Measured g factors and the tidal-wave description of transitional nuclei near A = 100
S.K. Chamoli, A.E. Stuchbery, S. Frauendorf, J. Sun, Y. Gu, R.F. Leslie, P.T. Moore, A. Wakhle, M.C. East, T. Kibédi, A.N. Wilson
aa r X i v : . [ nu c l - e x ] A p r SKC/002-ANU
Measured g factors and the tidal-wave description of transitional nuclei near A = 100 S.K. Chamoli, A.E. Stuchbery, S. Frauendorf, J. Sun, Y. Gu, R.F.Leslie, P.T. Moore, A. Wakhle, M.C. East, T. Kib´edi, and A.N. Wilson Department of Nuclear Physics, Research School of Physics and Engineering,Australian National University, Canberra, ACT 0200, Australia Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA (Dated: May 29, 2018)The transient-field technique has been used in both conventional kinematics and inverse kinematicsto measure the g factors of the 2 +1 states in the stable even isotopes of Ru, Pd and Cd. The statisticalprecision of the g (2 +1 ) values has been significantly improved, allowing a critical comparison withthe tidal-wave version of the cranking model recently proposed for transitional nuclei in this region. PACS numbers: 21.10.Ky, 27.60.+j, 25.70.De, 23.20.En
I. INTRODUCTION
The stable isotopes of Mo, Ru, Pd, and Cd,include some of the best examples of vibrational levelstructures, with − Cd, in particular, frequently be-ing cited as ‘textbook’ examples [1, 2]. Recent studiesindicate that the vibrational picture is reasonably goodat the two-phonon level in − Cd, but breaks down atthe three-phonon level, particularly for non-yrast states[3].The lower mass stable isotopes and neutron-deficientisotopes of these elements are near Zr , which is al-most double magic, and Sn , which is double magic.Having few valence nucleons, the level schemes there-fore show spherical structures and are accessible to shellmodel calculations [4]. By way of contrast with the spher-ical and vibrational structures in the low and interme-diate mass isotopes, the heavier isotopes, with neutronnumbers near midshell, make a transition to rotationalstructures. There has been considerable effort in recentyears to study the spectroscopy of isotopes in this re-gion produced either as fission fragments or as radioac-tive beams. Examples of experimental work relevant tothe present study are measurements of quadrupole mo-ments and B ( E
2) values in neutron-deficient Cd isotopesproduced as radioactive beams [5], and measurements of g factors in neutron-rich fission fragments, in which a re-duced magnitude for several neutron-rich nuclei was at-tributed to contributions from neutrons in the h / orbit[6, 7].On the experimental side, the present work focuses onmeasurements of the g factors of the first excited statesin all of the stable even isotopes of Ru, Pd and Cd bythe transient-field technique. The precision is improvedconsiderably compared with previous work.On the theoretical side, we use the tidal wave ap-proach for calculating the g factors in this transitionalregion. The model uses the fact that in semi-classicalapproximation the yrast states of vibrational nuclei cor-respond to quadrupole waves traveling over the surfaceof the nucleus like the tidal waves over the surface ofthe ocean. It has been demonstrated that the energies of the yrast states, as well as the B ( E
2) values of thetransitions between them, are very well described by thismodel for the even-even nuclei with 44 ≤ Z ≤
48 and54 ≤ N ≤
68 [8, 9]. The present work extends themodel to g factors, which allows an examination of theway in which the angular momentum is shared betweenthe protons and neutrons.The paper is arranged as follows: Section II reportsthe g -factor experiments. The measurements using con-ventional kinematics are described first (sect. II A),followed by the measurements using inverse kinematics(sect. II B). A summary and discussion of adopted exper-imental g factors in sect. II C completes the experimentalpart of the paper. The tidal wave model calculations ofthe g factors are presented in sect. III and the compari-son between theory and experiment is discussed in sect.IV. The conclusion follows (sect. V). II. TRANSIENT-FIELD g -FACTORMEASUREMENTS The g factors of the first excited states were measuredin all the stable even isotopes of Ru, Pd and Cd using thetransient-field technique and beams from the AustralianNational University 14UD Pelletron accelerator. Mea-surements on the Cd and Ru isotopes in ‘conventionalkinematics’ are described in sect. II A; those on the Ru,Pd and Cd isotopes in ‘inverse kinematics’ are describedin sect. II B. The experiments used the ANU HyperfineSpectrometer [10]. Experimental procedures were similarto those described elsewhere [11–17].Before describing the experiments we review some pro-cedures and terminology associated with the determina-tion of the experimental g factors from transient-field pre-cession measurements [11, 13, 14, 18].The observed transient-field precession, ∆Θ obs , is re-lated to the nuclear g factor, g , by∆Θ obs = gφ ( τ ) (1)where φ ( τ ) = − µ N ~ Z t e t i B TF ( v ( t ) , Z ) e − t/τ dt, (2)and µ N is the nuclear magneton; τ is the mean life of thenuclear state. The transient field strength B TF ( v ( t ) , Z )depends on the atomic number and velocity of the ionwithin the ferromagnetic layer of the target. It is oftenparametrized in the form B TF ( v, Z ) = a TF Z p Z ( v/v ) p v . (3)For fully magnetized iron hosts the Rutgers parametriza-tion gives a TF = 16 . p Z = 1 .
1, and p v = 0 .
45 [19].As can be seen from Eq. (2), φ is a function of τ . Inthe present work τ > t e is typically about0.5 ps, so the exponential factor in Eq. (2) remains nearunity; the observed precession is insensitive to τ , but notindependent of it, especially for the shorter-lived states.Furthermore, for each isotope the reaction kinematics,and slowing down of the ions in the ferromagnetic layer,are slightly different. It is useful to define the limitingcase of φ for τ → ∞ , namely φ ( ∞ ) = − µ N ~ Z t e t i B TF ( v ( t ) , Z ) dt. (4)In the following presentation of experimental results,the observed precession angles ∆Θ obs will always begiven. Depending on what is known about the transient-field strength for the particular combination of ion andferromagnetic host, it may be useful to present, in addi-tion, a set of corrected experimental precessions for eachisotope A , which reflect the relative g factors:∆Θ( A ) = ∆Θ obs ( A ) φ ( ∞ , A ref ) /φ ( τ, A ) , (5)where A ref denotes a chosen reference nuclide. The ratioof calculated φ values effectively removes the small differ-ences in the measured precession angles due to differencesin level lifetimes and reaction kinematics. In the presentwork φ ( ∞ , A ref ) /φ ( τ, A ) is near unity. It is independentof the chosen transient field scale parameter a TF , and isalso insensitive to any reasonable choice of p Z , and p v .In some cases it is appropriate to give experimental g factors relative to a reference g factor in the nucleus A ref , namely g ref ( A ref ): g ( A ) = g ref ( A ref ) ∆Θ obs ( A )∆Θ obs ( A ref ) φ ( τ ref , A ref ) φ ( τ, A ) , (6)where τ ref is the mean life of the reference state. A. Conventional kinematics
1. Cd isotopes
The g factors of the first 2 +1 states in , , , Cdwere measured simultaneously, relative to each other, us- ing the transient-field technique in conventional kinemat-ics. The experiment was similar to that on the Mo iso-topes [12]. Table I summarizes the relevant level proper-ties and reaction kinematics.States of interest were Coulomb excited using beams of95 MeV S. In the order encountered by the beam, thetarget consisted of layers of nat
