¹⁰⁶Pd: a typical spherical-like nucleus

The spherical nucleus, as well as its surface vibrational excitations, as one of the main paradigms of the collective excitations in nuclear structure, has been believed for decades. However, this traditional idea has been questioned recently (spherical nucleus puzzle) Garrett10 ; Heyde11 ; Garrett16 ; Heyde16 ; Garrett18 . Based on the experimental data from the Cd nuclei (Cd puzzle) Garrett08 ; Garrett12 ; Batchelder12 ; Garrett19 ; Garrett20 , especially on the observation that there is no the $0_{3}^{+}$ state at the three-phonon levels, I proposed the concept of a spherical-like nucleus Wang22 , which is a $\gamma$-soft rotational mode with spherical-like spectra for the low-lying part. The experimental analysis of the existence of the spherical-like spectra can be seen in Ref. Wang25 . The interacting boson model with SU(3) higher-order interactions (SU3-IBM) was proposed by me Wang22 to produce the lowest part of the spherical-like spectra successfully Wang22 ; Wang25 .

The SU3-IBM is a new extension of the interacting boson model (IBM) proposed by Arima and Iachello in 1975 Iachello87 ; Iachello75 . In the IBM, discussions of the collective behaviors are based on the three dynamical symmetry limits: the U(5) symmetry limit (spherical shape), the SU(3) symmetry limit (prolate shape), and the O(6) symmetry limit ($\gamma$-soft rotation) Casten10 ; Wang08 . However, in the SU3-IBM, the SU(3) symmetry is the most important, and dominates all the quadrupole deformations Wang22 ; Wang25 . Historically, SU(3) symmetry has played a very important role in the researches of nuclear structure Elliott581 ; Arima69 ; Hecht69 ; Draayer83 ; Deaayer87 ; Isacker85 ; Isacker00 ; zhang14 ; Bonatsos17 ; Kota20 . Now, it becomes more important, or even fundamental.

The SU3-IBM can be also used to explain the B(E2) anomaly Wang20 ; Zhang22 ; Wangtao ; Zhang24 ; Pan24 ; Zhang25 ; Zhang252 , to explain the prolate-oblate shape asymmetric evolution Fortunato11 ; Zhang12 ; Wang23 , to produce the features of the $\gamma$-soft nucleus ¹⁹⁶Pt at a better level WangPt ; ZhouPt , and to produce the E(5)-like spectra in ⁸²Kr Zhou23 . Recently this model can successfully produce the boson number odd-even effect in ^196-204Hg WangHg , which is a unique prediction of the SU3-IBM Zhang12 . Otsuka et al. argue that nuclei previously considered prolate shape should be rigid triaxial Otsuka19 ; Otsuka21 ; Otsuka24 . Recently, we have confirmed this result in ¹⁶⁶Er with the SU3-IBM ZhouEr . These results prove that the SU3-IBM can give more accurate descriptions of the collective behaviors in nuclei than previous theories, and can provide a unified picture for understanding the collective excitations in nuclear structure.

Fig. 1 depicts the spherical-like spectra for $N=7$ which is the boson number of ¹⁰⁶Pd. In Ref. WangPt , the spectra for $N=6$ is also shown. For the low-lying part up to the $12_{1}^{+}$ state, the spherical-like spectra are nearly the same, and resemble the phonon excitations of the spherical nucleus. The characteristics of the spectra are: (1) similar to the two-phonon excitations in the spherical nucleus, the $4_{1}^{+}$, $2_{2}^{+}$, $0_{2}^{+}$ triple states really exist (the first group states); (2) there is no the $0_{3}^{+}$ state near the $6_{1}^{+}$, $4_{2}^{+}$, $3_{1}^{+}$ and $2_{3}^{+}$ states (the second group states), which is discussed in detail in Wang25 ; (3) the $0_{3}^{+}$ state is near the $8_{1}^{+}$, $6_{2}^{+}$, $5_{1}^{+}$, $4_{3}^{+}$, $4_{4}^{+}$ and $2_{4}^{+}$ states (the third group states) whose energy is nearly twice the one of the $0_{2}^{+}$ state (two-times relationship). To the best of my knowledge, this spectra have never been found in previous models in nuclear structure.

