††thanks: deceased

and

# Phase transitions in N = 40, 60 and 90 nuclei

A. Prášek  P. Alexa  Department of Physics, VŠB – Technical University
Ostrava, 17\. listopadu 2172/15, CZ-708 00 Ostrava, Czech Republic  D.
Bonatsos  Institute of Nuclear and Particle Physics, N.C.S.R. Demokritos,
GR-15310 Aghia Paraskevi, Attiki, Greece  G. Thiamová  Universite Grenoble 1,
CNRS, LPSC, Institut Polytechnique de Grenoble, IN2P3, F-38026 Grenoble,
France  D. Petrellis and P. Veselý  Nuclear Physics Institute, Czech Academy
of Sciences, CZ-250 68 Řež near Prague, Czech Republic

###### Abstract

In this paper we focus on three mass regions where first-order phase
transitions occur, namely for $N=40$, 60 and 90. We investigate four isotopic
chains (Se, Zr, Mo and Nd) in the framework of microscopic Skyrme-Hartree-Fock
+ Bardeen-Cooper-Schrieffer calculations for 15 different parametrizations.
The microscopic calculations show the typical behavior expected for first-
order phase transitions. To find the best candidate for the critical point
phase transition we propose new microscopic position and occupation indices
calculated for positive-parity and negative-parity proton and neutron single-
quasiparticle states around the Fermi level. The microscopic calculations are
completed by macroscopic calculations within the Algebraic Collective Model
(ACM), and compared to the experimental data for 74Se, 102Mo and 150Nd,
considered to be the best candidates for the critical point nuclei.

###### pacs:

21.10.Re, 21.60.Ev, 23.20.Lv

##  I Introduction

Quantum phase transitions (QPT) [1, 2, 3] and critical point symmetries (CPS)
[4, 5, 6, 7] in even-even atomic nuclei belong to one of most studied topics
in nuclear structure physics for several decades. They are observed in chains
of isotopes in which the addition of two neutrons causes a radical change of
the nuclear structure, the sudden jump from vibrational behavior in 150Sm to
rotational behavior in 152Sm [5, 8] being a characteristic example. The
neutron number is used as a control parameter in these transitions [9], while
various collective quantities are used as the order parameter [10, 11, 12,
13].

The Ehrenfest classification of nuclear QPTs started in 1981 [14], with the
realization that within the parameter space of the Interacting Boson Model
[15, 16, 17, 18] a second-order QPT takes place between U(5) (spherical) [15]
and O(6) ($\gamma$-unstable, i.e., soft towards triaxial deformation) [17]
nuclei, while a first-order QPT occurs between U(5) (spherical) and SU(3)
(deformed) [16] nuclei. The critical point symmetries E(5) [4, 19] and X(5)
[5, 20, 21] have been introduced, in 2000 and 2001 respectively, for the
description of these QPTs within the framework of the Bohr collective
Hamiltonian [22, 23, 24] providing parameter independent (up to overall
scales) predictions for spectra and transition rates at the critical point of
these QPTs. The $N=90$ isotones 150Nd [25], 152Sm [8], 154Gd [26], 156Dy [27]
have been suggested as the best experimental manifestations of the X(5) CPS. A
$\gamma$-rigid version of X(5), called X(3), in which the $\gamma$ value has
been fixed to zero, has also been introduced [28].

Even before the introduction of the concept of CPS, the mechanism of the onset
of deformation within microscopic models was investigated in a series of
papers by Federman and Pittel [29, 30, 31, 32]. They pointed out the crucial
role played by the proton-neutron interaction in the creation of deformation,
in particular stressing the major role played by intruder neutron orbitals in
the development of deformation across the nuclear chart. In addition, the
concept of quasi-dynamical symmetries [33], i.e., symmetries persisting in the
presence of strong symmetry-breaking interactions, has been introduced for
both first-order [34] and second-order [35, 36] QPTs.

Recently, the connection between CPSs linked to QPTs and the effect of shape
coexistence (SC) [37, 38, 39, 40, 41] attracted considerable interest [42, 43,
44, 45]. Shape coexistence [37, 38, 39, 40, 41] is said to occur in nuclei in
which the ground state band and an excited $K=0$ band lie close in energy and
simultaneously exhibit radically different structures and symmetries, for
example one of them being spherical and the other one deformed, or both of
them being deformed, but one of them exhbiting a prolate (rugby-ball-like)
shape and the other one showing an oblate (pancake-like) one. The $N=60$
isotones 100Zr [42, 43, 44] and 98Sr [45] have been found as the best nuclei
in which the connection between CPS and QPT is observed. The relevant studies
have been carried out in the framework of the IBM with configuration mixing
(IBM-CM), which takes particle-hole (p-h) excitations into account [46, 47,
48, 49] and allows for mixing of the ground state configuration of $N$ bosons
with the excited 2p-2h configuration, described by $N+2$ bosons [50, 51]. It
should be noticed that the occurence of phase coexistence in the region of
transition between spherical and deformed shapes has been pointed out in the
IBM framework already in 1998 [52], while the structural similarity between
the $N=90$ and $N=60$ regions has been pointed out already in 1981 [53],
implying a common microscopic origin of the QPT from spherical to deformed
shapes in these two regions. Furthermore, it has been recently suggested [54,
55, 56] that 100Zr is the critical point of two intertwined QPTs, the one from
spherical to deformed shapes already mentioned and another one from normal
configurations (without particle-hole excitations) to intruder configurations
(involving particle-hole excitations).

In addition, a recent study [57] of the systematics of energy levels and
$B(E2)$ transition rates in the $N=40$ region around 74Se pointed out the
structural similarity of the $N=40$ region to the above mentioned $N=60$ and
$N=90$ regions. It has been shown that in all these three regions intruder
orbitals start participating to the onset of deformation. In particular,
neutron particle-hole excitations are found to lead to both SC and QPT from
spherical to prolate deformed shapes taking place in parallel.

The above findings suggest that the three regions with $N=90$, $N=60$, and
$N=40$ exhibit in a similar way a QPT from spherical to deformed shapes, based
on the microscopic picture of the onset of deformation due to increased
proton-neutron interaction, with SC triggered at the same time by the creation
of neutron particle-hole excitations (called proton-induced particle-hole
excitations in the framework of the dual shell mechanism [58] for SC developed
within the proxy-SU(3) approximation [59, 60, 61], corroborated through
covariant density functional theory calculations [62, 63]). Theoretical
calculations in these three regions, related to the presence of SC, have been
recently reviewed in [41].

However, despite the similar microscopic origins of the three regions, only
the critical isotones with $N=90$ can be described in terms of the parameter-
free predictions of the X(5) CPS [25, 8, 26, 27], while for $N=60$ and $N=40$
the data for spectra and $B(E2)$ transition rates are far away from the X(5)
predictions. It is therefore of interest to look for a more flexible
theoretical framework, possibly able to accommodate these three critical
regions simultaneously.

