Letter | Published:

# Proton superfluidity and charge radii in proton-rich calcium isotopes

## Abstract

One of the most important global properties of the atomic nucleus is its size. Experimentally determined nuclear charge radii carry unique information on the nuclear force and complex dynamics of protons and neutrons moving inside the nucleus. The intricate behaviour of charge radii along the chain of Ca isotopes, including the unexpectedly large charge radius of neutron-rich 52Ca, poses a daunting challenge for nuclear theory1. Here we present the measurements of the charge radii of proton-rich isotopes 36,37,38Ca, whose properties are impacted by the interplay between nuclear superfluidity and weak binding. Calculations carried out within nuclear density functional theory show that the combination of a novel interaction2 and a state-of-the-art theoretical method can successfully explain the behaviour of charge radii from the lightest to the heaviest Ca isotopes. Through this model, we show how the new data on 36,37,38Ca elucidate the nature of nucleonic pairing in weakly bound proton-rich isotopes.

## Main

At low excitation energies, atomic nuclei can be described as condensates of nucleonic (proton or neutron) pairs3,4,5. The resulting superfluid correlations profoundly influence the structure of nucleonic systems, such as finite nuclei and neutron stars6. In weakly bound nuclei, pairing correlations can be enhanced due to the proximity of the particle continuum, which affects the asymptotic properties of density distributions and can impact nuclear size7.

The interplay between nuclear size, pairing correlations and weak binding takes a particularly dramatic form in spatially extended, neutron-rich halo nuclei8, such as 6He or 11Li. Indeed, their very existence hinges on the pairing energy among the outermost neutrons. For weakly bound, proton-rich nuclei, in which the Coulomb barrier has a localizing effect on the loosely bound or slightly unbound protons, pairing gives rise to the phenomenon of two-proton radioactivity9. The asymmetry in the energy spectra between mirror nuclei is another effect governed by a competition between weak binding and charge radius10. However, little is known about charge radii and superfluidity in weakly bound proton-rich nuclei.

The charge radii of Ca isotopes are known1 up to 52Ca. They exhibit an intricate behaviour, with almost equal values in 40Ca and 48Ca, a local maximum at 44Ca, appreciable odd–even staggering, and a large value for 52Ca. This pattern poses a serious challenge for nuclear theory1,11,12,13. Below the lightest stable 40Ca isotope, the charge radius is known only to 39Ca (ref. 14), primarily due to the difficulty in the production of short-lived, proton-rich Ca nuclei.

Recent advances in radioactive beam production and manipulation techniques, namely fast in-flight production and separation15 followed by gas stopping16, have made it possible to develop good-quality, high-rate beams of short-lived, proton-rich Ca isotopes, including 36Ca with a half-life of T1/2 = 102(2) ms (ref. 17), which is sufficient for precise laser spectroscopy measurements. This production mechanism complements the capabilities of the isotope separator on line (ISOL) facilities (see Methods), where significant data on charge radii of selected elements have been obtained18,19, including the charge radius of neutron-rich 52Ca with a half-life of 4.6(3) s (ref. 20).

Our experiment was performed at the National Superconducting Cyclotron Laboratory at Michigan State University. A schematic of the complete set-up is shown in Fig. 1. Projectile-fragmentation reactions of a 140 MeV per nucleon 40Ca primary beam on a thin transmission target (beryllium, 356 μm in thickness) were used to produce 36–39Ca isotopes (see Methods). The desired Ca ions were separated from the primary beam and other reaction products through the A1900 Fragment separator15 based on magnetic rigidity selections. The selected fast Ca-ion beam was then injected into a gas cell stopper filled with a helium buffer gas16, where the ions were thermalized. Singly charged Ca ions of the desired mass were extracted at an energy of 30 keV after the stopping range selection in the cell, and the low-energy beam was transported to the beam cooler and laser spectroscopy (BECOLA) facility21,22.