The (1) and (2) features can be found in ¹²⁰Cd Batchelder12 and ¹¹²Cd Garrett19 ; Garrett20 experimentally, and can be well described by the SU3-IBM Wang22 ; Wang25 . Especially for the normal states of ^108-120Cd, a systematic fitting has been performed with a single Hamiltonian, and the anomalous evolution trend of the quadrupole moments of the $2_{1}^{+}$ states is first illustrated by the theory Wang25 . To the best of my knowledge, this phenomenon has not been mentioned in previous studies. However the higher levels $8_{1}^{+}$, $6_{2}^{+}$, $5_{1}^{+}$, $4_{3}^{+}$, $4_{4}^{+}$, $2_{4}^{+}$ and $0_{3}^{+}$ (the third group states) and $10_{1}^{+}$, $8_{1}^{+}$, $7_{1}^{+}$, $6_{3}^{+}$, $5_{2}^{+}$, $6_{4}^{+}$, $4_{5}^{+}$, $3_{2}^{+}$, $2_{5}^{+}$ and $0_{4}^{+}$ (the fourth group states) are not confirmed so far, and the feature of the energy positions of the $0^{+}$ states in the spherical-like spectra is also not found in actual nuclei.

Observing these features experimentally is extremely critical to confirm the existence of spherical-like nucleus. The lowest part (up to the $6_{1}^{+}$, $4_{2}^{+}$, $3_{1}^{+}$ and $2_{3}^{+}$ states) only gives the partial characteristics of the new pattern, which is not comprehensively enough, and other possibilities may exist. A direct comparison between experiments and theories is still needed and extremely critical. This can convince the researchers in the field of nuclear structure that there is indeed a new collective excitation, not far from the magic nuclei.

Due to the shape coexistence Heyde11 ; Heyde16 ; Garrett22 ; Bonatsos232 ; Leoni24 , identification of the $0^{+}$ states of the normal states of the spherical-like nucleus is a critical step. Fig. 2(a) shows the evolutional behaviors of the lowest six $0^{+}$ states in ^108-120Cd. In the experiments, the researchers have confirmed that two $0^{+}$ states in ^108-120Cd (red and blue colour) do not belong to the normal states, but are the bandheads of the first and second intruder states respectively Garrett19 ; Garrett20 (please also see the experimental analysis in Wang25 ). It is clear that the red and blue connected lines of the two $0^{+}$ states show the parabolic feature, a sign of the intruder state Heyde11 ; Heyde16 ; Garrett22 ; Bonatsos232 ; Leoni24 . ^100,102Cd have more $0^{+}$ states, but they are not easily identified for the multi-shape coexistence Garrett19 ; Garrett20 and the weak coupling with the normal states Garrett12 . In ^118,120Cd, the coupling interaction can be ignored Wang25 , but the higher $0^{+}$ states are absent. Therefore, in the Cd nuclei, the spherical-like spectra cannot be obtained explicitly.

Further analysis of the nearby nuclei is needed. The effects of the intruder states on the normal states can be ignored, and the experimental data should be more. I find that ¹⁰⁶Pd satisfies these conditions. The boson number is $N=7$, which is the same as the ones in ¹¹⁰Cd and ¹¹⁸Cd. In Fig. 2(b), the lowest six $0^{+}$ states in ¹¹⁰Cd, ¹¹⁸Cd and ¹⁰⁶Pd are shown. In ¹⁰⁶Pd, the $0_{3}^{+}$ intruder state Peters16 ; Marchini22 is much higher than the $0_{2}^{+}$ state, which is like the case in ¹¹⁸Cd. Thus the coupling interaction between the normal states and the intruder states can be also ignored in ¹⁰⁶Pd. Importantly, compared with ¹¹⁸Cd, the $0_{4}^{+}$ state does not belong to the normal states too, which is not mentioned in previous studies.

