# Spectroscopy of deeply bound orbitals in neutron-rich Ca isotopes

P. J. Li<EMAIL_ADDRESS>J. Lee<EMAIL_ADDRESS>P. Doornenbal  S. Chen
S. Wang  A. Obertelli  Y. Chazono  J. D. Holt  B. S. Hu  K. Ogata  Y. Utsuno
K. Yoshida  N. L. Achouri  H. Baba  F. Browne  D. Calvet  F. Château  N. Chiga
A. Corsi  M. L. Cortés  A. Delbart  J-M. Gheller  A. Giganon  A. Gillibert  C.
Hilaire  T. Isobe  T. Kobayashi  Y. Kubota  V. Lapoux  H. N. Liu  T.
Motobayashi  I. Murray  H. Otsu  V. Panin  N. Paul  W. Rodriguez  H. Sakurai
M. Sasano  D. Steppenbeck  L. Stuhl  Y. L. Sun  Y. Togano  T. Uesaka  K.
Wimmer  K. Yoneda  O. Aktas  T. Aumann  K. Boretzky  C. Caesar  L. X. Chung
F. Flavigny  S. Franchoo  I. Gasparic  R.-B. Gerst  J. Gibelin  K. I. Hahn  J.
Kahlbow  D. Kim  T. Koiwai  Y. Kondo  D. Körper  P. Koseoglou  C. Lehr  B. D.
Linh  T. Lokotko  M. MacCormick  K. Miki  K. Moschner  T. Nakamura  S. Y. Park
D. Rossi  E. Sahin  F. Schindler  H. Simon  P-A. Söderström  D. Sohler  S.
Takeuchi  H. Toernqvist  J. Tscheuschner  V. Vaquero  V. Wagner  V. Werner  X.
Xu  H. Yamada  D. Yan  Z. Yang  M. Yasuda  L. Zanetti

###### Abstract

The calcium isotopes are an ideal system to investigate the evolution of shell
structure and magic numbers. Although the properties of surface nucleons in
calcium have been well studied, probing the structure of deeply bound nucleons
remains a challenge. Here, we report on the first measurement of unbound
states in 53Ca and 55Ca, populated from 54,56Ca($p,pn$) reactions at a beam
energy of around 216 MeV/nucleon at the RIKEN Radioactive Isotopes Beam
Factory. The resonance properties, partial cross sections, and momentum
distributions of these unbound states were analyzed. Orbital angular momentum
$l$ assignments were extracted from momentum distributions based on
calculations using the distorted wave impulse approximation (DWIA) reaction
model. The resonances at excitation energies of 5516(41) keV in 53Ca and
6000(250) keV in 55Ca indicate a significant $l$ = 3 component, providing the
first experimental evidence for the $\nu 0f_{7/2}$ single-particle strength of
unbound hole states in the neutron-rich Ca isotopes. The observed excitation
energies and cross-sections point towards extremely localized and well
separated strength distributions, with some fragmentation for the $\nu
0f_{7/2}$ orbital in 55Ca. These results are in good agreement with
predictions from shell-model calculations using the effective GXPF1Bs
interaction and ab initio calculations and diverge markedly from the
experimental distributions in the nickel isotones at $Z=28$.

###### keywords:

unbound states, knockout reaction, single-particle strength, shell evolution

††journal: Physics Letters B

[alanzhou] organization=CAS Key Laboratory of High Precision Nuclear
Spectroscopy, Institute of Modern Physics, Chinese Academy of Sciences,
addressline=Lanzhou 730000, country=China [ahku] organization=Department of
Physics, The University of Hong Kong, addressline=Pokfulam, Hong Kong,
country=China [ariken] organization=RIKEN Nishina Center, addressline=2-1
Hirosawa, Wako, Saitama 351-0198, country=Japan [ayork] organization=School of
Physics, Engineering and Technology, University of York,
addressline=Heslington, York, YO10 5DD, country=UK [atudarmstadt]
organization=Institut für Kernphysik, Technische Universität Darmstadt,
addressline=64289 Darmstadt, country=Germany [acea] organization=IRFU, CEA,
Université Paris-Saclay, addressline=F-91191 Gif-sur-Yvette, country=France
[atriumf] organization=TRIUMF, addressline=4004 Wesbrook Mall, Vancouver,
British Columbia V6T 2A3, country=Canada [amcgill] organization=Department of
Physics, McGill University, addressline=Montréal, Quebec City H3A 2T8,
country=Canada [anccs] organization=National Center for Computational
Sciences, addressline=Oak Ridge National Laboratory, Oak Ridge, Tennessee
37831, country=USA [apd] organization=Physics Division, addressline=Oak Ridge
National Laboratory, Oak Ridge, Tennessee 37831, country=USA [akyushu]
organization=Department of Physics, Kyushu University, addressline=Fukuoka
819-0395, country=Japan [arcnp] organization=Research Center for Nuclear
Physics (RCNP), Osaka University, addressline=Ibaraki, country=Japan [ajaea]
organization=Advanced Science Research Center, Japan Atomic Energy Agency,
Tokai, addressline=Ibaraki 319-1195, country=Japan [acns] organization=Center
for Nuclear Study, University of Tokyo, addressline=RIKEN campus, Wako,
Saitama 351-0198, country=Japan [acaen] organization=Université de Caen
Normandie, ENSICAEN, CNRS/IN2P3, LPC Caen UMR6534, addressline=F-14000 Caen,
country=France [atohoku] organization=Department of Physics, Tohoku
University, addressline=Sendai 980-8578, country=Japan [aklbtme]
organization=Key Laboratory of Beam Technology of Ministry of Education,
addressline=College of Nuclear Science and Technology, Beijing Normal
University, Beijing 100875, country=China [akth] organization=Department of
Physics, Royal Institute of Technology, addressline=SE-10691 Stockholm,
country=Sweden [aijclab] organization=Université Paris-Saclay, CNRS/IN2P3,
IJCLab, addressline=F-91405 Orsay cedex, country=France [abro]
organization=Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS, PSL
Research University, Collège de France, addressline=Case 74, 4 Place Jussieu,
75005 Paris, country=France [ajaver] organization=Pontificia Universidad
Javeriana, Facultad de Ciencias, Departamento de Física, addressline=Bogotá,
country=Colombia [aunal] organization=Universidad Nacional de Colombia, Sede
Bogotá, Facultad de Ciencias, Departamento de Física, addressline=Bogotá
111321, country=Colombia [aut] organization=Department of Physics, University
of Tokyo, addressline=7-3-1 Hongo, Bunkyo, Tokyo 113-0033, country=Japan
[aatomki] organization=Institute for Nuclear Research, Atomki,
addressline=P.O. Box 51, Debrecen H-4001, country=Hungary [aibs]
organization=Institute for Basic Science, addressline=Daejeon 34126,
country=Korea [arikkyo] organization=Department of Physics, Rikkyo University,
addressline=3-34-1 Nishi-Ikebukuro, Toshima, Tokyo 172-8501, country=Japan
[agsi] organization=GSI Helmholtzzentrum für Schwerionenforschung GmbH,
addressline=Planckstr. 1, 64291 Darmstadt, country=Germany [ainst]
organization=Institute for Nuclear Science & Technology, VINATOM,
addressline=179 Hoang Quoc Viet, Cau Giay, Hanoi, country=Vietnam [azagreb]
organization=Ruđer Bošković Institute, addressline=Bijenička cesta 54, 10000
Zagreb, country=Croatia [akoeln] organization=Institut für Kernphysik,
Universität zu Köln, addressline=D-50937 Cologne, country=Germany [aewha]
organization=Ewha Womans University, addressline=Seoul 03760, country=Korea
[atitech] organization=Department of Physics, Tokyo Institute of Technology,
addressline=2-12-1 O-Okayama, Meguro, Tokyo, 152-8551, country=Japan [avarns]
organization=Vietnam Agency for Radiation and Nuclear Safety, addressline=113
Tran Duy Hung, Cau Giay, Hanoi, country=Vietnam [aoslo]
organization=Department of Physics, University of Oslo, addressline=N-0316
Oslo, country=Norway [aelinp] organization=Extreme Light Infrastructure-
Nuclear Physics (ELI-NP)/Horia Hulubei National Institute for Physics and
Nuclear Engineering (IFIN-HH), addressline=Str. Reactorulu, Măgurele 077125,
country=Romania [amadrid] organization=Instituto de Estructura de la Materia,
CSIC, addressline=E-28006 Madrid, country=Spain

