Collective excitations in 106Cd

Collective excitations in 106Cd

NUCLEAR PHYSICS A Nuclear Physics A571 (1994) 393-412 North-Holland Collective excitations in lo6Cd Dan Jerrestam *, B. Cederwall b, B. Fogelbergc, ...

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Nuclear Physics A571 (1994) 393-412 North-Holland

Collective excitations in lo6Cd Dan Jerrestam *, B. Cederwall b, B. Fogelbergc, A. Gizon d, J. Gizon d, L. Hildingsson b, E. Ideguchie, W. Klamra b, J. Kownacki f, F. LidCn b31,Tb. Lindbladb, S. Mitarai e, J. NybergB a Universityof Uppsala. NFL, Studsvik, S-61 I 82 Nykiping, Sweden and The Niels Bohr Institute, Rise, DK-4000 Roskilde, Denmark b Marine Siegbahn Institute of Physics, Frescativ. 24, S-10405 Stockholm, Sweden c University of Uppsala, NFL, Studsvik, S-61 I 82 Nykiiping, Sweden d Institut des Sciences Nucleaires. IN2P3-CNRS/Universite Joseph Fourier, 53 avenue des Martyrs, F-38026 Grenoble Cedex, France c Department of Physics, Faculty of Science, Kyushu University 33, Fukuoka 812, Japan f Institute of Experimental Physics, University of Warsaw, Warsaw, Poland B The Niels Bohr Institute, Rise, DK-4000 Roskilde, Denmark

Received 19 August 1993

Abstract High spin states in icsCd were populated by the reactions g4Zr(170, 5n) and 7aGe(34S, 4n) at 80 MeV and at 148 MeV, respectively. The y-decay was studied by y-spectroscopic methods using the Nordball multi-detector array. Protons and a-particles were detected in particle detector system, thus selecting the neutron channel. The experiment included y-ray yields, yy-coincidences and y-ray angular correlation measurements. Collective bands extending up _3 ns for to spin 26+, 20- and 21-, have been observed in loaCd. A new lifetime of 1 1+6 the 16+ state at 7 118.7 keV has been found. Both total Routhian surfaces and spin diabatic surfaces have been calculated and used for assigning quasiparticle configurations to the bands. The (+, 0) band is assigned as a vh:,,2 configuration below I = 16+ and at higher spins suggested to be built on a ng&2vh:,,2


The large hindrance


served for the decay from the 16+ state supports the latter assignment. With the alignment of the vh21,,2 pair the deformation is predicted to change from (c2, y) = (0.13, -2O ) to (0.17,4’ ). The configuration

of the negative parity bands is assigned as either a vht ,,2d$2

or a vhi 1,2 (g7/[email protected]/2) *, with (62,~) = (O.l4,9O).

NUCLEAR REACTIONS g4Zr(170, 5n) 80 MeV and 76Ge(34S, 4n) at 148 MeV. Measured EY, I,, yy-coinc., Calculated Compton

lcsCd deduced levels, J” experimental

Total Energy Surfaces, fixed configurations, suppressed





effects. Array of

Charged particle detector systems.

I Present address: Schuster Laboratory, University of Manchester,


Kingdom. 03759474/94/$07.00 @ 1994 - Elsevier Science B.V. All rights reserved SSDI0375-9474(93)E0647-Q

M 13 9PL, United

1. Introduction The nuclei in the mass A N 106 region have small /?z deformation at low spins and are very soft with respect to this deformation and to the triaxiality y. A richness of phenomena may occur in these A N 106 nuclei like single particle excitations, band termination, vibrational excitations, collective oblate and prolate excitations and superdeformed bands. Cadmium nuclei reveal interesting ~fo~ation about the com~tition of vibmtion~ and rotational structures and “‘Cd is in the transitional region between these extremes. The cadmium nuclei have been studied extensively during the seventies: lo4Cd [ 1,2], “‘Cd [1,3], “‘Cd [3,4], ro7Cd [3,5], “*Cd [6] and ‘09Cd [3]. An overview over the is found in ref. [ 71. During the collective states and the associated orbitals in ‘e41r06,108Cd last years some additional papers were published on rotational bands in “‘Cd [ 8 ], “‘Cd [ 9,101 and on the vibrational nuclei ’ i°Cd [ 111. P~ladium isotones show a similar behaviour as the cadmium nuclei and systematics in this region is presented in ref. 121. Superdeformed bands have been found in 104,‘05Pd [ 12 ] and ‘03Pd [ 13 ] and are predicted to occur in the cadmiums as well. Band termination has recently been suggested in iWRu [ 141 as expected for this mass re8ion [ 13,151. The orbitals closest to the Fermi surface for lo6Cd are the ds/2 (high 52), the g7/2 (low a) and the hii, (low 52 ) neutron orbitals. The proton orbitals below the Z = 50 shell gap belong to the g9/z parentage (high D ) for cadmium nuclei. Shape changes are likely to occur in “‘Cd when low Q orbitals of Yhti,z parentage become occupied and polarize the nuclei to larger defo~ation and to greater y. Another possible phenomenon is the shape change in the direction of smaller 62 due to ngglz alignment. At high spins, Z N 3Ofi, the available spin from the quasiparticles occupying orbitals in the last filled shells is expected to become exhausted and terminating states will occur. This will bring the nucleus towards spherical shape. The measurement of “‘Cd by Samuelson et al. 141 was with a high quality and almost complete for all the low spin states found so far. Yet, the high spins states could not be investigated since the angular momentum input was not sufficient and we have now extended the information on high spin states in lo6Cd using heavier beams and higher energies and a multidetector


