# 78Ni revealed as a doubly magic stronghold against nuclear deformation

R. Taniuchi, C. Santamaria, P. Doornenbal, A. Obertelli, K. Yoneda, G. Authelet, H. Baba, D. Calvet, F. Château, A. Corsi, et al.

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R. Taniuchi, C. Santamaria, P. Doornenbal, A. Obertelli, K. Yoneda, et al., 78Ni revealed as a doubly magic stronghold against nuclear deformation. Nature, 2019, 569 (7754), pp.53-58. 10.1038/s41586-019-1155-x. hal-02123351

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# ⁷⁸Ni revealed as a doubly magic stronghold against nuclear deformation

R. Taniuchi<fnref n="1" />,², C. Santamaria²,³, P. Doornenbal²*, A. Obertelli²,³,⁴, K. Yoneda², G. Authelet³, H. Baba², D. Calvet³, F. Château³, A. Corsi³, A. Delbart³, J.-M. Gheller³, A. Gillibert³, J. D. Holt⁵, T. Isobe², V. Lapoux³, M. Matsushita⁶, J. Menéndez⁶, S. Momiyama<fnref n="1" />,², T. Motobayashi², M. Niikura<fnref n="1" />, F. Nowacki⁷, K. Ogata⁸,⁹, H. Otsu², T. Otsuka<fnref n="1" />,²,⁶, C. Péron³, S. Péru¹⁰, A. Peyaud³, E. C. Pollacco³, A. Poves¹¹, J.-Y. Roussé³, H. Sakurai<fnref n="1" />,², A. Schwenk⁴,¹²,¹³, Y. Shiga²,¹⁴, J. Simonis⁴,¹²,¹⁵, S. R. Stroberg⁵,¹⁶, S. Takeuchi², Y. Tsunoda⁶, T. Uesaka², H. Wang², F. Browne¹⁷, L. X. Chung¹⁸, Z. Dombradi¹⁹, S. Franchoo²⁰, F. Giacoppo²¹, A. Gottardo²⁰, K. Hadyńska-Klęk²¹, Z. Korkulu¹⁹, S. Koyama<fnref n="1" />,², Y. Kubota²,⁶, J. Lee²², M. Lettmann⁴, C. Louchart⁴, R. Lozeva⁷,<fnref n="23" />, K. Matsui<fnref n="1" />,², T. Miyazaki<fnref n="1" />,², S. Nishimura², L. Olivier²⁰, S. Ota⁶, Z. Patel²⁴, E. Şahin²¹, C. Shand²⁴, P.-A. Söderström², I. Stefan²⁰, D. Steppenbeck⁶, T. Sumikama²⁵, D. Suzuki²⁰, Z. Vajta¹⁹, V. Werner⁴, J. Wu²,²⁶ & Z. Y. Xu²²

Nuclear magic numbers correspond to fully occupied energy shells of protons or neutrons inside atomic nuclei. Doubly magic nuclei, with magic numbers for both protons and neutrons, are spherical and extremely rare across the nuclear landscape. Although the sequence of magic numbers is well established for stable nuclei, experimental evidence has revealed modifications for nuclei with a large asymmetry between proton and neutron numbers. Here we provide a spectroscopic study of the doubly magic nucleus ⁷⁸Ni, which contains fourteen neutrons more than the heaviest stable nickel isotope. We provide direct evidence of its doubly magic nature, which is also predicted by ab initio calculations based on chiral effective-field theory interactions and the quasi-particle random-phase approximation. Our results also indicate the breakdown of the neutron magic number 50 and proton magic number 28 beyond this stronghold, caused by a competing deformed structure. State-of-the-art phenomenological shell-model calculations reproduce this shape coexistence, predicting a rapid transition from spherical to deformed ground states, with ⁷⁸Ni as the turning point.

Characterization of the few doubly magic nuclei, known and predicted, provides a benchmark for our knowledge of the fundamental forces that drive the evolution of shell closures with proton-to-neutron asymmetry<fnref n="1" />,². With reliable and globally applicable interactions, accurate predictions of the location of the two-neutron drip line and the limits of the nuclear landscape can be made³. These, in turn, are critical for nucleosynthesis models, which rely on nuclear structure inputs.

An initial characterization of magic is often provided by the excitation energy of the first $J^II = 2^+$ (J, angular momentum; II, parity) state, $E(2^+_1)$, as illustrated in Fig. 1 in a Segrè chart (a two-dimensional grid in which nuclei are arranged by their proton (Z) and neutron (N) numbers). Magic nucleon numbers (2, 8, 20, 28, 50, 82 and 126), which were first correctly reproduced theoretically for stable isotopes by introducing a strong spin-orbit interaction⁴,⁵, stand out in the chart, as excitation from the ground state requires promoting nucleons across major nuclear shells, and therefore more energy, given the large energy gaps involved. Further experimental observables, such as nuclear binding energies, charge radii or reduced γ-ray-transition probabilities, are indispensable for a comprehensive characterization of a nucleus. However, their acquisition is experimentally challenging.

With the extension of studies to unstable, radioactive isotopes (RIs) with a large neutron excess—also termed ‘exotic’ nuclei—magic numbers emerged as a local feature. In such nuclei, nuclear shell structure

changes, sometimes drastically, as a function of proton (neutron) number along isotonic (isotopic) chains, revealing interesting properties of the underlying nuclear forces. For instance, it was recognized that several canonical neutron magic numbers⁶⁻⁹, such as $N = 8, 20$ and $28$, disappear far from stability, while new ones have been claimed at $N = 16$ (ref. ¹⁰) and $N = 32, 34$ (refs <fnref n="1" />,²,¹¹).

Shifts of these magic numbers challenge nuclear theory, and certain cases can be explained by drifts of single-particle orbitals (SPOs) with varying nucleon number¹². The central potential of the nucleon-nucleon effective interaction and the tensor force contribute strongly to this evolution¹³,¹⁴. Also three-nucleon forces, which originate from the composite nature of nucleons, have a substantial impact¹⁵,¹⁶. So far, a coherent picture of the nuclear shell structure and its evolution towards the most neutron-rich nuclei remains elusive.

