Measurement of the electric quadrupole moment of 32Al

Physics Letters B 647 (2007) 93–97 www.elsevier.com/locate/physletb

Measurement of the electric quadrupole moment of 32Al D. Kameda a,∗ , H. Ueno a , K. Asahi a,b , M. Takemura b , A. Yoshimi a , T. Haseyama a , M. Uchida b , K. Shimada b , D. Nagae b , G. Kijima b , T. Arai b , K. Takase b , S. Suda b , T. Inoue b , J. Murata d , H. Kawamura d , Y. Kobayashi a , H. Watanabe c , M. Ishihara a a RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan b Department of Physics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan c Department of Nuclear Physics, Research School of Physical Sciences and Engineering, The Australian National University, Canberra AT 0200, Australia d Department of Physics, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima, Tokyo 171-8501, Japan

Received 6 November 2006; received in revised form 17 January 2007; accepted 23 January 2007 Available online 15 February 2007 Editor: D.F. Geesaman

Abstract The electric quadrupole moment Q for the ground state of 32 Al has been measured using the β-NMR technique. Spin-polarized 32 Al nuclei were obtained from the fragmentation of 40 Ar projectiles at E/A = 95 MeV/nucleon, and were implanted in a single crystal α-Al2 O3 stopper. The quadrupole moment was deduced from the measured quadrupole coupling constant. The obtained value, |Q(32 Alg.s. )| = 24(2) mb, is remarkably small as compared with other Al isotopes, but is well explained by shell model calculations within the sd shell. The present result indicates that 32 Al has a normal sd-shell structure, in contrast to the neighboring N = 19 isotone 31 Mg for which a strongly deformed intruder ground state has been reported to occur. © 2007 Elsevier B.V. All rights reserved. PACS: 21.10.Ky; 24.70.+s; 23.20.En; 21.60.Cs Keywords: Electric quadrupole moment of 32 Al; Projectile fragmentation; β-NMR; Nuclear deformation; Island of inversion

Among neutron-rich Ne, Na and Mg isotopes, those with neutron numbers around N = 20 are reported to show large deformations that cannot be explained by the conventional shell model assuming the N = 20 shell closure [1–4]. The anomalous deformations are reproduced by introducing configurations in which some neutrons are allowed to occupy the pf orbits via neutron excitations from the sd orbits [5–8]. Such a region where the sd-shell closure deteriorates and an intruder configuration dominates the ground-state structure is known as the “island of inversion” [8]. Studies of the electromagnetic moments of nuclei in the vicinity of the island [9–13] have played an important role in elucidating the process of transition from sd-normal to pf -intruder structures. In the case of neutron-rich Na isotopes, the magnetic moments μ of 30,31 Na,

* Corresponding author.

E-mail address: [email protected] (D. Kameda). 0370-2693/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2007.01.063

and the electric quadrupole moments Q of 29–31 Na were reported to significantly deviate from the conventional sd-shell model calculations, in contrast to the case of lighter Na isotopes where the μ and Q moments are reasonably reproduced within the sd shell [9,10]. Later, these differences were consistently treated in Monte Carlo shell model (MCSM) calculations including the pf orbits [14]. According to the MCSM calculations, the pf -intruder configuration dominates the ground state of 31 Na (N = 20) and 30 Na (N = 19), while it substantially mixes with the sd-normal configuration in the 29 Na (N = 18) ground state. The calculation on 29 Na further indicates that the value of the Q moment is appreciably affected by such a midway mixing of the pf orbits while that of the μ moment remains rather insensitive. The calculations in Ref. [14] also shed light on the role played by the effective energy gap between the sd and pf shells, that varies with Z and N and drives the variation of the nuclear wave function from the normal to intruding configurations.