Ag, 0.05 mg/cm thick,and nat Cd, 0.98 mg/cm thick, which had been evapo-rated onto an annealed iron foil, 2.64 mg/cm thick. Onthe back of the iron foil a 5.47 mg/cm thick layer ofnatural copper had already been evaporated. For addi-tional mechanical support, and improved thermal contactwith the cooled target mount, this multilayered targetwas pressed onto thicker ( ∼ µ m) copper foil using anevaporated layer of indium as adhesive. Coulomb excitedCd nuclei recoiled through the iron foil, where they ex-perienced the transient field, and then stopped in thenon-magnetic copper layer where they subsequently de-cayed. The thin Ag layer on the front of the target wasincluded to help protect the Cd layer, which has a lowmelting point of ∼ ◦ C. To minimize the effect of beamheating on the target, it was maintained at a tempera-ture of 6K by mounting it on a cryocooler (SumitomoRDK-408D). No deterioration of the Cd target layer wasobserved despite a high beam current of ∼ . ∼ γ -ray detector plane to magnetize theferromagnetic layer of the target. The direction of thisfield was reversed periodically to minimize systematic er-rors.Backscattered beam ions were detected in a pair of sil-icon photodiode detectors, 10 mm high by 9 mm wide,placed 3.8 mm from the beam axis in the vertical planeparallel to the target, and 16 mm upstream of the target;the average scattering angle was 151 ◦ . To measure thetransient-field precession, γ rays emitted in coincidencewith backscattered particles were observed in two 50%(relative efficiency) HPGe detectors and two 20% HPGedetectors placed at ± ◦ and ± ◦ to the beam axis,respectively. The target-detector distances were set sothat the detector crystals all subtended a half angle of18 ◦ . Figure 1 shows a coincidence γ -ray spectrum ob-served in the detector at +65 ◦ to the beam direction.Particle- γ angular correlations were measured in a se-quence of runs of about 75 min duration. The backward-placed Ge detectors were kept at ± ◦ , to normalize thenumber of counts, while the angular correlation was sam-pled with the two forward Ge detectors at ± ◦ , ± ◦ , ± ◦ , ± ◦ and ± ◦ , in turn. The measured angularcorrelations for the 2 +1 → +1 transitions are comparedwith the calculated angular correlations in Fig. 2.The transient-field precession angles, ∆Θ, were deter-mined by the usual procedures [11, 13, 18, 22]. Briefly,∆Θ = ǫ/S , where ǫ is the ‘effect’ and S ( θ γ ) =(1 /W ) dW /dθ , often referred to as the ‘slope’, is the loga-rithmic derivative of the angular correlation at the γ -raydetection angle, θ γ . Formally, ǫ = ( N ↓ − N ↑ ) / ( N ↓ TABLE I: Level properties and reaction kinematics for Cd and Ru isotopes recoiling in iron after Coulomb excitation by 95MeV S beams. E (2 +1 ) is the energy and τ (2 +1 ) is the mean life of the 2 +1 level [20]. h E i i and h E e i are the average energies withwhich the Cd ions enter into and exit from the iron foil. The corresponding ion velocities are h v i /v i and h v e /v i . The averageion velocity is h v/v i . v = c/
137 is the Bohr velocity. t Fe is the effective time for which the ions experience the transient fieldin the iron layer of the target. These quantities were calculated with the stopping powers of Ziegler et al. [21]. φ ( τ ) is thetransient-field precession per unit g factor calculated as described in the text.Isotope E (2 +1 ) τ (2 +1 ) h E i i h E e i h v i /v i h v e /v i h v/v i t Fe φ ( τ )(keV) (ps) (MeV) (MeV) (fs) (mrad) Cd 657 7.4 51.6 8.4 4.35 1.75 2.77 543 − . Cd 617 8.9 51.1 8.5 4.29 1.75 2.75 552 − . Cd 559 13.7 50.7 8.6 4.23 1.75 2.72 564 − . Cd 512 20.3 50.3 8.7 4.18 1.74 2.70 572 − . Ru 540 18.2 58.6 12.3 4.86 2.23 3.34 410 − . Ru 475 26.6 58.1 12.1 4.79 2.21 3.30 417 − . Ru 358 83.4 57.6 12.5 4.72 2.20 3.26 426 − . C d / / + + energy (keV)200 300 400 500 600 700 800 c oun t s Conventional Kinematics95 MeV 32S on natCd C d C d C d C d C d / / + + C d / / + + FIG. 1: Spectrum of γ -rays observed at +65 ◦ to the beam axis in coincidence with backscattered S beam ions. This spectrumrepresents all of the data for the field up direction in the detector at +65 ◦ , obtained during the precession measurement on theCd target. + N ↑ ), where N ↑ ( ↓ ) refers to the counts recorded forfield up (down) at + θ γ ; however the evaluation of ǫ fromthe experimental data proceeds via the formation of adouble ratio of counts recorded for field up and down ina pair of detectors at ± θ γ .Results of the precession measurements on the 2 +1 states of the even Cd isotopes are given in Table II. Dif-ferences in the ‘slopes’, S (65 ◦ ), stem mainly from dif-ferences in the small level of feeding intensity from thehigher excited states, especially the 4 +1 state. The effect of this feeding contribution on the extracted g factorswas evaluated as described in Ref. [15]. It was foundthat extreme values must be assumed for the magnitudeof g (4 +1 ) in order to make even a few percent change inthe precession observed for the 2 +1 state. The effect ofthe feeding contribution is therefore accurately includedin the present analysis by evaluating S for the fed (i.e.observed) angular correlation for the 2 +1 state.The absolute values of the g factors in the Cd iso-topes were determined by reference to Pd in the in-
TABLE II: Results of precession measurements and g factors for the 2 +1 states in Cd isotopes (conventional kinematics). | S (65 ◦ ) | is the logarithmic derivative of the angular correlation. ∆Θ F and ∆Θ B are the precession angles observed with the γ -ray detector pairs at ± ◦ and ± ◦ , respectively; h ∆Θ i obs is the average of these. ∆Θ has been corrected for smalldifferences in the reaction kinematics as the ions traverse the iron layer of the target, so that relative values of ∆Θ give relative g factors.Isotope | S (65 ◦ ) | ∆Θ F ∆Θ B h ∆Θ i obs ∆Θ g/g ( Cd) g a (rad − ) (mrad) (mrad) (mrad) (mrad) , Cd b − − − − Cd 2.74 − − − − Cd 2.71 − − − − Cd 2.69 − − − − Cd 2.70 − − − − a Assigned errors include the uncertainty in the transient-fieldcalibration. b Results for the composite 635 keV line, which includes both
Cd and
Cd, are included to show consistency with the mea-surements on these isotopes reported below. q (degrees)012345 W ( q )
2+ 0+
FIG. 2: Experimental and calculated angular correlationsfor even- A Cd isotopes following Coulomb excitation with 95MeV S beams. (Data for the different isotopes have beenoffset for presentation.) verse kinematics measurements described below. Specif-ically, the precessions measured (in inverse kinematics)for
Cd and
Cd beams, gave g ( Cd) = +0 . g ( Cd) = +0 . g factors in Table II are givenby g = ∆Θ /φ ( ∞ ) = ∆Θ / ( − . ± . Cd by Benczer-Koller et al. [23],in which the transient field strength was determined bythe Rutgers parametrization [19], gave g (2 +1 ; Cd) =+0 . Z ∼ −
48 traversing iron foils with velocities inthe range 2 v . v . v . For this reason, the values of φ ( τ ) shown in Table I were evaluated using the Rutgersparametrization [19] in Eq. (2).The g factors of several states in the two odd- A iso-topes, Cd and
Cd, were measured as a by-productof the measurement on the natural target. These resultswill be presented and discussed elsewhere [24].
