11institutetext: Departamento de Ciencias Integradas y Centro de Estudios
Avanzados en Física, Matemática y Computación, Universidad de Huelva, 21071
Huelva, Spain 22institutetext: Instituto Carlos I de Física Teórica y
Computacional, Universidad de Granada, Fuentenueva s/n, 18071 Granada, Spain

# Sr isotopes: the interplay between shape coexistence and quantum phase
transitions

E. Maya-Barbecho 11<EMAIL_ADDRESS>J.E. García-Ramos 1122
<EMAIL_ADDRESS>
###### Abstract

In this contribution we study the even-even Sr isotopes considering the
influence of particle-hole intruder states. We find that the rapid onset of
deformation at A $=98$ and onwards can be explained by the crossing of regular
and intruder states. We compare the systematics of Sr with the ones of Zr, Pt,
and Hg nuclei.

##  1 Introduction

The understanding of the appearance of nuclear deformation is still of major
interest in nuclear physics. The collective phenomena in the atomic nucleus is
the origin of the onset of deformation, but it is also the responsible of the
vibrational character of nuclei. The range of phenomena between both limits is
modulated by a fine balance between the pairing, which tends to produce
spherical shapes, and the quadrupole force, which makes the nucleus deformed.
In others words, there is a competition between the stabilizing effect of the
shell closure energy gap and the residual interaction between like and unlike
nucleons.

Nuclei near to a shell closure tends to be spherical and excitations are
relatively costly, specially the particle-hole ones, but, surprisingly, enough
of the energy of such excitations can be considerably reduced due to the
increase of the residual interaction because of the presence of a larger
number of effective valence nucleons. This effect is particularly important
when protons are near to a shell closure, while neutrons are at the middle of
a shell, or viceversa. In this case, a particle-hole excitation will enhance
the proton-neutron quadrupole interaction and it will considerable reduce the
energy of such an excitation, being possible to appear at very low energy,
even becoming the ground state of the system. Moreover, these configurations
are expected to be more deformed than the normal ones. We will refer to them
as intruder and regular configurations, respectively, and to the interplay
between them as configuration mixing heyde11 .

Along this contribution we will study the connection between shape coexistence
and the so called Quantum Phase Transition (QPT) Cejn09 . We will focus on the
case of Sr, but a detailed comparison with Zr, Pt, and Hg nuclei is included.

##  2 Shape coexistence and quantum phase transitions: faces of a single
phenomenon?

Shape coexistence in nuclear physics was first proposed by Morinaga in the
1950’s and nowadays becomes a concept that appears throughout the entire
nuclear landscape, especially in those nuclei at or near shell or sub-shell
closures heyde11 . Shape coexistence presents certain key experimental
features: an U shape pattern in the energy systematics of a set of states, the
lowering of certain excited $0^{+}$ states (intruder states), a rapid change
in the value of the spectroscopic quadrupole moments, and the existence of
strong E0 transitions. All of them show an almost symmetric trend with respect
to the corresponding mid-shell and they are enhanced at this place.

Shape coexistence can be understood in terms of two well-known theoretical
approaches: the spherical shell model and the mean field. According to the
shell model approach, the intruder configurations correspond to the promotion
of pairs of nucleons across the shell or subshell gap closures. The other
major approach is the mean-field theory where, roughly speaking, one obtains
an energy surface depending on certain deformation parameters. In this
landscape, the ground band will correspond to the deepest minimum (regular
states), but the other local minima can be interpreted as intruder
configurations, each of them owning to different deformations.

The concept of QPT Cejn09  implies the sudden change of the ground state
structure of the system as a function of a control parameter. Such a control
parameter can be in the case of the atomic nucleus, e.g., the neutron number.
Therefore, a QPT can appear in an isotopic chain where the ground state
deformation varies in an abrupt way when passing from isotope to isotope. Of
course, the latter should be considered as an approximation because a control
parameter should be, strictly speaking, continuous. A QPT is characterized by
the discontinuity of some derivative of the system energy. Assuming that the
two-neutron separation energy (S2n) is somehow proportional to the derivative
of the energy with respect to the neutron number, a first-order QPT involves a
discontinuity in S2n, while a second-order one a discontinuity in the
derivative of S2n. The even-even Sr isotope chain, as will be shown along the
text, corresponds to the case of a first-order QPT.