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By following the line of the N = 19 isotonic chain, one notes that both 30 Na (Z = 11) and 31 Mg (Z = 12) (recently, the magnetic moment and spin of the ground state have been measured [11]) exhibit features of strong deformation. Since the driving parameter for the occurrence of inversion should be varying gradually with Z, one would expect for the next N = 19 isotone, 32 Al (Z = 13), that dominance or mixing of the pf -intruding configuration might occur. In this respect we have previously measured the magnetic moment of 32 Al [13], and found that the observed μ moment was well reproduced by the full sd-shell model. This result indicates that the ground state of 32 Al hardly gives rise to the inversion to the fullest extent. On the other hand, we note that the Q moment may provide a more sensitive probe to the degree of nuclear deformation. In the present work, we thus measured the Q moment of 32 Al to further elucidate the collective aspect of the isotope. If intruder configurations mix with the sd-normal configurations, 32 Al may show an enhanced Q moment over the conventional sd-model prediction, as is observed in the case of 29 Na. The experiment was carried out at the RIKEN accelerator research facility, using the projectile-fragment separator RIPS [15]. The 32 Al beam was obtained from the fragmentation of 95 MeV/nucleon 40 Ar projectiles with a 0.37 g/cm2 thick Nb target. Typical intensity of the 40 Ar beam was 40 particle-nA at the target. The 32 Al fragments were separated from other projectile fragments by using two dipole magnets and a 444 mg/cm2 -thick aluminum degrader placed on the intermediate focal plane of RIPS. To obtain polarized 32 Al nuclei, the direction of the primary beam before the target was inclined by a beam swinger so that the fragments emitted at θ = 1.3◦ –5.7◦ from the primary beam axis were accepted by RIPS. A momentum region of width p/pc = ±3% centered at pc = 12.6 GeV/c, which corresponded to the peak in the yield, was selected. The spin-polarized 32 Al beam thus obtained had a polarization about 0.5% (see Fig. 3) and a typical intensity of 3.6 × 103 particles per second at the second focal plane, with isotopic purity of 85%. The polarized beam was introduced into the NMR apparatus located at the final focus of RIPS, where the 32 Al nuclei were implanted in a stopper of single crystal α-Al2 O3 (corundum). The static magnetic field B0 ∼ 0.5 T was applied to the stopper in the direction perpendicular to the reaction plane, so as to preserve the polarization. The temperature of the stopper was kept around 80 K to reduce the spin-lattice relaxation in the stopper. The experimental setup around the stopper was almost identical to that schematically shown in Fig. 1 of Ref. [13], except for the setting of the angle θc between the crystal c-axis and the B0 field as described later. In an α-Al2 O3 crystal, oxygen atoms form the h.c.p. structure, and 2/3 of the octahedral cages in it are occupied by aluminum atoms. There are, therefore, two possible locations for an 32 Al atom in the crystal, the one is the substitutional site of 27 Al with 2/3 probability, and the octahedral interstitial site with 1/3 probability. Thus, the major stopping site for 32 Al in α-Al O is expected to be the substitutional site of 2 3 27 Al. The electric field gradient tensor eq in the substitutional

site is well studied using stable nuclei 27 Al, and is reported to show a good axial symmetry about the c-axis of the h.c.p. lattice [16]. To determine the quadrupole coupling constant νQ of 32 Al in α-Al2 O3 , we applied the oscillating magnetic field to the stopper in the following scheme. According to the first order perturbation theory, the resonance frequencies f+ and f− for 32 Al (I = 1) in the Zeeman plus quadrupole interactions are given [17] by f± (νQ ) = ν0 ∓

3νQ 3 cos2 θc − 1 , 4 2

(1)

where the Larmor frequency ν0 = gμN B0 / h and the quadrupole coupling constant νQ = e2 qQ/ h are related to the nuclear g-factor g and quadrupole moment eQ, respectively. Here μN , h and e denote the nuclear magneton, the Planck’s constant and the electric charge, respectively. Note for later discussion, that the dependence of f± on νQ is by a factor of 2 weaker for θc = 90◦ than for θc = 0◦ . Now, suppose that a region νQi f

to νQ is searched. This is accomplished, according to Eq. (1), by sweeping the frequency f of the RF field over two differf ent frequency regions F± given by F+ : f+ (νQi ) − f+ (νQ ) and f

