# Low-spin particle/hole-core excitations in 41,47,49Ca isotopes
studied by cold-neutron capture reactions

S. Bottoni1,2, N. Cieplicka-Oryńczak3, S.Leoni1,2, B. Fornal3, G.Colò1,2, P.F.
Bortignon1,2, G. Bocchi1,2, D. Bazzacco4, G. Benzoni2, A. Blanc5, A.
Bracco1,2, S. Ceruti1,2, F .C .L. Crespi1,2, G. de France6, E. R. Gamba7,2,
Ł.W. Iskra2,3, M. Jentschel5, U. Köster5, C. Michelagnoli5, B. Million2, D.
Mengoni8,4, P. Mutti5, Y. Niu9, C. Porzio1,2, G. Simpson5, T. Soldner5, B.
Szpak3, A. Türler10, C.A. Ur11, W. Urban12 1Dipartimento di Fisica, Università
degli Studi di Milano, 20133 Milano, Italy  2INFN Sezione di Milano, 20133,
Milano, Italy  3Institute of Nuclear Physics, PAN, 31-342 Kraków, Poland
4INFN Sezione di Padova, 35131 Padova, Italy  5Institut Laue-Langevin, 38042
Grenoble CEDEX 9, France  6GANIL, BP 55027, 14076 Caen CEDEX 5, France  7Museo
Storico della Fisica e Centro di Studi e Ricerche Enrico Fermi, 00184 Roma,
Italy  8Dipartimento di Fisica e Astronomia, Università degli Studi di
Padova, 35131 Padova, Italy  9School of Nuclear Science and Technology,
Lanzhou University, Lanzhou 730000, China  10Universität Bern and Paul
Scherrer Institut, Villigen, Switzerland  11ELI-NP, Magurele-Bucharest,
Romania  12Faculty of Physics, Warsaw University, 00-681 Warsaw, Poland

###### Abstract

We present recent results on the structure of the one-valence-particle 41Ca
and 49Ca, and one-valence-hole 47Ca, nuclei. The isotopes of interest were
populated via the cold-neutron capture reactions 40Ca(n,$\gamma$),
48Ca(n,$\gamma$) and 46Ca(n,$\gamma$), respectively. The experiments were
performed at the Institut Laue-Langevin, within the EXILL campaign, which
employed a large array of HPGe detectors. The $\gamma$ decay and level schemes
of these nuclei were investigated by $\gamma$-ray coincidence relationships,
leading to the identification of 41, 10, and 6 new transitions in 41Ca, 47Ca,
and 49Ca, respectively. Branching ratios and intensities were extracted for
the $\gamma$ decay from each state, and $\gamma$-ray angular correlations were
performed to establish a number of transition multipolarities and mixing
ratios, thus helping in the spin assignment of the states. The experimental
findings are discussed along with microscopic, self-consistent beyond-mean-
field calculations performed with the Hybrid Configuration Mixing model, based
on a Skyrme SkX Hamiltonian. The latter suggests that a fraction of the low-
spin states of the 41Ca, 49Ca, and 47Ca nuclei is characterized by the
coexistence of either 2p-1h and 1p-2h excitations, or couplings between
single-particle/hole degrees of freedom and collective vibrations (phonons) of
the doubly-magic “core”.

###### pacs:

28.20.Np, 23.20.Lv, 23.20.En, 24.10.Cn

##  I INTRODUCTION

The structure of calcium isotopes between the doubly-magic 40Ca (N=20) and
48Ca (N=28) nuclei has been the subject of many experimental studies over the
past decades Yasuda  _et al._ (2010); Lui  _et al._ (2011); Montanari  _et
al._ (2011a, b, 2012); Garcia Ruiz  _et al._ (2015); Noji  _et al._ (2015);
Gade  _et al._ (2016); Hadyńska-Klȩk  _et al._ (2016); Riley  _et al._ (2016);
Crawford  _et al._ (2017); Talwar  _et al._ (2018); Hadyńska-Klȩk  _et al._
(2018). With six stable isotopes, calcium plays a crucial role in stellar
nucleosynthesis D. Bodansky and Fowler (1968); Truran (1972); Käppeler  _et
al._ (1985). The formation of Ca isotopes involves several astrophysical
processes, such as silicon- and oxygen-burning S. E. Woosley and Clayton
(1973), as well as s- and r-processes Käppeler  _et al._ (1982); Cameron
(1979), which generate, for example, the heaviest, symmetric N=Z stable
nucleus, i.e. 40Ca, and the lightest stable doubly-magic neutron-rich system,
namely 48Ca, in the nuclide chart. Moreover, the Z=20 isotopic chain contains
a rare cosmogenic radioactive nucleus, i.e. 41Ca, produced by neutron-capture
reactions on 40Ca induced by cosmic rays Kubik  _et al._ (1986).
In this context, nuclear structure studies along Ca isotopes are crucial to
understand, for instance, the evolution of single-particle states and
collectivity from symmetric to neutron-rich systems, which are properties
significantly affecting the reaction rates in stellar environments. Moreover,
new experimental results may serve as a benchmark for the most advanced
theoretical models, such as state-of-the-art shell model calculations Honma
_et al._ (2002, 2005); Utsuno  _et al._ (2012); Tsunoda  _et al._ (2014) and
ab initio approaches, employing chiral two- and three-nucleon interactions
Hebeler  _et al._ (2011); Holt  _et al._ (2012, 2014); Simonis  _et al._
(2016).
The low-lying structure of 40Ca is characterized by a 0+ state at 3.4 MeV, as
a first excited state - a clear signature of a robust double shell closure in
this nucleus - and a very collective octupole, 3- vibration at 3.7 MeV, with a
B(E3) of $\approx$ 30 W.u. Kibedi and R.H.Spear (2002). Moreover, in the spin
range 2$\hbar$-8$\hbar$, deformed and superdeformed bands have been observed
and associated to 4p-4h and 8p-8h excitations, respectively Ideguchi  _et al._
(2001). These features are gradually lost in mid-shell Ca nuclei, where
deformed structures take over spherical ones already at low energies, owing to
neutron p-h excitations across the pfg energy gap. This scenario changes again
in 48Ca, where the presence of a low-lying 0+ state at 4.3 MeV and a 3- phonon
with B(E3) $\approx$ 7 W.u. Burrows (2006) suggests the restoration of the
spherical symmetry, although weaker than in 40Ca.
In this framework, Ca nuclei one-particle or one-hole away from double shell
closures are of particular interest. These isotopes are ideal to investigate
the interplay between fermionic and bosonic degrees of freedom, as it occurs
in the coexistence and competition between pure p-h excitations and the so-
called particle/hole-vibration coupling Bohr and Mottelson . As a matter of
fact, the low-lying structure of one-valence-particle/hole nuclei is strongly
influenced by the collective phonons of the underlying “core”. On the other
hand, core excitations are perturbed and damped by the single particle/hole
motion and non-collective p-h excitations Hamamoto (1969, 1974). Therefore, a
comprehensive investigation of these mechanisms, moving along the Ca isotopic
chain, may significantly advance our understanding of the emergence of complex
phenomena, such as the quenching of spectroscopic factors and the anarmonicity
of vibrational spectra in this mass region.
In this paper, we present new experimental results in 41Ca, 47Ca, and 49Ca,
populated via cold neutron-capture reactions and studied by $\gamma$-ray
spectroscopy. Neutron-capture reactions induced by cold and thermal neutrons
populate the corresponding N+1 systems at the neutron separation energy Sn.
The spin of the capture level depends on the ground-state spin J of the target
nucleus and can only be J $\pm$ 1/2, being 1/2 the spin of the neutron. As a
consequence, neutron-induced reactions on even-even nuclei always proceed
through a single 1/2+ neutron-capture state. The $\gamma$-ray decay is
typically dominated by high-energy, E1 primary transitions, which
preferentially populate 1/2- and 3/2- states (based on $\gamma$-decay
selection rules), followed by secondary electromagnetic radiation of different
character and multipolarity.
In this context, it is clear that the combined use of neutron-capture
reactions and detectors with high energy resolution (e.g. HPGe crystals)
enables to perform an almost-complete $\gamma$-ray spectroscopy from the
neutron binding energy to the ground state, providing an exhaustive picture of
the low-spin structure of the nuclei of interest. The present experimental
results on the 41Ca, 47Ca, and 49Ca nuclei will be discussed in the framework
of the Hybrid Configuration Mixing Model Colò  _et al._ (2017); Bottoni  _et
al._ , with particular attention to the interplay between single-particle/hole
states and couplings with core excitations.
The paper is organized as follows: in Sec. II the experimental details will be
presented along with the different reactions performed; in Sec. III the
analysis of the data will be discussed, while in Sec.IV the experimental
results will be outlined in connection with theoretical interpretations.

##  II THE EXPERIMENT

Figure 1: (Color online) Relative $\gamma$-ray efficiency of the HPGe array
used in the 48Ca(n,$\gamma$) experiment. Experimental data obtained from the
35Cl(n,$\gamma$) reaction are displayed along with the fit function (red). The
efficiency of the full EXILL setup is also reported Jentschel  _et al._ (2017)
(see text for details).

