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g-factors in the ground state and γ-bands in 164,166,168Er
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A 600 (1996) 272-282
g-factors in the ground state and 5,-bands in 164,166,168Er E Brandolini", C. Cattaneo ", R.V. Ribas b, D. Bazzacco a, M. De Poli c, M. Ionescu-Bujor d, R Pavan a, C. Rossi-Alvarez a a Dipartimento di Fisica deU'Universita di Padova and Sezione di Padova, INFN, Padova, Italy b Univ. de S. Paulo, Inst. de Fisica, Lab. Pelletron, C.P. 66318, S. Paulo, Brazil c Laboratori Nazionali di Legnaro, INFN, Legnaro, Italy d Institute of Physics and Nuclear Engineering, Bucharest, Romania
Received 10 July 1995; revised 27 November 1995
Abstract The g-factors of the ground state band members up to the 10+ level and of the 2 + state in the y-vibrational band have been measured in 164'l~'16SErusing the Coulomb excitation transient field technique. In the g-band the g-factor values were found to be nearly constant as a function of spin. For the 2+ states of the y-band, values about 20% larger than those in the g-band have been determined. This latter fact is consistent with the known decay properties of the y-band. Keywords: Nuclear reactions: 164Er, t~Er, 168Er(saNi, 58Ni~), E = 165, 210, 225 MeV; measured
y(O, H, T) in polarized Gd, (partiele)-~, coin, following Coulomb excitation, l~,t66,168Er levels, deduced g-factors. Enriched targets, thin foil transient field 1MPACtechnique. Cranked Hattree-Foek-Bogoliuhnv, interacting boson, geometrical collective model, projected angular momentum shell models.
1. I n t r o d u c t i o n The ground state (g) bands in stable even-even rare earth isotopes have been extensively studied in the last two decades mainly in connection with the band crossing phenomena which have been observed for the first time in 162Er [ 1 ]. The first crossing in these nuclei was interpreted as due to the alignment o f a pair of i!3/2 neutrons. This description has been confirmed by the measurement o f the average g-factor after backbending [2] in the nearby lying 15SDy, as well as in other nuclei. An interesting question, faced both theoretically and experimentally, was to establish to which extent the g-band g-factor values are affected at low spins by configuration mixing with the s-band. To elucidate this aspect, a few years ago we have measured the 0375-9474/96/$15.00 t~) 1996 Elsevier Science B.V. All rights reserved SSDI 03 7 5-9474 ( 96 ) 00003-6
F. Brandolini et al./Nuclear Physics A 600 (1996) 272-282
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g-factors up to the 10+ level in the yrast band of the l~'16SEr and 156'15SGd nuclei [3], by using the transient field (TF) technique. No significant variation of the g-factors with spin has been found for all investigated nuclei. In particular, the g-factor reduction at low spin previously reported in 166Er [4] has been not confirmed by our measurement. One has to point out that a decrease of the g-factors at low-spin levels in the g-band could not be explained by the cranked shell model (CSM) [5] and the cranked HartreeFock-Bogoliubov (CHFB) [6] calculations, while former CHFB calculations [7] in fact predicted this behaviour for 166Er. The interest in this subject was remarkable in the subsequent years: extensive CHFB calculations of the g-band magnetic moments in the even-even rare earth nuclei [8] have predicted a rather general attenuation effect even at rather low spins. On the other side, calculations of the same type in Dy isotopes have reported a nearly constant behavior [9]. More recently, calculations based on the angular momentum projected shell model (AMPSM) [ 10] were performed for Dy isotopes and a substantial reduction of the g-factors was obtained only for states after the band crossing. The geometrical collective model (GCM) predicts also nearly constant values [ 11 ], but it has to be noted that this model, as all the collective ones, cannot account for quasiparticle alignment. In order to test the various theoretical approaches, more experimental data obtained with higher accuracy is needed. In the present work we report the first g-factor measurements in the g-band of 164Er. The interest for this study was due to the fact that the 164Er nucleus shows the backbending effect in the yrast band, while 166Er and 168Er are not backbender nuclei. Different mixing of the g- and s-bands could therefore be expected at low spins, which may affect the g-factor values. A second aspect, not explored in our previous study, concerns the g-factors of the 2 + "y-vibrational states. Due to the experimental difficulties, up to now only few values with large errors were reported in the literature for the g-factor of the 2+ states in the deformed rare earth nuclei [ 12]. Concerning the Er nuclei, low precision values were previously determined in 166Er and 16SEr [ 13,14]. In an extreme collective model, identical g-factors are predicted for the 2 + states in both the g- and ~,-band. On the other hand, in the frame of IBM-2, it has been recently proposed that a disparity in those values can be related to F-spin impurity [ 15]. To get further insights into these aspects, in the present work precise measurements of the g-factors of the 2 + states in 164'166'16SEr have also been performed. A deviation of the 2 + state g-factor from that of the g-band was established, which may open a new perspective in nuclear theory. Similar to our previous investigation [3] the measurements were performed by the thin-foil TF perturbed angular correlation method in multiple Coulomb excitation. Targets containing a mixture of enriched isotopes have been used in order to prevent systematic errors when comparing the results for different isotopes. The measurements are essentially relative ones, since the TF calibration can be greatly affected by systematic errors.
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Table 1 Targets used in the experiments. The isotope ratios were 1:1 for both targets
Target
Isotopes
Id. 1 2
Target (mg/em 2)
166,168Er 164'l~Er
1.5 1.5
Ferromagnet (mg/cm 2) 5.7 5.3
Ag backing (mg/cm 2)
Beamenergy (MeV)
25.0 25.0
225 160, 210, 225
2. Experimental procedure The states under study were populated using Coulomb excitation induced by a 58Ni beam at the LNL XTU Tandem. Bombarding energies of 165, 210 and 225 MeV were employed in order to get a selective population of the levels. The beam current was typically 2--4 pnA. The target consisted of three main layers: the proper target, the ferromagnetic layer of Gd and a metallic backing. The foils were attached to each other with a thin (0.2 mg/cm z) layer of In. In Table 1, a summary of the experimental conditions is reported. In order to avoid systematic errors when studying isotopic chains, targets with two isotopes were used. Gd has been used as ferromagnetic layer, since it gives rise to a precession effect about twice as large as that obtained using iron. The drawback is that it has to be kept at liquid nitrogen temperature in order to get the full magnetization. In order to avoid overlapping of the lines from the excitation of the Gd nuclei, 16°Gd enriched material was used. A metallic sample has been obtained by reducing Gd oxide and subsequently rolling to the desired thickness. The Gd foils were annealed following the procedures described in [3] and the magnetization achieved, checked with a double coil magnetometer, was found to be about 6.2/.to/atom. The external polarizing field, of about 0.4 T, was periodically inverted every few minutes. Four Ge detectors were located at 4-68° and 4-112 ° since quadrupole 7-transitions were detected. The backscattered projectiles were registered with a 4 × 8 cm parallel plate avalanche counter at 3.2 cm from the target, covering a 2 sr solid angle. The thickness of the Gd layers was chosen such that the exit velocity of the recoil nuclei was larger than about 2v0 in order to avoid stopping in the ferromagnetic layer. The perturbed angular correlation effect is defined according to the formula e = (v/~ - 1)/(Vr~+ 1) where p is the ratio of the coincidence rate in a given detector with the applied field in "up" and "down" direction of the external magnetic field. The effect in the case of a single excitation is related to the precession angle by = S A O , where S is the logarithmic derivative of the angular correlation. In the present case, where cascade transitions are observed, average quantities should be considered. The experimental slope S has been deduced by measuring the effect obtained with the polarizing magnetic field fixed in one direction and rotating the Ge assembly by 4-3°. This was the maximum angle for which no variation of 7-ray absorption was assured in our experimental setup. A more detailed description of the experimental method and setup can be found in the literature [ 16]. A typical sensitivity for this measurement was ,~ 150-200 mr/g.