At BECOLA, the ion beam was first introduced into a helium-buffer-gas-filled radiofrequency quadrupole cooler/buncher23. Ions were trapped in the cooler/buncher, further cooled (improving beam emittance) and periodically extracted at an approximate energy of 29,850 eV as ion beam bunches. The time spread of an ion bunch was set to ~1 μs (full-width at half-maximum) by adjusting the radiofrequency and release potential. The narrow time spread is critical for a high-sensitivity measurement as it reduces the background significantly24,25. The bunched ion beam was then overlapped with laser light and co-propagated for bunched-beam collinear laser spectroscopy.

Laser-induced fluorescence was detected using two sets of photon detection systems (see Methods). The 4s 2S1/2 ↔ 4p 2P3/2 transition in Ca ɪɪ at 393 nm was probed to measure the hyperfine spectrum. The timing between the ion bunches was optimized based on the lifetime of Ca isotopes to maximize the signal to noise ratio (see Methods). A sawtooth-pattern ramping voltage with discrete steps was applied to the photon detection system to vary the incoming ion beam velocity. Ions experienced a Doppler-shifted laser frequency and the hyperfine spectrum was scanned at a fixed laser frequency. Spectra of stable 40,44Ca, which were produced in an offline ion source, were measured periodically to be used as a reference of the isotope shift, and also for ion beam energy calibration purposes (see Methods). The obtained spectra of Ca isotopes are included in Fig. 1.

The 37,39Ca isotopes, which have a nuclear ground-state spin of I = 3/2, show hyperfine splittings with six allowed transitions, whereas single peaks were observed for the even Ca isotopes as they have I = 0. The hyperfine spectra were fitted using a pseudo-Voigt profile to reproduce lineshape broadening, due mostly to the fluctuation of the applied potential (see Methods). The resulting centroids of the hyperfine spectra of 36–39Ca were compared to the reference measurement of 40Ca, and isotope shifts δν40,A were extracted, which are summarized in Table 1.

The differential mean-square (m.s.) charge radius $$\delta \left\langle {r^2} \right\rangle ^{A,A^\prime } = \left\langle {r^2} \right\rangle ^{A^\prime } - \left\langle {r^2} \right\rangle ^A$$ can be obtained from the isotope shift between isotopes A and A′ through

$$\delta \nu ^{A,A^\prime } = \nu ^{A^\prime } - \nu ^A = k\frac{{m_{A^\prime } - m_A}}{{m_{A^\prime }m_A}} + F\delta \left\langle {r^2} \right\rangle ^{A,A^\prime }$$
(1)

where k and F are the mass and field shift coefficient, respectively, determined by the electronic configuration, and m is the atomic mass. The atomic factors of the 2S1/2 ↔ 2P3/2 transition in Ca ɪɪ used in the present study were determined as k = 409.35(42) GHz amu and F = −284.7(82) MHz fm−2 from a King plot analysis26. The obtained differential m.s. charge radii are summarized in Table 1 and the root-mean-square (r.m.s.) charge radii are shown in Fig. 2. The obtained 〈r2〉 of 39Ca is consistent with the known value14, and our experiment has reduced the uncertainty by a factor of three.

Overall, the charge radii decrease towards neutron-deficient isotopes across the neutron N = 20 shell closure, which makes a sharp contrast with the discontinuity seen at the N = 28 shell closure. Similar behaviour has been observed in the lighter Ar (ref. 27) and K (ref. 28) isotopic chains. The charge radii of the proton-rich Ca isotopes also exhibit an odd–even staggering, although the amplitude of this effect is smaller than around 44Ca.

