# At the borderline of shape coexistence: Mo and Ru

E. Maya-Barbecho1, S. Baid2, J.M. Arias2,3 and J.E. García-Ramos1,3
1Departamento de Ciencias Integradas y Centro de Estudios Avanzados en Física,
Matemática y Computación, Universidad de Huelva, 21071 Huelva, Spain
2Departamento de Física Atómica, Molecular y Nuclear, Facultad de Física,
Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, Spain
3Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada,
Fuentenueva s/n, 18071 Granada, Spain

###### Abstract

Background

Even-even isotopes of Mo ($Z=42$) and Ru ($Z=44$) are nuclei close to the
subshell closure at $Z=40$, where shape coexistence plays a significant role.
As a result, their spectroscopic properties are expected to resemble those of
Sr ($Z=38$) and Zr ($Z=40$). Exploring the evolution of these properties as
they move away from the subshell closure is of great interest.

Purpose

The purpose of this study is to reproduce the spectroscopic properties of
even-even ${}^{96-110}_{\phantom{961-}42}$Mo and
${}^{98-114}_{\phantom{961-}44}$Ru isotopes and to determine the influence of
shape coexistence.

Method

We have employed the interacting boson model with configuration mixing as the
framework to calculate all the observables for Mo and Ru isotopes. We have
considered two types of configurations: 0-particle-0-hole and
2-particle-2-hole excitations. The model parameters have been determined using
a least-squares fitting to match the excitation energies and the $B(E2)$
transition rates.

Results

We have obtained the excitation energies, $B(E2)$ values, two-neutron
separation energies, nuclear radii, and isotope shifts for the entire chain of
isotopes. Our theoretical results have shown good agreement with experimental
data. Furthermore, we have conducted a detailed analysis of the wave functions
and obtained the mean-field energy surfaces and the nuclear deformation
parameter, $\beta$, for all considered isotopes.

Conclusions

Our findings reveal that shape coexistence plays a significant role in Mo
isotopes, with the crossing of intruder and regular configurations occurring
at neutron number $60$ ($A=102$), which induces a quantum phase transition. In
contrast, in Ru isotopes, the intruder states have minimal influence,
remaining at higher energies. However, at neutron number $60$, also a quantum
phase transition occurs in Ru isotopes.

Mo isotopes, Ru isotopes, shape coexistence, intruder states, interacting
boson model

##  I Introduction

The subshell closure at $Z=40$ is known to exhibit rapid onset of deformation,
particularly around neutron number $60$, resulting from the filling of the
neutron $1g_{7/2}$ orbit, which interacts with the proton $1g_{9/2}$ one. This
phenomenon was well explained in Federman and Pittel (1977); Federman  _et
al._ (1979); Federman and Pittel (1979), and more recently in Togashi  _et
al._ (2016), providing insights into the origin of deformation in this mass
region. Generally, the shape of a nucleus arises from a delicate balance
between pairing and quadrupole nuclear interactions, favoring spherical and
deformed shapes, respectively. It is important not to overlook the role of
monopole interaction, which is responsible for the evolution of single-
particle energies.

The region around $Z=40$ is also known for the presence of states with
different shapes within a narrow energy range. This situation, known as shape
coexistence, was initially proposed in nuclear physics by Morinaga Morinaga
(1956) to explain the nature of the first excited state ($0^{+}$) in 16O,
which was assumed to be deformed while the ground state is obviously spherical
due to its doubly magic nature. These experimental findings were theoretically
confirmed in Brown (1964); Brown and Green (1966a, b), where particle-hole
excitations across the energy shell gap were allowed. A deformed band,
originated at low energy due to the quadrupole part of the nucleon-nucleon
interactions, emerged on top of the first excited $0^{+}$ state. The presence
of additional effective valence nucleons is crucial in explaining the
characteristic dropping of this deformed band, often referred to as the
intruder band. Another notable example of shape coexistence is observed
experimentally in 40Ca and is effectively described through shell model
calculations involving multi-particle multi-hole excitations Caurier  _et al._
(2007); Poves (2016, 2018).

The presence of shape coexistence is clearly manifested through experimental
observations of nuclear radii and isotope shifts. The phenomenon was first
identified in the odd-even staggering of the radii of mercury isotopes, which
indicated the coexistence of states with significantly different degrees of
deformation. Subsequent studies, such as the measurements of mercury radii in
the neutron-deficient region well beyond the mid-shell Marsh  _et al._ (2018),
have confirmed the role of shape coexistence in explaining nuclear structure
in various mass regions. This phenomenon is particularly prominent near shell
or subshell closures in protons (neutrons) with neutrons (protons) around the
mid-shell. Shape coexistence has been observed in light, medium, and heavy
nuclei Heyde  _et al._ (1983); Wood  _et al._ (1992); Heyde and Wood (2011);
Garrett  _et al._ (2022a).

From a theoretical perspective, shape coexistence is described using two
complementary approaches: self-consistent methods based on Hartree-Fock (HF)
or Hartree-Fock-Bogoliubov (HFB) theories, and the nuclear shell model. In the
region around $Z=40$, notable studies have been conducted using the
relativistic interaction PC-PK1, focusing on Kr, Sr, Zr, and Mo isotopes near
neutron number $60$ Xiang  _et al._ (2012). These studies revealed a rapid
evolution in Sr and Zr isotopes, while a more moderate evolution was observed
in Mo and Kr isotopes. Prolate-oblate shape coexistence was observed in 98Sr
and 100Zr. In another study Nomura  _et al._ (2016), even-even Ru, Mo, Zr, and
Sr isotopes were investigated using the HFB approach with a Gogny-D1M
interaction. The spectroscopic properties were obtained by mapping the energy
density functional into an interacting boson model with configuration mixing
(IBM-CM) energy surface. The Ru isotopes exhibited a smooth shape evolution
with no evidence of intruder bands. The Mo isotopes required two different
particle-hole configurations, resulting in a good reproduction of the yrast
band but with some deficiencies in describing the non-yrast band and inter-
band transitions. State-of-the-art calculations within the HFB framework were
carried out by Rodríguez-Guzmán  _et al._ (2010), allowing for the treatment
of axial and triaxial degrees of freedom on an equal footing. These
calculations, applied to Sr, Zr, and Mo isotopes, revealed an oblate shape for
Mo isotopes at neutron number $58$, gradually transitioning to a triaxial
shape as the neutron number increased. An island of triaxiality was evident
from neutron number $60$ up to $68$. Another study Abusara  _et al._ (2017)
investigated Mo and Ru isotopes using the relativistic-Hartree-Bogoliubov
formalism with density-dependent zero- and finite-range nucleon-nucleon
interactions, as well as a separable pairing. The results were in agreement
with other mean-field calculations. Additionally, the study Thakur  _et al._
(2021) employed the density-dependent meson exchange model DD-ME2 and density-
dependent point coupling models DD-PC1 and DD-PCX to explore the shape
evolution of Zr, Mo, and Ru isotopes, considering only axial situations. The
predictions indicated a spherical shape for the lightest Ru isotopes, nearly
degenerate prolate and oblate minima for 96-102Ru, a prolate and an oblate
degenerate minima in 104Ru, and an oblate shape for the heaviest Ru isotopes.
In the case of Mo isotopes, the lightest isotopes were predicted to have
spherical shapes, with nearly degenerate prolate and oblate minima for
94-100Mo.

Moving into the shell-model framework, it is important to note that the
description of the region around $Z=40$ is influenced by the simultaneous
occupation of neutron and proton spin-orbit partners. When the neutron
$1g_{7/2}$ orbital begins to be filled, the interaction with the proton
$1g_{9/2}$ orbital favors the existence of a deformed region in Zr and Sr
nuclei with a neutron number larger than $58$, and likely in Mo and Ru nuclei
as well. This concept has been recently applied to the Zr region Togashi  _et
al._ (2016) using the Monte Carlo Shell Model Otsuka  _et al._ (2001); Shimizu
_et al._ (2012, 2017), which has the capability to handle open shells.
However, the idea was first introduced in the seminal works of Federman and
Pittel Federman and Pittel (1977); Federman  _et al._ (1979); Federman and
Pittel (1979) where the simultaneous occupation of neutron-proton spin-orbit
partners was emphasized as crucial. Federman and their colleagues extensively
explored this mass region using a reduced model space consisting of the
$3s_{1/2}$, $2d_{3/2}$, and $1g_{7/2}$ neutron orbits, and the $2p_{1/2}$,
$1g_{9/2}$, and $2d_{5/2}$ proton orbits Federman  _et al._ (1984); Heyde  _et
al._ (1988); Etchegoyen  _et al._ (1989); Pittel  _et al._ (1993). More
recently, large-scale shell-model calculations have been performed for the
same mass region using more realistic valence spaces, as demonstrated in
studies such as Holt  _et al._ (2000) and Sieja  _et al._ (2009). In Coraggio
_et al._ (2022), a shell-model calculation was conducted for 100Mo and 100Ru,
starting from a realistic nucleon-nucleon potential and deriving the effective
shell-model Hamiltonian and decay operators within many-body perturbation
theory, with a focus on studying neutrinoless double-$\beta$ decay. In Bakshi
_et al._ (2022), the multi-quasiparticle triaxial projected shell model was
used to investigate the band structures of 98-106Ru isotopes, providing a
consistent description. In Jiang  _et al._ (2016), the odd-even and even-even
isotopes of 95-102Ru were studied using the nucleon pair approximation with a
phenomenological pairing plus quadrupole interaction, yielding good agreement
with experimental data.

Other works that provide insights into the nature of this mass region include
the following studies. In Giannatiempo  _et al._ (2016); Giannatiempo (2017),
the authors analyzed 98Ru within the framework of the IBM-2 Arima  _et al._
(1977) and concluded that a clear vibrational pattern is present. In Kisyov
_et al._ (2016), even-even and even-odd Ru isotopes were investigated using
the IBM Iachello and Arima (1987), revealing a transitional behavior. The
g-factors of Ru and Pd nuclei were calculated using the IBM-2 in Giannatiempo
(2013). The even-even 98-110Ru isotopes were studied using the affine
$\widehat{SU(1,1)}$ Lie Algebra in Jafarizadeh  _et al._ (2016). In Böyükata
_et al._ (2010), the $A=100$ region was described using the IBM-1 with a
single Hamiltonian featuring constant parameters. Although this work captured
the overall trends well, it could not reproduce the fine details of the
spectra, especially the rapid shape evolution observed around neutron number
$60$. In Lévai and Arias (2021), the even-even Mo isotopes were investigated
using a Bohr Hamiltonian with a sextic potential in the $\beta$ direction,
without dependence on $\gamma$. This approach provided a good description of
the spectra across the entire chain of isotopes, and it was concluded that
104Ru is the closest nucleus to the critical point symmetry E(5) Iachello
(2000). In Kern  _et al._ (1995), a large set of isotopes were studied to
identify good candidates for vibrational-like behavior, i.e., U(5) nuclei.
Among others, 100Mo and 98-104Ru were identified as suitable candidates. It is
important to note that this work is relatively old, and with the present
experimental knowledge, the conclusions may have evolved.

The present work extends our previous analysis of the $Z\approx 40$ and
$A\approx 100$ region García-Ramos  _et al._ (2003); García-Ramos and Heyde
(2019, 2020); Maya-Barbecho and García-Ramos (2022, ), particularly focusing
on the even-even Mo and Ru isotopes. Mo is an excellent candidate for studying
the influence of intruder states on the onset of deformation for $N\approx 60$
due to the rapid lowering of the energy of the $2_{1}^{+}$ state, a
significant increase in the ratio $E(4_{1}^{+})/E(2_{1}^{+})$, or a sudden
increase of the radius. However, Ru exhibits a smoother trend in these
observables, and the influence of intruder states seems to be minimal.
Nevertheless, the onset of deformation at $N\approx 60$, specifically in
104Ru, has been suggested to generate a relatively flat energy surface,
reminiscent of the concept of critical point symmetry E(5) Frank  _et al._
(2001).