The experiments were performed at Institut Lau-Langevin (ILL) in Grenoble,
within the EXILL experimental campaign Jentschel  _et al._ (2017). Neutron-
capture reactions were studied at the High Flux Reactor of ILL Moll (1974),
which delivers the most intense, continuous neutron beams worldwide for
scientific research.
In the present measurement, a high-efficiency, high-resolution composite HPGe
array was installed at the PF1B cold-neutron beam line Abele  _et al._ (2006),
where the neutron flux was about 108 neutrons cm-1 s-1, after collimation.
The array comprised 8 clover detectors from the EXOGAM setup Simpson  _et al._
(2000), 6 coaxial GASP detectors Alvarez (1993) and 2 ILL clover detectors,
providing a total photopeak efficiency of $\approx$ 6% at 1.3 MeV. Apart from
the ILL clovers, all the other HPGe detectors were equipped with BGO anti-
Compton shields for background suppression. In the case of the
48Ca(n,$\gamma$) experiment, the GASP and ILL detectors were replaced by 16
LaBr3:Ce fast scintillators from the FATIMA collaboration Roberts  _et al._
(2014) for lifetime measurements by using fast-timing techniques Regis  _et
al._ (2013). As a consequence, the $\gamma$-ray efficiency of this HPGe
detector configuration, comprising the EXOGAM clovers only, differs from the
one of the full EXILL setup. In Fig.1, the relative efficiency of the HPGe
array used in the 48Ca(n,$\gamma$) experiment is reported, arbitrary
normalized at $\approx$ 400 keV to the efficiency curve of the full EXILL
configuration Jentschel  _et al._ (2017). The experimental efficiency values
were determined up to 8 MeV using known $\gamma$-ray transitions in the 36Cl
nucleus, populated in the 35Cl(n,$\gamma$) reaction. At high energy, the
deviation of the present efficiency from the EXILL curve is consistent with
the reduced number of HPGe detectors in the setup.
The compact geometry of the 8 EXOGAM clovers, mounted in a symmetric, ring
configuration around the scattering chamber, was used to study $\gamma$-ray
angular correlations, with the aim of determining the multipolarity of the
detected radiation, thus constraining the spin and parity of the observed
states. All the possible angular combinations between crystals (11 angles from
0∘ to 90∘) were grouped into 3 angles only, i.e. 0∘, 45∘, and 90∘,
corresponding to the angles between clover detectors with respect to the
target position. This enabled to increase the statistics for $\gamma$-$\gamma$
coincidences, allowing us to perform angular-correlation studies also in cases
of weak transitions. Experimental data were fitted by using the analytic
function Morinaga (1976)

$W(\theta)=1+a_{22}q_{2}P_{2}(\cos\theta)+a_{44}q_{4}P_{4}(\cos\theta),$ (1)

where $a_{ii}$ are the multipole expansion coefficients, Pi(cos$\theta$) the
Legendre Polynomials, and qi the attenuation parameters which take into
account the finite size of the detectors. The latter were determined by
studying $\gamma$-ray angular correlations of known transitions of the 152Eu
$\gamma$-ray source, and were found to be q1=0.87 and q2=0.6.
The 41Ca and 47Ca nuclei were populated by (n,$\gamma$) reactions on an
enriched target. For this purpose, a 40.6-mg Ca(NO3)2 compound, enriched to
31.7% in 46Ca, was prepared at Paul Scherrer Institute, Switzerland. The
nitrate solution was directly dried in a 25 $\mu$m thin FEP (fluorinated
ethylene propylene) bag, which has negligible neutron-capture cross section.
It is important to note that 46Ca is the isotope with the second-lowest
natural relative isotopic abundance (only 0.004%), after 3He. A large fraction
of the target (60.5%) was composed by 40Ca, with the latter being the most
abundant Z=20 isotope. This allowed us to perform 46Ca(n,$\gamma$) and
40Ca(n,$\gamma$) reactions at the same time. On the other hand, the 49Ca
nucleus was populated by neutron-capture reactions on a 350-mg CaCO3 compound
target, enriched to 60.5% in 48Ca. Also in this case, traces of other Ca
isotopes were present in the sample.
The composition of the targets used in the current experiments, along with the
(n,$\gamma$) cross sections for thermal neutrons Mughabghab , are reported in
Tab. 1. The isotopes of interest for this work and the corresponding targets
are highlighted in bold.

Table 1: Isotopic composition of the Ca(NO3)2 and CaCO3 compounds used in the current experiment and corresponding (n,$\gamma$) cross sections Mughabghab . The (n,$\gamma$) columns show the percentage of capture reactions for a given isotope. The nuclei of interest for the the present work are marked in bold. Details of the FEP bag are also reported (see text for details). |  $\sigma$(n,$\gamma$) |  | Ca(NO3)2 |  | CaCO3 |
---|---|---|---|---|---|---
| [barn] |  | [atoms %] | (n,$\gamma$) | [atoms %] | (n,$\gamma$)
TARGET |  |  |  |  |  |
40Ca | 0.41 |  | 60.5 | 34% | 27.9 | 13%
42Ca | 0.68 |  | 0.63 | 1% | 0.3 | 0%
43Ca | 6.2 |  | 0.15 | 1% | 0.1 | 1%
44Ca | 0.88 |  | 5.35 | 6% | 2.5 | 2%
46Ca | 0.74 |  | 31.7 | 32% | 0.1 | 0%
48Ca | 1.09 |  | 1.57 | 2% | 69.2 | 83%
C | 3.84$\cdot 10^{-3}$ |  |  | 0% | 100 | 0%
N | 7.47$\cdot 10^{-2}$ |  | 200 | 20% |  | 0%
O | 2.24$\cdot 10^{-4}$ |  | 600 | 0% | 300 | 0%
FEP |  |  |  |  |  |
C | 3.84$\cdot 10^{-3}$ |  | 89 | 0% | 6 | 0%
F | 9.51$\cdot 10^{-3}$ |  | 178 | 2% | 13 | 0%

##  III DATA ANALYSIS

Data were acquired using fast, digital electronics in triggerless mode and the
analysis was performed by considering coincident events, built within a
200-ns, prompt time window. The good energy resolution and efficiency of the
HPGe array turned out to be essential to observe, with high accuracy, very
weak $\gamma$-ray decay paths, whereas the BGO Compton shields significantly
suppressed $\gamma$-ray coincidences with Compton-scattered radiation.
The level schemes and the $\gamma$-ray decays of the 41Ca, 47Ca, and 49Ca
nuclei were studied by using $\gamma$-$\gamma$ and triple-$\gamma$ coincidence
techniques. At first, very selective gates on primary, high-energy $\gamma$
transitions were used to identify secondary $\gamma$-ray cascades and to
locate new low-lying states. The energies of the latter were determined by
correcting the measured $\gamma$ energies by the recoil energy of the nucleus.
This is particularly crucial for high-energy transitions, considering the
relatively light mass of the isotopes studied in this work. As a second step,
gates on $\gamma$ transitions depopulating low-lying states enabled to
determine new decay paths from the neutron-capture level and to measure
precisely the value of the Sn neutron separation energy for all three nuclei.
The latter were obtained by considering all the possible combinations of
$\gamma$ rays decaying directly from the neutron-capture level. The Sn values
obtained in this work are presented in Tab 2, along with those ones reported
in the literature Gruppelaar and Spilling (1967); Cranston  _et al._ (1970);
Arnell  _et al._ (1969); Nesaraja and McCutchan (2016); Burrows (2007, 2008).

Table 2: Neutron separation energies (Sn) for the 41Ca, 49Ca, and 47Ca nuclei obtained in this work, compared with values reported in the literature Gruppelaar and Spilling (1967); Cranston  _et al._ (1970); Arnell  _et al._ (1969); Nesaraja and McCutchan (2016); Burrows (2007, 2008). Isotope | Sn [keV] | Sn [keV]
---|---|---
| (this work) | (literature)
41Ca | 8362.4(2) | 8362.8(2)
47Ca | 7275.4(2) | 7276.4(3)
49Ca | 5146.3(3) | 5146.5(2)
Figure 2: (Color online) Projection of the $\gamma$-$\gamma$ coincidence
matrix measured in the 40Ca(n,$\gamma$) reaction a), gated on the 4418-keV
transition of the 41Ca nucleus. Projections of the triple-$\gamma$ coincidence
matrix, gated on the 2010- and 1390-keV b), 1943- and 727-keV c), and 1943-
and 520-keV d) transitions. New $\gamma$ rays, observed for the first time,
are marked in red, while those already known in the literature (Gruppelaar and
Spilling, 1967; Cranston  _et al._ , 1970; Arnell  _et al._ , 1969; Nesaraja
and McCutchan, 2016; PGA, ) are marked in black. Transitions associated to
(n,$\gamma$) reactions on contaminants present in the target are labelled by
circles. First and second escape peaks for high-energy transitions are marked
by 1${}^{\text{st}}$ and 2${}^{\text{nd}}$, respectively.

The $\gamma$-ray intensities and branching ratios for each level were
evaluated using $\gamma$-$\gamma$ matrices, constructed considering all HPGe
detectors in the array. Gates were set on transitions feeding the level of
interest and relative intensities of deexciting transitions, with respect to a
given $\gamma$ ray in the level scheme, were extracted, taking into account
efficiency corrections. Branching ratios were determined by taking the ratio
of the intensity of a $\gamma$ transition to the summed intensity of all
transitions deexciting a given state. Concerning primary $\gamma$ rays, all
possible decay paths for each transition were considered, based on the
analysis of the $\gamma$-$\gamma$ coincidence matrix, and the intensity
balance was used to extract $\gamma$-ray intensities and branching ratios from
the neutron-capture state. Systematic errors associated to the partial angular
coverage of the detectors (i.e., $\approx$ 90%) have been taken into account,
by including a conservative 1% systematic error on the measured $\gamma$-ray
intensities. Uncertainties originating from efficiency correction (see Fig. 1)
were also considered. Possible uncertainties coming from self-absorption of
the targets and summing effects were not included, being negligible when
compared to other sources of error. In particular, the former can be excluded
due to the small samples used in the current experiment (see Sec. II), and the
latter was estimated to be lower than 10-4.

