Ground State Electromagnetic Moments of $^{37}$Ca

TL;DR

This paper reports the first measurement of the ground-state electromagnetic moments of $^{37}$Ca using collinear laser spectroscopy, and compares the results with shell model and ab initio calculations to understand nuclear structure.

Contribution

It provides the first experimental determination of the magnetic dipole and electric quadrupole moments of $^{37}$Ca and compares these with theoretical models.

Findings

Experimental moments agree with shell model calculations.

$^{36}$Ca shows a more robust closed-shell structure than $^{40}$Ca.

Magnetic moment of $^{39}$Ca suggests larger non-$sd$-shell components.

Abstract

The hyperfine coupling constants of neutron deficient $^{37}$Ca were deduced from the atomic hyperfine spectrum of the $4s~^2S_{1/2}$ $\leftrightarrow$ $4p~^2P_{3/2}$ transition in Ca II, measured using the collinear laser spectroscopy technique. The ground-state magnetic-dipole and spectroscopic electric-quadrupole moments were determined for the first time as $\mu = +0.7453(72) \mu_N$ and $Q = -15(11)$ $e^2$fm$^2$, respectively. The experimental values agree well with nuclear shell model calculations using the universal sd model-space Hamiltonians versions A and B (USDA/B) in the $sd$-model space with a 95\% probability of the canonical nucleon configuration. It is shown that the magnetic moment of $^{39}$Ca requires a larger non-$sd$-shell component than that of $^{37}$Ca for good agreement with the shell-model calculation, indicating a more robust closed sub-shell structure of $^{36}$Ca at the neutron number $N$ = 16 than $^{40}$Ca. The results are also compared to valence-space in-medium similarity renormalization group calculations based on chiral two- and three-nucleon interactions.

Figures

Tables

Table 1. Table 1: The obtained hf coupling constants of 37,39 Ca for the 4 ​ s 4 𝑠 4s   S 1 / 2 2 superscript subscript 𝑆 1 2 2 {}^{2}S_{1/2} and 4 ​ s 4 𝑠 4s   P 3 / 2 2 superscript subscript 𝑃 3 2 2 {}^{2}P_{3/2} states. The first and second parentheses contain uncertainties due to statistical and systematic errors, respectively. The systematic errors are from high voltage calibrations and the variation of the A hf superscript 𝐴 hf A^{\rm hf} -factor ratio from the literature value, which dominates the systematic error.

		$A^{\mathrm{hf}}$ (MHz)		$B^{\mathrm{hf}}$ (MHz)
$A$	$I^{\pi }$	${}_{}{}^{2}S_{1/2}^{}$	${}_{}{}^{2}P_{3/2}^{}$	${}_{}{}^{2}P_{3/2}^{}$
$37$	3/2+	+1064.5(103)(08)	+40.57(39)(27)	$-$22.9(163)(05)
$39$	3/2+	+1457.20(14)(34)	+55.53(9)(32)	+5.79(26)(32)

Table 2. Table 2: Results for the magnetic moments of 3/2 + states for Z = 20 𝑍 20 Z=20 ( 39 Ca and 37 Ca) and N = 20 𝑁 20 N=20 ( 39 K and 37 Cl). Numbers are given in the unit of μ N subscript 𝜇 𝑁 \mu_{N} except ⟨ s z ⟩ delimited-⟨⟩ subscript 𝑠 𝑧 \left<s_{z}\right> . The effective g 𝑔 g -factors are taken from the Table I of Richter  et al. ( 2008 ) for six-parameter form of the M1 operator. The μ 𝜇 \mu (IS) and μ 𝜇 \mu (IV) are defined as μ 𝜇 \mu (IS/IV) = μ 𝜇 \mu ( T 3 = + T subscript 𝑇 3 𝑇 T_{3}=+T ) ± plus-or-minus \pm μ 𝜇 \mu ( T 3 = − T subscript 𝑇 3 𝑇 T_{3}=-T ).