##  1 Introduction

Nuclear shell evolution towards the driplines poses a significant challenge in
modern nuclear physics, characterized by the quenching or collapse of
conventional magic numbers, and the reordering of single-particle orbitals,
potentially giving rise to the emergence of new magic numbers and/or sub-shell
closures. The calcium isotopes feature a robust proton $Z=20$ shell closure.
Hence, the doubly magic isotopes 40Ca and 48Ca as well as experimental
evidence of new neutron sub-shell closures at $N=32$ [1, 2] and 34 [3, 4, 5]
provide a unique testing ground to study shell evolution.

In the independent-particle shell model picture, nucleons occupy the single-
particle orbitals and are bound in an attractive mean-field induced by the
interactions among all nucleons. Ideally, occupation probabilities for the
single-particle and single-hole states in the vicinity of a doubly magic
nucleus should reflect the independent particle filling. However, near the
doubly magic nucleus 48Ca, a nearly 50$\%$ reduction in spectroscopic strength
has been observed in the $0f_{7/2}$ neutron hole state populated from one-
neutron knockout of 50Ca$\rightarrow$49Ca, in contrast to the
48Ca$\rightarrow$47Ca neutron knockout [6, 7]. These results point to the
fragmentation of angular momentum $l=3$ strength to higher-lying states and
cannot be reproduced by any shell model calculations. Also for a systematic
study of one neutron knockout reaction in 51-55Sc isotopes [8] the observed
spectroscopic strength of the $l=3$ contribution was approximately half of the
prediction from shell model calculations using the GXPF1A interaction [9],
indicating a weakening of the $N=28$ shell gap and a considerable
fragmentation of neutron $0f_{7/2}$ spectroscopic strength in neutron-rich
scandium isotopes ($N=30-34$). In contrast, a recent experiment on the
neutron-knockout from 52Ca employing the quasi-free ($p,pn$) scattering
reaction has revealed a spectroscopic factor of $C^{2}S=6.6(10)$ for the $\nu
0f_{7/2}$ hole state, suggesting a consistent shell structure in 48Ca and 52Ca
[2].

Given these experimental variations, further inspection regarding the
evolution of the $\nu 0f_{7/2}$ single-particle orbital in the neutron-rich Ca
isotopic chain has become prudent. Early ab-initio calculations using many-
body perturbation theory and the low-momentum interaction $V_{\text{low}k}$
obtained by evolving a chiral N3LO NN potential predict a decreasing and
eventually vanishing gap in the effective single particle energies (ESPEs)
between the $0f_{7/2}$ and $1p_{3/2}$ single particle energies (SPEs) for
calcium isotopes beyond 48Ca for calculations in the $pfg_{9/2}$ model space
[10, 11, 12]. Conversely, the phenomenological interactions GXPF1(A) [13, 14,
9] and KB3G [15] yield a significant gap of $\approx$5 MeV for 48Ca, which,
albeit decreasing, remains at approximately 3 MeV between 54Ca and 60Ca. A
similar trend is observed for the recent data driven, in-medium similarity
renormalization group (IMSRG) based interaction UFPA-CA [16].

Here, we report on the first invariant-mass measurement of unbound states in
53,55Ca, populated using one-neutron quasi-free scattering reactions at
relativistic energies. For the first time, the spectroscopic strength of the
$\nu 0f_{7/2}$ neutron-hole states in 53Ca and 55Ca has been extracted by
comparing experimental data to single-particle cross sections using DWIA
calculations. In doing so, experimental knowledge on bound states of the
isotopes 53,55Ca, acquired in recent one-neutron knockout studies [5, 17] and
an earlier $\beta$-decay investigation of 53K [18], is expanded significantly.
Our results show that the extracted spectroscopic strengths of the neutron
$1p_{1/2}$, $1p_{3/2}$, and $0f_{5/2}$ hole states are consistent with shell
model calculations based on the GXPF1Bs interaction [5] and the ab-initio
valence-space in-medium similarity renormalization group (VS-IMSRG) [19].