###  III.1 41Ca

Figure 3: (Color online) Level scheme of 41Ca, as measured in the current
experiment. Newly-observed $\gamma$-ray transitions and levels are reported in
red. Levels with tentative spin assignment are marked by dashed lines. Dashed
arrows indicate $\gamma$ rays with no firm placement in the level scheme
(i.e., the 4799- and 1622-keV cascade) or very weakly observed (see text for
details). Figure 3: -continued

The level scheme of the 41Ca nucleus was built by using both $\gamma$-$\gamma$
and $\gamma$-$\gamma$-$\gamma$ ray coincidence relationships, setting gates on
the most intense $\gamma$ rays. This was possible thanks to the high
statistics collected and the rather high level density between the neutron-
capture level and the ground state, which results in the emission of $\gamma$
rays with multiplicity greater than two. Fig. 2 presents examples of
$\gamma$-ray spectra measured in the current experiment, for the
40Ca(n,$\gamma$) reaction. The $\gamma$ rays observed here, for the first
time, are marked in red, while all other correspond to transitions reported in
the literature Gruppelaar and Spilling (1967); Cranston  _et al._ (1970);
Arnell  _et al._ (1969); Nesaraja and McCutchan (2016); PGA . Peaks coming
from transitions associated to (n,$\gamma$) reactions on contaminants present
in the target are labelled by circles. Panel a) presents the projection of the
$\gamma$-$\gamma$ coincidence matrix, obtained by setting a gate on the known
4418-keV, primary $\gamma$-ray transition, populating the 1/2- state at 3944
keV. Two new $\gamma$ rays with energies 544 keV and 1274 keV, depopulating
the 3944-keV, 1/2- state can be seen. Panels b), c), and d) show the
projections of the $\gamma$-$\gamma$-$\gamma$ coincidence matrix, gated on the
2010-1390-, 1943-727-, and 1943-520-keV combinations of $\gamma$ rays. In
these cases, all the spectra are almost background-free, enabling the
identification of very weak $\gamma$ rays. In particular, it is worth noting
the 1672-keV b) and the 2402-keV c) lines ($\approx$ 5 counts only),
depopulating the 1/2- state at 5072 keV.

Figure 4: (Color online) Panels a)-d): $\gamma$-ray angular correlation
between the 3/2- $\rightarrow$ 7/2-, 1943-keV transition and the 520-, 720-,
1671-, and 1151-keV $\gamma$ rays, depopulating the 3/2-, 1/2+, 1/2- states at
2462, 2670, and 3613 keV, respectively. Panels e)-h): angular correlations
between the 3/2- $\rightarrow$ 3/2-, 520-keV transition and the 1151-, 1482-,
2290-, 5900-keV lines depopulating the 1/2-, 1/2-, 1/2-, and 1/2+ states at
3613, 3944, 4753, and 8362 keV, respectively. Experimental fits are shown as
solid red lines, while theoretical predictions are displayed as dashed blue
lines (see Sec. III.1 and Tab. LABEL:tab:41Ca for details).

The level and $\gamma$-ray decay scheme of the 41Ca nucleus is presented in
Fig 3, where 41 new transitions and 2 new levels obtained in this work are
displayed in red (tentative levels and $\gamma$ rays are marked as dashed
lines). Of particular note are the two new levels at 3564- and 6374-keV of
excitation energy. In the decay scheme, the order of $\gamma$ rays was
assigned on the basis of previously-known levels, as well as on the
observation of parallel cascades. In the case of the 4799-1622-keV decay
chain, involving the newly-found 3564-keV level, the high energy transition
was tentatively assumed to depopulate directly the neutron-capture state.
However, since it is not possible to firmly constrain their right order, the
two transitions are displayed as dashed lines.
The scarce statistics collected for the $\gamma$ decays involving the new
levels did not allow for the study of $\gamma$-ray angular correlations,
therefore their (1/2±,3/2±) spin and parity is tentatively assigned on the
basis of the most probable $\gamma$-ray multipolarities. On the contrary,
angular correlations could be performed for a number of levels of known spin
and parity, as presented in Fig. 4. The top panels a)-d) show angular
correlations between the pure E2 1943-keV ground-state decay (3/2-
$\rightarrow$ 7/2-), and the 520-, 727-, 1671-, and 1151-keV transitions,
depopulating the 3/2-, 1/2+, 1/2- states at 2462 keV, 2670 keV, and 3613 keV,
respectively. The solid red curve corresponds to the experimental fit, which
enabled to determine the $\delta$ mixing ratios between the two most probable
multipolarities, by using a $\chi^{2}$ minimization procedure. Theoretical
predictions are also shown as dashed blue lines. In particular, a M1+E2
character, with $\delta$ = 0.13 (19), was found for the 520-keV, 3/2-
$\rightarrow$ 3/2- transition. This value is in agreement, within the error,
with the one reported in the literature ($\delta$ = 0.03 (12) Nesaraja and
McCutchan (2016)) and was used to extract angular correlations for $\gamma$
rays in coincidence with the 520-keV line, as displayed in the bottom panels
e)-h) of Fig. 4. These are the 1151-, 1482-, 2290-keV transitions depopulating
the 1/2-, 1/2-, and 3/2+ states at 3613, 3944 and 4753 keV, respectively, and
the 5900-keV, 1/2+ $\rightarrow$ 3/2- primary transition, for which a E1(+M2)
character was found ($\delta$ = 0.00(1)), confirming the expected dipole
nature of this high-energy, primary $\gamma$ ray.
The energies of levels and $\gamma$ rays, along with the $\gamma$-ray
multipolarities, mixing ratios, branching ratios, and $\gamma$-ray intensities
are presented in Tab. LABEL:tab:41Ca. New results obtained in this work are
marked by stars.

