculations are based on the Shell Model, and include all the odd-mass Cd isotopes with A = 113-129. The single-particle space is limited in the Z=40-50 and N=64-82 shells with the π0g_9/2, ν2s_1/2, ν1d_3/2 and ν0h_11/2 single-particle orbits. No further truncation is introduced. Degenerate single-neutron energies are adopted as in Ref. [9]. The two-body interaction is randomized within the two-body random ensemble (TBRE) [14-16]. In other words, the two-body interaction elements (denoted by $V_{ijkl}^J$) are all Gaussian random numbers constrained by

where the brackets of 〈〉 denote average, J labels the interaction rank, and the i, j, k, and l indexes can equal 1, 2, 3, and 4, corresponding to the g_9/2, s_1/2, d_3/2, and h_11/2 orbits, respectively. For example, $V_{1414}^J$ represents the pn interaction between the g_9/2 and h_11/2 orbits with rank J.

We generate 3 000 000 sets of random two-body interaction elements, and input them into the shell-model code [17] to calculate the Q and μ values of the ${11/2}_1^{-}$ states for all the ^{113 129}Cd isotopes. For the Q calculation, the effective charges are set as eπ=1.5e and eν=0.5e. For the μ calculation, the single-particle g factors are set as gπs=5.586×0.7μN, gπl=1μN, gνs=-3.826×0.7μN, and gνl=0, where the spin g factors are conventionally quenched with a factor of 0.7.

To quantitatively describe the Q (μ) systematics, we introduce the Pearson correlation coefficient [18] [denoted by ρQ (ρμ)] as a measure of the linear correlation between the Q (μ) values and neutron numbers, which has a value between ± 1, where 1, 0, and -1 correspond to perfect positive linear correlation, no linear correlation, and perfect negative linear correlation, respectively. Its magnitude represents the degree of linearity. For instance, the experimental Q values of the Cd Iπ=11/2^- isomers [9] give ρQ=0.997, which is very close to 1, and thus reflects the positive linear Q systematics as observed. In the following random-interaction analysis, we focus on the positive linear systematics with ρQ > 0.9 or ρμ > 0.9, because it agrees with the experimental reality [9].

For the random-interaction ensemble, the distribution of the calculated ρQ and ρμ is adopted to comprehensively describe the degree of Q and μ linearity, as denoted by P(ρQ) and P(ρμ), respectively. Before any further investigation, it should be noted that random numbers can also have a considerable possibility to accidentally exhibit linear systematics, which introduces unexpected interference into our analysis. To eliminate this interference, the calculated P(ρQ) and P(ρμ) should be normalized with the Pearson-coefficient distribution of 9 random numbers.¹ Such a distribution can be taken as the background of the current analysis, and is denoted by P_bkg. The P_bkg is calculated with 25 000 000 sets of 9 random numbers from the normal distribution, and is illustrated in Fig. 1(a).

Figure 1:Full-screen View

(Color online) P(ρQ) in the TBRE and TBRE+pairing, compared with P_bkg (see text for definition). Panel (a) provides a straightforward comparison. In panel (b), the P(ρQ) is normalized with the P_bkg. The error bar represents the statistic error.

3 Linear Q Systematics

The P(ρQ) from the TBRE is shown in Fig. 1(a), and is compared with the P_bkg. The TBRE P(ρQ) is totally different from the P_bkg, and maintains two major peaks around ρQ= ± 1. Thus, the Q values from the TBRE potentially exhibit linear systematics with dominant possibility. We normalize the TBRE P(ρQ) with the P_bkg as shown in Fig. 1(b), where two peaks of $P({\rho }_Q>0.9)/P_{\text{b}kg}\simeq 100$ and $P({\rho }_Q<-0.9)/P_{\text{b}kg}\simeq 50$ emerge, and thus quantitatively demonstrate the predominance of the linear Q systematics in the TBRE.

The linear Q systematics was previously attributed to the seniority scheme beyond the h_11/2 orbit [9]. This may also explain the predominance of the linear Q systematics in the TBRE, because a random-interaction ensemble favors seniority-like behavior [19, 20]. However, several arguments may challenge this explanation. Firstly, the predominance of the seniority scheme in random-interaction ensembles is based on the J=0-ground-state situation of doubly even nuclei, which is inapplicable to the odd-mass Cd isotopes. Secondly, the Cd isotopes under investigation are all non-magic nuclei, whose pairing collectivity is supposedly depressed by pn interaction. Thirdly, the TBRE with degenerate single-particle energies, as adopted in this case, does not necessarily favor the seniority scheme [20]. In other words, the seniority-scheme interpretation of the linear Q systematics in the TBRE is still controversial.