Therefore, in the present paper we investigate the three CPS regions within
different microscopic and macroscopic approaches, attempting to shed more
light on the nature of the CPS and possibly find a common framework for all of
them. In Sec. II empirical systematics of different spectral and transitional
signatures of CPS are discussed in the three regions of interest. In Sec. III
we turn to microscopic calculations based on the Skyrme-Hartree-Fock +
Bardeen-Cooper-Schrieffer (SHF + BCS) model to study potential energy curves
(PEC) as functions of quadrupole, octupole and hexadecapole deformations, and
introduce microscopic position and occupation indices for positive-parity and
negative-parity proton and neutron single-quasiparticle states around the
Fermi level. In Sec. IV we briefly introduce the Algebraic Collective Model
(ACM), which is then applied to 74Se, 102Mo and 150Nd in Sec. V.

##  II Empirical systematics

We are going to use the energy ratios

$R_{4/2}={E(4_{1}^{+})\over E(2_{1}^{+})},\quad R_{2/0}={E(2_{1}^{+})\over
E(0_{2}^{+})},\quad R_{2/2}={E(2_{\gamma}^{+})\over E(2_{1}^{+})},$ (1)

as well as the rates of change with the neutron number $N$ (which possess
twice the value of relevant mathematical derivative),

$\displaystyle{\mathrm{d}R_{4/2}\over\mathrm{d}N}(N)=R_{4/2}(N)-R_{4/2}(N-2),$
(2)
$\displaystyle{\mathrm{d}R_{2/2}\over\mathrm{d}N}(N)=R_{2/2}(N)-R_{2/2}(N-2).$

In addition we are going to use the transition rates $B(E2;2_{1}^{+}\to
0_{1}^{+})$, and their rate of change with the neutron number $N$ (which
possesses twice the value of relevant mathematical derivative),

${\mathrm{d}B(E2;2_{1}^{+}\to 0_{1}^{+})\over\mathrm{d}N}(N)=B(E2;2_{1}^{+}\to
0_{1}^{+})(N)-B(E2;2_{1}^{+}\to 0_{1}^{+})(N-2),$ (3)

The ratio $R_{4/2}$ is a well known indicator of collectivity [64], obtaining
values 2.0–2.4 for near-spherical nuclei, 2.4–3.0 for $\gamma$-unstable
nuclei, and 3.0–3.33 for deformed nuclei.

The transition rate $B(E2;2_{1}^{+}\to 0_{1}^{+})$ is known [65] to be
proportional to the square of the quadrupole deformation parameter $\beta$,
expressing the deviation of the nuclear shape from sphericity.

The ratio $R_{2/0}$ is known [57] to exhibit a maximum in the region of the
QPT from spherical to deformed nuclei.

Table 1: Energy ratios $R_{4/2}$, $R_{2/2}$, and transition rates
$B(E2;2_{1}^{+}\to 0_{1}^{+})$, as well as their rates of change as a function
of the neutron number $N$, ${\mathrm{d}R_{4/2}\over\mathrm{d}N}(N)$,
${\mathrm{d}R_{2/2}\over\mathrm{d}N}(N)$, and
${\mathrm{d}B(E2)\over\mathrm{d}N}(N)$, are listed for nuclei at $N=90$, 60,
42 and 40, lying near the maxima appearing in Figs. 1-4. In addition, the
energy ratios $R_{2/0}$ are shown for the isotones listed, labeled as
$R_{2/0}(N)$. The parameter-independent predictions of the critical point
symmetries X(5) [5, 68] and X(3) [28] are also shown for comparison. See Sec.
II for further discussion.
nuc. | $R_{4/2}$ | ${\mathrm{d}R_{4/2}\over\mathrm{d}N}(N)$ | $B(E2)$ | ${\mathrm{d}B(E2)\over\mathrm{d}N}(N)$ | $R_{2/0}(N)$ | $R_{2/2}$ | ${\mathrm{d}R_{2/2}\over\mathrm{d}N}(N)$
---|---|---|---|---|---|---|---
|  |  | W.u. | W.u. |  |  |
150Nd90 | 2.927 | 0.434 | 114.4 | 56.9 | 0.193 | 8.156 | 4.017
152Sm90 | 3.009 | 0.693 | 143.7 | 86.8 | 0.178 | 8.916 | 5.786
154Gd90 | 3.015 | 0.821 | 158.0 | 89.3 | 0.181 | 8.095 | 4.873
156Dy90 | 2.934 | 0.700 | 149.2 | 50.4 | 0.204 | 6.464 | 3.392
98Sr60 | 3.006 | 0.806 | 92.4 | 74.7 | 0.669 | 10.674 | 8.676
100Zr60 | 2.656 | 0.982 | 80.5 |  | 0.642 | 5.628 | 3.808
102Mo60 | 2.507 | 0.386 | 69.0 | 30.6 | 0.425 | 2.859 | 0.873
72Zn42 | 2.297 | 0.278 | 21.1 | 3.5 | 0.432 | 2.539 | 0.551
74Ge42 | 2.457 | 0.385 | 33.1 | 9.6 | 0.402 | 2.021 | 0.266
76Se42 | 2.380 | 0.232 | 45.1 | 6.4 | 0.498 | 2.175 | 0.176
74Se40 | 2.148 | 0.249 | 38.7 | 17.4 | 0.743 | 1.999 | 0.472
X(5) | 2.907 |  |  |  | 0.177 |  |
X(3) | 2.440 |  |  |  | 0.350 |  |

Figure 1: Experimental [66] energy ratios $R_{4/2}$ and their rate of change
$\mathrm{d}R_{4/2}/\mathrm{d}N$ with respect to the neutron number $N$ in the
$N=90$, 60, 40 regions. See Section II for further discussion.

Figure 2:  Experimental $B(E2;2_{1}^{+}\to 0_{1}^{+})$ transition rates [67]
and their rate of change $\mathrm{d}B(E2;2_{1}^{+}\to 0_{1}^{+})/\mathrm{d}N$
with respect to the neutron number $N$ in the $N=90$, 60, 40 regions. See
Section II for further discussion.

Figure 3:  Experimental [66] energy ratios $R_{2/0}$ in the $N=90$, 60, 40
regions. See Section II for further discussion.

Figure 4: Experimental [66] energy ratios $R_{2/2}$ and their rate of change
$\mathrm{d}R_{2/2}/\mathrm{d}N$ with respect to the neutron number $N$ in the
$N=90$, 60, 40 regions. See Section II for further discussion.

In Fig. 1 an abrupt increase of $R_{4/2}$ is seen at $N=90$ in the Nd-Sm-Gd-Dy
region, at $N=60$ in the Sr-Zr-Mo region, and at $N=42$ in the Zn-Ge-Se
region. This is corroborated by the rate of change of $R_{4/2}$ vs. $N$ [10]
within these series of isotopes.

In Fig. 2 an abrupt increase of $B(E2;2_{1}^{+}\to 0_{1}^{+})$ is seen around
$N=90$ in the Nd-Sm-Gd-Dy region, around $N=60$ in the Sr-Zr-Mo region, and
around $N=42$ in the Zn-Ge-Se region. This is corroborated by the rate of
change of $B(E2;2_{1}^{+}\to 0_{1}^{+})$ vs. $N$ [10] within these series of
isotopes.

