Quadrupole Moments of Neutron-Deficient $^{20, 21}$Na

The electric-quadrupole coupling constant of the ground states of the proton drip line nucleus $^{20}$Na($I^{\pi}$ = 2$^{+}$, $T_{1/2}$ = 447.9 ms) and the neutron-deficient nucleus $^{21}$Na($I^{\pi}$ = 3/2$^{+}$, $T_{1/2}$ = 22.49 s) in a hexagonal ZnO single crystal were precisely measured to be $|eqQ/h| = 690 \pm 12$ kHz and 939 $\pm$ 14 kHz, respectively, using the multi-frequency $\beta$-ray detecting nuclear magnetic resonance technique under presence of an electric-quadrupole interaction. A electric-quadrupole coupling constant of $^{27}$Na in the ZnO crystal was also measured to be $|eqQ/h| = 48.4 \pm 3.8$ kHz. The electric-quadrupole moments were extracted as $|Q(^{20}$Na)$|$ = 10.3 $\pm$ 0.8 $e$ fm$^2$ and $|Q(^{21}$Na)$|$ = 14.0 $\pm$ 1.1 $e$ fm$^2$, using the electric-coupling constant of $^{27}$Na and the known quadrupole moment of this nucleus as references. The present results are well explained by shell-model calculations in the full $sd$-shell model space.


Introduction
The spectroscopic electric-quadrupole moment (Q) provides a direct measure of the deviation of the charge distribution in a nucleus from spherical symmetry and thus is sensitive to details of the nuclear wavefunction. Q is often used in tests of various theories, which attempt to reproduce experimental data.
Shell-model calculations have been successful in predicting Q near the stability line in the nuclear chart, where the required effective interactions are well constrained by experimental data (for example, Ref. [1]). Proton and neutron effective charges are usually required in shell-model calculations to obtain Q.
Within a given shell-model space, the effective charges represent 2 ω, I π = 2 + excitations of the core nucleons to valence orbits [2] and reflect the virtual excitation of the isoscalar and isovector giant quadrupole resonances. Values of effective charges for sd-shell nuclei have been obtained from a systematic analysis of experimental E2 matrix elements [2,3].
When moving away from stability, however, some nuclei show significant disagreement between experiment and theory. The neutron-rich B isotopes, for example, show a reduction of the neutron effective charge [4] because the loosely-bound valence neutrons, far removed from the core, have less probability to excite the core than well bound neutrons. Such variation of the neutron effective charge can also be seen in the neutron-rich 16 N [5] and 18 N [6]. Neutron-deficient nuclei, especially those with small proton separation energies, may also be expected to show variation of effective charges. One such example is the Q of neutron-deficient 37 K [7], which requires an increased neutron effective charge and a decreased proton effective charge relative to the typical values of nuclei closer to stability in the sd shell. This is attributed to a substantial coupling to the isovector giant resonance beyond that indicated by the typical effective charges. Experimental Q of such neutron-deficient nuclei are still scarce, even in the sd shell. Additional systematic data of Q of neutron-deficient nuclei are important to further improve our knowledge of the exotic structure of drip line nuclei.
The Q of the neutron-rich Na isotopes have been measured via laser spectroscopy [8] and in β-ray detecting nuclear magnetic resonance (β NMR) [9] experiments. Intrusion of the f p shell across the N = 20 shell gap can be clearly seen in Q of ground states of neutron-rich Na isotopes beyond 29 Na in the deviation between experiment and shell model calculations restricted to the sd-shell model space [10]. Improved agreement is realized in the Monte Carlo Shell Model approach, where both the sdand f p-shell model spaces are considered [10]. Compared with the well studied neutron-rich Na isotopes, precise Q for the neutron-deficient Na isotopes are still lacking. The Q of the proton drip line nucleus 20 Na(I π = 2 + , T 1/2 = 447.9 ms) was reported in a figure in Ref. [11] but no experimental detail nor a value of Q was given. The Q of neutron-deficient nucleus 21 Na(I π = 3/2 + , T 1/2 = 22.49 s) disagrees with a shell-model calculation in the sd-shell model space [Q theor. ( 21 Na) = 11.6 e fm 2 (calculated using OXBASH shell model code [12]], although it has a large error [Q( 21 Na) = 6 ± 3 e fm 2 (reevaluated value using Ref. [8,13,14])].