No evidence for $\nu 0f_{7/2}$ strength below the neutron separation energy
$S_{n}$ was found in earlier experiments [5, 17], leading to the assumption
that it is associated with states above the neutron decay threshold. Our
findings reveal that the $0f_{7/2}$ neutron-hole states in 53,55Ca maintain a
nearly pure single-hole character at high excitation energies, demonstrating
remarkable agreement with the theoretical predictions. Differences are found
in the location of the $1p_{3/2}$ and $0f_{7/2}$ hole states, from which the
propagation of the $N=28$ shell gap can be inferred. By completing the trend
of spectroscopic strength distribution of the $pf$ SPEs up to 56Ca, crucial
insights into the evolution of shell structures in neutron-rich calcium
isotopes is provided.

##  2 Experimental Setup

The experiment was carried out at the Radioactive Isotope Beam Factory (RIBF)
operated by the RIKEN Nishina Center and the Center for Nuclear Study (CNS),
University of Tokyo. Secondary beam cocktails were produced by fragmentation
of a 345 MeV/nucleon 70Zn primary beam that impinged on a 10-mm-thick 9Be
target with a typical intensity of 240 pnA. The beams of interest were
selected and transported to the secondary target using the BigRIPS two-stage
fragment separator [20]. The beam particles were identified in BigRIPS by
their atomic number $Z$ and mass-to-charge ratio ($A/Q$) on an event-by-event
basis, shown in Fig. 1(a) of Ref. [21], extracted by their energy loss in a
set of plastic detectors and the B$\rho$-$\Delta$E-TOF method [22]. The 54Ca
and 56Ca beams bombarded the 151-mm-long liquid hydrogen (LH2) target of the
MINOS device [23] with an average energy of $\sim$250 MeV/nucleon to induce
quasi-free $(p,pn)$ scattering reactions. MINOS also included a 300-mm-long
time projection chamber (TPC) surrounding the LH2 target to reconstruct the
trajectories of the recoil protons with a position resolution of $\sim$2 mm
($\sigma$) in the beam direction [24]. Reaction residues were analyzed by the
SAMURAI spectrometer [25] with a magnetic field set to 2.7 T in the center.
The particle identification from SAMURAI is shown in Fig 1(b) of Ref. [21],
following a method similar as for the incoming beams. Hydrogen targets
simplify the knockout process by providing a straightforward proton probe,
leading to improved access to deeply bound states compared to traditional
nuclear targets like Be or C, as evidenced by ongoing efforts at facilities
such as RIBF and GSI[26, 27, 28, 29, 30].

Population to unbound states of 53,55Ca was followed by forward directed
neutron emission. These approximately beam-velocity neutrons were detected by
two large-acceptance plastic scintillator arrays, the NeuLAND demonstrator
[31] and NEBULA [25, 32], located 11.8 m and 14.85 m downstream of the target,
respectively. The NeuLAND demonstrator consisted of 400 modules ($5\times
5\times 250$ cm3) in 8 layers. The NEBULA array consisted of 120 neutron
detection modules ($12\times 12\times 180$ cm3), arranged in a two-wall
configuration. The neutron momenta were derived from their TOFs and the flight
paths between the reaction vertex and the hit position on the neutron
detectors. Neutron detection efficiencies of $\varepsilon_{n}\approx 37\%$ and
$\varepsilon_{nn}\approx 12\%$ for decay energies at 2 MeV were simulated with
the GEANT4 [33] framework.

The de-excitation $\gamma$ rays emitted from fragments were detected by the
DALI2+ array [34, 35], consisting of 226 NaI(Tl) crystals that surrounded the
MINOS device. An add-back procedure combining hits within 15 cm was applied
during the analysis to increase the photopeak efficiency. For a 2-MeV $\gamma$
ray emitted by particles moving at 60% of the speed-of-light, the photopeak
efficiency was $23\%$ with add-back, and the energy resolution after Doppler
correction was $75$ keV($\sigma$).

##  3 Results

Figure 1:  Doppler-corrected de-excitation $\gamma$-ray spectra of 52,54Ca
fragments following neutron decays. (a) and (b) show the total spectra for
52Ca and 54Ca, respectively. The fit function (red line) comprises a double-
exponential background (black dash dotted line) and simulated $\gamma$-ray
response functions of DALI2+ (blue lines). (c) and (d) show the two new
$\gamma$-ray transitions in the 52Ca spectra obtained in coincidence with the
closest 1427-keV transition and 3150-keV transition, respectively.  Figure 2:
(a) Relative energy spectrum of
${}^{54}\text{Ca}(p,pn)^{53}\text{Ca}\rightarrow$ ${}^{52}\text{Ca}+1n$;
Exclusive relative energy spectra in coincidence with the 2563-keV state (b)
and the ground state (c) in 52Ca. (d) Relative energy spectrum of
${}^{56}\text{Ca}(p,pn)^{55}\text{Ca}\rightarrow$ ${}^{54}\text{Ca}+1n$;
Exclusive relative energy spectra in coincidence with the 3680-keV state in
54Ca (e), and ground state (f). The overall fit function (red line) comprises
the sum of the simulated responses of NeuLAND + NEBULA and beamline detectors
(blue lines), and a non-resonant background (black long-dashed line).

Unbound states of 53,55Ca populated from 1$n$ knockout reactions
instantaneously decay into systems of neutrons plus calcium fragments. The
fragments may be in excited states, ensuing $\gamma$-ray de-excitation.
Accordingly, properties of unbound states were reconstructed by invariant-mass
spectroscopy. The excitation energy of an unbound state is written as

$\displaystyle E_{ex}=E_{rel}+S_{n}+(E_{\gamma}),$ (1)

where the $E_{rel}$ refers to the relative energy of the unbound system
reconstructed from the momenta of the fragment and neutron. $S_{n}$ represents
the one neutron separation energy, while $E_{\gamma}$ is the summed
$\gamma$-ray energy following the neutron emission, if any.