Charge radii of proton-rich Ca isotopes are expected to be impacted by nuclear superfluidity. Indeed, as seen in Fig. 3a, for 36–38Ca the 0d3/2 valence proton shell is weakly bound and the next 0f7/2 orbital lies above the Coulomb barrier. This means that pairing correlations due to the scattering of proton pairs from the bound states into the proton continuum are expected to leave an imprint on the properties of these nuclei. In the presence of low-lying continuum space, the standard Hartree–Fock (HF)+ Bardeen–Cooper–Schrieffer (BCS) treatment of nucleonic pairing becomes questionable, as the resulting particle and pair densities are not spatially localized and acquire an unphysical gas component because of the nonzero occupation probabilities of unbound single-particle states29. To analyse these effects in more detail, we used nuclear density functional theory with realistic energy density functionals30. As a typical representative of standard functionals, we used the Skyrme parametrization SV-min31, which employs density-dependent pairing. The performance of standard functionals on global trends of nuclear gross properties is excellent, but their simple pairing term fails to describe isotope shifts and odd–even staggering of charge radii in medium–light nuclei13. To overcome this hurdle, a nuclear density functional with a more elaborate pairing term was employed by Fayans et al.2,32. Recently, this functional was reoptimized13 by taking into account additional information on the charge radii of neutron-rich Ca. Since this work used the HF + BCS approximation to pairing, we shall refer to the resulting model as Fy(Δr, BCS).

As said above, the BCS approximation fails when applied to weakly bound nuclei. To gain some insight into the magnitude of the spurious gas effect in HF + BCS, Fig. 3c shows the occupations of single-proton states in 36Ca obtained in the Fy(Δr, BCS) model and Fig. 3b shows the corresponding single-proton r.m.s. radii. The ground-state of 36Ca is predicted to have a finite proton pairing gap, which results in nonzero occupations of unbound single-proton orbits above the Z = 20 shell closure. Since these states are formally not localized, their r.m.s. radii become unphysically large.

To overcome this problem, one has to resort to the full Hartree–Fock–Bogolyubov (HFB) formalism, which—for particle-bound nuclei—guarantees the spatial localization of nuclear densities29,33; hence preventing the spurious proton gas from appearing. To this end, we re-optimized the original Fy(Δr, BCS) model within the full HFB framework. The resulting energy density functional is Fy(Δr, HFB) (see Methods). To probe the impact of the charge radius of 36Ca obtained in this Letter on model predictions, we also developed a modified Fy(Δr, HFB) functional by including the constraint on δr240,36 to the dataset.

The single-proton radii of localized proton HFB canonical states29 in 36Ca obtained in Fy(Δr, HFB) are shown in Fig. 3b. The r.m.s. radii of unbound HFB orbits behave very regularly and do not show the marked increase which is seen for BCS orbits. This allows the HFB occupations of unbound single-proton states to stay slightly larger than the BCS occupations (see Fig. 3c).

Charge radii of Ca isotopes predicted by the SV-min (HFB), Fy(Δr, BCS) and Fy(Δr, HFB) models are shown in Fig. 2. The SV-min (HFB) results exhibit a characteristic monotonic dependence on A and a practically nonexistent odd–even effect1,12,13. Moreover, it is seen that this functional systematically predicts too large charge radii in the neutron-deficient Ca isotopes. Due to the presence of proton gas, the functional Fy(Δr, BCS) also dramatically overestimates the charge radii in 36,38Ca. On the other hand, both the standard and modified Fy(Δr, HFB) models provide a remarkably good overall description of the experimental data. Both models tend to slightly overestimate the odd–even staggering of charge radii, but this is a minor deficiency considering the scale of this effect. The unexpectedly large charge radius of 52Ca (ref. 1) is well explained by the Fy(Δr, HFB) models, and their description of A > 48 radii has been improved significantly.

The comparison between SV-min (HFB) and Fy(Δr, HFB) shows the importance of the non-standard density dependence of the Fayans functional, especially its novel pairing term. The ability to accommodate the radii along the Ca chain can be clearly associated with two particular features of the Fayans functional: mainly the gradient coupling in the pairing functional and a gradient term in the surface energy13. These gradient terms are necessary to reproduce the demanding δr248,44 and δr248,52 values.