If the normal states of ¹⁰⁶Pd show the spherical-like spectra, there must be a $0^{+}$ state with energy that is twice the one of the $0_{2}^{+}$ state (two-times relationship). I indeed find this $0^{+}$ state, see the $0_{5}^{+}$ state of ¹⁰⁶Pd in Fig. 2(b). The fit below will show that this is not a coincidence.

In the SU3-IBM only the U(5) symmetry limit and the SU(3) symmetry limit are considered. Various $\gamma$-softness can emerge from the SU3-IBM Wang22 . The Hamiltonian is as follows WangPt ; Wang25

where $c$, $\eta$, $\alpha$, $\beta$, $\gamma$, $\delta$ are fitting parameters. $\hat{Q}$ is the SU(3) quadrupole operator. $\Omega$ is $[\hat{L}\times\hat{Q}\times\hat{L}]^{(0)}$ and $\Lambda$ is $[(\hat{L}\times\hat{Q})^{(1)}\times(\hat{L}\times\hat{Q})^{(1)}]^{(0)}$. $\hat{C}_{2}[SU(3)]$, $\hat{C}_{3}[SU(3)]$ are the SU(3) second- and third-order Casimir operators, respectively.

The parameters of the spherical-like spectra in Fig. 1 are $\eta=0.5$, $\alpha=\frac{3N}{2N+3}$, $\beta=\gamma=\delta=0$, which is the simplest situation. To better fit the actual spectra characteristics in ¹⁰⁶Pd, these parameters should have some small changes. In this letter, the parameters are $\eta=0.47$, $\alpha=\frac{3N}{2N+3}$, $\beta=0$, $\gamma=1.728$, $\delta=1.34$ and $c=1177.42$ keV. The $\hat{L}^{2}$ interaction should be added, and its coefficient is 37.04 keV. The $\eta$ is determined to allow the energies of the $0^{+}$ states to agree with the experimental data. The $\gamma$ and $\delta$ are to match the value of $B(E2;0_{2}^{+}\rightarrow 2_{1}^{+})$.

Fig. 3 shows the experimental and theoretical results of the low-lying levels of the normal states in ¹⁰⁶Pd, up to the $10_{1}^{+}$ state and under 4000 keV. The degree of agreement between theory and experiment is striking. Most of the levels are almost uniformly located, except that the positions of the red levels are slightly higher. Experimentally, the three blue levels ($8^{+}$, $7^{+}$, $6^{+}$) have not been found yet, and are only marked according to the features of the level bands, but they can be believed to exist. Clearly, the third group levels $8_{1}^{+}$, $6_{2}^{+}$, $5_{1}^{+}$, $4_{3}^{+}$, $4_{4}^{+}$, $2_{4}^{+}$ and $0_{3}^{+}$ and the fourth group levels $10_{1}^{+}$, $8_{1}^{+}$, $7_{1}^{+}$, $6_{3}^{+}$, $5_{2}^{+}$, $6_{4}^{+}$, $4_{5}^{+}$, $3_{2}^{+}$, $2_{5}^{+}$ and $0_{4}^{+}$ really exist and can be easily found. (The order of labeling in the experiment agrees with Ref. Pd17 ) The theoretical five lowest $0^{+}$ states all have the experimental correspondences.

