# Emerging $\gamma$-softness in 196Pt in the SU3-IBM

Tao Wang<EMAIL_ADDRESS>College of Physics, Tonghua Normal
University, Tonghua 134000, People’s Republic of China  Bing-cheng He
<EMAIL_ADDRESS>Physics Division, Argonne National Laboratory, Lemont, Illinois
60439, USA  Chun-xiao Zhou<EMAIL_ADDRESS>College of Mathematics and
Physics Science, Hunan University of Arts and Science, Changde 415000,
People’s Republic of China  Dong-kang Li<EMAIL_ADDRESS>College of
Physics, Tonghua Normal University, Tonghua 134000, People’s Republic of China
Lorenzo Fortunato<EMAIL_ADDRESS>Dipartimento di Fisica e Astronomia
“G.Galilei”-Università di Padova, via Marzolo 8, I-35131 Padova, Italy  INFN-
Sez.di Padova, via Marzolo 8, I-35131 Padova, Italy

###### Abstract

Recently, it has been argued that a new $\gamma$-soft rotational spectrum
emerges in the interacting boson model with SU(3) higher-order interactions,
opening up new approaches to understand the $\gamma$-softness in realistic
nuclei. In a previous paper, $\gamma$-softness with degeneracy of the ground
and quasi-$\gamma$ bands is observed, which displays a O(5) partial dynamical
symmetry. In this paper, another special point connected with the middle
degenerate point is discussed, which is found to be related with the
properties of 196Pt. This emergent $\gamma$-softness has also been shown to be
important for understanding the prolate-oblate asymmetric shape phase
transition. The low-lying spectra, $B(E2)$ values and quadrupole moments in
196Pt are discussed showing that the new model can account for several
observed features.

##  I Introduction

Recently, an extension of the interacting boson model with SU(3) higher-order
interactions (SU3-IBM for short) was proposed to describe the spherical-like
$\gamma$-soft spectra in 110Cd Wang22 , to explain the puzzling $B(E2)$
anomaly Wang20 ; Zhang22 , to discuss the prolate-oblate asymmetric shape
phase transition in Hf-Hg region wang23 , and to provide an E(5)-like
description for 82Kr zhou23 . O(6) higher-order interactions were found to be
unable to explain the $B(E2)$ anomaly wangtao . These works imply that the
SU(3) symmetry dominates the deformation of a nucleus, and the
$\gamma$-softness in realistic nuclei is an emergent phenomenon, which has a
deep relationship with the SU(3) symmetry.

Cd isotopes show a new $\gamma$-soft rotational behavior, which is unexpected
in standard nuclear structure studies Garrett10 ; Garrett12 ; Batchelder12 ;
Heyde16 ; Garrett18 ; Garrett19 ; Garrett20 . The $B(E2)$ anomaly also seems
uncommon, which rejects conventional theoretical explanations, including the
interacting boson model (IBM-2) calculations based on the SkM∗ energy-density
functional and the symmetry-conserving configuration mixing (SCCM)
calculations 168Os ; 166W ; 172Pt ; 170Os . The successful explanation of
these two abnormal phenomena makes the new theory of SU3-IBM very attractive,
and further exploration of the applications of the theory becomes very
valuable, especially on various phenomena related to $\gamma$-softness in
nuclear spectra. What needs to be emphasized is that these two anomalous
phenomena cannot be explained by the interacting boson model with the O(6)
$\gamma$-softness wangtao . Recent studies on the prolate-oblate asymmetric
shape phase transition revealed that the key ingredient of the new model
SU3-IBM is to describe the oblate shape with the SU(3) third-order interaction
wang23 . $\gamma$-softness comes from the competition between the prolate
shape and the oblate shape, thus in the SU3-IBM the new $\gamma$-softness is
an emergent phenomenon.

The IBM provides an elegant approach to describe the low-lying collective
excited behaviors in nuclear structure Iachello87 . In the simplest IBM-1, the
basic building constituents are the $s$ and $d$ bosons with angular momentum
$l=0$ and $l=2$ respectively, and the collective states of a nucleus can be
spanned by the su(6) algebra. Up to two-body interactions, a consistent-$Q$
(CQ) Hamiltonian adopted in this model is Casten10 ; Wang08

$\hat{H}_{1}=c[(1-\eta)\hat{n}_{d}-\frac{\eta}{N}\hat{Q}_{\chi}\cdot\hat{Q}_{\chi}].$
(1)

Here $\hat{n}_{d}$ is the $d$-boson number operator,
$\hat{Q}_{\chi}=[d^{{\dagger}}\times\tilde{s}+s^{{\dagger}}\times\tilde{d}]^{(2)}+\chi[d^{{\dagger}}\times\tilde{d}]^{(2)}$
is the generalized quadrupole operator, $N$ is the total boson number, $c$ is
a scale parameter and $0\leq\eta\leq
1,-\frac{\sqrt{7}}{2}\leq\chi\leq\frac{\sqrt{7}}{2}$ are parameters that allow
to span a full range of different nuclear spectra. Although the formalism is
simple, it can describe the spherical ($\eta=0$, the U(5) limit), prolate
($\eta=1$, $\chi=-\frac{\sqrt{7}}{2}$, the SU(3) limit), oblate ($\eta=1$,
$\chi=\frac{\sqrt{7}}{2}$, the $\overline{\textrm{SU(3)}}$ case) and
$\gamma$-unstable ($\eta=1$, $\chi=0$, the O(6) limit) nuclei. This
Hamiltonian is extensively used in fitting realistic nuclear spectra and
discussing the shape phase transitions between different shapes Iachello87 ;
Casten10 ; Wang08 .