2. Ru isotopes
The g factors of the first 2 +1 states in , , Ruwere measured simultaneously, relative to each other,using the transient-field technique in conventional kine-matics and procedures very similar to those described insect. II A 1. States of interest were again Coulomb excitedusing beams of 95 MeV S. The target consisted of alayer of nat
Ru, 0.63 mg/cm thick, which had been sput-tered onto an annealed iron foil, 2.34 mg/cm thick. Theiron foil was then pressed onto a 12.5 µ m thick copper foilusing an evaporated layer of indium, ∼ thick,as adhesive. Coulomb excited Ru nuclei recoiled throughthe iron foil, where they experienced the transient field,and then stopped in the non-magnetic indium and copperlayers where they subsequently decayed. (Both indiumand copper have cubic crystalline structure so quadrupoleinteractions for the 2 + states of the Ru isotopes are neg-ligible in both host materials.)Save for the different target, the experiment was es-sentially identical to that on the Cd isotopes reportedin the previous section. The total beam time for theprecession measurement was about 60 hours. Table I in-cludes a summary of the relevant level properties andreaction kinematics. Figure 3 shows a coincidence γ -rayspectrum observed in the detector at +65 ◦ to the beamdirection. Although natural Ru contains Ru (5.5%), Ru (1.9%), Ru (12.7%), and
Ru (17.0%), along
TABLE III: Results of precession measurements and g factorsfor the 2 +1 states in Ru isotopes (conventional kinematics).The transient-field strength was calibrated using the Rutgersparametrization [19].Isotope | S (65 ◦ ) | h ∆Θ i g/g ( Ru) g a (rad − ) (mrad) Ru 2.72 − Ru 2.59 − Ru 2.50 − a No uncertainty in the transient-field calibration is included here.Adopted g factors, including uncertainties in the field strength, arepresented in sect. II C. with Ru (12.6%),
Ru (31.6%) and
Ru (18.7%),transient-field precessions could be obtained with mean-ingful precision only for the latter three isotopes.Particle- γ angular correlations were measured in a se-quence of runs of about 50 min duration. The two Ge de-tectors at negative angles were kept at − ◦ and − ◦ ,to normalize the number of counts, while the angular cor-relation was sampled at a sequence of angles in the pos-itive hemisphere with the other two Ge detectors. Theprocedure was then reversed: the detectors at positiveangles remained fixed while the detectors at negative an-gles were moved to a sequence of angles. The measuredangular correlations are compared with the calculatedangular correlations in Fig. 4.The absolute values of the g factors in the Ru iso-topes as presented in Table III were determined by useof the Rutgers parametrization [19], which was demon-strated in the previous section (II A 1) to be applicablefor ions traversing iron hosts under the conditions of thismeasurement. The uncertainties shown in Table III cor-respond to the uncertainties in the relative g factors. Un-certainties on the absolute values of the g factors will bediscussed in sect. II C below, where these measurementswill be combined with the measurements in inverse kine-matics. B. Inverse kinematics: Ru, Pd and Cd isotopes
As summarized in Table IV, transient-field measure-ments in inverse kinematics were performed on all of thestable even Ru and Pd isotopes, and , , , Cd, us-ing ∼ . A beams from the 14UD Pelletron. Thebeam intensities ranged from ∼ . Pd to ∼ .
03 pnA for Ru. Negative ion beams of the Ru andPd isotopes were produced from natural metal powderpressed into a standard copper cathode. CdO − beamswere produced from cadmium oxide - natural for the Cd and
Cd beams, and partially enriched for the
Cd and
Cd beams. Beams of MoO − ions wereobtained from a metallic Mo cathode in the presence ofO gas.For these inverse kinematics experiments the ANU Hy- TABLE IV: Summary of measurements in inverse kinematics. E B and I B are the beam energy and intensity.Beam E B I B Measurement a ; Purpose Duration(MeV) (enA) (h)Target I Ru
245 5 ǫ ; g ( Ru)/ g ( Pd) 2.5 Pd
230 5 ǫ ; B TF ( v ) 2 Pd
245 5 ǫ ; B TF ( v ) 2.25 Pd
260 5 ǫ ; B TF ( v ) 1.75 Cd
240 5 ǫ ; g ( Cd)/ g ( Pd) 5.5Target II Ru
240 1.5 ǫ ; g ( Ru) 10 Ru
240 0.5 ǫ ; g ( Ru) 25 Ru
240 4 ǫ ; g ( Ru) 4.5 Ru
240 3 ǫ ; g ( Ru) 1 Ru
240 5 ǫ ; g ( Ru) 2 Pd
245 0.8 ǫ ; g ( Pd) 8 Pd
245 6 ǫ ; g ( Pd) 2.2 Pd
245 8 ǫ ; TF calibration 3.3 Pd
245 6 ǫ , W (Θ) ; g ( Pd) 3.5 Pd
245 3 ǫ ; g ( Pd) 1.7 Cd
240 2 ǫ ; g ( Cd) 9.6 Cd
240 2 ǫ ; g ( Cd) 11 Cd
240 1.4 ǫ ; g ( , Cd)/ g ( Cd) 18Target III Mo
240 2.2 ǫ ; TF calibration 18 Ru
240 2.5 ǫ ; g ( Ru) 18 Ru
240 2.5 ǫ ; g ( Ru) 12 Ru
240 2.5 ǫ ; g ( Ru) 15 Ru
240 2.5 ǫ ; g ( Ru) 10 Pd
240 2 ǫ ; TF calibration 12 a ǫ : transient-field precession; W (Θ) : angular correlation. perfine Spectrometer was configured with a forward arrayof three particle detectors; the apparatus and experimen-tal procedures were similar to those in our recent workon the Fe isotopes [16, 17].The first two targets (labeled I and II) used for thesemeasurements consisted of C layers on 6.1 mg/cm thickcopper-backed gadolinium foils. After rolling and anneal-ing under vacuum, a thin 0.04 mg/cm layer of copperwas evaporated onto the beam-facing side (front) of thegadolinium foil to assist the adhesion of the C layer, anda thicker 5.5 mg/cm layer of copper was evaporated onthe back. The layer of carbon, 0.4 mg/cm thick, wasadded to the front of the target by applying a suspen-sion of carbon powder in isopropyl alcohol. Additionalcopper foil (4.5 mg/cm ) was placed behind the target tostop the beam. The target was cooled below 5 K, bothto minimize the effect of beam heating, and to maximizethe magnetization of the gadolinium layer of the target.Although they were nominally the same, Target I gavesomewhat larger precessions for Pd ions at 245 MeVthan Target II. We attribute this difference to variationsin the effective thickness of the target layers at the beamspot. It is also possible that the magnetization differs,despite the targets having been prepared from the same c oun t s energy (keV)200 300 400 500 600 70004000800012000 Conventional kinematics
95 MeV 32S on natRu R u R u R u R u R u R u + + R u R u + R u R u4 + + + + R u + + FIG. 3: Spectrum of γ -rays observed at +65 ◦ to the beam axis in coincidence with backscattered S beam ions. This spectrumrepresents the data for the field up direction in the detector at +65 ◦ , obtained during the precession measurement on thenatural Ru target. annealed gadolinium foil. An external magnetic field of0.09 T was applied perpendicular to the γ -ray detectionplane to magnetize the gadolinium layer of the target.This field was reversed reversed periodically throughoutthe measurements.Additional experiments were performed with a thirdtarget (labeled Target III) which consisted of enriched Mg, 0.45 mg/cm thick, evaporated onto gadolinium,3.2 mg/cm thick, which was followed by layers of nickel(0.01 mg/cm ) and copper (5.4 mg/cm ). This targethad previously been used for similar measurements atYale by Taylor et al. [25].Level properties of the beam ions and the reaction kine-matics for target ions ( C or Mg) scattered into theouter particle detectors (average scattering angle 20.7 ◦ )are summarized in Table V.The de-exciting γ rays from the Mo, Ru, Pd andCd isotopes were measured in coincidence