The paper is organized as follows. In Section II, the present experimental
knowledge on Mo and Ru nuclei is reviewed. In Section III, the theoretical
framework used in this work is presented, namely the IBM-CM (interacting boson
model with configuration mixing). The procedure for obtaining the fitting
parameters of the model will also be discussed. In Section IV, the correlation
energy gain is studied. It plays a crucial role in understanding the nuclear
structure and the interaction between nucleons. In Section V, a detailed
comparison between theory and experiment for excitation energies and E2
transition rates is presented. This analysis will provide insights into the
agreement between the theoretical predictions of the IBM-CM and the
experimental data. In Section VI, the wave functions of the nuclear states are
analyzed. The structure and configuration mixing of these states will be
examined, allowing for a deeper understanding of the underlying nuclear
dynamics. In Section VII, additional observables, including radii, isotopic
shifts, and two-neutron separation energies, are studied. These quantities
provide valuable information about the nuclear shapes, deformations, and
binding energies. In Section VIII, a calculation of the IBM mean-field energy
surfaces and an investigation of the deformations in Mo and Ru nuclei is
presented. In Section IX, an analysis of the possible existence of a quantum
phase transition in the studied nuclei is discussed. Finally, in Section X,
the summary and the conclusions of the paper are presented.

##  II Experimental data in the even-even Mo and Ru nuclei

The energy systematics of isotopes 92-110Mo below 3 MeV are presented in Fig.
1, which illustrates a significant increase in the density of states in the
lower part of the spectrum as the mass increases. Additionally, a
transformation from a vibrational-like pattern to a rotational one is
observed, beginning at 102Mo and continuing for heavier isotopes. Another
important observation is the decrease in energy of the first excited $0^{+}$
state, with the minimum also occurring at 102Mo. Throughout the entire chain,
the energies of the $4_{1}^{+}$ and $2_{2}^{+}$ states are relatively close,
almost degenerate in the case of 108Mo, indicating the presence of a certain
degree of $\gamma$ softness in the heavier isotopes.

Fig. 2 depicts the energy systematics of 94-114Ru isotopes. A clear
vibrational pattern is evident for isotopes 94-102Ru, which later evolves into
a rotational structure with some degree of $\gamma$ softness, as indicated by
the close proximity of the $4_{1}^{+}$ and $2_{2}^{+}$ states. Similarly, a
dropping $0^{+}$ state is observed, with a minimum energy at 102Ru.

Figure 1: The experimental energy level systematics of low-lying positive
parity states for the Mo isotopes are displayed, showing levels up to
approximately 3 MeV in energy. Levels with dashed blue lines likely correspond
to spherical shapes, while those in red represent deformed shapes (for more
details refer to the information in Section II). Figure 2: The same as Fig. 1
but for Ru isotopes.

For the comparison with theoretical calculations, we will consider the
evaluated experimental data from Nuclear Data Sheets publications for specific
isotopes: $A=96$ Abriola and Sonzogni (2008), $A=98$ Singh and Hu (2003); Chen
and Singh (2020), $A=100$ Singh (2008), $A=102$ DeFrenne (2009), $A=104$
Blachot (2007), $A=106$ De Frenne and Negret (2008), $A=108$ Blachot (2008),
$A=110$ DeFrenne (2009), $A=112$ Lalkovski and Kondev (2015), $A=114$ Blachot
(2012). In addition to these sources, we have incorporated the most up-to-date
references for certain isotopes as described below.

An excellent experimental overview of this mass region, including Mo and Ru
isotopes, can be found in Garrett  _et al._ (2022a) with updated references.
In Thomas  _et al._ (2016), $\gamma\gamma$ angular correlation experiments
were conducted to study the low-lying states of 96-98Mo, allowing the
determination of angular momenta and multiple mixing ratios. Detailed Coulomb-
excitation studies of 98Mo and 100Mo were performed in Zielińska  _et al._
(2002) and Wrzosek-Lipska  _et al._ (2012) respectively. In Ha  _et al._
(2020), 106-108-110Mo nuclei were investigated through $\beta$-delayed
$\gamma$-ray spectroscopy, and for the first time, the $0_{2}^{+}$ band in
108-110Mo was measured. The authors of Freeman  _et al._ (2017) measured
neutron and proton occupancy in 98Mo and 100Mo, revealing a clear change in
the filling of the proton g9/2 shell between the two isotopes.

For Ru isotopes, in Garrett  _et al._ (2020), the $0_{2}^{+}$ and $\gamma$
bands in 98Ru were observed using $\gamma$-ray spectroscopy following the
$\beta$-decay of 98Rh, as well as via the 100Ru(p, t) reaction. The
$0_{2}^{+}$ state is suggested to be an intruder state rather than a two-
phonon vibrational state, although the mean-field calculation presented in the
same work does not fully support this hypothesis. The lifetimes of states
$2_{1}^{+}$, $2_{2}^{+}$, and $4_{1}^{+}$ in 98Ru were measured in Radeck  _et
al._ (2012) in an attempt to resolve discrepancies observed in the literature
regarding the lifetime of the $4_{1}^{+}$ state. In Urban  _et al._ (2013),
the $0_{2}^{+}$ band in 102Ru was studied, along with an analysis of the
mixing between the $0_{1}^{+}$ and $0_{2}^{+}$ states to understand the
deformation evolution of the ground state in even-even Ru isotopes.
Measurement of $28$ E2 and $3$ M1 matrix elements involving $17$ low-lying
excited states in 104Ru was conducted using Coulomb excitation in J. Srebrny
and T. Czosnyka and Ch. Droste and S.G. Rohoziński and L. Próchniak and K.
Zajac and K. Pomorski and D. Cline and C.Y. Wu and A. Bäcklin and L.
Hasselgren and R.M. Diamond and D. Habs and H.J. Körner and F.S. Stephens and
C. Baktash and R.P. Kostecki (2006). The $g$-factor of the $2_{1}^{+}$ state
in 96-104Ru was obtained in Taylor  _et al._ (2011). More recently, in Garrett
_et al._ (2022b) a Coulomb excitation experiment was conducted for 102Ru
obtaining a little larger E2 matrix elements than the evaluated ones for the
transitions from $2_{1}^{+}$ and $2^{+}_{2}$ to the $0_{1}^{+}$ state, and
measuring for the first time the E2 matrix element of the transition
$2_{3}^{+}\rightarrow 0_{1}^{+}$. They concluded that the $0_{2}^{+}$ state,
with $\beta\approx 0.18$ is a little less deformed than the ground state, with
$\beta\approx 0.24$.

##  III The Interacting Boson Model with configuration mixing formalism

###  III.1 The formalism

The framework used in this work is the IBM-CM (interacting boson model with
configuration mixing). This model is an extension of the original IBM
(interacting boson model) proposed by Arima and Iachello Iachello and Arima
(1987). The IBM-CM allows for the simultaneous treatment of multiple boson
configurations corresponding to particle-hole excitations across a shell or
subshell closure Duval and Barrett (1981, 1982).

In this version of the model, known as IBM-1, no distinction is made between
proton and neutron bosons or particles and holes. For the study of Mo and Ru
nuclei, we consider the closure for protons of the $Z=40$ subshell, where the
regular states correspond to a 0h-2p (0 holes and 2 particles) proton
configuration for Mo and to a 0h-4p proton configuration for Ru, and the
intruder states correspond to a 2h-4p proton configuration for Mo and to a
2h-6p proton configuration for Ru. The number of valence neutrons is
determined considering a neutron closed shell at $N=50$. Hence, the number of
valence bosons, denoted as $N$, will be half of the sum of the valence
protons, which is $2$ for Mo and $4$ for Ru, plus half the number of valence
neutrons. The intruder configuration will have additionally $2$ bosons.
Therefore, the regular and intruder spaces will form a $[N]\oplus[N+2]$
Hilbert space.

The Hamiltonian of the system consists of two sectors: one corresponding to
the regular part, $[N]$, and another corresponding to the intruder part,
$[N+2]$. The total Hamiltonian is written as follows:

$\hat{H}=\hat{P}^{{\dagger}}_{N}\hat{H}^{N}_{\rm
ecqf}\hat{P}_{N}+\hat{P}^{{\dagger}}_{N+2}\left(\hat{H}^{N+2}_{\rm
ecqf}+\Delta^{N+2}\right)\hat{P}_{N+2}\ +\hat{V}_{\rm mix}^{N,N+2}~{},$ (1)

where $\hat{P}_{N}$ and $\hat{P}_{N+2}$ are projection operators onto the
$[N]$ and the $[N+2]$ boson subspaces, respectively,

$\hat{H}^{i}_{\rm
ecqf}=\varepsilon_{i}\hat{n}_{d}+\kappa^{\prime}_{i}\hat{L}\cdot\hat{L}+\kappa_{i}\hat{Q}(\chi_{i})\cdot\hat{Q}(\chi_{i})$
(2)

is the Hamiltonian of the extended consistent-Q formalism (ECQF), Warner and
Casten (1983); Lipas  _et al._ (1985) with $i=N,N+2$, $\hat{n}_{d}$ is the $d$
boson number operator,
$\hat{L}=\sqrt{10}\left[d^{\dagger}\times\tilde{d}\right]^{(1)}$ is the
angular momentum, and
$\hat{Q}(\chi)=\left[s^{\dagger}\times\tilde{d}+d^{\dagger}\times
s\right]^{(2)}+\chi\left[d^{\dagger}\times\tilde{d}\right]^{(2)}$ is the
quadrupole operator. Note that the ECQF corresponds to a simplified version of
the general IBM Hamiltonian. The parameter $\Delta^{N+2}$ represents the
energy needed to excite two proton particles across the $Z=40$ subshell gap,
resulting in 2p-2h excitations. The operator $\hat{V}_{\rm mix}^{N,N+2}$ is
the mixing between the $N$ and the $N+2$ configurations and is given by

$\hat{V}_{\rm mix}^{N,N+2}=\omega_{0}^{N,N+2}(s^{\dagger}\times
s^{\dagger}+s\times s)+\omega_{2}^{N,N+2}(d^{\dagger}\times
d^{\dagger}+\tilde{d}\times\tilde{d})^{(0)}.$ (3)

In this study, we assume that $\omega_{0}^{N,N+2}=\omega_{2}^{N,N+2}=\omega$,
where $\omega$ is a constant parameter.

The $E2$ transition operator is built with the same quadrupole operator that
appears in the Hamiltonian (2). It is defined as the sum of two contributions
that act separately in the regular and the intruder sectors without crossed
contributions,

$\hat{T}(E2)_{\mu}=\sum_{i=N,N+2}e_{i}\hat{P}_{i}^{\dagger}\hat{Q}_{\mu}(\chi_{i})\hat{P}_{i}~{}.$
(4)

The $e_{i}$ ($i=N,N+2$) are the effective boson charges and the parameters
$\chi_{i}$ take the same values that in the Hamiltonian (2). Note that the
operator cannot connect the regular with the intruder sector or viceversa.

The free parameters associated with the above operators need to be determined
in order to reproduce a set of excitation energies and transition rates, as
described in Section III.2.

This approach has been successfully employed in recent studies for Sr Maya-
Barbecho and García-Ramos (2022), Zr García-Ramos and Heyde (2019, 2020), Pt
García-Ramos and Heyde (2009); García-Ramos  _et al._ (2011), Hg García-Ramos
and Heyde (2014, 2015a) and Po isotopes García-Ramos and Heyde (2015b, c).