More than a decade ago, one of the authors (L. Fortunato) and his
collaborators generalized the simple formalism (1), and a cubic-$Q$
interaction is introduced as follows Fortunato11

$\hat{H}_{2}=c[(1-\eta)\hat{n}_{d}-\frac{\eta}{N}(\hat{Q}_{\chi}\cdot\hat{Q}_{\chi}+\frac{\kappa_{3}}{N}[\hat{Q}_{\chi}\times\hat{Q}_{\chi}\times\hat{Q}_{\chi}]^{(0)})],$
(2)

where $\kappa_{3}$ is the coefficient of the cubic term. In the SU(3) limit,
when $\chi=-\frac{\sqrt{7}}{2}$, the cubic interaction can describe an oblate
shape (SU(3) oblate), which is different from the previous
$\overline{\textrm{SU(3)}}$ oblate shape in Hamiltonian (1). This indicates
that the previous description of the oblate shape with
$\overline{\textrm{SU(3)}}$ symmetry can be replaced by the SU(3) symmetry and
a new evolutional path from the prolate shape to the oblate shape can be
established within only the SU(3) limit, see the bottom black line in Fig. 1.
Thus an analytically solvable prolate-oblate shape phase transitional
description within the SU(3) limit can be provided, see Ref. Zhang12 , which
offers a rare example for finite-$N$ first-order quantum shape transition. The
phase transitional point is also a degenerate point Zhang12 , which implies a
hidden symmetry Wang22 . This hidden symmetry is responsible for the whole new
progress in Wang22 ; Wang20 ; Zhang22 ; wang23 ; zhou23 ; wangtao . Moreover,
in this extended Hamiltonian $\hat{H}_{2}$, there is only a very tiny region
of rigid triaxiality in the large-$N$ limit at $\chi=-\frac{\sqrt{7}}{2}$ when
the parameter changes from the U(5) limit to the SU(3) degenerate point, see
the green line in Fig. 1.

These new results presented by Ref. Fortunato11 ; Zhang12  encourage us to
understand the existing experimental phenomena from a new perspective. Some
new and unexpected results have emerged recently. A new shape triangle can be
drawn (see Fig. 1), which is similar to the Casten triangle related to the
Hamiltonian (1) Casten06 . In the SU3-IBM new theory Wang22 , new
$\gamma$-soft triaxial rotation is found, which is different from the O(6)
$\gamma$-unrelated rotational mode in Hamiltonian (1). The shape transitional
behaviors from the U(5) limit to the SU(3) degenerate point was numerically
explored (green line in Fig. 1). The key observation is that, spherical-like
$\gamma$-soft triaxial rotational spectra actually exists (see Fig. 5 (a)),
which may be the candidate to solve the spherical nucleus puzzle Heyde11 ;
Heyde16 ; Garrett18 . Within the parameter region of the green line in Fig. 1,
we find the unexpected result that there is an accidental degeneracy of the
corresponding energy levels between the ground and quasi-$\gamma$ bands such
that they form an exactly degenerate multiplet, which represents a O(5)
partial dynamical symmetry. It should be pointed out that this partial
dynamical symmetry is not the same thing as the previous studies in Leviatan11
. It will be clear, from the ensuing discussion, that while this degenerate
multiplet corresponds to that found in the SO(5) symmetry with quantum number
$\tau=2$, the next one ($\tau=3$) is not exactly degenerate, a feature that is
often observed in actual nuclear spectra.

Figure 1: New shape triangle: the top point of the triangle presents the U(5)
limit, which is spherical shape. The two bottom points and the black line
between them are all within the SU(3) limit.The left bottom point presents the
SU(3) prolate shape, and the right one presents the SU(3) oblate shape.

Historically, higher-order interactions in IBM-1 were introduced to describe
$\gamma$-rigid triaxial deformation and interactions
$[d^{{\dagger}}d^{{\dagger}}d^{{\dagger}}]^{(L)}\cdot[\tilde{d}\tilde{d}\tilde{d}]^{(L)}$
can play a key role for triaxiality of the ground state Isacker81 ; Isacker84
. An important progress related with our works is investigating SU(3)
symmetry-conserving higher-order interactions Isacker85 . Subsequently, within
the SU(3) limit, an algebraic realization of the rigid asymmetric rotor was
established Isacker00 ; zhang14 . Recently, this realization has been used to
explain the $B(E2)$ anomaly Zhang22 . SU(3) third-order and fourth-order
interactions are also discussed in Ref. Rowe77 ; Draayer85 ; Leschber88 ;
Elliott86 ; Kota20 . Higher-order terms are also important in partial
dynamical symmetry Leviatan11 . Higher-order interaction
$(\hat{Q}_{0}\times\hat{Q}_{0}\times\hat{Q}_{0})^{(0)}$ can present a
rotational spectrum Isacker99 , where $\hat{Q}_{0}$ is the quadrupole operator
in the O(6) limit. This result was further studied by Rowe05 ; Dai12 . However
the O(6) symmetry was questioned in wangtao . In these series of new
developments Wang22 ; Wang20 ; Zhang22 ; wang23 ; zhou23 ; wangtao ;
Fortunato11 ; Zhang12 ; Isacker00 ; zhang14 , SU(3) higher-order interactions
begin to show an extremely important role, albeit at a phenomenological level.
These higher-order interactions have already been shown to be relevant to some
realistic anomalies in nuclear structure Wang22 ; Wang20 ; Zhang22 , so
introducing these terms is of practical significance.

The $\gamma$-soft shape was first described in Ref. Jean56 , where the
geometric Hamiltonian is not dependent on the $\gamma$ variable. In the IBM,
the $\gamma$-soft case can be described by the O(6) limit Iachello78 ;
Casten78  and the nucleus of 196Pt was the first candidate for the O(6)
spectra. However there was still some debates about it Fewell85 ; Casten87 .
In the IBM-2 Iachello87 , triaxial shape can be described even with up to two-
body interactions Bijker82 ; Dukelsky04 ; Caprio04 ; Caprio05 . Three-body
interactions
$[d^{{\dagger}}d^{{\dagger}}d^{{\dagger}}]^{(L)}\cdot[\tilde{d}\tilde{d}\tilde{d}]^{(L)}$
are also used in the IBM-2 to investigate the $\gamma$ triaxiality Nomura12 .
In the sdg-IBM, $l=4$ g bosons can be introduced and hexadecapole deformation
can be discussed Isacker10 . Except for the IBM, triaxial shapes are also
investigated by many existing nuclear models Casten10 ; Heyde11 ; Bohr75 ;
Ring80 ; Reinhard03 ; Zuker05 ; Ring11 ; Shimizu12 ; Nomura21 .