$A$		$Z=20$	$N=20$	$\mu$(IS)		$\mu$(IV)
	sp $g^{\mathrm{free}}$	+1.148	+0.124	+1.272	$-$0.600	+1.024
39	Exp.	+1.0217(1) Minamisono et al. (1976)	+0.3915073(1) Sahm and Schwenk (1974)	+1.4131(1)	$-$0.2284(3)	+0.6302(1)
	sp $g^{\mathrm{eff}}$	+0.930	+0.469	+1.399	$-$0.266	+0.461
	VS-IMSRG	+1.349	$-$0.035	+1.314	$-$0.488	+1.384
37	Exp.	+0.7453(72)	+0.6841236(4) Blaser et al. (1972)	+1.429(7)	$-$0.19(2)	+0.061(7)
	USDA-EM1	+0.770	+0.677	+1.447	$-$0.139	+0.093
	USDB-EM1	+0.754	+0.675	+1.429	$-$0.187	+0.079
	VS-IMSRG	+1.055	+0.290	+1.345	$-$0.409	+0.765

Table 3. Table 3: Wave function for 37 Ca(3/2 + ). The occupation for each configuration is shown in %.

($n$5, $n$1, $n$3)	USDA-EM1	USDB-EM1	VS-IMSRG
(621)	94.61	95.03	90.28
(603)	1.68	1.64	2.81
(612)	0.32	0.68	0.51
(423)	2.63	2.21	5.61
(513)	0.51	0.23	0.29
(522)	0.23	0.20	0.45
$\mu$ (${\mu }_N$)	0.770	0.754	1.055

Equations

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

Full text

Introduction —

A nucleus with finite spin possesses electromagnetic moments capable of providing critical information for the investigation of nuclear structure. Most notably, close to nuclei with magic numbers of nucleons (e.g. 2, 8, 20, 28, 50, 82…) such systems, due to their simple and robust structures, provide discerning comparisons between experiment and theory. One of the highlights of modern nuclear structure studies has been the disappearance of established magic numbers (essentially those seen in stable nuclei Mayer (1949); Haxel et al. (1949)) and the appearance of new magic numbers at extreme neutron-to-proton ratios Ozawa et al. (2000), for example the neutron number $N$ = 16 Kanungo et al. (2009); Hoffman et al. (2009).

The ground-state electromagnetic moment of 37Ca, which has one neutron added to the $N$ =16 36Ca nucleus in the vicinity of the proton dripline, was determined in the present study. The neutron occupies the $d_{3/2}$ orbital with $j_{<}\equiv l-1/2$ where $l$ is the orbital angular momentum. The spin-orbit partner $d_{5/2}$ with $j_{>}\equiv l+1/2$ is fully occupied. Here, the first-order core-polarization effect, first introduced by Arima and Horie Arima and Horie (1954a, b), is expected to play an important role on the magnetic moment due to collective M1 excitations between the spin-orbit partners. The counterpart is a 39Ca nucleus with one neutron hole in the doubly-closed 40Ca core in the $j_{<}\equiv l-1/2$ shell. In the single-particle model magnetic moments of 37Ca and 39Ca are equal, and take the Schmidt value. This provides a unique situation where the transition from the $JJ$ closed sub-shell 36Ca to the $LS$ doubly closed 40Ca configuration can be seen in nearby isotopes in a single element. The variation of structure around the 36Ca and 40Ca nuclei is investigated through the first order core-polarization model in the context of the universal sd model-space Hamiltonians versions A and B (USDA/B) and the chiral effective field theory.

Experiment — The radioactive ion beam of 37Ca ($I^{\pi}$ = 3/2+, $T_{1/2}$ = 175 ms) was produced via projectile-fragmentation reactions of a 40Ca primary beam on a Be target. The 40Ca beam was accelerated to 140 MeV/nucleon in the coupled cyclotrons at the National Superconducting Cyclotron Laboratory at Michigan State University. The 37Ca fragments were separated using the A1900 fragment separator Morrissey et al. (2003), thermalized in a He-filled gas cell Cooper et al. (2014) and extracted as singly-charged ions at an energy of 30 keV. The low-energy beam was then mass analyzed through a dipole magnet and transported to the BEam COoling and LAser spectroscopy (BECOLA) facility Minamisono et al. (2013); Rossi et al. (2014). The typical rate of 37Ca at the BECOLA facility was 103 ions/s.