Table 3: Initial and final states, $\gamma$-ray energies, multipolarities, mixing ratios, branching ratios, and $\gamma$-ray intensities normalized to the 1942.5-keV transition (100 units) of 41Ca, as observed in this work. New findings are marked by a star. Multipolarities and mixing ratios not measured in this work are also reported Gruppelaar and Spilling (1967); Cranston  _et al._ (1970); Arnell  _et al._ (1969); Nesaraja and McCutchan (2016). Ei [keV] | J${}^{\pi}_{i}$ | Ef [keV] | J${}^{\pi}_{f}$ | Eγ [keV] | Multipolarity | $\delta$ | BRγ | Iγ
---|---|---|---|---|---|---|---|---
1942.5(1) | 3/2- | 0 | 7/2- | 1942.5(1) | E2 |  | 1.0 | 100
2009.8(1) | 3/2+ | 0 | 7/2- | 2009.8(1) | M2+E3 | 0.16(2) | 1.0 | 9.527(235)
2462.3(1) | 3/2- | 2009.8(1) | 3/2+ | 452.6(2)* |  |  | 0.013(2) | 0.148(14)
|  | 1942.5(1) | 3/2- | 519.7(1) | M1+E2 | 0.13(19)* | 0.987(2) | 11.027(576)
2575.2(1) | 5/2- | 0 | 7/2- | 2575.2(1) | M1+E2 |  | 1.0 | 0.256(20)
2605.3(1) | 5/2+ | 0 | 7/2- | 2605.3(1) | E1+M2 | -0.03(1) | 1.0 | 0.756(29)
2669.9(1) | 1/2+ | 2009.8(1) | 3/2+ | 660.2(1) | M1 |  | 0.268(15) | 0.791(44)
|  | 1942.5(1) | 3/2- | 727.4(1) | E1(+M2) | -0.02(6)* | 0.732(15) | 2.164(113)
3049.2(1) | 3/2+ | 2669.9(1) | 1/2+ | 379.3(2) |  |  | 0.043(13) | 0.043(14)
|  | 2605.3(1) | 5/2+ | 444.0(1) |  |  | 0.350(23) | 0.355(21)
|  | 2009.8(1) | 3/2+ | 1039.4(1) | M1 |  | 0.377(26) | 0.382(27)
|  | 1942.5(1) | 3/2- | 1106.6(1) | E1 |  | 0.231(19) | 0.234(19)
3399.6(1) | 1/2+ | 2009.8(1) | 3/2+ | 1389.9(1) | M1+E2 |  | 1.0 | 2.168(125)
3525.3(1) | 3/2+ | 2669.9(1) | 1/2+ | 855.5(2) | M1+E2 | 0.22(7) | 0.090(13) | 0.070(10)
|  | 2605.3(1) | 5/2+ | 920.0(1)* |  |  | 0.101(10) | 0.079(8)
|  | 2009.8(1) | 3/2+ | 1515.5(1) |  |  | 0.357(27) | 0.281(27)
|  | 1942.5(1) | 3/2- | 1582.8(1) | (E1+M2) |  | 0.453(27) | 0.356(29)
3564.1(2)* | (1/2±,3/2±) | 1942.5(1) | 3/2- | 1621.6(2)* |  |  | 1.0 | 0.105(17)
3613.2(1) | 1/2- | 2669.9(1) | 1/2+ | 943.4(1) | (E1) |  | 0.118(9) | 0.383(28)
|  | 2462.3(1) | 3/2- | 1151.0(1) | M1+E2* | -0.49${}^{+11}_{-13}$* | 0.349(21) | 1.138(78)
|  | 1942.5(1) | 3/2- | 1670.6(1) | M1+E2* | 0.10${}^{+12}_{-10}$* | 0.533(22) | 1.736(122)
3730.5(1) | 3/2- | 2575.2(1) | 5/2- | 1155.3(2) |  |  | 0.249(29) | 0.102(13)
|  | 2462.3(1) | 3/2- | 1268.4(1) | M1+E2 |  | 0.428(35) | 0.176(15)
|  | 1942.5(1) | 3/2- | 1787.7(1) | M1+E2 |  | 0.181(30) | 0.075(14)
|  | 0 | 7/2- | 3729.8(5) | (E2) |  | 0.142(39) | 0.058(18)
3846.0(1) | 1/2+ | 2009.8(1) | 3/2+ | 1836.3(1) |  |  | 0.779(18) | 0.949(60)
|  | 1942.5(1) | 3/2- | 1903.3(1) |  |  | 0.221(18) | 0.269(23)
3943.8(1) | 1/2- | 3399.6(1) | 1/2+ | 544.1(1)* |  |  | 0.003(1) | 0.057(10)
|  | 2669.9(1) | 1/2+ | 1274.0(1)* |  |  | 0.025(2) | 0.495(34)
|  | 2462.3(1) | 3/2- | 1481.6(1) | M1+E2* | 0.09(2)* | 0.067(6) | 1.322(91)
|  | 1942.5(1) | 3/2- | 2001.2(1) |  |  | 0.905(8) | 17.9(13)
4416.7(1) | 3/2+ | 2605.3(1) | 5/2+ | 1811.7(2) |  |  | 0.217(19) | 0.127(12)
|  | 2009.8(1) | 3/2+ | 2407.3(2) |  |  | 0.563(26) | 0.330(28)
|  | 1942.5(1) | 3/2- | 2473.6(1)* |  |  | 0.220(19) | 0.129(11)
4603.2(1) | 3/2- | 3049.2(1) | 3/2+ | 1555.1(6)* |  |  | 0.013(3) | 0.037(8)
|  | 2669.9(1) | 1/2+ | 1933.3(1)* |  |  | 0.112(11) | 0.325(30)
|  | 2462.3(1) | 3/2- | 2140.9(2)* |  |  | 0.022(4) | 0.064(10)
|  | 2009.8(1) | 3/2+ | 2593.4(1)* |  |  | 0.209(32) | 0.606(114)
|  | 1942.5(1) | 3/2- | 2660.8(2) |  |  | 0.644(31) | 1.865(127)
4728.0(1) | 3/2+ | 2009.8(1) | 3/2+ | 2718.2(2) |  |  | 1.0 | 0.198(21)
4752.7(1) | 1/2- | 3399.6(1) | 1/2+ | 1353.1(1)* |  |  | 0.035(3) | 0.269(19)
|  | 3049.2(1) | 3/2+ | 1703.4(2)* |  |  | 0.021(2) | 0.157(16)
|  | 2462.3(1) | 3/2- | 2290.4(1) | M1+E2* | -0.04(1)* | 0.270(17) | 2.047(141)
|  | 2009.8(1) | 3/2+ | 2743.0(1)* |  |  | 0.040(3) | 0.307(23)
|  | 1942.5(1) | 3/2- | 2810.1(1) |  |  | 0.634(19) | 4.812(306)
4777.8(1) | 3/2+ | 3525.3(1) | 3/2+ | 1252.5(3)* |  |  | 0.049(6) | 0.109(13)
|  | 3399.6(1) | 1/2+ | 1378.2(2)* |  |  | 0.035(3) | 0.078(7)
|  | 3049.2(1) | 3/2+ | 1728.6(1)* |  |  | 0.101(9) | 0.226(20)
|  | 2605.3(1) | 5/2+ | 2172.6(1)* |  |  | 0.104(10) | 0.232(87)
|  | 2009.8(1) | 3/2+ | 2767.9(1) |  |  | 0.646(20) | 1.448(104)
|  | 1942.5(1) | 3/2- | 2835.3(2)* |  |  | 0.067(7) | 0.150(15)
5011.8(1) | 1/2+ | 3399.6(1) | 1/2+ | 1612.2(2) |  |  | 0.284(22) | 0.115(8)
|  | 2009.8(1) | 3/2+ | 3002.0(1) |  |  | 0.716(22) | 0.290(23)
5071.8(1) | 1/2- | 3399.6(1) | 1/2+ | 1672.2(2)* |  |  | 0.399(38) | 0.096(11)
|  | 2669.9(1) | 1/2+ | 2401.9(1)* |  |  | 0.601(38) | 0.144(16)
5449.8(3) | 1/2- | 2462.3(1) | 3/2- | 2987.5(3)* |  |  | 1.0 | 0.176(16)
5467.2(2) | 3/2- | 2575.2(1) | 5/2- | 2892.2(2)* |  |  | 0.377(42) | 0.127(19)
|  | 2462.3(1) | 3/2- | 3004.9(4)* |  |  | 0.117(22) | 0.040(8)
|  | 0 | 7/2- | 5467.2(2) |  |  | 0.506(41) | 0.171(18)
5669.4(2) | 3/2- | 2462.3(1) | 3/2- | 3207.4(2)* |  |  | 0.322(43) | 0.052(8)
|  | 0 | 7/2- | 5669.2(2) |  |  | 0.678(43) | 0.110(13)
5703.2(2) | 1/2- | 2669.9(1) | 1/2+ | 3033.3(2)* |  |  | 1.0 | 0.041(4)
5888.8(1) | 1/2- | 3730.5(1) | 3/2- | 2158.3(1)* |  |  | 1.0 | 0.057(6)
6373.8(2)* | (1/2±,3/2±) | 3730.5(1) | 3/2- | 2643.1(5)* |  |  | 0.380(40) | 0.070(10)
|  | 2462.3(1) | 3/2- | 3910.7(10)* |  |  | 0.176(33) | 0.033(7)
|  | 1942.5(1) | 3/2- | 4431.2(2)* |  |  | 0.444(40) | 0.082(9)
7190.4(2) | 1/2- | 2462.3(1) | 3/2- | 4728.2(2)* |  |  | 1.0 | 0.027(7)
8362.4(2) | 1/2+ | 7190.4(2) | 1/2- | 1171.7(2)* |  |  | 0.0003(1) | 0.029(4)
|  | 6373.8(2)* | (1/2±,3/2±) | 1989.5(4)* |  |  | 0.0018(1) | 0.198(10)
|  | 5888.8(1) | 1/2- | 2473.6(1)* |  |  | 0.0005(1) | 0.050(7)
|  | 5703.2(2) | 1/2- | 2659.0(3)* |  |  | 0.0004(1) | 0.039(4)
|  | 5669.4(2) | 3/2- | 2693.4(2)* |  |  | 0.0019(2) | 0.206(12)
|  | 5467.2(2) | 3/2- | 2894.9(2)* |  |  | 0.0035(2) | 0.382(19)
|  | 5449.8(3) | 1/2- | 2912.7(1)* |  |  | 0.0019(2) | 0.201(21)
|  | 5071.8(1) | 1/2- | 3290.9(2)* |  |  | 0.0021(2) | 0.221(21)
|  | 5011.8(1) | 1/2+ | 3350.8(1) |  |  | 0.0043(3) | 0.458(25)
|  | 4777.8(1) | 3/2+ | 3584.9(1) |  |  | 0.0250(16) | 2.695(123)
|  | 4752.7(1) | 1/2- | 3609.9(1) |  |  | 0.0761(52) | 8.204(441)
|  | 4728.0(1) | 3/2+ | 3634.9(1)* |  |  | 0.0024(2) | 0.261(18)
|  | 4603.2(1) | 3/2- | 3759.6(1) |  |  | 0.0308(21) | 3.317(166)
|  | 4416.7(1) | 3/2+ | 3945.8(1) |  |  | 0.0059(4) | 0.635(30)
|  | 3943.8(1) | 1/2- | 4418.4(1) |  |  | 0.1911(131) | 20.6(13)
|  | 3846.0(1) | 1/2+ | 4516.4(1) |  |  | 0.0138(10) | 1.484(82)
|  | 3730.5(1) | 3/2- | 4631.7(5)* |  |  | 0.0020(2) | 0.212(13)
|  | 3613.2(1) | 1/2- | 4749.2(1) |  |  | 0.0343(24) | 3.703(194)
|  | 3564.1(2) | (1/2±,3/2±) | 4798.5(2)* |  |  | 0.0007(1) | 0.075(7)
|  | 3525.3(1) | 3/2+ | 4836.9(3) |  |  | 0.0078(6) | 0.840(45)
|  | 3399.6(1) | 1/2+ | 4962.9(1) |  |  | 0.0176(15) | 1.895(139)
|  | 3049.2(1) | 3/2+ | 5313.3(1) |  |  | 0.0051(4) | 0.547(29)
|  | 2669.9(1) | 1/2+ | 5692.5(1) |  |  | 0.0174(15) | 1.872(135)
|  | 2462.3(1) | 3/2- | 5900.0(2) | E1(+M2)* | 0.00(1)* | 0.0703(65) | 7.577(651)
|  | 2009.8(1) | 3/2+ | 6352.6(1) |  |  | 0.0074(8) | 0.800(78)
|  | 1942.5(1) | 3/2- | 6419.7(1) |  |  | 0.4759(247) | 51.3(49)

###  III.2 47Ca

Projections of the $\gamma$-$\gamma$ coincidence matrix for the 47Ca nucleus
are presented in Fig. 5, where new $\gamma$ rays are shown in red, while
transitions reported in the literature are marked in black (Cranston  _et al._
, 1970; Arnell  _et al._ , 1969; Burrows, 2007; PGA, ). The spectrum in panel
a) is obtained by gating on the 565-keV transition. Of particular interest are
the 4697-, 4676-, and 2825-keV $\gamma$ rays, which depopulate directly the
neutron-capture level, feeding previously-known low-lying states at 2578, 2599
and 4450 keV, respectively. Panel b) shows a spectrum obtained by gating on
the 862-keV line in which new $\gamma$ rays at 1182 keV and 1933 keV,
populating the state at 2876 keV, can be seen. Finally, panel c) presents the
spectrum measured in coincidence with the 3218-keV transition. In the picture,
besides the 1182 keV transition discussed above, new $\gamma$ rays with
energies 1458 keV and 1479 keV are visible. These feed the 1/2+ and 3/2+
states at 2599 keV and 2578 keV, respectively, from the state at 4057 keV.
The level and $\gamma$-decay scheme of the 47Ca nucleus, obtained in this
work, is presented in Fig.6 (left), with new $\gamma$ rays shown in red. In
this case, $\gamma$-ray angular correlations allowed to further characterize a
number of transitions in terms of multipolarity and mixing ratios, enabling to
firmly assign the spin and parity of the states involved in the decays, as
presented in Fig.6 (right). Experimental fits are shown as solid red lines,
while theoretical predictions are displayed as dashed blue lines. Angular
correlations are performed against the 3/2- $\rightarrow$ 7/2- decay to the
ground state, since the 2013-keV $\gamma$ ray has a pure E2 character. Panel
a) shows the angular correlation for the 2044-keV line, depopulating the state
at 4057 keV. The results suggest a M1+E2 character for this transition, with a
mixing ratio $\delta$=0.66${}^{+0.20}_{-0.11}$. The shape of the angular
correlation is compatible with a 3/2- spin-parity assignment for the 4057-keV
state. Panel b) presents similar results for the 2795-keV $\gamma$-ray. The
experimental fit indicates a M1+E2 character, with a mixing ratio
$\delta$=0.58${}^{+0.43}_{-0.14}$. In this case, the angular correlation is
well reproduced assuming a 1/2- spin-parity assignment for the 4808-keV state.
Finally, panel c) shows the angular correlation for the 4400-keV, primary
transition. A dominant E1 character is found, with a M2 mixing with
$\delta$=-0.23(6). Moreover, the shape of the angular correlation suggests a
3/2- spin-parity assignment for the 2876-keV state. The energies of levels and
$\gamma$ rays, along with the $\gamma$-ray multipolarities, mixing ratios,
branching ratios, and $\gamma$-ray intensities are presented in Tab. 4, where
new results are marked by stars.

