application/xmlFirst measurement and shell model interpretation of the g factor of the 21+ state in self-conjugate radioactive 44TiS. SchielkeK.-H. SpeidelO. KennJ. LeskeN. GemeinM. OfferY.Y. SharonL. ZamickJ. GerberP. Maier-Komorg factor44Ti, α-transfer reaction40Ca beamInverse kinematicsTransient fieldPhysics Letters B 567 (2003) 153-158. doi:10.1016/j.physletb.2003.06.027journalPhysics Letters BCopyright © 2003 Elsevier B.V. All rights reserved.Elsevier B.V.0370-26935673-414 August 20032003-08-14153-15815315810.1016/j.physletb.2003.06.027http://dx.doi.org/10.1016/j.physletb.2003.06.027doi:10.1016/j.physletb.2003.06.027http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2014-01-01T00:14:32ZSCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physicshttp://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBodyhttp://creativecommons.org/licenses/by/3.0/

Journals

S300.3

PLB19998S0370-2693(03)00917-110.1016/j.physletb.2003.06.027Elsevier B.V.Experiments

Fig. 1

Level scheme with relevant γ transitions. The lifetimes of the 21+, 41+ and 22+ states are results of the present work (see text).

Fig. 2

Particle spectrum observed in coincidence with all γ-rays (upper spectrum). The lower spectrum refers to a gate on the 44Ti(21+→01+)γ transition only. Gates for γ-coincidence spectra are indicated for 44Ti and 40Ca, respectively (see text).

Fig. 3

γ-coincidence spectrum observed with the 0° Ge detector with the particle gate on 44Ti events (see Fig. 2).

Table 1

Summary of the slope of the measured angular correlation, the experimental precession angle and the deduced g factor and lifetimes. Comparison to earlier lifetime data [10] is shown

Ex (MeV)τ (ps)|S(65°)|Φexp (mrad)g(21+)

[10]

present21+: 1.0834.5(12)3.97(28)0.423(55)17.6(4.9)+0.52(15)41+: 2.4540.60(10)0.65(6)–––22+: 2.5311.40(20)1.65(30)–––Table 2

Experimental data in comparison with results from fp shell model calculations using the effective interactions KB3 and FPD6

QuantityExperimentalKB3FPD6(f7/2)4full fp shell(f7/2)4full fp shellE(21+) (MeV)1.0830.5701.3930.9431.296E(41+) (MeV)2.4541.5762.5491.9412.495g(21+)+0.52(15)+0.554+0.532+0.554+0.514Q(21+) (eb)–+0.147−0.096+0.135−0.218B(E2;01+→21+) (e2b2)0.069(5)0.0320.0580.0310.070B(E2;21+→41+) (e2b2)0.047(4)0.0150.0270.0140.034B(E2;01+→22+) (e2b2)0.0006(1)0.00050.00220.00020.00002B(E2;21+→22+) (e2b2)0.0057(10)0.00080.01510.00170.0060First measurement and shell model interpretation of the g factor of the 21+ state in self-conjugate radioactive 44Ti

Schielke

K.-H.

Speidel

speidel@iskp.uni-bonn.de

Kenn

Leske

Gemein

Offer

Y.Y.

Sharon

Zamick

Gerber

Maier-Komor

aHelmholtz-Institut für Strahlen- und Kernphysik, Universität Bonn, D-53115 Bonn, GermanybDepartment of Physics & Astronomy, Rutgers University, New Brunswick, NJ 08903, USAcInstitut de Recherches Subatomiques, F-67037 Strasbourg, FrancedPhysik-Department, Technische Universität München, James-Franck-Str., D-85748 Garching, Germany

Editor: V. Metag

Abstract

The g factor of the 21+ state in radioactive 44Ti has been measured for the first time with the technique of α transfer to 40Ca beams in inverse kinematics in combination with transient magnetic fields, yielding the value, g(21+)=+0.52(15). In addition, the lifetimes of the 21+, τ=3.97(28) ps, and the 41+ states, τ=0.65(6) ps, were redetermined with higher precision using the Doppler shift attenuation method. The deduced B(E2)'s and the g factor were well explained by a full fp shell model calculation using the FPD6 effective NN interaction. The g factor can also be accounted for by a simple rotational model (g=Z/A). However, if one also considers the B(E2)'s and the E(41+)/E(21+) ratios, then an imperfect vibrator picture gives better agreement with the data.