In Fig. 3 maxima of the energy ratio $R_{2/0}$ are seen around $N=88$ in the
Nd-Sm-Gd-Dy region, around $N=58$ in the Sr-Zr-Mo region, and around $N=40$ in
the Zn-Ge-Se region. In other words, the maxima appear two neutrons earlier in
relation to the maximal changes of $R_{4/2}$ and $B(E2;2_{1}^{+}\to
0_{1}^{+})$ seen in Figs. 1 and 2.

In Fig. 4 an abrupt increase of $R_{2/2}$ is seen around $N=90$ in the Nd-Sm-
Gd-Dy region, around $N=60$ in the Sr-Zr-Mo region, and around $N=42$ in the
Zn-Ge-Se region. This is corroborated by the rate of change of $R_{2/2}$ vs.
$N$ [10] within these series of isotopes.

In Table 1 the values of these quantities for the nuclei close to the maxima
are reported. We see that while the three regions exhibit in Figs. 1-4 very
similar behavior, the numerical values in each region differ. In particular

a) The $R_{4/2}$ ratio exhibits near-rotational values in the $N=90$ region,
intermediate values in the $N=60$ region, and much lower, near-vibrational
values in the $N=40$ region.

b) In qualitative agreement with a), the $B(E2;2_{1}^{+}\to 0_{1}^{+})$
transition rate exhibits higher values in the $N=90$ region, intermediate
values in the $N=60$ region, and lower values in the $N=40$ region. This is
expected, since the $B(E2;2_{1}^{+}\to 0_{1}^{+})$ transition rate is
proportional to the square of the quadrupole deformation $\beta$ [65], despite
the fact that part of the difference is due to the factor $A^{4/3}$
accompanying $\beta^{2}$ in the relevant equation [65].

c) The $R_{2/0}$ values exhibit the opposite trend, namely lower values in the
$N=90$ region, evolving to higher values in the $N=40$ region, in agreement to
the well known fact [66] that the $0_{2}^{+}$ state raises to very high values
as a nucleus approaches the rotational limit of $R_{4/2}=10/3$.

d) The $R_{2/2}$ ratio exhibits higher values in the $N=90$ region,
intermediate values in the $N=60$ region, and lower values in the $N=40$
region, corroborating the structural similarity between the ground state band
and the $\gamma$-band [69].

These observations indicate that the parameter-independent values of
$R_{4/2}=2.904$ and $R_{2/0}=0.177$ [5, 68] characterizing the X(5) critical
point symmetry [5] apply very well to the $N=90$ isotones, but they do not
apply in the $N=60$ and $N=40$ regions, for which a more flexible model
appears to be required. The values seen in the $N=40$ region, in particular,
are quite similar to the parameter-independent values of $R_{4/2}=2.44$ and
$R_{2/0}=0.35$ seen in X(3) [141], the $\gamma$-rigid analog of X(5).

The $N=90$ isotones 150Nd [25], 152Sm [8], 154Gd [26], 156Dy [27] are well
established examples of the X(5) critical point symmetry.

In the $N=60$ region, 104Mo62 has initially been suggested [70] as an X(5)
candidate, based on its ratio $R_{4/2}=2.917$, which is very close to the X(5)
value of 2.904. In addition, its ratio $R_{2/0}=0.217$, is also very close to
the X(5) value of 0.177. However, it was later disregarded, since its $B(E2)$s
within the ground state band have been found [71] to exhibit a deformed
behavior. Figs. 1-4 suggest that the $N=60$ isotones 98Sr, 100Zr, 102Mo are
better candidates for the critical point of the spherical to deformed QPT in
the $Z=40$ region.

It should be noticed that Figs. 1-4 shed light on the nature of ground state
QPTs [11, 12, 3]. According to the Ehrenfest classification, a first order
phase transition occurs when the first derivative of a physical quantity
(serving as the order parameter) exhibits a discontinuity with respect to the
control parameter [11, 12]. The ground state spectra and $B(E2)$ transition
rates shown in Figs. 1 and 2 suggest the occurrence of a first order ground
state QPT within the ground state band for nuclei around $N=90$, 60, 42, since
the rate of change of these quantities vs. the neutron number $N$ exhibits a
discontinuity, while the quantities themselves show an increase with $N$,
which becomes abrupt at the critical point of the QPT. The same behavior is
seen in Fig. 4, suggesting a similar behavior for the $\gamma$-band. However,
this is not the case for the ratio $R_{2/0}$ (Fig. 3), which presents itself a
maximum around the critical point. This difference is rooted to the fact that
the ground state band and the $\gamma$-band tend to have similar structures
[69], while the nature of the $\beta$-band has been a point of controversy
[72, 73, 74, 75], with the first excited $0^{+}$ state being able to
correspond to several different physical situations and not necessarily to the
band-head of the $\beta$-band [76].

##  III Microscopic mean field calculations

The regions of interest have been investigated within several different mean-
field frameworks. Non-relativistic calculations using the Skyrme interaction
have been performed for the Nd-Sm-Gd-Dy [77] and Zr [78, 79] isotopes, while
the Gogny interaction has been used in the Nd-Sm-Gd-Dy [80, 81, 82], Zr [83]
and Se [84] regions. Relativistic mean field calculations have been performed
for the Nd-Sm-Gd-Dy [85, 86, 87] and Sr-Zr-Mo [88, 89, 90, 91] isotopes. A
5-dimensional quadrupole collective Hamiltonian with parameters determined
from relativistic mean field calculations has been used in the Nd-Sm-Gd-Dy
[92, 93, 94] and Sr-Zr-Mo [95] isotopes. In addition, an IBM Hamiltonian with
parameters determined by covariant density functional theory calculations has
been applied to the Nd-Sm-Gd-Dy [96, 97, 98, 99], Sr-Zr-Mo [100], and Se [101]
regions.

In the present section we are trying to accommodate all three regions of
$N=90$, $N=60$ and $N=40$ within the same theoretical framework.

Self-consistent mean-field methods based e.g. on the most widely used Skyrme-
Hartree-Fock + Bardeen-Cooper-Schrieffer (SHF + BCS) calculations [102]
represent a microscopic possibility to investigate phase transitions in finite
nuclei. One would expect to observe a typical bump structure in the potential
energy curves (PEC) calculations as a function of the quadrupole deformation
parameter $\beta_{2}$ [96, 97] that manifests itself in a wider flat region
around the minimum of the PEC compared to neigboring nuclei. In addition, the
presence of opposite-parity intruder states in both proton and neutron single-
quasiparticle spectra close to the Fermi level represents another microscopic
signature of the phase transition [29, 30, 31, 32].

Because there exist plenty of SHF functional parametrizations we investigate
here the whole family of parametrizations based on the SV-bas one (15
parametrizations) [103]. In calculations using the axial SHF code SKYAX [104]
with a density dependent $\delta$-force interaction in the pairing channel
[105] single-particle levels up to 75 MeV were taken into account (8
oscillator shells). We investigated PECs as a function of quadrupole
$\beta_{2}$, octupole $\beta_{3}$ and hexadecapole $\beta_{4}$ deformations
for four isotopic chains around $N=40$, 60 and 90 (68Se – 80Se, 92Zr – 104Zr,
94Mo – 106Mo, 144Nd – 156Nd).