At BECOLA, the 37Ca beam was first injected into a radio frequency quadrupole (RFQ) cooler and buncher Barquest et al. (2017). The ion beam was trapped, cooled (improving the emittance) and extracted at an approximate energy of 29850 eV as ion bunches for the bunched-beam collinear laser spectroscopy Campbell et al. (2002); Nieminen et al. (2002). Laser-induced fluorescence measurements were performed on the $4s~{}^{2}S_{1/2}$ $\leftrightarrow$ $4p~{}^{2}P_{3/2}$ transition in Ca II at 393 nm to measure the hyperfine (hf) spectrum. The ion-beam bunch was extracted from the RFQ every 330 ms and the bunch width (full width at half-maximum) was set to $\sim$1 $\mu$s without degrading the typical resolution of $\sim$80 MHz of the hf spectrum. A Sirah Matisse TS Ti:sapphire ring laser was used to produce 787 nm light that was subsequently frequency doubled to 393 nm light using a SpectraPhysics Wavetrain. The Matisse was stabilized using a HighFinesse WSU-30 wavelength meter, calibrated with a frequency-stabilized He-Ne laser. The laser-light power was stabilized at 300 $\mu$W, which gave a maximum signal to noise ratio, using a laser power controller beo (2018). Two identical photon detectors were used in series along the beam line to detect the resonant fluorescence. A scanning voltage was applied to the light collection section to vary the incoming ion beam velocity so that the Doppler-shifted laser frequency could be tuned into resonance with the hf transitions.

Experimental results — The obtained hf spectrum of 37Ca is shown in Figure 1.

The hf spectrum of 39Ca(3/2+, 859.6 ms) was also measured in the present study. The 39Ca beam was produced in a similar procedure as 37Ca, and the hf spectrum was measured with the same laser power as 37Ca to be used as a line shape reference in the fitting of 37Ca. There are six allowed hf transitions between the ${}^{2}S_{1/2}$ and ${}^{2}P_{3/2}$ states since the nuclear spins of 37,39Ca are $I$ = 3/2. The shift of a hf level is given by

[TABLE]

where $A^{\rm hf}$ and $B^{\rm hf}$ are the magnetic and quadrupole hf coupling constants, respectively, $K=F(F+1)-I(I+1)-J(J+1)$, $I$ is the nuclear spin, $J$ is the total electronic angular momentum and $\bm{F=I+J}$. The hf coupling constants are defined as $A^{\rm hf}=\mu B_{0}/IJ$ and $B^{\rm hf}=eQV_{zz}$. Here, $\mu$ and $Q$ are the magnetic-dipole and spectroscopic electric-quadrupole moments of the nucleus, respectively, $B_{0}$ and $V_{zz}$ are the magnetic field and the electric field gradient, respectively, generated by orbital electrons at the position of the nucleus and $e$ is the elementary charge. The $B_{0}$ and $V_{zz}$ are isotope-independent, assuming a point-like nucleus.