2. Experimental methods 2.1. THE EXPERIMENTS Excited states of ‘06Cd were populated using the 94Zr f “0, 5n) ‘@Cd at 80 MeV and the 76Ge(34S, 4n) at 148 MeV reactions. The targets consisted of highly enriched selfsupporting 0.7 mg/cm2 and 1.O mg/cm’ foils, respectively. Beams were produced by the tandem accelerator and the booster at the Niels Bohr Institute. The y-rays were detected by 15 Compton suppressed Ge detectors mounted in the NORDBALL frame [ 16 1 in the oxygen run. For the run utilizing the sulphur beam 19 Compton suppressed Ge detectors and a Low Energy Photon detector (LEP) were used in conjunction with a BaFz-ball.

D. Jerrestam et al. / to6Cd



and alphas were detected in a fast-slow plastic scintillator



system called Hystrix [ 17 ] and in the silicon based detector system SiBall [ 18 ] for the “0 and the 34S beams, respectively. A particle identification signal from the charged particle systems made it possible to distinguish between reaction channels involving proton and alpha particle emission and combinations thereof. The (v/c)-ratio in our setup was estimated to be around 2.0% and 2.5% for the majority of y-rays belonging to lo6Cd detected in the two experiments. Due to Doppler broadening the average experimental resolution of the y-ray detectors was better in the oxygen run and yield AEbt,,,, sz 5 keV at 1 MeV. The detection range of y-rays was set up to 0.01-4.0 MeV when data from the two experiments were combined. 2.2. DATA COLLECTION AND ANALYSIS

The events stored on tape by the acquisition system [ 191 consisted of events with two or higher folds of Compton suppressed y-rays detected in the Ge detectors or by one or more y-rays detected in the Ge detectors in coincidence with a y-ray detected in the LEP. It was required that at least two Ge detectors should have triggered within a time window of approximately 130 ns. Pile up events in the Ge detectors were rejected. If also charged particles were detected in the particle detector system, the resulting particle identification signal was stored. Time information existed only for the sulphur beam and was obtained from timing pulses from the BaFz-ball and from the individual Ge’s extending up to 500 ns. The energy signals from the Ge detectors were gain matched and stored pairwise into symmetrized matrices. The energy from the LEP was taken as the x-value when storing LEP-Ge coincidences into a matrix. Subtraction procedures were used for the different reaction channels connected with the emission of charged particles and the xn-channel, following the methods described in refs. [ 8,171. This was done to subtract the smaller contribution of the charged particle reaction channels from the matrix associated with no particles detected and was needed since the limited solid angles and efftciencies of the particle detector systems lead to events with undetected charged particles. Examples of gated spectra from such a matrix are shown in Fig. 1. After gainmatching and applying the subtraction procedure mentioned above, the matrix associated with the xn-channel from the experiment using the oxygen beam contained 47.9x lo6 events and in the same reaction the isotopes lo6Cd and “‘Cd were produced with approximately the same cross-section. For the sulphur beam reaction we acquired around 600 million events in the xn-channel and for this reaction lo6Cd was produced with a cross-section twice that of any other nuclei. 2.3. LIFETIMES

A search for lifetimes in the nanosecond region was done using first a sort of Ge-energies versus the individual times of the Ge’s and secondly by double gating on the energies of the pair of Ge’s. The latter was done to ensure that the lifetime was at the level the

s de~yed into and out of, since gates on most si e y-rays wem gates on doublets s. The times for the i~~~d~ Ge’s were fmm the time di~eren~es between aft r from the fust of the 60 ~a~~~b~ e~eme~ts triage . From the ~~fet~rn~ a~~~s~s we did estab~i~ a new ha~~~fecrf 1I $ ns for the X6+state at 7 118.7 keV.