### Doppler-shift corrected $\gamma$-ray spectra

As exclusive final state and momentum distributions require knowledge on
coincident $\gamma$-ray decay, they are presented first. Figure 1 displays the
Doppler-corrected $\gamma$-ray spectra of ${}^{52,54}\text{Ca}$ fragments. For
52Ca, peaks at 2563, 1427, 1961, 3150, and 3388 keV are found, consistent with
the transitions reported from a $\beta$-decay study [18]. In addition, two new
$\gamma$-ray transitions are observed at 553(9) and 950(17) keV in this work,
resulting in tentative level placements at 4543(10) and 4100(17) keV,
according to the $\gamma$-$\gamma$ coincidence analysis, shown in Fig. 1(c,
d). The population percentages for 2563, 3150, 3990, 4100, 4543, and 5951 keV
states in 52Ca, extracted from Fig. 1(a), are as follows: 51.0(28), 19.6(23),
0.0(46), 7.1(13), 8.3(10), and 14.0(20) $\%$, respectively.

For 54Ca, $\gamma$-ray energies at 459(12), 1187(23), 1667(41), and 2052(26)
keV were extracted from Fig. 1(b), consistent with Refs. [3, 21]. Considering
the error bars of the present work, the obtained energies of 54Ca have been
adopted from Ref. [21] for further analysis. The $\gamma$-ray spectra were
fitted with a double-exponential background and simulated DALI2+ response
functions. In Fig.1(b), the $\gamma$-ray spectrum was almost entirely
described by the response function of populating the 3680-keV state on top of
exponential background. The $\gamma$-ray intensity of the 3680-keV state and
possibly the 2466-keV state excludes the possibility of a direct feeding to
the 2021-keV state. Consequently, in the decomposition of the $E_{rel}$
spectra of ${}^{54}\text{Ca}+1n$, feedings to the 2021-keV state are not
considered.

###  $E_{rel}$ spectra

The relative energy spectra of the unbound 53,55Ca system following the
${}^{54,56}\text{Ca}(p,pn)$ reaction are shown in Fig. 2. The top panels (a)
and (d) show the inclusive $E_{rel}$ spectra of the unbound states, whereas
the other panels depict the exclusive $E_{rel}$ spectra in coincidence with
final states of 52,54Ca fragments. The $E_{\gamma}$ spectra were analyzed to
ascertain the contributions of fragments’ final states, in coincidence with a
fixed $E_{rel}$-bin, spanning the entire $E_{rel}$ range of [0, 5] MeV. Panel
(b) and (e) represent examples of the exclusive $E_{rel}$ spectra obtained in
coincidence with the 2563-keV state of 52Ca and the 3680-keV state of 54Ca,
respectively. Contributions to the ground states of the 52,54Ca fragments,
shown in Figs. 2(c) and (f), were extracted by subtraction of all
contributions to excited states from the inclusive spectra.

The $E_{rel}$ spectra were fitted with response functions using Breit-Wigner
distributions [36], folded with the detector response obtained from a GEANT4
simulation of the NeuLAND + NEBULA array and beam-line detectors. A non-
resonant background was considered, for which the shape was obtained from
event mixing [37, 38] and an amplitude fitted together with the response
functions of the resonances. The exclusive $E_{rel}$ spectra, i.e.,
differences when gating on the bound $\gamma$-decaying states of 52,54Ca, help
distinguishing the overlapping peaks and prepare the resonance candidates for
the fit of the inclusive $E_{rel}$ spectra. The exclusive cross-sections to
the unbound states of 53,55Ca were derived from the amplitudes of the Breit-
Wigner peaks corrected with the simulated 1$n$-detection efficiency. Note that
$\sim 8\%$ of events originating from the inelastic excitation process, as
discussed in detail in Refs. [2, 5], were subtracted from the relative energy
spectra and momentum distributions. The fit results are summarized in
supplemental Fig.3 and Fig. 4, and Table 1 in Ref. [39].

Figure 3:  Partial level scheme of unbound states in 53Ca following the
${}^{54}\text{Ca}(p,pn)$ reaction. The position and width of the unbound
states were reconstructed from a detailed analysis of decay neutrons in
coincidence with $\gamma$ rays. Arrow widths represent the relative intensity.
Bound states of 53Ca were omitted. Figure 4:  Same as Fig. 3, but for unbound
states in 55Ca following the ${}^{56}\text{Ca}(p,pn)$ reaction.

The overall fit of the 52Ca+$1n$ relative energy spectrum, shown in Fig. 2(a),
is characterized by numerous overlapping peaks, extracted from the exclusive
spectra as resonance candidates illustrated in Fig. 2(b, c) and its
supplemental Fig. 1 in Ref. [39]. Among the observed transitions,
approximately 75$\%$ of the measured cross-sections align with the 52Ca ground
state, 13$\%$ with the 2563-keV state, and 12$\%$ with remaining states above
the 2${}^{+}_{1}$ state of 52Ca. Of these, two resonances at 980(110) and
2326(10) keV were identified to be in coincidence with the 52Ca ground state
[Fig. 2(c)], corresponding respectively to states at 4170, and 5516 keV; four
resonances at 277(42), 1189(87), 1716(27), and 3680(140) keV were in
coincidence with the 2563-keV state [Fig. 2(b)], corresponding respectively to
states at 6030, 6942, 7416 and 9430 keV; further details regarding the
remaining four resonances, which were in coincidence with higher excited
states, can be found in Ref. [39]. Interestingly, the dominant peak at
$2326(10)$ keV, decaying directly to the ground state of 52Ca via 1$n$
emission, exhibited a width of $\Gamma=253(25)$ keV, approximately aligning
with the single-particle width of $\Gamma_{r}^{l=3}=220$ keV for $f$-wave
neutron emission[40].

The $E_{rel}$ spectrum following the ${}^{56}\text{Ca}(p,pn)^{55}\text{Ca}$
reaction, as shown in Fig. 2(d), can be clearly described by three resonance
peaks added on a non-resonant background. Due to low statistics, weaker states
with low cross sections could not be effectively observed in this data. A
resonance at 1404(41) keV was observed in coincidence with the ground state of
54Ca, placed at a state of 2970 keV. Resonance peaks at 2089(80) keV and
664(34) keV decay to the 2466-keV state and 3680-keV state of 54Ca,
respectively. Given the uncertainties, the latter two resonances can be
treated as decays from a state at an energy of 6000(250) keV. This state is
above the two neutron separation energy of $S_{2n}=5405(167)$ keV [4, 41],
which can also decay by 2$n$ emission. Accordingly, its cross-section in Table
1 [39] is considered as the lower limit. The limited statistics of the
53Ca$+2n$ channel prevented a detailed analysis of this state. Hence, the
upper cross-section limit for this state was estimated in the $E_{rel}$ range
of [0-1] MeV, considering larger error bars from limited statistics and
interference from a resonant-like structure beyond 1 MeV, and subsequently
corrected with 2$n$ detection efficiency. These cross-sections are provided in
Table 1 and used for extracting the spectroscopic factors $C^{2}S$.