In the selenium chain all parametrizations predict similar PECs, the
parametrization SV-mas07 gives slightly different results (see Fig. 5). In the
quadrupole deformation PECs one observes phase transition from oblate to
prolate shape at $N=38$-44 (see Fig. 6). For $N=40$ SC is predicted with one
spherical and one oblate minimum close in energy. A wider flat region is
observed for $N=38$ and 40 (SV-bas) and for $N=40$ and 42 (SV-mas07). Octupole
deformation does not play any significant role, as expected. Hexadecapole
deformation remains positive and close to zero ($\beta_{4}=0$ for $N=40$ and
42).

Figure 5: PEC as a function of $\beta_{2}$ for 74Se (a) and 76Se (b) for all
investigated parametrizations. The parametrization SV-mas07 (violet dashed)
gives a slightly different PEC compared to the others.

Figure 6: PEC as a function of $\beta_{2}$ for the selenium isotopes for two
SHF parametrizations, SV-bas (a) and SV-mas07 (b). Flat-bottomed PECs are
indicated by thicker curves.

In the zirconium chain all parametrizations again predict similar quadrupole-
deformation PECs with a more pronounced minima for the SV-tls parametrization
(see Figs. 7 and 8). Interestingly, for the SV-bas one observes three minima,
oblate, spherical and prolate for $N\geq 58$, the oblate one and the spherical
one being almost degenerate for $N=60$ (the flattest PEC). For SV-tls, the
flat-bottomed PECs were found for $N=58$-60. Octupole deformation is again not
important. Hexadecapole deformation is around $\beta_{4}=0$.

Figure 7: PEC as a function of $\beta_{2}$ for 98Zr (a) and 100Zr (b) for all
investigated parametrizations. The parametrization SV-tls (dark green dashed)
gives a more pronounced oblate minimum and the lowest energy differences for
different $\beta_{2}$ values.

Figure 8: PEC as a function of $\beta_{2}$ for the zirconium isotopes for two
SHF parametrizations, SV-bas (a) and SV-tls (b). Flat-bottomed PECs are
indicated by thicker curves.

Figure 9: PEC as a function of $\beta_{2}$ for 100Mo (a) and 102Mo (b) for all
investigated parametrizations. The parametrization SV-tls (dark green dashed)
gives a more pronounced oblate minimum and the lowest energy differences for
different $\beta_{2}$ values.

Figure 10: PEC as a function of $\beta_{2}$ for the molybdenum isotopes for
two SHF parametrizations, SV-bas (a) and SV-tls (b). Flat-bottomed PECs are
indicated by thicker curves.

The molybdenum chain behaves similarly (see Figs. 9 and 10). The flat-bottomed
PECs were found for $N=60$ (SV-bas) and $N=58$ (SV-tls).

Figure 11: PEC as a function of $\beta_{2}$ for the neodymium isotopes for two
SHF parametrizations, SV-bas (a) and SV-mas07 (b). Flat-bottomed PECs are
indicated by thicker curves. Figure 12: PEC as a function of $\beta_{2}$ for
150Nd for all investigated parametrizations. The parametrization SV-mas07
(violet dashed) gives more pronounced minima.

In the neodymium chain one observes a gradual development of prolate
deformation (see Fig. 11) with a wider flat region around the prolate
quadrupole deformation minimum for $N=90$ for all parametrizations (see Fig.
12). Octupole deformation does not play any significant role, in the region
$N\leq 90$ octupole deformation softness occurs. Hexadecapole deformation
increases from $\beta_{4}=0$ to $\beta_{4}=0.22$ along the chain with a PEC
wider flat region for $N=60$ (see Fig. 13).

Figure 13: PEC as a function of $\beta_{3}$ (a) and $\beta_{4}$ (b) for the
neodymium isotopes for SV-bas parametrization.

To check if triaxiality plays an important role in the investigated regions we
have to use Sky3D code [106]. Results for the parametrization SV-bas are
presented in Table 2. Because the codes use different definitions of
$\beta_{2}$, $\beta_{\mathrm{2SKYAX}}$ is recalculated according to the
$\beta_{2}$ definition in the Sky3D code. One can see that both values of
$\beta_{2}$ coincide and the triaxiality is not important in the investigated
regions.

Table 2: Comparison of quadrupole deformations, $\beta_{\mathrm{2SKYAX}}$ and $\beta_{\mathrm{2Sky3D}}$, $\gamma_{\mathrm{{Sky3D}}}$, calculated using SKYAX and Sky3D codes for the parametrization SV-bas. nucleus | $\beta_{\mathrm{2SKYAX}}$ | $\beta_{\mathrm{2Sky3D}}$ | $\gamma_{\mathrm{Sky3D}}$
---|---|---|---
68Se | -0.150 | 0.147 | 60.0
70Se | -0.126 | 0.132 | 60.0
72Se | -0.102 | 0.108 | 60.0
74Se | 0.000 | 0.000 | 22.4
76Se | 0.000 | 0.003 | 52.4
78Se | 0.053 | 0.063 | 0.1
80Se | 0.090 | 0.086 | 0.0
92Zr | 0.000 | 0.002 | 17.8
94Zr | 0.000 | 0.004 | 7.2
96Zr | 0.000 | 0.007 | 10.5
98Zr | 0.000 | 0.004 | 53.2
100Zr | -0.108 | 0.005 | 56.8
102Zr | -0.114 | 0.122 | 59.6
104Zr | -0.121 | 0.003 | 57.0
94Mo | 0.000 | 0.003 | 5.4
96Mo | 0.000 | 0.009 | 2.8
98Mo | 0.000 | 0.012 | 4.6
100Mo | 0.000 | 0.009 | 41.3
102Mo | -0.114 | 0.118 | 57.1
104Mo | -0.120 | 0.126 | 57.9
106Mo | -0.126 | 0.129 | 59.4
144Nd | 0.048 | 0.033 | 7.8
146Nd | 0.089 | 0.089 | 0.2
148Nd | 0.119 | 0.125 | 0.0
150Nd | 0.179 | 0.203 | 0.0
152Nd | 0.204 | 0.214 | 0.0
154Nd | 0.210 | 0.219 | 0.0
156Nd | 0.216 | 0.225 | 0.0

Figure 14: Position index $I$ for the selenium (a) and zirconium (b) chains
(SV-bas parametrization).

Figure 15: Position index $I$ for the molybdenum (a) and neodymium (b) chains
(SV-bas parametrization).