D. Jerrestam et al. / Io6Cd



The geometry of the NORDBALL provided us with angular correlation information. In the NORDBALL the Ge detectors are positioned in four rings around the beam-axis with the angles 8 = 37”, 79”, 101” and 143“ with respect to the beam. Pairs of y-rays that contained one y-ray detected in the 37” or the 143” rings and with the other y-ray detected in the 79“ or 10 lo rings were sorted into a matrix. The x-axis was chosen as the energy of the y-rays from the 37” ( 143” ) rings. The yields of a y-ray were extracted using gates set upon the adjacent y-ray in the level scheme on both axes. The ratio of these intensities, the DC0 ratio, was compared with the calculated ratio for a stretched E2 or a 1AZ I< 1 transition. In the following sections, assignment of a y-ray as a stretched E2 transition is done if the DC0 ratio is close to 1.0 and if there is no strong evidence of a dipole assignment. A more detailed discussion of the multipolarity assignments and the ambiguities for transitions is found in ref. [ 8 1.

3. Bands in l”Cd The proposed extended level scheme of “‘Cd is shown in Fig. 2 and in Fig. 3 with the y-rays grouped into bands labelled 1 to 3. Most of the observed y-rays are listed in Table 1. The intensities and energies listed are obtained with fits on gated spectra of the 632.7 keV transition, the 807.3 keV transition and on the 1028.5 keV transition. The very weak 58 1.3 keV y-ray is not included in the level scheme even though it was observed in some gates. The level scheme is extended above the 15- state in band 1 and the 14- state in band 2. For band 3 the level scheme is to a great extent new above the 3044.2 keV level, however some y-rays are known from other experiments. Some of the transitions in band 3 were observed previously [20] and grouped into a band labelled B in ref. [ 201. Yet, we believe that most of these transitions are incorrectly placed. We instead suggest that some of these y-rays are part of the yrast (IL = + , a = 0) band, labelled 3 in Fig. 3 with the 1O+ bandhead at 48 16.3 keV level. Starting from spin 13- in band C (our band 1) we observe the continuation of the (-, 1) band up to spin 21-. Most of the transitions in band 1 and 2 are known from ref. [4]. The ordering of the levels and the DCO-ratios are obtained from the two separate runs described above. The errors on the listed y-ray energies are less than 0.2 keV if the error is not explicitly

given. Fits were done on the gated spectra of all the y-rays in the E2-

cascades to obtain the intensities of the coincident y-rays in the gates, thus establishing the order of the y-rays. Angular correlation information on the bands labelled 1 and 2 in Fig. 2 establishes the negative parity bands up to the 19- and to the 18- levels. The single remaining transitions in the two bands, the 1467 keV and the 1446 keV y-rays, form a natural continuation of the bands and are supported by the coincidence relations. The DCO-ratios suggest that the y-rays 807.3, 892.3, 602.8, 980.7, 1150.8 and 1311.1 form a band from lO+ to 22+ feeding through a complex decay into the yrast band at


D. Jerrestam et al. / lwCd


1467 lo+


1367.2 7517.4




6264.6 1060.6





Fig. 2. The extended level scheme of ‘%Zd deduced from our experiment for the negative parity bands.

the 8: and to the 6: and 6: states. Two additional transitions, 1487 and 1633, may be up to the 26+ state. The 1028.5 keV transitions feeding out from the 48 16.3 keV level must be an E2 transition considering the decay out to the 6+-states and the E2 multipolarity is supported by the measured DCO-ratio.

the continuation

D. Jerrestam et al. / “%d


3 __ _



(243__ ,i



t 1311.1

t 1150.8



-!802.8 i882.3 + 807.3

t 1026.5 3787.8 433.0







Fig. 3. The extended level scheme of *06Cd for the positive parity band.


D. Jerrestam et al. / lwCd



Listed are some ofthe y-rays observed in the 2+-O+ gate, 632.7 keV, in ‘%d. The given intensities are extracted from the same gate and normalized to the 4+-2+ transition y-ray energy








171.1 187.7 226.1 241(l) 269.3 423.4 433.0(2) 524.7 541.0 552.5 581.3 592.8 598.5 602.8(2) 604(l) 610.8 622.6(3) 634.9 645.5 691.0 695( 1) 703.5(3) 753.8 807.3(2) 861.2 861.2 862.7 (8) 889.7 892.3(3) 980.7(2) 997.9(2) 1009.2(2) 1008.8 1028.5(2) 1050.6(3) 1077(l) 1138(l) 1142.5(5) 1149(2) 1150.8(2) 1252.6(5) 1284.9(10) 1288.9(4) 1295.6(10)