f

F− : f− (νQi ) − f− (νQ ). If νQ is within the region νQi to νQ , a frequency sweep over the region F− interchanges the populations between the magnetic sublevels of m = −1 and m = 0, by means of the adiabatic fast passage (AFP) technique [17]. In the next step, the resulting population in m = 0 is interchanged with the population in m = +1 by a sweep over the region F+ . Finally a sweep over F− again interchanges the populations between m = −1 and m = 0, thus completing the total population reversal. Each step of sweep was made in a 2 ms duration. Consequently, the sequence of sweeps F− –F+ –F− was thus carried out in a 6 ms duration. To fulfill an adiabatic condition for the population reversal, each frequency sweep was restricted in a small range due to limitation of the RF power. Thus, the measurement was performed in two steps in order to determine the quadrupole coupling constant. In the first step, the stopper was aligned at θc = 90◦ , and a rough search for the resonance was made in a wide νQ region. In the second step, νQ was precisely determined with a stopper angle θc = 0◦ , by exploiting the stronger sensitivity of f± to νQ for the θc = 0◦ case [see Eq. (1)]. The RF field application was controlled by an arbitrary waveform generator. Beta rays emitted from 32 Al nuclei were detected by using plastic scintillator telescopes located above and below the stopper. The angular distribution of β rays emitted from polarized nuclei is expressed as W (θ ) = 1 + v/c · Aβ P cos θ where P and Aβ denote the degree of polarization and the β-ray asymmetry factor, respectively. (For the β decay of 32 Al, Aβ ∼ = −0.85.) θ is the β-ray emission angle measured from the axis of polarization. The ratio of the β-ray velocity v to the speed of light c was approximated as v/c ≈ 1. In the β-NMR technique, the resonance is detected as a change of the asymmetry in W (θ ). If the full reversal of polarization (P → −P ) is achieved, the up/down ratio R = W (0◦ )/W (180◦ ) of the β-ray counting rates

D. Kameda et al. / Physics Letters B 647 (2007) 93–97

Fig. 1. Time chart for measurement of β-ray asymmetry change. Tbeam , Trf and Tcount denote the time periods for the beam implantation, RF-field application and β-ray counting, respectively. In the present experiment, Tbeam = Tcount = 46 ms and Trf = 6 ms. “RF-i” (i = 1, 2, . . . , n) indicates a cycle of β-asymmetry measurement with the sweep over the ith νQ region, while the cycle without RF field is denoted by “RF-off”. The number of cycles in the set, for example, took n = 7 in Fig. 3(b).

changes as R=

1 − Aβ P 1 + Aβ P → R = , 1 − Aβ P 1 + Aβ P

(2)

and consequently the double ratio R  /R deviates from unity as R  (1 − Aβ P )2 ∼ = = 1 − 4Aβ P , R (1 + Aβ P )2

(3)

when the AFP condition is fulfilled. Fig. 1 shows the time chart for the each process described above, the beam implantation, RF-field application and β-ray counting. The beam was pulsed with beam-on and off periods of 46 ms and 58 ms, respectively. These time scales were chosen so as to optimize the β-ray counts under the condition of the given decay half-life of 32 Al t1/2 = 35(5) ms [18] and the limited β-counting time that should exclude the RF-application period as well as the beam-on period. In the beam-off period, an RF field was first applied to the stopper to reverse the direction of polarization of 32 Al. In Fig. 1, the applied RF field is represented by F− F+ F− . The β-rays were counted during the succeeding period of 46 ms to obtain the β-ray up/down ratio R  in Eq. (2). During the remaining 6 ms, the same RF field was applied again to restore the original direction of polarization. In the following n − 1 cycles, the above procedure was repeated with the RF fields for the respective νQ regions, and then a cycle without RF fields was performed to obtain the up/down ratio R. The set of n + 1 cycles was repeated until sufficient statistics was obtained. Such repetition should serve to reduce possible systematic effects on R  and R arising e.g. from fluctuations of the beam profile at the stopper. Typical β-ray yield under this sequence was 100 counts per second. Fig. 2 shows the obtained β-ray asymmetry change 1−R  /R as a function of the assumed value of quadrupole coupling con-

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Fig. 2. β-Detected quadrupole resonance spectrum obtained for 32 Al implanted in an α-Al2 O3 stopper with its c-axis aligned perpendicular to an external field B0 . The ordinate represents the observed β-ray asymmetry which is defined by Eq. (3) in the text, while the abscissa is the quadrupole coupling constant νQ . The vertical bar attached to each data point (solid circle) indicates the statistical error, and the horizontal bar shows the region of νQ (of width  = 274 kHz) over which the frequency sweep was executed. The frequencies f± of RF fields corresponding to the sweep [see Eq. (1)] are indicated in two horizontal scales located below the plot.