To investigate the relationship between the seniority scheme and linear Q systematics in random-interaction ensembles, we introduce an extra pairing force between like nucleons into the TBRE as

where the pairing strength V₀ is equal to 0.01 to match the magnitude of the TBRE elements. Such a generated random-interaction ensemble is denoted by TBRE+pairing, where the low-lying nuclear states are supposed to be predominately governed by the seniority scheme. In Fig. 1, the P(ρQ) of the TBRE+pairing is presented. It can be seen that it has a shape similar to that of the TBRE; however, the TBRE+pairing P(ρQ > 0.9) is twice of the TBRE one. Thus, the seniority scheme indeed reinforces the positive linear Q systematics as claimed in Ref. [9].

To demonstrate the interaction properties of the positive linear Q systematics, we calculate the average values of $V_{ijkl}^J$, which can reproduce ρQ > 0.9 in the TBRE and TBRE+pairing. These average values are presented as $\langle V_{ijkl}^J\rangle$ in Fig. 2 following the order defined in Table 1. It can be seen that ρQ > 0.9 does not require strong attractive $V_{iijj}^{J=0}$ elements, i.e., the pairing in the TBRE, which means that the pairing may be inessential for the positive linear Q systematics. On the other hand, the TBRE $\langle V_{1414}^J\rangle$ values with ρQ > 0.9 obviously follow the parabolic rule [21] as analytically expressed by

where A, B, and C are constant factors independent of J. A schematic parabola, in blue, is illustrated in Fig. 2 to emphasize the regularity. The parabolic evolution of $V_{1414}^J$ implies the quadrupole nature of the pn interaction between the g_9/2 and h_11/2 orbits [22]. Thus, even if the seniority scheme does not present, the positive linear Q systematics is still evident with quadrupole-like pn interaction in the TBRE. On the other hand, the ρQ > 0.9 value in the TBRE+pairing presents a similar $\langle V_{1414}^J\rangle$ evolution to that in the TBRE. This means that, the quadrupole nature of the pn interaction should be maintained for the positive linear Q systematics, irrespectively of whether the seniority scheme presents or not. Moreover, the theoretical reproduction of the linear Q systematics is always beyond a quadrupole perturbation [10,12]. Therefore, it can be concluded that, in random-interaction ensembles, the quadrupole-like pn interaction is responsible for the predominance of the positive linear Q systematics, and the seniority scheme is not a determinant but a boost for this predominance, at least in the TBRE.

Figure 2:Full-screen View

(Color online) $\langle V_{ijkl}^J\rangle$ for ρQ > 0.9 in the TBRE and TBRE+pairing. All the data are organized in the order defined in Table 1. The gray area highlights the pn interaction between the g_9/2 and h_11/2 orbits as labeled by $V_{1414}^J$, where a schematic parabolic curve of Eq. (3) is provided as a visual guide. The $V_{iijj}^{J=0}$ elements in the TBRE+pairing are most attractive, because of the additionally introduced pairing force defined in Eq. (2). The dashed line represents the ensemble average of the TBRE. The error bar represents the statistic error.

4 Linear μ Systematics

In Fig. 3 the P(ρμ) in the TBRE and TBRE+pairing, normalized with P_bkg, is presented. The TBRE P(ρμ)/P_bkg is always close to 1, which demonstrates that it is not characterized by the linear μ systematics. In the TBRE+pairing ensemble, the pairing significantly increases the P(|ρμ|>0.9), corresponding to the predominance of the linear μ systematics. Thus, the pairing, is essential for the linear μ systematics. However, the P(|ρμ|>0.9)/P_bkg values in the TBRE+pairing are still smaller than the P(|ρQ|>0.9)/P_bkg in both TBRE and TBRE+pairing [see Fig. 1(b)], which means that, the linear μ systematics is always less evident than the Q systematics in random-interaction ensembles, similar to the realistic situation.