###  III.2 The fitting procedure: energy spectra and absolute $B(E2)$ reduced
transition probabilities

Table 1: Hamiltonian and $\hat{T}(E2)$ parameters resulting from the study of Mo isotopes in the present work. All quantities have dimensions of energy (given in keV), except $\chi_{N}$ and $\chi_{N+2}$, which are dimensionless, and $e_{N}$ and $e_{N+2}$, which are given in units of $\sqrt{\text{W.u.}}$. It should be noted that the value of $\Delta^{N+2}=1500$ keV is fixed for all isotopes. Nucleus | $\varepsilon_{N}$ | $\kappa_{N}$ | $\chi_{N}$ | $\kappa^{\prime}_{N}$ | $\varepsilon_{N+2}$ | $\kappa_{N+2}$ | $\chi_{N+2}$ | $\kappa^{\prime}_{N+2}$ | $\omega$ | $e_{N}$ | $e_{N+2}$
---|---|---|---|---|---|---|---|---|---|---|---
96Mo | 695.84 | 0.00 | 1.50 | 15.00 | 191.56 | -9.96 | 1.12 | 13.69 | 45.0 | 2.48 | 4.00
98Mo | 813.16 | 0.00 | -1.70 | -5.00 | 873.21 | -25.08 | 1.50 | -5.00 | 45.0 | 2.73 | -0.87
100Mo | 517.29 | -2.00 | -1.49 | 10.00 | 408.52 | -21.98 | 0.05 | 4.34 | 45.0 | 2.24 | 2.88
102Mo | 470.23 | -4.93 | -1.50 | -5.00 | 446.78 | -35.00 | 0.14 | 1.08 | 15.0 | 4.00 | -2.15
104Mo | 150.00 | -10.00 | -1.34 | 7.02 | 450.04 | -35.00 | 0.49 | -0.63 | 15.0 | 2.11 | -2.05
106Mo | 294.38 | -15.00 | -0.58 | 0.02 | 263.83 | -29.90 | 0.37 | 7.68 | 15.0 | 1.90 | 1.93
108Mo | 294.38111Hamiltonian parameters for the regular sector taken from 106Mo. | -15.00111Hamiltonian parameters for the regular sector taken from 106Mo. | -0.58111Hamiltonian parameters for the regular sector taken from 106Mo. | 0.02111Hamiltonian parameters for the regular sector taken from 106Mo. | 203.58 | -28.85 | 0.12 | 8.98 | 15.0 | 1.90222$\hat{T}(E2)$ regular parameter taken from 106Mo. | 2.12
110Mo | 294.38111Hamiltonian parameters for the regular sector taken from 106Mo. | -15.00111Hamiltonian parameters for the regular sector taken from 106Mo. | -0.58222$\hat{T}(E2)$ regular parameter taken from 106Mo. | 0.02111Hamiltonian parameters for the regular sector taken from 106Mo. | 200.00 | -31.79 | 0.01 | 7.58 | 15.0 | 1.90222$\hat{T}(E2)$ regular parameter taken from 106Mo. | 2.12333$\hat{T}(E2)$ intruder parameter taken from 108Mo.
Table 2: Hamiltonian and $\hat{T}(E2)$ parameters resulting from the study of Ru isotopes in the present work. All quantities have dimensions of energy (given in keV), except $e_{N}$ and $e_{N+2}$, which are given in units of $\sqrt{\text{W.u.}}$. It should be noted that the values $\chi_{N}=\chi_{N+2}=0$, $\kappa^{\prime}_{N+2}=0$ keV, $\omega=15$ keV and $\Delta^{N+2}=2200$ keV were fixed for all isotopes. Nucleus | $\varepsilon_{N}$ | $\kappa_{N}$ | $\kappa^{\prime}_{N}$ | $\varepsilon_{N+2}$ | $\kappa_{N+2}$ | $e_{N}$ | $e_{N+2}$
---|---|---|---|---|---|---|---
98Ru | 683.60 | -15.79 | -1.28 | 410.72111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru. | -24.08111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru. | 2.55 | 4.00111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru.
100Ru | 546.85 | -19.89 | 8.60 | 410.72 | -24.08 | 2.34 | 4.00
102Ru | 535.28 | -20.46 | 4.75 | 410.72111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru. | -24.08111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru. | 2.27 | 4.00111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru.
104Ru | 412.75 | -24.34 | 7.03 | 410.72111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru. | -24.08111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru. | 2.11 | 4.00111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru.
106Ru | 360.60 | -26.37 | 5.67 | 410.72111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru. | -24.08111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru. | 1.96 | 4.00111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru.
108Ru | 298.53 | -29.19 | 7.60 | 410.72111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru. | -24.08111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru. | 1.67 | 4.00111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru.
110Ru | 250.00 | -30.00 | 9.55 | 410.72111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru. | -24.08111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru. | 1.55 | 4.00111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru.
112Ru | 250.00 | -30.00 | 6.43 | 410.72111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru. | -24.08111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru. | 1.76 | 4.00111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru.
114Ru | 299.33 | -34.25 | 5.55 | 410.72111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru. | -24.08111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru. | 1.80 | 4.00111Hamiltonian and $\hat{T}(E2)$ parameters for the intruder sector taken from 100Ru.

In this section, we describe how the parameters of the Hamiltonian (1), (2),
and (3), as well as the effective charges of the $\hat{T}(E2)$ transition
operator (4), were determined.

We focus on studying the even-even isotopes 96-110Mo and 98-114Ru, covering a
large portion of the neutron shell $50-82$. We exclude nuclei very close to
the neutron shell closure due to the limited reliability of IBM results for
those cases.

The goal of the fitting procedure is to achieve a satisfactory overall
agreement with the available excitation energies and $B(E2)$ reduced
transition probabilities. A standard $\chi^{2}$ method is used to determine
the values of the parameters appearing in the Hamiltonian and the
$\hat{T}(E2)$ operator, following the approach described in García-Ramos and
Heyde (2009, 2014, 2015b, 2019). In general, there are $13$ parameters
involved, but the number may be smaller for most nuclei. We impose the
constraint that the parameters should vary smoothly from one isotope to
another. Additionally, we strive to keep as many parameters as possible at
constant values, particularly the parameters $\Delta^{N+2}$ and $\omega$.

The resulting parameter values for the IBM-CM Hamiltonian and $\hat{T}(E2)$
operator are presented in Tables 1 and 2 for Mo and Ru, respectively. In these
tables, certain parameters could not be determined unambiguously from the
available experimental information. Specifically, the parameters corresponding
to the intruder sector of most Ru isotopes and the parameters of the regular
sector of 108-110Mo. For Ru, only a few $0^{+}$ states in 100Ru could be
identified as intruder members, but there is no strong evidence of other
intruder states in the rest of the isotope chain. Therefore, we have to assume
the same intruder parameters obtained for 100Ru for the entire Ru chain. As a
result, the description of Ru intruder states should be considered only as
approximate. In the case of 108-110Mo, no evidence of regular states exists,
so the regular parameters of 106Mo are used for those isotopes.

It is worth noting the smooth variation or constancy of certain parameters,
such as $\chi_{N}=\chi_{N+2}=0$ and $\kappa^{\prime}_{N+2}=0$ keV in the Ru
chain. Thus, both configurations in Ru exhibit $\gamma$-unstability. The value
of $\omega$ is constant in Ru and in the majority of Mo isotopes, with
$\omega=15$ keV, except for 96-100Mo where $\omega=45$ keV. A value of
$\Delta^{N+2}=1500$ keV is employed for the entire Mo chain, which is
compatible with the values used for Zr ($\Delta^{N+2}=820-3200$ keV) and Sr
($\Delta^{N+2}=1360-1900$ keV). For the entire Ru chain, a value of
$\Delta^{N+2}=2200$ keV is considered, but it should be noted that this value
is only constrained by the experimental information of 100Ru.

|
---|---
Figure 3: Absolute energy of the lowest unperturbed regular (red) and intruder
(green) 0${}^{+}_{1}$ states for 96-110Mo (a) and 98-114Ru (b). Thin dashed
lines correspond to the reference lines for regular (red) and intruder (green)
configurations (see text).

##  IV Correlation energy and unperturbed energy spectra

The position of intruder states is generally expected to be at higher energy
compared to regular states due to the creation of a 2p-2h excitation across
the shell gap. The parameter $\Delta^{N+2}=1500$ keV for Mo and
$\Delta^{N+2}=2200$ keV for Ru represents the energy needed for this
excitation. However, in practice, the energy of the intruder states is
corrected by the pairing energy gain resulting from the formation of two extra
$0^{+}$ pairs Heyde  _et al._ (1985, 1987).

The presence of extra bosons leads to a reduction in energy for the considered
configuration. As a result, the lowest energies are expected to appear around
the mid-shell, and the reduction in energy will be more significant for the
intruder configuration due to the larger number of bosons (two units larger).
Therefore, the actual energies of intruder states may differ from the initial
expectation based solely on the shell gap energy.

|
---|---
Figure 4:  Energy spectra for the Mo isotopes (panel (a)) and Ru isotopes
(panel (b)), obtained from the IBM-CM Hamiltonian presented in Table 1 and 2,
respectively. For these calculations, the mixing term in the Hamiltonian has
been switched off. For each angular momentum, the energy levels of the two
lowest-lying regular and intruder states are displayed. The regular states are
represented by full, while the intruder states are shown with dashed lines.

To gain a better understanding of the energy systematics in the regular and
intruder configurations, we can examine the absolute energies of the lowest
regular and intruder $0^{+}$ states by suppressing the mixing term in the IBM-
CM Hamiltonian. In this analysis, we consider pure states where the reference
regular energy corresponds to $0$, while the intruder energy corresponds to
$\Delta^{N+2}$. In Fig. 3, panel (a) depicts the energy curves for Mo
isotopes, while panel (b) shows the energy curves for Ru isotopes. In the case
of Mo isotopes, a notable observation is the crossing of the regular and
intruder energies between $A=100$ and $A=102$, corresponding to a neutron
number of $60$. From $A=102$ onwards, the intruder configuration becomes the
ground state. The correlation energy of the regular configuration increases
gradually, while the intruder configuration exhibits a much more rapid change.
This behavior can be attributed to the larger number of bosons in the intruder
configuration and to have $|\kappa_{N+2}|>|\kappa_{N}|$. The minima of both
curves occur precisely at the mid-shell region.

Turning to the case of Ru isotopes, we observe that the regular configuration
remains the lowest energy state throughout the isotopic chain. Notably, the
gain in correlation energy is more significant for the regular state. However,
it is important to note that the Hamiltonian parameters for the intruder
sector are kept constant and are the same as those used for 100Ru. Therefore,
these results should be interpreted with caution, considering the potential
limitations of using the same Hamiltonian parameters in the intruder sector
for all Ru isotopes.

Overall, these energy correlation plots provide insights into the relative
energies of the regular and intruder configurations in Mo and Ru isotopes,
highlighting their different behaviors and the impact of correlation energy on
their ordering.

To gain an initial understanding of the distribution of intruder and regular
states in the spectrum, we can examine the unperturbed spectra of Mo and Ru
isotopes, with the regular ground state as the reference. Fig. 4 illustrates
these spectra, with panel (a) representing Mo isotopes and panel (b)
representing Ru isotopes. Full lines for the regular states and dashed lines
for the intruder ones.

In the case of Mo isotopes (panel (a)), we observe that the regular
configuration exhibits a more vibrational character for the lighter isotopes.
However, as the mass increases, it switches to a more rotational or O(6)
behavior. This transition is evident from the reduction in excitation energies
of the $2^{+}$ and $4^{+}$ states. On the other hand, the intruder
configuration demonstrates a more prominent rotational character throughout
the isotopic chain, which becomes more evident for isotopes with $A>100$.

In the case of Ru isotopes (panel (b)), the intruder configuration is
considerably higher in energy, with a minimum occurring at $A=98-102$. Beyond
this region, the energy of the intruder configuration increases rapidly with
increasing mass. The regular configuration, on the other hand, displays a
clear vibrational pattern in the lighter isotopes. It is possible to easily
identify the two-phonon triplet ($4^{+}$, $2^{+}$, $0^{+}$) or the three-
phonon quintuplet ($6^{+}$, $4^{+}$, $3^{+}$, $2^{+}$, $0^{+}$). However, in
the heavier isotopes, the presence of a $\gamma$-unstable structure becomes
evident. The clear manifestation of this structure is seen in the seniority
two doublet ($2^{+}$ and $4^{+}$) and the seniority three quartet ($6^{+}$,
$4^{+}$, $3^{+}$, and $0^{+}$). Notably, at $A=104$ (neutron number $60$),
there is a transition point where the system switches from one structure to
the other.

Overall, these unperturbed spectra provide valuable insights into the nature
of intruder and regular states in Mo and Ru isotopes, highlighting the
vibrational and rotational characteristics and the presence of clear
structural patterns in the heavier isotopes.

|
---|---
Figure 5: Excitation energies (up to $E\approx 3.0$ MeV) for Mo in panel (a)
with the experimental data and in panel (b) with the IBM-CM theoretical
results. Same information for Ru isotopes, in panel (c) experimental data and
in panel (d) IBM-CM theoretical results. Only two excited states (if known
experimentally) per angular momentum are plotted.

##  V Detailed comparison for energy spectra and $B(E2)$ transition rates

|
---|---
Figure 6: Comparison of the absolute $B(E2)$ transition probabilities along
the yrast band, given in W.u. Panels (a) and (c) corresponds to known
experimental data for Mo and Ru, respectively and panels (b) and (d) to the
theoretical IBM-CM results, also for Mo and Ru, respectively.