To track the phase transitions on a single-quasiparticle level, one has to
investigate a relative position of single-quasiparticle positive and negative
parity proton and neutron states with respect to the Fermi level. We,
therefore, define new four position indices $I(\nu,\pi)$ for proton positive
($\nu=p$, $\pi=+$), proton negative ($\nu=p$, $\pi=-$), neutron positive
($\nu=n$, $\pi=+$), and neutron negative ($\nu=n$, $\pi=-$) parity states.
Values of $I(\nu,\pi)$ are found from
$I(\nu,\pi)=\sum_{i(\nu,\pi)}(0.5-|v^{2}_{i}-0.5|)$, where $v^{2}_{i}$ is the
pairing occupation probability of the state $i$ and we sum over all model
states of defined $\nu$ and $\pi$. Because $v^{2}=0.5$ exactly at the Fermi
level, the contribution of the levels close to the Fermi level to the position
index approaches the maximum value of 0.5, whereas the contribution of the
levels far away from the Fermi level approaches to zero. The position index
also depends on the strength of the pairing force. If it is low, the indices
approach zero, too. In the limit of zero pairing $v^{2}_{i}=0$ or 1 for all
states $i$ and $I(\nu,\pi)=0$. Parity enables to separate intruder states: In
the selenium isotopes ($Z=34$) both proton and neutron intruder states have
positive parity, while in the zirconium ($Z=40$) and molybdenum ($Z=42$)
isotopes proton intruder states have positive parity and neutron intruder
states have negative parity. In the neodymium ($Z=60$) isotopes, instead,
proton intruder states have negative parity and neutron intruder states have
positive parity. Results for selenium, zirconium, molybdenum and neodymium
chains are presented in Figs. 14 and 15 for the SV-bas parametrization.

In selenium isotopes protons and neutrons occupy the same shell and
$I(p,+)=I(n,+)$ between $N=38$ and 40 and around $N=44$. $I(p,-)=I(n,-)$
symmetrically around $N=40$ for $N=36$ and $N=44$. In zirconium isotopes
$I(p,-)=I(n,-)$ for $N=56$ and neutron intruder position index $I(n,-)$
reaches its maximum for $N=60$, i.e., these states are closer to the Fermi
level. In molybdenum isotopes $I(p,-)=I(n,-)$ again close to $N=56$ and the
neutron intruder position index $I(n,-)$ gradually increases up to $N=60$.

In neodymium isotopes, both $I(p,+)=I(n,+)$ and $I(p,-)=I(n,-)$ between $N=88$
and 90 where the phase transition should occur. The position indices for
intruder proton negative-parity states and for proton positive-parity states
are almost the same. This means that almost equal number of proton negative
and positive parity states lie close to the Fermi level, with a slight
preference for the intruder states (higher position index) before the phase
transition point. Close to the phase transition point a crossing of both
indices occurs and after it, the proton intruder states are more distant from
the Fermi level and, simultaneously, the proton pairing gets weaker. A common
feature of both regions, $N=60$ and $N=90$, namely a gradual increase of
$I(n,+)$ and $I(n,-)$ and a gradual decrease of $I(p,+)$ and $I(p,-)$ is more
pronounced in the case of neodymium. In addition, in both regions, the
intruder-neutron position index approaches its maximum for $N=60$ and 90,
respectively, indicating a maximum proximity of neutron intruder states to the
Fermi level.

In the case of Nd, the position indices for intruder negative- and normal
positive-parity states are almost the same. This means that almost equal
number of proton negative and positive parity states lie close to the Fermi
level, with a slightly preferable intruder states (higher position index)
before the phase transition point. At the phase transition point a crossing of
the both indices occurs and after it, normal proton states are close to the
Fermi level and the intruder states are more distant and, simultaneously, the
proton pairing gets weaker. Similar situation occurs for Mo, except the
weakening of the pairing interaction.

Figure 16: Occupation index $i$ for the proton (a) and neutron (b) states
closest to the Fermi level in the selenium chain (SV-bas parametrization).

Figure 17: Occupation index $i$ for the proton (a) and neutron (b) states
closest to the Fermi level in the zirconium chain (SV-bas parametrization).

Figure 18: Occupation index $i$ for the proton (a) and neutron (b) states
closest to the Fermi level in the molybdenum chain (SV-bas parametrization).

Figure 19: Occupation index $i$ as a function of the neutron number for the
proton (a) and neutron (b) states closest to the Fermi level in the neodymium
chain (SV-bas parametrization).

If we look closely at the single-quasiparticle (qp) states close to the Fermi
level obtained in our SHF+BCS calculations, we can identify microscopic
signatures of the CPS on the single-particle level. Here, we restrict
ourselves to the SV-bas parametrization only, because for other
parametrizations similar results were obtained. The states resulting from the
SHF+BCS calculations are characterized by the projection of the total angular
momentum $j_{z}$ on the symmetry axis and their parity $\pi$. In Figs. 16–19
the occupation index $i=0.5-v^{2}$ for the qp states closest to the Fermi
levels for the Se, Zr, Mo and Nd chains is shown (the index $i$ approaches 0
when the qp state aproaches the Fermi level and is positive for particle
states and negative for hole states; in addition it aproaches 0.5 for particle
and -0.5 for hole states if the qp states are distant from the Fermi level or
the pairing interation is weak).

One can observe a typical proton-qp level crossing around the phase transition
points: for the Se chain $1/2^{-}$ gets occupied while $5/2^{-}$ unoccupied,
for the Zr and Mo chains the intruder $9/2^{+}$ gets occupied while $1/2^{-}$
unoccupied, for the Mo chain $7/2^{-}$ gets occupied in addition, and for the
Nd chain $3/2^{+}$ and the intruder $1/2^{-}$ get occupied while $5/2^{+}$ and
the intruder $3/2^{-}$ unoccupied and the proton pairing gets weaker.

For the Se chain and $N=40$ the neutron pairing gets weaker, at $N=38$
neutron-qp states $1/2^{-}$ and intruder $3/2^{+}$ cross the Fermi level and
the intruder $9/2^{+}$ lies closer to the Fermi level and later between $N=42$
and 44, $5/2^{-}$, $3/2^{-}$ and the intruder $5/2^{+}$ cross the Fermi level.
For the Zr and Mo chains, $1/2^{+}$ and intruders $3/2^{-}$ and $7/2^{-}$
cross the Fermi level and become occupied, while for Mo only $5/2^{+}$ becomes
unoccupied. In the case of the Nd chain, $3/2^{-}$ and the intruder $1/2^{+}$
get occupied around $N=90$.

To summarize, we have found three possible microscopic signatures of
criticality (QPT): (i) flatness of PECs, (ii) position index for neutron
intruder states approaching a maximum value, and (iii) occupation index of
selected proton qp-levels along the isotopic chain changes its sign (i.e. some
particle states become hole states and v.v. at the QPT point).

To bridge from the microscopic calculations within the SHF+BCS model towards
macroscopic calculations within the Algebraic Collective Model (ACM), we can
use the procedure described in Ref. [107]. In macroscopic calculations we need
a potential $V_{\mathrm{ACM}}$ as a function of two deformation parameters,
$\beta$ and $\gamma$. It can be constructed from the microscopic PEC as a
function of $\beta_{2}$, $V(\beta_{2})$ and expanded up to the first order
term in $\cos{3\gamma}$:

$V(\beta,\gamma)=\frac{1}{2}[V(\beta_{2})+V(-\beta_{2})]+\frac{1}{2}[V(\beta_{2})-V(-\beta_{2})]\cos{3\gamma}$
(4)

The potential obtained from Eq. 4 is composed of two parts, a
$\gamma$-independent symmetric part and a $\gamma$-dependent antisymmetric
part. The results for the investigated $N=40$, 60 and 90 nuclei are shown in
Figs. 20 and 21 for two parametrizations, SV-bas and SV-tls. Qualitatively, we
get a $\gamma$-independent well for the symmetric part of the potential and a
decreasing antisymmetric $\gamma$-dependent potential for low and medium
quadrupole deformations.