E Brandolini et al./Nuclear Physics A 600 (1996) 272-282 I-Z
275
25000
~Ni (225 MeV) on '°4"°SEr
o
t
20000
15000
Coulomb excitation
~{
0
~ ~.
200
400
600
800 ENERGY (keV)
Fig. 1. Typical coincidence y-my spectrum taken at 68 ° in the case of the l~ErJ66Er measurement,
3. Experimental results and data analysis An example of coincidence y-ray spectrum is shown in Fig. 1 for t64Er and 166Er nuclei. As seen in the figure, the spectra are very clean and the population of the 2 + states in the y-band is sufficient to allow an accurate analysis of the transition to the ground state. The measured effects, as well as the experimental and calculated slopes are shown in Table 2. In this table, the direct population Pair defined as the fraction of the transition yield due to the direct feeding of the decaying level is also shown. This quantity is a measure of the sensitivity for the g-factor determination of the corresponding level. It has to be noted that the effects observed for the l~'16SEr target were smaller than those obtained with the 164'16~Er target, and also than the similar ones determined in our previous experiment [3] performed at 210 MeV bombarding energy. Since the bulk magnetic properties were good, this was related to a poor thermal contact between the cooling system and the target, which increased the temperature and hence reduced the Gd magnetization. This is a concrete example of how one could get systematic errors when relying on a standard calibration. The results are however not affected when relative g-factors are derived. In order to obtain reliable precession angles from the measured effects a good knowledge of the angular correlation slope is required. Very accurate calculations of the angular distribution can be done using the code COULEX [17] if a good set of E2 transition matrix elements are available. Fortunately a detailed analysis of Coulomb excitation in these nuclei has been done recently [ 18,19], and a large set of experimental matrix elements have been obtained. In particular good agreement has been obtained in 166Er with the rotation-vibration model assuming an asymmetry parameter y = 12.7 °. In order to prove the internal consistency of the description, the observed peak intensi-
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Table 2 Experimental effects, experirnental slopes, calculated slopes and percentage of direct populations (see text)
E~m
1'~
*(%o)
165
4+ 6+ 8+ 4+ 6+ 8+ 10+ 2~ 4+ 6+ 8+ 10+ 2~
737(20) 557(21) 507(74) 466(14) 446(10) 378(18) 266(44) 2020(130) 392(15) 384(15) 369(24) 288(53) 1640(200)
210
225
&~p.
S~.
e~r
e(%o)
0.79(2) 0.78(3) 0.72(4) 0.71(11) 3.23(40)
1.044 0.851 0.774 0.898 0.814 0.755 0.719 3.305 0.843 0.795 0.743 0.710 3.281
75.6 86.2 88.9 33.3 56.7 76.5 95.1 100.0 21.9 39.3 65.7 81.5 100.0
624(24) 514(21) 476(74) 382(20) 400(15) 316(16) 283(44) 2030(130) 318(21) 310(15) 318(24) 296(53) 1670(200)
I~Er 225
4+ 6+ 8+ 10+ 2~
249(15) a 239(11) 212(14) 182(32) 1100(110)
0.76(2) 0.77(3) 0.70(4) 0.67(10) 3.50(41)
&~p.
S~.
P~
0.75(2) 0.77(3) 0.70(4) 0.65(10) 3.35(35)
1.028 0.842 0.771 0.886 0.803 0.745 0.714 3.328 0.797 0.783 0.735 0.702 3.292
75.1 86.2 90.3 25.0 56.7 74.5 96.3 100.0 18.7 41.6 63.1 82.4 100.0
0.797 0.783 0.735 0.704 3.292
18.7 41.6 63.1 82.4 100.0
I~Er 0.838 0.782 0.734 0.705 3.484
18.3 36.3 62.6 82.3 100.0
249(15) a 242(10) 206(12) 175(28) 1060(100)
0.75(2) 0.76(3) 0.73(4) 0.72(11) 3.24(38)
a For the 166"168Ertarget the effects for the 4 + states could not be measured independently, since the deexciting "F-lines overlap.
ties were compared to those calculated with the code COULEX. It turned out that the symmetric rotor model adopted in the analysis of previous measurements in 166Er and 16SEr [ 13,14] did not properly reproduce the observed intensities. The intensities observed in the present measurement for 166Er and 16SEr are well reproduced using the experimental matrix elements deduced in Refs. [ 18,19]. This is a worthwhile caution when adopting calculated slopes in the analysis, even if it is known that the angular distributions are not as strongly dependent on the values of the matrix elements as the intensities are. In the case of 164Er, similar deviations in intensity calculated with rotational matrix elements were found as for 16~Er, and therefore matrix elements proportional to those for 166Er were taken. When using calculated slopes one has to carefully verify the good adherence of the target layers, since a gap between them could give rise to a reduction due to vacuum deorientation. The presence of a gap between the layers can be efficiently monitored by the line shape of suitable transitions. The best sensitivity is obtained for the 10+ states, which have lifetimes comparable with the stopping time. In this case, even a small detachment would result in an evident deviation of the experimental line shape from the predicted one. An example is shown in Fig. 2, where one sees that a very good agreement is obtained using the tabulated lifetime and generating the line shape with a specific code [20].
E Brandolini et al./Nuclear Physics A 600 (1996) 272-282
277
m 1000 z
o
80O
~66EF
600
10"--->8"
/
400 200 0
I
I
I
438
440
442
r
I
444 446 448 E N E R G Y (keY)
Fig. 2. Experimentaland predictedline shape with ~"= 2.5 ps for the decay of the 10+ state in 166Er in the 166,168 E F m e a s u r e m e n t .