11.7(11) 13.9(9) 6.0(8) 0.4b 11.1(8) 5.2(6) 2.6(5) 8.4(4) 10.4(7) 17.3(9) Q 10.0(7) 13.0(8) Q 17.7(13) 8.5(10) 6.1(9) 20.9(12) 39.9(15) 8.5(9) 4.0(8) <5.2(l) 3.7(7) 21.5(10) 9.4(15) 87.6(15) 3.0(12) 25.7(25) 12.5 (20) 8.6(7) 38.1(14) 27.1(18) 7.0(18) 6.8(9) 15.2(9) 5.4(8)


0.4i-J 1Oa 1.7b 1.38

2.1* 1.6b 0.7b

1.4b Norma 9.2a

3.9a 1.28

5.4(8) 69.5(8) 7.0(20) 3.3(8) 1.0(4) 1.5(9)

1.68 5.58 1.3b

3679.0 3507.9 2330.7 4816.3 3679.0 3507.9 3787.8 2629.3 3044.2 3044.2 3084.5 3084.5 4106.4 7118.7 3787.8 2104.6 4816.3 3679.0 4324.5 3320.3 4816.3 3787.8 3084.5 5623.6 4967.6 1493.9 3354.7 5214.2 6515.9 8099.4 2491.8 2503.1 5976.4 4816.3 6264.8 4121 4816.3 7119 4193.6 9250 7517.4 3787.8 8408 3787.8





1.08(5) 1.13(6)

U=Vl 3507.9 3320.3 2104.6 4575.3 3409.8 3084.5 3354.7 2104.6 2503.1 2491.8 2503.1 2491.8 3507.9 6515.9 3184 1493.9 4193.6 3044.2 3679.0 2629.3 4121 3084.5 2330.7 4816.3 4106.4 632.7 2491.8 4324.5 5623.6 7118.7 1493.9 1493.9 4967.6 3787.8 5214.2 3044.2 3679.0 5976.4 3044.2 8099.4 6264.8 2503.1 7119 2491.8

E2 Ml E2 El Ml El E2 E2 Ml Ml E2 E2

1.00(7) 0.42(6) 0.52(9)

Ml Ml El E2 Ml Ml Ml E2 E2 E2 E2

1.00(5) 0.69(33)

E2 E2 E2 E2 E2 E2 E2 E2

1.02(12) 1.05(9) 0.92(12) 1.05(5) 1.08(8) 1.20(7) 1.12(12) 0.99(10) 2.06(30)



E2 E2 E2 E2

l.OO(ll) 0.99(9) 1.11(12) 1.10(14)

0.97(7) 0.91(10) 0.58(5) 1.00(25) 1.07(10)

l.Ol(4) 0.43( 10)

1.07(12) 1.05(4)


D. Jerrestam et al. / ‘06Cd TABLE

1 -continued

y-ray energy








1311.1(7) 1367.2(S) 1426.3 1446(l) 1467(l) 1471.8 1487(l) 1531(l) 1633(l)

2.9(S) 3.4(6) 3.9(5) 0.6(3) 2.8(10) 5.4(6) 2.8(10)


10561 8884.6 2920.2 9854 10352 2104.6 12048 4575 13681

1.6= 0.4= 1.1a




9250 7517.4 1493.9 8408 8884.6 632.7 10561 3044.2 12048

E2 E2 El

0.98( 14) 1.10(22) 0.92(10)

(E2) (E2) E2



(E2) (E2)

B Intensities from a gate on the 807 keV y-ray. b Intensities from a gate on the 1028 keV y-ray assuming that the 3787.8 keV level was only fed by the same y-ray.



The aligned angular momenta and the Routhians for bands 1 and 2 for “‘Cd are shown in Fig. 4 and in Fig. 5 as a function of the rotational frequency, fiw. The Harris parameters used for reference are chosen as JO = 8.9h2 MeV-’ and JI = 15.7h3 MeV4 and are used for all nuclei discussed in this paper. We have also arbitrarily selected the 8+ at 3787.8 and the 6: at 2491.8 keV as part of the (+, 0) band to visualize the evolution of the band, though the band is feeding out to many states from the lO+ at 48 16.3 keV. Comparisons are made with the neighbouring palladium nucleus io4Pd, with the same number of neutrons, but having two protons less, and with *‘*Cd having two more neutrons. Both ‘@‘Pd and “*Cd have more particles outside the closed shells than “‘Cd and are thus expected to be more collective. Also the structure in terms of the Nilsson orbitals favours a higher degree of collectivity in these neighbouring nuclei. The fewer protons in l”Pd decrease the occupation of high Q proton orbitals of gq/2 parentage. A similar deformation driving effect occurs when going from to6Cd to “*Cd, since the low Q orbitals of ht 112parentage then are brought closer to the Fermi surface. Inspection of the plots of the aligned amount of angular momentum and Routhians of the ( + , 0) bands of these nuclei in Fig. 6 also shows a more regular behaviour for the neigbouring nuclei than for “%d. A comparison of this type is nevertheless of substantial value because it gives some hints regarding e.g. the nature of the aligned quasiparticle states. 4.1. THE POSITIVE