Figure 3:Full-screen View

(Color online) P(ρμ) in the TBRE and TBRE+pairing, normalized with P_bkg. The dashed line highlights the exact P_bkg. The error bar represents the statistic error.

As mentioned in Ref. [9], the h_11/2 seniority scheme can only provide a constant μ value independent of the neutron number. Configuration mixing with proton excitation has to be introduced to reproduce the linear μ systematics [12]. To demonstrate the property of such configuration mixing in the TBRE+pairing, we also calculate the average values of $V_{ijkl}^J$, which can reproduce ρμ > 0.9 in the TBRE+pairing, and present them as $\langle V_{ijkl}^J\rangle$ in Fig. 4, following the order defined in Table 1. Similar to the positive linear Q systematics, the positive linear μ systematics also requires some specific structure of $\langle V_{1414}^J\rangle$, which is responsible for the configuration mixing with proton excitation.

Figure 4:Full-screen View

As in Fig. 2, but for ρμ > 0.9 in the TBRE+pairing.

Let’s take a close look at the $\langle V_{1414}^J\rangle$ with ρμ > 0.9 in the TBRE+pairing, as shown in Fig. 5(a). The $\langle V_{1414}^{J=\text{e}ven}\rangle$ evolution presents an obvious parabolic feature against J, while the odd-J evolution seems less regulated. Such an odd-even difference also characterizes the δ force between the g_9/2 proton and h_11/2 neutron. More specifically, the even-J interaction element of the δ force has only the T=0 contribution, according to Eq. (4.15) in Ref. [22], which also follows a parabolic evolution due to the short-range property of the δ force. The odd-J interaction element, on the other hand, includes both the T=0 and T=1 components. The T=1 component diminishes as J → J_max because of the Pauli’s principle, which is totally different from the T=0 evolution. Thus, the odd-J interaction elements present a more complicated evolution than the even-J ones.

Figure 5:Full-screen View

(Color online) $\langle V_{1414}^J\rangle$ for ρμ > 0.9 in the TBRE+pairing. The solid lines represent the fitting to the δ force (see text for detail). Panel (a) provides an overall description, where the $\langle V_{1414}^{J=\text{e}ven}\rangle$ and $\langle V_{1414}^{J=\text{o}dd}\rangle$ follow different regularities, and thus, are presented separately. Panel (b) presents the $\langle V_{1414}^{J=\text{o}dd}\rangle$ excluding the T=0 component of the δ force with the same S^T=0 as for $\langle V_{1414}^{J=\text{e}ven}\rangle$. The error bar represents the statistic error.

To quantitatively demonstrate the pn-interaction property of the positive linear μ systematics in the TBRE+pairing, the $\langle V_{1414}^J\rangle$ in Fig. 5 are further fitted to a unified δ force. Analytically, the even-J interaction element of the δ force between the g_9/2 proton and h_11/2 neutron is written as

and the odd-J interaction element is written as

with

and

where S^T=0 and S^T=1 are the interaction strengths for T=0 and T=1 components, respectively. With S^T=0 and a unified interaction offset as the fitting variables, the $\langle V_{1414}^{J=\text{e}ven}\rangle$ is fitted to Eq. (4) as shown in Fig. 5(a). It can be seen that Eq. (4) describes the $\langle V_{1414}^{J=\text{e}ven}\rangle$ evolution suitably, and the best-fit S^T=0=-0.55(9). If the $\langle V_{1414}^{J=\text{o}dd}\rangle$ is characterized by the same δ force as the $\langle V_{1414}^{J=\text{e}ven}\rangle$, the T=0 component in the $\langle V_{1414}^{J=\text{o}dd}\rangle$ should have a strength of S^T=0=-0.55. Such a T=0 component is excluded in the $\langle V_{1414}^{J=\text{o}dd}\rangle$ according to Eq. (6), and the residual $\langle V_{1414}^{J=\text{o}dd}\rangle -V_{\delta }^{T=0}$ is presented in Fig. 5(b), which should correspond to the T=1 component of a δ force governed by Eq. (7). Thus, the residues are fitted to Eq. (7) with S^T=1 and a unified interaction offset as the fitting variables. The best-fit S^T=1=0.30(6), and the corresponding fitting curve appropriately describes the plots in Fig. 5(b). Therefore, it is concluded that a unified δ force with S^T=0=-0.55 and S^T=1=0.3 characterizes the pn interaction for the positive linear μ systematics in the TBRE+pairing.