with forward-scattered target ions detected by an array of three sil-icon photodiode detectors downstream from the target,arranged in a vertical stack as described in [16]. For themeasurements in inverse kinematics, two 50% efficientHPGe detectors and two 12.7 cm by 12.7 cm NaI de-tectors were placed in pairs at ± ◦ and ± ◦ to thebeam axis, respectively. The target-detector distanceswere again set such that the detector crystals all sub-tended a half angle of 18 ◦ . Figure 5 shows examples ofcoincidence γ -ray spectra observed in the Ge detectorat +65 ◦ to the beam direction and the NaI detector at +115 ◦ to the beam.Angular correlations were measured for Pd beamsat 245 MeV, in a sequence of runs of about 20 min du-ration. The backward-placed NaI detectors were keptat ± ◦ , to normalize the number of counts, while theangular correlation was sampled with the two forwardGe detectors at ± . ◦ , ± ◦ , ± ◦ , ± ◦ and ± ◦ , inturn. The results of these measurements are comparedwith the calculated angular correlations in Fig. 6. Cal-culated angular correlations were used for the analysis ofthe precession data. At the relatively low beam energyof ∼ . S (65 ◦ ) = − . − for the outer particle counters, and S (65 ◦ ) = − . − for the center detector.The velocity dependence of Pd ions traversing gadolin-ium was investigated through measurements on 230, 245and 260 MeV Pd ions traversing Target I. The resultsare summarized in Table VI. In these measurements thePd ions sample the transient field over a velocity rangethat extends beyond that covered in the g -factor mea-surements. Between the three runs at different beamenergies there is an overlap of the velocity range sam-pled by the Pd ions in consecutive measurements dueto the difference in scattering conditions for the center( h θ C i = 0 ◦ ) and outer ( h θ O i = 20 . ◦ ) particle detec-tors. Agreement between the observed precessions forthe measurements that span similar velocity ranges is TABLE V: Level properties and reaction kinematics for Mo, Ru, Pd and Cd beam ions recoiling in gadolinium after Coulombexcitation on target ions in inverse kinematics during the g -factor measurements. The reaction kinematics are shown forrecoiling C (Target I and II) or Mg (Target III) ions detected in the outer particle counters (average scattering angle 20.7 ◦ ).See Table I and text.Isotope E B E (2 +1 ) τ (2 +1 ) h E i i h E e i h v i /v i h v e /v i h v/v i t Gd (MeV) (keV) (ps) (MeV) (MeV) (fs)Targets I and II Ru 240 832.6 4.1 136.5 29.7 7.57 3.53 5.38 576 Ru 652.4 8.8 137.6 30.9 7.52 3.56 5.34 614
Ru 539.6 18.2 138.8 32.0 7.48 3.60 5.32 638
Ru 475.1 26.6 139.9 33.3 7.44 3.63 5.32 644
Ru 358.0 83.4 141.0 33.5 7.39 3.65 5.31 653
Pd 245 556.4 16.6 142.4 32.3 7.49 3.57 5.32 637
Pd 555.8 14.4 143.6 33.6 7.44 3.61 5.33 633
Pd 511.9 17.6 144.7 34.8 7.42 3.64 5.33 633
Pd 434.0 34.7 145.8 35.9 7.38 3.66 5.32 647
Pd 373.8 63.5 146.9 37.1 7.34 3.69 5.31 652
Cd 240 632.6 9.8 140.6 31.4 7.31 3.46 5.16 643
Cd 633.0 9.3 141.7 32.6 7.27 3.49 5.17 641
Cd 617.5 8.9 143.9 34.9 7.19 3.54 5.18 638
Cd 558.5 13.7 144.8 35.9 7.16 3.56 5.16 652Target III Mo 240 787.4 5.1 89.3 37.8 6.06 3.94 4.93 367 Ru 832.6 4.1 87.1 34.9 6.05 3.83 4.86 368
Ru 539.6 18.2 90.1 37.7 6.03 3.90 4.88 385
Ru 475.1 26.6 91.6 39.1 6.02 3.93 4.89 385
Ru 358.0 83.4 93.1 40.1 6.01 3.96 4.90 386
Pd 511.9 17.6 94.1 40.3 5.98 3.92 4.87 384 excellent. Fig. 7 compares the velocity dependence ofthe average transient field strength for Pd ions travers-ing gadolinium with that of the Rutgers parametrization.While the field strength parameter a TF has to be scaledup by a factor of about 1.4 for Target I, the data arenevertheless consistent with a v . dependence over therange applicable for the present g -factor measurements.To evaluate relative g factors, we have therefore used theRutgers parametrization to determine the scaling ratiosdescribed in sect. II, which are needed to correct fordifferences in the velocity range over which the differentisotopes experience the transient field.The absolute scale of the experimental g factors hasbeen set here by reference to previous measurementson Pd by the external field and radioactivity tech-niques [26, 27]. We have followed the recommendationof Johansson et al. [26] and disregarded the measure-ments that used iron hosts, which gave evidence of havingslightly reduced hyperfine fields. The adopted g factor isthen the weighted average of their measurement using acobalt host [26] and an earlier external field measurementwith which it agrees [27]. After making a small correctionfor a more recent level lifetime ( τ = 17 . g = +0 . Pd includes iron-hostdata [29, 30]. If these data are included in the average(after being adjusted to correspond to the same adopted lifetime), the resultant value, g = +0 . g factor results are summarized in Ta-ble VII. Figure 8 shows a comparison of the present andprevious data for relative g factors in the Pd isotopes.The previous measurements used the transient-field tech-nique in conventional kinematics whereas the presentmeasurements use inverse kinematics. The consistencyof the data in Fig. 8 is important because transient-fieldmeasurements in inverse kinematics, like those reportedhere, are reaching a high level of statistical precision, tothe point where uncontrolled systematic effects associ-ated with changing the beam species might become evi-dent. C. Adopted g factors This section gives a summary of the present and pre-vious g (2 +1 ) values in Mo, Ru, Pd and Cd isotopes.Table VIII summarizes the previous data on the Moisotopes from Refs. [7, 12], along with the values adoptedfor the following comparison with theory. The new mea-surement for g (2 +1 ) in Mo, relative to
Pd, is includedin the summary of adopted g factors.Table IX summarizes the g factors in the Ru isotopes.Further explanation is required concerning the methodused to combine the results of the three independent sets TABLE VI: Reaction kinematics, precessions and average transient-field strengths for
Pd. See Tables I and II and text. h θ carbon i is the average scattering angle of the carbon target ions. E B h θ carbon i h E i i h E e i h v i /v i h v e /v i h v/v i t Gd ∆Θ obs h B TF i (MeV) ( ◦ ) (MeV) (MeV) (fs) (mrad) (ktesla)230 0 127.1 27.1 6.89 3.18 4.80 713 − . .
23) 3.47(19)230 20.7 135.7 31.0 7.12 3.40 5.04 682 − . .
01) 3.84(18)245 0 136.6 31.6 7.14 3.43 5.07 676 − . .
86) 3.84(17)245 20.7 145.8 35.9 7.38 3.66 5.32 647 − . .
34) 3.89(22)260 0 146.1 36.2 7.38 3.67 5.33 644 − . .
01) 3.87(19)260 20.7 156.0 41.2 7.63 3.92 5.59 616 − . .
18) 3.34(21) -180 -120 -60 0 60 120 1800.00.51.01.50.0 W ( q ) q (degrees) W ( q ) W ( q ) FIG. 4: (Color online) Experimental and calculated angularcorrelations for even- A Ru isotopes following Coulomb exci-tation with 95 MeV S beams. Different symbols representthe four γ -ray detectors. of measurements on the Ru isotopes (one in conventionalkinematics and two in inverse kinematics). On one hand,the two measurements in inverse kinematics with gadolin-ium as the ferromagnetic host were calibrated relative tomeasurements on Pd (and Mo) performed with the
TABLE VII: Results of g -factor measurements in inverse kine-matics. h ∆Θ i obs is the average observed experimental preces-sion angle. The g factors are referenced to g (2 +1 ; Pd =+0 .