Although 196Pt seems to adapt to the description in terms of the O(6)
symmetry, some noticeable deviations still exist and cannot be described at a
satisfactory level in the IBM. The first drawback is that it has a large
electric quadrupole moment Fewell85 , $Q_{2_{1}^{+}}$=0.62(8), pointing
towards the oblate side. The second is that the staggering feature of $\gamma$
band breaks the O(5) symmetry, which seems to be intermediate between the
$\gamma$-soft and $\gamma$-rigid. The third is the positions of the
$0_{2}^{+}$, $0_{3}^{+}$, $0_{4}^{+}$ states, which can not be reproduced well
Leviatan09 .

Recently two important results in the SU3-IBM are also found wang23 ; zhou23 .
In wang23 , the SU(3)-IBM is used to describe the prolate-oblate shape phase
transition with an asymmetric way, which can well explain the shape
transitions from 180Hf to 200Hg including the nucleus 196Pt. It was found
that, in this shape phase transition, another special point that shows
accidental degeneracy features, can be located near the middle of the
degenerate line and it can be used to describe the properties of 196Pt wang23
. In zhou23 , a shape transitional behavior like the one from the U(5) limit
to the O(6) limit in IBM is found by introducing the SU(3) fourth-order
interaction, which can describe the E(5)-like $\gamma$-softness in 82Kr.

Following the ideas of the previous researches, further exploration of the
applications of the SU3-IBM is necessary. The 196Pt is the focus. The three
obvious deficiencies discussed above can be well overcome simultaneously. This
nucleus is usually regarded as a typical example with O(6) symmetry in
Hamiltonian (1). We have found that the new model can give a more reasonable
description and our work provides a new understanding for $\gamma$-softness in
196Pt and other similar nuclei, which is related with the SU(3) symmetry. Thus
these results in the SU3-IBM (Wang22 ; Wang20 ; Zhang22 ; wang23 ; zhou23 ;
wangtao  and this paper) together confirm the validity of the new idea.

##  II Hamiltonian

In the SU3-IBM, the $d$ boson number operator $\hat{n}_{d}$ must be included,
which can describe a spherical shape. This is vital for the pairing
interaction between the valence nucleons. Other interacting terms are all
SU(3) conserving invariants. The traditional second-order interactions can
describe the prolate shape. Ref. Fortunato11  pointed out that the third-order
Casimir operator can describe the oblate shape. Other higher-order
interactions should be considered in some peculiar phenomena, such as $B(E2)$
anomaly and some unusual experimental data which can not be described by
previous theories Wang20 ; Zhang22 . In zhou23 , the square of the second-
order Casimir operator is found to be vital for the $\gamma$-softness of the
realistic nuclei. In this paper, the third-order invariant operator and the
square of the second-order invariant operator are introduced into the
interactions, like in Ref. zhou23  (the fourth-order interaction is only a
supplementary term here). Although this is a simple formalism in the SU3-IBM,
it shows many new interesting phenomena. Thus the Hamiltonian discussed in
this paper is

$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle
c\left[(1-\eta)\hat{n}_{d}+\eta\left(-\frac{\hat{C}_{2}[\textrm{SU(3)}]}{2N}\right.\right.$
(3)
$\displaystyle\left.\left.+\kappa\frac{\hat{C}_{3}[\textrm{SU(3)}]}{2N^{2}}+\xi\frac{\hat{C}_{2}^{2}[\textrm{SU(3)}]}{2N^{3}}\right)\right],$

where $0\leq\eta\leq 1$, _c_ is a global energy scale parameter,_N_ is the
boson number, $\kappa$ is the coefficient of the cubic term,
$\kappa=\frac{9\kappa_{{}_{3}}}{2\sqrt{35}}$, $\xi$ is the coefficient of the
fourth-order interaction, and $\hat{C}_{2}[\textrm{SU(3)}]$ and
$\hat{C}_{3}[\textrm{SU(3)}]$ are the second-order and third-order SU(3)
Casimir operators separately. If the fourth-order term is not considered, the
Hamiltonian (3) can be described by the new shape triangle in Fig. 1.

In the SU(3) limit the two Casimir operators can be related with the
quadrupole second or third-order interactions as following

$\hat{C}_{2}[\textrm{SU(3)}]=2\hat{Q}\cdot\hat{Q}+\frac{3}{4}\hat{L}\cdot\hat{L},$
(4)
$\hat{C}_{3}[\textrm{SU(3)}]=-\frac{4}{9}\sqrt{35}[\hat{Q}\times\hat{Q}\times\hat{Q}]^{(0)}-\frac{\sqrt{15}}{2}[\hat{L}\times\hat{Q}\times\hat{L}]^{(0)}.$
(5)

For a given SU(3) irrep $(\lambda,\mu)$, the eigenvalues of the two Casimir
operators under the group chain
$\textrm{U(6)}\supset\textrm{SU(3)}\supset\textrm{O(3)}$ are given as

$\langle\hat{C}_{2}[\textrm{SU(3)}]\rangle=\lambda^{2}+\mu^{2}+\lambda\mu+3\lambda+3\mu,$
(6)
$\langle\hat{C}_{3}[\textrm{SU(3)}]\rangle=\frac{1}{9}(\lambda-\mu)(2\lambda+\mu+3)(\lambda+2\mu+3).$
(7)

If $\kappa=\frac{3N}{2N+3}$, the second term in Hamiltonian (3) describes the
SU(3) degenerate point ($\xi=0$). It should be noticed that the location of
the SU(3) degenerate point along the variable $\kappa$ is related to the boson
number $N$ wang23 . For 196Pt, $N=6$, it is found at $\kappa=1.2$. In the
large-$N$ limit, $\kappa\rightarrow 1.5$. At this degenerate point, the SU(3)
irreps satisfying the condition $\lambda+2\mu=2N$ are all degenerate.

Figure 2: Partial low-lying level evolution along the green line in Fig. 1 for
$N=6$.

##  III $\gamma$-soft spectra for the point $B$

Figure 3: Level evolution of the $4_{1}^{+}$, $2_{2}^{+}$ and $0_{2}^{+}$
states for the parameter $\kappa$ from 1.0 to 1.6 when $\eta=0.5$, $\xi=0$ and
$N=6$. Two crossing points $A$ and $B$, where accidental degeneracy occurs,
can be observed.