Q( 20 Na) and Q( 21 Na) have been precisely determined in this study using the multi-frequency β-NMR technique under presence of an electric-quadrupole interaction. Preliminary results were reported in Ref. [15]. The systematic behavior of Q(Na) across the full neutron sd shell, including the present results of Q( 20 Na) and Q( 21 Na), is discussed.

EXPERIMENT
The experiments were performed at the radioactive beam facility ISAC-I at TRIUMF. The electric-quadrupole coupling constants, eqQ/h, of 20,21,27 Na were measured in separate runs. The experimental procedure in the measurement of 20 Na is explained below. Similar procedures were taken in the measurements of 21,27 Na and the minor changes in conditions among the three measurements are summarized in Table 1. The 500 MeV proton beam from the TRIUMF cyclotron was used to bombard a thick SiC production target, which was coupled to a surface ion source. 20 Na singly-charged ions were extracted at an energy of 40.8 keV and mass separated. The pure 20 Na beam was transported to the polarizer beam line [16] in the ISAC-I experimental hall. The 20 Na ions first passed through a Na vapor and were neutralized by charge exchange reactions. Collinear laser pumping was used to polarize the Na atoms by the D 1 transition (3s 2S 1/2 ↔ 3p 2P 1/2 ) and circularly polarized light [17]. Both of the ground state hyperfine levels (3s 2S 1/2 F = I + 1/2 and I − 1/2) were pumped to achieve high polarization (see Table 1) using side band frequencies produced by an electro-optic modulator (EOM), a technique that was successfully employed in the past [18]. The collinear laser light was generated by a Coherent 899-21 frequency-stabilized ring-dye-laser pumped by a 7-W argon-ion laser. The polarized atoms were then re-ionized in a He-gas target to be deflected to the β-NMR station.
The polarized 20 Na ions were delivered to the β-NMR apparatus and implanted into a hexagonal ZnO single crystal. The implantation depth was ∼ 500Å.
An external dipole-magnetic field of B 0 = 0.5286 ± 0.0005 T was applied parallel to the direction of polarization to maintain the polarization in ZnO and to make the magnetic sublevels split (Zeeman splitting). The 20 Na nucleus decays mainly to the first excited state in daughter nucleus 20 Ne by emitting β + rays with a half-life of 447.9 ms. The maximum β-ray energy is 11.23 MeV.
β rays from the stopped 20 Na were detected by a set of plastic scintillation counters placed at 0 • (u) and 180 • (d) relative to the external field direction.
The counting rate is asymmetric between u and d counters for a polarized source. The angular distribution, depends on the asymmetry parameter A s , the polarization P and the angle ϕ between the direction of the momentum of the β ray emitted in the decay and the polarization axis. The A s for the β decay of 20 Na to the first excited state in 20 Ne is A s = 1/3. The initial magnitude (P 0 ) and spin-lattice relaxation time (T 1 ) of the nuclear polarization of 20 Na were measured in this study from the asymmetric β-ray angular distribution. P 0 = 37 ± 1% and T 1 = 9 ± 0.5 s were obtained for 20 Na in ZnO. The preliminary results of production of polarization were reported in Ref. [19]. The long T 1 relative to the half-life introduced virtually no significant loss of polarization in the NMR measurement.