The new charge radii data on proton-rich Ca isotopes demonstrated the need for the careful treatment of proton superfluid correlations in the presence of low-lying continuum states. It is remarkable that the new Fy(Δr, HFB) models, based on the HFB formalism and the Fayans energy density functional, developed with the focus on 36–38Ca, also improve the description of neutron-rich isotopes all the way to, and including, 52Ca. It is worth noting that the experimental information to constrain theoretical models of nucleonic pairing is very limited, and is primarily based on binding-energy relationships. This work shows that data on charge radii of very proton-rich nuclei are invaluable for pinning down elusive pairing interactions in nuclei. Finally, let us note that the charge radii of heavier 100–130Cd (Z = 48) isotopes are also well reproduced by the Fayans functional34. The success of the Fayans model when applied to the long-standing puzzle of the Ca charge radii, together with its demonstrated applicability to other isotopic chains, is an important step towards developing a global nuclear model.

## Methods

### Isotope production technique

In the present study, a projectile-fragmentation reaction was used to produce the isotopes in-flight. This reaction is characterized by the reaction products being emitted in a narrow forward solid angle at nearly the primary-beam velocity. This makes the fast and efficient selection of fragments possible without regard to their chemical properties by using a magnetic spectrometer.

In the ISOL technique used at other facilities, rare isotopes are extracted from thick targets bombarded by light ions. High-quality, high-intensity beams are available at low energy (~60 keV) for certain elements. The ISOL production method, however, suffers from a serious limitation due to long release times from thick targets. This can lead to large decay losses for nuclides that have long diffusion and/or effusion times and with short half-lives at the limits of the nuclear chart.

The limitation has been partly overcome by the Ion Guide at an Isotope Separator On Line (IGISOL) approach36, where thin targets and a supersonic helium jet to extract reaction products immediately from production to the beam line were used, allowing refractory and short-lived isotopes to be studied. The drawback, however, is the intensities that can be obtained.

### The photon detection system

Two sets of photon detection system were used to detect the resonant fluorescence; each detector comprised a light collection section and a photomultiplier tube. The collection sections contained an ellipsoidal reflector, which focuses the signal tighter than background light, and a following compound parabolic concentrator, which further eliminates background based on its wider incidence angles.

### The laser system

A Sirah Matisse TS Ti:Sapphire ring laser was used to produce 787 nm light, and a SpectraPhysics WaveTrain generated the 393 nm light by frequency doubling the 787 nm light. The Ti:Sapphire laser frequency was locked using a HighFinesse WSU-30 wavelength meter calibrated by a frequency-stabilized He-Ne laser. Laser power at the interaction region was stabilized at 300 μW using a laser power controller37.

### Optimization of bunching period

The bunching period was chosen to maximize the signal-to-noise ratio for each isotope. An increased bunching period Tb reduces the background as B(Tb)  1/Tb for a given time spread of an ion bunch. The signal is proportional to the number of trapped ions, which depends on the ion rate and the decay loss, as $$S\left( {T_{\rm b}} \right) \propto (1{\mathrm{/}}\lambda )\left( {1 - {\rm e}^{ - \lambda T_{\rm b}}} \right){\mathrm{/}}T_{\rm b}$$, where λ is the decay constant. The signal-to-noise ratio (SNR) can then be evaluated as

$${\mathrm{SNR}} = \frac{{S\left( {T_{\rm b}} \right)}}{{\sqrt {B\left( {T_{\rm b}} \right)} }} \propto \frac{{(1{\mathrm{/}}\lambda )\left( {1 - {\rm e}^{ - \lambda T_{\rm b}}} \right)}}{{\sqrt {T_{\rm b}} }}$$
(2)

The SNR is maximized at Tb ≈ 1.3/λ ≈ 1.8 × T1/2. This condition does not depend on the ion rate, as long as the space charge of the trapped ions does not disturb the emittance of the released ion bunches. The bunching periods of 36,37Ca were chosen directly on the basis of the lifetime; however, due to the increased rates, the bunching periods of 38,39Ca were shortened to avoid space charge effects. The bunching periods and approximate rates for each isotope are summarized in Table 1.