Figure 5: (Color online) Projections of the $\gamma$-$\gamma$ coincidence
matrix measured in the 46Ca(n,$\gamma$) reaction. Gates on the 565-keV a),
862-keV b), and 3218-keV c) $\gamma$ rays are presented, with new transitions,
observed for the first time, displayed in red. First and second escape peaks
for high-energy transitions are marked by 1${}^{\text{st}}$ and
2${}^{\text{nd}}$, respectively. Figure 6: (Color online) (Left) Level scheme
of 47Ca as measured in the current experiment. Newly-observed $\gamma$-ray
transitions are reported in red. (Right) Angular correlations in 47Ca which
enabled to pin down the multipolarity of the 2044-keV a), 2795-keV b), and
4400-keV c) transitions (see Sec. III.2 for discussion). Experimental data are
presented as dots along with experimental fits (solid red line), while
theoretical predictions are shown as dashed blue line.

###  III.3 49Ca

The level scheme of the 49Ca nucleus obtained in this work is shown on the
left side of Fig. 7. New $\gamma$ rays are marked in red, with tentative
transitions displayed as dashed lines. The known $\gamma$ rays reported in the
literature are shown in black (Arnell  _et al._ , 1969; Burrows, 2008; PGA, ).
An example of $\gamma$-ray spectrum, gated on the 1/2- $\rightarrow$ 3/2-,
2023-keV transition, is presented on the right side of Fig. 7. The most
intense $\gamma$-ray, namely the 3123-keV, corresponds to the direct
populations of the 1/2- state at 2023 keV, from the neutron capture level. Two
new $\gamma$ lines with energies 1074 keV and 2049 keV are also present. The
first corresponds to the primary transition populating the 3/2- state at 4072
keV, while the latter is the 3/2- $\rightarrow$ 1/2- decay, depopulating the
4072-keV state. Finally, the 2249-keV line depopulating the 4272-keV level and
reported in Arnell  _et al._ (1969); Burrows (2008) was not observed in the
present work, suggesting a branch from this level to the 2023-keV state
$\leq$0.001 %. It is important to note that the 48Ca sample used in the
current measurement was contaminated by other nuclei with a non-negligible
cross section for neutron capture. This is the case, for example, of the 113Cd
nucleus, the neutron-capture cross section of which is about 20 kb. Therefore,
even if present in small quantity, the $\gamma$-ray decay of the 114Cd isotope
is rather strong. In Fig. 7, $\gamma$ rays corresponding to (n,$\gamma$)
reactions on target contaminants are marked by circles.
The energies of levels and $\gamma$ rays, along with branching ratios and
$\gamma$-ray intensities are presented in Tab. 5. New results obtained in this
work are marked by stars, while $\gamma$-ray multipolarities are taken from
(Arnell  _et al._ , 1969; Burrows, 2008). It is important to note that in the
case of the 5146-keV transition, the procedure described in Sec. III to
extract branching ratios cannot be applied, since the $\gamma$ ray feeds
directly the ground state. According to the literature (Arnell  _et al._ ,
1969; Burrows, 2008), the 5146-keV transition is 3 times larger than the 3123-

Table 4: Initial and final states, $\gamma$-ray energies, multipolarities, mixing ratios, branching ratios, and $\gamma$-ray intensities normalized to the 2013.2-keV transition (100 units) of 47Ca, as observed in this work. New findings are marked by a star. Multipolarities and mixing ratios not measured in this work are also reported Cranston  _et al._ (1970); Arnell  _et al._ (1969); Burrows (2007). Ei [keV] | J${}^{\pi}_{i}$ | Ef [keV] | J${}^{\pi}_{f}$ | Eγ [keV] | Multipolarity | $\delta$ | BRγ | Iγ
---|---|---|---|---|---|---|---|---
2013.2(1) | 3/2- | 0 | 7/2- | 2013.2(1) | E2 |  | 1.0 | 100
2578.3(1) | 3/2+ | 2013.2(1) | 3/2- | 565.1(1) | (E1) |  | 1.0 | 0.327(14)
2599.0(1) | 1/2+ | 2013.2(1) | 3/2- | 585.8(1) | (E1) |  | 1.0 | 0.956(33)
2875.6(3) | 3/2-* | 2013.2(1) | 3/2- | 862.4(1) |  |  | 1.0 | 10.922(730)
4057.3(2) | 3/2-* | 2875.6(3) | 3/2- | 1182.4(1)* |  |  | 0.216(15) | 1.952(116)
|  | 2599.0(1) | 1/2+ | 1458.0(2)* |  |  | 0.060(5) | 0.538(39)
|  | 2578.3(1) | 3/2+ | 1478.6(1)* |  |  | 0.011(2) | 0.103(15)
|  | 2013.2(1) | 3/2- | 2043.9(1) | M1+E2* | 0.66${}^{+0.20}_{-0.11}$* | 0.713(18) | 6.444(469)
4450.2(2) | (1/2-,3/2-) | 2578.3(1) | 3/2+ | 1871.9(2)* |  |  | 0.583(54) | 0.202(14)
|  | 2013.2(1) | 3/2- | 2437.3(2)* |  |  | 0.417(54) | 0.144(31)
4808.2(3) | 1/2-* | 2875.6(3) | 3/2- | 1932.6(2)* |  |  | 0.052(18) | 0.096(35)
|  | 2599.0(1) | 1/2+ | 2209.1(2)* |  |  | 0.087(9) | 0.160(14)
|  | 2013.2(1) | 3/2- | 2794.9(1) | M1+E2* | 0.58${}^{+0.43}_{-0.14}$* | 0.861(20) | 1.591(106)
7275.4(2) | 1/2+ | 4808.2(3) | 1/2- | 2467.1(1) |  |  | 0.0164(16) | 1.640(111)
|  | 4450.2(2) | (1/2-,3/2-) | 2825.1(2)* |  |  | 0.0040(4) | 0.398(29)
|  | 4057.3(2) | 3/2- | 3218.3(1) |  |  | 0.0754(70) | 7.550(502)
|  | 2875.6(3) | 3/2- | 4400.1(1) | E1+M2* | -0.23(6)* | 0.0989(95) | 9.903(716)
|  | 2599.0(1) | 1/2+ | 4675.7(1)* |  |  | 0.0021(5) | 0.213(45)
|  | 2578.3(1) | 3/2+ | 4697.1(1)* |  |  | 0.0009(6) | 0.086(57)
|  | 2013.2(1) | 3/2- | 5261.8(1) |  |  | 0.8024(155) | 80.4(70)
Figure 7: (Left) Level scheme of 49Ca, as obtained in this work, with new transitions displayed in red. (Right) Projection of the $\gamma$-$\gamma$ coincidence matrix gated on the 2023-keV transition in 49Ca, showing two new $\gamma$ rays at 1074 keV and 2049 keV. $\gamma$ rays coming from (n,$\gamma$) reactions on target contaminants are marked by circles (see Sec. III.3 for details). The first escape peak for the 3123-keV transition is marked by 1${}^{\text{st}}$. Table 5: Initial and final states, $\gamma$-ray energies, branching ratios, and $\gamma$-ray intensities normalized to the 5145.9-keV transition (100 units) of 49Ca, as observed in this work. New findings are marked by a star. Multipolarities are taken from Arnell  _et al._ (1969); Burrows (2008). Ei [keV] | J${}^{\pi}_{i}$ | Ef [keV] | J${}^{\pi}_{f}$ | Eγ [keV] | Multipolarity |  | $\delta$ | BRγ | Iγ
---|---|---|---|---|---|---|---|---|---
2023.2(1) | 1/2- | 0 | 3/2- | 2023.2(1) | (M1,E2) |  |  | 1.0 | 33.8(22)
3861.2(1) | (1/2-,3/2-) | 0 | 3/2- | 3861.2(1) |  |  |  | 1.0 | 0.582(110)
4072.4(2) | 3/2- | 2023.2(1) | 1/2- | 2049.1(2)* |  |  |  | 0.047(18) | 0.020(15)
|  | 0 | 3/2- | 4072.0(1) | (M1,E2) |  |  | 0.953(18) | 0.406(184)
4260.9(3) | 1/2- | 0 | 3/2- | 4260.9(3)* |  |  |  | 1.0 | 0.181(104)
4272.0(1) | 1/2- | 0 | 3/2- | 4272.0(1) | (M1,E2) |  |  | 1.0 | 0.090(121)
5146.3(3) | 1/2+ | 4272.4(2) | 1/2- | 874.2(3)* |  |  |  | 0.0006(10) | 0.086(129)
|  | 4260.9(3) | 1/2- | 885.2(3)* |  |  |  | 0.0013(9) | 0.173(118)
|  | 4072.4(2) | 3/2- | 1073.8(1)* |  |  |  | 0.0022(9) | 0.300(125)
|  | 3861.2(1) | (1/2-,3/2-) | 1286.1(2)* |  |  |  | 0.0042(9) | 0.566(118)
|  | 2023.2(1) | 1/2- | 3123.4(1) |  |  |  | 0.2472(121) | 33.2(22)
|  | 0 | 3/2- | 5145.9(1) |  |  |  | 0.7444(120) | 100

-keV transition, therefore such a value was adopted to properly normalize the branching ratios for the $\gamma$ rays depopulating the capture state.