Slope measurements were performed at 225 MeV projectile bombarding energy mainly as an apparatus check. One has to note that the effects corresponding to the rotation of the y-detector assembly by an angle of 3 ° were comparable with the experimental effects caused by the precession of the TF perturbed angular correlations. For a precise determination of S(0) in all experimental runs long measuring time would have been required in order to get a high accuracy in the determined gfactors. The precession data were analyzed with the code MAGMO [21], which fits simultaneously all the y-transitions in several independent measurements, having the g-factors of the relevant levels as free parameters. Both the multiple Coulomb excitation, with the particular geometry of the experiment and the complex decay of the set of levels are handled by the code. In order to describe the transient field in Gd, the Chalk River parametrization [22] Bay = 27.5v/voZ e x p ( - O . 1 3 5 v / v o ) ( k T ) is employed by the program. For the 166'!68Er target the normalization of the TF field was left as a free parameter. The deduced g-factors are reported in Table 3, along with the data reported in previous measurements. The quoted errors are only statistical, except for those of the 4 + state g-factors which include also some uncertainty associated to a small relaxation of these rather long lived states, as shown by comparing the experimental slopes with the calculated ones (see Table 2). Relative gfactor values, which are independent of the field calibration, are also reported in Table 3. One may point out the higher precision achieved in the present experiment with respect to the previous TF measurements done at Canberra [ 13,14]. This is due to the use of a more efficient particle detector, as well as of Gd as ferromagnetic medium.
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Table 3 Summary of the experimental g-factors Nucleus g(2 +) d
Iv
Li~time (ps)
LNL present work
LNL a adopted
J~Er 0.343(8)
4+ 6+ 8+ 10+ 2~ 4+ 6+ 8+ 10+ 2~ 4+ 6+ 8+ 10+ 2~
124
0.341(20) 0.314(15) 0.340(16) 0.318(34) 0.404(30) 0.285(20) 0.287(15) 0.278(22) 0.280(37) 0.371(24) 0.293(30) 0.301(20) 0.305(26) 0.310(44) 0.387(30)
0.341(20) 0.314(15) 0.340(16) 0.318(34) 0.404(30) 0.285(20) 0.290(12) 0.282(15) 0.275(25) 0.371(24) 0.293(30) f 0.325(16) 0.301(18) 0.302(31) 0.387(30)
~Er 0.320(4)
~6SEr 0.329(7)
3.7 i.4 2.7 170 24.2 6.0 2.5 4.7 164 16.7 5.2 2.1 3.9
LNL g~t 1.18(8) 1.08(7) 1.17(7) 1.10(13) 1.39(12) 0.98(8) 1 0.97(7) 0.95(9) 1.28(10) 1.01(11) 1.12(7) 1.04(8) 1.04(11) 1.33(12)
Canberra b
0.267(32) 0.236(46) 0.203(74) 0.280(50)
• e e ¢
Bonn c
0.315(6) 0.258(11) 0.262(47)
0.328(46) 0.321(63) 0.322(80) 0.361(69)
a Adopted g-factor values include previous LNL data [3]. b Refs. [13,14]. c Ref. 14]. d Ref. [ 12]. e Renormalized to the best data for the 4+ state. f J68Er 4+ g-factor was scaled from that of 166Er according to the ratios of the 2+ g-factors.
4. Discussion 4.1. g-factors in the g-band The experimental g-factors o f the g-band levels, obtained in our measurements, are displayed in Fig. 3, together with theoretical calculations. One notices a rather constant behaviour o f the measured g-factors with spin in all the investigated nuclei. Compared to the 2 ÷ g-factors measured with M6ssbaner techniques (shown in the first column o f Table 3) the g-band g-factors are about 5 - 1 0 % lower. Therefore one can presume that in our experimental conditions the effective "IF was correspondingly smaller than the adopted parametrization. In order to better show the g-factors' behaviour as a function o f the neutron number and spin, relative g-factor values are presented in Table 4 taking as reference 166Er. These values are independent o f the field calibration and were obtained for each target in a separate analysis o f the ratios o f the observed effects. They are in substantial agreement with those o f Table 3 deduced from the general fit. One sees that the g-factor ratios are constant up to spin 10 +, indicating that the different structure o f the three nuclides does not give rise to relevant consequences. Most o f the theoretical predictions for the spin dependence o f the g-factors in the yrast band are based on CHFB calculations [ 6 - 8 ] . The calculated values for the t64'166'i68Er
E Brandolini et al./Nuclear Physics A 600 (1996) 272-282 I
i
I
I
!
r
]
i
279
!
0.5 1 @4Ei,-
0
(3
I
0.4 0.3 0.2
0
C
0.1
I
i
I
i
0.5
,66Er 0.4 0.3 b d 0.2 .....
0.1
--
0
0.5
:6~Er 0.4 0.3 0.2 0.1 I
[
I
2
4
6
' 8
10
i
12
14
16
18 20
22
Spin
Fig. 3. The spin dependence of the g-factors in g-bands. The theoretical curves are obtained within CHFB (curves (a) I6], (b) I7] and (e) I81) and AMPSM (curve (d) [24]) models.