An ideal framework to deal with QPTs is the interacting boson model (IBM)
iach87  which is a paradigmatic example of a nuclear algebraic model. In that
framework, a QPT can be understood considering a Hamiltonian that is a
combination of two given symmetries, combined through a control parameter,
$H=x\ H_{\mbox{sym1}}+(1-x)\ H_{\mbox{sym2}}$ that allows to go from one to
the other limit. The presence of a QPT is revealed by the existence of a
critical value of the control parameter, xc, for which the underlying
structure of the system (phase) passes from symmetry 1 to symmetry 2.

The IBM can be also used to deal with shape coexistence, allowing to define a
framework where different particle-hole excitations are present in the
spectrum. The formalism is known as IBM with configuration mixing, IBM-CM in
short duval81 ; duval82 .

Is there any connection between shape coexistence and QPT? First, one should
note several similarities, namely, both phenomena involve a rapid change in
the structure of certain set of states, either ground or excited states and,
in both cases, at the mean-field level, several minima coexist. In the case of
shape coexistence, the regular and the intruder configurations can be thought
as having independent energy surfaces that interact among them, especially
when their minima are close in energy, while under a QPT a single energy
surface exists where two or more minima are present. The evolution of the
relative position of the minima as a function of the neutron number is at the
origin of the change of the nuclear deformation and it can be understood in
terms of either shape coexistence or a QPT. On the other hand, from a quantum
point of view clear differences should exist between both approaches. The most
obvious one is the existence of a larger Hilbert space when two configurations
are present, because of the extra nucleons that occupy the single particle
levels. This extra set of levels presents a different energy systematics than
the regular one because for them the effective nuclear interaction is
different. Moreover, the intruder configuration is expected to be found at low
energy when the number of valence nucleons is large, i.e. around mid shell,
therefore a parabolic-like energy systematics for the intruder states appears
frequently. However, the situation can be blurred due to the interaction
between intruder and regular configurations or because their crossing Garc11 .
When a QPT exist without the presence of multiple particle-hole excitations, a
larger density of states can be also observed at low energy as for
configuration mixing, but it never happens around mid-shell Garc20 . The
remaining open question is whether a QPT can be induced by shape coexistence,
which we will try to answer along the rest of the contribution.

##  3 Sr and Zr isotopes versus Pt and Hg nuclei

The regions around the sub-shell closure Z $=40$ are, together with the area
around Z $=82$, two of the best examples of shape coexistence in nuclei. In Hg
and Pt nuclei, two types of configurations show up with a clear presence of
low lying $0^{+}$ states, combined with a parabolic-like shape in the spectra
of Hg, whereas this behavior is not so obvious in Pt. On the other hand, Zr
and Sr are known to exhibit the faster change in deformation in an isotope
chain at N $=60$, with two particle-hole configuration coexisting and crossing
at this point.

Figure 1: S2n and $E(4_{1}^{+})/E(2_{1}^{+})$ as a function of the neutron
number, $N$, for Sr, Zr, Pt and Hg isotopes.

From the point of view of a QPT, there are two key observables that can be
used as indicators of its presence, namely, the two-neutron separation energy,
S2n, that will experience a discontinuity at a first-order transition point
and the ratio $E(4_{1}^{+})/E(2_{1}^{+})$ which provides a hint for the
appearance of deformation or, in a qualitative way, it is connected with the
order parameter of the QPT. In Fig. 1, we present the systematics of S2n and
$E(4_{1}^{+})/E(2_{1}^{+})$, for Sr and Zr, on the one hand and, for Pt and
Hg, on the other. Regarding the value of S2n, one notices a clear change of
slope in Sr and Zr, but a fully linear behavior in Hg and Pt. On the other
hand, $E(4_{1}^{+})/E(2_{1}^{+})$ shows for Sr and Zr the typical rapid change
that is expected in a QPT, passing from a value close to $2$ (even $1.5$ for
96-98Zr) to another close to $3.3$. However, in Pt and Hg the interpretation
of the systematics is not so evident and the physics involved in these cases
seems to be quite different from the one of Sr and Zr.