### Energy calibration with stable 40,44Ca

The ion's rest-frame resonant laser frequency can be obtained from the set laser frequency and the total potential applied to the ion beam, which is the sum of the potentials applied to the cooler/buncher and the photon detection system. The total potential makes the largest contribution to the uncertainty in the determination of the charge radius, since the potential at which ions are released from the cooler/buncher is not directly known and the uncertainty is estimated to be 10 V. To eliminate this uncertainty, a well-known isotope shift between stable 40Ca and 44Ca isotopes (δν40,44 = 850.231(65) MHz26) was used to calibrate the total potential. This calibration is the predominant source of systematic uncertainty. The spectroscopy of 40,44Ca was performed using ions produced by an offline discharge plasma ion source38 and introduced into the cooler/buncher. The calibration was performed every several hours throughout the experiment to monitor and correct for the time-dependent variation of the total potential, and was also affected by fluctuations of the laser frequency. By examining the variations of the calibration throughout the experiment, the systematic uncertainty due to both the laser frequency and total potential was determined.

### Hyperfine spectra fitting procedure

All peaks were fitted using an asymmetric pseudo-Voigt function39. For 36,37Ca, the lineshape (asymmetry, which was negligibly small, and Lorentz fraction) of the peaks was constrained to match those obtained from the 40Ca reference; however the linewidth was allowed to vary due to the longer running time. The 40,44Ca peaks were typically ~75 MHz wide, whereas 36,37Ca, which required the longest running times, had linewidths of ~80 MHz. In 39Ca (I = 3/2), all six peaks of the hyperfine spectra were fitted with a common lineshape and width, and their positions were obtained by fitting the three hyperfine coupling constants $$\left( {A_{{\mathrm{lower}}}^{{\mathrm{HF}}},A_{{\mathrm{upper}}}^{{\mathrm{HF}}},B_{{\mathrm{upper}}}^{{\mathrm{HF}}}} \right)$$, with the upper and lower levels shifted by

$${\mathrm{\Delta }}E = \frac{K}{2}A^{{\mathrm{HF}}} + \frac{{3K(K + 1) - 4I(I + 1)J(J + 1)}}{{8I(2I - 1)J(2J - 1)}}B^{{\mathrm{HF}}}$$
(3)

where K = F(F + 1) − I(I + 1) − J(J + 1), F is the quantum number defined by the vector F = I + J, and I and J are the nuclear and atomic spins respectively. The height of each peak was allowed to vary freely in 39Ca and, because of the excellent statistics, the relative intensity of each peak was used as a constraint in the fit of the 37Ca spectrum (also I = 3/2).

### Density functional theory calculations

Our calculations were carried out with the original SV-min (ref. 31) and Fy(Δr, BCS) (ref. 13) energy density functionals, as well as the new functionals Fy(Δr, HFB) developed in this work. In general, our optimization methodology closely followed that of ref. 13. Namely, by carefully selecting the experimental datasets constraining theoretical models, we generate functionals to study questions pertaining to different observables. The functional SV-min was optimized to the basic dataset of ref. 31 containing 224 experimental data points, including binding energies, diffraction radii, surface thickness, charge radii and spin–orbit splitting. The functional Fy(Δr, BCS) was constrained by the basic dataset of SV-min as well as by additional data on differential charge radii of Ca isotopes (δr248,40, δr248,44, δr252,48). Pairing information is included here in terms of neutron and proton odd–even binding energy staggering $${\mathrm{\Delta }}_E^{(3)}$$ extracted from three-point binding-energy differences.

Unlike in Fy(Δr, BCS), which was developed under the HF+BCS protocol, pairing correlations in Fy(Δr, HFB) are treated within the full HFB framework using the techniques of ref. 40 and without employing the gap stabilization method41. Moreover, to constrain the pairing term, three-point binding energy differences $${\mathrm{\Delta }}_E^{\mathrm{ee}}$$ involving ground states of even–even nuclei13 were employed rather than $${\mathrm{\Delta }}_E^{(3)}$$. As discussed in ref. 42, for open-shell systems, $${\mathrm{\Delta }}_E^{\mathrm{ee}}$$ is related to the pairing rotational moment of inertia and is superior to $${\mathrm{\Delta }}_E^{(3)}$$, which involves properties of odd-mass systems that depend on poorly-known time-odd fields impacting individual orbits occupied by an odd nucleon. We have checked within the BCS method that the fit to even–even gaps produces practically the same pairing gaps as the fit to $${\mathrm{\Delta }}_E^{(3)}$$ used in Fy(Δr, BCS). The dataset of modified Fy(Δr, HFB) also includes the value of δr240,36. For the purpose of this work, the functional SV-min has also been re-optimized in the HFB variant; the resulting model is called SV-min (HFB), again, using here $${\mathrm{\Delta }}_E^{\mathrm{ee}}$$ as pairing information.