PACS

21.10.Ky25.70.Hi27.40.+z

Keywords

g factor44Ti, α-transfer reaction40Ca beamInverse kinematicsTransient field

Introduction

The particular interest in N=Z nuclei lies in the feature that both protons and neutrons occupy the same orbitals; hence isospin symmetry as well as neutron–proton pair correlations are the dominant features of the nuclear structure [1,2]. 44Ti with N=Z=22 is such a nucleus, with two valence protons and two valence neutrons in the 1f7/2 shell outside the doubly-magic 40Ca (N=Z=20) core.The lowest 2+1 state in 44Ti, like its 0+1 ground state, has isospin T=0. Hence in measurements of the static magnetic and quadrupole moments of this excited state, as well as of the B(E2;0+1→21+), we are picking out the isoscalar values. Up to now one could obtain a handle on, say, isoscalar g factors only by studying the ground state magnetic moments of odd−A mirror nuclei. It was found in such studies [3] that the deviation from the Schmidt value is much smaller for isoscalar moments than for isovector moments.It is a well-known fact that magnetic moments and lifetimes of nuclear states are sensitive to the detailed composition of the nuclear wave functions. Since the spin g factors of protons and neutrons are different in sign and magnitude (gsp=+5.586, gsn=−3.826), such measurements enable one to determine the relative proton or neutron contributions in the nuclear state in question. This unique feature of magnetic moments and lifetimes has been used in many nuclei for testing nuclear model predictions [4], and it was shown that experimental g factor data need in general to be of few-percent precision to distinguish between the predictions of different models. For measuring magnetic moments of levels with ps lifetimes, where magnetic fields of kilo Tesla strengths are required, transient magnetic fields (TF) constitute at present the sole available experimental technique. As the field strength increases with ion velocity, the nuclear states to be studied should be populated in reactions for which the resulting nuclei emerge with high kinetic energies [4]. This condition is well satisfied by the technique of projectile Coulomb excitation in inverse kinematics: the nuclei of interest are fast projectiles (generally provided by an accelerator) which collide with light target nuclei, resulting in strong kinematic focussing and high ion velocities in the forward direction. As a consequence, highly efficient detection of target nuclei and coincident de-excitation γ rays of the projectiles is achieved. In particular, this technique was applied in recent measurements of g factors of the 21+ and 41+ states for the stable titanium nuclei 46,48,50Ti, utilizing these isotopes as beams and carbon as a target [5–7]. The g factor trends observed were rather well explained within the framework of large-scale shell model calculations using an fp shell model space and a modified Kuo–Brown effective interaction. Inadequacies in the numerical agreement between the calculated and experimental results have been attributed to possible 40Ca core excitations which were excluded for computational reasons.In the present case of 44Ti, projectile excitation would require a radioactive beam, a general approach which is presently being pursued in several laboratories. Due to the unavailability of such a beam an alternative technique was applied, which incorporates the merits of the inverse kinematics, as mentioned above, but is based on a particle-transfer reaction to stable beam nuclei. In several former measurements with projectile Coulomb excitation using carbon targets, α transfer was found to be a particularly strong reaction channel and has therefore been applied to g factor experiments. In this way g(21+) values were measured for radioactive 62Zn obtained in α transfer to a 58Ni beam [8] and also for 54Cr using a 50Ti beam [9]. In both cases, the 21+ states were predominantly populated. This state selectivity is characteristic for the transfer mechanism and is distinctly different from that of a fusion/evaporation reaction in which high energy states are populated, feeding the low-lying states via cascade transitions. The reduced feeding in the case of α transfer ensures a clean measurement of the precession, as it is almost exclusively associated with that of the 21+ state itself.2

Experimental details

In the current experiment, a beam of isotopically pure 40Ca was accelerated to an energy of 95 MeV at the Cologne tandem accelerator providing intensities of (2–3) pnA on a multilayered target. The target consisted of 0.45 mg/cm2natC, deposited on 3.82 mg/cm2 Gd, which was evaporated on a 1.0 mg/cm2 Ta foil, backed by a 3.48 mg/cm2 Cu layer. For the Gd evaporation, the tantalum substrate was kept at a temperature of 800 K to ensure good magnetic properties of gadolinium [11]. Besides Coulomb excitation of the 40Ca projectiles in collisions with carbon nuclei, essentially to the first 3− state at 3.736 MeV, strong α transfer occurred in the 12C(40Ca,8Be)44Ti, whereby the 44Ti(21+) state of interest was predominantly populated; weak excitations of the 41+, 22+ and 31− states were also observed (see Fig. 3). The relevant level scheme of 44Ti is shown in Fig. 1

. The residual nuclei 40Ca and 44Ti from Coulomb excitation and α transfer, respectively, both move through the Gd layer at high velocities in the direction of the primary 40Ca beam for spin precessions. These nuclei came to rest in the hyperfine interaction-free environment of the copper backing.