Fig. 2 presents the partial low-lying level evolutions from the U(5) limit to
the SU(3) degenerate point for $N=6$ as a function of $\eta$. The choice of
the boson number $N=6$ correponds to 196Pt. (In the previous paper Wang22 ,
$N=7$ is discussed for the 110Cd.) It’s clear that the four lowest $0^{+}$
states are all degenerate if $\eta=1.0$. The key finding is that the
$4_{1}^{+}$ state and the $2_{2}^{+}$ state are degenerate, as well as the
triplet of states $6_{1}^{+}$, $4_{2}^{+}$, $3_{1}^{+}$. This degeneracy can
hold for some higher levels, which indicates a O(5) partial dynamical
symmetry. Unfortunately the reason for this degeneracy is still unknown. This
unexpected $\gamma$-softness is found in Ref. Wang22 .

Figure 4: Partial low-lying level evolution along the blue line in Fig. 1 for
$N=6$ when (a) $\xi=0$ and (b) $\xi=0.05$.

The SU(3) degenerate point is found at $\kappa=1.2$. A further exploration
around this degenerate point is undoubtedly useful for understanding the new
$\gamma$-softness. Fig. 3 plots level evolution of the $4_{1}^{+}$,
$2_{2}^{+}$ and $0_{2}^{+}$ states for the parameter $\kappa$ from 1.0 to 1.6
when $\eta=0.5$, $\xi=0$ and $N=6$. Obviously there are two crossing points
between the $4_{1}^{+}$ and $2_{2}^{+}$ states at $\kappa_{A}=1.188$ and
$\kappa_{B}=1.404$. The position relationships of these three states are very
important for understanding the $\gamma$-softness in realistic nuclei. The
left one is the point $A$, and the right point is denoted by $B$, which is the
special point discussed in this paper. The location of the point $B$ is
$\kappa_{B}=1.404$. Point $A$ is biased toward the prolate side and point $B$
toward the oblate side. In the $\gamma$-soft region between the two points,
point $B$ is closest to the oblate shape. 196Pt is a $\gamma$-soft nucleus
with large positive quadruple moment, so it is natural to investigate whether
the spectrum at point $B$ can be used to describe this nucleus. It should be
noticed that the value of $\kappa_{A}=1.188$ is somewhat smaller than the
value of the SU(3) degenerate point 1.2, thus the real degenerate line having
the O(5) partial dynamical symmetry between the U(5) limit and the SU(3)
degenerate point is not the directly connected green line in Fig. 1
Fortunato11 , which is also discussed in wang23 . There is a sudden shape
change through the SU(3) degenerate point from the prolate shape to the oblate
shape Zhang12 . However in the large-$N$ limit Fortunato11 , the point $A$ is
the critical point between the prolate shape and the $\gamma$-rigid triaxial
shape, while the point $B$ is the critical point between the $\gamma$-rigid
triaxial shape and the oblate shape. Thus the positions of these points are
different from each other wang23 . However, for small $N$, this connected line
is a good approximation wang23 . In Fig. 1, the blue line passes through the
point $B$. It should be noticed that, for any $N\geq 4$, this special point
$B$ exists.

Figure 5: Energy spectra of (a) the point $A$, (b) the cousin point $B$ and
(c) adding the fourth-order interaction when $\xi=0.05$ for $N=6$. Figure 6:
Level evolution of the $0_{2}^{+}$ and $0_{3}^{+}$ states for the parameter
$\xi$ from 0 to 0.30 when $\eta=0.5$, $\kappa=1.404$ and $N=6$.

Fig. 4 (a) presents the partial low-lying level evolutions along the blue line
via the point $B$. The degeneracy between the ground band and the
$\gamma$-band are somewhat broken, which can be also seen in Fig. 5 (b) for
the energy spectra of the point $B$. Compared with Fig. 2, the levels of
$0_{2}^{+}$ and $2_{3}^{+}$ states move up a little and $6_{1}^{+}$,
$4_{2}^{+}$, $3_{1}^{+}$ and $0_{2}^{+}$ are nearly degenerate. Thus the low-
lying part of the spectra is similar to that shown in the O(6) limit
Iachello87 . For point $B$, the energies of $0_{2}^{+}$ and $0_{3}^{+}$ states
are 1.1637c and 2.1472c in Fig. 4 (a). The ratio of the two states
$R^{\prime}=E_{0_{3}^{+}}/E_{0_{2}^{+}}=1.845$ is much larger than the
experimental one 1.236. In Ref. zhou23 , it is found that, the introduction of
the fourth-order interaction $\hat{C}_{2}^{2}[\textrm{SU(3)}]$ can reduce the
energy difference between the $0_{2}^{+}$ and $0_{3}^{+}$ states, even to
zero. Fig. 6 presents the level evolutions of $0_{2}^{+}$ and $0_{3}^{+}$
states when $\xi$ increases from 0 to 0.3 for $\eta=0.5$ and $\kappa=1.404$.
It should be noticed that the fourth-order interaction is only a supplementary
term here wang23 , so the value of $\xi$ is small. The distance between the
two states reduces first and then gets bigger. The minimum value is around
0.144, at which $R^{\prime}=1.188$, smaller than the experimental value. In
this paper, for a better fitting of the quadruple moment, we select
$\xi=0.05$. Fig. 4 (b) presents the partial low-lying level evolutions when
$\eta$ varies from 0 to 1 for $\kappa=1.404$ and $\xi=0.05$. The most
significant change is that the positions of the $0_{2}^{+}$ state increases
from 1.16c to 1.47c. The spectra of the position can be seen in Fig. 5 (c),
which seems somewhat irregular. But the low-lying part looks very similar to
the spectra of the realistic $\gamma$-soft nucleus. For comparison, Fig. 5 (a)
presents the spectra of the point $A$ for $N=6$, which is not shown in Wang22
and seems different from the two figures below. In Fig. 5 (a) the low-lying
part is very similar to the vibrational spectra of a rigid spherical nucleus.
The key difference is that some bands ($0_{4}^{+}$, $2_{8}^{+}$, $0_{6}^{+}$
as bandheads) are greatly elevated. When moving from the point $A$ to the
point $B$, then with the adding of fourth-order term, the degeneracies are
gradually broken and the familiar $\gamma$-soft feature emerges, while the
regularity of the spectra is weakened.