Figure 20: Symmetric (a) and antisymmetric (b) parts of the potential derived
from the microscopic PECs for the SV-bas parametrization and the investigated
$N=40$, 60 and 90 nuclei.

Figure 21: Symmetric (a) and antisymmetric (b) parts of the potential derived
from the microscopic PECs for the SV-tls parametrization and the investigated
$N=40$, 60 and 90 nuclei.

##  IV Algebraic collective model

The Algebraic Collective Model (ACM) [108, 109, 110], introduced as a
computationally tractable version of the collective model of Bohr and
Mottelson (BMM) [23, 24] restricted to quadrupole rotational and vibrational
degrees of freedom, is characterized by a well defined algebraic structure.
Unlike the conventional U(5)$\supset$SO(5)$\supset$SO(3) dynamical subgroup
chain used, for example, in the Frankfurt program [112, 111], as well as in
the IBM [15, 18], the ACM makes use of the subgroup chain [113, 114, 115, 116,
117]

${\rm SU}(1,1)\times{\rm SO}(5)\supset{\rm U}(1)\times{\rm SO}(3)\supset{\rm
SO}(2)$ (5)

to define basis wave functions as products of $\beta$ wave functions and SO(5)
spherical harmonics.

Several advantages result from this choice of the dynamical subgroup chain:

(i) With the now available SO(5) Clebsch-Gordan coefficients [118, 119], and
explicit expressions for SO(5) reduced matrix elements, matrix elements of BMM
operators can be calculated analytically [109].

(ii) By appropriate choices of SU(1,1) modified oscillator representations,
the $\beta$ basis wave functions range [109] from those of the
U(5)$\supset$SO(5) harmonic vibrational model to those of the rigid $\beta$
wave function of the SO(5)-invariant model of Wilets and Jean [120].

(iii) With these SU(1,1) representations, collective model calculations
converge an order of magnitude more rapidly for deformed nuclei than in
U(5)$\supset$SO(5) bases [110].

Thus, the ACM combines the advantages of the BMM and the IBM and makes
collective model calculations a simple routine procedure [118, 121]. A
pedagogical treatment of the geometrical and algebraic foundations of the ACM
can be found in the book by Rowe and Wood [122].

Recall, that both E(5) [4] and X(5) [5] critical point symmetries correspond
to special solutions of the Bohr Hamiltonian [22], in which an infinite square
well potential in the quadrupole ($\beta$) degree of freedom is assumed. The
question of whether the assumption of a flat potential in both models is
justified has been raised and investigated using various methods. Limiting
ourselves to the regions of interest, we mention the following cases.

a) The need for a potential with a non-flat bottom in 152Sm within the
geometric collective model has been early realized [123]. A sextic potential
possessing two minima has been used in the Bohr Hamiltonian for the
description of some Nd [124], Mo [125], and Se [126] isotopes. A potential
with two minima (spherical and deformed) has been recently employed in the
Bohr Hamiltonian for the description of the Zr isotopes [127, 128]. The
existence of a bump in the PES of good experimental X(5) cases might be
related to the confined $\beta$-soft (CBS) rotor model [129], employing an
infinite square well potential displaced from zero, as well as to the
relevance of the Davidson potential
$\beta^{2}+\frac{\beta_{0}^{4}}{\beta^{2}}$ [130, 131], where $\beta_{0}$ is
the minimum of the potential.

b) Potential energy surfaces (PES) for the $N=90$ isotones, calculated in the
relativistic mean field (RMF) framework [96, 97], also exhibit a shape with
two minima. An alternative approach [132, 98], allowing for efficient
calculation of spectra and transition rates, is based on the use of an IBM
Hamiltonian, the parameters of which are determined by fitting the IBM PES to
the PES obtained from RMF calculations. The results of these studies do not
predict flat potential energy surfaces (PES) for the $N=90$ isotones [96, 97,
98, 99], which are the best experimental manifestations of X(5), while a
similar picture occurs also in the $N=60$ [100] and $N=40$ [101] isotones.

c) Shell model calculations in the regions of interest have been rather
limited, because of obvious computational hurdles. However, recent
calculations [133] for the Se isotopes have pointed out the importance of the
tensor force [134, 135] in shaping up the nuclear properties. While the
central force has a minimum at zero, the tensor force possesses a minimum at
non-zero distance from the center (see Fig. 1 of [133]), its shape roughly
resembling a Kratzer potential [136, 137]. Monte-Carlo shell model
calculations have recently reached the Zr isotopes, studying the QPT appearing
in them at $N=60$ [138, 139].

From the above the importance of having a flexible potential within the ACM to
address these questions becomes evident.

In our implementation of the ACM, observables of interest are formed by taking
sums of products of several generating operators, that are SO(3) invariant,
such as $\hat{\beta^{2}}$, $\hat{\beta^{-2}}$,
$\beta\frac{\mathrm{d}}{\mathrm{d}\beta}$, $\Delta^{2}$. The Hamiltonians that
can be analysed are polynomials in these operators with coefficients that may
be arbitrary real numbers or functions of the quantum numbers on which they
operate.

For the purposes of our study, the starting point will be the Hamiltonian in
the form

$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle
x_{1}{\nabla^{2}}+x_{2}+x_{3}\beta^{2}+x_{4}\beta^{4}+\frac{x_{5}}{\beta^{2}}+x_{6}\beta\cos{3\gamma}+x_{7}\beta^{3}\cos{3\gamma}+x_{8}\beta^{5}\cos{3\gamma}+\frac{x_{9}}{\beta}\cos{3\gamma}+x_{10}\cos^{2}{3\gamma}$
(6) $\displaystyle+$ $\displaystyle
x_{11}\beta^{2}\cos^{2}{3\gamma}+x_{12}\beta^{4}\cos^{2}{3\gamma}+\frac{x_{13}}{\beta^{2}}\cos^{2}{3\gamma}+\frac{x_{14}}{\hbar^{2}}[\hat{\pi}\otimes\hat{q}\otimes\hat{\pi}]_{0},$

where

$\nabla^{2}=\frac{1}{\beta^{4}}\frac{\partial}{\partial\beta}\beta^{4}\frac{\partial}{\partial\beta}+\frac{1}{\beta^{2}}\hat{\Lambda}$
(7)

is the Laplacian on the 5-dimensional collective model space and
$\hat{\Lambda}$ is the $SO(5)$ angular momentum operator [122]. Such a
Hamiltonian, expressed in terms of the quadrupole deformation parameters
$\beta$ and $\gamma$ serves as a useful starting point for a description of a
wide range of nuclear collective spectra. As the above Hamiltonian is a
rational function of the basic observables $\hat{q}$ and $\hat{\pi}$, its
matrix elements are obtained efficiently via available analytic expressions.