stant νQ , with θc = 90◦ . The corresponding frequencies f± which were actually applied to the stopper are shown in the two horizontal scales below the plot. Horizontal bars attached to the data points represent the searched νQ regions with the sweep f width (≡ |νQ − νQi |) = 274 kHz. The resonance was observed in a region around νQ = 200–500 kHz. In the next step, we switched to the θc = 0◦ setup and performed measurements to acquire finer frequency spectra as shown in Fig. 3. The resonance was clearly observed in a region around νQ = 400 kHz as shown in Fig. 3(a). To determine the resonance frequency more precisely, we carried out measurements with a narrow sweep width  = 103 kHz. The result is shown in Fig. 3(b). Here, the Larmor frequency was taken as ν0 = 7440.3 kHz which was determined from the magnitude of B0 field and the g-factor of 32 Al. The quadrupole coupling constant νQ was determined from a fitting analysis of the two spectra shown in Fig. 3. For an RF f sweep over the region νQi to νQ , the change in the β-asymmetry 1 − R  /R (the quantity plotted in Figs. 2 and 3) should be given by  F(νc , ; νQ , σ, p) = p G(ν; νQ , σ )R(νc − ν, ) dν,   (ν − νQ )2 1 exp − , G(ν; νQ , σ ) = √ 2σ 2 2πσ

(4)

where the center νc and width  of the sweep are defined f f as νc ≡ (νQi + νQ )/2 and  ≡ |νQ − νQi |. In this expression, R(ν, ) represents the probability for a 32 Al spin to be reversed by AFP with a sweep region of width  and center displacement ν ≡ νc − ν from the resonance ν. The function

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Fig. 4. Comparison of the experimental Q moments (solid circles) and the USD shell model calculations (open squares) for aluminum isotopes. The vertical bar attached to each solid circle denotes the experimental uncertainty. The data were taken from Refs. [21–23]. In the calculation, the conventional effective charges (ep , en ) = (1.3, 0.5) [28] were employed.

Fig. 3. β-Detected quadrupole resonance spectra obtained for 32 Al in an α-Al2 O3 stopper with its c-axis aligned parallel to an external magnetic field B0 . See also the caption of Fig. 2. Widths of the νQ sweep were  = 137 kHz (a) and  = 103 kHz (b).

R was determined numerically from simulations of the AFP spin reversal process taking the actual conditions for RF fields into account. Another factor G(ν; νQ , σ ) phenomenologically takes into account the distribution of the resonance frequency ν around its mean value νQ , and is expressed as a Gaussian of width σ . The remaining factor p = 4Aβ P is the asymmetry change to be obtained when the spin reversal is fully achieved. The fittings of the function F were carried out by varying the parameters νQ , σ and p so as to best reproduce the data in Figs. 3(a) and 3(b) at the same time. Solid curves in Figs. 3(a) and 3(b) indicate the best fit functions thus obtained by the least-χ 2 method. Thus, the quadrupole coupling constant was determined as νQ = 407 ± 22 kHz. The uncertainty in the Larmor frequency was evaluated to be δν0 = 19 kHz based on the errors for the 32 Al g-factor and the B0 field. Here we employed a more precise value of |g| = 1.951(5) than the earlier reported value of |g| = 1.959(9) [13], which we have recently obtained using a single crystal Si stopper [19]. (Very recently, the g-factor of 32 Al has been reported to be |g| = 1.9516(22) [20] which is in good agreement with the employed g-factor.) The error in the B0 value was measured to be 0.04% from proton-NMR measurements during the same beam time, which is an order of magnitude smaller than the error in the g-factor. Note that the width of the sweep