Several positive parity states with J” up to 12+ are known from previous works [ 4,7]. An irregular band, based on the ground state and extending to 12+ was proposed by Samuelson et al. [4] This band is however not the yrast one above the 6+ level, other

402 (+,O)


in Cd and Pd


v (+,O) A (+,a) q (+,O)

"Wd ‘W?d fo4Pd


0. I





Fig. 4. The experimental aligned spin, i, as a function of the rotational frequency hw in loaCd.

levels with .J” = 8+,IO+and 12+ are present at lower, or substanti~ly lower energies, than the proposed band members. A detailed discussion of the lower lying levels of ‘OaCd is given in ref. [4], where a calculation within a two-quasiparticle + slightly deformed




-5 0.0


0.4 hw[Me?$



Fig. 5. The experimental Routhians e’ as a function of the rotational frequency frw in roaCd.


D. Jerrestam et al. / lo6Cd




in fo6Cd

m (0) v C-91) A C-,0)



Fig. 6. The experimental aligned spin, i, in the ( +, 0) band as a function of the rotational frequency fro in lo4Pd, lo6Cd and lo8Cd.

rotor model is also presented. The model predicts three 8+ and three 1O+ states from the simplest combinations of the neutron g7/2, ds,r, h, 1/2 and the proton g9/2 orbitals, at low energies. The quasiparticle structure of several experimentally observed levels have been deduced in ref. [ 41 and other previous works, see also ref. [ 71, and references quoted by these authors. The fact that we now observe the ( + , 0) band up to relatively high angular momenta permits additional conclusions regarding the structure of some levels. The identity of the u (htt,r)’ lO+ state is of special significance. This aligned state is of importance for the evolution of the ground bands in the region. The gain of about 10 units of aligned spin in l”Pd and “*Cd, see Fig. 6, is taken [6] to be caused by the alignment of a htl,r neutron pair at the band crossing. The gain of aligned spin in ‘06Cd is even larger, about 15 units, by that requiring an additional source of alignment. Samuelson et al. [4] suggested that the 4436.1 and 5241.1 keV levels were the lO+ and 12+ members of the (h11,2)2 band. These levels and the associated y-ray decay are also observed in our experiments, but we could not extend the band to higher spins. We propose that our band 3 is a better candidate from a stronger population and the strong in-band transitions. Additional support for this proposal is given by the level energy. The corresponding level in “‘Cd is found at 4153 keV. The systematic of single quasineutron states in the Sn isotopes indicates that the energy should be about 600 keV higher in “‘Cd, in fair agreement with our observation of 48 16.3 keV. Still, only the lo+, 12+ and 14+ levels of the ( +, 0) band are likely to have the


D. Jerrestam et al. / lwCd

(hi 1,~)’ two-quasineutron structure. Another band crossing occurs at the 16+ level, which is isomeric with a half life around ten ns. The depopulating in-band transition is thus 30-50 times slower than the Weisskopf estimate. This is also the point where the amount of aligned spin reaches its highest observed value in io6Cd, see Fig. 4. Evidently the 7118.7 keV 16+ level must have a four-quasiparticle structure since no pair of orbitals can couple to such a high angular momentum. A coupling of one of the almost degenerate 6+ states near 2.5 MeV with the lO+ (h1i,2)2 state yields the correct spin and a very reasonable excitation energy. The 6+ states can be described as being largely (ugr,z)’ and (vg7,2, vds,z) or superpositions thereof. However, also a coupling of the lO+ (hii,2)2 with a pair of gs,z protons yields correct spin and energy. The lowest 8+ levels in the “‘Cd and “*Cd nuclei are assigned as ( r(g$2) and two g9,2 protons coupling to 6+ would then have an energy around 2.5 MeV thus making up a total energy close to the observed 16+ state. A rotational band built on configurations involving the proton g9,2 orbital are expected to be strongly coupled with a large spacing between the individual levels, if not the band becomes semidecoupled from the influence of the hII, neutrons. Clearly the ( + , 0) band above 16+ is not strongly coupled. Using these likely assignments for the 16+ level, only the latter may explain the retardation by the great difference in K values between a (ng9,2)2, (Vht1,2)2 configuration with K = 6 and a (vhlt,z)2 configuration with K = 0. There is also another isomeric (ag9,2)2,(vd5,2)2 12+ state at 4659.8 keV in “‘Cd 7which in analog to the 16+ state may be hindered from K-forbiddenness. Another argument for an yrast configuration at 16+ involving two rtg9,2 orbitals is the low energy associated with such proton configurations, that is the 8+ at 3044.2 and the 12+ at 4659.8 are yrast. It is thus likely that our observed band 3 is the yrast one from spin 16 and upwards and is built upon the (Icgg,2)2, (vh1i,2)2 configuration. The greater collectivity of the ‘@‘Pdand “‘Cd nuclei is supported by the lower crossing frequencies in the (0, + ) bands. These have crossing frequencies between the vacuum band and the hf,,, band at fro x 0.39 MeV for IwPd and hw z 0.42 MeV “sCd 0.40 MeV from ref. [ 91) which is to be compared with Tto zz 0.47 MeV for ‘%d.