The isotope ⁷⁸Ni (28 protons and 50 neutrons) is a unique case, included in all motivations for planned and operating next-generation RI-beam in-flight facilities, such as the Radioactive Isotope Beam Factory (RIBF) in Japan, the Facility for Rare Isotope Beams (FRIB) in the USA and the Facility for Antiproton and Ion Research (FAIR) in Germany. Predictions regarding the neutron drip-line location³ of even-even nuclei, for which the two-neutron separation energy becomes negative (also shown in Fig. 1), have revealed that ⁷⁸Ni is the only neutron-rich doubly magic nucleus presently lacking

<footnote n="1">¹Department of Physics, The University of Tokyo, Tokyo, Japan. ²RIKEN Nishina Center, Saitama, Japan. ³IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France. ⁴Institut für Kernphysik, Technische Universität Darmstadt, Darmstadt, Germany. ⁵TRIUMF, Vancouver, British Columbia, Canada. ⁶Center for Nuclear Study, The University of Tokyo, Saitama, Japan. ⁷IPHC, CNRS/IN2P3, Université de Strasbourg, Strasbourg, France. ⁸Research Center for Nuclear Physics, Osaka University, Ibaraki, Japan. ⁹Department of Physics, Osaka City University, Osaka, Japan. ¹⁰CEA, DAM, DIF, Arpajon, France. ¹¹Departamento de Física Teórica and IFT-UAM/CSIC, Universidad Autónoma de Madrid, Madrid, Spain. ¹²ExtreMe Matter Institute (EMMI), GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt, Germany. ¹³Max-Planck-Institut für Kernphysik, Heidelberg, Germany. ¹⁴Department of Physics, Rikkyo University, Tokyo, Japan. ¹⁵Institut für Kernphysik and PRISMA Cluster of Excellence, Johannes Gutenberg-Universität Mainz, Mainz, Germany. ¹⁶Physics Department, Reed College, Portland, OR, USA. ¹⁷School of Computing, Engineering and Mathematics, University of Brighton, Brighton, UK. ¹⁸Institute for Nuclear Science & Technology, VINATOM, Hanoi, Vietnam. ¹⁹MTA Atomki, Debrecen, Hungary. ²⁰Institut de Physique Nucléaire, CNRS/IN2P3, Université Paris-Saclay, Orsay, France. ²¹Department of Physics, University of Oslo, Oslo, Norway. ²²Department of Physics, The University of Hong Kong, Hong Kong, Hong Kong.</footnote>

<footnote n="23">²³CSNSM, CNRS/IN2P3, Université Paris-Saclay, Orsay, France. ²⁴Department of Physics, University of Surrey, Guildford, UK. ²⁵Department of Physics, Tohoku University, Sendai, Japan. ²⁶State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing, China. *e-mail: pieter@ribf.riken.jp</footnote>
<figure><img src="image_2.png" /><figcaption>**Fig. 1** | Experimental $E(2_1^+)$ systematics of the even–even nuclear landscape. Shown are known $E(2_1^+)$ of even–even isotopes$^{40}$ and the value for $^{78}$Ni obtained in the present study. Canonical magic numbers are indicated by dashed lines and doubly magic nuclei are labelled. $^{68}$Ni, for</figcaption></figure>

which the number of neutrons, *N* = 40, matches the harmonic-oscillator shell closure, is also marked. The predicted two-neutron drip line and its uncertainties$^3$ are shown in blue.

spectroscopic information on excited states that can be reached with current and next-generation facilities.

After the first production$^{17}$ of $^{78}$Ni, enormous efforts have been made to investigate its structure. Previous measurements indirectly inferred persistent *N* = 50 (refs 18–22) and *Z* = 28 (refs 23–25) shell closures at $^{78}$Ni. This notion has been supported theoretically by ab initio predictions$^{26}$. Conversely, studies of $^{66}$Cr (*Z* = 24) and $^{70,72}$Fe (*Z* = 26) revealed constantly low $2_1^+$ excitation energies, which question the *N* = 50 shell closure for proton numbers below *Z* = 28 (refs 27,28). Likewise, several studies supported a reduction of the *Z* = 28 proton shell gap towards and beyond *N* = 50 (refs 29–33). A vanishing of the proton and neutron shell closures would be accompanied by an onset of deformation, implying considerable consequences, such as shape coexistence and gain in nuclear binding energy. The former signifies the occurrence of several quantum states of different shapes lying close and low in energy; the latter slants the two-neutron drip line and accordingly shifts the limits of nuclear existence towards heavier isotopes. Hitherto, no ultimate conclusion on the magic character of $^{78}$Ni has existed. Here, we provide first direct evidence from in-beam γ-ray spectroscopy in prompt coincidence with proton knockout reactions of fast RI beams.

## Production of RI beams

The experiment was carried out at RIBF, which combines three injectors with four coupled cyclotrons. Neutron-rich RI beams were produced by induced relativistic in-flight fission of a $^{238}$U primary beam with energy of 345 MeV per nucleon on a 3-mm-thick beryllium production target, located at the F0 focus of the BigRIPS fragment separator$^{34}$ shown in Fig. 2a. $^{79}$Cu and $^{80}$Zn particles, produced at rates of 5 and 290 particles per second, respectively, were identified on an event-by-event basis from focal plane F3 to F7, before being guided to the MINOS reaction target system$^{35}$ (see Fig. 2b) located at F8, with a remaining energy of approximately 250 MeV per nucleon (61% of the speed of light).

## γ-ray detection after secondary reaction

MINOS was composed of a 102(1)-mm-thick (all uncertainties correspond to one standard deviation, s.d.) liquid-hydrogen target and a time-projection chamber to reconstruct the reaction vertices. This allowed to overcome inaccuracies in the Doppler reconstruction that originated from the thick target (see Methods for details). The γ-ray spectrometer DALI2$^{36}$ surrounded MINOS to detect prompt de-excitation γ-rays with high efficiency. Secondary-reaction species were subsequently identified with the ZeroDegree spectrometer from F8 to F11. An overview of the facility and the experimental setup, including all the focal points, is provided in Fig. 2 together with obtained particle identification plots.

## γ-rays from the $^{79}$Cu(*p*, 2*p*) $^{78}$Ni reaction

The total reaction cross-sections for the production of $^{78}$Ni following the $^{79}$Cu(*p*, 2*p*)$^{78}$Ni and $^{80}$Zn(*p*, 3*p*)$^{78}$Ni reactions were 1.70(42) and 0.016(6) mb, respectively, yielding 937 and 815 events. The energies of coincident prompt γ-rays were corrected for the Doppler shift in the spectra shown in Fig. 3. For the (*p*, 2*p*) reaction channel, the most intense γ-ray transition was observed at 2,600(33) keV with a significance level (S.L.) of 7.5, and was tentatively assigned to the $2_1^+ \rightarrow 0_{gs}^+$ decay of $^{78}$Ni. Four weaker transitions, located at 583(10) keV, 1,103(14) keV, 1,540(25) keV and 2,110(48) keV, were identified. All decay strengths (corrected for the γ-ray-energy-dependent detection efficiency of DALI2) and confidence levels are summarized in Extended Data Table 1 and Extended Data Fig. 2. Sufficient statistics allowed us to establish prompt coincidences between the 2,600-keV transition and all the weaker transitions (Fig. 3b), as well as a possible coincidence between the 583-keV transition and the 2,110-keV transition (Extended Data Fig. 1b). Conversely, no coincidence was observed between the 583-keV and 1,103-keV transitions (Extended Data Fig. 1b, c). These are therefore considered to decay independently into the proposed $2_1^+$ state. On account of its correspondence to the theoretical descriptions discussed below, the 583-keV line is tentatively assigned to the $4_1^+ \rightarrow 2_1^+$ transition. This leads to a ratio of $R_{4/2} = 1.22(2)$ between $E(2_1^+)$ and $E(4_1^+)$, which is comparable to those of the well known doubly magic nuclei $^{40}$Ca (1.35), $^{48}$Ca (1.18), $^{56}$Ni (1.45), $^{132}$Sn (1.09) and $^{208}$Pb (1.06).