##  V Numerical results obtained within the ACM

In the numerical application of the ACM to the studied nuclei we fitted the
parameters of the Hamiltonian of Eq. (6) to the experimentally known low-lying
members of the ground, beta and gamma bands in 150Nd, 102Mo and 74Se. Both
theoretical and experimental spectra are shown in Figs. 22–24. As for the
100Zr, the used ACM model does not give satisfactory results. In an attempt to
describe the very low energy of the $\beta$ bandhead ($R_{\beta/2_{1}}=1.56$),
the structure of the ground and excited bands is severely disturbed. We intend
to address this issue in our future studies.

Figure 22: Comparison of experimental (left) and ACM (right) spectra for
150Nd. The energy difference of the $2^{+}_{1}$ and $0^{+}_{1}$ states is
normalized to the experimental value in 150Nd. The $B(E2)$ transition rates
are expressed in units of the $B(E2,2_{1}^{+}\rightarrow 0_{1}^{+})=1$
transition. The parameters used in the ACM calculations are displayed in the
right corner. Figure 23: Comparison of experimental (left) and ACM (right)
spectra for 102Mo. The energy difference of the $2^{+}_{1}$ and $0^{+}_{1}$
states is normalized to the experimental value in 102Mo. The $B(E2)$
transition rates are expressed in units of the $B(E2,2_{1}^{+}\rightarrow
0_{1}^{+})=1$ transition. The parameters used in the ACM calculations are
displayed in the right corner. Figure 24: Comparison of experimental (left)
and ACM (right) spectra for 74Se. The energy difference of the $2^{+}_{1}$ and
$0^{+}_{1}$ states is normalized to the experimental value in 74Se. The
$B(E2)$ transition rates are expressed in units of the
$B(E2,2_{1}^{+}\rightarrow 0_{1}^{+})=1$ transition. The parameters used in
the ACM calculations are displayed in the right corner.

When fitting the experimental spectra, we take into account the fact that the
first excited $K=0$ state band is not necessarily the $\beta$ band [76]. There
are still many questions about the true nature of this state [72, 73, 74, 75].
Thus, we focus the comparison to the ground state band and to the first
excited $2^{+}$ states, which are undoubtedly the $\gamma$ bandheads. In doing
so, we try to achieve the best possible description of the energy of the first
excited $K=0$ state, whose low energy is a signature of the shape phase
coexistence, but it is challenging, especially in case of 74Se. As a result of
the attempt to describe the low-lying $K=0$ states we observe centrifugal
stretching in the ground band, all the more prominent when the energy of the
first excited $K=0$ state decreases.

Even if the three nuclei are candidates for critical point nuclei of a first-
order QPT, we have already emphasized that the energy conditions in them are
very different. For example, the ratio $R_{42}$ decreases from 2.93 in 150Nd
to 2.51 in 102Mo and 2.15 in 74Se. This is very strongly reflected by the
value of the parameter $x_{3}$ which increases from 150Nd to 74Se.

For 150Nd the agreement of the experimental and theoretical spectrum is very
good. Along with the energies, the moments of inertia of the bands are also
relatively well described. The $\beta$ bandhead is significantly lower than
the $\gamma$ bandhead, but relatively high compared to the other two studied
nuclei, the ratio $R_{\beta 2}=E(0_{2})/E(2_{1})$ being 5.2. Obtaining a low-
lying $\beta$ band is generally a big problem mainly for lighter nuclei but in
this case the energies of both $\beta$ and $\gamma$ bandheads are well fitted
by the model. We also see that the ratio $B(E2,0^{+}_{2}\rightarrow
2^{+}_{1})/B(E2,2^{+}_{1}\rightarrow 0^{+}_{1})$ is 0.7 to be compared to the
experimental value of 0.37. A big value of this ratio (above 0.3) is believed
to be a signature of the QPT.

In case of 102Mo the ratio $R_{\beta 2}=E(0_{2})/E(2_{1})$ is about 2.4. In an
attempt to fit such a low-lying state a pronounced centrifugal stretching in
the ground state band is hard to avoid. The ratio $B(E2,0^{+}_{2}\rightarrow
2^{+}_{1})/B(E2,2^{+}_{1}\rightarrow 0^{+}_{1})$=1 is in an excellent
agreement with the experimental value of 0.95.

74Se is the lightest studied isotope with $R_{42}$ having a near-vibrational
value of 2.15. Thus to get an overall good agreement with the experimetal
spectrum a higher value of the parameter $x_{3}=-1$ is needed. Some
centrifugal stretching is again observed in the ground band, but the energies
and moments of inertia of the $\gamma$ band are very good. The energy of the
$\beta$ bandhead is less satisfactory, being about twice as high as its
experimental counterpart. Consequently, the theoretical ratio
$B(E2,0^{+}_{2}\rightarrow 2^{+}_{1})/B(E2,2^{+}_{1}\rightarrow 0^{+}_{1})$=1
is much smaller than the experimental value of 7.

Let us comment briefly on some of the parameters used in the calculation. All
in all, the parameters $x_{5}$ and $x_{8}$ allow to reduce the ratios
$R_{42}$, $R_{\beta 2}$, $R_{\gamma 2}=E(2_{\gamma})/E(2_{1})$ and
$R_{2(\beta)2}=E(2_{\beta})/E(2_{1})$, the effect of the parameter $x_{8}$
being more important (see Fig. 25). We may also notice that while in 150Nd the
$2^{+}$ member of the $\gamma$ band is the third excited $2^{+}$ state, the
situation changes in 102Mo where the $2^{+}$ member of the $\gamma$ band is
the second excited $2^{+}$ state. (The second or third excited $2^{+}$ state
is identified as a $\gamma$ bandhead based on the basis of a stronger
$B(E2,3^{+}_{1}\rightarrow 2^{+}_{2,3})$ transition). The relative position of
the $2_{\gamma}$ and $2_{\beta}$ states is strongly sensitive to the parameter
$x_{9}$, as is illustrated in Fig. 25. It can be seen that from a certain
value of this parameter, the relative position of the $2_{\gamma}$ and
$2_{\beta}$ states changes. Therefore, the largest value of $x_{9}$ for 150Nd
(among the three studied cases) is obvious. It should be noted that the term
proportional to parameter $x_{10}$ in Hamiltonian of Eq. (6) can induce
triaxiality. This term is not needed for 150Nd, where microscopic calculations
result in an effective $\gamma$ value of $0^{\circ}$ (see Table 2). In the
case of 102Mo and 74Se, a small parameter value of $x_{10}=0.5$ leads to an
overall improvement of the fit of the experimental spectra. This seems to be
consistent with microscopic calculations, where, especially in the case of
74Se, the effective value of the parameter $\gamma=22^{\circ}$.

Figure 25: Ratios $R_{42}$, $R_{\beta 2}$, and $R_{\gamma 2}$ as functions of
the ACM parameter $x_{8}$ (a) and ratios $R_{42}$, $R_{\beta 2}$, and
$R_{\gamma 2}$ as functions of the ACM parameter $x_{9}$ (b).