Table I shows the absolute $B(E2)$ values for $E2$ transitions from the low-lying states in ¹⁰⁶Pd. Qualitatively, the results of the theory and experiment fit at a good level, and there is no complete inconsistency. For the strong $E2$ transitions, they almost fit very well. The value of $B(E2;0_{2}^{+}\rightarrow 2_{2}^{+})$ (19(+7-3) W.u.) seems smaller than the theoretical result (84.6 W.u.), but in ¹⁰⁸Pd, it’s 47(+5-11) W.u. and in ^112-116Cd, they are 99(16), 127(16) and even $3.0\times 10^{4}$(8) W.u. respectively. Overall, the new theory fits better than the IBM-2 Giannatiempo98 ; Giannatiempo18 . In the IBM-2, the value of $B(E2;2_{3}^{+}\rightarrow 0_{2}^{+})$ (12.3 keV) is much smaller than the experimental result (39(4) W.u.). In ^108,110Pd, they are 35(+14-15) and 160(40) respectively. Clearly the values of $B(E2;4_{2}^{+}\rightarrow 3_{1}^{+})$, $B(E2;2_{5}^{+}\rightarrow 3_{1}^{+})$, $B(E2;2_{5}^{+}\rightarrow 2_{3}^{+})$ and $B(E2;0_{5}^{+}\rightarrow 2_{3}^{+})$ are all smaller. I look forward to more precise experiments with ¹⁰⁶Pd. Table II shows the summing of $B(E2;2_{i}^{+}\rightarrow 0_{2}^{+})$, $B(E2;2_{i}^{+}\rightarrow 2_{2}^{+})$, $B(E2;4_{i}^{+}\rightarrow 2_{2}^{+})$, $B(E2;3_{i}^{+}\rightarrow 2_{2}^{+})$, $B(E2;2_{i}^{+}\rightarrow 4_{1}^{+})$, $B(E2;4_{i}^{+}\rightarrow 4_{1}^{+})$, $B(E2;3_{i}^{+}\rightarrow 4_{1}^{+})$, which are also discussed in Ref. Pd17 . Clearly, the new theory fits the best. The quadrupole moments are sensitive to the wave functions of the low-lying states. Table III shows the quadrupole moments of the $2_{1}^{+}$, $4_{1}^{+}$, $6_{1}^{+}$, $2_{2}^{+}$ and $2_{3}^{+}$ states. Clearly the new theory also agrees with the experimental result very well, and better than the IBM-2. From Table I-III, the deviation of the IBM-2 is about 1.36 times that of the new theory.

From the results of these fits, it can be seen that the spherical-like spectra do exist and do better than the IBM-2. The IBM-2 is insufficient to understand the systematic behaviors in the Cd nuclei Garrett08 ; Garrett12 , the B(E2) anomaly Garahn16 and the prolate-oblate shape asymmetric evolutions Wang23 . So it is insufficient for understanding the collective behaviour of atomic nuclei. In conclusion, we can obtain the key conclusions of this letter. This fit directly confirms the existence of the spherical-like spectra and denies the possibility of the existence of the spherical nucleus in Cd-Nd nuclei region.

Table IV shows the possible two-times relationships of the $0^{+}$ states in other candidates with spherical-like spectra. The possible $0_{3}^{+}$ state, whose energy is nearly twice the one of the $0_{2}^{+}$ state, are shown. Its energy is also near the ones of $5_{1}^{+}$ and $8_{1}^{+}$ states. These will be discussed in future, and with the configuration mixing calculations Garrett08 ; Garrett12 .

A conclusion can now be given that ¹⁰⁶Cd is indeed the spherical-like nucleus. The levels of the states up to $10_{1}^{+}$ state under 4000 keV are all verified. The theoretical results of the B(E2) values and the quadrupole moments fit with the experimental data at a good level, and outperforms the IBM-2. To better include more energy levels, the SU3-IBM-2 distinguishing protons and neutrons will be considered. Combined with the fitting results of the SU3-IBM to date, it can be believed that this offers a more accurate description of the collective behaviors in atomic nuclei. Our understanding of the shape evolutions in nuclear structure is undergoing a fundamental shift, just like the descriptions of Heyde and Wood: “The emerging picture of nuclear shapes is that quadrupole deformation is fundamental to achieving a unified view of nuclear structure.” and “a shift in perspective is needed: sphericity is special case of deformation.” Heyde16 .