In this section, we compare the theoretical calculations with the experimental
data up to an excitation energy of approximately $3$ MeV. Fig. 5 presents a
detailed comparison between the experimental excitation energies of Mo (panel
(a)) and their corresponding theoretical values (panel (b)). Similarly, it
compares the experimental excitation energies of Ru (panel (c)) with their
theoretical counterparts (panel (d)). It is important to note that, throughout
the entire chain, the theoretical $2_{1}^{+}$ excitation energy closely
matches the experimental data. Consequently, we utilize this experimental
energy as a reference to normalize the theoretical spectra (refer to Section
3.2 of Ref. García-Ramos and Heyde (2009) for more details).

Beginning with the Mo isotopes, the spectra of the lighter isotopes exhibit
the expected characteristics for nuclei near the neutron number $50$ shell
closure. Additionally, there is a sudden increase in the excitation energies
of certain states attributed to the presence of the neutron number $56$
subshell closure. Subsequently, the spectra evolve smoothly towards a more
collective behavior, characterized by a compressed spectrum. The IBM-CM
accurately reproduces all of these features, particularly the position of the
$0_{2}^{+}$ state along the entire chain.

In the case of Ru isotopes, the theoretical energies quantitatively reproduce
the experimental ones, clearly depicting a transition from a purely
vibrational-like spectrum (where the one-, two-, and three-phonon states are
easily identified) to an O(6) spectrum. However, it should be noted that the
experimental levels are more scattered compared to the theoretical ones. There
is a notable discrepancy observed for the $0_{3}^{+}$ state in 106Ru, which
appears much higher in energy than predicted by the theoretical model. It is
worth mentioning that this particular state was not included in the fitting
procedure. Additionally, a slight discrepancy is observed for the $0_{2}^{+}$
state in the case of 108-110Ru, where its energy is slightly higher than the
corresponding experimental value.

|
---|---
Figure 7: Comparison of the few non-yrast intraband absolute $B(E2)$
transition probabilities in Mo and Ru, given in W.u. Panels (a) and (c)
correspond to the known experimental data for Mo and Ru, respectively, panel
(b) and (d) to the theoretical IBM-CM results, also for Mo and Ru,
respectively.

The information regarding $B(E2)$ transitions is presented through a series of
tables and figures. Tables LABEL:tab-be2-Mo and LABEL:tab-be2-Ru provide a
detailed comparison between the known experimental values and the
corresponding theoretical ones for Mo and Ru, respectively. Additionally,
figures 6 and 7 illustrate selected intra- and interband transitions,
respectively, highlighting the results for both Mo and Ru.

Overall, there is a satisfactory agreement between the theoretical predictions
and experimental data, with only a few specific discrepancies. Notably, in
98Mo, the model predicts a smaller $B(E2;2_{2}^{+}\rightarrow 2_{1}^{+})$
value than observed experimentally. Additionally, it is worth mentioning the
nearly equal $B(E2)$ values for transitions within the yrast band. Generally,
in both chains of isotopes, the natural increase of $B(E2)$ values with
angular momentum, reaching a maximum at mid-shell, is accurately reproduced by
the model.

The reliable reproduction of the transition rates serves as a stringent test
for the model. Therefore, the agreement obtained between theory and experiment
demonstrates the reliability of the presented calculations, particularly in
cases where experimental information is more abundant.

Figure 8: Experimental excitation energies and absolute $B(E2)$ transition rates (given in W.u.) for selected states in 100-108Mo. Figure 9: The same as Fig. 8 but for the theoretical IBM-CM results.  Table 3: Comparison of the experimental absolute $B(E2)$ values (given in W.u.) with the IBM-CM Hamiltonian results for Mo isotopes. Data are taken from the Nuclear Data Sheets Abriola and Sonzogni (2008); Singh and Hu (2003); Chen and Singh (2020); Singh (2008); DeFrenne (2009); Blachot (2007); De Frenne and Negret (2008); Blachot (2008); DeFrenne (2009); Lalkovski and Kondev (2015); Blachot (2012) complemented with references presented in section II.  Isotope |  Transition |  Experiment |  IBM-CM
---|---|---|---
96Mo |  $2_{1}^{+}\rightarrow 0_{1}^{+}$ |  20.7(4) |  20.7
|  $0_{2}^{+}\rightarrow 2_{1}^{+}$ |  51(7) |  40
|  $2_{2}^{+}\rightarrow 2_{1}^{+}$ |  16.4(24) |  27.5
|  $2_{2}^{+}\rightarrow 0_{1}^{+}$ |  1.10(11) |  0.14
|  $2_{3}^{+}\rightarrow 2_{1}^{+}$ |  $<28$ |  5
|  $2_{3}^{+}\rightarrow 0_{1}^{+}$ |  $<0.18$ |  0.32
|  $2_{4}^{+}\rightarrow 2_{1}^{+}$ |  0.43(20) |  0.07
|  $2_{4}^{+}\rightarrow 0_{1}^{+}$ |  0.080(11) |  0.022
|  $4_{1}^{+}\rightarrow 2_{1}^{+}$ |  41(7) |  34
|  $4_{2}^{+}\rightarrow 4_{1}^{+}$ |  0.18$(^{+9}_{-10})$ |  14
|  $4_{2}^{+}\rightarrow 2_{2}^{+}$ |  22.0$(^{+6}_{-10})$ |  56.4
|  $4_{2}^{+}\rightarrow 2_{1}^{+}$ |  1.9$(^{+5}_{-9})$ |  0.0
|  $4_{3}^{+}\rightarrow 2_{2}^{+}$ |  $<72$ |  7
|  $4_{3}^{+}\rightarrow 2_{1}^{+}$ |  $<0.72$ |  0.08
|  $2_{5}^{+}\rightarrow 3_{1}^{+}$ |  140$(^{+30}_{-40})$ |  115
|  $2_{5}^{+}\rightarrow 2_{3}^{+}$ |  4$(^{+8}_{-4})$ |  12
|  $6_{1}^{+}\rightarrow 4_{1}^{+}$ |  $<290$ |  79
|  $6_{2}^{+}\rightarrow 4_{3}^{+}$ |  $<52$ |  96
|  $6_{2}^{+}\rightarrow 4_{2}^{+}$ |  $<1.1$ |  1.8
|  $6_{2}^{+}\rightarrow 4_{1}^{+}$ |  $<47$ |  0.5
|  $6_{2}^{+}\rightarrow 6_{1}^{+}$ |  $<65$ |  17
|  $3_{1}^{+}\rightarrow 2_{2}^{+}$ |  $<1.8$ |  25
|  $3_{1}^{+}\rightarrow 2_{1}^{+}$ |  $<1.3$ |  0.06
|  $5_{1}^{+}\rightarrow 4_{3}^{+}$ |  $<1700$ |  20
|  $5_{1}^{+}\rightarrow 3_{1}^{+}$ |  $<2000$ |  69
|  $5_{1}^{+}\rightarrow 4_{2}^{+}$ |  $<100$ |  3
|  $3_{3}^{+}\rightarrow 4_{1}^{+}$ |  2.7$(^{+1.7}_{-2.7})$ |  0.1
|  $3_{3}^{+}\rightarrow 2_{3}^{+}$ |  4.1$(^{+2.4}_{-4.1})$ |  0.1
|  $3_{3}^{+}\rightarrow 2_{1}^{+}$ |  0.19$(^{+10}_{-19})$ |  0.00
98Mo |  $2_{1}^{+}\rightarrow 0_{1}^{+}$ |  20.1(4) |  19
|  $2_{1}^{+}\rightarrow 0_{2}^{+}$ |  9.7$(^{+10}_{-25})$ |  6.5
|  $4_{1}^{+}\rightarrow 2_{1}^{+}$ |  42.3$(^{+9}_{-8})$ |  23.5
|  $6_{1}^{+}\rightarrow 4_{1}^{+}$ |  10.1(4) |  23.3
|  $2_{2}^{+}\rightarrow 0_{1}^{+}$ |  1.02$(^{+15}_{-12})$ |  7.90
|  $2_{2}^{+}\rightarrow 0_{2}^{+}$ |  2.3$(^{+5}_{-4})$ |  2.3
|  $2_{2}^{+}\rightarrow 2_{1}^{+}$ |  48$(^{+9}_{-8})$ |  2
|  $4_{1}^{+}\rightarrow 2_{2}^{+}$ |  15.2$(^{+33}_{-30})$ |  16.6
|  $2_{3}^{+}\rightarrow 4_{1}^{+}$ |  14(4) |  5
|  $2_{3}^{+}\rightarrow 2_{2}^{+}$ |  $<22$ |  3.5
|  $2_{3}^{+}\rightarrow 2_{1}^{+}$ |  3.0(7) |  30.9
|  $2_{3}^{+}\rightarrow 0_{2}^{+}$ |  7.5$(^{+6}_{-5})$ |  0.0
|  $2_{3}^{+}\rightarrow 0_{1}^{+}$ |  0.032$(^{+7}_{-6})$ |  0.228
100Mo |  $2_{1}^{+}\rightarrow 0_{1}^{+}$ |  37.6(9) |  37.5
|  $0_{2}^{+}\rightarrow 2_{1}^{+}$ |  89(3) |  76
|  $4_{1}^{+}\rightarrow 2_{1}^{+}$ |  69(6) |  71
|  $6_{1}^{+}\rightarrow 4_{1}^{+}$ |  94$(^{+16}_{-12})$ |  100
|  $2_{2}^{+}\rightarrow 0_{1}^{+}$ |  0.62(6) |  0.03
|  $2_{2}^{+}\rightarrow 0_{2}^{+}$ |  5.7$(^{+14}_{-11})$ |  2.2
|  $2_{2}^{+}\rightarrow 2_{1}^{+}$ |  52(7) |  70
|  $2_{3}^{+}\rightarrow 4_{1}^{+}$ |  36$(^{+34}_{-20})$ |  42
|  $2_{3}^{+}\rightarrow 0_{2}^{+}$ |  15$(^{+5}_{-3})$ |  70
|  $2_{3}^{+}\rightarrow 2_{1}^{+}$ |  0.28$(^{+15}_{-9})$ |  0.11
|  $4_{2}^{+}\rightarrow 2_{2}^{+}$ |  30$(^{+7}_{-5})$ |  21
|  $8_{1}^{+}\rightarrow 6_{1}^{+}$ |  122$(^{+23}_{-17})$ |  121
102Mo |  $2_{1}^{+}\rightarrow 0_{1}^{+}$ |  74(9) |  74
|  $0_{2}^{+}\rightarrow 2_{1}^{+}$ |  70(30) |  30
|  $4_{1}^{+}\rightarrow 2_{1}^{+}$ |  89(18) |  105
104Mo |  $2_{1}^{+}\rightarrow 0_{1}^{+}$ |  92(6) |  92
|  $4_{1}^{+}\rightarrow 2_{1}^{+}$ |  110(4) |  133
|  $6_{1}^{+}\rightarrow 4_{1}^{+}$ |  109(4) |  144
|  $8_{1}^{+}\rightarrow 6_{1}^{+}$ |  81(4) |  138
106Mo |  $2_{1}^{+}\rightarrow 0_{1}^{+}$ |  102.3(25) |  102.3
|  $4_{1}^{+}\rightarrow 2_{1}^{+}$ |  140 (30) |  146
|  $6_{1}^{+}\rightarrow 4_{1}^{+}$ |  130(60) |  159
|  $8_{1}^{+}\rightarrow 6_{1}^{+}$ |  89(12) |  160
|  $10_{1}^{+}\rightarrow 8_{1}^{+}$ |  93(13) |  66
108Mo |  $2_{1}^{+}\rightarrow 0_{1}^{+}$ |  140(90) |  140
Table 4: Same as Table LABEL:tab-be2-Mo but for Ru isotopes. Isotope |  Transition |  Experiment |  IBM-CM
---|---|---|---
98Ru |  $2_{1}^{+}\rightarrow 0_{1}^{+}$ |  29.8(10) |  29.8
|  $0_{2}^{+}\rightarrow 2_{1}^{+}$ |  42 $(^{+12}_{-11})$ |  33
|  $4_{1}^{+}\rightarrow 2_{1}^{+}$ |  57(4) |  43
|  $6_{1}^{+}\rightarrow 4_{1}^{+}$ |  12.8$(^{+17}_{-14})$ |  40.9
|  $2_{2}^{+}\rightarrow 0_{1}^{+}$ |  1.04$(^{+17}_{-14})$ |  0.00
|  $2_{2}^{+}\rightarrow 2_{1}^{+}$ |  46$(^{+7}_{-6})$ |  43
|  $8_{1}^{+}\rightarrow 6_{1}^{+}$ |  $2.5(^{+5}_{-3})$ |  26
|  $10_{1}^{+}\rightarrow 8_{1}^{+}$ |  1.27$(^{+31}_{-23})$ |  0.43
100Ru |  $2_{1}^{+}\rightarrow 0_{1}^{+}$ |  35.7(3) |  35.8
|  $0_{2}^{+}\rightarrow 2_{1}^{+}$ |  35(5) |  32
|  $4_{1}^{+}\rightarrow 2_{1}^{+}$ |  52(4) |  52
|  $2_{2}^{+}\rightarrow 2_{1}^{+}$ |  31(6) |  52
|  $2_{2}^{+}\rightarrow 0_{1}^{+}$ |  2.0(4) |  0.0
|  $2_{3}^{+}\rightarrow 4_{1}^{+}$ |  17(5) |  11
|  $2_{3}^{+}\rightarrow 0_{2}^{+}$ |  37$(^{+8}_{-9})$ |  23
|  $2_{3}^{+}\rightarrow 2_{1}^{+}$ |  1.24$(^{+0.43}_{-0.53})$ |  0.00
|  $2_{3}^{+}\rightarrow 0_{1}^{+}$ |  0.43(10) |  0.02
|  $2_{4}^{+}\rightarrow 0_{3}^{+}$ |  270$(^{+60}_{-50})$ |  196
|  $2_{4}^{+}\rightarrow 4_{1}^{+}$ |  1.9$(^{+0.6}_{-0.5})$ |  0.2
|  $2_{4}^{+}\rightarrow 0_{2}^{+}$ |  1.9(5) |  0.0
|  $4_{2}^{+}\rightarrow 2_{2}^{+}$ |  41$(^{+27}_{-21})$ |  29
|  $4_{2}^{+}\rightarrow 4_{1}^{+}$ |  27$(^{+18}_{-14})$ |  26
|  $4_{2}^{+}\rightarrow 2_{1}^{+}$ |  1.9$(^{+13}_{-10})$ |  0.0
|  $4_{3}^{+}\rightarrow 2_{3}^{+}$ |  77$(^{+32}_{-29})$ |  7
|  $4_{3}^{+}\rightarrow 4_{1}^{+}$ |  1.8$(^{+9}_{-8})$ |  0
|  $4_{3}^{+}\rightarrow 2_{1}^{+}$ |  0.9(4) |  0.3
|  $3_{1}^{+}\rightarrow 2_{2}^{+}$ |  9.6$(^{+46}_{-41})$ |  39.2
|  $3_{1}^{+}\rightarrow 4_{1}^{+}$ |  15(5) |  16
|  $3_{1}^{+}\rightarrow 2_{1}^{+}$ |  3.9$(^{+13}_{-12})$ |  0
102Ru |  $2_{1}^{+}\rightarrow 0_{1}^{+}$ |  44.6(7) |  44.6
|  $0_{2}^{+}\rightarrow 2_{1}^{+}$ |  35(6) |  36
|  $4_{1}^{+}\rightarrow 2_{1}^{+}$ |  66(11) |  66
|  $6_{1}^{+}\rightarrow 4_{1}^{+}$ |  68(25) |  74
|  $2_{2}^{+}\rightarrow 0_{1}^{+}$ |  1.14(15) |  0.00
|  $2_{2}^{+}\rightarrow 2_{1}^{+}$ |  $32(5)$ |  66
|  $8_{1}^{+}\rightarrow 6_{1}^{+}$ |  56(19) |  70
|  $10_{1}^{+}\rightarrow 8_{1}^{+}$ |  57(21) |  58
104Ru |  $2_{1}^{+}\rightarrow 0_{1}^{+}$ |  57.9(11) |  57.1
|  $0_{2}^{+}\rightarrow 2_{1}^{+}$ |  25(3) |  21
|  $4_{1}^{+}\rightarrow 2_{1}^{+}$ |  83(9) |  81
|  $2_{2}^{+}\rightarrow 0_{1}^{+}$ |  2.8(5) |  0.0
|  $2_{2}^{+}\rightarrow 2_{1}^{+}$ |  55(6) |  81
106Ru |  $2_{1}^{+}\rightarrow 0_{1}^{+}$ |  66(10) |  66
108Ru |  $2_{1}^{+}\rightarrow 0_{1}^{+}$ |  62(6) |  62
|  $4_{1}^{+}\rightarrow 2_{1}^{+}$ |  102(8) |  85
|  $2_{2}^{+}\rightarrow 0_{1}^{+}$ |  0.5(4) |  0.0
|  $2_{3}^{+}\rightarrow 4_{1}^{+}$ |  0.08(7) |  0.00
|  $2_{3}^{+}\rightarrow 0_{1}^{+}$ |  0.005(3) |  0.000
110Ru |  $2_{1}^{+}\rightarrow 0_{1}^{+}$ |  66(5) |  66
|  $4_{1}^{+}\rightarrow 2_{1}^{+}$ |  86(10) |  91
|  $6_{1}^{+}\rightarrow 4_{1}^{+}$ |  120(50) |  101
|  $2_{2}^{+}\rightarrow 0_{1}^{+}$ |  0.6(3) |  0.0
112Ru |  $2_{1}^{+}\rightarrow 0_{1}^{+}$ |  70(7) |  70
|  $8_{1}^{+}\rightarrow 6_{1}^{+}$ |  82(13) |  106
|  $10_{1}^{+}\rightarrow 8_{1}^{+}$ |  85(13) |  100
|  $7_{1}^{+}\rightarrow 5_{1}^{+}$ |  83(12) |  68

