N=16 magicity revealed at the proton drip-line through the study of 35Ca

TL;DR

This study reports the first measurement of the mass and shell gap of $^{35}$Ca, revealing a significant N=16 magicity at the proton drip-line and confirming $^{36}$Ca as doubly-magic through experimental and theoretical analysis.

Contribution

First experimental determination of the mass and shell gap of $^{35}$Ca, demonstrating N=16 magicity at the proton drip-line and comparing it with other doubly-magic nuclei.

Findings

Large N=16 shell gap of 4.61 MeV in $^{36}$Ca.

$^{36}$Ca identified as a doubly-magic nucleus.

Shell model calculations underestimate the N=16 gap by 840 keV.

Abstract

The last proton bound calcium isotope $^{35}$Ca has been studied for the first time, using the $^{37}$Ca($p, t$)$^{35}$Ca two neutron transfer reaction. The radioactive $^{37}$Ca nuclei, produced by the LISE spectrometer at GANIL, interacted with the protons of the liquid hydrogen target CRYPTA, to produce tritons $t$ that were detected in the MUST2 detector array, in coincidence with the heavy residues Ca or Ar. The atomic mass of $^{35}$Ca and the energy of its first 3/2$^+$ state are reported. A large $N=16$ gap of 4.61(11) MeV is deduced from the mass measurement, which together with other measured properties, makes $^{36}$Ca a doubly-magic nucleus. The $N = 16$ shell gaps in $^{36}$Ca and $^{24}$O are of similar amplitude, at both edges of the valley of stability. This feature is discussed in terms of nuclear forces involved, within state-of-the-art shell model calculations. Even though the global agreement with data is quite convincing, the calculations underestimate the size of the $N = 16$ gap in 36Ca by 840(110) keV.

Full text

Proton rich nuclei, Shell Model, Shell Evolution $sdpf$-shell spectroscopy, Level schemes and transition probabilities.

Introduction. Magic nuclei, corresponding to special numbers of neutrons and/or protons for which shell gaps are large, feature an enhanced stability as compared to others. Some of the “classical” magic numbers, well identified in stable nuclei (2, 8, 20, 28, 50, 82 and 126), are found to collapse in exotic regions of the chart of nuclides under the combined actions of nuclear forces and correlations (see, e.g. Sorl08 ; Gade15 ; Otsu20 ; Nowa21 ). In fact, in the last decades, many experimental and theoretical efforts have proven the disappearance of magic numbers in neutron rich nuclei such as $N=8$ Iwas00 ; Pain06 ; Imai09 ; Krie12 ; Meha12 ; Mors18 ; Chen18 , $N=20$ Guil84 ; Moto95 ; Iwas01 ; Yana03 ; Terr08 ; Craw16 ; Door16 ; Lica19 or $N=28$ Bast07 ; Forc10 ; Take12 ; Sorl13 ; Rile19 ; Long21 and also the appearance of new shell closures such as $N=16$ Ozaw00 ; Kanu09 ; Hoff09 ; Tsho12 , $N=32$ Huck85 ; Wien13 ; Rose15 or $N=34$ Step13 ; Mich18 ; Iimu23 .

Considering the recently discovered $N=32$ and $34$ neutron (sub-)shell closures Huck85 ; Wien13 ; Rose15 ; Step13 ; Mich18 ; Iimu23 together with the well established ones at $N=20$ and $N=28$, the Ca isotopes exhibit thus far the largest number of identified magic nuclei within a single isotopic chain. On the neutron-deficient side of the Ca chain, the high excitation energy of the first 2+ and 1+ states in 36Ca, as well as their large neutron-removal spectroscopic factors Lal22 , point to a significant sub-shell closure at $N=16$, the size of which remains to be determined.

The magic number $N=16$ has been identified only around 24O until now, in replacement of $N=20$ Otsu01 , which vanishes around 28O Reve20 ; Bagc20 . This $N=16$ magicity was inferred from the combined information of the drop in interaction cross sections Ozaw00 , the high excitation energy of the first 2+ state in 24O Hoff09 and its small quadrupole deformation Tsho12 .

In this letter, we report on the first measurement of the atomic mass and first excited state of the last proton bound Ca isotope 35Ca, produced in the 37Ca$(p,t)^{35}$Ca reaction, evidencing the magicity of $N=16$ close to the proton drip-line.