nuclei, normalized to the g-factors of the 2 + states measured with Mi~ssbauer techniques are displayed in Fig. 3 (curves (a)-(c)). These calculations differ in some details, like particle number projection, the choice of the microscopic hamiltonian, the description of the pairing interaction and the choice of the parameters. The wide variation of the predictions of these calculations, especially for spins below the crossing frequency, is probably due to the specific set of parameters used, in particular the spherical singleparticle energies [8]. At higher frequencies, above the i13/2 neutron pair alignment, the Table 4 Relative g-factor values as function of the spin. The 4 + state value for the ~6SEr could not be extracted since the de,exciting 'F-line overlapped with that of 166~A" Target
2+
4+
6+
8+
I0 +
164 166 168
1.107(3) 1 1.03(3)
1.14(3) 1
1.15(3) 1 1.02(5)
1.16(4) 1 1.06(7)
0.97(11) 1 1.10(18)
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E Brandoliniet al./Nuclear PhysicsA 600 (1996)272-282
predictions of most of these calculations are in agreement, at least qualitatively, while at low spins there are strong differences. Recently g-band g-factor calculations have been performed in the angular momentum projected shell model. In this model [23], the hamiltonian (a Q • Q plus monopole and quadrupole pairing force) is diagonalized in a base including selected 0, 2 and 4 quasi-particle configurations. The rotational symmetry is restored by the angular momentum projection technique. The absolute g-factor values calculated by this model for 164'166'168Ernuclei [ 10,24] (curves (d) in Fig. 3) show only a small reduction for the levels before the crossing, in agreement with the experimental data. 4.2. g-factors of the 2~ states
The ratio g(2+)/g(6~) determined in the present work resulted to be 1.29(10), 1.28(11) and 1.19(11) for t64A66A68Er respectively. As the g-factor values are essentially constant for the low g-band levels and therefore nearly equal to that of the 6 + state, from the above ratios a significant variation of the g-factor of the 2+-vibrational states relative to the g-band g-factor is clearly evidenced. Previously the g(2+)/g(6+) values of 1.05(19) and 1.10(14) for 166Er and 16SEr, respectively, have been reported [13,14], which agree within errors with our present data but also with a ratio of 1. Preliminary data of the present work concerning the g(2+)/g(6+) ratios in Er isotopes were previously communicated, together with similar values found in the 162'164Dy isotopes [25]. The presently found disparity between the measured g(2 + )/g(6 +) ratios and unity can be related to the microscopic composition of the y-band in the frame of the rotational model. It is well known that the y-band excitation energy in the rare earths shows an irregular behavior as a function of Z, which is correlated with the microscopic composition of the state [26]. Substituting our ratios in the formula g = gR + (gr -- g R ) K 2 / I ( I + 1),
where for gR the experimental g-factor of the 2 + state in the g-band measured by the M6ssbauer technique is taken, one derives gr - gR = 0.12(3). This value is consistent with the known electromagnetic properties of the y-band in Er nuclei. In fact, taking the known E2/M1 intensity ratios for the y-band in 166Er [27] and Q0 = 7.5 b, from the formula for A1 = 1 transitions 0.93. E . Q0 [gK - g R I = [ 6 1 x / ( l - 1 ) ( I + 1) ' where the multipole mixing parameter 8 is in module about 1.6 and the energy is given in MeV, one gets Igr-gR[ = 0.11 (2). From similar data in 168Er [ 28 ] a value of 0.13 ( 1) is obtained. Triaxiality is neglected in these last considerations. The gr value can in principle be evaluated on the basis of p-h configurations contributing to the y-band, but detailed microscopic calculations are not available and standard RPA calculations [26] can not intrinsically predict a relevant difference between gr and gR [6].
E Brandoliniet al./Nuclear PhysicsA 600 (1996) 272-282
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GCM predicts, in the y-band, values of 8 of the correct size and with a positive sign, in agreement with the previous analysis, while the ratio g(2~)/g(2~) of 1.06 is rather smaller than our experimental value [28]. One has to mention that the model assumes a different deformation for neutrons and proton fluids, caused by the pairing interaction. In the frame of the raM-2, precise experimental values of the g(2+)/g(2~) ratios are considered of particular importance, in order to establish the contribution of Fspin admixtures [15]. However, a value smaller than unity is predicted in t66Er in disagreement with the present experiment. The critical role in M1 properties is played by the vector parameter Xv, which is proportional to fl~r - fir. Here fl,r and fl~ are the deformation parameters of the proton and neutron boson fluid respectively [ 15], which are roughly proportional to those defined in GCM [29]. The ratio g(2+)/g(2 +) has been found to be specifically sensitive to this parameter [30]. In order to get IBM-2 consistent with GCM, and consequently with our data, Xv must be taken positive and not about - 0 . 7 as in Ref. [ 15]. This important difference between the models was already observed some years ago [31 ]. It must be noted that microscopically a positive value for Xv is favoured [32], but a negative one has so far been assumed in order to reproduce the negative sign of 8 for most y ~ g transitions. Recently M1 experimental matrix elements in transitional even Os isotopes have been described with IBM-2 [29]. In that work the ratio g(2-~)/g(2~) is larger than unity in 188Os, but this fact is accompanied by a positive value of 8 for the 2~- ~ 2~- transition. This is not true in Er isotopes. In conclusion one has to stress once more that not only the size, but also the sign of the M1 matrix elements are decisive to discriminate among nuclear models. Therefore precise measurements of g-factor in the y-band as well of 8 in the y ~ y and y ~ g band transitions appear to be very important.
Note added
After completing the analysis of this work [25], we were informed that from the mixing ratio, measured in the angular correlation of the 5 + ~ 4 + transition in the 166Er y-band, a value of ( g r - g R ) = 0.090(13) has been deduced [33], which agrees in magnitude and sign with the value obtained from our g-factor measurements.
Acknowledgements Thanks are due to A. Buscemi, S. Martini and R. Zanon for the help on the experiment preparation, to Dr R. Pengo and G. Manente for contribution to the target preparation. One of us (R.V.R.) would also like to thanks the Istituto Nazionale di Fisica Nucleare (Italy) and the Funda~o de Amparo ~t Pesquisa de S,~o Paulo (Brazil) for the grants received during his stay in Italy.