### Momentum distributions

The momentum distributions of the reaction products were reconstructed using
the beam and fragment velocities at the reaction vertex and the scattering
angles calculated from the position measurements. Exclusive parallel momentum
distributions (PMDs) for unbound states of 53Ca and 55Ca are presented in Fig.
5, extracted by fitting the $E_{rel}$ spectra within a fixed PMD-bin, allowing
to separate the contributions of the overlapping resonances. For PMDs, a
resolution of 40 MeV/c $(\sigma)$ was evaluated from the unreacted 54,56Ca
beams.

The orbital angular momentum $l$ of the removed neutron can be deduced from
the momentum distribution in neutron knockout reactions. Theoretical
predictions for single-particle cross-sections and momentum distributions of
neutron removal from $l=1$ and 3 orbitals were calculated using the distorted
wave impulse approximation (DWIA) model [42, 43] and then compared with
experimental results. DWIA calculations have been successfully applied in the
quasi-free scattering studies for neutron-rich nuclei in this region [2, 5,
21, 44, 45, 46]. The calculated PMDs have been folded with the experimental
resolution, also taking into account the energy spread induced by the thick
target.

Figure 5:  Exclusive parallel momentum distributions for (a) 5516-keV and (c)
4170-keV states of ${}^{53}\text{Ca}$, (b) 6000-keV and (d) 2970-keV states of
${}^{55}\text{Ca}$. Solid curves are the calculated DWIA predictions assuming
an 1$n$ knockout from $p$ and $f$ orbitals and have been normalized to
experimental cross-sections. Panel (e) and (f) display PMDs for all the
unbound states of ${}^{53}\text{Ca}$ and ${}^{55}\text{Ca}$, respectively,
compared with a linear fit (black, solid lines) between p-curve (red, dashed)
and f-curve (blue, dashed). Error bars represent statistical errors.

The PMD of the 5516-keV state in 53Ca [Fig. 5(a)] was well reproduced by the
$f$-orbital distribution from the DWIA calculation, providing evidence for the
$l=3$ assignment of this state. In addition, fitting of the PMD of the
4170-keV state [Fig. 5(c)] yielded a reduced $\chi^{2}$ of 0.5 for the
$f$-curve, and 2.6 for the $p$-curve, favoring the $l=3$ assignment of neutron
removal from the $f$-orbital. The 6000-keV state in 55Ca was tentatively
assigned to $l=3$. The PMD of this state, shown in Fig. 5(b), exhibits a wide
distribution and is better fitted with the $f$-curve, yielding a reduced
$\chi^{2}$ value of 0.6. In contrast, the fit using a $p$-curve yields a
higher reduced $\chi^{2}$ of 1.6, suggesting a preference for the $l=3$
assignment. The PMD of the 2970-keV state of 55Ca [Fig. 5(d)] was consistent
with a $p$-wave neutron knockout, supporting the $l=1$ assignment of this
state. In supplemental Fig. 4 [39], the PMD fitting indicates a preference of
$l=3$ for the 6030-keV state and $l=1$ for the 7416-keV and 7658-keV states.
Due to limited statistics, other exclusive PMDs did not yield meaningful
results. More details can be found in Ref. [39].

The PMDs for the entire $E_{rel}$ strength indicated a dominance of $f$-wave
neutron removal. Fig. 5(e) and (f) demonstrate linear fits incorporating the
$p$-wave and $f$-wave components. The best fit with relative contributions of
two components allows for determining the cross-sections for a neutron
knockout from the $l=1$ and 3 orbitals. The inclusive PMDs of all the unbound
states of 53Ca can be effectively described by the $f$-wave curve.
Specifically, the extracted cross-sections of 1.6(12) mb and 47.8(29) mb
correspond to the $p$-wave and $f$-wave components, respectively. Compared to
the total cross-section of 39.4(23) mb for the $7/2^{-}$ states listed in
Table 1, the extracted $f$-components fall outside the accepted error margins.
This departure primarily results from the underestimation of the
$p$-components, attributed to the inadequate fit in the PMDs of [0, 120] MeV
regions. Additionally, the relatively high $S_{2n}$ allows the potential
contribution from knockout processes within the $sd$ shell, which is beyond
the scope of this study. In Fig. 5(f), the total cross-section extracted for
all the unbound states of 55Ca amounts to 46.7(44) mb, in accordance with the
cross-sections of the $f$ and $p$ components as listed in Table 1, accounting
for the non-negligible background demonstrated in Fig 2(d). In the fitting
processes, the ratio between the $p$-wave and $f$-wave components has been
constrained using the cross-sections in Table 1, resulting in a fit with a
reduced $\chi^{2}$ of 1.05. Allowing this ratio to vary freely improved the
fit to a reduced $\chi^{2}$ of 0.84, albeit with the $p$-component
contribution decreasing by approximately 50$\%$. Importantly, the extracted
cross-sections remain consistent with Table 1’s results and fall within the
associated error bars.

### Spectroscopic factors

Spectroscopic factors $C^{2}S$ were obtained by comparing the measured cross-
sections to single particle cross-sections $\sigma_{sp}$ of the DWIA
calculations, shown in Table 1. A systematic uncertainty of 15$\%$ was
considered for the calculated $\sigma_{sp}$ [43]. Quenching factors were not
applied due to the proximity of $C^{2}S$ to the independent particle
expectation of $2j+1$ value and the additional uncertainties associated with
their application, given the debated isospin dependence[47] and contradictory
results using hydrogen versus traditional carbon or beryllium targets[48, 49].

