ISSN 1063-7788, Physics of Atomic Nuclei, 2007, Vol. 70, No. 8, pp. 1330–1335. c© Pleiades Publishing, Ltd., 2007.

NUCLEITheory

Magnetic Moments: Study of the Interplay between Single-Particleand Collective Excitations*

N. Benczer-Koller1)**, G. J. Kumbartzki1), P. Boutachkov1), A. Escuderos1),Y. Y. Sharon1), L. Zamick1), E. A. Stefanova1), 2), S. J. Q. Robinson3), and V. Werner4)

Received October 31, 2006

Abstract—Both theoretical descriptions of nuclei and experimental techniques for measuring magneticmoments have become very sophisticated. Yet, in general, good agreement between calculations andmeasurements still eludes us. Highlights are presented of new measurements of g factors of mixed-symmetry states in 92,94Zr, of the 4+

1 state in 70Ge, and of the 2+1 states in some S and Ar isotopes.

PACS numbers: 21.10.Ky, 21.60.Cs, 21.60.Ev, 27.30.+t, 27.40.+z, 27.50.+e, 27.60.+jDOI: 10.1134/S1063778807080029

1. INTRODUCTION

The interplay between single-particle and collec-tive excitations is very important in the lighter nucleias well as in heavier ones. The measurement of mag-netic moments of nuclear states allows for the estab-lishment of constraints on the theoretical descriptionof these states. In particular, the fact that single-particle neutrons and protons have different magneticmoments, both in sign and magnitude, helps deter-mine the microscopic structure of these states. A re-alistic description is, of course, more complex, and theinterplay between collective and single-particle exci-tations results in magnetic moments that lie some-where in between these two model extremes. There-fore, it becomes important for the experimental datato be as precise as possible so as to select amongpossible theoretical models.

In this presentation, three examples will be dis-played that show the effectiveness of magnetic mo-ments when microscopic information on specificstates’ wave functions is desired. The recent mea-surements in 92,94Zr mixed-symmetry states, in N =38 isotones, and in the A = 38, 40 isotopes of Ar andS will be discussed.

∗The text was submitted by the authors in English.1)Department of Physics and Astronomy, Rutgers University,

New Brunswick, USA.2)Institute for Nuclear Research and Nuclear Energy, Sofia,

Bulgaria.3)Department of Geology and Physics, University of Southern

Indiana, Evansville, USA.4)Wright Nuclear Structure Laboratory, Yale University, New

Haven, USA.**E-mail: nkoller@physics.rutgers.edu

2. EXPERIMENTAL TECHNIQUE

Recently, new techniques have been developed totake advantage of the radioactive beams that areslowly becoming available. These techniques makeuse of the transient field that is felt by swift nu-clei traversing ferromagnetic materials [1]. The nu-clei of interest are usually, but not always, excitedby Coulomb excitation. The exceptions involve caseswhere α or other nucleon transfer reactions occur. Inrecent experiments, projectiles have been excited ininverse kinematics, resulting in excited probe nucleiwith relatively large velocities [2]. This technique ex-ploits the facts that the transient field is higher forhigh-velocity ions and that the interacting target ionsare focused by the kinematics in the forward direction,where they can be easily detected, thus increasing theeffectiveness of the method.

The technique has been described at length in sev-eral papers over the last few years [2–7]. A schematiclayout of the experimental setup used in the inversekinematics geometry experiments is shown in Fig. 1.

3. THE 92,94Zr ISOTOPES

The Zr isotopes span a region of nuclei where theprotons fill the Z = 40 shell and, as N increases, theneutrons gradually fill the d5/2 shell. The magneticmoments of the first 2+ and 4+ states in 92,94Zr arenegative [6]. The Schmidt values for d5/2 neutronsand protons are gπ(d5/2) = +1.92 and gν(d5/2) =−0.77, respectively. Therefore the measured g factorssuggest that proton excitation from the core is lessimportant than was observed in the semimagic Caisotopes [8, 9].