## γ-rays from $^{80}$Zn(*p*, 3*p*) $^{78}$Ni

Although similar numbers of (*p*, 2*p*) and (*p*, 3*p*) events were detected, the findings were largely different. A transition remained visible at 1,067(17) keV in the (*p*, 3*p*) spectrum (Fig. 3d), but no other prominent peak was observed in the energy range up to 2,600 keV, with transitions reported for the (*p*, 2*p*) channel having an S.L. of only around 1. A surprising additional strength above the $2_1^+ \rightarrow 0_{gs}^+$ decay revealed a transition at 2,910(43) keV (S.L. = 3.5), which is either weakly or not populated in the (*p*, 2*p*) channel. This cannot be interpreted as a decay into the $2_1^+$ state owing to the low intensity of the peak at 2,600 keV. Instead, it is ascribed to the decay of a second $2_1^+$ state to the $0_+$ ground state. This level placement is further corroborated by the spectrum for γ-ray-detection multiplicities of $M_γ < 4$ (Fig. 3c), which shows an enhanced peak-to-total ratio of decays from low-lying levels—in this case the 2,910-keV transition. By applying similar arguments, the 1,067-keV transition has too large intensity to correspond to the 1,103-keV transition observed in the (*p*, 2*p*) channel, and is therefore considered to feed the 2,910-keV level. Taking all these observations into account, the level scheme shown in Fig. 4b is proposed for $^{78}$Ni.
<figure><img src="image_3.png" /></figure>

**Fig. 2** | Layout of the experimental equipment and particle identification plots of isotopes. **a**, Schematic of the two final cyclotron stages (intermediate-stage ring cyclotron, IRC, and superconducting ring cyclotron, SRC) of the RIBF facility, along with the BigRIPS and ZeroDegree fragment separators. Sets of dipole and superconducting quadrupole triplet magnets are shown in blue and grey, respectively. Reaction residues cover a flight length of 126 m between creation at the F0 production target and the final focal point, F11. Schematic, courtesy of N. Miyauchi. **b**, Layout of MINOS (grey) and the surrounding DALI2 γ-ray spectrometer (blue) located at the F8 focal plane. Incoming secondary $^{80}$Zn and $^{79}$Cu beams (blue arrow) react and $^{78}$Ni exits the target (red arrow). **c**, Components of the radioactive beam accepted by the BigRIPS fragment separator. Events corresponding to $^{79}$Cu (red ellipse) and $^{80}$Zn (dashed red ellipse) are indicated. **d**, Reaction products accepted by ZeroDegree. $^{78}$Ni is indicated by the red ellipse. **e**, **f** and **g** share the colour scale in **d**, which indicates the number of events for different isotopes transmitted through BigRIPS and ZeroDegree.

## Doubly magic $^{78}$Ni

The energies $E(2_1^+)$ along the chain of nickel isotopes (Fig. 4a) exhibit a local maximum at $^{68}$Ni, which is attributed to the $N=40$ harmonic-oscillator shell closure. This assessment is reinforced by the isotope's low quadrupole collectivity$^{37}$; however, mass measurements identified this as a very localized feature$^{38}$, and ref. 39 pointed out that quadrupole collectivity is dominated by neutron excitations. Beyond the harmonic-oscillator shell closure, in the independent-particle model, neutrons fill the $\nu 0g_{9/2}$ SPO ($\nu$ and $\pi$ denote neutron and proton orbitals, respectively) with little impact on $E(2_1^+)$, until a steep rise is observed for our value at $N=50$. In fact, the measured excitation energy of 2.6 MeV at that neutron number is essentially as high as the excitation energy of 2.7 MeV reported for the doubly magic $^{56}$Ni ($Z=N=28$)$^{40}$—thus providing first direct experimental evidence for a comparable magic character. To gain further insight into the structural evolution of neutron-rich nickel isotopes, their $E(2_1^+)$ values are compared with state-of-the-art theoretical calculations in Fig. 4a. Large-scale shell model (LSSM) calculations provided the two shell-model Hamiltonians outlined in refs. 28,41, with a transition at $N=44$ from the LNPS to the PFSDG-U Hamiltonian. In particular, the LSSM calculations for $^{78}$Ni included the full proton *pf* and neutron *sdg* harmonic-oscillator shells into their model space, which is crucial to pick up emerging quadrupole collectivity from quasi-degenerated neutron SPOs$^{28}$.

The Monte Carlo shell model (MCSM)$^{42}$ allows us to incorporate more SPOs into the calculation. For the MCSM predictions presented here, the A3DA-m Hamiltonian$^{43}$ was employed for $^{64-76}$Ni, which

<figure><img src="image_4.png" /></figure>

**Fig. 3** | Doppler-corrected γ-ray energy spectra. **a**, De-excitation γ-rays measured in coincidence with $^{78}$Ni following (*p*, 2*p*) reactions. **b**, Prompt coincidences with the 2,600-keV transition following (*p*, 2*p*) reactions. The spectrum reveals coincidences between the 2,600-keV transition and other low-lying transitions. The coincidence range is illustrated by the hatched area. **c**, De-excitation γ-rays measured in coincidence with $^{78}$Ni following (*p*, 3*p*) reactions. To reduce events with multiple γ-ray hits from Compton scattering, the γ-ray detection multiplicity, $M_\gamma$, was restricted to values below 4. As a consequence, the visibility of the 2,910-keV transition is enhanced. **d**, Same as **c**, but for $M_\gamma < 6$. Observed transitions are indicated by their energies in **a** and **c**. Simulated response curves of the γ-ray detector for the individual transitions are illustrated by the light-blue solid (S.L. $\ge$ 3) and dotted (S.L. < 3) lines, whereas the fitted double-exponential background is shown by the dark-blue dashed line. The magenta solid lines show the sum of the background and individual transitions. Vertical error bars correspond to 1 s.d.