The ZnO was chosen for implantation to measure eqQ/h because polarization of Na isotope is well maintained in the ZnO with long relaxation time [19] relative to their lifetimes. The ZnO had its c-axis set perpendicular to the external magnetic field. The electric-field gradient, q, in ZnO is axially symmetric (asymmetry parameter η = 0) and parallel to the c-axis. The Hamiltonian of the electromagnetic interaction between nuclear moments and external fields [20] in this condition is given by where µ is the magnetic moment, H 0 is the magnetic field, I is the nuclear spin and I z is the third component of the spin operator. The magnetic sublevel energies, E m , of the Na ions implanted in ZnO are given by where m is the magnetic quantum number, g is the nuclear g factor, ν Q = 3eqQ/{2I(2I − 1)h} is the normalized electric-quadrupole frequency and θ is the angle between the c-axis and the external magnetic field. Eq. (3) is given to first-order in eqQ/h, taking the electric-quadrupole interaction as a perturbation to the main magnetic interaction. The first term in Eq. (3) involves the 2I + 1 magnetic sublevels separated by a fixed energy value due solely to the magnetic interaction (Zeeman splitting). These sublevels are further shifted by the electric-quadrupole interaction and the energy spacing between adjacent sublevels is no longer constant. The 2I separate transition frequencies that appear due to the electric-quadrupole interaction are determined as The transition frequencies correspond to the energy difference between two adjacent levels in Eq. An asymmetry change, A s P , in the β-decay angular distribution is obtained as the NMR signal: The double ratio, R, is defined by β-ray countings, W (θ), as where the subscript, off(on), stands for without(with) radio frequency (RF) applied. An adiabatic fast passage (AFP) method [21] was used in the RF application for NMR. AFP inverts the direction of initial polarization (P → −P ) and effectively doubles the signal size over a depolarization method for a more efficient NMR measurement.
The NMR signal can be maximized and therefore effectively searched by applying all the 2I transition frequencies in Eq. (4)  is a constant. This multi-frequency β-NMR technique is discussed in detail elsewhere [22,23].
A schematic of the multi-frequency β-NMR system is shown in Fig. 1. A computer controlled RF generating system produces an RF signal, which is sent to a 300 W amplifier. The amplified signal is applied to an RF coil, which is part of an LC resonance circuit. The circuit includes an impedance matching transformer and a bank of six selectable variable capacitors. After 10 ms RF time for 20 Na, another frequency is generated by the RF generating system and sent to the same LC resonance circuit. A different capacitor, which has been tuned to the second frequency to satisfy the LC resonance condition, is selected by the fast switching relay system. The system ensures sufficient power for any set of four transition frequencies for 20 Na (I = 2) over the expected search region of eqQ/h. The switching time between two signals is 3 ms.
Since an RF corresponds to a transition of populations between adjacent m − 1 and m, the AFP interchanges the a m−1 and a m . The total inversion of polarization (P → −P ) was achieved by applying RF as shown in Fig. 2 for 20 Na (I = 2). After 10 sequential applications of transition frequencies, the direction of initial polarization is inverted, where P is defined as P = I m=−I a m m/I. The typical inversion efficiency, defined by P ′ = αP 0 , was α ∼ −0.81, where P ′ is the inverted polarization. Each applied RF was frequency modulated with a modulation width, FM = ± 20 kHz, for AFP. The FM is also to cover a certain region of eqQ/h for an effective search for resonance. The  Table 1.
Multi-frequency β-NMR spectra were measured both with a positive-helicity (σ + ) and a negative-helicity (σ − ) laser light. An NMR signal, 2A s P av , was defined as: to maximizes the NMR signal obtained.
The second error is the systematic error in determining the centroid. Possible variations of the centroid due to the sub resonance were considered in the systematic error based on the errors of q s /q m , k s /k m and σ s /σ m . One tenth of the σ m was also included in the systematic error [24] for uncertainties caused by the line width of main resonance due to the FM. The latter dominates the present systematic error.
Q may be extracted from these eqQ/h together with an eqQ/h of a Na isotope in ZnO, whose Q is already known as the electric-field gradient is identical among Na isotopes. The eqQ/h of 27 Na in ZnO was measured for this purpose.
The Q is precisely known in a separate measurement as Q ( Possible variations of the centroid, due to the sub resonance as well as a one tenth of the σ m were included in the systematic error. The Q of Na isotope with a mass number A can now be extracted as Q( 20 Na) and Q( 21 Na) were precisely extracted from Eq. (11) and Q( 27 Na) [9] as |Q( 20 Na)| = 10.3 ± 0.8 e fm 2 , The statistical and systematic errors were added quadratically. The results are summarized in Table 2 together with the previously known Q and the shell-model predictions discussed in the following section.

DISCUSSION
The present Q( 20 Na) is consistent with and as precise as the previous report ± 0.015 MHz [13]), and Q of 23 Na (Q( 23 Na) = 10.6 ± 0.1 e fm 2 [14]). The large error in the previous Q( 21 Na) is due to the large error in ν 0 (F = 3 ↔ 2), which is essentially a systematic error caused by the detuning of the laser frequency from the exact resonance [8]. The deviation between present and previous Q( 21 Na) may reside in the systematic error. It is noted that the similar discrepancy between Q measured by β-NMR [9] and optical [8] technique is systematically seen in other Na isotopes, which was shown in Fig. 5 of Ref. [9] and also in Fig. 6.