Experimental techniques. The 37Ca nuclei were produced at about 50 MeV/nucleon by fragmentation reactions of a 95 MeV/nucleon 40Ca20+ beam, with an average intensity of about 2 e$\mu$A, in a 2-mm thick 9Be target. They were selected by the LISE3/GANIL spectrometer Ann , leading to a purity of 20% and mean rate of 3$\times$103 pps. They were identified by means of their time-of-flight (TOF) measurement between one of the two low-pressure multiwire proportional chambers, CATS cats , and the cyclotron radio frequency. The two CATS detectors, placed at a relative distance of 51 cm, were also used to track the ions before their interaction with protons of the liquid hydrogen (LH) cryogenic target CRYPTA Koy20 of 9.7 mg cm*-2* at its center, placed at a distance of 67.8 cm downstream of the second CATS detector.

After their interaction with the protons of the LH target, the outgoing ions were detected by a Zero Degree Detection (ZDD) system, composed of an ionization chamber (IC), used for their $Z$ identification, a set of two $XY$ drift chambers (DC), to determine their outgoing angles, and a thick plastic scintillator, mostly used for time-of-flight measurements. The energy and angle of the outgoing triton from the transfer reactions were determined by a set of six MUST2 telescopes must2 , covering angles between 3 and 37∘. Each of them is composed of a 300-$\mu$m thick 10x10 cm double-sided silicon strip detector (DSSSD) with 128 strips on each side backed by sixteen 4-cm thick CsI crystals. The triton identification was performed using their combined energy loss, $\Delta E$, measured in the DSSSD and residual energy, $E$, measured in the CsI crystals.

The DSSSDs were calibrated strip by strip, using a triple-alpha source placed at the target position, covering an energy range from 5 to 6 MeV. The CsI detectors were calibrated using the kinematics of the tritons originating from $(p,t)$ reactions with incoming 38Ca and 36Ar nuclei, transmitted in another spectrometer setting, and for which $Q$-values are accurately known from precise mass measurements of the nuclei involved : 38Ca mass38Ca , 36Ca mass36Ca , 36Ar mass36Ar and 34Ar mass34Ar . With this calibration, atomic masses and uncertainties can be determined from the weighted mean and standard deviation of four independent $Q$-value measurements performed in the four MUST2 telescopes, located at the closest distance from the target. A similar procedure was applied successfully in Refs. Lal21 ; PhD to determine the atomic mass of 36Ca using the 37Ca($p,d$)36Ca reaction. Moreover, a further confirmation of the method and of the calibration is obtained from the good agreement between the mass excess of 35K, $\Delta M$ = -11205(110) keV, deduced in the present work using the known mass of 37K mass35K and the precise measurement of $\Delta M$ = -11172.9(5) mass35K in a penning trap.

Results. The excitation energy $E_{x}$ of 35Ca, produced by the 37Ca($p,t$)35Ca reaction, was deduced using the missing-mass method from the measurement of the energy and angle of the recoiling triton, detected in MUST2, and an incoming 37Ca identified and tracked in CATS. The 35Ca nucleus can be produced in a bound or unbound state. As the decay product 34K after one proton emission is unbound, the full excitation energy spectrum of 35Ca can be obtained when gating on Ca Fig. 1 a) or Ar Fig. 1 b) isotopes, identified through the measurement of their energy losses in the IC of the ZDD. The red lines in Fig. 1 show the best fit obtained using multiple Gaussian functions plus a background contribution (green dashed line), generated by interactions of the beam particles with the windows of the LH2 target and determined in a dedicated run with an empty target. The width of each peak used in the fit is constrained by simulations performed with the nptool package Matta2016 , the reliability of which is checked from the observed widths of isolated peaks in the reference reactions. The simulated width also matches the ground state value of 700 keV (sigma) in 35Ca, which is found to be the only bound state of 35Ca, as shown in Fig. 1a).

The $Q$-value of the 37Ca($p,t$)35Ca reaction has been determined from the energy of the ground state peak and the precise mass value of 37Ca mass38Ca . This leads to a mass excess of $\Delta$M(35Ca) = 4777(105) keV. Half of the uncertainty emerges from systematic effects such as the propagation of errors on the measured angle and energy of the tritons and on the energy calibration of the CsI detectors. The other half arises from the low statistics, about 15 counts per detector.