encompasses the full *pf* shell as well as the $0g_{9/2}$ and $1d_{5/2}$ SPOs for protons and neutrons. To enable more detailed calculations for $^{78}$Ni, the A3DA-m Hamiltonian was extended to the full proton and neutron *pf* and *sdg* shells. The mean-field-based quasi-particle random-phase approximation (QRPA)$^{44}$ calculations used the Gogny D1M effective force$^{45}$. Finally, we present new ab initio results based on the valence-space formulation$^{46}$ of the in-medium similarity renormalization group (IM-SRG)$^{47}$ and the coupled-cluster (CC) method$^{26}$, both of which use two- and three-nucleon interactions derived from chiral effective-field theory$^{48}$. All theoretical calculations describe well the pattern of $E(2_1^+)$ along the chain of nickel isotopes in Fig. 3a, notably the large enhancement at $N=50$, thus confirming that $^{78}$Ni is a doubly magic nucleus.

## Two different shapes emerge

Complete predictions for the low-lying-level structure of $^{78}$Ni are presented in Fig. 4b next to the proposed experimental level scheme. The LSSM and MCSM calculations similarly predict competing spherical (quadrupole deformation parameter, $\beta \approx 0$) and prolate deformed ($\beta \approx 0.3$) intruder configurations, with one discrepancy: whereas LSSM
<figure><img src="image_5.png" /><figcaption>**Fig. 4** | Comparison of theoretical predictions with experimental data.
a, Experimental $E(2_1^+)$ values$^{40}$ for even–even nickel isotopes are compared with calculations using phenomenological shell-model interactions (LSSM$^{28,41}$ and MCSM$^{43}$), the beyond-mean-field approach (QRPA), the ab initio approach (IM-SRG and CC$^{26}$), as a function of neutron number, N. The present result for $^{78}$Ni is indicated by the red star. b, Experimental level scheme compared with detailed theoretical calculations for $^{78}$Ni.</figcaption></figure>

puts the deformed $2^+$ state at 2.88 MeV, and thus below the spherical $2^+$ state at 3.15 MeV, for MCSM the deformed $2^+$ state at 2.89 MeV lies above the spherical $2^+$ state at 2.57 MeV. The respective $4^+$ states are located approximately 0.5–0.7 MeV above the $2^+$ states, hence justifying the tentative spin assignment for the experimental level at 3.18 MeV. In addition, the calculated deformed $0^+$ states are located approximately 0.25 MeV below their respective deformed $2^+$ states. A possible unobserved transition from the deformed $2^+$ state to the deformed $0^+$ state is expected to be several orders of magnitude weaker than direct decays to the ground state owing to the large energy difference of the latter. It is further noted that restricting the MCSM calculations to the A3DA-m Hamiltonian$^{43}$ puts the first $2^+$ state at 2.89 MeV and the second $2^+$ state at 4.72 MeV, strikingly demonstrating the necessity for the inclusion of the full neutron sdg shell to properly characterize low-lying deformed configurations$^{28}$.

It is important to stress the structural differences between the spherical and deformed configurations, specifically the average number of particle-hole excitations above $Z = 28$ ($n_{p-h}^{\pi}$) and $N = 50$ ($n_{p-h}^{\nu}$) for the $0^+$, $2^+$ and $4^+$ states. With MCSM, these numbers are $0.4 \lesssim n_{p-h}^{\pi} \lesssim 0.9$ and $0.7 \lesssim n_{p-h}^{\nu} \lesssim 1.7$ for the three spherical states, whereas for the deformed states the respective values are $n_{p-h}^{\pi} \approx 2.5$ and $n_{p-h}^{\nu} \approx 2.7$. Similar values are obtained with LSSM. Recollecting the different levels populated from the (*p*, 2*p*) and (*p*, 3*p*) reactions, the calculated spectroscopic factors, which quantify the overlap between final and initial state in single-nucleon knockout reactions, can help to unravel the nature of levels populated. Inspection of these spectroscopic factors from $^{79}$Cu to final states in $^{78}$Ni with the LSSM and MCSM Hamiltonians shows that spherical configurations are strongly favoured. In Fig. 5 experimental partial cross-sections to the individual levels are compared with calculated ones obtained within the distorted-wave impulse approximation (DWIA) formalism$^{49}$ folded with the shell-model spectroscopic factors (see Methods). Although the assumptions of the DWIA and shell-model spectroscopic-factor calculations are not fully consistent, their combination provides a qualitative picture that can be compared with experimental results. We note that for the removal of a single proton from the $\pi0f_{7/2}$ or $\pi0f_{5/2}$ SPOs, the calculated cross-sections, which are weakly dependent on the projectile and excitation energy, are about 1 mb. The bulk of spectroscopic strength, which originates from the removal of a $\pi0f_{7/2}$ proton, is calculated at energies of 4–5 MeV, with an abrupt increase that is not observed experimentally. However, this energy is close or beyond the evaluated

Transitions with S.L. $\ge 5$ are represented by solid arrows and dotted arrows correspond to S.L. $< 5$. The 1.54-MeV transition is not placed, whereas the 2.11-MeV transition has S.L. $< 3$. The evaluated neutron separation energy, $S_n$, and its error$^{50}$ (1 s.d.) are indicated by the orange dashed line and shading, respectively. Predicted deformed states are indicated by dashed lines. For convenience, theoretical predictions show only the two lowest $0^+$, $2^+$ and $4^+$ spin-parity values.

neutron separation energy$^{50}$, $S_n = 5.16(78)$ MeV. Owing to the large uncertainty of $S_n$, quantitative comparisons between the experimental and theoretical total reaction cross-sections were not feasible, but we note that LSSM places the average of the distribution lower than MCSM. Good agreement between experiment and theory is observed for the ground state, which corresponds predominantly to the removal of a $\pi0f_{5/2}$ proton, and the strikingly low direct population of the observed low-lying levels, which is illustrated by the similar integrated cross-sections in Fig. 5. So far, no theoretical framework can predict microscopic (*p*, 3*p*) cross-sections. It must be stressed, however, that the calculated two-nucleon overlaps between the ground state in $^{80}$Zn and the excited states in $^{78}$Ni also favour the population of spherical final states.