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References [1] A. Johnson, H. Ryde and S.M. Hjorth, Nucl. Phys. A 179 (1972) 753. [2] G. Seiler-Clark, D. pare, H. Emling, A. Balanda, H. Grein, E. Grosse, K. Kulessa, D. Schwalm, H.J. Wollersheim, H. Hass, G.J. Kumbartz~ and K.H. Speidel, Nucl. Phys. A 399 (1983) 211. [31 E Brandolini, P. Pavan, D. Bazzacco, C. Rossi-Alvarez, R.V. Ribas, M. De Poli and A.M.I. Haque, Phys. Rev. C 45 (1992) 1549. 141 A. Alzner, E. Bodenstedt, B. Gemtinden, J. van den Hoff and H. Reif, Z. Phys. A 322 (1985) 467. [5] Y.S. Chen and S. Frauendorf, Nucl. Phys. A 393 (1983) 135. [6] K. Sugawara-Tanabe and K. Tanabe, Phys. Lett. B 207 (1988) 1243. [7] A. Ansari, E. Wrist and K. Mrihihans, Nucl. Phys. A 414 (1984) 215. [8] M. Saha and S. Sen, Nuci. phys. A 552 (1993) 37. [9] M.L. Cescato, Y. Sun and P. Ring, Nucl. Phys. A 533 (1991) 455. [10] Y. Sun and J.L. Egidio, Nucl. Phys. A 580 (1994) 1. [ 11 ] D. Troltenier, J.A. Mahran and W. Greiner, Z. phys. A 348 (1994) 1. [ 12] P. Raghavan, At. Nucl. Data Tables 42 (1989) 189. [ 13] C.E. Domn, H.H. Bolotin, A.E. Stuchbery and A.P. Byme, Z. Phys. A 325 (1986) 285. [ 14] C.E. Doran, A.E. Stuchbery, H.H. Bolotin, A.P. Byrae and G.J. Lampard, Phys. Rev. C 40 (1989) 2035. [ 15] J.N. Ginocchio, W. Frank and P. yon Brentano, Nucl. Phys. A 541 (1992) 211. [ 16] D. Bazzacco, E Brandolini, K. L6wenich, P. Pavan, C. Rossi-Alvarez, R. Zannoni and M. De Poli, phys. Rev. C 33 (1986) 1783. [ 17] A. Winther and J. De Boer, in: Coulomb Excitation, eds. K. Alder and A. Winther (Academic Press, New York, 1966) p. 303. [ 18] C. Fablander, I. Thorslund, B. Vamestig, A. Bitcklin, L.E. Svennson, D. Disdier, L. Kraus, I. Linck, N. Schnlz, J. Pedersen and D. Cline, Nucl. phys. A 537 (1992) 183. [19] B. Kotlinski, D. Cline, A. Bticklin, K.G. Helmet, A.E. Kavka, W.I. ICeman, E.G. Vogt, C.Y. Wu, R.M. Diamond, A.O. Macchiavelli and M.A. Deleplanque, Nucl. phys. A 517 (1990) 365. [20] K. L6wenich, 1986 LNL Ann. Report, LNL-INFN (PEP) 002/887, p. 200. [21] R.V. Ribas, Nucl. Inst. Meth. Phys. Res. 328 (1993) 553. [22] O. Htiusser, H.R. Andrews, D. Horn, M.A. Lone, P. Tams, P. Skensved, R.M. Diamond, M.A. Deleplanque, E.L. Dines, A.O. Macchiavelli and ES. Stephens, Nucl. Phys. A 412 (1984) 141. 123] K. Ham and Y. Sun, Nucl. Phys. A 529 (1991) 445. [24] K. Ham and Y. Sun, Int. J. Modem Phys. E (to appear). [25] F. Brandolini, Proc. Conf. on Perpectives for the Interacting Boson Model, eds. R.E Casten, A. Vitturi, A.B. Balantiken, B.R. Barrett, J.N. Ginocchio, G. Maino and T. Otsuka (World Scientific, Singapore, 1995) p. 647. [261 D.R. Bes, P. Federman, E. Maqueda and A. Zuker, Nucl. Phys. 65 (1965) 1. [27[ C.W. Reich and J.E. Cline, Nucl. Phys. A 159 (1970) 181. [281 K. Schreckenbach and W. Gelletly, Phys. Lett. B 94 (1980) 298. [291 S. Kuyucak and A.E. Stuchbery, Phys. Lett. B 348 (1995) 315. [30] P. van lsacker, P.O. Lipas, K. Hemaki, I Koivistoinen, D.D. Warner, Nucl. Phys. A 476 (1988) 301. [31 ] A.E.L. Dieperink, O. Seholten and D.D. Warner, Nucl. Phys. A 469 (1987) 173. [32] T. Otsuka, A. Arima and E Iachello, Nucl. Phys. A 309 (1978) 1. [ 33 } E. Bodenstedt, private communication.
Nuclear Physics A 600 (1996) 272-282
g-factors in the ground state and 5,-bands in 164,166,168Er E Brandolini", C. Cattaneo ", R.V. Ribas b, D. Bazzacco a, M. De Poli c, M. Ionescu-Bujor d, R Pavan a, C. Rossi-Alvarez a a Dipartimento di Fisica deU'Universita di Padova and Sezione di Padova, INFN, Padova, Italy b Univ. de S. Paulo, Inst. de Fisica, Lab. Pelletron, C.P. 66318, S. Paulo, Brazil c Laboratori Nazionali di Legnaro, INFN, Legnaro, Italy d Institute of Physics and Nuclear Engineering, Bucharest, Romania
Received 10 July 1995; revised 27 November 1995
Abstract The g-factors of the ground state band members up to the 10+ level and of the 2 + state in the y-vibrational band have been measured in 164'l~'16SErusing the Coulomb excitation transient field technique. In the g-band the g-factor values were found to be nearly constant as a function of spin. For the 2+ states of the y-band, values about 20% larger than those in the g-band have been determined. This latter fact is consistent with the known decay properties of the y-band. Keywords: Nuclear reactions: 164Er, t~Er, 168Er(saNi, 58Ni~), E = 165, 210, 225 MeV; measured
y(O, H, T) in polarized Gd, (partiele)-~, coin, following Coulomb excitation, l~,t66,168Er levels, deduced g-factors. Enriched targets, thin foil transient field 1MPACtechnique. Cranked Hattree-Foek-Bogoliuhnv, interacting boson, geometrical collective model, projected angular momentum shell models.
1. I n t r o d u c t i o n The ground state (g) bands in stable even-even rare earth isotopes have been extensively studied in the last two decades mainly in connection with the band crossing phenomena which have been observed for the first time in 162Er [ 1 ]. The first crossing in these nuclei was interpreted as due to the alignment o f a pair of i!3/2 neutrons. This description has been confirmed by the measurement o f the average g-factor after backbending [2] in the nearby lying 15SDy, as well as in other nuclei. An interesting question, faced both theoretically and experimentally, was to establish to which extent the g-band g-factor values are affected at low spins by configuration mixing with the s-band. To elucidate this aspect, a few years ago we have measured the 0375-9474/96/$15.00 t~) 1996 Elsevier Science B.V. All rights reserved SSDI 03 7 5-9474 ( 96 ) 00003-6
F. Brandolini et al./Nuclear Physics A 600 (1996) 272-282
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g-factors up to the 10+ level in the yrast band of the l~'16SEr and 156'15SGd nuclei [3], by using the transient field (TF) technique. No significant variation of the g-factors with spin has been found for all investigated nuclei. In particular, the g-factor reduction at low spin previously reported in 166Er [4] has been not confirmed by our measurement. One has to point out that a decrease of the g-factors at low-spin levels in the g-band could not be explained by the cranked shell model (CSM) [5] and the cranked HartreeFock-Bogoliubov (CHFB) [6] calculations, while former CHFB calculations [7] in fact predicted this behaviour for 166Er. The interest in this subject was remarkable in the subsequent years: extensive CHFB calculations of the g-band magnetic moments in the even-even rare earth nuclei [8] have predicted a rather general attenuation effect even at rather low spins. On the other side, calculations of the same type in Dy isotopes have reported a nearly constant behavior [9]. More recently, calculations based on the angular momentum projected shell model (AMPSM) [ 10] were performed for Dy isotopes and a substantial reduction of the g-factors was obtained only for states after the band crossing. The geometrical collective model (GCM) predicts also nearly constant values [ 11 ], but it has to be noted that this model, as all the collective ones, cannot account for quasiparticle alignment. In order to test the various theoretical approaches, more experimental data obtained with higher accuracy is needed. In the present work we report the first g-factor measurements in the g-band of 164Er. The interest for this study was due to the fact that the 164Er nucleus shows the backbending effect in the yrast band, while 166Er and 168Er are not backbender nuclei. Different mixing of the g- and s-bands could therefore be expected at low spins, which may affect the g-factor values. A second aspect, not explored in our previous study, concerns the g-factors of the 2 + "y-vibrational states. Due to the experimental difficulties, up to now only few values with large errors were reported in the literature for the g-factor of the 2+ states in the deformed rare earth nuclei [ 12]. Concerning the Er nuclei, low precision values were previously determined in 166Er and 16SEr [ 13,14]. In an extreme collective model, identical g-factors are predicted for the 2 + states in both the g- and ~,-band. On the other hand, in the frame of IBM-2, it has been recently proposed that a disparity in those values can be related to F-spin impurity [ 15]. To get further insights into these aspects, in the present work precise measurements of the g-factors of the 2 + states in 164'166'16SEr have also been performed. A deviation of the 2 + state g-factor from that of the g-band was established, which may open a new perspective in nuclear theory. Similar to our previous investigation [3] the measurements were performed by the thin-foil TF perturbed angular correlation method in multiple Coulomb excitation. Targets containing a mixture of enriched isotopes have been used in order to prevent systematic errors when comparing the results for different isotopes. The measurements are essentially relative ones, since the TF calibration can be greatly affected by systematic errors.