The atomic mass of the ground state of 35Ca, which has isospin components $T_{Z}$ = -5/2, $T=5/2$ and spin $J^{\pi}=1/2^{+}$, can also be estimated from the Isobaric Multiplet Mass Equation (IMME) in its quadratic form with $T_{Z}$. A first attempt to derive the mass excess of 35Ca from the IMME, leading to, $\Delta$M(35Ca) = 4453(60) keV, was obtained in Ays85 , which deviates significantly from our value. However, by taking accurate atomic masses of all nuclei involved (that were not available in 1985) and the energy of the $T$ = 5/2 isobaric analogue state in 35K proposed by Ref. Tri99 , we find $\Delta$M(35Ca$)=4624(50)$ keV. This value agrees within one sigma with the present measurement. Conversely, the atomic mass of 35Ca extrapolated in the last Atomic Mass Evaluation compilation, 5190(200) keV Wang21 , is 2$\sigma$ away from our result.

The differential cross sections corresponding to the g.s. and to the first excited state of 35Ca are shown in Fig. 1c,d). They have been obtained from the distribution of center-of-mass angles, after normalization of its amplitude using the number of incident nuclei, the density of protons in the target, as well as the detection efficiencies (intrinsic and geometrical) of the experimental setup. As the shape of the angular distribution is characteristic of the transferred angular momentum $L$, DWBA calculations were performed with the code FRESCO fresco assuming $L=0$ or $L=2$ angular momentum transfer, using the optical parameters given in Lal22Sup . Both calculations of Fig. 1c,d) consider only one dominant reaction channel, that are, for $L=0$ (in green), the direct removal of a pair of neutrons from the $2s_{1/2}$ orbital and, for $L=2$ (in blue), the direct removal of one neutron from the $2s_{1/2}$ and the other from the $1d_{3/2}$ orbital.

In spite of the limited statistics, it is seen that the shape of the ground state angular distribution of Fig. 1c) is much better fitted when assuming an $L=2$ transferred momentum ($\chi^{2}$/ndf = 10.2/7), rather than an $L=0$ one ($\chi^{2}$/ndf = 27.1/7). This $L=2$ transfer from the 3/2+ g.s. of 37Ca, corresponds to the removal of one neutron from the $2s_{1/2}$ and the other from the $1d_{3/2}$ orbital, leaving a single neutron in the $2s_{1/2}$ orbital. Therefore, the g.s. of 35Ca has a spin and parity of 1/2+, which is in agreement with the established 1/2+ g.s. spin value of the mirror nucleus 35P.

Excited states of 35Ca are visible in the excitation energy spectrum of Fig. 1b), gated on outgoing Ar. There, the number of contributions used in the fit is guided by the statistical tests of the $\chi^{2}$ and the $p$-value, as well as the number of (3/2+ and 5/2+) states populated in the two-proton transfer quasi-mirror reaction 37Cl(11B,13N)35P Orr88 . The clear rising edge at about 2 MeV in Fig. 1b) indicates the presence of the first excited state, which is found at 2.24(33) MeV. However, due to the high density of states from 3 MeV onward and the present energy resolution, different fit functions lead to very similar $\chi^{2}/ndf$ (see Fig. 4.38 of Ref. PhD ). This precludes a conclusion about the number of higher excited states populated and their exact energies. It also significantly contributes to increase the uncertainty of the energy of the first excited state determined in this work.

In the mirror nucleus 35P, the first excited state 3/2+ at 2.3866(5) MeV Wie08 has been strongly populated in the two-proton transfer reaction 37Cl(11Be,13N)35P Orr88 , supporting its tentative spin assignment in 35Ca. Given the large uncertainty on the centroid of the 3/2+, one cannot bring valuable conclusions on the MED between the two nuclei.

The differential cross section of the 2.2 MeV excited state, shown in Fig. 1d), was extracted using a condition on the excitation energy $1.5<E_{x}<3.2$ MeV and requiring an Ar isotope in the ZDD. The data are better fitted with an $L=0$ transferred momentum ($\chi^{2}$/ndf = 20.4/7) than with $L=2$ ($\chi^{2}$/ndf = 29.2/7). This favors spin and parity of $3/2^{+}$, with two holes in the neutron $2s_{1/2}$ orbital and one neutron in the $1d_{3/2}$ one. Given the present energy resolution, one cannot exclude the contamination from a higher excited state, such as a $5/2^{+}$ state likely arising from the neutron $1d_{5/2}$ removal (as found at around 3.8 MeV in the mirror nucleus), that may account for the local maximum at about 45∘.