Figure 2: a), c), e), and g) panels: IBM value of $\beta$ for the unperturbed
regular, $[N]$, and intruder configurations, $[N+2]$, and full calculation,
IBM-CM. b), d), f), and h) panels: ground state energy for the unperturbed
regular and the intruder configurations and for the full calculation. The
green thin dashed line stands for the energy needed to create a particle-hole
excitation and the red thin dotted one stands for the reference level.

To have a more clear view of the analogies and differences between both
situations, we present in Fig. 2 some relevant IBM calculations based on
Maya22  for Sr, Garc20  for Zr, Garc09  for Pt, and Garc14b  for Hg. In
particular, the results correspond to the value of the IBM deformation
parameter $\beta$ for the regular ([N]), the intruder configuration ([N+2])
and the complete system (IBM-CM) (panels a), c), e), and g)). Moreover, the
unperturbed energy for the lowest regular, intruder, and ground state (IBM-CM)
are depicted in panels b), d), f), and h). Concerning Sr and Zr, the
distinctive feature is the crossing of the regular and the intruder
configurations at A $=98$ and $100$, respectively, corresponding to N $=60$,
inducing a change in the slope of the ground state energy and also a rapid
onset of deformation, which has a dramatic effect on the trend of the radii
Garc19 ; Maya22 . In the case of Pt and Hg, both configurations remain quite
close, without crossing in the case of Hg, and crossing before and after the
mid-shell (N $=104$) in the case of Pt. The ground state energy is hardly
modified, but the deformation parameter $\beta$ passed from the less deformed
configuration (the regular one) to the most deformed one (the intruder one)
around mid-shell. The crossing or the close approaching of regular and
intruder configurations in Pt and Hg, respectively, have also a noticeable
influence in the radii of these nuclei, as shown in Garc11 ; Garc14b . In
summary, in Sr, Zr and Pt nuclei, a intruder state becomes the ground state
when approaching the mid-shell, while it never happens in Hg. Note that in Pt,
due to the strong mixing between both configurations Garc09 , no dramatic
effect are observed in the spectra, although the energy crossing exists.

Figure 3: Axial IBM-CM mean-field energy surfaces for 92-102Sr isotopes. Red
dotted line stands for the unperturbed regular configuration, [N], green
dashed one for the unperturbed intruder configuration, [N+2], while the black
continuous line for the full IBM-CM calculation.

The above features have direct consequences in the systematics of S2n and
$E(4_{1}^{+})/E(2_{1}^{+})$. In the case of Sr and Zr, what it is observed in
Fig. 1 can be easily understood in terms of the crossing of configurations.
Indeed, the change in the nature of the ground state produces a sudden change
in the slope of the energy systematics and therefore a discontinuity in the
value of S2n. On the other hand, because the crossing is so fast, not only the
ground state presents an intruder character, but also the rest of low-lying
members of the yrast band involved in the energy ratio
$E(4_{1}^{+})/E(2_{1}^{+})$, which makes that for $N<60$ all the states
present a vibrational character, while for $N\geq 60$ a rotational one
appears. On the other hand, in the case of Hg, because the ground state
presents all the way a regular character and, moreover, there is no
interaction between intruder and regular sector, it can be understood the
linear tendency of the S2n even at the mid-shell. Besides, the crossing
between $2_{1}^{+}$ and $4_{1}^{+}$ regular and intruder states can explain
the drop of the ratio $E(4_{1}^{+})/E(2_{1}^{+})$ around the mid-shell.
Finally, to explain the systematics of Pt in Fig.1 is a challenge. First, the
crossing of regular and intruder $0^{+}$ energies would suggest a
discontinuity in S2n as the observed one in Sr and Zr. However, experimental
values show a fully linear systematics. The reason for such a behavior is the
strong mixing between the regular and the intruder sectors (see Garc09 ). Note
that according to Garc14a  and figure Fig. 2, the deformation changes as it
should be in a QPT, however the strong mixing somehow hides its effect in both
the S2n and $E(4_{1}^{+})/E(2_{1}^{+})$ observables.