In Figs. 26-28 potentials used in the ACM fits for 150Nd, 102Mo and 74Se are
plotted. Their similarity to the microscopic symmetric and antisymmetric
potentials (Figs. 20-21) for low and medium quadrupole deformation values is
clearly visible, especially for 74Se and 102Mo, where the symmetric parts
($V_{1}(\beta)$) exhibit a minimum at low or medium deformations while the
antisymmetric parts ($V_{2}(\beta)$) are descreasing. The potentials presented
for 102Mo and 74Se are shown without the term $x_{10}\cos^{2}{3\gamma}$ in Eq.
(6) for simplicity, and thus under the assumption of zero triaxiality. The
cases of 150Nd and 102Mo are qualitatively very similar. The upper left panel
(the symmetric potential part not depending on $\cos{3\gamma}$ and containing
positive even powers of $\beta$) looks like a Davidson potential, widely used
in the context of QPTs [131], except that it does not go to infinity at the
origin. The upper middle panel (the antisymmetric potential part that is
$\cos{3\gamma}$ dependent and contains positive odd powers of $\beta$) again
looks like a Davidson potential. Their sum (upper right panel) indicates that
the result is dominated by the even powers. In the lower left and middle
panels we see that the negative even powers of $\beta$ lead to minus infinity
at the origin, while the negative odd powers lead to plus infinity at the
origin. The final result (lower right panel) indicates that the negative even
powers dominate at the origin, but the negative odd powers dominate a little
further to the right. It is clear that it is shaped up by the Davidson
potential of the positive even powers, the minus infinity potential
contributed by the negative even powers near the origin, and the plus infinity
potential contributed by the negative odd powers a little further to the
right. The final result resembles very much the confined beta-soft (CBS) model
of Pietralla et al., which uses an infinite square well potential displaced
from the origin. This has been used for both deformed nuclei [129] and gamma-
soft nuclei [141]. In other words, the main contribution of the negative odd
powers is to produce the displacement from the origin.

In the case of 74Se it seems that the negative powers are not needed, since no
displacement from the origin seems to be required. The total potential is
dominated by the positive even powers of $\beta$. A Davidson potential is
created by the positive even powers, but it is very shallow and looks flat.
The same holds for the total potential. Thus in the case of 74Se the total
potential looks very similar to the square well potential starting from the
origin, which is used in the $E(5)$ and $X(5)$ CPS. The only difference is
that the right wall of the potential does not rise abruptly to infinity, but
rises more smoothly, resembling the sloped wall potential used by Caprio [142]
in the $E(5)$ and $X(5)$ frameworks. The main effect of the sloped wall is
that it allows the beta band to go lower in energy. It also allows for
$R_{4/2}$ ratios lower than the 2.91 value predicted by $X(5)$. For example,
in Fig. 6 of Ref. [142] one can see that the spectra and $B(E2)$s for 150Nd
are well reproduced by a sloped well which has $R_{4/2}=2.667$ (see also Fig.
5 of Ref. [142] for sloped wall results producing $R_{4/2}=2.643$). However,
74Se has the experimental value of $R_{4/2}=2.148$, thus this sloped wall
alone cannot describe it.

Figure 26: Symmetric (left) and antisymmetric (middle) parts of the ACM
potential and the total ACM potential (right) obtained from the fit to the
experimental spectra of 150Nd. Potential terms diverging for $\beta=0$ are
excluded in the upper panel and included in the lower panel. Figure 27:
Symmetric (left) and antisymmetric (middle) parts of the ACM potential and the
total ACM potential (right) obtained from the fit to the experimental spectra
of 102Mo. Potential terms diverging for $\beta=0$ are excluded in the upper
panel and included in the lower panel. Figure 28: Symmetric (left) and
antisymmetric (middle) parts of the ACM potential and the total ACM potential
(right) obtained from the fit to the experimental spectra of 74Se.

##  VI Conclusions

In the present investigation three regions of the nuclear chart ($N=40$, 60,
90), in which a QPT from spherical to deformed nuclei is observed, are studied
on equal footing in three different ways, a) by consideration of the empirical
systematics of spectra and $B(E2)$ transition rates, b) through microscopic
mean field calculations using the Skyrme-Hartree-Fock + BCS approach, and c)
in the framework of the Bohr Hamiltonian, using the Algebraic Collective
Model. Several similarities and differences among these three regions are
pointed out.

The empirical systematics of spectra and $B(E2)$ transition rates indicate the
uniform occurrence of a first-order QPT in all three regions. In contrast,
they also show that the numerical values of the various indicators of
collectivity differ in the three regions, showing values corresponding to high
deformation at $N=90$, intermediate deformation at $N=60$, and low deformation
at $N=40$. From this point of view, the $N=90$ QPT appears to be close to the
parameter-independent predictions given by the X(5) critical point symmetry,
while the $N=40$ region appears to be closer to the X(3) CPS.

The potential energy curves obtained through microscopic Skyrme-Hartree-Fock +
BCS calculations for several different parametrizations provide a flat-
bottomed potential at non-zero prolate deformation in the $N=90$ region, while
in the $N=60$ region some degree of flatness is obtained on the oblate side.
In the $N=40$ region signs of flatness appear around zero deformation.

In the framework of the Algebraic Collective Model, the free parameters in
front of the various terms in the Bohr Hamiltonian are fitted to the data. The
resulting potential energy curves in the $N=90$ and $N=60$ regions resemble a
Davidson potential possessing a deformed minimum, being at the same time
displaced from the origin, in analogy to the confined $\beta$-soft (CBS)
model, applicable in deformed and transitional nuclei. In contrast, the PEC
obtained in the $N=40$ region resembles a flat potential starting from the
origin, as expected within the E(5) and X(5) CPSs, albeit with a sloped right
wall instead of a vertical one. The radically different picture obtained for
$N=40$ can be attributed to the fact that valence protons and neutrons in this
case occupy the same major shell.

Several open questions call for further investigations. It has been shown that
the symmetric and antisymmetric parts of the microscopic PECs and the ACM
potential behave in a similar way at least for low and middle $\beta_{2}$
values. To establish a closer link between these two approaches one has to
take into account also vibrational and rotational zero-point corrections in
the PEC calculations and include higher order terms in $\beta_{2}$ and
$\cos{3\gamma}$ to obtain a more realistic ACM potential. The rigid kinetic
term in the ACM Hamiltonian should be also replaced by a deformation dependent
($\beta_{2}$ and $\cos{3\gamma}$) term derived from the microscopic mass
tensor that might solve the problem of rotational band stretching in 100Zr.

The preference for prolate shapes in the $N=90$ region and for oblate shapes
in the $N=60$ region should be further investigated. Since spectra alone do
not provide differentiation between prolate and oblate shapes, relevant
investigations should be focused on the determination of the sign of the
relevant quadrupole moments, which is opposite for oblate nuclei in relation
to prolate nuclei. In light and medium-mass nuclei, in which the valence
protons and neutrons occupy the same major shell, the influence of the SU(4)
Wigner supermultiplet symmetry should be taken into account [143].

###### Acknowledgements.

This work was supported by the project SP2023/25 financed by the Czech
Ministry of Education, Youth and Sports and by the GAČR grant 22-14497S. This
paper is dedicated to the memory of Adam Prášek, an excellent student of VŠB-
Technical University of Ostrava who worked on the ACM calculations and passed
away in Grenoble Alpes on October 1, 2023.

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