Discussion.- The one-neutron separation energy of 36Ca, $S_{n}(^{36}$Ca) = 19.331(110) MeV, was determined by using the present mass measurement of 35Ca and the known value of $\Delta M(^{36}$Ca) = -6483.6 (56) keV mass36Ca . The top panel of Fig. 2 shows the experimental $S_{n}$ values along the calcium isotopic chain, including the new value of 36Ca. A pronounced decrease of $S_{n}$ is seen, on top of the odd-even oscillations, after having passed a (sub-) shell gap, for $N$ = 16, 20 and 28, as well as $N$=32 and 34 Wien13 ; Mich18 .

The bottom part of Fig. 2 displays differences of one-neutron separation energies, $\Delta S_{n}(N)=S_{n}(N)-S_{n}(N+1)$ for even-even Ca isotopes. At closed shells, where the effect of pairing is significantly reduced, $\Delta S_{n}(N)$ is directly related to the amplitude of the shell gap. The maximum of magicity along the Ca isotopic chain is reached at $N=20$ ($\sim 7.20$ MeV) for the self-conjugate 40Ca nucleus. At $N=16$, a shell gap of 4.61(11) MeV is obtained, the size of which is very similar to the one at $N=28$ ($\sim 4.80$ MeV), significantly larger than that at $N=32$ ($\sim 2.82$ MeV) and twice at large as at $N=$34 ($\sim 2.28$ MeV). This provides a strong evidence of the magicity at $N=16$, corroborated by the systematics of first $2^{+}$ excited states, also presented in the bottom part of Fig. 2, which follows the same trend as the $\Delta S_{n}(N)$ values. Note that the spacing between these two curves ($\Delta S_{n}(N)$ and $2^{+}$) is weaker at $N=32$ and $N=34$ than for other magic shells. This is likely due to the fact that the $2^{+}$ states at low energy are more of a pure neutron origin and coincide with the amplitude of the neutron gap, while those at higher energy combine neutron and proton excitations and are more subject to correlations.

Shell-model (SM) calculations have been carried out with the Antoine rmp code using the sdpf valence space below $A=41$ and the pf one from $A=41$ onward. The nuclear, isospin conserving parts, are given by the sdpf-u-mix interaction caurier2014 and by the pfsdg-u interaction pfsdg , respectively. The two-body matrix elements of the Coulomb interaction are computed with harmonic oscillator wave functions with $\hbar\omega=41A^{-1/3}-25A^{-2/3}$. The Coulomb corrections to the single-particle energies are taken from the experimental spectra of the $A$ = 17 and $A$ = 41 mirror nuclei.

The theoretical $S_{n}(N)$ and $\Delta S_{n}(N)$ values are shown in red in Fig. 2. The overall $S_{n}$ trend is found to be well reproduced by SM calculations. In particular, the theoretical $\Delta S_{n}(N)$ values are in good agreement with the experimental ones at $N=20,28,32$ and 34. The predicted amplitude of the $N=16$ shell gap, $\Delta S_{n}=3.77$ MeV, is however 840(110) keV lower than the experimental value of 4.61(11) MeV. Shell gaps of 3.84 and 4.00 MeV, predicted by the USDA and USDB interactions USD , are closer but still smaller than the experimental value. In fact this discrepancy can be attributed to a residual defect of the USD family of interactions (notice that USD is the sd part of sdpf-u-mix) that produce a somewhat (20%) smaller T=1 $(1s_{1/2})^{2}$ monopole interaction. The $\Delta S_{n}$ value at $N=16$ measured in this work is compatible to the one of $\Delta S_{p}$ at $Z=16$ in 36S of $4.709$ MeV, indicating that the mirror symmetry conserves the size of the 16 gap. Our SM calculations underestimate the $Z=16$ gap in 36S by the same amount.