Figs. 8 and 9 display the detailed excitation energies up to approximately $3$
MeV and the $B(E2)$ transition rates for both experimental data and
theoretical results. These figures focus on a selected set of Mo isotopes,
namely 100-108Mo, where the coexistence of two bands is most evident. The
separation into bands has been performed by first considering the yrast band
and then grouping the remaining levels around the $0^{+}$ or $2^{+}$
bandheads, or according to an O(6) scheme.

In the case of 100Mo, the yrast band exhibits a predominantly vibrational
character, which is accurately reproduced by the IBM-CM calculation. However,
the remaining states cannot be easily grouped in any obvious manner. The main
features of the $B(E2)$ transition rates are generally well reproduced, except
for the $2_{3}^{+}\rightarrow 0_{2}^{+}$ transition, where the theoretical
model predicts a larger value than observed experimentally. Moving on to
102Mo, the yrast band begins to deviate from the harmonic behavior in terms of
both energies and $B(E2)$ values. Nevertheless, the energies and $B(E2)$
values are correctly reproduced by the theoretical calculations. In the cases
of 104Mo and 106Mo, the spectra appear quite similar, and the IBM-CM
calculations accurately reproduce them. However, based on the analysis
presented in Section VI, the $0_{2}^{+}$ state is regular in the former case
while is intruder in the latter, and the $2_{2}^{+}$ state is fully mixed in
the former case but is intruder in the latter. Lastly, in the isotope 108Mo,
part of the spectra exhibit an O(6) character, although there are no known
experimental $B(E2)$ values available for comparison. Table LABEL:tab-be2-Mo
provides a comparison between the relevant $B(E2)$ values and their
corresponding theoretical predictions, which complements the information
provided in the former figures.

Figs. 10 and 11 illustrate the experimental and theoretical spectra for the
range of 100-108Ru isotopes. The overall agreement between experiment and
theory is remarkable, with no significant discrepancies observed. As
demonstrated in Fig. 5, nuclei up to 104Ru exhibit a vibrational-like
structure, allowing for the identification of different members of one-, two-,
three-, and even four-phonon multiplets. Starting from 106Ru, a transition
towards an O(6)-like structure becomes evident, with 104Ru being the critical
point of this transition. Another noteworthy feature is the absence of
intruder states, except for the $0_{4}^{+}$ and $0_{5}^{+}$ states in 100Ru
(not shown in the figure). This highlights the consistency of the theoretical
model in reproducing the experimental spectra. Table LABEL:tab-be2-Ru provides
a comparison between the relevant $B(E2)$ values and their corresponding
theoretical predictions for Ru isotopes, which complements the information
provided in the former figures.

Figure 10: Experimental excitation energies and absolute $B(E2)$ transition
rates (given in W.u.) for selected states in 100-108Ru. Figure 11: The same as
Fig. 10 but for theoretical results.

##  VI Wave function structure

In this section, we investigate the structure of wave functions, specifically
focusing on the fraction lying in the regular sector, denoted as $[N]$. The
wave function, within the IBM-CM, can be written as

$\displaystyle\Psi(k,JM)$ $\displaystyle=$
$\displaystyle\sum_{i}a^{k}_{i}(J;N)\psi((sd)^{N}_{i};JM)$ (5)
$\displaystyle+$ $\displaystyle\
\sum_{j}b^{k}_{j}(J;N+2)\psi((sd)^{N+2}_{j};JM)~{},$

where $k$ refers to the different states with a given $J$, while $i$, and $j$
run over the bases of the $[N]$ and $[N+2]$ sectors, respectively. The weight
of the wave function contained within the $[N]$-boson subspace, can then be
defined as the sum of the squared amplitudes,

$w^{k}(J,N)\equiv\sum_{i}\mid a^{k}_{i}(J;N)\mid^{2}.$ (6)

Fig. 12 illustrates $w^{k}(J)$ for the first two states of each angular
momentum, with the full line representing the first state and the dashed line
representing the second state.

For Mo isotopes, the ground state undergoes a rapid transition from an almost
entirely regular structure up to $A=100$, to a fully intruder one from $A=102$
onwards. This trend is also observed for the $2_{1}^{+}$, $4_{1}^{+}$,
$6_{1}^{+}$, and $8_{1}^{+}$ states, with the exception of $A=96$, where the
$6_{1}^{+}$ and $8_{1}^{+}$ states correspond to intruder configurations. No
clear trend can be observed for odd angular momenta, both for the first and
second members. The second $0^{+}$ state predominantly exhibits an intruder
character in most cases, except for $A=102-104$, where it is fully regular. On
the other hand, the second $2^{+}$ state shows significant mixing for
$A=96-100$ and $104$, while being almost purely intruder for the remaining
cases.

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Figure 12: Regular content of the Mo (panels (a) and (b)) and Ru (panels (c)
and (d)) for the two lowest-lying states for each $J$ value (full lines with
closed symbols correspond with the first state while dashed lines correspond
with the second state) resulting from the IBM-CM calculation.

In the case of Ru isotopes, the observed trend is relatively straightforward.
For the first member of all angular momentum states a regular wave function is
predominant. However, for the second member, in some cases it undergoes a
transition from an intruder character to a regular one starting in the
majority of cases from $A=104$ and onwards.

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Figure 13: The energy systematics of the four lowest states below $4$ MeV for
both Mo and Ru isotopes. The panels (a)-(h) represent Mo, while panels (i)-(p)
correspond to Ru. Each panel corresponds to a specific angular momentum: (a)
and (i) for $J=0$, (b) and (j) for $J=2$, (c) and (k) for $J=3$, (d) and (l)
for $J=4$, (e) and (m) for $J=5$, (f) and (n) for $J=6$, (g) and (o) for
$J=7$, and (h) and (p) for $J=8$. The size of each symbol is proportional to
the fraction of the wave function lying in the regular sector. The dot for the
state $0_{1}^{+}$ in 98Ru corresponds to $100\%$.

The representation of Fig. 12 can become cumbersome, especially in cases like
panel (b) where the levels cross each other. To provide a clearer
visualization, it is more effective to combine the fraction of the wave
function within the regular sector with the excitation energy of the states.
In Fig. 13, we present the regular content of the first four states per
angular momentum along with their corresponding excitation energies. The size
of each dot associated with a state is proportional to the regular content of
its wave function. To provide a reference point, the size of the dot for the
$0_{1}^{+}$ states in 98Ru (panel (i)) corresponds to $100\%$ of regular
content. In the case of Mo isotopes, it is evident how the regular content of
the $0_{1}^{+}$ state transitions from the ground state in 96-100Mo to the
first and second excited $0^{+}$ states in 102-104Mo and 106-110Mo,
respectively. Similarly, for the $2^{+}$ and $4^{+}$ states, the regular
content transitions from the first member in 96-102Mo to the second or third
member in 104-110Mo. For angular momenta $6^{+}$ and $8^{+}$, the regular
content is mainly concentrated in the second and third members throughout the
isotopic chain. In contrast, the situation in Ru is much simpler, with the
majority of states belonging to the regular sector, as indicated by their
significant regular content.