A cross-section of 28.6(15) mb was determined for the 5516-keV state in 53Ca,
yielding a $C^{2}S$ of 6.5(3). This observation suggests the presence of a
nearly pure $\nu 0f_{7/2}$ neutron-hole state, aligning with an extreme
single-particle description. Two neighboring states at 4170 keV and 6030 keV
are tentatively assigned to a spin-parity of $7/2^{-}$, based on the exclusive
PMD fitting. Their cross-sections are 8.1(16) mb and 2.7(8) mb, resulting in
$C^{2}S$ values of 1.8(3) and 0.6(2), respectively. In Ref. [5], the ground
state of 53Ca was tentatively assigned to a spin-parity of 1/2-, associated
with $\nu p_{1/2}$ neutron knockout with a $C^{2}S$ of 2.2(2); and the
2220-keV state is tentatively assigned to a spin-parity of 3/2- associated
with $\nu p_{3/2}$ neutron knockout with a $C^{2}S$ of 3.1(2). Concerning the
7416-keV and 7658-keV states, favoring the $l=1$ in the PMD fitting, a
plausible assignment of a spin-parity of $3/2^{-}$ is proposed, yielding
$C^{2}S$ values of 0.9(3) and 0.3(1), respectively. However, the unfavorable
orbital assignment can not be entirely excluded due to the error bars.
Spectroscopic factors could not be determined for other unbound states in
53Ca, as their respective PMD and accordingly $l$-value could not be
determined.

For 55Ca, only two bound states have been identified from one-neutron
knockout: The ground state was tentatively assigned to a spin-parity of
$5/2^{-}$ with a $C^{2}S$ of $2.0^{+1.6}_{-0.9}$, and the 673-keV state
tentatively assigned to a spin-parity of $1/2^{-}$ with a $C^{2}S$ of
$3.0^{+1.0}_{-0.6}$ [17]. The present PMDs favor a spin parity assignment of
$3/2^{-}$ for the 2970-keV state. Accordingly, the calculated single particle
cross section for an $l=1$ states is 5.72 mb, resulting in an $C^{2}S$
extracted to be 3.7(8). Similarly, the 6000-keV state can be attributed to
spin parity $7/2^{-}$ and a $C^{2}S$ estimated to be 4.8-5.9(9). Here, the
lower limit originates from the 1$n$ emission data, while an upper limit was
estimated by taking into account the 2$n$ emission.

Table 1:  Identified unbound neutron-hole states populated from
${}^{54,56}\text{Ca}(p,pn)$ reactions. Excitation energies, cross sections,
and spectroscopic factor ($C^{2}S$) are compared with theoretical values from
shell model calculations using the GXPF1Bs interaction and ab initio VS-IMSRG
calculations employing the chiral NN+3N interactions [1.8/2.0 (EM) and
$\Delta$N2LO${}_{\text{GO}}$]. Spin-parity assignments are based on
systematics and theoretical calculations. The experimental $C^{2}S$ were
deduced by dividing the measured cross sections by the calculated single
particle cross-section from the DWIA framework. Level energies are given in
keV and cross-sections in mb.

|  |  |  | DWIA | GXPF1Bs |  $pf$1.8/2.0(EM) |  $pf$$\Delta$N2LO${}_{\text{GO}}$ |  $pfg$1.8/2.0(EM) |  $pfg$$\Delta$N2LO${}_{\text{GO}}$
---|---|---|---|---|---|---|---|---|---
E | $\sigma^{exp}_{-1n}$ | $C^{2}S_{exp}$ | $J^{\pi}$ | $\sigma^{sp}$ |  E |  $C^{2}S$ |  E |  $C^{2}S$ |  E |  $C^{2}S$ |  E |  $C^{2}S$ |  E |  $C^{2}S$
${}^{54}\text{Ca}(p,pn)^{53}\text{Ca}$$\rightarrow$${}^{52}\text{Ca}+1n$
4170(120) | 8.1(16) | $1.8(3)(3)^{1}$ | $7/2^{-}$ | 4.59 |  49854 |  0.44 |  70424 |  0.44 |  72934 |  0.24 |  72164 |  0.14 |  63874 |  0.64
5516(41) | 28.6(15) | $6.5(3)(10)^{1}$ | $7/2^{-}$ | 4.38 |  5225 |  7.1 |  6756 |  7.3 |  7053 |  7.3 |  6454 |  7.5 |  6633 |  6.8
6030(58) | 2.7(8) | $0.6(2)(1)^{1}$ | $7/2^{-}$ | 4.31 |  |  |  |  |  |  |  |  |  |
${}^{56}\text{Ca}(p,pn)^{55}\text{Ca}$$\rightarrow$${}^{54}\text{Ca}+1n$
2970(170) | 20.9(48) | $3.7(8)(5)^{1}$ | $3/2^{-}$ | 5.72 |  3152 |  3.3 |  3975 |  3.7 |  3007 |  3.7 |  3728 |  3.6 |  2805 |  3.6
6000(250) |  19.7(34)2 | $4.8(8)(7)^{1}$ | $7/2^{-}$ | 4.10 |  5789 |  3.8 |  8023 |  6.8 |  7681 |  7.1 |  7344 |  4.6 |  7000 |  6.2
6000(250) |  24.2(36)3 | $5.9(9)(8)^{1}$ | $7/2^{-}$ | 4.10 |  59274 |  1.54 |  90084 |  0.64 |  83534 |  0.34 |  74314 |  2.14 |  69724 |  1.04

1\. The $1^{st}$ and $2^{nd}$ parentheses represent the statistical error and
the systematic error originating from the uncertainty of $\sigma_{sp}$[43],
respectively.
2\. Lower limits obtained from the 1n emission data.
3\. Upper limits estimated from the 2n emission data.
4\. The second largest $C^{2}S$ value in the calculations.

##  4 Discussion

The observation of $0f_{7/2}$ neutron-hole states in the quasi-free scattering
reaction 54,56Ca$(p,pn)$53,55Ca reveals a shell gap of approximately 3 MeV
between the $0f_{7/2}$ and $1p_{3/2}$ orbitals, with large spectroscopic
factors indicating the robustness of the $N=28$ shell closure. Hence, a large
neutron excess in the Ca isotopes seems not to result in $N=28$ shell gap
degradation. Neutrons in the deeply bound $0f_{7/2}$ shell are arranged as in
the extreme single particle picture, providing further evidence for the
magicity of $N=32$ and $N=34$. In contrast, the results for 55Ca demonstrate
more pronounced fragmentation in the deeply bound $\nu 0f_{7/2}$ orbital.
These findings on location and occupation of the deeply bound orbitals offer
an unprecedented opportunity to test state-of-the art theories up to high
excitation energies beyond the neutron separation energies.