Theoretical calculations were performed using OXBASH shell model code [12] in the full sd-shell model space from A = 20 to 27 with the USDA interaction [1] and Woods-Saxson single-particle wave functions. Calculations above A = 28 were not included, although Q are known up to A = 31 [11,25]. The contribution from 2p-2h intruder configuration across the N = 20 shell gap becomes important above 29 Na, and these nuclei therefore do not compare well with shell-model calculations in the sd-shell model space [10]. Theoretical Q were calculated with Q theory = e p Q p 0 + e n Q n 0 , where Q  Table 3 and shown in Fig. 6, where signs of present results (the solid circles) were taken from the calculations. Calculations with the USD [29] or the USDB for by the USDA interaction [1], which is determined by fitting energy levels of only neutron-rich nuclei in the sd shell. The present Q( 21 Na) reconciles the discrepancy between the previous Q( 21 Na) and the theoretical calculation, and completes the account of the systematic behavior of Q of Na isotopes across the neutron 1d 5/2 and 2s 1/2 shells except Q( 24 Na), which is not measured yet.
No variation of effective charges were required to realize agreement between the shell-model calculations in the sd-shell model space and experiment. This was most surprising for the case of 20 Na, which lies at the proton drip line.
Nuclei adjacent to both the proton and neutron drip lines have been shown to require significantly different e p and e n values to reproduce experimental Q. The neutron-rich B isotopes, as discussed in section 1, show a reduction of the neutron effective charge [4] and neutron-deficient nucleus 37 K requires an increased neutron effective charge and a decreased proton effective charge [7].
The isovector part, e pol , of effective charges was varied to investigate its contribution to the present Q, keeping the isoscalar part, e pol , the same as the one determined from the E2 matrix elements between low-lying states [3], which are sensitive to the e  Table 3.
Results of shell-model calculations from A = 20 to 25 with valence nucleons restricted in the 1d 5/2 shell are shown in Fig. 6 in the dotted-dashed line.
In the calculation, the USDA interaction and Woods-Saxson single-particle wave functions were used. The calculated Q are systematically smaller than experimental values and indicate an importance to include the 2s 1/2 and 1d 3/2 shells for configuration mixing within the sd-shell model space. By including the 2s 1/2 and 1d 3/2 shells in the calculation, each component of the wave function contributes coherently to Q and adds up to the theoretical Q, which is discussed above and shown in the solid line in Fig. 6. This is again an indication of the collective nature of Q of Na isotopes (A = 20 ∼ 24). The small Q of 25,26,27 Na are well explained in the framework of the single-particle picture as follows. The 3 valence protons, occupying half the 1d 5/2 shell, yield zero Q [30]. The 6 neutrons in 25 Na, that fully occupies the 1d 5/2 shell, couples to 0 + and the 7th and 8th valence neutrons in 26,27 Na occupy the 2s 1/2 shell.
A ratio of Q between 20 Na and 21 Na can be precisely determined from the present eqQ/h [Eqs. (8) and (9) The ratio clearly indicates that the Q( 21 Na) is larger than Q( 20 Na). The ratio is consistent with the trend of the Q predicted by shell-model as seen in   Fig. 1. Schematic illustration of the multi-frequency β-NMR system. The rf coil was placed in a vacuum chamber, which is not drawn here.      Table 3 for the values. The solid line shows the shell model results with the USDA interaction [1]. Table 1 Experimental conditions in the eqQ/h measurements of 20,21,27 Na. P 0 is the initial polarization, T 1 is the relaxation time of polarization, E 0 is the maximum β-decay energy of the largest branch, B r , and A s is the β-decay asymmetry parameter integrated over measured branches.  Q theory (e fm 2 ) [12] +9.9 +11.6 −1.1 Table 3 Shell-model calculations of Q of Na isotopes. Q theory were obtained using e p = 1.3e and e n = 0.5e. Signs of Q exp. are given when the sign can be determined by the experiment.