##  VII Study of other observables: radii, isotopic shifts, and two-neutron
separation energies

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Figure 14: Charge mean-square radii for the Mo (a) and Ru nuclei (c) and
isotopic shift for the Mo (b) and Ru nuclei (d). The data are taken from
Angeli and Marinova (2013). Lines with dots for theoretical values and dots
with error bars for experimental data. For Mo, $\langle r^{2}\rangle_{ref}$ is
fixed to reproduce the value of $\langle r^{2}\rangle$ in 96Mo, while for Ru
to reproduce 96Ru.

###  VII.1 Radii and isotopic shifts

The nuclear charge radii is an experimental observable and its analysis
provides direct information on the presence of deformation in nuclei. In our
case, we anticipate to obtain from such analysis some indication on the onset
of deformation around neutron number $60$. Specifically, we expect to observe
a kink in the isotope shift at this point. In this section, we will compare
the theoretical values for radii and isotope shifts predicted by the IBM-CM
with the experimental data Angeli and Marinova (2013).

The value of the nucleus’ radius calculated using the IBM is closely
associated with the matrix element of the $\hat{n}_{d}$ operator for the
ground state. This value should be combined with a linear trend that depends
on the number of bosons, which can be easily explained in terms of the liquid
drop model. Additionally, in the case of the IBM-CM, it is necessary to
consider both the regular and intruder configurations. In summary, in the IBM-
CM the nuclear radius can be expressed as,

$r^{2}=r_{c}^{2}+\hat{P}^{{\dagger}}_{N}(\gamma_{N}\hat{N}+\beta_{N}\hat{n}_{d})\hat{P}_{N}+\hat{P}^{{\dagger}}_{N+2}(\gamma_{N+2}\hat{N}+\beta_{N+2}\hat{n}_{d})\hat{P}_{N+2}.$
(7)

The appearing parameters are common for the entire chain of isotopes and are
fixed to best reproduce the experimental data, which are referred to 108Mo and
104Ru, respectively. The resulting values for Mo are $\gamma_{N}=0.221$ fm2,
$\beta_{N}=-0.627$ fm2, $\gamma_{N+2}=0.215$ fm2, and $\beta_{N+2}=-0.024$
fm2, while for Ru, the values are $\gamma_{N}=0.248$ fm2 and $\beta_{N}=0.018$
fm2. In the case of Ru, there is no dependence on the intruder part. This
approach closely follows the methodology of a previous work Zerguine  _et al._
(2012), considering only a single configuration.

In the case of Mo, the sudden increase in radius at 102Mo is correctly
captured, along with the overall trend. However, the experimental and
theoretical results do not coincide within the error bars, similar to what has
been observed in other studies García-Ramos  _et al._ (2011); García-Ramos and
Heyde (2014, 2015b, 2019); Maya-Barbecho and García-Ramos (2022). As a matter
of fact, the onset of deformation is predicted more strongly than observed.
This tiny discrepancy could be connected with the prediction of an equal
degree of deformation for 104-108Ru while experimentally there is a more
gradual variation. For Ru, where only the regular contribution is required,
the linear trend of the radii is accurately reproduced, and no abrupt changes
are observed. Note the different scale between panels (b) and (d).

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Figure 15: Comparison of experimental and theoretical two-neutron separation
energies in Mo (panel (a)) and Ru (panel (b)) isotopes.

###  VII.2 Two-neutron separation energies

The definition of the S2n involves the value of the binding energy of two
neighboring nuclei separated by two mass units, having the same value of Z, as
expressed by the equation,

$S_{2n}(A)=BE(A)-BE(A-2).$ (8)

where $BE$ represents the binding energy of the nucleus, considered as
positive. In the case of the IBM, an additional contribution depending on the
number of neutrons and the square of the number of neutrons needs to be added
to the calculated binding energy (see Fossion  _et al._ (2002)). This
introduces an extra linear term into the S2n value. Therefore, the S2n can be
expressed as:

$S_{2n}(A)={\cal A}+{\cal B}A+BE^{lo}(A)-BE^{lo}(A-2),$ (9)

where $BE^{lo}$ represents the “local” binding energy derived from the IBM
Hamiltonian, and the coefficients ${\cal A}$ and ${\cal B}$ are assumed to be
constant for an isotopic chain Fossion  _et al._ (2002). In the case of IBM-CM
calculations, we anticipate that the effective number of bosons, or
equivalently the mass number, for the ground state will be influenced by the
presence of intruder states. To account for this effect, we propose as an
ansatz,

$S_{2n}(A)={\cal A}+{\cal B}(A+2(1-w))+BE^{lo}(A)-BE^{lo}(A-2),$ (10)

where $w=w^{1}(0)$ (see Eq. (6)). The values of ${\cal A}$ and ${\cal B}$ are
determined, once the $BE^{lo}$’s are known, through a least-squares fit to the
experimental values of S2n, as explained in detail in Fossion  _et al._
(2002); García-Ramos and Heyde (2014, 2019). In our case, the obtained values
are ${\cal A}=55.2$ MeV and ${\cal B}=-0.407$ MeV for Mo and ${\cal A}=65.7$
MeV and ${\cal B}=-0.499$ MeV for Ru.

In Fig. 15, the comparison between experimental and theoretical results is
presented, highlighting the excellent agreement observed in both isotopic
chains around neutron number $60$ ($A=102$ in Mo and $A=104$ in Ru). It is
important to note that only the first portion of the neutron shell was
utilized to determine the parameters ${\cal A}$ and ${\cal B}$. For the Ru
isotopes, a clear linear trend is observed, which is accurately reproduced by
the model. In the case of the Mo isotopes, a slight flattening is observed at
$A=102$, corresponding to neutron number $60$, although the IBM-CM calculation
predominantly exhibits a linear trend. This specific point corresponds to the
intersection of the intruder and regular configurations. It should be noted
that in the Mo isotopes, the value $A=108$ lies beyond the midpoint of the
shell, even though the same linear portion as in the first half-shell is
considered. The observed discrepancy in 110Mo arises from the discussion on
the values of the linear coefficients in the first and second parts of the
shell, as outlined in Fossion  _et al._ (2002).

Figure 16: Axial symmetry energy for 96-110Mo, corresponding to the IBM-CM
Hamiltonian provided in Table 1. The full configuration mixing calculation
(full black line) is shown together with the unperturbed calculations for the
regular sector (red dotted line) and for the intruder configuration (green
dashed line). Figure 17: Matrix coherent-state calculation in the
$\beta-\gamma$ plane for 96-110Mo, corresponding to the IBM-CM Hamiltonian
provided in Table 1. The red dot marks the position of the absolute minimum.

##  VIII Nuclear deformation and mean-field energy surfaces

One of the goals of this work is to analyze the onset of deformation around
neutron number $60$. However, it is important to note that deformation itself
is not directly an observable. Nevertheless, the IBM provides various
approaches for calculating the deformation of a nucleus. The most common
approach is the IBM mean-field method, which allows for the calculation of an
energy functional based on deformation parameters Ginocchio and Kirson (1980);
Dieperink and Scholten (1980); Dieperink  _et al._ (1980). In the case of the
IBM-CM, an expanded formalism was necessary to simultaneously describe both
regular and intruder configurations, which involved the introduction of a
matrix coherent-state method Frank  _et al._ (2002, 2004); Morales  _et al._
(2008). Detailed descriptions of the method and its application to Pt, Hg, Po,
Zr, and Sr isotopes can be found in references García-Ramos  _et al._ (2014);
García-Ramos and Heyde (2014, 2015b, 2019); Maya-Barbecho and García-Ramos
(2022).

Fig. 16 shows the axial energy surfaces and Fig. 17 shows the full
$\beta-\gamma$ plane energy surfaces for Mo isotopes. In the axial case, the
unperturbed calculations for regular (red dotted line) and intruder (green
dashed line) configurations are presented, along with the full calculations
(black solid line). It is evident that the intruder configuration evolves from
a spherical shape to a flat surface in 100Mo, transitioning into an oblate
deformed shape and eventually becoming nearly $\gamma$-unstable. This
configuration gains correlation energy more rapidly than the regular
configuration. On the other hand, the regular configuration evolves slowly
from a spherical shape to a shallow prolate deformed energy surface. The full
calculation clearly depicts the transition from a regular to an intruder
ground state.

When analyzing the energy surfaces in the $\beta-\gamma$ plane (Fig. 17), the
transition from a spherical to an oblate shape can be observed in 102Mo. A
secondary prolate minimum appears in 104Mo, and the nuclei become almost
$\gamma$-unstable in 108-110Mo. Previous HFB calculations using a Gogny
interaction Rodríguez-Guzmán  _et al._ (2010); Nomura  _et al._ (2016) exhibit
similar energy surfaces to the present results, although there 100Mo is
already prolate deformed. From 102Mo onwards, the shape is predominantly
oblate, with triaxial minima and shallow valleys in the $\gamma$ direction.
The coexistence of two minima, one oblate and the other prolate, is present in
104Mo. Note that the situation is similar to the case of Sr where the
coexistence of a prolate and an oblate minimum exists Maya-Barbecho and
García-Ramos (2022), but differs from the Zr nuclei where an spherical and a
prolate minimum coexist García-Ramos and Heyde (2020).

In Fig. 18, the axial energy surfaces for Ru isotopes are shown and the full
$\beta-\gamma$ plane energy surfaces are displayed in Fig. 19. In the axial
case (Fig. 18), both the unperturbed and full calculations are presented. It
is noteworthy that the intruder configuration remains well separated from the
regular configuration throughout, and this is consistent with the full
calculation. The nuclei start out as spherical but gradually evolve into
flatter shapes, reaching full flatness at 104Ru. Subsequently, they transform
into $\gamma$-unstable shapes, with the deepest minimum occurring at 110Ru.

When examining the energy surfaces in the $\beta-\gamma$ plane (Fig. 19),
similar conclusions can be drawn. The flattest energy surface is observed at
104Ru, which serves as a boundary between the spherical and $\gamma$-unstable
shapes. In a previous study Nomura  _et al._ (2016), it was found that the
lightest Ru isotopes exhibit a slightly prolate shape, while a very shallow
triaxial minimum is observed in 104-106Ru. In 108-114Ru, oblate minima are
obtained, albeit with a very flat $\gamma$ direction, and in certain cases,
they become almost triaxial.

Figure 18: Same as Fig. 16 but for 98-114Ru and Table 2. Figure 19: Same as
Fig. 17 but for 98-114Ru and Table 2.

To gain a clearer understanding on the evolution of nuclear shape, it is
useful to represent the value of $\beta$ corresponding to the minimum in the
energy functional (indicated by red dots in Figs. 17 and 19). It is important
to note that the $\beta$ variable defined in the IBM does not directly
correspond to the one used in the collective model. However, there exists a
linear relationship Ginocchio and Kirson (1980) connecting both variables.
Therefore, the variables presented should be understood as being proportional
to the Bohr-Mottelson ones. Note that positive values corresponds to prolate
shapes while negative to oblate.

Fig. 20 illustrates the IBM $\beta$ values for Mo and Ru isotopes in panels
(a) and (b), respectively. The values correspond to the full calculation (IBM-
CM) as well as the regular ($[N]$) and intruder ($[N+2]$) configurations,
assuming no interaction exists between the two configurations.

In the case of Mo isotopes (panel (a)), the regular configuration is spherical
in the range $A=96-102$, transitioning to a prolate shape for $A=104-110$. On
the other hand, the intruder configuration is spherical until $A=100$, but
then it rapidly increases its value, becoming oblate. The full calculation
exhibits a similar behavior to the intruder configuration, transitioning
rapidly from spherical to prolate deformation around $A=102$. It is worth
noting that although the obtained deformation for $A=110$ is oblate, the
potential is almost $\gamma$-unstable. Therefore, the sign of $\beta$ for
110Mo is unimportant. In both configurations, the deformation develops quite
rapidly.

Turning to the case of Ru isotopes (panel (b)), the regular configuration, as
well as the full calculation, corresponds to spherical shapes for 98-102Ru.
However, they undergo a sudden transition to $\gamma$-unstable deformation for
104-114Ru, with the maximum deformation occurring at the mid-shell. The
intruder configuration exhibits a similar behavior, but it is important to
remember that the intruder Hamiltonian remains fixed throughout the entire
isotope chain.

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Figure 20: Value of $\beta$ extracted from the IBM-CM energy surface for Mo
(panel (a)) and Ru (panel (b)) isotopes. $[N]$, $[N+2]$, and IBM-CM correspond
to a pure regular configuration, a pure intruder configuration, and to the
complete calculation.