Figure 6: Panel (a) and (g) depict the measured cross-sections for 53Ca and
55Ca as a function of excitation energy. The cross-sections to bound states
below $S_{1n}$ were deduced from $\gamma$-ray spectroscopy measurements[5,
17]. The cross-sections to the unbound states (this work) were obtained from
invariant mass spectroscopy. Identified neutron-hole states are colored
accordingly to the orbital of the removed neutron. The other states with no
conclusive PMDs are labeled "not assigned". Panel(b-f) are the theoretical
predictions of energies and cross-sections. Predictions associated with the
$\nu 0g_{9/2}$ orbital are scaled by a factor of 5.

The experimental cross-sections are confronted with theoretical predictions
obtained by combining the $\sigma_{sp}$ with the $C^{2}S$ values from the
shell model or ab-initio calculations, shown in Table 1 and Fig. 6. Shell
model calculations were performed using the latest GXPF1Bs Hamiltonian [5]
from the GXPF1 family of effective nucleon-nucleon interactions [9] in the
$pf$ model space. These calculations have also been effectively employed to
interpret the bound states results for 54Ca$(p,pn)$53Ca [5] and
56Ca$(p,pn)$55Ca [17] reactions. The VS-IMSRG [19, 50, 51, 52] calculations
were carried out employing two different sets of chiral NN+3N interactions:
1.8/2.0 (EM) [53, 54] and $\Delta$N2LO${}_{\text{GO}}$ [55], derived from
chiral effective field theory. The former interaction has proven its
effectiveness in numerous studies [56, 57, 58], while the latter interaction,
optimized using properties of A$\leq$4 nuclei and nuclear matter, exhibits
remarkable capability to reproduce binding energies and charge radii from
$A=16$ to $A=132$ [55]. The chosen model space used an 40Ca core with valence
neutrons in the $pf$ or $pfg_{9/2}$ shell using a multi-shell variant of the
VS-IMSRG [59].

As shown in Fig. 6, the experimental cross-sections in panel (a,g) are in good
agreement with theoretical predictions in panels (b-f,h-l) which are derived
from both SM and ab-initio calculations. The SM predictions using the
effective GXPF1Bs Hamiltonian are remarkably consistent with the experimental
results for both bound states [5, 17] and the unbound states observed in this
work. Calculated energy and $C^{2}S$ values strongly reinforce the
experimental spin assignments.

As for the VS-IMSRG calculations, the predicted cross-sections of major
neutron-hole states are in line with the data. However, the position of the
$0f_{7/2}$ hole-state is predicted to be approximately 1 MeV higher for 53Ca,
and between 1 and 2 MeV higher for 55Ca. More generally, utilization of the
1.8/2.0 (EM) interaction leads to higher energies for the $1p_{1/2}$,
$1p_{3/2}$ and $0f_{7/2}$ neutron-hole states. Conversely, using the
$\Delta$N2LO${}_{\text{GO}}$ interaction leads to predicted energies for the
$1p_{1/2}$ and $1p_{3/2}$ neutron-hole states within 200-keV difference from
the experimental data, but results in a shell gap larger than 4 MeV between
the $0p_{3/2}$ and $0f_{7/2}$ orbitals. Another interesting observation is
that model space expansion from the $pf$ shell to the $pf+g_{9/2}$ orbitals
leads to an improved agreement between the predictions and experimental
results. Specifically, the location of the $0f_{7/2}$ hole state is lowered by
about 500 keV for 53Ca and 1 MeV for 55Ca. Furthermore, the $0f_{7/2}$
strength fragments into two major states in panel (j) and (l) for the isotope
55Ca, while a single major hole state is evident in panel (d) and (f) for
53Ca, and thus consistent with the GXPF1Bs Hamiltonian. As the two states in
55Ca are predicted close in energy, they may have remained unobserved due to
their width and/or the experimental resolution. Further improvements are
needed in the VS-IMSRG framework, including the consideration of continuum
effects on unbound states[60]. It is noteworthy that these effects are almost
negligible for the $7/2^{-}$ states due to its significant centrifugal
barrier. The GXPF1Bs calculations, based on an effective interaction, do not
explicitly include continuum effects. Ref. [61] demonstrates that there is no
significant difference between effective interactions and those with explicit
continuum coupling, such as the Gamow shell model.

Figure 7: (1) Schematic illustration of shell evolution from Ni to Ca for
neutron $pf$ orbitals. The wavy line implies the attractive interaction
between the proton $0f_{7/2}$ and the neutron $0f_{5/2}$ orbits. Adopted from
Ref [62]. (2) Normalized spectroscopic factors of neutron-hole states as a
function of excitation energy in the Ca($Z=20$) and Ni($Z=28$) isotopes,
populated from neutron removal reaction. Data of Ca isotopes are taken from
neutron knockout reactions [6, 2, 5, 17] and the present work (unbound
states); Data of Ni isotopes are taken from $(p,d)$ or
$(^{3}\text{He},\alpha)$ transfer reactions [63, 64]. Values where
$C^{2}S/(2J+1)<0.05$ have been multiplied by a factor of 5 and are displayed
with hatching in the color corresponding to the respective orbital.