The bandheads of the negative parity bands are at 6- and at 9-. The 7- at 3409.8 keV is not a member of band 1, since the B (E2), from lifetime measurements [ 71, for the 269.3 keV y-ray, 9- +7-, is small: 5.3:::; Weisskopf units. The 187.7 keV transition, 8- +6-, is in contrast to the 269.3 keV transition a member of the negative parity bands with a much larger B (E2)-value: 37 f 13 W.U. [ 71. From the lifetime measurements it is concluded that these bands are built on combinations of an almost pure uhli,2 quasiparticle with either a up?,2 or a vds,z quasiparticle. This would result in, with y O”, that the favoured signatures of the orbitals of the v (h I 1j2) ’ ( d5,2)’ configuration is responsible of the (-, 0) band and similar is the v (hl 1,2) ’ (g7,2) ’ configuration lowest in energy for the (-, 1) band. Another possibility is that these bands are built on one quasiparticle occupying the


D. Jerrestam et al. / lo6Cd

uh,rlz orbital and one mixed u ([email protected]/2) quasiparticle. We would then expect that these band should have some interband transitions, which are not observed. The admixture of ug7/2 with udslz should become larger with increasing rotational frequency, which make both of the above described



An assignment

with a more

pure u (hr i/2 )’ (g7,2 ) ’ configuration may also may be a good description of these negative parity bands at y -O”, where the (-, 1) would be the favoured signature, since the (--, 1) band is the yrare band of the two.

5. Calculations Two different calculations on ‘06Cd are presented below, one with a Woods-Saxon potential and the other using universal parameters for the Nilsson orbitals. The previous is expected to have higher precision for the results but can presently only use the cranking frequency, )[email protected], as one of the evolution parameters instead of the observable quantity spin. The other model manages to transform from ho to spin and allows a more detailed comparison.

5.1. WOODS-SAXON Total Routhian Surfaces (TRSs) calculations have been performed for “%d using a deformed Woods-Saxon potential and the Strutinsky shell correction formalism with a monopole pairing interaction as described in refs. [21,22]. In the calculations the total Routhian of the nucleus is minimized with respect to the deformation parameters /32, /& and y at different rotational frequencies and for different configurations. Each TRS has well defined parity and signature but no other quantum numbers are conserved. Accordingly the minima in the TRSs define the conditions of stability for the occupied quasiparticle configurations. Following the description in ref. [23] the minima in the lowest TRSs for “‘Cd could be assigned to the relevant quasiparticle configurations. Fig. 7 shows the calculated lowest TRSs for the ground state, at hw = 0.57 MeV and at fiw = 0.70 MeV corresponding to J+ = 0, 18 and 24 h in the minima. In the ground state the nuclear potential is extremely soft, particularly with respect to the prolate collective and prolate non-collective degrees of freedom. No pronounced nonspherical minimum is present which is typical for vibrational or quasivibrational nuclei. When cranking the nucleus, deformed minima occur based on the occupation of different aligned two-quasiparticle configurations. The vibrational-like ground band will be crossed by the uh211,2 quasiparticle configuration at approximately Aw = 0.30 MeV. This is the configuration occupied in the deformed minimum in the TRSs in Fig. 7 at ho = 0.57 MeV with (/&,y) = (0.16,6”). The equilibrium deformation of the uhfl,, configuration is induced by the shape driving properties of the low-Q hf,,, neutrons that favour a large &deformation and positive y. The aligned uh:,,2 configuration is thus a natural candidate for the rotational band structure built on the lO+ state. We suggest


Fig. 7. Total Routhian

D. Jerrestam et al. / loaCd

Surfaces for lo6Cd calculated for the ground state, at fro= at &IJ = 0.70 MeV.