393 and assigned uncertainties that reflect the uncertain-ties in the measured precession angles. Adopted g factors,which include uncertainties in the reference g factor, are pre-sented in sect. II C.Nuclide h ∆Θ i obs g a (mrad)Target I Ru − . . Pd − . . Cd − . . Ru − . . Ru − . . Ru − . . Ru − . . Ru − . . Pd − . . Pd − . . Pd − . . Pd − . . Pd − . . Cd − . . Cd − . . Cd − . . Mo − . . Ru − . . Ru − . . Ru − . . Ru − . . Pd − . . same target. On the other hand, the measurement in con-ventional kinematics, with an iron host, was calibrated(sect. II A 2) using the Rutgers parametrization [19]. Inthis situation it is difficult to properly combine the datasets and propagate the errors by the usual procedure ofworking with one or two g -factor ratios. Nevertheless,the mathematical relationships relating the data sets aresimple. Energy (keV)0 200 400 600 800 C oun t s Cd = ke V g E Ge spectrum
Energy (keV)0 200 400 600 800 C oun t s Ru = ke V g E Ge spectrum
Energy (keV)0 200 400 600 800 C oun t s NaI spectrum
Energy (keV)0 200 400 600 800 C oun t s Pd = ke V g E Ge spectrum
Energy (keV)0 200 400 600 800 C oun t s NaI spectrum
Energy (keV)0 200 400 600 800 C oun t s N a I s p ec t r u m FIG. 5: Examples of γ -ray spectra observed at +65 ◦ and +115 ◦ to the beam axis in coincidence with C ions forward scatteredinto the outer particle counters. These spectra represent all of the data for the field up direction, obtained during the precessionmeasurements on Ru,
Pd and
Cd using Target II.
The procedure adopted to combine these measure-ments began by writing down the relationships betweenall of the experimental ‘knowns’, namely the measuredprecession angles and previously determined g factor val-ues, and the experimental ‘unknowns’, i.e. the transient-field strengths and the g factors to be extracted fromthe data. To illustrate the procedure, it is most con-venient to work with corrected experimental precessionangles, ∆Θ( A ), as defined in Eq. (5) and the integraltransient-field strengths, φ ( ∞ ), as defined in Eq. (4). The6 precession measurements on Target II are then relatedby ∆Θ II ( A ) = g ( A ) φ II , where A denotes the 6 even-even nuclei − Ru and
Pd; φ II is the same for allmeasurements on Target II. Similarly, the 6 precessionmeasurements on Target III are related by ∆Θ III ( A ) = g ( A ) φ III , where A now denotes , − Ru, Mo and
Pd. The 3 experimental precessions measured inconventional kinematics are related by ∆Θ conv ( A ) = g ( A ) φ conv , where A denotes − Ru. With these dataalone, only ratios of g factors can be obtained. Addi-tional data must be used to obtain the absolute valuesof the g factors or, alternatively, the values of φ II , φ III and φ conv . As noted above, for this purpose we haveused the previously measured g factors in both Pd and Mo, along with the Rutgers parametrization [19] to de-termine φ conv . The parametrization of the field strength,and hence φ conv , was assigned a 10% uncertainty (i.e. a TF = 16 . ± . g factors, the integrated transient-fieldstrengths, and the previous data to be used for field cal-ibration, were then used as the basis for a chi-squaredfit to determine the Ru g factors. The fitting proceduregives the correct average values for the Ru g factors andtheir associated experimental uncertainties, including theuncertainty in the transient-field calibration.The data included in the fit were: the precession datafor all three measurements, the previous experimental g factors in Pd and Mo, and the scale parameter of thetransient field for the Rutgers parametrization applied toiron hosts. There were therefore 18 data values in the fit(15 ∆Θ values, two independently measured g factors andone transient-field scale factor). Ten parameters wereextracted from the fit: the five g factors for the evenRu isotopes between Ru and
Ru, the transient-fieldstrengths in the three measurements, and the g factors of Pd and Mo. Note that new values of the latter two g factors were output fit parameters while their previousexperimental values were input data for the fit.The chi-squared per degree of freedom was 0.95. Theresults of this fit procedure and the consistency of thethree data sets are shown in Fig. 9. The field calibra-tion, and hence the absolute values of the g factors, wasdetermined predominantly by the previous g factor in Pd. Indeed, the value of g (2 + ) in Pd returned bythe fit did not differ significantly in either magnitude or0 q (degrees)00.511.500.511.52 W ( q ) CenterTop & Bottom108Pd 2+ 0+
FIG. 6: Measured and calculated angular correlations for
Pd in inverse kinematics. The upper panel is for the centerparticle detector (average C scattering angle 0 ◦ ); the lowerpanel is for the top and bottom (outer) particle detectors (av-erage C scattering angle 20.7 ◦ ). precision from the adopted previous value. For Mo,however, an improved g (2 + ) value was obtained, and isreported in Table VIII. It is worth noting that the tran-sient field strength for Ru in iron hosts agreed with theRutgers parametrization [19] to within 5%, whereas forthe measurements on gadolinium hosts the transient fieldstrength was of order 30% stronger than the predictionof the Rutgers parametrization.As can be seen from Table IX the present g factor for Ru is 20% higher than that obtained by Johansson etal. from perturbed angular correlation measurements ona radioactive source alloyed with an iron host [26]. Theyassumed a static hyperfine field of B hf = 50 . ± . g -factor measurementimplies that in Ref. [26] the effective static field for Ruimpurities in iron was B hf = − ± et al. [35] who took B hf = − ± et al. [6, 7] adopted a hyperfine field for Ru iniron of B hf = − . ± . g factors of the neutron-rich isotopes[6, 7] by about 20%, these values are reported withoutadjustment in Table IX.Finally, it is to be noted that Taylor et al. [25] per-formed transient-field g -factor measurements on the Ruisotopes in parallel with the present work. Their adopted g factors agree within the experimental uncertainties B TF ( k T e s l a ) FIG. 7: (Color online) Measured transient-field strength for
Pd ions traversing the gadolinium layer of Target I. Dot-ted horizontal lines indicate the velocity range over which thetransient field is sampled. The average transient-field strengthis plotted at the average ion velocity. Open symbols corre-spond to the detection of C ions in the central detectorwhile filled symbols indicate detection in the outer detectorswith an average scattering angle of 20.7 ◦ . The solid line showsthe v . velocity dependence of the Rutgers parametrization[19] scaled to best fit these data. g ( + )
102 104 106 108 110A present - inverse kinematicsprevious - conventional kinematics
Pd isotopes0.40.30.20.10.0
FIG. 8: Comparison of transient-field measurements on thePd isotopes. The present measurements in inverse kinemat-ics are compared with previous measurements in conventionalkinematics [31–34]. The data are shown relative to the presentadopted g (2 + ) value in Pd. with those reported here.1 g ( + ) A96 98 100 102 104 1060.00.20.40.60.8
Ru isotopes
FIG. 9: (Color online) Comparison of g factors from the threemeasurements on the Ru isotopes, calibration values from theliterature, and the average values obtained from the global fit.See text.TABLE VIII: g (2 +1 ) values in the Mo isotopes.Nuclide g (2 + )Previous Present Adopted Mo +1.15(14) a - +1.15(14) Mo +0.308(43) a - +0.308(43) Mo +0.394(31) a - +0.394(31) Mo +0.483(36) a +0.485(29) +0.485(29) Mo +0.471(33) a - +0.471(33) Mo +0.42(7) b - +0.42(7) Mo +0.27(2) c - +0.27(2) Mo +0.21(2) c - +0.21(2) Mo +0.5(3) c - +0.5(3) a Adopted value in [12]. b [36]. c [7]. III. TIDAL WAVE APPROACH FOR g -FACTORCALCULATIONS As noted in the Introduction, the tidal-wave modeluses the fact that in the semi-classical approximationthe yrast states of vibrational nuclei correspond toquadrupole waves traveling over the surface of the nu-cleus, like the tidal waves over the surface of the ocean.In the frame of reference that co-rotates with the wavethe quadrupole deformation is static - like that of a ro-tor. The difference between the vibrator and rotor isthat the angular momentum of an ideal rotor is gener-ated by increasing the angular velocity, whereas in thecase of the ideal vibrator the angular velocity is constant(equal to half the vibrational frequency) and angular mo-mentum is generated by increasing the amplitude of the
TABLE IX: g (2 +1 ) values in the Ru isotopes.Nuclide g (2 + )Previous Present Adopted Ru +0.445(28) +0.445(28) Ru +0.4(3) a +0.408(32) +0.408(32) Ru +0.46(5) b +0.429(23) +0.429(23) Ru +0.37(8) b +0.354(21) c +0.453(23) +0.453(23) Ru +0.41(5) a +0.406(21) +0.406(21) Ru +0.28(13) d +0.28(13) e Ru +0.28(4) d +0.28(4) e Ru +0.42(6) d +0.42(6) e Ru +0.44(9) d +0.44(9) ea From [37]. b From [35], updated for lifetime in [20]. c From [26], updated for lifetime in [20]. d From [7]. e As discussed in the text, there may be evidence that these valuesshould be increased by a factor of 1.2.