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Table 1 Targets used in the experiments. The isotope ratios were 1:1 for both targets
Target
Isotopes
Id. 1 2
Target (mg/em 2)
166,168Er 164'l~Er
1.5 1.5
Ferromagnet (mg/cm 2) 5.7 5.3
Ag backing (mg/cm 2)
Beamenergy (MeV)
25.0 25.0
225 160, 210, 225
2. Experimental procedure The states under study were populated using Coulomb excitation induced by a 58Ni beam at the LNL XTU Tandem. Bombarding energies of 165, 210 and 225 MeV were employed in order to get a selective population of the levels. The beam current was typically 2--4 pnA. The target consisted of three main layers: the proper target, the ferromagnetic layer of Gd and a metallic backing. The foils were attached to each other with a thin (0.2 mg/cm z) layer of In. In Table 1, a summary of the experimental conditions is reported. In order to avoid systematic errors when studying isotopic chains, targets with two isotopes were used. Gd has been used as ferromagnetic layer, since it gives rise to a precession effect about twice as large as that obtained using iron. The drawback is that it has to be kept at liquid nitrogen temperature in order to get the full magnetization. In order to avoid overlapping of the lines from the excitation of the Gd nuclei, 16°Gd enriched material was used. A metallic sample has been obtained by reducing Gd oxide and subsequently rolling to the desired thickness. The Gd foils were annealed following the procedures described in [3] and the magnetization achieved, checked with a double coil magnetometer, was found to be about 6.2/.to/atom. The external polarizing field, of about 0.4 T, was periodically inverted every few minutes. Four Ge detectors were located at 4-68° and 4-112 ° since quadrupole 7-transitions were detected. The backscattered projectiles were registered with a 4 × 8 cm parallel plate avalanche counter at 3.2 cm from the target, covering a 2 sr solid angle. The thickness of the Gd layers was chosen such that the exit velocity of the recoil nuclei was larger than about 2v0 in order to avoid stopping in the ferromagnetic layer. The perturbed angular correlation effect is defined according to the formula e = (v/~ - 1)/(Vr~+ 1) where p is the ratio of the coincidence rate in a given detector with the applied field in "up" and "down" direction of the external magnetic field. The effect in the case of a single excitation is related to the precession angle by = S A O , where S is the logarithmic derivative of the angular correlation. In the present case, where cascade transitions are observed, average quantities should be considered. The experimental slope S has been deduced by measuring the effect obtained with the polarizing magnetic field fixed in one direction and rotating the Ge assembly by 4-3°. This was the maximum angle for which no variation of 7-ray absorption was assured in our experimental setup. A more detailed description of the experimental method and setup can be found in the literature [ 16]. A typical sensitivity for this measurement was ,~ 150-200 mr/g.
E Brandolini et al./Nuclear Physics A 600 (1996) 272-282 I-Z
275
25000
~Ni (225 MeV) on '°4"°SEr
o
t
20000
15000
Coulomb excitation
~{
0
~ ~.
200
400
600
800 ENERGY (keV)
Fig. 1. Typical coincidence y-my spectrum taken at 68 ° in the case of the l~ErJ66Er measurement,
3. Experimental results and data analysis An example of coincidence y-ray spectrum is shown in Fig. 1 for t64Er and 166Er nuclei. As seen in the figure, the spectra are very clean and the population of the 2 + states in the y-band is sufficient to allow an accurate analysis of the transition to the ground state. The measured effects, as well as the experimental and calculated slopes are shown in Table 2. In this table, the direct population Pair defined as the fraction of the transition yield due to the direct feeding of the decaying level is also shown. This quantity is a measure of the sensitivity for the g-factor determination of the corresponding level. It has to be noted that the effects observed for the l~'16SEr target were smaller than those obtained with the 164'16~Er target, and also than the similar ones determined in our previous experiment [3] performed at 210 MeV bombarding energy. Since the bulk magnetic properties were good, this was related to a poor thermal contact between the cooling system and the target, which increased the temperature and hence reduced the Gd magnetization. This is a concrete example of how one could get systematic errors when relying on a standard calibration. The results are however not affected when relative g-factors are derived. In order to obtain reliable precession angles from the measured effects a good knowledge of the angular correlation slope is required. Very accurate calculations of the angular distribution can be done using the code COULEX [17] if a good set of E2 transition matrix elements are available. Fortunately a detailed analysis of Coulomb excitation in these nuclei has been done recently [ 18,19], and a large set of experimental matrix elements have been obtained. In particular good agreement has been obtained in 166Er with the rotation-vibration model assuming an asymmetry parameter y = 12.7 °. In order to prove the internal consistency of the description, the observed peak intensi-
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Table 2 Experimental effects, experirnental slopes, calculated slopes and percentage of direct populations (see text)
E~m
1'~
*(%o)
165
4+ 6+ 8+ 4+ 6+ 8+ 10+ 2~ 4+ 6+ 8+ 10+ 2~
737(20) 557(21) 507(74) 466(14) 446(10) 378(18) 266(44) 2020(130) 392(15) 384(15) 369(24) 288(53) 1640(200)
210
225
&~p.
S~.
e~r
e(%o)
0.79(2) 0.78(3) 0.72(4) 0.71(11) 3.23(40)
1.044 0.851 0.774 0.898 0.814 0.755 0.719 3.305 0.843 0.795 0.743 0.710 3.281
75.6 86.2 88.9 33.3 56.7 76.5 95.1 100.0 21.9 39.3 65.7 81.5 100.0
624(24) 514(21) 476(74) 382(20) 400(15) 316(16) 283(44) 2030(130) 318(21) 310(15) 318(24) 296(53) 1670(200)
I~Er 225
4+ 6+ 8+ 10+ 2~
249(15) a 239(11) 212(14) 182(32) 1100(110)
0.76(2) 0.77(3) 0.70(4) 0.67(10) 3.50(41)
&~p.
S~.