0.57 MeV and

that we observe the head of a band that quickly evolves into a four-quasiparticle coniiguration, see the plot of alignment versus ho. The initial alignment of the bandhead of the Vh:i,2 configuration is N 1Oh and the alignment then increases rapidly with frequency to N 15h in the band. In Fig. 8 the quasiparticle energies for neutrons and protons are plotted as a function of ho for a fixed deformation with (0.16,6” ), which corresponds to the deformation of the aligned Vh:,,2 configuration discussed above. The possible candidates at ho N 0.6 MeV for the additional alignment are: alignment of other hii,z or “g,@d5,z” neutron orbitals or a pair of ngg/z orbitals. However, the TRS calculations predict that the deformation parameters drift towards smaller p2 and larger y and that the spin contribution from protons will only increase a little, so most of the additional aligned angular momentum comes from neutrons. That is to say, the TRS calculations suggest that the neutrons are the source of the additional -5h alignment just after the initial Vhf,,2 alignment, and then at Z N 25h the protons may contribute. Furthermore, the TRSs suggest that a sizeable decrease of the deformation occurs after the alignment of the uh:,,2 as shown in Fig. 7. The observed negative-parity rotational structure can, in the TRS calculations, be explained as built on the deformed bandheads that will occur when coupling the favoured signature of the lowest hi 112neutron orbital to either the 5/2+ [ 4 13 ] (ug7/2 ) or 3/2+ 14 11 I (Vdj,z) orbitals. These configurations will be strongly mixed and the predicted deformations are close to (p2, y ) = (0.14,8’). For the above described two-quasineutron configurations is also the shape predicted to evolve towards smaller /3z with increasing

il. Jerrestam


g s

et ai / I&Cd


Bp0.160fl,=0.OQO~6.02=48 1MCd (%,a) : solid=C+,+112), [email protected]~trlL% d&-dotted’+,+llt),



& -& -1.0 s 0


















Fig. 8. Calculated quasiparticle

diagrams for lo6Cd as a function of Rw with the fixed deformation (Pz,v) = (o.16,6°).


D. Jerrestum et al. / lo6Cd

rotational frequency in a similar way as for the ( +, 0) band. However, it is important to note that the potential parameters used in the TRS calculations are, although “universal”, intended for nuclei with deformed ground states and the predicted energies of different configurations relative to the ground state are less certain in this transitional region.



Spin diabatic surfaces (SDS) have been calculated for “‘Cd and “*Cd to get deformation parameters as a function of spin using the modified oscillator with pairing included [ 24,251. Standard parameters for K and p has been selected from ref. [ 151. Also SDS have been calculated using fixed configurations and where the minimized energies with respect to (~2, y ) for a given spin have been selected. These minimized bands are shown in Fig. 9 and Fig. 10 for “‘Cd and “*Cd, respectively, together with the experimental data (filled squares, circles and triangles). Some of these surfaces showed a different dependence on the deformation parameters as a function of spin to that of the TRS calculation. This was the case for both the negative parity bands and the (+, 0) band after the Vh:,,, alignment where the TRS calculations predicted a decrease of the deformation in contrast to the SDS calculation which did not suggest any decrease below spin 30h. The calculated deformations from the SDS calculation were mostly around 20% greater


in fo6Cd

Experimental . (+,0)

: g;

data g

0 (0.13,-2) q

0 + A v

(O.lO,-63 (0.17,4) (0.16,7) (0.14,9) (0.14.9)

+, 0) vacuum I (+,O TT(+)~ (+, 0) ” ( -p (+,O TT+)&I(-)2 v(+ /J(v(+)(-)











Fig. 9. Calculated and experimental energies (shown with tilled squares, circles and triangles) of bands in loaCd as a function of spin. The indicated deformations for the calculated bands are an average, since the deformations were allowed to vary between the spins. A reference energy has been subtracted using the function 0.0167 x I (I + 1) [MeV]. See text for further details.

LX Jerrestam et al. / l”Cd



Experimental *


: pyj



0 (0.14,-7) +,O)Vacuum D (O.iO,-60I {+,0)lT(+)2

0 (0. I 7,12) + (0.18,18) A (0.19.2) 0 (0.19,3)

(+.a) v(-p (+,O n(+)G(-)2 v(+) t -) v(+)(-)


Fig. 10. Calculated and experimental energies (shown with filled squares, circles and triangles) of bands in lo8Cd as a function of spin. See previous figure caption for explanation.

than those of the TRS calculation.

Observe that in the Figure caption for Fig. 9 and Fig. 10 the average (~2, y) is given, not /3 and y. This larger deformation was not anticipated and does not correspond to experimental data on “‘Cd, where the bands indicated with squares and triangles have a measured fi equal to 0.16 and 0.09-o. 14, respectively [ lo]. A more detailed inspection of Fig. 10 for “‘Cd reveal great similarities between the experimental data and the calculated bands. The calculated bands are the lowest one for the experimental range of the (+, 0) band and up to spin 21I? for the negative parity bands. Above these spins there may be other ~on~gurations that come lower down in energy than those displayed. The (+, 0) band is well described by a band built on an aligned pair of v (h,,,2)’ quasiparticles feeding into the ground state band and the calculation does almost reproduce the bandcrossing at I = 8+, where the calculated v (h,r,2)’ band seems to be N 0.2-0.4 MeV below the excitation energy for reproducing the experimental energies at the crossing. The yrast band at 8+, unfilled squares, are built on n (g9,2)2 excitations and corresponds to an observed 8+ level. The calculated negative parity bands are dramatically varying at the lowest spins with respect to (~2, y) and at 8- and 9- the associated lower deformations may be unphysical. We have not tried to calculate the energy dependence of the negative parity bands after the suggested alignment of a pair of hl i/2-neutrons, but to our surprise the alignment could qualitatively be reproduced within the same configuration: v (hr 1,~)’ (“g7,[email protected],2”)1. We do not have any explanation for this. Turning to ‘%d and Fig. 9 for the experimental (filled squares) and calculated (un-