##  IV $B(E2)$ values and quadrupole moments for the point$B$

Figure 7: The evolution of the $B(E2;2_{1}^{+}\rightarrow 0_{1}^{+})$ (green
real line), $B(E2;0_{2}^{+}\rightarrow 2_{1}^{+})$ (blue real line),
$B(E2;0_{2}^{+}\rightarrow 2_{2}^{+})$ (blue dashed line),
$B(E2;0_{3}^{+}\rightarrow 2_{1}^{+})$ (red real line), and
$B(E2;0_{3}^{+}\rightarrow 2_{2}^{+})$ (red dashed line) as the function of
$\eta$ when $\kappa=1.404$, and (a) $\xi=0$, (b) $\xi=0.05$ for $N=6$.

The $B(E2)$ values are vital for understanding the collective behaviors. In
the common experiences of nuclear structure studies, we often expect a
definite relationship between the energy spectra and the corresponding $B(E2)$
values, especially in the IBM. However this relation may lead to the wrong
conclusions. In the spherical nucleus puzzle Garrett12 ; Heyde16 ; Garrett18 ,
the energy spectra of the Cd isotopes are similar to the ones of the rigid
spherical vibrations, but the $B(E2)$ values are experimentally found to
violate the expectations. Thus new perspectives on the shape evolution from
the magic number nucleus to the deformation need to be developed. In the
$B(E2)$ anomaly 168Os ; 166W ; 172Pt ; 170Os , this case becomes more obvious.
From the level evolutions of the Pt-Os-W isotopes with neutron number, the
energy spectra of 172Pt, 168,170Os, 166W seem normal, but their $B(E2)$ values
completely exceed expectations. Thus collective behaviors cannot be determined
solely by the energy spectra. Various nuclear spectroscopic methods are needed
Garrett18 .

Figure 8: The evolution of the quadrupole moment of the $2^{+}_{1}$ state
along the blue line in Fig. 1 for $N=6$ when $\xi=0$ (the solid blue line) and
$\xi=0.05$ (the solid red line).

For understanding $\gamma$-softness, the $B(E2)$ values are also necessary.
Especially when the new $\gamma$-softness is provided Wang22 , how to
distinguish the different $\gamma$ softness becomes more and more important in
the description of the realistic nuclei properties. The $E2$ operator is
defined as

$\hat{T}(E2)=e\hat{Q},$ (8)

where $e$ is the boson effective charge. The evolution of the
$B(E2;2_{1}^{+}\rightarrow 0_{1}^{+})$, $B(E2;0_{2}^{+}\rightarrow
2_{1}^{+})$, $B(E2;0_{2}^{+}\rightarrow 2_{2}^{+})$,
$B(E2;0_{3}^{+}\rightarrow 2_{1}^{+})$, and $B(E2;0_{3}^{+}\rightarrow
2_{2}^{+})$ values are plotted along the blue line in Fig. 1 for $N=6$. In
Fig. 7 (a) the $B(E2;2_{1}^{+}\rightarrow 0_{1}^{+})$ value is nearly the
same. For $\eta=1.0$, it describes an oblate shape Zhang12 , thus the
$B(E2;2_{1}^{+}\rightarrow 0_{1}^{+})$ value is suppressed. With the
increasing of $\eta$, the values of $B(E2;0_{2}^{+}\rightarrow 2_{1}^{+})$ and
$B(E2;0_{3}^{+}\rightarrow 2_{2}^{+})$ get reduced while the ones of the
$B(E2;0_{2}^{+}\rightarrow 2_{2}^{+})$ becomes larger. The trends are similar
to the ones along the green line with degeneracy and the $\gamma$-softness can
emerge. When the fourth-order interaction is introduced in Fig. 7 (b), at
$\eta=0.5$, the value of $B(E2;0_{2}^{+}\rightarrow 2_{2}^{+})$ can be
reduced.

Fig. 8 shows the quadrupole moments of the $2^{+}_{1}$ state for $\xi=0$ (the
solid blue line) and $\xi=0.05$ (the solid red line). It is shown that, for
the blue line, when $\eta\geq 0.372$, the value becomes positive, which means
an oblate deformation. For the red line, it bends to the oblate side.

Figure 9: Energy spectra of (a) 196Pt, and of the Hamiltonian in Eq. (3) at
the point $B$ (b) and when adding the fourth-order interaction when $\xi=0.05$
for $N=6$ (c).

##  V Theoretical fitting of 196Pt

Without considering other higher-order interactions in the SU(3) limit, the
properties of the point $B$ or adding the fourth-order interaction are used to
fit the structure of 196Pt. Although the precision needs to be improved, the
fitting results seem excellent. For $\xi=0$ at point $B$, the overall energy
parameter $c$ in $\hat{H}$ is 0.9753 MeV to make the energy value of the
$0_{2}^{+}$ state equal to the experimental one. The $L^{2}$ term is also
added to fit the $2_{1}^{+}$ state, which is 0.00803 MeV. The theoretical
spectra of point $B$ shown in Fig. 9 (b) are compared with the experimental
data shown in Fig. 9 (a). The theory and the experiment correspond well
qualitatively, and we can see that the position relationships of each energy
level are also consistent. The rotational-like $\gamma$-band in 196Pt is an
interesting problem, and the theoretical spectra have similar structures.
$0_{4}^{+}$ and $0_{5}^{+}$ states also fit well. The main drawback is that
the $0_{3}^{+}$, $2_{5}^{+}$ and $2_{6}^{+}$ are somewhat higher, that is, the
energy difference between the $0_{3}^{+}$ and $0_{2}^{+}$ states is somewhat
larger than the experimental result. This is the typical feature of the new
$\gamma$-soft rotational mode Wang22 . For $\xi=0.05$, better fitting results
can be obtained, where the characteristics of the $\gamma$ band are consistent
with the actual situation and the energies of the $0_{3}^{+}$ and $0_{4}^{+}$
states are also reduced. For reducing the energies of the higher-levels, Pan
_et al._ presented a new method to provide an excellent fitting result for
194Pt Pan18 , which may be used to improve the fitting precision in the
SU3-IBM.