The theoretical values of the excitation energy of the $3/2^{+}$ state in 35Ca, $E_{x}$=2.38 MeV obtained with the sdpf-u-mix interaction, is in good agreement with the experimental ones of $E_{x}=2.24(33)$ MeV, further supporting its spin-parity assignment. The associated MED between 35Ca and 35P is predicted to be of -300 keV, compatible with the shifts of about $-250$ keV of the 1+ and 2${}^{+}_{1}$ states of the 36Ca - 36S mirror pair Lal22 , suggesting a similar origin. Shell model calculation predicts the next shell closure in the Ca isotopes to be at $N=14$, with a large gap of 5.46 MeV in unbound 34Ca.

The $N=16$ gap is almost equally large at both edges of the nuclear chart: it amounts to $\Delta S_{n}=4.61(11)$ MeV in 36Ca, which lies close to the proton drip line, and to $\Delta S_{n}=4.94(20)$ MeV in 24O, which is the last bound oxygen isotope. By applying the $A^{-1/3}$ factor related to the overall compression of the level spacing in atomic nuclei with increasing $A$, one finds that the $N=16$ gap is larger in 36Ca than in 24O, viz. 4.61 $\times$ (36/24)1/3= 5.26(12) MeV in 36Ca as compared to 4.94(20) in 24O.

Fig 3 shows the evolution of the effective single particles energies (ESPE) of the neutrons $1d_{5/2}$ (blue), $2s_{1/2}$ (black) and $1d_{3/2}$ (green) orbitals at $N=16$, computed with the sdpf-u-mix interaction, as a function of the proton number. The ESPE of the $1d_{5/2}$ and $1d_{3/2}$ orbitals, given relative to the $2s_{1/2}$ one, directly allows to determine the predicted amplitudes of the $N=14$ and $N=16$ gaps, respectively. Starting from 24O, with the addition of 6 protons to the $1d_{5/2}$ orbit, the attractive tensor interaction between the two spin-orbit partners reduces the predicted single-particle $N=16$ gap by about 2.5 MeV at $Z=14$. The addition of only two more protons to the $2s_{1/2}$ orbit restores the gap at $Z=16$, while adding 4 more to the $1d_{3/2}$ proton orbital, brings it back close to the value in 24O.

This predicted evolution of the $N=16$ gap (green line) is compared in Fig 3 to experimental $\Delta S_{n}$ values (red dashed line). The $\Delta S_{n}$ values, that corresponds to experimental correlated gaps, are systematically larger than the size of $N=16$ derived from ESPE. However, the overall trend is similar. It is worth noting that, in spite of a very large gap, 32S does not show significant signs of magicity because of the enhancement of the pairing and quadrupole correlations for this $N=Z$ nucleus.

Conclusion. The 37Ca($p,t$)35Ca reaction was performed to obtain the first measurement of the atomic mass of 35Ca, $\Delta M(^{35}$Ca) = 4.777(105) keV, as well as the excitation energy of its first excited state at 2.24(33) MeV. The measured differential cross sections together with our shell model calculations support a spin parity of $1/2^{+}$ for the ground state and $3/2^{+}$ for the first excited state. The atomic mass was used to infer the amplitude of the gap at $N=16$ in 36Ca, 4.61(11) MeV, which is very similar to the one of $N=28$, significantly larger than that at $N=32$ ($\sim 2.82$ MeV) and twice at large as at $N=$34 ($\sim 2.28$ MeV). This result corroborates the arguments in favor of double magicity of 36Ca from the observation of high-energy $2^{+}$ and $1^{+}$ states and their large neutron-removal spectroscopic factors $C^{2}S$ obtained from 37Ca($p,d$)36Ca Lal22 . The magicity of 36Ca is further confirmed by its small charge radius Mill19 , the smallest among all Ca isotopes.

The amplitude of the $N=16$ gap in 36Ca is also comparable to the one found in 24O, described as a doubly-magic nucleus (see, e.g. Hoff09 ; Tsho12 ). The fact that $N=16$ magicity is strongly present at both edges of the nuclear binding gives strong constraints to the proton-neutron interactions involved when adding 12 protons to the $sd$ shells. Thus far, the use of state-of-the-art shell-model interactions underestimate the gap derived from $\Delta S_{n}$ by about 0.61 (USDB) to 0.84 MeV (sdpf-u-mix). The SM calculation carried out with both interactions predicts the next shell closure of the Ca isotopic chain to be at $N=14$ in the doubly-magic and unbound 34Ca, which is the mirror of the bubble nucleus 34Si Muts17 .