1330

MAGNETIC MOMENTS 1331

Table 1. Comparison of experimental and calculated val-ues of g factors of 2+ and 4+ states in 92,94Zr (the experi-mental values are from [6, 21, 22] and the calculations for92Zr are from [18]; no calculations have yet been carriedout for 94Zr)

g(Jπ)92Zr 94Zr

Exp. SM QPM Exp.

g(2+1 ) −0.180 (10) −0.08 −0.11 −0.329 (15)

g(2+2 ) +0.0 (6) +1.07 +0.72 +0.6 (2)

g(4+1 ) −0.50 (10) −0.38 −0.32 −0.8 (4)

g(4+2 ) +1.15 +0.12

However, in this region of nearly spherical nuclei,other substructure manifestations are becoming evi-dent. The analog of isospin is introduced in the IBM-2 model through a quantum number called F spin.The wave functions can be written as combinationsof symmetric and antisymmetric proton–neutron bo-son pairs. The states with at least one antisymmetricboson pair are called mixed-symmetry states.

The first evidence of mixed-symmetry states wasestablished by initial experiments on 94Mo [10–13]and on other isotopes close to magic shells. These ex-periments have provided the evidence favoring the ex-istence of F-spin symmetries, thus confirming majorpredictions of the proton–neutron interacting bosonmodel (IBM-2) [14]. Further studies of 92Zr [15] and94Zr [16] have tried to establish the limits of the regionin the vicinity of N = 52 where the F-spin conceptcan be applied.

The description obtained for the 2+2 states from

the measured B(E2) and B(M1) values, for exam-

Beam

Excitation Ferromagnetic layerlayer

Stoppinglayer

GeParticledetector

B

TF

Fig. 1. Experimental geometrical setup for inverse kine-matics experiments. The beam is Coulomb excited bynuclei in the first layer of the target. The exciting targetnuclei recoil and are detected in a particle detector. Theexcited beam ions traverse the ferromagnetic layer andstop in the last target layer. γ rays are detected in coin-cidence with the recoiling target ions.

ple, implies a breaking of the F-spin symmetry inthose states. These observations led to considerabletheoretical work on 92Zr using a truncated shellmodel (SM) as well as a quasiparticle–phonon model(QPM) [17, 18]. The calculations have been extendedto include predictions for the magnetic moments ofthese states. The results of the calculations for 92Zrare displayed in Table 1. Calculations for 94Zr havenot yet been concluded.

The g factor experiments were carried out at theWright Nuclear Structure Laboratory at Yale Uni-versity on beams of 92,94Zr, in inverse kinematicscoupled to Coulomb excitation on a C target. Beamenergies of 275 to 290 MeV were used. Four Cloverdetectors were used for the detection of the γ rays, andeither solar cells or Canberra PIP detectors, 100 μmthick, were used to detect the recoil carbon ions atforward angles of 0◦ ≤ θ ≤ 33◦. Because of the ex-tremely short lifetimes of the mixed-symmetry states,namely, τ (92Zr; 2+

1 ) = 138(14) fs [19] and τ (94Zr;2+1 ) = 252(45) fs [20], the measured effects are very

small. Preliminary results are shown in Table 1 andFig. 2. These experiments confirm the assumption ofsignificant proton contributions to the structure of the2+2 state in 94Zr. In 92Zr, however, the neutrons com-

pete favorably with the proton excitations, yielding asmall g factor.

–0.8

90

Factor

A

1.6

92

1.2

0.8

0.4

0

–0.4

91 93 95 9694

3

–

2

+1

2

+2

4

+

Fig. 2. Experimental g factors of low-lying 2+, 4+ and3− states in the 90,92,94,96Zr isotopes. The g factors forthe semimagic 90,96Zr isotopes are dominated by protonconfigurations. The g factors of the 2+

1 and 4+1 states in

the 92,94Zr isotopes reflect strong neutron configurations.The g factors of the 2+

2 states of the 92,94Zr isotopes showthe mixed-symmetry nature of their configurations.