To compute odd-A nuclei, we applied the standard uniform filling approximation to blocking43,44, in which an unpaired nucleon is put with equal probability in each of the degenerate magnetic sub-states. To find the ground state, blocked calculations were carried out for all orbitals near the Fermi energy and the blocked state with lowest energy was selected.

It is to be noted that the excellent description of charge radii in the Ca chain with 39 ≤ A ≤ 50 was obtained in previous calculations with FaNDF0, DF3 and DF3-a Fayans functionals32,45,46. The functionals Fy(Δr, BCS) and Fy(Δr, HFB) used here derive from FaNDF0.

Finally, it has recently been suggested that the differences in the charge radii of mirror nuclei can be used to constrain the neutron equation of state for use in astrophysics47. Our theoretical results for 36–38Ca show that without an advanced approach to proton pairing in weakly bound proton-rich mirror partners, it will be difficult to arrive at precise predictions.

## Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

This work was supported in part by the National Science Foundation, Grant No. PHY-15-65546; the US Department of Energy, National Nuclear Security Administration, Grant No. DE-NA0002924; the US Department of Energy, Office of Science, Office of Nuclear Physics, Grant Nos. DE-SC0013365, DE-SC0018083 and DE-AC05-00OR22725 with UT-Battelle, LLC; the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Projektnummer 279384907 – SFB 1245; and the German Federal Ministry of Education and Research (BMBF), Grant No. 05P12RFFTG.

## Author information

### Affiliations

1. #### National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI, USA

• A. J. Miller
• , K. Minamisono
• , D. Garand
• , J. D. Lantis
• , C. Sumithrarachchi
•  & J. Watkins
2. #### Department of Physics and Astronomy, Michigan State University, East Lansing, MI, USA

• A. J. Miller
• , K. Minamisono
•  & W. Nazarewicz
3. #### Department of Chemistry, Augustana University, Sioux Falls, SD, USA

• A. Klose
• , C. Kujawa
•  & S. V. Pineda
4. #### Department of Chemistry, Michigan State University, East Lansing, MI, USA

• J. D. Lantis
•  & P. F. Mantica
5. #### Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA

• Y. Liu

• B. Maaß
• , W. Nörtershäuser
• , D. M. Rossi
•  & F. Sommer
7. #### Facility for Rare Isotope Beams, Michigan State University, East Lansing, MI, USA

• P. F. Mantica
•  & W. Nazarewicz
8. #### Institut für Theoretische Physik, Universität Erlangen, Erlangen, Germany

• P.-G. Reinhard
9. #### TRIUMF, Vancouver, British Columbia, Canada

• A. Teigelhöfer

### Contributions

A.J.M., K.M., A.K., D.G., J.D.L., Y.L., P.F.M., S.V.P., D.M.R., F.S., C.S. and A.T. performed the experiment. A.J.M., C.K., B.M., W. Nörtershäuser and J.W. designed and installed the upgraded photon detection system. A.J.M. performed data analysis and discussed with K.M., A.K., W. Nörtershäuser and D.M.R. W. Nazarewicz and P.-G.R. performed theoretical analysis. A.J.M., K.M., W. Nazarewicz and P.-G.R. prepared the figures. A.J.M., K.M., W. Nazarewicz and P.-G.R. prepared the manuscript. All authors discussed the results and contributed to the manuscript at all stages.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to K. Minamisono.