The de-excitation γ rays were measured in coincidence with forward scattered ions, either carbon ions or 2α particles from the decay of 8Be. Both types of ions pass through the target layers and an additional Ta foil and are detected in a 100 μm Si counter placed at 0° relative to the beam axis. The Ta foil between target and particle detector served as beam stopper. The Si detector, subtending an angle of ±30°, was operated with a low bias of 3–5 V (instead of the nominal 40 V). This enabled a better separation of the 2α particles associated with 44Ti (and other light charged particles such as protons) from the heavier carbon ions due to their different stopping behaviour in a thus reduced depletion layer of the Si detector. This separation procedure was already successfully applied in several earlier measurements (see, e.g., [6] and Fig. 2

Four 12.7 cm × 12.7 cm NaI(Tl) scintillators and a Ge detector with a relative efficiency of 40% were used for γ detection. Coincident particle and γ spectra are shown in Figs. 2 and 3

. The Ge detector placed at 0°, served as a monitor for contaminant lines in the energy region of interest and to measure nuclear lifetimes via the Doppler Shift Attenuation Method (DSAM).

Detailed (2α–γ)-angular correlations W(Θγ) have been measured for determining the slope S=[1/W(Θγ)dW(Θγ)/dΘγ] in the rest frame of the γ-emitting nuclei at angles Θγ=65° where the experimental sensitivity to the spin precessions is optimal. Precession angles, Φexp, were determined in the normal way via counting rate ratios R for ‘up’ and ‘down’ directions of the external field with detector pairs placed symmetrically to the beam direction. The precession angles were derived [4] as: (1)Φexp=1SR−1R+1=gμNℏ∫tintoutBTFvion(t)e−t/τdt, where g is the g factor of the 21+ state and BTF the transient field acting for the time interval (tout−tin) that the ions spend in the gadolinium layer; the exponential accounts for the decay of the 21+ state with lifetime τ.The lifetimes of the 21+, 41+ and 22+ states were redetermined from measured lineshapes of the three γ lines using the DSAM technique with the 0° Ge detector. The high ion velocities result in high sensitivity for the lifetimes in the ps range. The Doppler-broadenend lineshapes were fitted for the reaction kinematics applying stopping powers [12] to Monte Carlo simulations including the second order Doppler effect as well as the finite size and energy resolution of the Ge detector. Feeding from higher states was also taken into account. The computer code LINESHAPE [13] was used in the analysis.3

Results and discussion

The g factor of the 21+ state was derived from the experimental precession angle Φexp by determining the effective TF on the basis of the linear parametrization [4]: (2)BTF(vion)=Gbeam·Blin with (3)Blin=a(Gd)·Zion·(vion/v0), where the strength parameter a(Gd)=17(1) Tesla, v0=e2/ℏ, and Gbeam=0.90(5) is the attenuation factor of the TF strength induced by the calcium beam used [4].The precession and lifetime data with the deduced g factor for 44Ti are summarized in Table 1

. The new lifetime values are in good agreement with the earlier data but are of higher or comparable precision.

The g(21+) factor and the B(E2) values deduced from the lifetimes of the 21+, 41+ and 22+ states have been interpreted within the framework of the nuclear shell model. The calculations were carried out using the shell model code OXBASH [14] and both the KB3 [15] and the FPD6 [16] effective NN-interactions. In these calculations 44Ti is considered to consist of an inert 40Ca core with four valence nucleons, two protons and two neutrons. The results are compared with the experimental data in Table 2

. We also calculated a value for the unmeasured quadrupole moment of the 21+ state, that can be compared to corresponding measured values for neighbouring nuclei ranging from −0.14(7) eb for 44Ca to −0.21(6) eb for 46Ti [17].