A different way of seen the evolution of the nuclear deformation is to study
the mean-field energy surfaces in a chain of isotopes, which can be calculated
also using the IBM-CM formalism Frank02 . This is depicted in Fig. 3 for Sr
isotopes, where the axial energy surfaces for the regular and the intruder
configurations, as well as the one for the complete system are presented.
Concerning the regular configuration (red dotted line), it presents a smooth
evolution being spherical for A $=92-96$, becoming very flat and finally
axially oblate deformed for A $=98-102$. Regarding the intruder configuration
(green dashed line), it starts being also spherical for A $=92$ and $94$, but
rapidly becomes axially prolate deformed from A $=96$ and onwards. Finally,
the IBM-CM full calculation (black full line), which essentially follows the
lowest of the intruder and regular curves, starts also with a spherical
minimum, A $=92-96$ and also becomes an axially prolate deformed one for A
$=98-102$. A very relevant feature is that both configurations evolve
separately from a spherical to a deformed shape, more rapidly in the case of
the intruder configuration. Therefore, most probably in both configurations
separately, regular and intruder, a QPT is present. Moreover, the fact that
both configurations cross at A $=98$ makes much more abrupt the change of
deformation, also inducing a QPT in the whole system. The QPT present in the
separated regular and the intruder configurations is called type-I QPT Gavr19
and involves the existence of a single configuration energy surface where
several minima are present. On the other hand, for the whole energy surface a
type-II QPT Gavr19  appears because two energy surfaces that cross are
present. A type-II QPT is expected to induce much more abrupt changes than the
type-I and it will own the typical phenomenology of a first order QPT. Note
that the type-I QPT is also able to reproduce the S2n trend of Zr isotopes, as
was shown in Garc05 , although the type-II QPT provides a more consistent
framework Garc20  to explain the rapid onset of deformation. It is worthy to
mention that this is not the only mechanism to induce a discontinuity in the
S2n, as happens, for instance, in the rare-earth region at $N\approx 90$.

##  4 Conclusions

In this contribution, we have studied the onset of deformation in Sr on the
light of the coexistence of several particle-hole configurations. We have
compared with Zr isotopes that is another example of $Z\approx 40$ nucleus and
that presents a common trend regarding many observables and with Pt and Hg
nuclei that are paradigmatic examples of nuclei where shape coexistence plays
a major role in $Z\approx 82$. With the latter case notable differences exist.

The origin of the onset of deformation in $Z\approx 40$ region can be
understood in terms of the residual proton-neutron interactions resulting in
major modifications in the occupation of the 1g9/2 proton and 1g7/2 neutron
orbitals. This approach suggested in Federman and Pittel Fede77 ; Fede79b
emphasizes the importance of the simultaneous occupation of neutrons and
protons spin-orbit partners. In this work, we have tried to define an
effective approach for the Federman-Pittel mechanism using the IBM-CM.

In the framework of the IBM, the rapid changes observed in transitional
regions, as is the case of Sr, can be understood in terms of a Quantum Phase
Transition (QPT), however, the use of configuration mixing, IBM-CM, explains
in a more consistent way the very abrupt changes in terms of the crossing of
two configurations, calling to this situation type-II QPT. Note that the
crossing does not always creates a QPT, as happens in the case of Pt, where
the strong interaction between the involved particle-hole configurations
hinder the observation of abrupt changes.

Sr, together with Zr, isotopes are very clear examples of QPTs that can be
explained in terms of the crossing of two configurations, in other words,
examples of Type-II QPTs.

##  5 Acknowledgements

This work was partially supported by grant PID2019-104002GB-C21 funded by
MCIN/AEI/10.13039/50110001103 and “ERDF A way of making Europe”. Resources
supporting this work were provided by the CEAFMC and Universidad de Huelva
High Performance Computer (HPC@UHU) funded by ERDF/MINECO project
UNHU-15CE-2848.

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