TABLE X: g (2 +1 ) values in the Pd isotopes. Uncertaintiesin square brackets are relative errors for the sequence of iso-topes; errors in round brackets include the uncertainty in thetransient-field calibration.Nuclide g (2 + )previous present adopted Pd +0.401 [26] a +0.418[21] 0.411(30) Pd +0.419 [0.23] a +0.461[18] 0.446(30) Pd +0.393 [0] a +0.393[0] 0.393(23) Pd +0.339 [15] a +0.347[14] 0.343(22) Pd +0.308 [16] a +0.350[14] 0.333(21) Pd Pd +0.24(13) b +0.24(13) Pd +0.2(1) b +0.2(1) a Average of relative g -factor measurements in [31–34], normalizedto the adopted value for Pd. b From [7]. The result for
Pd corresponds to τ = 117(6) ps[38, 39]. TABLE XI: g (2 +1 ) values in the Cd isotopes.Nuclide g (2 + )previous a present and adopted Cd 0.40 (0.10) +0.393(31)
Cd 0.34 (0.09) +0.389(31)
Cd 0.285 (0.055) b +0.407(29) Cd 0.32 (0.08) +0.360(24)
Cd 0.29 (0.07) +0.325(21)
Cd 0.30 (0.07) +0.296(24) a From [31]. b Calibration value adopted in [31]. ≤ Z ≤
48 and 54 ≤ N ≤
68 are very welldescribed by this model. Individual differences betweenthe nuclides, which reflect the response of the nucleonicorbits at the Fermi surface to rotation, are reproduced.The B ( E
2) values of the transitions between the yrastlevels are also well accounted for, including their linearincrease with angular momentum in vibrational nuclei.In this section, the extension of the model to calculate g factors is described.The details of the tidal wave approach are presentedin Refs. [8, 9]. In essence, the self-consistent crankingmodel is applied to nuclei that are spherical or weaklydeformed in their ground states. The calculations arebased on the Tilted-Axis-Cranking (TAC) version of theCranking model as described in Ref. [40]. The casesconsidered correspond to a rotation of the nucleus abouta principal axis. We start from the rotating mean field h ′ = h nilsson ( ǫ, γ ) + ∆( P + + P ) − ωJ x − λN, (7)which consists of quadrupole deformed Nilsson potential h nilsson [41], combined with a monopole pair field P + andfixed pair potential ∆. The chemical potential λ is fixedfor each deformation such that the particle number iscorrect for ω = 0.The calculations of the g factors are performed on agrid of triaxial quadrupole deformations while the angu-lar frequency ω is fixed at every grid point by the condi-tion J = h ω ( J ) | J x | ω ( J ) i + J c , (8)which is facilitated by linear interpolation between dis-crete ω grid points. A small correction is applied to theangular momentum J c = 100 MeV − ε sin ( γ − π/ ω. (9)About half of it takes into account the coupling betweenthe oscillator shells and another half is expected to comefrom quadrupole pairing, both being neglected in theCranking calculations. For the study of the 2 +1 -states J = 2 was set. The calculations for J = 4 have alsobeen carried out. The diabetic tracing technique, as de-scribed in Ref. [40], reliably prevented sudden changesof the quasiparticle configuration. The tracing was per-formed by using moderate steps, ∆ ω = 0 .
05 MeV, andcomparing the overlap of configurations step by step.The total energies are calculated by means of theStrutinsky method (SCTAC in [40]): E ( J, ǫ, γ ) = E LD ( ǫ, γ ) − ˜ E ( ǫ, γ ) ++ h ω ( J ) | h ′ | ω ( J ) i − h ω = 0 | h ′ | ω = 0 i + ω ( J ) J. (10) After minimizing the energy E ( J, ǫ, γ ) to obtain the equi-librium deformation parameters, ǫ e and γ e , the magneticmoment is calculated. µ = h ω ( J ) , ǫ e , γ e | µ x | ω ( J ) , ǫ e , γ e i , (11)where µ x = µ N ( J x,p + ( η . − S x,p − η . S x,n ) . (12)The spin contributions to the single-particle magneticmoments were evaluated with a common attenuation fac-tor, η = 0 .
7. The possibility that the correction term tothe angular momentum, J c , contributes to the magneticmoment was disregarded. The g factor is given by g ( J ) = µ ( J ) J . (13)The g factors turned out to be sensitive to the choiceof ∆ p and ∆ n . (See e.g. Ref. [42] for a general discussionon the sensitivity of g factors to pairing.) For the tran-sitional nuclei around A = 100, the experimental pair-ing parameters ∆, as calculated from the even-odd massdifferences, fluctuate considerably with the particle num-bers. Using these experimental values in the calculationstranslates into fluctuations of the g factors that are incontradiction to experiment. The even-odd mass differ-ences do not only reflect the pairing strength but are alsosensitive to the level spacing and deformation changes,which may be the major source of the fluctuations. Forthis reason we adopted constant values of ∆, which aresomewhat smaller than the average experimental valuesobtained by means of the four-point formula. The pairgap parameters ∆ p = ∆ n = 1 . p = ∆ n =1.2 MeV for the Pdand Cd isotopes. The results for ∆ p = ∆ n =1.1 MeV arealso shown for several Pd and Cd isotopes.In all cases we have reported g factors for the nucleardeformation of minimum calculated energy. For the heav-iest isotopes of Mo and Ru the Strutinsky method pre-dicts that the oblate minimum is lowest, with a close byprolate minimum. As with other mean field approaches,the inaccuracies in the prolate-oblate energy differenceare of the order of a few hundred keV. In both Moand
Ru there are small energy differences along thegamma degree of freedom. A detailed examination ofthe dependence of the g factors on the nuclear shape insuch cases is beyond the scope of the present report. Theeffect is expected to be secondary compared to the effectsof pairing, which affect the calculated shapes as well asthe g factors.The results of the calculations are listed in Tab. XIIand compared with the experimental data in Fig 10. Byapplying the same calculation, the energies, B ( E
2) val-ues, and g factors, can be obtained for the yrast sequencesin the considered nuclei (see also Refs. [8, 9]). The goodagreement with experimental data demonstrates the ap-plicability of the tidal wave approach.3
52 56 60 64 680.00.10.20.30.40.50.60.7 g ( + ) Experiment Calculation Mo N
52 56 60 64 680.00.10.20.30.40.50.60.7 g ( + ) N Experiment Calculation Ru
56 60 64 68 720.00.10.20.30.40.50.60.7 Pd g ( + ) N ExperimentCalculation with ∆ =1.1MeVCalculation with ∆ =1.2MeV
56 60 64 68 720.00.10.20.30.40.50.60.7 g ( + ) N Cd ExperimentCalculation with ∆ =1.1MeVCalculation with ∆ =1.2MeV FIG. 10: (Color online) g (2 + ) calculations compared withexperiment. The green line shows Z/A . TABLE XII: Deformations and calculated g factors.Nuclide deformation 0 + deformation 2 + g (2 + ) g (4 + ) ǫ γ ǫ γ Mo Mo < Mo Mo Mo Mo Mo Mo Mo − .