Table I lists the $B(E2)$ values of some low-lying states in 196Pt, the point
$B$ (Res.1), adding the fourth-order term (Res.2), the O(6) partial dynamical
symmetry model (PDS) Leviatan09  and the modified soft-rotor model (MSR) Li22
. These three models are all related to higher-order interactions in IBM. In
the O(6) partial dynamical symmetry, one three-body interactions that is
partially solvable in O(6) symmetry can be constructed, which can mix the
$\Sigma=4$ and $\Sigma=2$, but it does not change the case $\Sigma=6$ for
$N=6$ ($\Sigma$ is the O(6) label). In the modified soft-rotor model, the
higher-order interactions are used to fit the Pt isotopes, which is inspired
by the O(6) higher-order symmetry description of 194Pt Pan18 . For Res.1, this
$\gamma$-soft description of the point $B$ can show a good consistency with
the experimental data qualitatively sheet07 . From the overall fitting
results, it looks like somewhat worse than the other two theories, but the
result is still good considering that the parameters of the point $B$ are not
adjustable. When the fourth-order interactions are introduced for $\xi=0.05$,
the fit can be greatly improved.

Figure 10: The staggering parameter $S(J)$ for 196Pt and various models. Table 1:  Absolute $B(E2)$ values in W.u. for $E2$ transitions from the low-lying states in 196Pt, Res.1 for the cousin point $B$, Res.2 for adding the fourth-order interaction when $\xi=0.05$, the PDS model and the MSR model with effective charge $e=2.385$ (W.u.)1/2 for Res.1 and $e=2.623$ (W.u.)1/2 for Res.2. $L_{i}~{}~{}L_{f}$ | Exp.a | Res.1 | Res.2 | PDSb | MSRc
---|---|---|---|---|---
$2_{1}^{+}~{}~{}0_{1}^{+}$ | 40.60(20) | 40.6 | 40.6 | 40.6 | 40.6
$2_{2}^{+}~{}~{}2_{1}^{+}$ | 54(+11 -12) | 60.3 | 43.2 | 53.0 | 45.3
$~{}~{}~{}~{}~{}~{}0_{1}^{+}$ | $<7.8\times 10^{-6}$ | 2.9 | 5.05 | 0.27 | 1.4
$4_{1}^{+}~{}~{}2_{1}^{+}$ | 60.0(9) | 52.6 | 51.4 | 53.0 | 53.6
$0_{2}^{+}~{}~{}2_{1}^{+}$ | 2.8(15) | 3.6 | 0.122 | 0.44 | 3.1
$~{}~{}~{}~{}~{}~{}2_{2}^{+}$ | 18(10) | 100.0 | 79.5 | 54.1 | 69.6
$6_{1}^{+}~{}~{}4_{1}^{+}$ | 73(+4 -73) | 52.1 | 45.3 | 54.1 | 54.7
$4_{2}^{+}~{}~{}4_{1}^{+}$ | 17(6) | 34.4 | 27.4 | 25.8 | 18.1
$~{}~{}~{}~{}~{}~{}2_{1}^{+}$ | 0.56(+12 -17) | 0.65 | 1.34 | 0.14 | 0.69
$~{}~{}~{}~{}~{}~{}2_{2}^{+}$ | 29(+6 -29) | 38.0 | 31.2 | 28.3 | 28.8
$2_{3}^{+}~{}~{}4_{1}^{+}$ | 0.13(12) | 2.34 | 0.23 | 0.059 | 0.60
$~{}~{}~{}~{}~{}~{}2_{2}^{+}$ | 0.26(23) | 3.93 | 3.47 | 0.19 | 0.27
$~{}~{}~{}~{}~{}~{}0_{1}^{+}$ | 0.0025(24) | 0.025 | 0.02 | 0 | 0.064
$~{}~{}~{}~{}~{}~{}0_{2}^{+}$ | 5(5) | 38.3 | 36.5 | 17.6 | 0.003
$0_{3}^{+}~{}~{}2_{1}^{+}$ | $<5.0$ | 0.88 | 6.82 | 0 | 1.05
$~{}~{}~{}~{}~{}~{}2_{2}^{+}$ | $<0.41$ | 0.015 | 0.114 | 0.41 | 0.009
$6_{2}^{+}~{}~{}6_{1}^{+}$ | 16(5) | 23.2 | 17.7 | 15.3 | 9.5
$~{}~{}~{}~{}~{}~{}4_{1}^{+}$ | 0.48(14) | 0.29 | 0.77 | 0.12 | 0.50
$~{}~{}~{}~{}~{}~{}4_{2}^{+}$ | 49(13) | 42.1 | 31.9 | 32.7 | 32.9
$8_{1}^{+}~{}~{}6_{1}^{+}$ | 78(+10 -78) | 49.1 | 29.3 |  | 47.6
aFrom Ref.sheet07  |  |  |  |  |
bFrom Ref.Leviatan09  |  |  |  |  |
cFrom Ref.Li22  |  |  |  |  |
Table 2:  $R^{\prime}$ and $R$ in 196Pt and various models. | Exp.a | Res.1 | Res.2 | MSRc | CQc | O(6)b | PDSb
---|---|---|---|---|---|---|---
$R^{\prime}$ | 1.236 | 1.845 | 1.440 | 1.131 | 1.281 | 1.543 | 1.786
$R$ | -0.701 | -0.896 | -0.864 | -0.192 | -0.286 | -0.289 | -0.629
aFrom Ref.sheet07  |  |  |  |  |  |  |
bFrom Ref.Leviatan09  |  |  |  |  |  |  |
cFrom Ref.Li22  |  |  |  |  |  |  |
Table 3:  Quadrupole moments in eb and average variance $\Delta Q$ for some low-lying states in 196Pt and various models. | Exp.a | Res.1 | Res.2 | MSRb | CQb
---|---|---|---|---|---
$Q(2_{1}^{+})$ | +0.62 | +0.30 | +0.70 | +0.50 | +0.40
$Q(2_{2}^{+})$ | -0.39 | -0.48 | -0.81 | -0.47 | -0.37
$Q(4_{1}^{+})$ | +1.03 | +0.036 | +0.63 | +0.66 | +0.40
$Q(6_{1}^{+})$ | -0.18 | -0.42 | +0.35 | +0.76 | +0.38
$\Delta Q$ |  | 0.54 | 0.39 | 0.51 | 0.44
aFrom Ref.sheet07  |  |  |  |  |
bFrom Ref.Li22  |  |  |  |  |

To better substantiate this conclusion, a quantitative analysis is mandatory.
The first quantity that we might study is the staggering parameter $S(J)$ in
$\gamma$ band energies Casten91 ; Casten07  defined as

$S(J)=\frac{(E_{J}-E_{J-1})-(E_{J-1}-E_{J-2})}{E_{2_{1}^{+}}},$ (9)

which quantifies how adjacent levels within a $\gamma$ band are grouped. Fig.
10 presents the $S(J)$ for $J=4,5,6$ in experimental data, Res.1 and Res.2 in
the SU3-IBM, MSR Li22 , CQ Li22  and O(6) symmetry Leviatan09 . Black squares
are the experimental results. CQ shows the typical $\gamma$-soft feature of
strong staggering. O(6), MSR and Res.1 display similar trends, while Res.2
gives the best fitting.