A Voigt profile Stancik and Brauns (2008) was used in the fitting of the hf spectrum. All six peaks of the 39Ca hf spectrum were fitted with a common line shape and width, and free parameters of ground and excited states hf coupling constants, Lorentz fraction and intensities of each peak. The high ion-beam rate allowed the reliable determination of these parameters with high precision. The 37Ca hf spectrum was fitted with the relative peak intensities and Lorentz fraction constrained to those determined in the 39Ca analysis. Also the ratio between $A^{\rm hf}(^{2}S_{1/2})$ and $A^{\rm hf}(^{2}P_{3/2})$ was fixed to 26.24(4) deduced from the 39Ca fit. The obtained hf coupling constants are summarized in Table 1. It is noted that the $A^{\rm hf}$-factor ratio was determined in the previous measurement as 25.92(3) Garcia Ruiz et al. (2015) and deviates from the present ratio. The reason is not known. Variation in fitted hyperfine coupling constant due to the deviation of $A^{\rm hf}$-factor ratios was taken into account as systematic uncertainties. It is also noted that the $A^{\rm hf}$ and $B^{\rm hf}$ parameters are highly correlated in the fit. The statistical error for the results of 37Ca in Table 1 takes the correlation into account to be conservative, and used in the present analysis. Without the correlation, statistical errors of the $A^{\rm hf}(^{2}S_{1/2})$ and $B^{\rm hf}(^{2}P_{3/2})$ of 37Ca are 6.3 MHz and 8.4 MHz, respectively. The quadratic sum of statistical and systematical uncertainty was taken as total uncertainty in the following discussion.

Unknown nuclear moments may be deduced from hf coupling constants using a reference nucleus of the same element, whose hf coupling constants for the same electronic level, nuclear spin and electromagnetic moments are known. A simple ratio of hf coupling constants derives nuclear moments as $\mu=\mu_{\rm R}\frac{A^{\rm hf}}{A^{\rm hf}_{\rm R}}\frac{I}{I_{\rm R}}$ and $Q=Q_{\rm R}\frac{B^{\rm hf}}{B^{\rm hf}_{\rm R}}$, where the subscript R indicates a reference nucleus. In the present study, 43Ca($I$ = 7/2) was employed as a reference, and $A^{\rm hf}(^{2}S_{1/2})$ = $-$806.40207160(8) MHz Arbes et al. (1994) and $\mu$ = $-$1.317643(7)$\mu_{N}$ Lutz et al. (1973) were used to extract the magnetic moment. A theoretical value of $eV_{zz}$ = 1.513(7) MHz/fm2 Sahoo (2009) was used for the extraction of $Q$ since a sufficiently precise measurement of $B^{\rm hf}_{\rm R}(^{2}P_{3/2})$ does not exist.

The spectroscopic quadrupole moments were determined to be $Q(^{37}{\rm Ca})=-15(11)~{}e^{2}{\rm fm}^{2}$ and $Q(^{39}{\rm Ca})=+3.82(27)~{}e^{2}{\rm fm}^{2}$. The present $Q$(39Ca) is consistent with the previous value Matsuta et al. (1999) determined using the $\beta$-NMR technique, and three times more precise than the previous value with experimental determination of the prolate deformation (the positive sign). The $Q$(37Ca) is determined for the first time in the present study including its oblate deformation (the negative sign). In the shell model calculation with the USDB interaction, discussed later in this Letter, effective charges, $e_{\rm p}$ = 1.5 and $e_{\rm n}$ = 0.5, gives $Q$(37Ca) = $-$2.6 $e^{2}$fm2. The agreement is fair, but no further discussion is made here because the present value has a large uncertainty due to the low signal-to-noise ratio to resolve the ${}^{2}P_{3/2}$ splitting in the hf spectrum.

The magnetic moment of 37Ca was determined from the $A^{\rm hf}$(${}^{2}S_{1/2}$) to be $\mu(^{37}{\rm Ca})=+0.7453(72)~{}\mu_{N}$. The result is summarized in Table 2. It is noted that the hf anomaly is neglected in the extraction of the magnetic moment. The hf anomaly, ${}^{1}\Delta^{2}$, is caused by the difference of the nuclear magnetization distribution Bohr and Weisskopf (1950) between two isotopes 1 and 2, and is given by $A^{\rm hf}_{1}/A^{\rm hf}_{2}\approx g_{1}/g_{2}\left(1+^{1}\Delta^{2}\right)$, where the $g$ factor is defined as $g=\mu/I$. For 39,43Ca, there exist independent measurements of $A^{\rm hf}$(43Ca) Arbes et al. (1994), $g$(43Ca) Lutz et al. (1973) and $g$(39Ca) Minamisono et al. (1976). The hf anomaly can be deduced together with the present value of $A^{\rm hf}$(39Ca) for the ${}^{2}S_{1/2}$ state as ${}^{43}\Delta^{39}=+0.0012(3)$. The hf anomaly between 37Ca and 43Ca is expected to be similar to ${}^{43}\Delta^{39}$, and the contribution to $\mu$(37Ca) is negligible compared to the experimental uncertainty.