##  IV COMPARISON WITH THEORY

The experimental excitation energy spectra of 41Ca, 47Ca and 49Ca have been
partially compared with theoretical calculations performed in the framework of
the Hybrid Configuration Mixing Model (HCM) (Colò  _et al._ , 2017; Bottoni
_et al._ , ). The model was designed to microscopically describe one-valence-
particle/hole nuclei with respect to an even-even, doubly-magic “core” with
mass A and it is based on a Hamiltonian of Skyrme type, which for the
particle-core coupling case reads

$\displaystyle H=H_{0}+V,$ $\displaystyle
H_{0}=\sum_{jm}\epsilon_{j}a_{jm}^{{\dagger}}a_{jm}+\sum_{NJM}\hbar\omega_{NJ}\Gamma^{{\dagger}}_{JM}\Gamma_{JM},$
(2) $\displaystyle V=\sum_{\begin{subarray}{c}jm\\\
j^{\prime}m^{\prime}\end{subarray}}\sum_{NJM}h(jm;j^{\prime}m^{\prime},NJM)a_{jm}[a^{{\dagger}}_{j^{\prime}m^{\prime}}\otimes\Gamma^{{\dagger}}_{JM}]_{jm}.$

$H_{0}$ is the mean-field solution corresponding to Hartree-Fock (HF) particle
states and Random Phase Approximation (RPA) excitations of the core calculated
according to Ref. Coló  _et al._ (2013), with $a^{{\dagger}}$ and
$\Gamma^{{\dagger}}$ being the usual fermion-creator and boson-creator
operators, respectively. V is the coupling between single-particle states and
core excitations (see Ref. Colò  _et al._ (2010) for details). A similar
expression holds for the hole-core coupling case Bottoni  _et al._ (2018).
Consequently, the model accounts for both single-particle/hole states and
couplings with core excitations, predicting either particle/hole-phonon
coupled states or 2p-1h and 2h-1p shell-model-like excitations, or hybrid
mixtures, for the A+1 and A-1 systems, respectively. It is important to note
that the orthonormality and the completeness of basis states are properly
taken into account by eliminating, from the model space, those spurious
configurations which violate the Pauli principle. The wave functions
$\ket{\Psi_{n}}$ for each state are then written in terms of the original
basis $\ket{\alpha}$ as

$\ket{\Psi_{n}}=\sum_{\alpha}\xi_{n}(\alpha)\ket{\alpha},$ (3)

where $\xi_{n}$ are the amplitudes of each component. In the case of pure
single-particle/hole states, $\xi_{n}^{2}$ can be interpreted as the
spectroscopic factor.

Table 6: Experimental energy and B(E3; $3^{-}\rightarrow 0^{+}$) value of the ${3^{-}}$ phonon in the 40Ca nucleus Kibedi and R.H.Spear (2002), compared with RPA calculations performed with the SkX (Alex Brown, 1998) and SLy5 Chabanat  _et al._ (1998) Skyrme interactions. Calculations are done according to Coló  _et al._ (2013).  |  |  | E${}_{3^{-}}$ [MeV] |  |  | B(E3; $3^{-}\rightarrow 0^{+}$) [W.u.]
---|---|---|---|---|---|---
EXP |  |  | 3.74 |  |  | 27.7(30)
SkX |  |  | 2.95 |  |  | 17.0
SLy5 |  |  | 3.67 |  |  | 21.5

Particular attention was given to the choice of the Skyrme interaction. In
this work, calculations performed with the SkX parametrization Alex Brown
(1998) are presented for all nuclei. This interaction was fitted on binding
energies, charge radii but also single-particle energies of many doubly-magic
isotopes, resulting in an effective mass m*/m $\approx$ 1\. Yet, different
parametrizations were also tested. Calculations done with the SLy5 interaction
(m*/m $\approx$ 0.7) better reproduce, for instance, the properties of the 3-
phonon in the 40Ca nucleus (see Tab 6). As explained in Ref. Alex Brown
(1998), single-particle energies predicted by the SkX interaction specifically
for this nucleus quantitatively differ from the experimental values. Such an
effect can be ascribed to proton-neutron correlations, which are particularly
enhanced in N=Z systems. In 40Ca, 4p-4h and 8p-8h excitations start playing a
crucial role even at low energies (Ideguchi  _et al._ , 2001), thus affecting
the shell structure. This aspect is less pronounced in neutron-rich systems,
where excitations of the neutron excess dominate over those of the symmetric
core.

Table 7: RPA results for 40Ca core excitations used in the HCM calculations (see text for details), showing spins, energies and main composition of the wave function, along with the squared X RPA forward amplitudes. Only components with X2 $\geq$ 0.1 are displayed. B(E$\lambda$; J${}^{\pi}_{n}\rightarrow$ 0${}^{+}_{\text{g.s.}}$) values for the $1^{-}$ and $3^{-}$ states are also reported. |  | J${}^{\pi}_{n}$ |  | E [keV] |  | main w. f. |  | B(E$\lambda$; J${}^{\pi}_{n}\rightarrow$ 0${}^{+}_{\text{g.s.}}$)
---|---|---|---|---|---|---|---|---
|  |  |  |  |  | composition |  | [W.u.]
40Ca |  |  |  |  |  |  |  |
|  | 1${}^{-}_{1}$ |  | 7290 |  |  $\pi d_{5/2}^{-1}f_{7/2}$(0.11) |  | 2.06$\cdot$10-3
|  |  |  |  |  |  $\pi d_{3/2}^{-1}p_{3/2}$(0.33) |  |
|  |  |  |  |  |  $\nu d_{5/2}^{-1}f_{7/2}$(0.10) |  |
|  |  |  |  |  |  $\nu d_{3/2}^{-1}p_{3/2}$(0.24) |  |
|  | 2${}^{-}_{1}$ |  | 4193 |  |  $\pi d_{3/2}^{-1}f_{7/2}$(0.52) |  |
|  |  |  |  |  |  $\nu d_{3/2}^{-1}f_{7/2}$(0.45) |  |
|  | 2${}^{-}_{2}$ |  | 6549 |  |  $\nu d_{3/2}^{-1}f_{7/2}$(0.51) |  |
|  |  |  |  |  |  $\pi d_{3/2}^{-1}f_{7/2}$(0.44) |  |
|  | 3${}^{-}_{1}$ |  | 2947 |  |  $\pi d_{3/2}^{-1}f_{7/2}$(0.30) |  | 16.95
|  |  |  |  |  |  $\nu d_{3/2}^{-1}f_{7/2}$(0.27) |  |
|  |  |  |  |  |  $\pi s_{1/2}^{-1}f_{7/2}$(0.23) |  |
|  |  |  |  |  |  $\nu s_{1/2}^{-1}f_{7/2}$(0.21) |  |
|  | 3${}^{-}_{2}$ |  | 5399 |  |  $\pi d_{3/2}^{-1}f_{7/2}$(0.62) |  | 0.52
|  |  |  |  |  |  $\pi s_{1/2}^{-1}f_{7/2}$(0.32) |  |
|  | 3${}^{-}_{3}$ |  | 5613 |  |  $\nu d_{3/2}^{-1}f_{7/2}$(0.60) |  | 1.07
|  |  |  |  |  |  $\nu s_{1/2}^{-1}f_{7/2}$(0.32) |  |
|  | 3${}^{-}_{4}$ |  | 6881 |  |  $\nu s_{1/2}^{-1}f_{7/2}$(0.44) |  | 0.28
|  |  |  |  |  |  $\pi s_{1/2}^{-1}f_{7/2}$(0.36) |  |
|  | 4${}^{-}_{1}$ |  | 7837 |  |  $\pi d_{3/2}^{-1}f_{7/2}$(0.89) |  |
|  | 4${}^{-}_{2}$ |  | 7837 |  |  $\nu d_{3/2}^{-1}f_{7/2}$(0.87) |  |
|  | 4${}^{-}_{3}$ |  | 7837 |  |  $\pi s_{1/2}^{-1}f_{7/2}$(0.65) |  |
|  |  |  |  |  |  $\nu s_{1/2}^{-1}f_{7/2}$(0.34) |  |
|  | 4${}^{-}_{4}$ |  | 7837 |  |  $\nu s_{1/2}^{-1}f_{7/2}$(0.63) |  |
|  |  |  |  |  |  $\pi s_{1/2}^{-1}f_{7/2}$(0.30) |  |
|  | 5${}^{-}_{1}$ |  | 4998 |  |  $\pi d_{3/2}^{-1}f_{7/2}$(0.59) |  |
|  |  |  |  |  |  $\nu d_{3/2}^{-1}f_{7/2}$(0.41) |  |
|  | 5${}^{-}_{2}$ |  | 6341 |  |  $\nu d_{3/2}^{-1}f_{7/2}$(0.40) |  |
|  |  |  |  |  |  $\pi d_{3/2}^{-1}f_{7/2}$ (0.40) |  |
Table 8: RPA results for 48Ca core excitations used in the HCM calculations (see text for details), showing spins, energies and main composition of the wave function, along with the squared X RPA forward amplitudes. Only components with X2 $\geq$ 0.1 are displayed. B(E$\lambda$; J${}^{\pi}_{n}\rightarrow$ 0${}^{+}_{\text{g.s.}}$) values for the $2^{+}$ and $3^{-}$ states are also reported. |  | J${}^{\pi}_{n}$ |  | E [keV] |  | main w. f. |  | B(E$\lambda$; J${}^{\pi}_{n}\rightarrow$ 0${}^{+}_{\text{g.s.}}$)
---|---|---|---|---|---|---|---|---
|  |  |  |  |  | composition |  | [W.u.]
48Ca |  |  |  |  |  |  |  |
|  | 2${}^{+}_{1}$ |  | 2868 |  |  $\nu f_{7/2}^{-1}p_{3/2}$(0.98) |  | 1.31
|  | 3${}^{+}_{1}$ |  | 3225 |  |  $\nu f_{7/2}^{-1}p_{3/2}$(0.99) |  |
|  | 3${}^{+}_{2}$ |  | 5016 |  |  $\nu f_{7/2}^{-1}p_{3/2}$(0.98) |  |
|  | 3${}^{-}_{1}$ |  | 4434 |  |  $\pi s_{1/2}^{-1}f_{7/2}$(0.76) |  | 6.77
|  |  |  |  |  |  $\pi d_{3/2}^{-1}f_{7/2}$(0.19) |  |
|  | 3${}^{-}_{2}$ |  | 5372 |  |  $\pi d_{3/2}^{-1}f_{7/2}$(0.79) |  |
|  |  |  |  |  |  $\pi s_{1/2}^{-1}f_{7/2}$(0.20) |  |
|  | 4${}^{+}_{1}$ |  | 3124 |  |  $\nu f_{7/2}^{-1}p_{3/2}$(1.00) |  |
|  | 4${}^{+}_{2}$ |  | 4703 |  |  $\nu f_{7/2}^{-1}p_{3/2}$(0.98) |  |
|  | 4${}^{-}_{1}$ |  | 5111 |  |  $\pi s_{1/2}^{-1}f_{7/2}$(0.92) |  |
|  | 5${}^{+}_{1}$ |  | 3509 |  |  $\nu f_{7/2}^{-1}p_{3/2}$(1.00) |  |