The second is on the positions of the $0_{2}^{+}$, $0_{3}^{+}$ and $0_{4}^{+}$
states in the spectra. $R^{\prime}=E_{0_{3}^{+}}/E_{0_{2}^{+}}$ is the energy
ratio between the $0_{2}^{+}$ and $0_{3}^{+}$ states.
$R=E_{0_{4}^{+}}/E_{0_{3}^{+}}-2$ is used in Leviatan09 . Table II presents
the $R^{\prime}$ and $R$ in experimental data, Res.1, Res.2, MSR Li22 , CQ
Li22 , O(6) symmetry Leviatan09  and PDS Leviatan09  . For $R^{\prime}$,
Res.2, MSR and CQ can offer reasonable results. For $R$, Re.1, Res.2 and PDS
can give the best results that are consistent with the experiment data. In
Leviatan09 , although the introduction of higher-order interaction can fit the
$R$ well, it increases $R^{\prime}$. Res.2 is the only theory that makes both
values more consistent. What needs to be mentioned is that the $0_{4}^{+}$
sate in 194Pt is an intruder state Pan18 , thus the experimental $0_{5}^{+}$
(or $0_{4}^{+}$) state in 196Pt may be also an intruder state.

The experimental data about the quadrupole moments are rare, which are very
sensitive to specific nuclear structure models. Table III presents the
quadrupole moments of some low-lying states in 196Pt, which contain the
results of Res.1, Res.2, the MSR model Li22  and CQ Li22 . The average
variance of the fit $\Delta Q$ are calculated. Our results are more consistent
with the experimental data. The quadrupole moment of these states in the
$O(6)$ limit are all zero with $\Delta Q=0.64$. The quadrupole moments can be
a very useful indicator for judging the success of different nuclear models.

The results of these three quantitative calculations favour the descritption
in terms of SU3-IBM over previous theories.

##  VI Some discussions about the emerging $\gamma$-softness

Figure 11: Partial energy spectra in 120Cd normal states, 196Pt and 132Ba
ensdf . The energy of the $0_{3}^{+}$ state in the 120Cd normal states is
predicted by the authors.

The ppherical nucleus puzzle Garrett08 ; Garrett10 ; Garrett12 ; Batchelder12
; Heyde16 ; Heyde16 ; Heyde16 ; Garrett18 ; Garrett19 ; Garrett20  and $B(E2)$
anomaly 168Os ; 166W ; 172Pt ; 170Os  dramatically change our view of nuclear
structure and shape evolutions. These two peculiar phenomena occur in the
collective modes of nuclei with boson number $N<10$. This means that the
emergence of collective motions in realistic nuclei will be a more complex
problem. Even for typical large-deformation nuclei as 154Sm and 166Er, new
studies have found that the structure of 154Sm shows the coexistence of
prolate and triaxial shapes, while the deformed shape with a strong triaxial
instability is demonstrated on 166Er, which has shown to be related with self-
organization mechanism Otsuka19 ; Otsuka22 . Holding an obstinate simple view
of collective motions in nuclei seems very undesirable. Further experimental
information on the two puzzles may lead us to get a more accurate description
on nuclear structures.

This article can be seen as an extension of the previous works Wang22 ; wang23
. In these papers, the SU3-IBM is proposed, which is inspired by the
interesting findings in Ref. Fortunato11 ; Zhang12 . When only the U(5) limit
and the SU(3) limit are concerned, $\gamma$-soft rotational mode can also
occur as an emergent phenomenon if the SU(3) higher-order interactions are
introduced into the common formalism. This is very different from the
$\gamma$-softness related to the O(6) limit. In the traditional IBM, O(6)
symmetry is exactly solvable, and $\gamma$-unstable spectra can be expected.
Especially the SU3-IBM can describe the $B(E2)$ anomaly, which make the model
particularly useful.

Fig. 11 presents partial energy spectra of 120Cd, 196Pt and 132Ba, for their
boson number are all $N=6$. Ref. Batchelder12  obtains the spectra of 120Cd ,
in which the $0_{3}^{+}$ state at the three-phonon level is absent. The energy
of the $0_{3}^{+}$ state of 120Cd is predicted by our theory, which is around
2300 KeV, or even higher. Thus this is really a new $\gamma$-soft rotational
mode Garrett10 ; Garrett12 ; Garrett18 , despite it looks much like the rigid
spherical vibrational excitation mode. If we consider that the IBM can truly
describe the collective excitations in realistic nuclei, the only way to
describe the new $\gamma$-soft behaviors is to introduce the SU(3) higher-
order interactions. The theory works better than expected Wang22 ; Wang20 ;
Zhang22 .

196Pt and 132Ba are two typical traditional $\gamma$-soft nuclei. In the IBM,
this kind of $\gamma$-softness is related to the O(6) symmetry, or O(5)
symmetry, such as E(5) critical point description Iachello00 ; Zhangyu22 . In
fact, this case is not related to the deformation $\gamma$ parameter. The
SU3-IBM follows such a principle that triaxiality arises from the competition
between prolate shape and oblate shape. In this paper, we show that the new
emerging $\gamma$-softness can be also related to the traditional
$\gamma$-soft spectra and their $B(E2)$ behaviors. When other SU(3) higher-
order interactions are introduced, the fitting of spectra appears excellent.
This will be further discussed in following papers.