Discussion — The magnetic moments of 37Ca and 37Cl with one particle in the $d_{3/2}$ shell, and their counterparts 39Ca and 39K with one hole in the $d_{3/2}$ shell provide a unique opportunity to study the first-order core-polarization model for the nucleon configuration mixing Arima and Horie (1954a, b). The first-order corrections are important for closed-shell nuclei, where the $j_{>}=l+1/2$ component of the spin-orbit pair is mostly filled, and the $j_{<}=l-1/2$ component is mostly empty, which we will call the $JJ$ closed-shell configuration. The 36Ca wavefunction is dominated by the $(d_{5/2})^{6}(s_{1/2})^{2}$ $JJ$-type configuration for neutrons, where the $d_{5/2}$ orbital is filled and the $d_{3/2}$ orbital is empty. When both $j_{>}$ and $j_{<}$ orbitals are mostly filled, which we will call the $LS$ closed-shell configuration, the core-polarization effect is small. The 40Ca wavefunction is dominated by the $(d_{5/2})^{6}(s_{1/2})^{2}(d_{3/2})^{4}$ $LS$-type configuration for neutrons. It is noted that both 36Ca and 40Ca have an $LS$ closed shell for protons. The observable associated with this change from $JJ$ to $LS$ closed shell configurations is the $\mu$(37Ca), which has one particle outside of the 36Ca neutron $JJ$ core, relative to that of 39Ca with one hole inside the 40Ca neutron $LS$ core. We can also observe the similar transition in their mirror $\mu$(37Cl) (one particle outside of the 36S proton $JJ$ core) relative to that of 39K (one hole inside the 40Ca proton $LS$ core).

For the calculations we use the $sd$-shell model space with the USDA and USDB Hamiltonians Brown and Richter (2006). The magnetic moment (M1) operator is defined as ${\bm{\mu}}=g_{l}\left<{\bm{l}}\right>+g_{s}\left<{\bm{s}}\right>+g_{p}\left<[{\bm{Y}_{2}},{\bm{s}}]\right>$, where $l$, $s$, $p$ represent the orbital angular momentum, spin and tensor terms, respectively. The results of the calculations are summarized in Table 2. The free nucleon $g$ factors ($g_{l}^{\rm p}$ = 1, $g_{l}^{\rm n}$ = 0, $g_{s}^{\rm p}$ = 5.586, $g_{s}^{\rm n}$ = $-$3.826) were used for single-particle (Schmidt) values denoted as “sp $g^{\rm free}$”. All other calculations were performed with the effective $g$ factors that are obtained from a six-parameter fit to other magnetic moments in the $A$ = 16 - 40 mass region Richter et al. (2008). The results labeled USDA-EM1 and USDB-EM1 are given in Table 2 and discussed in this paper.

The first-order core polarization is contained within the $sd$-shell model space for present calculations, and the effective $g$-factors reflect higher-order corrections due to correlations beyond the $sd$ model space and meson-exchange currents Richter et al. (2008). For the $d_{3/2}$ orbital the effective single-particle magnetic moment for $A$ = 37 or single-hole magnetic moment for $A$ = 39 denoted as “sp $g^{\rm eff}$” for neutrons (protons) is 0.930$\mu_{N}$ (0.469$\mu_{N}$) compared to the single-particle value of 1.148$\mu_{N}$ (0.124$\mu_{N}$), and the variation indicates the contribution from the higher-order corrections through $g^{\rm eff}$.