F− and F+ is given by 3/4 · (3 cos2 θc − 1)/2 and the uncertainty in ν0 should be smaller than half of it, which took a value 39 kHz in Fig. 3(b). Otherwise, the spectral line shape would be smeared due to a possible mismatch of the swept regions with the real resonance frequencies. Other uncertainties such as due to the settings of frequency and θc were not important in determining νQ . In fact, the uncertainty of θc , δθc ≈ 1◦ , corresponds to the error of 0.06% on νQ . As a result, the magnitude of the quadrupole coupling constant for 32 Al was obtained as |νQ (32 Al)| = 407 ± 34 kHz. The sign of νQ was not determined in the present measurement. The Q moment was deduced from the relation |Q(32 Al)| = |Q(27 Al) · νQ (32 Al)/νQ (27 Al)|, where Q(27 Al) and νQ (27 Al) denote the Q moment for 27 Al and the quadrupole coupling constant for 27 Al in α-Al2 O3 , respectively. Thus, by inserting Q(27 Al) = 140.2(10) mb [21] and νQ (27 Al) = 2389(2) kHz [24], the electric quadrupole moment for the ground state of 32 Al was obtained as |Q(32 Alg.s. )| = 24(2) mb. In Fig. 4, the present result is compared with the Q moments for other Al isotopes. The sign of the Q moment of 32 Al is assumed to be positive in Fig. 4 according to the shell model calculations described below. The open squares indicate the results of sd-shell model calculations with the USD effective interaction [25] using the shell-model code OXBASH [26]. In the calculation, the radial integral part of the E2 operator was evaluated using the harmonic oscillator basis [27]. The standard values of effective charges (ep , en ) = (1.3, 0.5) for the sd-shell nuclei [28] were employed. It was earlier shown that the E2 effective charges ep and en may decrease as the N/Z ratio increases [29–31]. The use of ep and en values as evaluated through Eqs. (8) and (9) of Ref. [30] leads to a smaller calculated value, Qcalc (32 Al) = 21 mb, as compared to Qcalc (32 Al) = 27 mb obtained with the standard ep and en . This however does not much affect the argument below. Fig. 4 shows that the sd-shell model calculations well reproduce the systematic behavior of the Q moments of the Al isotopes although the data for N = 16–18, 20 are absent. This

D. Kameda et al. / Physics Letters B 647 (2007) 93–97

agreement provides clear evidence that the ground state of 32 Al is dominated by sd configurations. In fact, the Monte Carlo shell model (MCSM) calculations with an sdpf model space [32] show that sd configurations dominate the 32 Al ground state by ∼ 90% (the Q moment of 32 Al calculated by MCSM is almost identical to that by the USD calculation). Here we further notice that the Q moment of 32 Al is excessively small among those of the Al isotopes with natures of sd-shell nuclei. In terms of the sd-shell model the ground-state wave function for 32 Al π + is dominated by the |(πd5/2 )−1 ⊗ (νd3/2 )−1 I =1 configuration, where extra smallness of Q arises due to reduced values of geometrical factors that enter in the coupling of angular momenta forming the I = 1 nuclear spin. The normal nature of the ground state of 32 Al thus supported by the present work is in sharp contrast to the large deformation reported on the lighter N = 19 isotones 30 Na [9,10,14] and 31 Mg [11,33]. This indicates that there occurs a sudden structural change between the two adjacent N = 19 isotones, 32 Al and 31 Mg, a situation which is different from the case of the sodium isotope chain where the transitional isotope, 29 Na, exhibits a midway mixing of pf intruder configuration [14,34]. In contrast to the ground state of 32 Al, the level structure including an isomer [35–37] is not reproduced by the USD shell model calculations, suggesting the pf intrusion in the low-lying excited levels of 32 Al. In addition, the ground state of 33 Al (N = 20) is reported to show predominant sd-character in the β-decay [38], while a non-negligible contribution of intruder configurations is suggested by the magnetic moment measurement [20]. Furthermore, the presence of the low-lying 2p–2h state in 33 Al is pointed out by the inelastic scattering [39]. The coexistence of the sd-normal (spherical) and pf -intruder (deformed) states is suggested in 34 Al [40]. Although further investigation is needed, these observations have provided indication for the reduction of the sd–pf shell gap in the neutron-rich Al isotopes. In summary, taking advantage of spin-polarized radioactiveion beam, we have measured the electric quadrupole moment for the ground state of 32 Al using the β-NMR technique. The obtained Q moment |Q(32 Alg.s. )| = 24(2) mb is an order of magnitude smaller than the known Q moments of the other Al isotopes, but is quite compatible with the conventional shell model calculations within the sd-model space. The result provides clear evidence that the ground state of 32 Al is a normal sdshell state. From the viewpoint of the shell evolution along the neutron-rich N = 19 isotones 30 Na, 31 Mg and 32 Al, the present result suggests that the inversion between the amplitudes of sd and pf configurations occurs quite suddenly between 32 Al and 31 Mg with a drastic change of nuclear shapes. Acknowledgements The authors are grateful to staffs at the RIKEN Ring Cyclotron for their support during the running of the experiment.

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They would like to thank Dr. E. Yagi for useful help and advice with the X-ray diffraction analysis of the α-Al2 O3 sample. The authors D.K. and T.H. are grateful for the Special Postdoctoral Researcher Program in RIKEN. This work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture. The experiment was performed at RIKEN under the Experimental-ProgramNo. R398n. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

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