Lastly, we will also calculate the value of the deformation in an almost
model-independent manner using the work of Kumar, Cline, and their colleagues
Kumar (1972); Cline (1986). This method allows the extraction of deformation
using the experimental information coming from Coulomb excitation, which is a
powerful tool for accessing information about the shape of a nucleus. The key
concept is to utilize the notion of the “equivalent ellipsoid” for a given
nucleus. This ellipsoid is defined as uniformly charged with the same charge,
possessing the same $\left\langle r^{2}\right\rangle$ and $E2$ moments as the
original nucleus characterized by a specific eigenstate.

By analyzing measured data from various transitions obtained through Coulomb
excitation techniques, it is possible to extract the values of collective
model variables, namely $\beta$ and $\gamma$, for a given state. This approach
provides valuable experimental insights into the deformation of nuclei.

Table 5: Values of deformation parameters, $q^{2}$, $\beta$ and $\gamma$, extracted from the quadrupole shape invariants, together with the value of $w^{k}$ ([N] content, see Eq. (6)), for Mo and Ru isotopes. State | Iso |  $q^{2}$ ($e^{2}b^{2}$) |  $\beta$ |  $\gamma$ (deg) | $w^{k}$ |  Iso |  $q^{2}$ ($e^{2}b^{2}$) |  $\beta$ |  $\gamma$ (deg) | $w^{k}$
---|---|---|---|---|---|---|---|---|---|---
0${}_{\text{1}}^{+}$ |  96Mo | 0.28 | 0.17 | 48 | 0.971 |  98Ru | 0.40 | 0.20 | 30 | 0.997
0${}_{\text{2}}^{+}$ |  | 1.63 | 0.42 | 50 | 0.158 |  | 0.35 | 0.18 | 30 | 0.988
0${}_{\text{3}}^{+}$ |  | 0.62 | 0.26 | 46 | 0.579 |  | 2.05 | 0.45 | 30 | 0.012
0${}_{\text{1}}^{+}$ |  98Mo | 0.37 | 0.20 | 14 | 0.881 |  100Ru | 0.50 | 0.22 | 30 | 0.996
0${}_{\text{2}}^{+}$ |  | 0.12 | 0.11 | 14 | 0.122 |  | 0.40 | 0.19 | 30 | 0.995
0${}_{\text{3}}^{+}$ |  | 0.49 | 0.23 | 0 | 0.730 |  | 2.76 | 0.51 | 30 | 0.006
0${}_{\text{1}}^{+}$ |  100Mo | 0.53 | 0.23 | 25 | 0.825 |  102Ru | 0.64 | 0.24 | 30 | 0.994
0${}_{\text{2}}^{+}$ |  | 1.22 | 0.35 | 31 | 0.241 |  | 0.54 | 0.22 | 30 | 0.992
0${}_{\text{3}}^{+}$ |  | 0.64 | 0.26 | 13 | 0.763 |  | 3.60 | 0.57 | 30 | 0.010
0${}_{\text{1}}^{+}$ |  102Mo | 1.16 | 0.34 | 26 | 0.018 |  104Ru | 0.84 | 0.27 | 30 | 0.994
0${}_{\text{2}}^{+}$ |  | 1.48 | 0.39 | 18 | 0.968 |  | 0.56 | 0.22 | 30 | 0.996
0${}_{\text{3}}^{+}$ |  | 0.90 | 0.30 | 23 | 0.020 |  | 0.73 | 0.26 | 30 | 0.996
0${}_{\text{1}}^{+}$ |  104Mo | 1.45 | 0.38 | 43 | 0.013 |  106Ru | 1.00 | 0.29 | 30 | 0.994
0${}_{\text{2}}^{+}$ |  | 1.35 | 0.36 | 55 | 0.946 |  | 0.62 | 0.23 | 30 | 0.995
0${}_{\text{3}}^{+}$ |  | 1.01 | 0.32 | 44 | 0.430 |  | 0.86 | 0.27 | 30 | 0.996
0${}_{\text{1}}^{+}$ |  106Mo | 1.61 | 0.39 | 43 | 0.010 |  108Ru | 0.96 | 0.29 | 30 | 0.995
0${}_{\text{2}}^{+}$ |  | 1.16 | 0.33 | 30 | 0.297 |  | 0.83 | 0.27 | 30 | 0.997
0${}_{\text{3}}^{+}$ |  | 0.99 | 0.31 | 24 | 0.694 |  | 0.61 | 0.23 | 30 | 0.995
0${}_{\text{1}}^{+}$ |  108Mo | 2.23 | 0.46 | 36 | 0.008 |  110Ru | 1.05 | 0.30 | 30 | 0.996
0${}_{\text{2}}^{+}$ |  | 1.95 | 0.43 | 28 | 0.016 |  | 0.92 | 0.28 | 30 | 0.997
0${}_{\text{3}}^{+}$ |  | 1.29 | 0.35 | 27 | 0.603 |  | 0.71 | 0.24 | 30 | 0.995
0${}_{\text{1}}^{+}$ |  110Mo | 1.91 | 0.42 | 30 | 0.013 |  112Ru | 1.14 | 0.30 | 30 | 0.996
0${}_{\text{2}}^{+}$ |  | 1.68 | 0.39 | 30 | 0.012 |  | 0.97 | 0.28 | 30 | 0.998
0${}_{\text{3}}^{+}$ |  | 0.96 | 0.30 | 21 | 0.910 |  | 0.73 | 0.24 | 30 | 0.996
|  |  |  |  |  |  114Ru | 0.98 | 0.28 | 30 | 0.997
|  |  |  |  |  |  | 0.82 | 0.25 | 30 | 0.998
|  |  |  |  |  |  | 0.60 | 0.22 | 30 | 0.997

The procedure for calculating the deformation parameters involves the use of
quadrupole shape invariants. Focusing specifically on the $0^{+}$ states, the
following equations define these shape invariants (see Poves  _et al._ (2020)
for an application of the method),

$\displaystyle q_{2,i}$ $\displaystyle=$ $\displaystyle\sqrt{5}\left\langle
0_{i}^{+}\left|[\hat{Q}\times\hat{Q}]^{(0)}\right|0_{i}^{+}\right\rangle,$
(11) $\displaystyle q_{3,i}$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{35}{2}}\left\langle
0_{i}^{+}\mid[\hat{Q}\times\hat{Q}\times\hat{Q}]^{(0)}\mid
0_{i}^{+}\right\rangle.$ (12)

The deformation parameters are directly related to those of the triaxial rigid
rotor, denoted as $q$ and $\delta$, respectively:

$q=\sqrt{q_{2}},$ (13)

$\delta=\frac{60}{\pi}\arccos\frac{q_{3}}{q_{2}^{3/2}},$ (14)

where $\delta$ coincides with the parameter $\gamma$ of the Bohr-Mottelson
model up to a first order approximation.

The deformation parameter $\beta$ can also be obtained from the quadrupole
shape invariant (11) (see, e.g., references J. Srebrny and T. Czosnyka and Ch.
Droste and S.G. Rohoziński and L. Próchniak and K. Zajac and K. Pomorski and
D. Cline and C.Y. Wu and A. Bäcklin and L. Hasselgren and R.M. Diamond and D.
Habs and H.J. Körner and F.S. Stephens and C. Baktash and R.P. Kostecki
(2006); Clément  _et al._ (2007); Wrzosek-Lipska  _et al._ (2012)),

$\beta=\dfrac{4\pi\sqrt{q_{2}}}{3Zer_{0}^{2}A^{2/3}},$ (15)

where $e$ is the proton charge and $r_{0}=1.2$ fm.

The theoretical values of $\beta$, $\gamma$, $q^{2}$, and the fraction of wave
function belonging to the regular sector, $w^{k}$ (see Eq. (6)), are presented
in Table 5 for each $0_{1}^{+}$, $0_{2}^{+}$, and $0_{3}^{+}$ state across the
entire chains of Mo and Ru isotopes. This table reveals the coexistence of
different deformations within the same nucleus, with the regular states
typically exhibiting less deformation compared to the intruder states.

In the case of Mo isotopes, the intruder states generally display a
significant oblate deformation that increases with the mass number. However,
an exception is observed in 98Mo, where a low deformation is observed. This is
consistent with the “collapse” of the B(E2) values observed in its yrast band,
possibly related to the closure of the neutron number $56$ subshell. The
deformation of the first regular state steadily increases until 100Mo, but for
102Mo, there is a sudden increase, and the deformation remains relatively
constant for the remaining isotopes. Regarding the value of $\gamma$, clear
conclusions can only be drawn in a few cases, such as 96Mo and 98Mo, where the
states are oblate and prolate, respectively. In other cases, the deformation
is compatible with triaxiality.

Moving on to the Ru isotopes, the situation is relatively straightforward,
with a steady increase in deformation for the regular states, with a fixed
$\gamma$ value of $30$ degrees throughout. Only in two cases are intruder
states observed, exhibiting a large deformation.

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Figure 21: Value of the deformation, $\beta$, extracted from the value of the
quadrupole shape invariants for Mo (panel(a)) and Ru (panel (b)). Panel (c) is
also for Mo but here the $\beta$ values for the first regular and the first
intruder $0^{+}$ states are shown.

As a complement to Table 5, the trend of the value of $\beta$ for the first
two $0^{+}$ states in Mo and Ru isotopes is depicted in Fig. 21. In Ru
isotopes (panel (b)), the behavior is relatively straightforward, with a
constant increase in $\beta$ up to around the mid-shell region, reaching a
value of approximately $0.30$. This trend is similar to the one shown in Fig.
20. Both states belong to the regular sector. In the case of Mo isotopes
(panel (a)), the situation is more complex, especially for the $0_{2}^{+}$
state. The $0_{1}^{+}$ state exhibits a rapid increase in $\beta$ at $A=102$,
rising from $0.23$ to $0.34$, and then showing a more gradual increase up to
the mid-shell region where it reaches its maximum value of $0.42$. In
addition, it is also of interest to plot the value of $\beta$ for the first
$0^{+}$ state belonging to the regular sector ($0^{+}{\text{reg}}$) and the
first $0^{+}$ state belonging to the intruder sector ($0^{+}{\text{int}}$) in
Mo isotopes, as shown in panel (c) (note that in Ru only regular states are
observed). It can be observed that the regular state exhibits a rapid increase
in deformation at $A=102$ and subsequently shows a steady decrease. On the
other hand, the intruder state maintains a relatively high and constant value
of $\beta$, except for the aforementioned exception observed at $A=98$.

##  IX The quest of a quantum phase transition

Figure 22: Values for key QPT/shape coexistence observables for Kr, Sr, Zr,
Mo, and Ru isotopes as a function of neutron number. (a) $E(2_{1}^{+})$, (b)
$E(0_{2}^{+})$, (c) $E(4_{1}^{+})/E(2_{1}^{+})$, (d) S2n, and (e) $\langle
r^{2}\rangle-\langle r^{2}_{ref}\rangle$ (the reference values are different
for each isotope chain to better distinguish the data).

The potential relationship between the phenomenon of shape coexistence and the
existence of Quantum Phase Transitions (QPTs) has been the subject of recent
investigations García-Ramos and Heyde (2020); Gavrielov  _et al._ (2019a, b,
2022a). In brief, a QPT occurs in systems where the ground state’s structure
undergoes a sudden change when a control parameter varies slightly around a
specific value Sachdev (2011). It is important to note that this phenomenon
occurs at absolute zero temperature. The presence of a QPT is generally
associated with a combination of Hamiltonians possessing different symmetries.
Specifically, we can consider a scenario where two Hamiltonians, each
associated with a particular symmetry (A or B), are combined. Consequently,
the Hamiltonian of the system can be expressed as follows:

$\hat{H}=(1-x)\hat{H}_{A}+x\hat{H}_{B}.$ (16)

This formulation allows us to investigate the interplay between different
symmetries, A and B, by adjusting the parameter $x$, which determines the
relative weight or contribution of each symmetry to the overall Hamiltonian.

In most cases, the symmetry involved in the QPT can be considered as a
dynamical symmetry Iachello  _et al._ (1998). The QPT occurs at a critical
value $x=x_{c}$, where the wave function undergoes an abrupt transition from
having symmetry A to having symmetry B, even though the full Hamiltonian does
not possess any specific symmetry except at $x=0$ or $1$. This phenomenon is
closely connected to the concept of quasidynamical symmetry proposed by D.
Rowe in Rowe (2004). The existence of a QPT also implies a sudden change in
the so-called order parameter, which vanishes in the symmetric phase and takes
a nonzero value in the broken phase Sachdev (2011). Thus, the order parameter
carries information about the symmetry of the system’s ground state.