Nickel isotopes represent the neutron-rich frontier for ab initio calculations. For the IM-SRG results in Fig. 4b (for details see Methods), we calculate $E(2_1^+) = 3.35$ MeV and $E(4_1^+) = 3.75$ MeV for $^{78}$Ni using a proton *pf* and a neutron *sdg* SPO valence-space Hamiltonian. This value of $E(2_1^+)$ is several hundred kiloelectronvolts higher than the present measurement. Nevertheless, it is in good agreement with experimental trends across the chain and also predicts a sharp decrease in $E(2_1^+)$ for $^{80}$Ni. The average numbers of proton and neutron excitations for the $2_1^+$ state are $n_{p-h}^{\pi} = 0.9$ and $n_{p-h}^{\nu} = 1.3$, analogous to the LSSM and MCSM results, and the partial cross-sections (Fig. 5d) follow a similar trend. However, a stark difference is found for the second $2^+$ state, which lies at 5.81 MeV and is spherical. This deficiency is not unexpected, because ab initio methods are often built on truncations in allowed particle-hole excitations and thus fail to capture very collective features sufficiently. In fact, the $E(2_1^+)$ value for $^{78}$Ni only varies by about 100 keV when using several two- and three-nucleon interactions, so missing particle-hole excitations are probably the main uncertainty in the IM-SRG calculations. In particular, our IM-SRG results agree with the ab initio CC predictions of ref. 26 at the level of single and double correlations when using the same Hamiltonian. When triple correlations are further included, the $E(2_1^+)$ value for $^{78}$Ni calculated with CC is found to be in good agreement with the present measurement.

In the case of QRPA calculations, 65% of the $2_1^+$ wavefunction of $^{78}$Ni is composed of neutron excitations from the $\nu0g_{9/2}$ to the $\nu1d_{5/2}$ orbital across the $N = 50$ shell gap, whereas 28% are proton excitations from the $\pi0f_{7/2}$ to the $\pi0f_{5/2}$ orbitals. In this approach, the neutron shell gap at $N = 50$ is robust enough so that protons across the $Z = 28$ shell gap
<figure><img src="image_6.png" /><figcaption>**Fig. 5** | **Experimental and calculated partial cross-sections for the $^{79}\text{Cu}(p, 2p)^{78}\text{Ni}$ reaction.** a, Measured partial cross-sections (magenta vertical bars) and the associated errors (1 s.d.). The evaluated neutron separation energy, $S_n$, and its error (1 s.d.) are indicated. b, Partial cross-sections predicted by LSSM. c, MCSM prediction. d, IM-SRG prediction. In b–d, contributions of proton knockout from the $\pi0f_{5/2}$ and $\pi0f_{7/2}$ SPOs</figcaption></figure>

contribute considerably to the excitation. Similarly, in the neighbouring even–even isotopes $^{76}\text{Ni}$ and $^{80}\text{Ni}$, the $2^1\text{S}$ excitation stems from neutron excitations in the $\nu0g_{9/2}$ (80% of the wavefunction) and $\nu1d_{5/2}$ (72% of the wavefunction with an occupancy of 0.3 neutrons) SPOs, respectively.

In conclusion, we have provided the first direct experimental evidence for the preservation of the Z = 28 and N = 50 shell closures in $^{78}\text{Ni}$ by employing a dedicated setup to study very neutron-rich nuclei via in-beam γ-ray spectroscopy. A low-lying, second $2^1\text{S}$ state conforms with the notion of competing spherical and deformed configurations, which is to be verified by future measurements²⁸ on $^{76}\text{Fe}$ and $^{74}\text{Cr}$. Similarly, we predict a breakdown of the proton Z = 28 shell closure that favours prolate deformed ground states for heavier nickel isotopes, which establishes $^{78}\text{Ni}$ as a doubly magic stronghold before deformation prevails in more exotic nuclei. Understanding these structures is crucial for r-process nucleosynthesis, as the onset of deformation may shift the drip line towards heavier isotopes. Driving mechanisms for this evolution from spherical to deformed nuclei are yet to be fully understood at the ab initio level, which motivates future mass measurements to validate the predicted collapse of canonical shell closures.

are indicated by red and blue vertical bars, respectively. Integrated cross-sections and errors (1 s.d.) are shown as the running total functions of the excitation energy for the experimental and the theoretical predictions. Contributions from the $\pi1p_{3/2}$ SPO amount to less than 0.05 mb and have been omitted. Circles along the integrated cross-sections correspond to observed or calculated levels.

## Online content

Any methods, additional references, Nature Research reporting summaries, source data, statements of data availability and associated accession codes are available at https://doi.org/10.1038/s41586-019-1155-x.

Received: 16 November 2018; Accepted: 15 March 2019; Published online 1 May 2019.

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**Acknowledgements** We thank the staff of the RIKEN Nishina Center accelerator complex for providing a stable and high-intensity uranium beam and the BigRIPS team for the smooth operation of the secondary beams. We also thank N. Miyauchi for 3D schematic of the RIBF facility shown in Fig. 2a. The development of MINOS and the core MINOS team have been supported by the European Research Council through ERC grant number MINOS-258567. R.T. was supported by JSPS Grant-in-Aid for JSPS Research Fellows JP14J08717. A.O. was supported by JSPS long-term fellowship L-13520 at the RIKEN Nishina Center. C. Santamaria was supported by the IPA programme at the RIKEN Nishina Center. J.D.H. acknowledges support by the National Research Council of Canada and NSERC. This work was supported in part by the ERC through grant number 307986 STRONGINT, the DFG under grant SFB 1245 and the BMBF under contract number 05P18RDFN1. The MCSM calculations were performed on the K computer at RIKEN AICS (hp160211, hp170230, hp180179). J.M., T.O. and Y.T. acknowledge support from MEXT as 'Priority Issue on post-K computer' (Elucidation of the Fundamental Laws and Evolution of the Universe) and JICFUS. J.M. and K.O. were supported by Grant-in-Aid for Scientific Research JP18K03639 (J.M.) and JP16K05352 (K.O.). A. Poves acknowledges support by Mineco (Spain) grants FPA2014-57916 and Severo Ochoa Program SEV-2016-0597. This work was supported in part by the DFG through the Cluster of Excellence PRISMA. L.X.C. was supported by Vietnam MOST through the Physics Development Program grant TLCN.25/18. Z.D., Z.K. and Z.V. acknowledge support from the GINOP-2.3.3-15-2016-00034 project. M.L. C.L and V.V. acknowledge support from the German BMBF through grants 05P15RDNF1 and 05P12RDNF8.