PHYSICS OF ATOMIC NUCLEI Vol. 70 No. 8 2007

1332 BENCZER-KOLLER et al.

4. THE N = 38 ISOTOPES

The 6830Zn and 70

32Ge nuclei have two or four pro-tons, respectively, and ten neutrons outside the dou-bly magic 56Ni core. These nucleons occupy variousconfigurations in the well-studied fp shell and, inprinciple, the structures of the low-excited states areamenable to SM calculations [23].

Two questions arise: (a) Does the g9/2 shell playa role in the low-lying excitations? (b) Is the coreexcited as was suggested by recent magnetic momentmeasurements in the Ca isotopes [8, 9]?

The overall issue actually reduces to the selectionof the active nucleon valence space and of the effectiveNN interactions.

Recently, the Bonn group measured the magneticmoment of the N = 38, τ = 1.18(8) ps, 4+

1 statein 68Zn and obtained g(68Zn; 4+

1 ) = −0.37(17) [24].A priori, a negative g factor is a surprise since it doesnot fit the systematics in that region. The magneticmoments of the 4+

1 states in the nearby isotopes 64Znand 66Zn are indeed positive, as expected [24, 25].

The Bonn–Oslo group also performed new large-scale SM calculations for 68Zn based on a 56Ni coreand a 1p3/20f5/21p1/20g9/2 model space. The inclu-sion of the neutron 0g9/2 orbital was expected toreduce the calculated value of the g factor of the 4+

1state and perhaps even drive it to a negative sign.Calculated B(E2) values for the decay transitionsof the 2+

1 and 4+1 states were obtained that are in

good agreement with the experimental data. How-ever, even the smallest g(4+

1 ) value of +0.002 thatwas obtained with the large SM calculations ([24]as corrected in [25]) is still positive and disagreeswith the measurement. Other SM calculations [24,25] also fail to explain the measured large negativemagnetic moment of the 4+

1 state in 68Zn.

Table 2. Comparison of (preliminary) experimental andcalculated values of g factors of low-lying states in 70Ge(the theoretical results for the 2+

1 and 2+2 states are

from [25], while the theoretical result for the 4+1 state is

from [30])

g Exp. KB3 FPD6 GXPF1 GXPF1A

g(2+1 ) +0.50(4)∗ +0.528 +0.769 +0.397 +0.343

+0.47(3) [32]

+0.43(12) [29]

g(2+2 ) +0.4(6) [29] +0.678 +0.880 +0.745 +0.896

g(4+1 ) +0.5(2)∗ +0.470 +0.913 +0.431 +0.431

∗ This work.

It is important to determine whether the puzzle ofthe 4+

1 state in 68Zn also applies to the corresponding4+1 state in the isotone 70Ge which has two additional

protons. Therefore, the g factor of the 4+1 state in 70Ge

was measured. Experiments were carried out at theWright Nuclear Structure Laboratory, Yale Univer-sity, on beams of 70Ge at 190 or 225 MeV. The tran-sient field technique, in inverse kinematics, coupledto Coulomb excitation, was used for the experiments.Two different targets, consisting of a thin layer of ei-ther 26Mg or natural carbon, deposited on gadoliniumfoils backed by a copper layer, were used. Four Cloverdetectors were used for the detection of the γ rays.A rectangular solar cell (whose geometry enhancesthe particle-γ angular correlation) was used to recordthe Mg or C recoil ions. Preliminary results yield thefollowing g factor; g(70Ge; 4+

1 ) = +0.5(2) [26, 27].At the same time, the g factor of the 2+

1 state in70Ge was remeasured, for comparison with previousresults and for checking the calibration of the tran-sient field. The result was g(70Ge; 2+

1 ) = +0.50(4), inagreement with former work [28].