The experimental magnetic moment for 37Ca is in excellent agreement with USDA/B-EM1 calculations. The wavefunctions for 37Ca are given in Table 3 in terms of the percent probabilities for the six allowed partitions. The partitions are given in terms of the number of neutrons that occupy each orbital ($n$5, $n$1, $n$3) as in $(d_{5/2})^{n5}(s_{1/2})^{n1}(d_{3/2})^{n3}$. The partitions that are important for the magnetic moment are (621), (603) and (522). The magnetic moments for the (621) and (603) partitions are just the single-particle value of 0.930$\mu_{N}$. The interference of the (621) and (522) partitions decreases the magnetic moment to 0.760$\mu_{N}$/0.750$\mu_{N}$ for USDA/B-EM1. The addition of the other four partitions give a final result of 0.770$\mu_{N}$/0.754$\mu_{N}$ for USDA/B-EM1. These are in excellent agreement with experiment.

The mixing of the (522) partition with the $JJ$ closed-shell partition (620) plus one neutron particle in the $d_{3/2}$, (621), is the core-polarization effect, where one neutron is moved from the $d_{5/2}$ to the $d_{3/2}$ orbital Arima and Horie (1954a, b). On the other hand, the $LS$ closed-shell partition (604) plus one neutron hole in the $d_{3/2}$, (603), has no core-polarization correction. This is also true for the $LS$ closed shell partition (624) plus one neutron hole in $d_{3/2}$ (623) that is the $sd$-shell configuration for 39Ca. Thus, the magnetic moment is sensitive to the mixing between the $s_{1/2}$ and $d_{3/2}$ orbitals, via the relative amounts of the (621) and (603) partitions. The agreement with experiment indicates that 37Ca is dominated by the (621) partition that represents 36Ca in a $(d_{5/2})^{6}(s_{1/2})^{2}$ closed sub-shell configuration plus one neutron in the $d_{3/2}$ orbital.

The same situation occurs for the magnetic moment of the mirror nucleus 37Cl. The effective single-particle value of 0.469$\mu_{N}$ is increased by the core-polarization to 0.675$\mu_{N}$, which is also in good agreement with the experimental value.

As the configuration is varied from the $JJ$ type for 37Ca to the $LS$ type for 39Ca, the magnetic moment is expected to increase from 0.754$\mu_{N}$ (USDB) to 0.930$\mu_{N}$ (sp $g^{\rm eff}$). The experimental values increase from 0.7453(72)$\mu_{N}$ to 1.0217(1)$\mu_{N}$ Minamisono et al. (1976). The agreement of 37Ca is excellent but the calculation for 39Ca underestimates the experimental value as shown in Figure 2. This increase can be understood by observing that the ground state $I$ = 3/2 of 37Ca can be obtained with the excitation of neutrons from the $d_{5/2}$ to the $d_{3/2}$ shell, whereas for the $I$ = 3/2 state of 39Ca such excitation is prohibited in the $sd$-shell model space. A similar behavior can be seen in their mirror magnetic moments. Going from 37Cl to 39K we expect a reduction of the magnetic moment from 0.675$\mu_{N}$ to 0.469$\mu_{N}$, whereas the experimental values decrease from 0.6841236(4)$\mu_{N}$ Blaser et al. (1972) to 0.3915073(1)$\mu_{N}$ Sahm and Schwenk (1974).