In previous discussions, spectra and low-lying $B(E2)$ values of various
$\gamma$-soft rotational modes in different nuclear structure theories are
very similar, so distinguishing various $\gamma$-softness in different
theories is becoming extremely important. New $\gamma$-softness in the normal
states of Cd nuclei can be only described by the SU3-IBM. However the
spherical nucleus puzzle is still full of debates Garrett19 ; Garrett20 ;
Nomura18 ; Leviatan18 . A complete description of 110,112Cd including
configuration mixing, like Ref. Ramos11 ; Gavrielov19 ; Ramos19 ; Ramos20 ;
Gavrielov22 , is in progress. $B(E2)$ values between higher levels are useful,
such as the $0_{3}^{+}$, $2_{4}^{+}$ and $2_{5}^{+}$ states. Quadrupole
moments of the low-lying states may provide great value in distinguishing
among the various models.

Though a lot of theoretical work still needs to be done, we expect that many
observations in $\gamma-$soft nuclei might be described by the SU3-IBM in a
unified way. E(5)-like $\gamma$-softness in 82Kr is found in the new model
zhou23 . Each interaction in the SU3-IBM has a clear geometric meaning, and
different from common considerations in microscopic theory, such as SD-pair
shell model Luo98 ; Zhao00 . In the microscopic theory, various deformations
resulting from the proton-neutron interactions can give rise to different
deformation shapes, including the $\gamma$-soft rotational mode Luo ; He19 .
In the SU3-IBM, this is not so, and specific interaction corresponds to
certain shape, and the $\gamma$-softness is an emergent phenomenon, which can
not be expected before numerical calculations. How to understand this point is
an interesting problem. From the existing results, the SU3-IBM is closer to
the geometric collective model Bohr75 . It seems important to investigate
further the emerging $\gamma$-softness in the geometric model Fortunato05 ;
Fortunato16 .

Since the seminal works by Elliott Elliott1 ; Elliott2 ; Harvey , the SU(3)
symmetry has played a key role for the description of the rotational spectra
from the perspective of a microscopic shell model Kota20 ; Draayer1 ; Draayer2
; Draayer3 ; Draayer4 ; Draayer5 ; Rowe1 ; Rowe2 ; Draayer6 ; Draayer7 ;
Zuker1 ; Zuker2 ; Bonatsos171 ; Bonatsos23 . It was also found that the SU(3)
symmetry plays a key role in the description of shell-like quartetting of
nucleons, and the SU(3) third-order Casimir operator
$\hat{C}_{3}[\textrm{SU(3)}]$ is needed to describe the experimental spectra
in order to distinguish between the prolate and oblate shapes Cseh1 ; Cseh2 ;
Cseh3 ; Cseh4 . In the previous studies, the SU(3) symmetry has been related
to the prolate shape. In our new findings (Wang22 ; Wang20 ; Zhang22 ; wang23
; zhou23 ; wangtao  and this paper), the SU(3) symmetry actually dominates all
the quadrupole deformations. In particular, in this model, one can obtain
large quadrupole moments, even with spectra that resemble a gamma-soft
situation, a feature that cannot be obtained with partial dynamical symmetry
models. We expect that this new perspective can be further used in the SU(3)
shell model Draayer4 ; Draayer5 .

Along the dashed red line through the $A$ and $B$ point, shape phase
transition from the prolate shape to the oblate shape has been studied wang23
, and this is an asymmetric evolution, which is different from the symmetric
one in $\hat{H}_{1}$ Werner01 ; Jolie03 . The study of the properties of 196Pt
in this paper further supports this perspective and a more detailed
investigation on the prolate-oblate shape phase transition is also needed.

In addition, in Ref. zhou23 , the E(5)-like energy spectrum was also found in
the new model, where there are also some new findings pointing to the
$\gamma$-softness, which are further used to understand the 196Pt. We look
forward to further explore the relationship between the $\gamma$-softness and
the rigid triaxial-rotor mode zhou .

##  VII Conclusions

Recently, the interacting boson model with SU(3) higher-order interactions
(SU3-IBM) was proposed Wang22 ; Wang20 ; Zhang22  to resolve the spherical
nucleus puzzle Garrett10 ; Garrett12 ; Batchelder12 ; Heyde16 ; Garrett18 ;
Garrett19 ; Garrett20  and the $B(E2)$ anomaly 168Os ; 166W ; 172Pt ; 170Os .
Although this model can be obtained by only considering the U(5) limit and the
SU(3) limit, the $\gamma$-soft rotational mode can emerge with (quasi)-O(5)
partial dynamical symmetry. These results are extending our view of the
interacting boson model (IBM). The successful descriptions of the two abnormal
phenomena is also changing our view of nuclear structure and shape evolution.
As said by Heyde and Wood:“sphericity is a special case of deformation.” and
“the reference frame must be fundamentally one of a deformed many-body
system.” Heyde16 . Along the same lines, with the present paper we show that
also $\gamma$-softness is a special case of deformation. The emerging
$\gamma$-softness may play a key role in the shift of perspective.

Following the previous studies Wang22 ; wang23 , the emerging $\gamma$-soft
rotational mode can be explored to explain the properties of 196Pt. In our
studies, a special point, which is near the middle point of the degenerate
line connected the U(5) limit and the SU(3) degenerate point, is explored. The
purpose of this paper is only to explore the relationship between the emerging
$\gamma$-softness and the $\gamma$-soft properties in realistic nuclei.
Further detailed fitting will be done in future when other SU(3) higher-order
interactions are introduced.

Further investigation of the $\gamma$-rigid triaxiality is also important in
the SU3-IBM. This is a delicate topic Nomura12 . The phase diagram of the
SU3-IBM will be given in future, and it can offer meaningful guidance to the
rigid triaxiality. 6-$d$ interaction may be also valuable Isacker84 .
Distinguishing between the different $\gamma$-softness and discussing the
differences between the $\gamma$-softness and the rigid triaxiality are topics
that require further exploration.

Finally, a direct discussion and fitting of the new model on the oblate
nuclei, such as 196-204Hg, is no doubt extremely important for understanding
the SU3-IBM and for establishing the relationship between the new model and
the SU(3) shell model.

##  VIII ACKNOWLEDGMENT

This research is supported by the Educational Department of Jilin Province,
China (JJKH20210526KJ). C.-x.Z. gratefully acknowledges support from the
Project Supported by Scientific Research Fund of Hunan Provincial Education
Department, China (21A0427).

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