P~
0.75(2) 0.77(3) 0.70(4) 0.65(10) 3.35(35)
1.028 0.842 0.771 0.886 0.803 0.745 0.714 3.328 0.797 0.783 0.735 0.702 3.292
75.1 86.2 90.3 25.0 56.7 74.5 96.3 100.0 18.7 41.6 63.1 82.4 100.0
0.797 0.783 0.735 0.704 3.292
18.7 41.6 63.1 82.4 100.0
I~Er 0.838 0.782 0.734 0.705 3.484
18.3 36.3 62.6 82.3 100.0
249(15) a 242(10) 206(12) 175(28) 1060(100)
0.75(2) 0.76(3) 0.73(4) 0.72(11) 3.24(38)
a For the 166"168Ertarget the effects for the 4 + states could not be measured independently, since the deexciting "F-lines overlap.
ties were compared to those calculated with the code COULEX. It turned out that the symmetric rotor model adopted in the analysis of previous measurements in 166Er and 16SEr [ 13,14] did not properly reproduce the observed intensities. The intensities observed in the present measurement for 166Er and 16SEr are well reproduced using the experimental matrix elements deduced in Refs. [ 18,19]. This is a worthwhile caution when adopting calculated slopes in the analysis, even if it is known that the angular distributions are not as strongly dependent on the values of the matrix elements as the intensities are. In the case of 164Er, similar deviations in intensity calculated with rotational matrix elements were found as for 16~Er, and therefore matrix elements proportional to those for 166Er were taken. When using calculated slopes one has to carefully verify the good adherence of the target layers, since a gap between them could give rise to a reduction due to vacuum deorientation. The presence of a gap between the layers can be efficiently monitored by the line shape of suitable transitions. The best sensitivity is obtained for the 10+ states, which have lifetimes comparable with the stopping time. In this case, even a small detachment would result in an evident deviation of the experimental line shape from the predicted one. An example is shown in Fig. 2, where one sees that a very good agreement is obtained using the tabulated lifetime and generating the line shape with a specific code [20].
E Brandolini et al./Nuclear Physics A 600 (1996) 272-282
277
m 1000 z
o
80O
~66EF
600
10"--->8"
/
400 200 0
I
I
I
438
440
442
r
I
444 446 448 E N E R G Y (keY)
Fig. 2. Experimentaland predictedline shape with ~"= 2.5 ps for the decay of the 10+ state in 166Er in the 166,168 E F m e a s u r e m e n t .
Slope measurements were performed at 225 MeV projectile bombarding energy mainly as an apparatus check. One has to note that the effects corresponding to the rotation of the y-detector assembly by an angle of 3 ° were comparable with the experimental effects caused by the precession of the TF perturbed angular correlations. For a precise determination of S(0) in all experimental runs long measuring time would have been required in order to get a high accuracy in the determined gfactors. The precession data were analyzed with the code MAGMO [21], which fits simultaneously all the y-transitions in several independent measurements, having the g-factors of the relevant levels as free parameters. Both the multiple Coulomb excitation, with the particular geometry of the experiment and the complex decay of the set of levels are handled by the code. In order to describe the transient field in Gd, the Chalk River parametrization [22] Bay = 27.5v/voZ e x p ( - O . 1 3 5 v / v o ) ( k T ) is employed by the program. For the 166'!68Er target the normalization of the TF field was left as a free parameter. The deduced g-factors are reported in Table 3, along with the data reported in previous measurements. The quoted errors are only statistical, except for those of the 4 + state g-factors which include also some uncertainty associated to a small relaxation of these rather long lived states, as shown by comparing the experimental slopes with the calculated ones (see Table 2). Relative gfactor values, which are independent of the field calibration, are also reported in Table 3. One may point out the higher precision achieved in the present experiment with respect to the previous TF measurements done at Canberra [ 13,14]. This is due to the use of a more efficient particle detector, as well as of Gd as ferromagnetic medium.
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E Brandolini et al./Nuclear Physics A 600 (1996) 272-282
Table 3 Summary of the experimental g-factors Nucleus g(2 +) d
Iv
Li~time (ps)
LNL present work
LNL a adopted
J~Er 0.343(8)
4+ 6+ 8+ 10+ 2~ 4+ 6+ 8+ 10+ 2~ 4+ 6+ 8+ 10+ 2~
124
0.341(20) 0.314(15) 0.340(16) 0.318(34) 0.404(30) 0.285(20) 0.287(15) 0.278(22) 0.280(37) 0.371(24) 0.293(30) 0.301(20) 0.305(26) 0.310(44) 0.387(30)
0.341(20) 0.314(15) 0.340(16) 0.318(34) 0.404(30) 0.285(20) 0.290(12) 0.282(15) 0.275(25) 0.371(24) 0.293(30) f 0.325(16) 0.301(18) 0.302(31) 0.387(30)
~Er 0.320(4)
~6SEr 0.329(7)
3.7 i.4 2.7 170 24.2 6.0 2.5 4.7 164 16.7 5.2 2.1 3.9
LNL g~t 1.18(8) 1.08(7) 1.17(7) 1.10(13) 1.39(12) 0.98(8) 1 0.97(7) 0.95(9) 1.28(10) 1.01(11) 1.12(7) 1.04(8) 1.04(11) 1.33(12)
Canberra b
0.267(32) 0.236(46) 0.203(74) 0.280(50)
• e e ¢
Bonn c
0.315(6) 0.258(11) 0.262(47)
0.328(46) 0.321(63) 0.322(80) 0.361(69)
a Adopted g-factor values include previous LNL data [3]. b Refs. [13,14]. c Ref. 14]. d Ref. [ 12]. e Renormalized to the best data for the 4+ state. f J68Er 4+ g-factor was scaled from that of 166Er according to the ratios of the 2+ g-factors.
4. Discussion 4.1. g-factors in the g-band The experimental g-factors o f the g-band levels, obtained in our measurements, are displayed in Fig. 3, together with theoretical calculations. One notices a rather constant behaviour o f the measured g-factors with spin in all the investigated nuclei. Compared to the 2 ÷ g-factors measured with M6ssbaner techniques (shown in the first column o f Table 3) the g-band g-factors are about 5 - 1 0 % lower. Therefore one can presume that in our experimental conditions the effective "IF was correspondingly smaller than the adopted parametrization. In order to better show the g-factors' behaviour as a function o f the neutron number and spin, relative g-factor values are presented in Table 4 taking as reference 166Er. These values are independent o f the field calibration and were obtained for each target in a separate analysis o f the ratios o f the observed effects. They are in substantial agreement with those o f Table 3 deduced from the general fit. One sees that the g-factor ratios are constant up to spin 10 +, indicating that the different structure o f the three nuclides does not give rise to relevant consequences. Most o f the theoretical predictions for the spin dependence o f the g-factors in the yrast band are based on CHFB calculations [ 6 - 8 ] . The calculated values for the t64'166'i68Er
E Brandolini et al./Nuclear Physics A 600 (1996) 272-282 I
i
I
I
!
r
]
i
279
!