**Author contributions** R.T. performed offline data analyses and GEANT4 simulations and prepared the figures; P.D. and A.O. designed the experiment; R.T., C. Santamaria, P.D., A.O., J.D.H., J.M., F.N., K.O., T.O., A.S. and Y.T. wrote the manuscript; R.T., C. Santamaria, P.D., A.O., G.A., D.C., F.C., A.C., A.D., J.-M.G., A.G., V.L., M.M., S.M., M.N., C.P., A. Peyaud, E.C.P., J.-Y.R., Y.S., S.T. and H.W. were responsible for setting up the liquid-hydrogen target, the vertex reconstruction system, MINOS and the γ-ray detector array, DALI2; R.T., C. Santamaria, H.B., D.C., A.C. and T.I. were responsible for the data acquisition system and analysis software; R.T., C. Santamaria, P.D., A.O., G.A., H.B., D.C., F.C., A.C., A.D., J.-M.G., A.G., T.I., V.L., M.M., S.M., M.N., H.O., C.P., A. Peyaud, E.C.P., J.-Y.R., Y.S., S.T., H.W., F.B., L.X.C., Z.D., S.F., F.G., A.G., K.H., Z.K., S.K., J.L., M.L., C.L., R.L., K.N., T. Miyazaki, S.N., L.O., S.O., Z.P., E.S., C. Shand, P.-A.S., I.S., D. Steppenbeck, T.S., D. Suzuki, Z.V., V.W., J.W. and Z.Y.X. checked the data accumulation online and maintained operation of the experiment; P.D., A.O., K.Y., T. Motobayashi, H.S. and T.U. supervised the participants; F.N. and A. Poves performed the LSSM calculations; T.O. and Y.T. performed the MCSM calculations; S.P. performed the QRPA calculations; J.D.H., J.M., A.S., J.S. and S.R.S. performed the IM-SRG calculations; K.O. performed the DWIA calculations. All authors discussed the results and commented on the manuscript.

**Competing interests** The authors declare no competing interests.

**Additional information**

*Extended data* is available for this paper at https://doi.org/10.1038/s41586-019-1155-x.

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© The Author(s), under exclusive licence to Springer Nature Limited 2019
METHODS

**Beam production.** The experiment was performed at RIBF in Japan, which is operated by the RIKEN Nishina Center and the Center for Nuclear Study of the University of Tokyo. A $^{238}$U primary beam with an intensity of $7.5 \times 10^{10}$ particles per second was accelerated by four coupled cyclotrons$^{51}$ to an energy of 345 MeV per nucleon and impinged on a 3-mm-thick beryllium production target located at the F0 focus of the BigRIPS two-stage fragment separator$^{34}$ shown in Fig. 2a. The isotopes of interest were first purified with BigRIPS from focal plane F0 to F2 according to their magnetic rigidity and energy loss. Particles then traversed BigRIPS from F3 to F7 at approximately 60% of the speed of light and were identified by time-of-flight, magnetic-rigidity and energy-loss measurements on an event-by-event basis$^{52,53}$. Secondary beams with rates of 5 particles per second for $^{79}$Cu and 290 particles per second for $^{80}$Zn were directed onto the MINOS reaction target system.

**Secondary reactions.** MINOS$^{35}$ consisted of a tube of 102(1) mm length and 38 mm inner diameter filled with liquid hydrogen, coupled with a time-projection chamber (TPC). Secondary-reaction products were selected and identified with the ZeroDegree spectrometer, following the same approach as for BigRIPS. Emitted prompt γ-rays were detected with DALI2$^{36}$, which consisted of 186 NaI(Tl) scintillators arranged in a geometry that covered polar angles from 12° to 96° around the MINOS system and achieved a photo-peak efficiency of 26% at 1 MeV. Figure 2b shows schematics of DALI2 and MINOS. Around 30% energy loss inside the secondary target resulted in a velocity highly dependent on the reaction vertex position. Therefore, an annular TPC surrounded the cryogenic target to determine the vertex position following (p, 2p) and (p, 3p) knockout reactions.

**Data acquisition.** The general data acquisition system at the RIBF facility$^{53}$ was coupled with the dedicated MINOS electronics system$^{54,55}$, which was developed to read out the roughly 5,000 channels comprising the pad plane of the TPC.

**Vertex tracking and γ-ray energy reconstruction.** Observed γ-ray energies in the laboratory system were corrected for their Doppler shift, as described in ref. 56. For this reconstruction, the vertex position was determined using the TPC, by tracking the protons in the (p, 2p) and (p, 3p) knockout reactions of interest via an iterative tracking algorithm using a Hough transform method, as detailed in ref. 57. This process allowed a vertex resolution of less than 5 mm full-width at half-maximum. The velocity at the incident point was determined assuming energy loss inside the liquid-hydrogen target as calculated with the ATIMA code (https://web-docs.gsi.de/~weick/atima/).

Compton scattering of γ-rays resulted in partial energy deposition between neighbouring crystals of DALI2. When γ-rays were observed in detectors within a 15-cm distance from each other, the energy was summed and the detector with the highest energy deposition determined the position for the Doppler reconstruction.

The γ-ray detector response functions were simulated with the GEANT4 framework$^{58}$ for fitting and comparison with the experimental data. Stationary sources of $^{60}$Co, $^{88}$Y and $^{137}$Cs were used for γ-ray energy calibration of the array.

**Energy reconstruction of peaks.** Five and six peak candidates were investigated in the (p, 2p) and (p, 3p) channels, respectively. The energy values and their corresponding errors and intensities were determined by maximizing the likelihood values of the fits of 10-keV-binned spectra with all combinations of transition energies, assuming a multivariate Gaussian probability density function. The S.L. for the existence of each peak was deduced from the probability of the likelihood ratio between the null hypothesis and the alternative hypothesis. Obtained results are summarized in Extended Data Table 1 and Extended Data Fig. 2.

**LSSM calculation.** Following ref. $^{28}$, LSSM calculations were carried out in a valence space comprising the *pf* shell orbitals for protons and the *sdg* shell orbitals for neutrons, with an effective interaction based on realistic matrix elements, for which the monopole part was adjusted to reproduce the experimental evolution of the regulating gaps in the model space. Besides the comparison shown in Fig. 4, the spectroscopic qualities of this interaction have been assessed in recent studies$^{25,59-62}$. To generate the theoretical spectroscopic strength of one-proton removal from $^{79}$Cu to final states in $^{78}$Ni, the Lanczos strength-function method$^{63,64}$ was applied to the LSSM calculations. In practice, the removal operator $\tilde{a}_j$ was applied on the initial $^{79}$Cu ground state to generate the sum-rule state $\Sigma = \tilde{a}_j \left| \begin{array}{c} 5/2 \\ 3/2 \end{array} \right>$.