The simplest approach is an (f7/2)4 configuration which, however, is obviously inadequate as evident from the table. With that configuration both interactions usually underestimate the B(E2)'s by more than a factor of two (suggesting that not enough collectivity is provided), and predict a positive quadrupole moment Q(21+). The only quantity that is rather well reproduced, for reasons to be explained later, is the g(21+), where both interactions yield an identical value of g(21+)=+0.554.The situation is greatly improved by full fp shell calculations, in which the four valence nucleons can occupy the 1f7/2, 2p3/2, 1f5/2 and 2p1/2 orbitals. As shown in Table 2, the calculations with the FPD6 interaction account very well for the experimental data without requiring any admixture, into the 21+ wave function, of particle-hole excitations from the 40Ca core. Similarly, almost as good results were obtained with the KB3 interaction. The FPD6 full fp shell calculation, however, underestimates the extremely small B(E2;01+→22+) by an order of magnitude, whereas the B(E2;21+→22+) is very well reproduced. On the other hand, in a pure vibrational model the (22+→01+) transition would be a forbidden two-phonon transition implying a vanishing B(E2).It is worthwhile attempting to understand why the simple (f7/2)4 configuration accounts as well for the g(21+) as the full fp shell calculations. To this end, the individual components of the 21+ wave function obtained in the full fp calculations have been examined. In the FPD6 calculation the major component intensities are (f7/2)4 (26%), (f7/2)3(p3/2)1 (24%) and (f7/2)2(p3/2)2 (10%); in the KB3 calculations the same components have intensities of 54%, 15% and 7%, respectively. Evidently the p3/2 orbital plays a more important role here than do the f5/2 and the p1/2 orbitals, a feature which was also found for 44Ca [18,19]. One then finds that the other two main configurations in the 21+ wave function (see above) have g(21+) values close to that for the (f7/2)4 for which g(21+)=+0.554; for both interactions, g(21+)=+0.519 for (f7/2)3(p3/2)1 and +0.575 for (f7/2)2(p3/2)2. Since all of these values are very close to the experimental value a distinction between the two interactions would require a precision of the experimental value on a 1% level, which, however, cannot be achieved with the present technique. On the other hand, the most important perturbing configuration, (f7/2)3(p3/2)1, has a very different Q(21+) value than the (f7/2)4. With FPD6 one obtains: Q(21+)=+0.135 eb for (f7/2)4 and −0.150 eb for (f7/2)3(p3/2)1; with KB3 similar values are obtained. Thus the (f7/2)3(p3/2)1 configuration plays an important role in the full fp shell calculations in giving rise to the expected negative sign of Q(21+).The g(21+) can be calculated analytically for the (f7/2)4 configuration using an expression (Eq. (28) in Ref. [20]) derived by McCullen, Bayman and Zamick for configurations of nucleons in a single j shell. For N=Z nuclei (e.g., 44Ti) this expression simplifies in the f7/2 shell to (4)g=gp+gn2Schmidt=1.655+(−0.547)2=+0.554 independent of the details of the wave function.It is interesting to note that simple collective model formulae also explain well some of the properties of the 21+ state in 44Ti. The g factor of the collective model [21], g=Z/A=+0.5, is in good agreement with the experimental result. Furthermore, from the experimental B(E2;01+→21+)=0.069 e2b2 one derives an intrinsic quadrupole moment Q0=[16π/5B(E2)/e2]1/2 and finally Q(21+)=−2/7Q0=|−0.235| eb close in magnitude to the result of the full fp shell calculation with FPD6 (Table 2). Moreover, from the B(E2) value a deformation β=0.28 can be deduced that is consistent with the values of 0.25 for 44Ca and 0.32 for 42Ti and 46Ti [22].The observed value of the ratio B(E2;21+→41+)/B(E2;01+→21+) is 0.68. All four shell model calculations (see Table 2) predict this ratio to be between 0.45 and 0.49. The collective predictions for this ratio are 0.51 in the rotational model but 0.72 in the vibrational model, closer to what is observed. The ratio of the experimental excitation energies E(41+)/E(21+) is 2.27, again quite close to the pure vibrational value of 2.In summary, we note that the g(21+) measurement is accounted for by all four shell model calculations in Table 2 and the Z/A collective value. However, only the FPD6 full fp shell calculations fully account for our newly measured, more precise, B(E2;01+→21+) value of 0.069(5) e2b2 as well as the B(E2;21+→22+) of 0.0057(10) e2b2.

Acknowledgements

The authors are grateful to the operators of the tandem accelerator. They acknowledge the support of the BMBF, the Deutsche Forschungsgemeinschaft, the US Department of Energy under Grant DE-FG02-95ER-40940, and a Stockton College Sabbatical Grant to Y.Y.S.

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