New calculations [29, 30] were carried out byRobinson, Escuderos, Sharon, and Zamick for 70Gewithin the same SM space that was used in earliercalculations for 68Ge [31]. The calculations werebased on an inert core of 40Ca plus valence protonsand neutrons in the full fp shell. These calculationsutilized the SM codes OXBASH and ANTOINE.Four effective interactions were used, KB3, FPD6,GXPF1, and GXPF1A. Free nucleon g factors andeffective charges of eπ = 1.5e and eν = 0.5e wereutilized. These calculations differ from calculationsin [24] which were based on a 56Ni core plus the1p3/20f5/21p1/20g9/2 model space.

In the simplest SM picture for 70Ge, the 4+1 state

has 12 protons outside the presumed closed core of40Ca, filling completely the f7/2 and p3/2 subshells.In this model, the protons do not contribute to themagnetic moment. The 18 neutrons outside the 40Caare equivalent to two neutron holes in the f5/2p1/2

orbitals. The f7/2 and p3/2 orbitals are closed for theneutrons. In order to form a 4+ state, the two neutronholes both need to be in the f5/2 orbital. The g factorof that configuration is g(f5/2)ν = +0.547. The cal-culations described above for the KB3, GXPF1, andGXPF1A interactions show that the configurationdenoted above is indeed the dominant one. The fullcalculation yields the result shown in Table 2. Withthe FPD6 interaction, the above configuration is notdominant. Proton excitations from the p3/2 subshellplay an important role, yielding a larger, positive, gfactor.

PHYSICS OF ATOMIC NUCLEI Vol. 70 No. 8 2007

MAGNETIC MOMENTS 1333

Factor

–0.520

Neutron number

1.5

24

1.0

0.5

0

22 20 2422

Sulfur isotopes (

Z

= 16) Argon isotopes (

Z

= 18)

Z

/

A Z

/

A

Fig. 3. Experimental g factors of the 2+1 states in 38,40S and 38,40Ar compared with Z/A and with predictions from SM

calculations.

The resulting calculated g factors for 70Ge to-gether with the preliminary experimental g factors areshown in Table 2. The g factor of the 2+

1 state has beencorrected for feeding contributions from the decay ofthe 4+

1 state.None of the calculations [24, 25] were able to

explain the unexpected negative g factor of the 4+1

state in 68Zn. But the new calculations [27, 29, 30]for 70Ge do agree with the experimental values of theg factors and with the current theoretical views of theregion near N = 38. A remeasurement of the g factorof the 4+

1 state in 68Zn is desirable.

5. Ar AND S ISOTOPES

Measurements on light nuclei have been critical inadvancing the understanding of how the addition ofneutrons beyond the stability region affects the order-ing and structure of closed shells. In particular, exper-iments have been carried out on the stable 38,40Ar [33,34] and on the radioactive 38,40S isotopes [35] andcorresponding SM calculations have been performed.

The experiments on the S isotopes are new inso far as they are the first carried out on radioactivebeams at intermediate velocities. They were per-formed by the Michigan State University (MSU)and Australian National University (ANU) groupsat the National Superconducting Cyclotron Lab-oratory (NSCL) at MSU [35]. Secondary beamsof 38,40S were produced from 140-MeV/nucleonprimary beams directed onto a ∼1-mg/cm2 9Befragmentation target located at the entrance of theA1900 fragment separator. The secondary beamenergy was reduced to ∼40 MeV/nucleon. The target

consisted of a 355-mg/cm2 Au layer backed by a110-mg/cm2 iron ferromagnetic layer. A plastic scin-tillator phoswich was placed downstream to detectrecoil ions and the γ rays were recorded in fourteenHPGe detectors from the segmented germaniumarray (SEGA).

The technique of transient fields in inverse kine-matics, usually applied to low-energy Coulomb-excited nuclei, was extended in [35] to higher energy,lower Z, ions and the transient field was appropriatelycalibrated [36, 37]. The g factors of the first excited2+1 states of 38S and 40S were obtained: g(38S; 2+

1 ) =+0.13(5) and g(40S; 2+

1 ) = −0.01(6).