The 40Ca and 48Ca, RPA core excitations, used in this work for HCM
calculations, are reported in Tabs. 7 and 8. Along with spins and energies,
the main components of the wave functions and the B(E$\lambda$;
J${}^{\pi}_{n}\rightarrow$ 0${}^{+}_{\text{g.s.}}$) values for low-spin states
are presented for both collective phonons and non-collective excitations.

Table 9: Results of the Hybrid Configuration Mixing Model calculations for different states in 41Ca, 47Ca, and 49Ca. The main components $\ket{\alpha}$ of the wave functions are reported, along with the corresponding squared amplitudes $\xi_{n}^{2}(\alpha)$, considering only contributions with $\xi_{n}^{2}(\alpha)\geq$ 0.05 (see Sec. IV for details and discussion).  |  |  |  | Jπ |  |  |  | E [keV] |  |  |  | $\ket{\alpha}$ |  |  |  | $\xi_{n}^{2}(\alpha)$ |  |  |  |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
41Ca |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
|  |  |  | 7/2- |  |  |  | 0 |  |  |  | $\nu 1f_{7/2}$ |  |  |  | 0.99 |  |  |  |
|  |  |  | 11/2+ |  |  |  | 2368 |  |  |  | $\nu 1f_{7/2}\otimes 3^{-}_{1}$ |  |  |  | 0.40 |  |  |  |
|  |  |  | 5/2+ |  |  |  | 2429 |  |  |  | $\nu 1f_{7/2}\otimes 3^{-}_{1}$ |  |  |  | 0.35 |  |  |  |
|  |  |  |  |  |  |  |  |  |  |  | $\nu 1f_{7/2}\otimes 3^{-}_{3}$ |  |  |  | 0.08 |  |  |  |
|  |  |  | 9/2+ |  |  |  | 2444 |  |  |  | $\nu 1f_{7/2}\otimes 3^{-}_{1}$ |  |  |  | 0.32 |  |  |  |
|  |  |  | 1/2+ |  |  |  | 2468 |  |  |  | $\nu 1f_{7/2}\otimes 3^{-}_{1}$ |  |  |  | 0.33 |  |  |  |
|  |  |  |  |  |  |  |  |  |  |  | $\nu 1f_{7/2}\otimes 4^{-}_{2}$ |  |  |  | 0.10 |  |  |  |
|  |  |  | 7/2+ |  |  |  | 2604 |  |  |  | $\nu 1f_{7/2}\otimes 3^{-}_{1}$ |  |  |  | 0.31 |  |  |  |
|  |  |  | 3/2+ |  |  |  | 2608 |  |  |  | $\nu 1f_{7/2}\otimes 3^{-}_{1}$ |  |  |  | 0.28 |  |  |  |
|  |  |  |  |  |  |  |  |  |  |  | $\nu 1f_{7/2}\otimes 4^{-}_{2}$ |  |  |  | 0.13 |  |  |  |
|  |  |  | 13/2+ |  |  |  | 2785 |  |  |  | $\nu 1f_{7/2}\otimes 3^{-}_{1}$ |  |  |  | 0.10 |  |  |  |
|  |  |  |  |  |  |  |  |  |  |  | $\nu 1f_{7/2}\otimes 4^{-}_{2}$ |  |  |  | 0.22 |  |  |  |
47Ca |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
|  |  |  | 7/2- |  |  |  | 0 |  |  |  | $\nu 1f_{7/2}^{-1}$ |  |  |  | 0.99 |  |  |  |
|  |  |  | 3/2- |  |  |  | 1640 |  |  |  | $\nu 1f_{7/2}^{-1}\otimes(\nu 1f_{7/2}^{-1}1p_{3/2})_{2^{+}_{1}}$ |  |  |  | 0.12 |  |  |  |
|  |  |  |  |  |  |  |  |  |  |  | $\nu 1f_{7/2}^{-1}\otimes(\nu 1f_{7/2}^{-1}1p_{3/2})_{4^{+}_{1}}$ |  |  |  | 0.29 |  |  |  |
|  |  |  | 1/2+ |  |  |  | 3329 |  |  |  | $\nu 2s_{1/2}^{-1}$ |  |  |  | 0.27 |  |  |  |
|  |  |  |  |  |  |  |  |  |  |  | $\nu 1f_{7/2}^{-1}\otimes 3^{-}_{1}$ |  |  |  | 0.68 |  |  |  |
|  |  |  | 3/2+ |  |  |  | 3629 |  |  |  | $\nu 2d_{3/2}^{-1}$ |  |  |  | 0.43 |  |  |  |
|  |  |  |  |  |  |  |  |  |  |  | $\nu 1f_{7/2}^{-1}\otimes 3^{-}_{1}$ |  |  |  | 0.53 |  |  |  |
|  |  |  | 5/2+ |  |  |  | 4238 |  |  |  | $\nu 1f_{7/2}^{-1}\otimes 3^{-}_{1}$ |  |  |  | 0.93 |  |  |  |
|  |  |  | 7/2+,9/2+ |  |  |  | 4369 |  |  |  | $\nu 1f_{7/2}^{-1}\otimes 3^{-}_{1}$ |  |  |  | 0.96 |  |  |  |
|  |  |  | 11/2+ |  |  |  | 4416 |  |  |  | $\nu 1f_{7/2}^{-1}\otimes 3^{-}_{1}$ |  |  |  | 0.96 |  |  |  |
|  |  |  | 13/2+ |  |  |  | 4441 |  |  |  | $\nu 1f_{7/2}^{-1}\otimes 3^{-}_{1}$ |  |  |  | 0.96 |  |  |  |
49Ca |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
|  |  |  | 3/2- |  |  |  | 0 |  |  |  | $\nu 1p_{3/2}$ |  |  |  | 0.92 |  |  |  |
|  |  |  | 1/2- |  |  |  | 1835 |  |  |  | $\nu 1p_{1/2}$ |  |  |  | 0.99 |  |  |  |
|  |  |  | 9/2+ |  |  |  | 4296 |  |  |  | $\nu 1g_{9/2}$ |  |  |  | 0.11 |  |  |  |
|  |  |  |  |  |  |  |  |  |  |  | $\nu 1p_{3/2}\otimes 3^{-}_{1}$ |  |  |  | 0.83 |  |  |  |
|  |  |  | 3/2+,5/2+,7/2+ |  |  |  | 4652 |  |  |  | $\nu 1p_{3/2}\otimes 3^{-}_{1}$ |  |  |  | 0.94 |  |  |  |

Calculations for the 41Ca nucleus were performed by assuming a 40Ca core and
including neutron single-particle states of the $pfg_{9/2}$ shell and the sd
levels above the N=50 shell gap. In the cases of the 47Ca and 49Ca isotopes, a
48Ca core was taken with the full hole space for the former and by including
the $pf_{7/2}g_{9/2}$ orbitals for the latter. In the first case, 40Ca core
excitations up to 8 MeV and angular momentum L=8 were considered, while for
48Ca core excitations up to 6 MeV and L=8 were taken into account. Core
excitations in 40Ca are located, in general, at higher excitation energy than
in 48Ca. We note that the 8 MeV, L=8 cutoffs in 40Ca select only negative-
parity states, as positive-parity ones are predicted to be even higher.
Fig. 8 shows a comparison between predictions from the HCM model, with the SkX
Skyrme interaction, and experimental low-spin yrast states of 41Ca (left),
47Ca (middle) and 49Ca (right), obtained in this work. In the following, the
comparison will be limited to the energy of the states, since very limited
mixing ratios information is experimentally available for transitions
depopulating such states. For detailed comparison in terms of selected B(E3)
values we refer to Ref. Montanari  _et al._ (2012).” The results of the
calculations are also summarized in Tab. 9, for what concerns state energies
and dominant wave function components. It is important to stress that similar
predictions are obtained by using the SLy5 interaction: small differences are
observed in the energies of the levels (within $\approx$ 300 keV on average),
whereas wave function compositions are almost independent from the choice of
the interaction.
The 41Ca ground state is predicted to have a pure f7/2, single-neutron nature.
In the case of excited states, the comparison between experimental results and
theoretical predictions is limited to positive-parity states below 4 MeV (see
Fig 8). In this region, the HCM model predicts a multiplet of states with spin
1/2+, 3/2+ and 5/2+ at 2468 keV, 2608 keV, and 2429 keV, respectively. These
states arise mainly from the coupling between a f7/2 neutron and the octupole
3- vibration of the 40Ca core, with contributions from couplings with other
phonons (see Tab. 9). A good correspondence in terms of level ordering and
energy spacing with the lowest experimental 1/2+, 3/2+ and 5/2+ states is
found. However, the calculated multiplet is located $\approx$ 200-400 keV
below the experimental one. This difference in energy might also be related to
the 3- octupole vibration of 40Ca, which is predicted $\approx$ 800 keV below
the experimental value.