QPTs can be classified in a manner similar to the phase transitions that occur
in macroscopic systems at non-zero temperature, employing the Erhenfest
classification Landau and Lifshitz (1969). Based on this classical
classification, QPTs can be categorized as first-order and second-order (or
continuous) QPTs Landau and Lifshitz (1969). In a first-order QPT (where the
first derivative of the ground state energy with respect to the control
parameter exhibits a discontinuity), there exists a narrow region around
$x_{c}$ where both states with different symmetries, A and B, coexist. In the
case of a second-order QPT (according to the Erhenfest classification, where
the second derivative of the ground state energy with respect to the control
parameter displays a discontinuity), there is no coexistence of symmetries
around the critical region $x_{c}$.

A crucial characteristic of a QPT is that it leads to the degeneracy of a set
of states and the compression of the spectra García-Ramos  _et al._ (2001);
García-Ramos and Heyde (2020). In fact, in the thermodynamic limit, the
excitation energy of the first excited state approaches zero at the critical
point.

When studying QPTs in nuclear systems, one encounters challenges related to
the finite size of the system and the approximate nature of the control
parameter, often identified with the nuclear mass or neutron number. As a
result, all the main characteristics of a QPT can only be observed in an
approximate manner within a nucleus, and the abrupt changes are typically
smoothed out Iachello  _et al._ (1998). In Figure 22, we present experimental
data for Kr, Zr, Sr, Mo, and Ru, including the values of $E(2_{1}^{+})$,
$E(0_{2}^{+})$, $E(4_{1}^{+})/E(2_{1}^{+})$, S2n, and $\langle
r^{2}\rangle$.These quantities serve as indicators for the presence of a QPT
or the existence of shape coexistence. These nuclei are located near the
subshell closure at $Z=40$ and are close to the neutron number $60$, where a
rapid onset of deformation is observed.

In panel (a), it is evident that in all nuclei, the energy of the $2_{1}^{+}$
state decreases, indicating the appearance of deformation, particularly from
neutron number $60$ onwards. However, notable differences exist between Sr and
Zr, where the drop is very abrupt, Mo, which exhibits a slower decrease, and
Ru or Kr, where the transition is smoother. The behavior in Sr and Zr has been
interpreted in terms of the crossing of two different particle-hole
configurations García-Ramos and Heyde (2019); Maya-Barbecho and García-Ramos
(2022) or as a first-order QPT García-Ramos and Heyde (2020); García-Ramos
_et al._ (2005).

Moving to panel (b) and considering the energy systematics of the $0_{2}^{+}$
state, a distinct minimum is observed for Sr and Zr at neutron number $60$,
while Ru and Mo exhibit a relatively flat energy trend around this neutron
number. In Kr, it is not possible to extract a clean conclusion. The deep
minimum in Sr and Zr again suggests the crossing of two configurations or the
existence of a QPT, while the smoother behavior in Mo and Ru implies a slower
evolution.

Panel (c) depicts the ratio $E(4_{1}^{+})/E(2_{1}^{+})$, which serves as a
clear indicator of the onset of deformation. A value around $2$ corresponds to
a spherical nucleus, $2.5$ to a $\gamma$-unstable rotor, and $3.3$ to a rigid
rotor. Sr and Zr clearly exhibit an evolution from sphericity to a rigid
rotor, Ru and Kr indicate an evolution into a $\gamma$-unstable rotor, and Mo
lies in between both cases. This observable can also be considered as an
approximate order parameter, behaving as a first-order QPT in Sr and Zr, and
potentially as a second-order QPT in Mo (less clear in Ru).

In panel (d), the two-neutron separation energy, S2n, is displayed. This
observable, which can be understood as the derivative of the binding energy,
is a smoking gun for the presence of a QPT. The sudden change in its slope
around neutron number $60$ in Sr and Zr suggests the existence of a first-
order QPT, while Mo exhibits a small perturbation indicating a possible
second-order QPT. No departure from the linear trend is observed in Ru and Kr.

Finally, panel (e) presents the mean-square radii. Note that the origin has
been shifted differently for each isotope for clarity. Once again, a sudden
increase in the radius is observed in Sr and Zr around neutron number $60$,
while Mo shows a less drastic increase and Ru and Kr exhibit a linear trend.

Based on the information presented in Fig. 22, it is reasonable to assume that
Sr and Zr undergo a first-order QPT around neutron number $60$, Mo undergoes a
second-order QPT, while Ru exhibits a smooth evolution without any abrupt
changes. In Sr, Zr, and Mo, the ratio $E(4_{1}^{+})/E(2_{1}^{+})$ can be
considered as an order parameter, indicating the presence of a QPT, while S2n
points towards the existence of a discontinuity in the first or second
derivative of the binding energy.

Alternatively, all the observed features in Figure 22 can also be explained in
terms of the crossing of two configurations. The increase in
$E(4_{1}^{+})/E(2_{1}^{+})$ and the decrease in $E(2_{1}^{+})$ can be easily
explained by the crossing of a spherical and a deformed configuration. The
drop in the energy of the $0_{2}^{+}$ state can be attributed to the presence
of an intruder configuration that gains correlation energy more rapidly than
the regular configuration. Finally, the deviation from the linear trend in S2n
and mean-square radius can also be explained by the crossing of two
configurations.

Therefore, both the QPTs and the crossing of two configurations can provide
explanations for the observed phenomena in the figure.

In previous works, the observed QPT in Zr has been explained in terms of the
crossing of two weakly interacting configurations Gavrielov  _et al._ (2019b);
García-Ramos and Heyde (2020); Gavrielov  _et al._ (2022a), providing a
straightforward explanation for the behavior of S2n. A recent analysis has
also been conducted for Sr nuclei Maya-Barbecho and García-Ramos  with similar
conclusions. Moreover, in Zr, the different configurations, especially the
intruder one, undergo their own QPT. The concept of “Type I” and “Type II”
QPTs was introduced in Gavrielov  _et al._ (2019b, 2022a), referring to QPTs
occurring within a single configuration or involving two configurations,
respectively. In the case of Mo, it has been shown in Sec. IV that two
configurations indeed cross, although the observed changes in S2n are less
abrupt compared to Zr. This situation is reminiscent of the behavior observed
in Pt García-Ramos and Heyde (2009), where a crossing of two configurations
occurs without inducing a QPT. It is worth mentioning that the interaction
between configurations was quite strong in Pt, which resulted in the
suppression of the QPT. The mixing between configurations is controlled by the
parameter $\omega$, with values of 15 keV, 15 keV, and 50 keV for Zr, Mo, and
Pt, respectively. However, the interaction between configurations depends on
the matrix element $\langle 0^{+}_{\rm reg}|\hat{V}_{\rm mix}|0^{+}_{\rm
int}\rangle$, which has values around the region where the configurations
cross, namely 80 keV, 250 keV, and 250 keV for Zr, Mo, and Pt, respectively.
Therefore, it is evident that the case of Mo resembles Pt, where two
configurations cross but a strong interaction between them hinders the
presence of abrupt changes in the spectrum. Nevertheless, Mo still exhibits
some key elements of a QPT (see Fig. 22). When considering the unperturbed
configurations, both rapidly transition from a spherical to a deformed shape,
either oblate (regular) or prolate (intruder). According to Figs. 16 and 20,
the regular and intruder configurations undergo a “Type I” QPT, while the
ground state undergoes a “Type II” QPT around $A=102$.

The situation in Ru isotopes is indeed different, as the evolution of the
ground state is fully determined by a single configuration. According to Figs.
18 and 19, the energy surface of Ru isotopes is initially spherical for the
lighter ones, but it starts to flatten and becomes fully flat at $A=104$
(neutron number $60$). From this point onwards, a $\gamma$-unstable
deformation develops. Based on this behavior, Ru isotopes undergo a “Type I”
QPT of second order, as indicated by Fig. 22 and as proposed in Frank  _et
al._ (2001).

##  X Summary and conclusions

This work has focused on analyzing the even-even
${}^{96-110}_{~{}~{}~{}~{}~{}42}$Mo and ${}^{98-114}_{~{}~{}~{}~{}~{}44}$Ru
isotopes using the interacting boson model with configuration mixing (IBM-CM).
Initially, the parameters of the Hamiltonian and the $\hat{T}(E2)$ operators
were determined through a least-squares fitting process using available
experimental excitation energies and $B(E2)$ values. Additionally, radii and
two-neutron separation energies were computed and compared to experimental
data. Subsequently, a detailed analysis was conducted on the wave function of
the states, their deformation, and their mean-field energy surfaces. Finally,
the potential interplay between shape coexistence and quantum phase transition
was explored.

The primary objective of this study is to determine the extent of the shape
coexistence region around $Z=40$. In the case of Sr and Zr isotopes, the
influence of intruder states becomes evident at neutron number $60$, where
there is a crossing between the regular and intruder configurations in the
ground state. Therefore, it is natural to investigate whether this phenomenon
persists up to $Z=42-44$. Previous research has already examined the case of
Kr ($Z=36$), concluding that the intruder configuration no longer plays a
significant role in these nuclei and a rather smooth systematics is observed
Mărginean  _et al._ (2009); Albers  _et al._ (2012), although in Flavigny  _et
al._ (2017) the authors claim that in 98-100Kr, intruder states can play a
significant role. However, the situation in the other direction remains
unclear.

One of the main conclusions drawn from this study is that shape coexistence
plays a significant role in Mo isotopes, where a crossing between the intruder
and regular configurations occurs at neutron number $60$ in the ground state.
However, in Ru isotopes, the intruder states appear at much higher energies
and do not affect the behavior of the low-lying states. Therefore, it seems
that the boundary of the shape coexistence phenomenon lies within the Mo
isotopes.

The region under investigation is also known for the presence of quantum phase
transitions (QPTs). Based on the experimental information, it is challenging
to assert the existence of QPTs in Mo and Ru nuclei. However, other indirect
features indicate the presence of QPTs in both nuclei, albeit of different
nature. In Mo isotopes, the existence of a configuration crossing suggests a
first-order QPT (“Type II” QPT) similar to that observed in Sr and Zr
isotopes. However, the strong interaction between configurations smooths out
the systematics, resembling the situation in Pt nuclei. Hence, it can be
stated that in Mo isotopes, a “Type II” QPT exists, but it is not of first
order, because the observed changes in the key observables shown in Fig. 22
are rather smooth. In Ru isotopes, all the analyzed observables (refer to Fig.
22) indicate a gradual trend. However, the mean-field analysis of the nuclei
suggests the presence of a “Type I” second-order QPT at neutron number $60$,
as proposed in Frank  _et al._ (2001). Thus, neighboring nuclei exhibit QPTs,
but with different origins. In Mo isotopes, the QPT is induced by shape
coexistence (“Type II” QPT), whereas in Ru isotopes, it involves a single
configuration (“Type I” QPT). In both cases, the QPT is of second order.

It is important to acknowledge the limitations of the obtained results,
primarily stemming from the lack of experimental data on transition rates
involving states of the regular and intruder configurations. This absence of
information makes it challenging to precisely determine certain parameters in
the model operators. Fortunately, the study of isotopes with $Z=40$ is
currently an active area of experimental research, which holds promise for
obtaining additional data in the future.

A potential future extension of this work would involve investigating odd-even
nuclei, as demonstrated in a compelling manner for Nb isotopes in the
reference Gavrielov  _et al._ (2022b). This would represent the next step in
configuration mixing studies and could provide further insights and
understanding of the phenomena of shape coexistence and QPTs.

##  XI Acknowledgment

One of the authors (JEGR) wants to thank K. Heyde for his continuous advice
and support. This work was partially supported by the Consejería de Economía,
Conocimiento, Empresas y Universidad de la Junta de Andalucía (Spain) under
Groups FQM-160 and FQM-370, and under projects P20-00617, P20-01247 and
US-1380840; also through the projects PID2019-104002GB-C21,
PID2019-104002GB-C22, and PID2020-114687GB-I00 funded by
MCIN/AEI/10.13039/50110001103 and “ERDF A way of making Europe”. Resources
supporting this work were provided by the CEAFMC and the Universidad de Huelva
High Performance Computer (HPC@UHU) funded by ERDF/MINECO project
UNHU-15CE-2848.

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