225 0 − .
228 0 0.334 Ru Ru Ru Ru Ru Ru Ru Ru Ru − .
210 0 − .
210 0 0.301 Pd Pd Pd Pd Pd Pd Pd Pd Cd Cd Cd Cd Cd Cd Cd Cd IV. DISCUSSIONA. g factor trends in the Tidal Wave Model The N -dependence of g factors in transitional nucleihas been a challenge to theory. The main reason is thatthe g factors are sensitive to the underlying single par-ticle composition of the collective quadrupole degree offreedom. The collective states of transitional nuclei havebeen mostly described in the frame work of phenomeno-logical collective models such as the Bohr Hamiltonian[1] and the Interacting Boson Model [2], which do notspecify the fermionic structure of the collective mode.On this level, one simply assumes that only the pro-tons are responsible for the the current that generatesthe magnetic moment, i.e. g = Z/A . Sambataro andDieperink [43] addressed the experimental deviations of4the g factors of the transitional isotopes of Ru to Tein the framework of proton-neutron Interacting BosonModel. Since then, there has been little progress towardsa deeper microscopic-based interpretation. We approachthe problem anew within the tidal wave model [8, 9],which is completely microscopic. It describes the yraststates by means of the self-consistent Cranking model,which allows one to calculate the magnetic moment di-rectly from the nucleonic currents. In applying it to nu-clei that are spherical or slightly deformed in their groundstate, one has to numerically diagonalize the quasiparti-cle Routhian, Eq. (7). Only for well deformed nuclei isthe perturbative expression, as given in Ref. [44], ap-plicable. The calculations in Refs. [8, 9] indicate thatthe nuclides in the considered transitional region are an-harmonic vibrators or very soft rotors. The deforma-tion of the 2 + state increases with the number of va-lence proton holes below, and neutron particles above,the Z = N = 50 shell. It ranges from ǫ ∼ . ∼ .
25 for the rotational nuclei , , Mo. For each isotope chain, the deformation in-creases with the neutron number. Fig. 10 shows that thecalculated g factor trends are in overall good agreementwith experiment. In particular, the deviations from thevalue Z/A are well accounted for.The Z - and N - dependence of the g factors will nowbe discussed. Looking at the microscopic contributions tothe angular momentum, it is seen that a few quasiparticlelevels near the Fermi surface contribute most to the totalangular momentum. The g factors generally decreasealong the isotope chains. This decrease is the result ofthe fractional increase of J n , the angular momentumcontribution from the h / neutrons. To elucidate thisobservation, Fig. 11 compares the calculated g factorswith the simple approximation g ( J ) ≈ (1 . J p − . J n ) / , (14)where J p and J n are the angular momenta calculatedby means of the Cranking model for the proton N = 4and the neutron N = 5 shells, respectively. This approx-imation assumes that (i) only the g / proton holes andthe h / neutrons contribute to the magnetic moment,and (ii) the expression g = g l ± η g s l + 1 , (15)which is valid for the spherical shells, can be used to setthe nucleon g factors. The contribution from the neutron N = 4 shell is assumed to be zero because it is generatedby the d / and d / orbitals, which have opposite g fac-tors. Fig. 11 compares the approximation of Eq. (14)with the full calculation. It is seen that the approxi-mation qualitatively accounts for the N dependence ofthe g factors. The separate contributions of the variousoscillator shells, N , to the total magnetic moment werecalculated by means of Eq. (12). It turned out that thecontributions of the N = 3 proton and neutron shells, and of the N = 4 neutron shell, are negligible. The er-rors of the approximation in Eq. (14) are therefore dueto the use of the single-nucleon g factors for spherical j -shells. Nevertheless, the simplified expression, Eq. (14),allows one to understand the N and Z dependence of theexcited-state g factors.As an example, the behavior of the Mo isotopes wasexamined in greater detail. Fig. 12 shows the composi-tion of the angular momentum of the 2 + state. Withincreasing numbers of valence neutrons, the g / pro-ton fraction, J p , remains nearly constant for the isotopechain, whereas the h / neutron fraction, J n , increases,which causes the decrease of the g factor. As illustratedin Fig. 13, the increased neutron numbers push up theFermi surface, and with increased deformations the Fermilevel moves into the lower half of the Nilsson orbits with h / parentage, which generate J n . Since the h / neutrons have g = − .
24, they progressively reduce themagnetic moment. In the Cd isotopes the proton frac-tion, J p , is smaller because there are only two g / holes.The progressive occupation of the h / neutron orbitalsgenerates the J n fraction seen in Fig. 12, which causesthe decrease of the g factor clearly seen in Fig. 11 for thesimplified expression, Eq. (14).For the Cd isotopes, the deformation of the 2 + state isabout 0.1. As seen in Fig. 13, the Fermi level reaches the h / states only at N = 64. However one has to keepin mind that the smaller deformation implies a higherangular frequency of the tidal wave, which lowers the h / orbitals relative to the positive parity orbitals. Forthis reason, the occupation of the h / orbitals startsalready at N = 60. Hence, in the studied region, theneutrons in the h / orbit are primarily responsible forthe drop of the g factors with increasing neutron number.This observation agrees with the inference of Smith et al. [6] based on their measured g factors for several neutron-rich isotopes. Moreover, it has long been known that thestrong increase of J n in well deformed nuclei, caused bythe rotational alignment of the i / and j / neutrons,reduces the g factors below Z/A [45].The g factors of the Cd isotopes are underestimated byabout 20%. Reducing the proton pair gap ∆ p by about10%, while keeping the neutron gap ∆ n unchanged,would increase the proton fraction, J p , relative to theneutron fraction, J n , such that the g factors have thecorrect magnitude. The N dependent trend will not bechanged. A reduced ∆ p for Z = 48 appears reasonable,because there are only two proton holes to generate paircorrelations. The other isotopes have more proton holes,which generate stronger the pair correlations.The “canonical” estimate g = Z/A for collectivequadrupole excitations is based on the assumptions that(i) the ratio J p / ( J p + J n ) = Z/A and (ii) that the spincontributions of the protons and neutrons cancel. As-sumption (i) is rather poor for the Cd isotopes, whichare almost semi magic. It becomes better for Mo iso-topes, which are situated further into the open shells.Assumption (ii) is not justified for the high-spin intruder5
52 56 60 64 680.00.10.20.30.40.50.60.7 Mo g ( + ) Full Approximation N
56 60 64 680.00.10.20.30.40.50.60.7
Full Approximation Cd g ( + ) N FIG. 11: (Color online) Comparison between calculated g fac-tors. Black squares show the full calculation as described inthe text and shown in Fig. 10. Red dots show the approxima-tion of Eq. (14), which takes into account only the magneticmoments generated by the g / protons and the h / neu-trons. orbitals g / and h / , which almost completely gen-erate the magnetic moment. Although (i) and (ii) be-come more valid assumptions for increasing numbers ofvalence nucleons, the differences between the g factors ofthe nucleonic orbitals near the Fermi surface remain no-ticeable in the Z and N dependence of nuclear g factors(cf. Ref. [45]). B. N = 50 , cases: , Mo, Ru The tidal wave approach does not work for the N = 50spherical nucleus Mo. The ground state configurationis not able to generate enough angular momentum toreach J = 2. By recognizing the fact that the neutronsof Mo complete a shell, the lowest 2 + configurationis obtained as the two quasi-proton configuration withangular momentum projection of 2. Its deformation isfound to be zero. The calculated g factor of 1.32 is closeto the value 1.43 for the spherical g / proton orbital.