This non-physical state carried the entire one-proton-removal spectroscopic strength and was used as the initial state for a Lanczos diagonalization procedure. At the end of the procedure, the matrix elements were simply the components of $\Sigma$ in the Lanczos eigenstates basis. Even if the diagonalization was not fully complete, this procedure converged rapidly to an approximate strength distribution. To handle more easily the large dimension of the involved basis without losing any physics insight, the proton and neutron gaps were slightly reduced to project the results of ref. $^{28}$ into a smaller valence space containing all excitations across $Z = 28$ and $N = 50$ with up to seven particle-hole excitations. We checked that the orbital occupancies of $^{78}$Ni low-lying states and $^{79}$Cu ground-state wavefunctions remained identical to the ones obtained in the configuration space of ref. $^{28}$.

**MCSM calculation.** The MCSM is one of the most advanced computational methods for nuclear many-body systems. This work presents MCSM calculations performed on the K computer at the RIKEN Advanced Institute for Computational Science (AICS), Japan. By exploiting the advantages of quantum Monte Carlo variational and matrix-diagonalization methods, this approach circumvents the diagonalization of a $> 5 \times 10^{20}$-dimensional Hamiltonian matrix. Using the doubly magic nucleus $^{40}$Ca as an inert core, 8 protons and up to 32 neutrons were left to actively interact in a much larger model space than that used in conventional configuration interaction calculations. The A3DA-m interaction$^{43}$ in the model space of six SPOs (pf shell, $0g_{9/2}$ and $1d_{5/2}$ orbitals) for both protons and neutrons was used for $^{64-76}$Ni. This interaction was extended to a model space of nine SPOs (pf and sdg shells) for $^{78,80}$Ni, where upper SPOs in the sdg shell become more important. The MCSM eigenstate is represented as a superposition of MCSM basis vectors with the appropriate projection onto spin and parity. Each MCSM basis vector is a Slater determinant formed by mixed single-particle states, where the mixing amplitudes are optimized by quantum Monte Carlo and variational methods. This Slater determinant has intrinsic quadrupole moments, which can be expressed in terms of a set of $\beta_2$ and $\gamma$ deformation parameters. This property can be used to study the intrinsic shape of the calculated state.

The MCSM calculations with the A3DA-m interaction reveal properties of Ni and neighbouring isotopes. The shape evolution of nickel isotopes including shape coexistence has been described theoretically$^{43}$ and the calculations have been compared with experiments for $^{66}$Ni (ref. $^{65}$), $^{68}$Ni (refs $^{66,67}$) and $^{70}$Ni (refs $^{68,69}$). Nuclei near $N = 50$ are described well by calculations for $^{77}$Cu (ref. $^{23}$) and $^{79}$Cu (ref. $^{24}$). **IM-SRG calculations.** The IM-SRG$^{47,70}$ transforms the many-body Hamiltonian, $\hat{H}$, to a diagonal or block-diagonal form, $\hat{H} = U\hat{H}U^{-1}$, via a unitary transformation, $U = e^{\Omega}$, where the anti-Hermitian generator $\Omega$ builds in the off-diagonal correlations from the original $\hat{H}$ (ref. $^{71}$). The IM-SRG starts from a single-reference ground-state configuration, $|\Phi_0\rangle$ (such as Hartree-Fock), at the flow parameter $s = 0$ and connects this state to the fully correlated ground state, $|\Psi_0\rangle$, as $s \to \infty$, where the unitary transformation is also built up as a continuous sequence of unitary transformations, $U(s)$. With no truncations in the IM-SRG flow equations, this gives the exact ground-state energy, but in the IM-SRG(2) approximation used here, all operators are truncated at the two-body level. In the valence-space IM-SRG formulation$^{46,72-74}$, the unitary transformation is constructed to additionally decouple the valence space, leading to a valence-space Hamiltonian, which is subsequently diagonalized within the valence space using large-scale shell-model methods. In this work, this encompasses a $^{60}$Ca core and a Hamiltonian composed of the proton *pf* and neutron *sdg* SPOs.

Here, we use a particular set of two- and three-nucleon interactions (labelled 1.8/2.0 (EM)$^{75}$), which was fitted to few-body systems but predicts ground-state energies very well up to $^{100}$Sn (ref. $^{76}$). The calculations adequately reproduce the $E(2_1^+)$ trend from $^{70}$Ni to the steep rise in excitation energy for $^{78}$Ni. In addition, the 1.8/2.0 (EM) interaction within the IM-SRG framework systematically predicts experimental binding energies and $E(2_1^+)$ values of lighter nuclei up to the nickel isotopes, including high $E(2_1^+)$ values at known shell closures$^{77}$.

The shell-model diagonalization of nickel isotopes up to $^{72}$Ni is exact in the valence space. Because of the large dimensions of the Hamiltonian matrix, we calculate$^{74,78}$ and $^{78}$Ni limiting the number of particle-hole excitations across the proton $f_{7/2}$ orbital ($Z = 28$) and the neutron $g_{9/2}$ orbital to $7p-7h$. Similarly, for $^{76}$Ni and $^{80}$Ni up to $6p-6h$ and $5p-5h$ excitations were included, respectively. The convergence of the results in terms of *np*-nh excitations suggests uncertainties of about 10 keV for $^{74,76}$Ni and 50 keV for $^{78,80}$Ni. The spectroscopic factors shown in Fig. 5d are calculated by diagonalizing $^{79}$Cu and including up to $6p-6h$ excitations. At this level, the convergence of the largest spectroscopic factors is better than 5%. Therefore, these uncertainties are smaller than the uncertainties from the input two- and three-nucleon interactions, as well as the IM-SRG truncations.

**QRPA calculation.** In the presented QRPA approach, the second-order matrix was fully diagonalized$^{44,78}$, in agreement with the variational principle applied in Hartree-Fock-Bogoliubov (HFB) mean-field calculations. Eigenvalues and eigenvectors provided excitation energies and wavefunctions for all vibrational excited states described as coherent states of two quasi-particle excitations above the ground state. Transition probabilities from the ground state to all described excited states could be calculated for all parities and multipolarities and for any intrinsic deformation of the ground state. Not only spherical, but also axially symmetric deformed shapes were considered. QRPA and the underlying HFB equations were solved in a finite harmonic-oscillator basis with cylindrical coordinates including eleven major shells. To preserve symmetry restoration, the same curvature of the harmonic oscillator was given to both the radial and symmetric axis directions. This curvature was chosen for each considered nucleus to minimize the HFB binding energy. The oscillator parameter, $\hbar\omega$ ($h$, reduced Planck constant; $\omega$, frequency), as defined in a previous study$^{79}$, was found to be 9.9 MeV for $^{78}$Ni.