filled circles, diamonds

D. Jerrestam et al. / lwCd and crosses)

( +, 0) band it is obvious

that the bandcrossing

at spin 1O+ is not reproduced. This may be due to that the calculated v (hi 1,~)’ band is around 1 MeV too low, that is the position of the vhit,z orbital is wrong using the K and ,U from ref. [ 151. One can speculate that the calculated gain for occupying one Yhit,s-orbital, with a low Q, in ‘06Cd lowers a rotational band with an energy excess of x 0.5 MeV. This excess and a similar but counteractive effect for the occupation of a pair of aligned wg9/2 orbitals would then make the description of high spin bands in “‘Cd more understandable, since then the ( +, 0) band could be described by first aligning a pair of vhrt,Z quasiparticles and then a pair of ag9lz quasiparticles above spin 14fi. The calculated negative parity bands will then shift 0.5 MeV up in energy and will be yrast in the same spin region as for the experimental bands. The effect on the calculated oblate collective 72(g9,2)2 band indicated by untilled squares will probably be different from the prolate collective bands and may not necessarily be lowered by 1 MeV. This speculation on the excess lowering of bands occupying the I/htt,z quasiparticles also may be extended to “*Cd. The effect in “*Cd is expected to be smaller since the vhti,z orbital is much closer to the Fermi surface and is also supported by the better agreement between calculated and experimental data on the bandcrossing in the ( + , 0) band at 8+. Here is the difference only 0.2-0.4 MeV compared to the I MeV difference for lo6Cd. This assignment of the n(g9,2)‘@ v (hii,z)’ configuration to the ( +, 0) band in ‘06Cd is in good agreement with the systematics of energies of the bandhead at 16+ and is also justified from the greatly hindered transition from this bandhead. The TRS calculation does not reproduce the behaviour of the (+, 0) band above the lO+ state and suggest other sources for alignment, but we do prefer the result from the SDS at the high spins since these are more clear and better explain the data from our experiment.

6. Conclusions Three collective bands extending up to spin 21-, 20- and 26+, have been observed in “‘Cd. Spin assignments of the levels are based on DCO-ratios, intensities observed in y-energy gates and decay out to lower spin states. The negative parity bands can be described with a u (h, i/2) ’ ( dS,z) ’ quasiparticle configuration for the favoured signatures of the (-, 0) band and v (hi 1,~) i (gr/z ) i for the favoured signatures of the (-, 1) band. Another complementary description may be with the v (hii,2 )’ (“g7/[email protected],2”)’ quasiparticle configuration. The deformation stays rather constant in the SDS calculation with (62, y ) = (0.14, 9” ). In the TRS calculation the initial deformation is around (/3, y) = (0.14,8”) and decreases at the highest spins. The high spin states of the ( + , 0) band in lo6Cd may be described by the occupation of a pair of hit/Z neutrons at fiw -0.47 MeV followed by the alignment of an additional pair of g9/2 protons. The assignment to the latter source of alignments needs an adjustment of the calculated energy gain with 0.5 MeV with the occupation of a uhiil~ quasiparticle orbital and a counteractive effect for the pair of aligned protons for prolate collective

D. Jerrestam et al. / loaCd

bands in the SDS calculation.

The deformation


is changed from (EZ,y ) = (0.13, -2” )

for the ground state band to (0.17,4” ) for the uhi,,z band and finally to (0.18,7“ ) for the n (g9,2 ) ’ x v (hi i/z ) ’ band. Moreover, this assignment is also supported by systematics of the energies of quasiparticle orbitals in Sn nuclei, bandheads and by the large hindrance observed for the decay from the 16+ state. The University of Copenhagen is gratefully acknowledged by Dan Jerrestam for the financial support supplied. The staff at TAL/NBI is to be thanked for the assistance given. We are also grateful for comments from W. Nazarewicz and R. Wyss. Two of us (A.G., J.G.) also acknowledge the financial support obtained within the French-Swedish exchange programme of CNRS-NFR.

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