The isoscalar $\mu$(IS) and isovector $\mu$(IV) parts of a magnetic moment are also evaluated, which are deduced as $\mu$(IS/IV) = $\mu$($T_{3}=+T$) $\pm$ $\mu$($T_{3}=-T$) with the isospin $T_{3}$ = +1/2 for protons. The isoscalar spin expectation value $\left<s_{z}\right>$ was evaluated as $\mu$(IS) = $I$ + 0.38$\left<s_{z}\right>$ Sachs (1946); Sugimoto (1969) and also listed in Table 2. The change in $\mu$(IS) from the single-particle value of 1.272 to those deduced from experimental moments of 1.4131(1) for $A=39$ and 1.429(7) for $A=37$, is mainly due to the change of $g^{\rm free}$ to $g^{\rm eff}$. This can be seen in Figure 2 as the “sp $g^{\rm eff}$” value (blue bar) for $\mu$(IS) well explains the experimental values for $A$ = 37 and 39. On the other hand, there is a large reduction in $\mu$(IV) coming both from the change of $g^{\rm free}$ to $g^{\rm eff}$ and from the core-polarization within the $sd$-model space. The $\mu$(IV) is reduced to be near zero for $A=37$ and the shell model calculation reproduces the experiment, however for $A=39$ the discrepancy is large. The good agreement for 37Ca and 37Cl confirm the importance of core-polarization for the $JJ$ closed-shell nuclei. The larger deviation between theory and experiment for 39Ca and 39K than those for 37Ca and 37Cl indicates that additional non-$sd$-shell components of 40Ca are larger than those of 36Ca and 36S. It is also noted that magnetic moments of heavier 41,43,45Ca suggest large nucleon excitations across the $sd$ shell around the neutron number $N$ = 20 Garcia Ruiz et al. (2015). In this regard it appears that the 36Ca (36S) nucleus may be a better closed sub-shell nucleus at $N$ = 16 ($Z$ = 16) than the 40Ca nucleus.

We also calculate magnetic moments of $A$ = 37 and 39 pairs using the valence-space formulation of the ab initio in-medium similarity renormalization group (VS-IMSRG) Tsukiyama et al. (2012); Bogner et al. (2014); Stroberg et al. (2017, 2016). In this approach we consistently transform the $M1$ operator and no effective $g$ factors were used Parzuchowski et al. (2017); Henderson et al. (2018). The 1.8/2.0 chiral interaction defined in Refs. Hebeler et al. (2011); Simonis et al. (2016, 2017) was taken as the initial two- and three-nucleon potentials within a harmonic oscillator basis of 13 major shells, a frequency $\hbar\omega=16$ MeV, operators truncated at the two-body level and $sd$-shell model space with a 16O core. The results are summarized in Table 2 and 3. The calculations for 37,39Ca overestimate the experimental values, but have the canonical nucleon configuration of 90% and confirm a closed sub-shell structure of 36Ca. Compared to the USDA/B-EM1 calculations, the VS-IMSRG agrees with the dominance of (620) partition for 36Ca. However, the amount of the (522) partition that gives the core-polarization correction is a factor of two larger. The deviation is likely due to meson-exchange currents Bacca and Pastore (2014), which are not included in the present VS-IMSRG calculations, but are included indirectly through the effective $g$ factors in the USDA/B-EM1 calculations.

Summary – Bunched-beam collinear laser spectroscopy was performed to determine electromagnetic moments of 37Ca to probe the closed sub-shell nature of the 36Ca nucleus. Shell-model calculations were performed in the $sd$-shell-model space with the USDA/B-EM1 interaction and effective $g$ factors. The calculated $\mu$(37Ca) reproduces the experimental value with a 95% probability of the canonical nucleon configuration, yet the first-order core polarization effect within the $sd$ shell is critical for the agreement. The calculated value for $\mu$(39Ca), which has one neutron hole inside the 40Ca core, shows poor agreement with the experimental value compared to that of $\mu$(37Ca). A similar behavior can be seen in magnetic moments of the mirror partners 37Cl and 39K. This indicates that additional non $sd$-shell components of 40Ca are larger than those of 36Ca. The 36Ca nucleus appears to be a better closed sub-shell nucleus at $N$ = 16 than 40Ca as represented by the USDA/B-EM1 Hamiltonian. The ab-initio VS-IMSRG calculations give reasonable agreement with experimental $\mu$(39,37Ca) and confirms a closed sub-shell structure of 36Ca.