56 60 640.00.20.40.60.81.01.21.41.61.82.0 Mo N=5 NeutronN=4 ProtonN=4 Neutron N J N=3 Proton
56 60 64 680.00.20.40.60.81.01.21.41.61.82.0 N J Cd N=5 NeutronN=4 NeutronN=4 Proton
FIG. 12: (Color online) Composition of the angular momen-tum of the 2 + states of the Mo and Cd isotopes. The fractionof J = 2 of each oscillator shell as obtained from the Crankingmodel is shown. The distance between the lower frame andthe circles (red curve) is the fraction of the protons in the N = 3 oscillator shell, the distance between the circles andsquares (red and black curves) the N = 4 proton fraction,the distance between the squares and triangles (black andblue curves) the N = 4 neutron fraction, and the distancebetween the triangles (blue curve) and the upper frame the N = 5 neutron fraction. The contribution of the N = 5 pro-tons is negligible. The distance between the lower frame andorange curve with no symbols is the proton fraction accord-ing to the IBM-II boson counting rule calculated by means ofEq. (16). The neutron fraction is the distance between thiscurve and the upper frame. The straight green line shows2 Z/A . The N = 52 nuclide Mo also turns out to be not quiteamenable to the tidal wave approach. Assuming a defor-mation of ǫ = 0 .
09, one can generate 2 units of angularmomentum at a frequency that is consistent with the en-ergy of the 2 + state. However, this deformation is notstable and the equilibrium deformation lies at a smallervalue. As seen in Fig. 12, J p =0.8, which generatesthe magnetic moment. The neutron fraction, J n = 1 . d / and g / orbits(see Fig. 13) which have single-particle moments of op-posite sign. The resulting g factor of 0.58 is much larger6 ene r g y ( M e V ) e d g d s h FIG. 13: (Color online) Nilsson diagram of Molybdenum neu-trons close to the N = 50 shell, the dash lines represent neg-ative parity and belong to h / levels. The neutron numbersof Mo isotopes are shown at the calculated equilibrium de-formation, which illustrates the intrusion of the h / shellamong the positive parity orbits for N <
64, and explains theimpact of h / orbit on the calculated g factors. than the experimental value of 0 . ± . et al. [4] carried out shell model cal-culations, finding g ≈ .
2. They state that J n origi-nates mainly from the d / orbital. Assuming that ourvalue of J n = 1 . d / or-bital and using g ( d / ) = − .
52 from Eq. (15), one finds g (2 + ) = (1 . × . − . × . / .
26, which is closeto experiment. As compared with the shell model, in ourcalculations only pairing and quadrupole correlations aretaken into account. It seems that the quadrupole cor-relations are over estimated, which causes an increased g factor through admixtures of the g / orbital.In the case of the N =52 nuclide Ru, the tidal waveapproach gives a finite value of ǫ = 0 .
13 and a good es-timate for the frequency of the 2 + state. The calculated g factor is too large for the same reason as for Mo, how-ever the discrepancy is less, which is likely a consequenceof the increasing configuration mixing with the numberof valence nucleons and deformation. C. g factors of the 4 + states The tidal wave model can predict the g factors ofhigher excited yrast states above the 2 + state, and anumber of such predictions have been included in Ta-ble XII. Unfortunately the experiments to measurethese g factors are challenging and experimental dataare scarce. To our knowledge the only measurementto date is very recent work by G¨urdal et al. [46] on Pd, where g (4 + ) = 0 . ± .
09 was obtained, relativeto g (2 + ) = 0 . ± . g -factor ratio is consistent with the presentpredictions. Similar experiments on the heavier isotopessuch as Pd could decisively detect if g (4 + ) < . g (2 + ) as predicted by the tidal wave model calculations. D. Comparison with Interacting Boson Model
The present study gives new insights into the relativeangular momentum carried by protons and neutrons inthe transitional nuclei near A = 100, which can be com-pared and contrasted with the detailed analysis of Sam-bataro and Dieperink [43] based on the interacting bo-son model. These workers studied the g factors in thisregion in the framework of IBM-II, which distinguishesbetween proton and neutron bosons. The model param-eters were fitted to the energies and B ( E
2) values ofthe lowest collective quadrupole excitations. They foundthat, to a good approximation, J p ∝ N p and J n ∝ N n ,where N p and N n are the number of proton and neutronbosons, respectively. According to the IBM-II countingrule, these numbers are equal to one half of the numberof valence proton holes and one half of the number ofvalence neutrons relative to Z = N = 50, respectively,i.e. for J = J p + J n = 2 one has J p = 2 N p N p + N n = 2 50 − Z − Z + N −
50 = 2 50 − ZN − Z . (16)Sambataro and Dieperink assigned effective g factorsto the proton and neutron valence systems, g p ( N p ) and g n ( N n ), which were assumed to depend on N p and N n .Considering g p ( N p ) and g n ( N n ) as free parameters, theyfitted experimental g factors. They found that the re-sulting g p ( N p ) and g n ( N n ) values change smoothly withthe boson numbers, and claimed that they qualitativelycorrelate with the g factors of the valence neutrons andprotons that constitute the collective quadrupole mode.Fig. 12 compares the IBM-II values of J p and J n with our values, which are also are generated by thevalence particles and holes. As seen, the IBM-II doesnot track closely with the proton-neutron ratio of theangular momentum obtained from our microscopic cal-culation. More important, it does not provide any infor-mation about the composition of the proton and neutronfractions, which is decisive for the calculation of the mag-netic moments. These deficiencies are overcome by intro-ducing effective N p - and N n -dependent boson g factors,which are then adjusted to the experiment. In contrast,our discussion above demonstrates that the N - and Z -dependence of the measured g factors is well understoodin terms of the microscopic J p and J n fractions and con-stant g factors for the g / proton holes and h / neu-trons. V. CONCLUSION
The g factors of the first excited 2 + states in all of thestable even isotopes of Mo, Ru, Pd and Cd have beenstudied experimentally and theoretically. An extensive7set of measurements, using variations of the transient-field technique, was completed to ensure that the dataset is internally consistent, i.e. the relative g (2 +1 ) valuesare accurate both within and between the isotope chains.Absolute values of the g factors were set relative to g (2 +1 )in Pd. The experimental precision has been improvedconsiderably.The data have been compared in detail with the tidalwave version of the cranking model. We conclude thatthe tidal wave approach gives a convincing description ofthe mass-dependent g -factor systematics in vibrationaland transitional nuclei. Moreover the g factors reveal theproton-neutron composition of the collective quadrupolemode. In comparison with previous work in this regionbased on the proton-neutron interacting boson model(IBM-II), the tidal wave model is more solidly based onthe underlying single-particle structure. It is found thatthe simple IBM-II counting rule based on the valenceproton fraction gives only a rough guide. In particular,the individuality of the valence nucleons (especially theirsingle-particle g factors) must be considered explicitly.Looking to the future, on the experimental side itis feasible to test the spin-dependent predictions of the tidal-wave model in a number of cases. In terms of im-proving the theory, it has been noted that the g factorsare very sensitive to the relative strength of neutron andproton pairing. More accurate predictions than thosepresented here will require a more sophisticated, self con-sistent treatment of pairing. Acknowledgments
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