Figure 8: Experimental low-spin states in the 41Ca, 47Ca, and 49Ca nuclei,
compared with theoretical calculations performed with the HCM model, using the
SkX Skyrme interaction. Higher-spin levels observed in Ref. Montanari  _et
al._ (2012) are shown as dashed lines (see text for details) .

Calculations for the 47Ca nucleus are presented in the middle of Fig. 8. These
are the first results obtained with the HCM model for a valence-hole system in
this mass region. The ground state of 47Ca is calculated as a neutron
f${}^{-1}_{7/2}$ configuration. Concerning the 3/2- negative-parity state, it
is predicted to be the coupling between a f${}^{-1}_{7/2}$ neutron hole and
non-collective 1p-1h excitations of the 48Ca core, namely
(f${}^{-1}_{7/2}$p3/2)${}_{2^{+}_{1}}$ and
(f${}^{-1}_{7/2}$p3/2)${}_{4^{+}_{1}}$. It is interesting to note that the
wave function composition is similar to the one obtained by shell model
calculations in the full fpg space, i.e., [$\nu f^{-2}_{7/2}p_{3/2}$] Poves
_et al._ (2001). The positive-parity 1/2+ and 3/2+ states are instead
suggested to be members of the $\nu$f${}^{-1}_{7/2}$ $\otimes$ 3- multiplet
and are predicted about 1 MeV higher than in experiments. These results
indicate that the low-spin structure of the 47Ca nucleus is more complex and
probably contains more configurations than those included in the present HCM
model.
Finally, the level scheme of the 49Ca nucleus is presented on the right side
of Fig. 8. Its ground state is predicted to be pure, with a $\nu$p3/2
configuration, as well as the 1/2- state which is suggested to be a p1/2, pure
single-neutron state. The latter is in good agreement with the experimental
energy. The 9/2+ state is instead calculated as the lowest member of the
$\nu$p${3/2}$ $\otimes$ 3- multiplet.
All the calculated members of the $\nu$f7/2 $\otimes$ 3- ,
$\nu$f${}^{-1}_{7/2}$ $\otimes$ 3- and $\nu$p${3/2}$ $\otimes$ 3- multiplets
in 41Ca, 47Ca and 49Ca, respectively, are presented in Fig. 8 and their wave
function compositions are given in Tab. 9. These states in 47Ca and 49Ca were
already investigated by this collaboration in Ref. Montanari  _et al._ (2012)
and interpreted using the perturbative particle-vibration coupling approach
Bohr and Mottelson . Similar studies were also performed in the case of the
65Cu and 67Cu nuclei, where Ni cores were considered Bocchi  _et al._ (2014);
Niţă  _et al._ (2014). This neutron-rich region around Z=28 is characterized
by the coexistence of different nuclear shapes Crider  _et al._ (2016);
Morales  _et al._ (2016, 2017); Leoni  _et al._ (2017); Mărginean  _et al._
(2020); Porzio  _et al._ , the emergence of which is intimately related to
proton-neutron correlations and shell structure. In this sense, a microscopic
description of the many facets of nuclear excitations is much desirable and
the Hybrid Configuration Mixing model presented in this work provides a step
forward in this direction.
In the case of 41Ca, low-lying positive-parity states up to 13/2+ are
predicted to be members of the multiplet, located around 2500 keV of
excitation energy, in a range of $\approx$ 400 keV. Despite the fact that the
$\nu$f7/2 $\otimes$ 3- coupling is the dominant component in their wave
function, contributions from couplings with other core excitations are
significantly present. In the case of 47Ca, in addition to the 1/2+ and 3/2+
states discussed above, the higher spin states between 5/2+ and 13/2+ are
displayed and compared to experimental energies obtained in previous works
Montanari  _et al._ (2012) and shown by dashed lines. The HCM model also
predicts the B(E3; (13/2+,11/2+) $\rightarrow$ 7/2-) = 6.7 W.u., which is in
agreement, within the error, with the 7.4(19) W.u. experimental value reported
in Ref. Montanari  _et al._ (2012). For 49Ca, the 3/2+, 5/2+, 7/2+, and 9/2+
states, members of the $\nu$p${3/2}$ $\otimes$ 3- multiplet, are also shown,
and compared to the 9/2+ state located at 4017 keV (dashed line), which is the
only one known experimentally, as reported in several works Montanari  _et
al._ (2011a, 2012); Gade  _et al._ (2016); Crawford  _et al._ (2017). On the
other hand, the HCM model predicts the 3/2+, 5/2+, 7/2+ to be degenerate at
4652 keV. This is due to the absence of the $d_{3/2}$, $d_{5/2}$, and
$g_{7/2}$ orbitals in the configuration space. Nevertheless, the B(E3; 9/2+
$\rightarrow$ 3/2-) is calculated to be 5.2 W.u., which is in fair agreement
with the experimental value of 7.9(20) W.u. obtained by lifetime measurements,
as reported in Ref. Montanari  _et al._ (2012)
The 47Ca and 49Ca isotopes were also recently investigated in neutron knockout
Crawford  _et al._ (2017) and neutron pickup experiments Gade  _et al._
(2016), where the strength of the $\nu$1f7/2 in 47Ca and the relative strength
of the $\nu$1fg/2 and $\nu$1f5/2 in 49Ca were extracted. Experimental data
were compared with large-scale shell-model calculations using the GXPF1
effective interaction in the sd+fp+sdg model space, as well as NN+3N, ab
initio calculations in the pf and pfg9/2 model space (see Gade  _et al._
(2016); Crawford  _et al._ (2017) and references therein). In the case of
47Ca, the strength of the $\nu$1f7/2 orbital is found to be concentrated in
the ground state, with a measured (2J+1)C2S spectroscopic factor of
9.3(${}^{+1.1}_{-1.3})_{\text{stat}}$($\pm$1.9)sys. This is qualitatively
reproduced by shell-model calculations, which predict (2J+1)C2S=7.7 and
(2J+1)C2S=6.7-7.0 for the GXPF1 and NN+3N interactions, respectively. Similar
results are obtain in the present work with the HCM model, which estimates
(2J+1)C2S=7.9.
For 49Ca, results for the C2S spectroscopic factor for the 9/2+ state are
summarized in Tab. 10. It can be seen that the experimental results point to a
rather small value for the strength of the $\nu$1g/2 orbital at 4296 keV,
consistently with the complex octupole-coupled nature of the 9/2+ state. This
quenching is more pronounced in the case of the (d,p) measurement Uozumi  _et
al._ (1994), which is well reproduced by the HCM calculations here presented.
On the other hand, shell-model calculations (see Gade  _et al._ (2016))
predict a larger spectroscopic factor, which sits in between the two
experimental results.

Table 10: Experimental and theoretical (see Gade  _et al._ (2016) and references therein) C2S spectroscopic factors for the 9/2+ at 4296 keV in 49Ca.  |  |  |  | Experiment |  |  |  | C2S |  |  |  |
---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  | (d,p) Uozumi  _et al._ (1994) |  |  |  | 0.14 |  |  |  |
|  |  |  |  12C+48Ca Gade  _et al._ (2016) |  |  |  | 0.27(1) |  |  |  |
|  |  |  | Theory |  |  |  | C2S |  |  |  |
|  |  |  | HCM |  |  |  | 0.11 |  |  |  |
|  |  |  | GXPF1 Gade  _et al._ (2016) |  |  |  | 0.42 |  |  |  |

##  V CONCLUSIONS

In conclusion, the low-spin structure of the 41Ca, 47Ca, and 49Ca nuclei was
investigated in the EXILL experimental campaign, following $\gamma$-ray
spectroscopy of neutron-capture reactions on Ca targets. New levels,
$\gamma$-ray transitions and $\gamma$-ray branching ratios and intensities
were reported and a number of transitions were characterized by $\gamma$-ray
angular correlations, enabling to extract multiploraties and mixing ratios and
to assign spins and parities to the states involved in the decays.
Portions of the level schemes below 5 MeV were compared with theoretical
calculations performed with the Hybrid Configuration Mixing Model. Despite
some discrepancies, the model indicates the coexistence, at low energy, of
single-particle/hole states and coupled configurations with collective and
non-collective excitations of the doubly-magic core for all the nuclei studied
in this work. Moreover, experimental results and beyond-mean-filed
calculations obtained in this work by the HCM model for the 47Ca and 49Ca
nuclei were compared with other measurements, as well as shell model and ab
intio calculations. Similar results are observed, although the HCM model
better reproduces the quenching of spectroscopic factor for the 9/2+ state in
49Ca, pointing to the impact of long-range correlations, such as couplings
with phonons, upon the structure of nuclear excitations.
Overall, it appears that Ca isotopes provide a fundamental play ground for
state-of-the-art theories, which, in this mass region, tend to converge to
similar results, making Ca nuclei a cornerstone for a comprehensive
description of nuclear structure.
This work is also an important benchmark for the Hybrid Configuration Mixing
Model, here discussed, which becomes a powerful tool to compute the complex
structure of isotopes in heavier mass regions, such as the neutron-rich region
around the doubly-magic 132Sn nucleus Bocchi  _et al._ (2016); Bottoni  _et
al._ (2019). Indeed, in these heavy systems, shell model calculations and ab
initio methods have severe difficulties in dealing with collective excitations
of the core, due to the diverging dimension of the model space, thus
resulting, up to now, in a limited description of complex excitations.

##  VI ACKNOWLEDGMENTS

This work was supported by the Italian Istituto Nazionale di Fisica Nucleare
and by the Polish National Science Centre under Contract Nos.
2014/14/M/ST2/00738 and 2013/08/M/ST2/00257. The authors would like to thank
the technical services of the ILL, LPSC and GANIL for supporting the EXILL
campaign, as well as the EXOGAM collaboration and the INFN Legnaro for
providing HPGe detectors.

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