The experiments on the Ar isotopes were alsocarried out with the technique of transient fields ininverse kinematics. Experiments at the LawrenceBerkeley National Laboratory on 80-MeV40Ar beams [33] yielded g(40Ar; 2+

1 ) = −0.015(42).Experiments at the Munich tandem accelerator [34]were carried out by α transfer reactions on beams of32S and 34S, yielding g(36Ar; 2+

1 ) = +0.52(18) andg(38Ar; 2+

1 ) = +0.24(12).

The data were compared to SM calculations for40Ar [33] and 38,40S [35] within the (full sd)π (fullfp)ν space without requiring a collective excitation ofprotons and neutrons out of the 40Ca core into the fpshell, as was necessary to explain the 42,44Ca data [8,9]. For 36,38Ar, only the sd shell was involved. Thecalculations predict excitation energies and B(E2)values as well as g factors. The g factors are displayedboth in Fig. 3 and in Table 3, while the B(E2) valuesare shown in Table 3. In general, there is reasonable

PHYSICS OF ATOMIC NUCLEI Vol. 70 No. 8 2007

1334 BENCZER-KOLLER et al.

Table 3. Comparison of experimental and calculated values of g(2+1 ) factors of low-lying states in 38,40Ar and 38,40S

isotopes (the g factor for 36Ar and the theoretical quadrupole moments Q are shown for completeness; the SM calculationresults were presented in [30, 35])

Isotopeg B(E2) ↑, e2 b2 Q, e fm2

Exp. SM [35] SM [30] Exp. [38] SM [30] SM [30]38S +0.13(5) [35] −0.003 −0.0026 0.0235 0.0297 –10.2440S −0.01(6) [35] +0.035 +0.0353 0.0334 0.0518 –20.2136Ar +0.52(18) [34]38Ar +0.24(12) [34] +0.3085 0.0130 0.0170 +4.2040Ar −0.015(42) [33] −0.200 −0.2004 0.0330 0.0250 +8.28

agreement between theory and experiment for theexcitation energies, B(E2) values, and g factors.

The two nuclei 36S and 38Ar are N = 20 isotones.The addition of two f7/2 neutrons, to form 38S and40Ar, respectively, would drive the g factor to negativevalues. The observed 38S and 40Ar g factors, bothvery small, imply proton excitations within the sdshell. The cancellation [33–35] between the protonand neutron contributions to the g factor results insmall net magnetic moments for the 2+

1 states ofboth 38S and 40Ar. So the Z/A value expected forg factors in normal deformed nuclei is not achieved.However, collectivity can still be present, particularlyin the form of a vibration about a spherical shape for40Ar. The measurement of the quadrupole momentsof these nuclei could shed more light on their nucleonconfigurations.

6. CONCLUSIONS

The measurement of magnetic moments of short-lived low-lying states of nuclei provides critical mea-surements from which the microscopic details of thewave functions can be determined. The accuracy ob-tained by the recent measurements, which apply thetechnique of inverse kinematics and low as well asmedium energy Coulomb excitation, constrains sig-nificantly the theoretical models that are in currentuse. This experimental technique has also been usedto measure magnetic moments of very high excitedstates in particular superdeformed bands, magneticmoments of states excited in nuclei produced as ra-dioactive beams, and magnetic moments of sparselypopulated states.

ACKNOWLEDGMENTS

This work was supported in part by the USNational Science Foundation for Rutgers University

and the US Department of Energy under contractnos. DE-FG02-91ER-40609, DE-FG52-05NA25929, and DE-FG02-88ER40417 for YaleUniversity. A.E. acknowledges support from theSecretarıa de Estado de Educacion Universidades(Spain). Y.Y.S. is grateful for a Stockton CollegeCareer Development Grant. S.J.Q.R. is grateful forthe support from a University of Southern IndianaPott College Research Grant Award.

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