0.5 1 @4Ei,-
0
(3
I
0.4 0.3 0.2
0
C
0.1
I
i
I
i
0.5
,66Er 0.4 0.3 b d 0.2 .....
0.1
--
0
0.5
:6~Er 0.4 0.3 0.2 0.1 I
[
I
2
4
6
' 8
10
i
12
14
16
18 20
22
Spin
Fig. 3. The spin dependence of the g-factors in g-bands. The theoretical curves are obtained within CHFB (curves (a) I6], (b) I7] and (e) I81) and AMPSM (curve (d) [24]) models.
nuclei, normalized to the g-factors of the 2 + states measured with Mi~ssbauer techniques are displayed in Fig. 3 (curves (a)-(c)). These calculations differ in some details, like particle number projection, the choice of the microscopic hamiltonian, the description of the pairing interaction and the choice of the parameters. The wide variation of the predictions of these calculations, especially for spins below the crossing frequency, is probably due to the specific set of parameters used, in particular the spherical singleparticle energies [8]. At higher frequencies, above the i13/2 neutron pair alignment, the Table 4 Relative g-factor values as function of the spin. The 4 + state value for the ~6SEr could not be extracted since the de,exciting 'F-line overlapped with that of 166~A" Target
2+
4+
6+
8+
I0 +
164 166 168
1.107(3) 1 1.03(3)
1.14(3) 1
1.15(3) 1 1.02(5)
1.16(4) 1 1.06(7)
0.97(11) 1 1.10(18)
280
E Brandoliniet al./Nuclear PhysicsA 600 (1996)272-282
predictions of most of these calculations are in agreement, at least qualitatively, while at low spins there are strong differences. Recently g-band g-factor calculations have been performed in the angular momentum projected shell model. In this model [23], the hamiltonian (a Q • Q plus monopole and quadrupole pairing force) is diagonalized in a base including selected 0, 2 and 4 quasi-particle configurations. The rotational symmetry is restored by the angular momentum projection technique. The absolute g-factor values calculated by this model for 164'166'168Ernuclei [ 10,24] (curves (d) in Fig. 3) show only a small reduction for the levels before the crossing, in agreement with the experimental data. 4.2. g-factors of the 2~ states
The ratio g(2+)/g(6~) determined in the present work resulted to be 1.29(10), 1.28(11) and 1.19(11) for t64A66A68Er respectively. As the g-factor values are essentially constant for the low g-band levels and therefore nearly equal to that of the 6 + state, from the above ratios a significant variation of the g-factor of the 2+-vibrational states relative to the g-band g-factor is clearly evidenced. Previously the g(2+)/g(6+) values of 1.05(19) and 1.10(14) for 166Er and 16SEr, respectively, have been reported [13,14], which agree within errors with our present data but also with a ratio of 1. Preliminary data of the present work concerning the g(2+)/g(6+) ratios in Er isotopes were previously communicated, together with similar values found in the 162'164Dy isotopes [25]. The presently found disparity between the measured g(2 + )/g(6 +) ratios and unity can be related to the microscopic composition of the y-band in the frame of the rotational model. It is well known that the y-band excitation energy in the rare earths shows an irregular behavior as a function of Z, which is correlated with the microscopic composition of the state [26]. Substituting our ratios in the formula g = gR + (gr -- g R ) K 2 / I ( I + 1),
where for gR the experimental g-factor of the 2 + state in the g-band measured by the M6ssbauer technique is taken, one derives gr - gR = 0.12(3). This value is consistent with the known electromagnetic properties of the y-band in Er nuclei. In fact, taking the known E2/M1 intensity ratios for the y-band in 166Er [27] and Q0 = 7.5 b, from the formula for A1 = 1 transitions 0.93. E . Q0 [gK - g R I = [ 6 1 x / ( l - 1 ) ( I + 1) ' where the multipole mixing parameter 8 is in module about 1.6 and the energy is given in MeV, one gets Igr-gR[ = 0.11 (2). From similar data in 168Er [ 28 ] a value of 0.13 ( 1) is obtained. Triaxiality is neglected in these last considerations. The gr value can in principle be evaluated on the basis of p-h configurations contributing to the y-band, but detailed microscopic calculations are not available and standard RPA calculations [26] can not intrinsically predict a relevant difference between gr and gR [6].
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GCM predicts, in the y-band, values of 8 of the correct size and with a positive sign, in agreement with the previous analysis, while the ratio g(2~)/g(2~) of 1.06 is rather smaller than our experimental value [28]. One has to mention that the model assumes a different deformation for neutrons and proton fluids, caused by the pairing interaction. In the frame of the raM-2, precise experimental values of the g(2+)/g(2~) ratios are considered of particular importance, in order to establish the contribution of Fspin admixtures [15]. However, a value smaller than unity is predicted in t66Er in disagreement with the present experiment. The critical role in M1 properties is played by the vector parameter Xv, which is proportional to fl~r - fir. Here fl,r and fl~ are the deformation parameters of the proton and neutron boson fluid respectively [ 15], which are roughly proportional to those defined in GCM [29]. The ratio g(2+)/g(2 +) has been found to be specifically sensitive to this parameter [30]. In order to get IBM-2 consistent with GCM, and consequently with our data, Xv must be taken positive and not about - 0 . 7 as in Ref. [ 15]. This important difference between the models was already observed some years ago [31 ]. It must be noted that microscopically a positive value for Xv is favoured [32], but a negative one has so far been assumed in order to reproduce the negative sign of 8 for most y ~ g transitions. Recently M1 experimental matrix elements in transitional even Os isotopes have been described with IBM-2 [29]. In that work the ratio g(2-~)/g(2~) is larger than unity in 188Os, but this fact is accompanied by a positive value of 8 for the 2~- ~ 2~- transition. This is not true in Er isotopes. In conclusion one has to stress once more that not only the size, but also the sign of the M1 matrix elements are decisive to discriminate among nuclear models. Therefore precise measurements of g-factor in the y-band as well of 8 in the y ~ y and y ~ g band transitions appear to be very important.
Note added
After completing the analysis of this work [25], we were informed that from the mixing ratio, measured in the angular correlation of the 5 + ~ 4 + transition in the 166Er y-band, a value of ( g r - g R ) = 0.090(13) has been deduced [33], which agrees in magnitude and sign with the value obtained from our g-factor measurements.
Acknowledgements Thanks are due to A. Buscemi, S. Martini and R. Zanon for the help on the experiment preparation, to Dr R. Pengo and G. Manente for contribution to the target preparation. One of us (R.V.R.) would also like to thanks the Istituto Nazionale di Fisica Nucleare (Italy) and the Funda~o de Amparo ~t Pesquisa de S,~o Paulo (Brazil) for the grants received during his stay in Italy.
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