According to the best-fit S^T=0 < 0 and S^T=1 > 0, the positive linear μ systematics of the TBRE+pairing favors an attractive T=0 pn interaction and a repulsive T=1 one, and the T=0 component should be stronger than the T=1 one. On the other hand, the realistic pn interaction is also constructed with attractive T=0 and repulsive T=1 components (the T=1 pairing between protons and neutrons is still attractive, but does not apply here), and the T=0 strength is normally stronger [22]. This structural similarity between the pn interaction with ρμ > 0.9 in the TBRE+pairing and the realistic pn interaction suggests that the linear μ systematics may share a same microscopic mechanism in both TBRE+pairing ensemble and realistic nucleus.

5 Examination with Sn Isotopes

According to Sections 3 and 4, the pn interaction is more crucial than the h_11/2 seniority scheme for positive linear Q and μ systematics, in random-interaction ensembles. It would be useful to investigate the impact of pn interaction on Q and μ systematics in a realistic nuclear system. Tin isotopes are chosen as the standard, because they are typical semi-magic nuclei with no pn interaction between the valence nucleons, and thus provide the best platform to study the effect of the absence of pn-interaction on the linearity of the ${11/2}_1^{-}$ Q and μ systematics in a realistic nuclear system.

A plot of the Q and μ systematics of the ${11/2}_1^{-}$ states of the ^{115 131}Sn isotopes is shown in Fig. 6. In Panel (a), the Q systematics seems disorganized, especially near the N=64 and N=82 shell closures, even though the h_11/2 seniority scheme should be mostly robust near the shell closure. Fig. 6(b) presents the near constant (instead of linear) μ systematics of the Sn isotopes around the Schmitt estimation of the neutron h_11/2 orbit, which also supports the h_11/2 seniority scheme for Sn isotopes. Therefore, the absence of linear Q systematics in the Sn isotope chain indicates that the linear Q and μ systematics in Cd isotopes cannot be totally explained by the seniority scheme or pairing. The pn interaction may be also essential for both linear Q and μ systematics even in realistic nuclear system.

Figure 6:Full-screen View

Experimental Q and μ of the ${11/2}_1^{-}$ states of the Sn isotopes versus the mass numbers. All the data is taken from Ref. [23], except the Q value of ¹²⁷Sn, which is taken from Ref. [24], and the μ value of ¹¹⁷Sn, which is taken from Ref. [25]. The uncertainty in the sign of the Q values of ^{119, 115}Sn is represented by open squares in Panel (a). The Schmitt estimation of the neutron h_11/2 orbit is represented by a dashed line to demonstrate the constant μ value of pure h_11/2 seniority scheme. The N=64 and N=82 shell closures are highlighted.

6 Summary

To summarize, random interactions predominantly reproduce the linear Q and μ systematics of the Iπ=11/2^- isomers of the Cd isotopes as already reported in previous experimental work [9]. The importance of the pn interaction in such linear systematics is emphasized in such a Monte Carlo study.

The predominance of the linear Q systematics can be enhanced by the seniority scheme as expected. However, even without introducing seniority scheme, such predominance can still be perceived with a quadrupole-like pn interaction. We also note that the quadrupole collectivity is already emphasized in previous studies on the linear Q systematics [10, 12], which could further support this observation. For the linear μ systematics, the δ-force-like pn interaction and seniority scheme are both essential, and the pn interaction is further required to exhibit a similar structure to realistic nuclear interaction. Therefore, if the δ force plus quadrupole pn interaction is adopted in the shell-model calculation, the Q and μ linear systematics are both expected. A schematic calculation with S^T=0=0.55, S^T=1=-0.3 and χ=-0.1 for pn interaction and S^T=1=-1.0 for pp and nn interaction was performed, where χ is the strength of the quadrupole interaction. The resultant ρQ=0.947 and ρμ=0.833, agree with expectation.

The effect of the pn correlation on the Q and μ systematics in a realistic nuclear system was also discussed. The experimental data of the ${11/2}_1^{-}$ states along the Sn isotope chain also suggests that the pn correlation is the major driving force of both the linear Q and μ systematics even in a realistic nuclear system.