All HFB quasi-particle states were used to generate the two-quasi-particle excitation set. This means that no additional cut in energy, occupation probabilities
or isospin was introduced. The same effective interaction, namely, the D1M param- eter set⁴⁵ of the Gogny analytical form, was used to solve both the HFB and the QRPA equations in all particle–hole, particle–particle and hole–hole channels. Because the D1M effective interaction was fitted without exchange and pairing parts of the Coulomb interaction, only the direct term of the Coulomb field was included in the HFB as well as in the QRPA matrix elements. As concluded in a previous study⁸⁰, this term should have an impact only in case of large proton pairing energy, which is not the case here where Z = 28 is a closed proton shell. The two-body centre-of-mass correction term was also neglected in the present study, as its impact is expected to be very small.

The QRPA approach provides good agreement with the experimental $E(2_1^+)$ results, as shown in Fig. 4. We note that the $E(2_1^+)$ value of ⁶⁴Ni has been omitted because its spherical shape is degenerated with an oblate deformation (see figure 42 of ref. ⁴⁴) and is therefore not suitable for QRPA calculations.

**DWIA calculation.** DWIA describes a $(p, 2p)$ process as a proton–proton scatter- ing inside a target nucleus, A. The transition amplitude, T, whose absolute square is proportional to the reaction probability, is given by

$$T = \langle \chi_1 \chi_2 | v | \chi_0 \phi_N \rangle$$

where χ represents the scattering waves of the incoming nucleon ($\chi_0$) and the two outgoing nucleons ($\chi_1$ and $\chi_2$), $\phi_N$ is a bound-state wavefunction of the nucleon that is knocked out and v is the proton–proton interaction that causes the $(p, 2p)$ transition. In the plane-wave limit, T becomes a product of the Fourier transforms of v and $\phi_N$, which allows one to interpret a $(p, 2p)$ cross-section as a snapshot of the momentum distribution of the struck proton inside A. In reality, distortion effects such as deflection and absorption exist, which are well under control by using distorted waves for the incoming proton and outgoing protons. $(p, 2p)$ reac- tions have been established as an alternative to $(e, e'p)$ reactions. More details can be found in ref. ⁴⁹.

In the present work, we constructed each χ by a microscopic G-matrix fold- ing-model calculation—that is, the proton–proton interaction in infinite nuclear matter was folded by the nuclear density of A. We adopted the Melbourne G-matrix⁸¹ for the former and the Bohr–Mottelson density⁸² for the latter. The microscopic p–A potential was shown to describe proton-scattering observables for various nuclei in a very wide range of energies⁸¹. The Bohr–Mottelson parameter was used also for $\phi_N$. The correction to the nonlocality of χ and $\phi_N$ was accounted for by the prescription proposed by Perey and Buck⁸³. As for the transition inter- action v, the Franey–Love parameterization⁸⁴ was employed. All the central, spin- orbit and tensor parts were included. In Fig. 5, spectroscopic strengths from more deeply bound SPOs are not illustrated because they are negligibly small.

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All of the relevant data that support the findings of this study are available from the corresponding author upon reasonable request.

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<figure><img src="image_7.png" /></figure>

**Extended Data Fig. 1** | Energy spectra of prompt γ-ray coincidences with <sup>79</sup>Cu(<i>p</i>, 2<i>p</i>)<sup>78</sup>Ni reactions. **a**, As in Fig. 3a, but with a binning of 40 keV, which allows us to resolve the transition at 583 keV. **b**, γ-ray spectrum in coincidence with the 583-keV transition. The expected intensities for coincidences with the 583-keV transition are indicated by the simulated lineshapes with and without the 1,103-keV transition,

shown by the blue dashed and magenta solid lines, respectively. The results reveal no coincidence between the 583- and 1,103-keV transitions. **c**, γ-ray spectrum in coincidence with the 1,103-keV transition. The hypothesis of no coincidence between the 583- and 1,103-keV transitions is corroborated. Coincidence ranges are illustrated by the shaded areas in **b** and **c**.
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

Extended Data Fig. 2 | Evolution of peak significance and fitted intensities as a function of γ-ray multiplicity. a, b, Number of emitted

γ-rays obtained for the fitted individual transitions. c, d, S.L. of individual transitions for the $^{79}\text{Cu}(p, 2p)^{78}\text{Ni}$ (c) and $^{80}\text{Zn}(p, 3p)^{78}\text{Ni}$ (d) reactions.
<table><caption>Extended Data Table 1 | Observed γ-ray transition energies, relative intensities and S.L. for the $^{79}\text{Cu}(p, 2p)^{78}\text{Ni}$ and $^{80}\text{Zn}(p, 3p)^{78}\text{Ni}$ reaction channels</caption><thead><tr><th>Reaction</th><th>E<sub>γ</sub> (keV)</th><th>I<sub>rel</sub> (%)</th><th>Significance level</th></tr></thead><tbody><tr><td rowspan="6"><sup>79</sup>Cu(p, 2p)<sup>78</sup>Ni</td><td>583(10)</td><td>49(11)</td><td>5.3 σ</td></tr><tr><td>1,103(14)</td><td>49(12)</td><td>4.3 σ</td></tr><tr><td>1,540(25)</td><td>28(11)</td><td>2.6 σ</td></tr><tr><td>2,110(48)</td><td>33(13)</td><td>2.9 σ</td></tr><tr><td>2,600(33)</td><td>100(15)</td><td>7.5 σ</td></tr><tr><td>2,910(43)*</td><td>—</td><td>1.2 σ</td></tr><tr><td rowspan="6"><sup>80</sup>Zn(p, 3p)<sup>78</sup>Ni</td><td>581(16)</td><td>46(19)</td><td>1.8 σ</td></tr><tr><td>1,067(17)</td><td>84(26)</td><td>3.5 σ</td></tr><tr><td>1,610(240)</td><td>25(23)</td><td>1.2 σ</td></tr><tr><td>2,000(40)</td><td>66(25)</td><td>1.4 σ</td></tr><tr><td>2,720(240)<sup>†</sup></td><td>48(30)</td><td>1.4 σ</td></tr><tr><td>2,910(43)<sup>†</sup></td><td>100(29)</td><td>3.7 σ</td></tr></tbody></table>

$E_γ$ de-excitation energies, with statistical errors (1 s.d.) determined by maximizing likelihoods in probability density functions. $I_{rel}$, relative intensity with respect to the most intense transition for each reaction. Shown are the S.L. values for γ-ray detection multiplicity of $M_γ < 6$.

*The S.L. was tested with the obtained energy of 2,910 keV from the (p, 3p) reaction.

†The 2,600 keV transition observed in the (p, 2p) reaction was not observed distinctly in the (p, 3p) reaction, resulting in a likelihood maximum at 2,720(240) keV. The energy and the S.L. of the transition at 2,910(43) keV were determined by fixing the neighbouring transition to 2,600 keV.