From the systematics of the $C^{2}S$ in the Ca isotopes, shown in Fig. 7(b),
the $0f_{7/2}$ strength maintains a nearly pure neutron-hole character in
reaction residues populated by neutron knockout from the doubly magic nuclei
48,52,54Ca. These results are particularly intriguing, as it provides further
evidence for the $N=32,34$ sub-shell closures. Stronger fragmentation of the
$0f_{7/2}$ neutron hole state strength in 55Ca is expected, given the
difficulty in maintaining a pure single-hole character at approximately 6 MeV.
The experiment did not observe significant population to higher-lying
$0f_{7/2}$ hole states, possibly due to their low cross sections or decay via
2$n$ emission channel with limited statistics. In contrast, previous results
for 49Ca, finding a considerable reduction of $\nu 0f_{7/2}$ strength and a
spectroscopic factor of 3.4(4) for the $7/2^{-}$ state at 3357 keV [6],
demonstrate a slight deviation from current experimental findings. These
results could not be reproduced by any calculations, and suggest a larger
degree of fragmentation compared to 55Ca. It is also stressed that for ab-
initio calculations, a considerable fragmentation occurs for 55Ca only with
the inclusion of the $0g_{9/2}$ orbital. In contrast, for the GXPF1 family of
interactions this fragmentation is seen already with a model space restricted
to the $pf$-shell.

Figure 7(1) illustrates the evolution of the neutron $pf$ orbitals from nickel
to calcium isotopes. In Ca isotopes where $\pi 0f_{7/2}$ is unoccupied, the
vanishing $j_{>}-j_{<}$ proton-neutron coupling shifts the $\nu 0f_{5/2}$
orbital above the $\nu 1p_{1/2}$ orbital, leaving a gap at $N=32$ and forming
another gap at $N=34$ [62, 65]. Figure 7(2) presents a comparative analysis of
the spectroscopic strengths of neutron-hole states in the calcium and nickel
isotopes, populated from one neutron removal reaction. This analysis
establishes the trend in spectroscopic strength distribution and allows us to
examine the impact of the fully occupied or empty $\pi 0f_{7/2}$ shell on the
neutron $pf$-shell configuration. Its occupation has significant consequences
on the strength distribution of deeply bound single-particle states in the
$pf$ orbitals. For the $N$ = 27 isotones, the ground states of 47Ca and 55Ni
exhibit a similar pattern, featuring a nearly pure neutron-hole structure that
aligns with the independent particle shell picture. Already for $N$ = 29
isotones, differences between the calcium and nickel emerge: While the ground
state of 57Ni shows half the $C^{2}S$ compared to the ground state of 49Ca,
more strength associated with the $\nu 0f_{5/2}$ orbital is found for the
other low-lying states. For neutron numbers $N=$31 and larger, the calcium
isotopes keep the relatively large spacing between the four orbitals in the
neutron $pf$-shell, and the neutron-hole states maintain a nearly pure hole
nature with limited fragmentation. In contrast, the occupation of the $\pi
0f_{7/2}$ orbital in the nickel isotopes leads to level compression between
$1p_{1/2}$, $1p_{3/2}$, and $0f_{5/2}$ orbitals (<1 MeV), promoting cross-
shell excitations, and further strength fragmentation for the $pf$ orbitals.

##  5 Conclusion

In summary, unbound states of 53Ca and 55Ca were measured for the first time
using the quasi-free scattering reactions 54,56Ca($p,pn$), respectively,
providing intriguing results on the shell evolution in the Ca isotopes near
$N=34$. Resonance energies, exclusive cross sections, and momentum
distributions have been extracted, resulting in the observation of eight
unbound states in 53Ca and two unbound states in 55Ca. The 5516-keV state in
53Ca was found to have a momentum distribution of $l=3$ and was tentatively
assigned to have a spin-parity of $7/2^{-}$ with a $C^{2}S$ of 6.5(3),
indicating a nearly pure neutron-hole state. For 55Ca, the 6000-keV state was
tentatively assigned a spin-parity of $7/2^{-}$ with a $C^{2}S$ of 4.8-5.9(9),
while the 2970-keV state was tentatively assigned a spin-parity of $3/2^{-}$
with a $C^{2}S$ of 3.7(8). Our findings are supported by theoretical
predictions using the DWIA framework with calculations either utilizing the
GXPF1Bs Hamiltonian or the VS-IMSRG method employing the 1.8/2.0(EM) and
$\Delta$N2LO${}_{\text{GO}}$ interactions. Notably, calculations using the
GXPF1Bs interaction exhibited a remarkable consistency with the experimental
data, both in terms of resonance energies and $C^{2}S$ values. A shell gap of
approximately 3 MeV between the $0f_{7/2}$ and $1p_{3/2}$ orbitals is found in
neutron-rich calcium isotopes, which was deduced from the neutron-hole states
with high $C^{2}S$ values, indicating the robustness of the $N=28$ shell
closure, even with the deeply bound $\nu 0f_{7/2}$ orbital. The persistence of
the $0f_{7/2}$ single-particle strength up to 54Ca, along with its
fragmentation in 56Ca, underscores the absence of the possible $0f_{5/2}$ and
$0g_{9/2}$ components in the closed-shell nucleus of 54Ca, providing further
evidence for the $N=34$ magicity.

In view that the considerable reduction of $\nu 0f_{7/2}$ strength to the
neutron-hole state in 49Ca was observed in the neutron knockout from 50Ca on a
beryllium target at lower velocities, it is desirable to study the entire
calcium isotopic change with a solid and a liquid hydrogen target at the same
energies in order to further elucidate differences in the reaction
mechanism[26, 27, 28, 29, 30].

## Acknowledgments

We would like to express our gratitude to the RIKEN Nishina Center accelerator
staff for providing the stable and high-intensity primary beam and to the
BigRIPS team for operating the secondary beams. P.J.L. acknowledges the
support from China Postdoctoral Science Foundation(Grant No. YJ20210186). J.L.
acknowledges the support from Research Grants Council (RGC) of Hong Kong with
grant of Early Career Scheme (ECS-27303915). S.C. acknowledges the support of
the IPA program at RIKEN Nishina Center. F.B. acknowledges the support of the
Special Postdoctoral Researcher Program. The VS-IMSRG calculations were
performed with support from NSERC under grants SAPIN-2018-00027 and RGPAS-
2018-522453, the Arthur B. McDonald Canadian Astroparticle Physics Research
Institute, and The Digital Research Alliance of Canada. T.A. acknowledges
funding by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) - Project-ID 279384907 - SFB 1245. Y.T. acknowledges the support
from JSPS Grant-in-Aid for Scientific Research Grants No. JP21H01114. D.S. was
supported by the National Research, Development and Innovation Fund of Hungary
(NKFIH), financed by the project with contract no. TKP2021-NKTA-42 and under
the K18 funding scheme with project no. K128947.

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