Multiphonon vibrational states in 106,108Pd

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A584 (1995) 547-572

Multiphonon vibrational states in

l°6'l°sPd

L.E. Svensson a, C. Fahlander a, L. Hasselgren ~', A. B~icklin ~', L. Westerberg a, D. Cline b, T. Czosnyka b,l, C.Y. Wu b, R.M. Diamond c, H. Kluge c,2 ~' Department of Radiation Sciences and The St,edberg Laboratory, Uppsala Unirersity, Box 535, S-751 21 Uppsala, Sweden b Nuclear Structure Research Laboratory, Unicersity of Rochester, Rochester, N Y 14627, USA c Lawrence Berkeley Laboratory, Berkeley', CA 94720, USA

Received 7 September 1994

Abstract The nuclei l°6pd and l°Spd have been Coulomb excited using beams of J~'O, 5SNi and 2°spb. The data determined 25 and 31 E2 matrix elements in m6Pd and 1°sPd, respectively. The experimental data are qualitatively in agreement with the spherical-harmonic quadrupole vibrational model, but quantitatively there are large discrepancies, even for the states generally accepted to be the two-phonon vibrational states. A sum-rule method was applied to the data, the result of which implies that the vibrational quadrupole strengths of the 0 + and 4 + two-phonon states are fragmented and shared between several E2 matrix elements. The E2 matrix elements imply appreciable quadrupole triaxiality with a y-centroid of about 20°. A systematic comparison is made with neighbouring Pd-isotopes and with l l4cd" Keywords: NUCLEAR REACTIONS l°6'l°Spd(160, 1°O), E = 48 MeV; u~6'l°~pd(SSNi,5~Ni), E = 165.5 MeV; l°6A°spd(2°sPb, 2°spb), E = 878 MeV; measured y y , py(0), ppy(0) follow-

ing Coulomb excitation, l°8pd levels; l°6'l°Spd deduced B(E2), quadrupole moments. H)2-~()pd, l l4cd systematics. Enriched targets.

I. Introduction T h e P d - n u c l e i are four p r o t o n holes away from the Z = 50 shell closure. It is c o m m o n l y accepted that they are n e a r - s p h e r i c a l in their g r o u n d states, a n d that

1Permanent address: Heavy Ion Laboratory, Warsaw University, Warsaw, Poland. 2 Permanent address: Hahn-Meitner Institute, D-14091 Berlin 39, Germany. 0375-9474/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9474(94)00514-1

548

L.E. Svensson et al. / Nuclear Physics A584 (1995) 547-572

the structure of their lowest-lying states is dominated by vibrational degrees of freedom as reflected in the typical vibrational 0 ÷, 2 ÷, and 4 ÷ triplet of states at about twice the energy of the first 2 + state. These 'two-phonon' states are connected to the 'one-phonon' 2 ÷ state with B(E2) values of about 30 to 90 Weisskopf units (W.u.). It is not possible to identify clearly a quintuplet of states, that could be related to the three-phonon vibration, since the states above the two-phonon multiplet are very much spread in energy. The aim of the present work is to investigate to what extent the 'two-phonon' and higher-lying states have a vibrational character, and to what extent it is necessary to invoke other collective degrees of freedom, such as rotation of static quadrupole deformation that may be triaxial and y-unstable. Measurements of the transitional and diagonal E2 matrix elements involving the low-lying states are needed to elucidate such questions. This paper is concerned primarily with l°6pd and l°8pd. Only five B(E2) values from their 'one-' and 'two-phonon' states, and the quadrupole moments of their first excited 2 + states, were known previously [1-3], most of them measured by Coulomb excitation [4-9]. We have performed heavy-ion Coulomb excitation on both these nuclei using 160-, 58Ni- and 2°8Pb-ion beams. As a result diagonal and transitional E2 matrix elements between the 'two-', 'three-' and 'four-phonon' states in each nucleus have been determined. The experimental data are compared with previous data on 1°2'1°4pd [10], 11°Pd [11] and ll4Cd [12].

2. Experiments 2.1. 160 experiments Beams of 48 MeV 160-ions from the EN tandem accelerator at The Svedberg laboratory in Uppsala were used to Coulomb excite l°6pd and l°8pd. The 160-ions were stopped in thick Pd-foils, enriched to more than 95% in l°6pd and l°sPd. A Ge-detector was placed 9 mm from the target and operated in coincidence with an annular 30.5 cm long NaI-detector with 25.4 cm outer diameter surrounding the target chamber and the Ge-detector [13]. Through gating either on the 2~-~ 0~ transition energy, or on the energy of the sum of the cascade of transitions from the 02~, 2 2, + 4~- triplet of states in the NaI-spectrum, it was possible to detect weak transitions in the Ge-spectrum populating these states. It is easy to verify background or contaminating lines by comparison of spectra between these two gates and between the different isotopes. The intensities of the y-rays detected in the Ge-detector were corrected for angular-correlation, summing and solid-angle effects as described in [13]. To ensure that only the electromagnetic interaction is important, the bombarding energy must be chosen such that the classical separation distance between the surfaces of the two colliding nuclei exceed 5 fm [14]. In the 160 experiment the energy was about 5 MeV higher than the safe energy for ions scattered backwards from the target surface. To allow for the uncertainty in the angular distribution and in the Coulomb-nuclear-interference effects, the error in the yields from the 160 experiment was increased by a weighted error of

L.E. St'ensson et al. /Nuclear Physics A584 (1995) 547-572

549

15%. Ge-spectra obtained with a gate on the 2~-~ 0~- transition energy in the NaI-spectrum is shown in Fig. 1 for l°6pd, and in Fig. 2 for l°sPd.

2.2. SSNi experiments In the 5SNi experiments, a 165.5 MeV 5SNi-beam from the MP tandem Van dc Graaff accelerator at the University of Rochester was used. This energy is below the safe energy at all scattering angles. The targets were self-supporting and had thicknesses of 1.01 m g / c m 2 (98.48% enrichment) and 0.92 m g / c m 2 (98.88% enrichment) for l°6pd and l°sPd, respectively. The scattered SSNi-ions were detected by four circular Si-detectors and one annular Si-detector. In the ~°~'Pd experiment one extra particle detector was used. The scattering angular ranges arc given in Table 1. All detectors were mounted at a distance of 6.4 cm from thc target except for the annular detector, which was placed 2.9 cm from the target. To prevent 6-electrons from entering the detectors, all circular detectors were covered by thin Ni- or Cu-foils. In detectors C and F the 5SNi-foil was 2.6 and 2.25 m g / c m 2 thick to prevent the recoiling nuclei from entering the detector. In the detector placed at an angle of 45 ° with respect to the incoming beam, it was possible to discriminate between recoils with energies from 49 to 88 MeV, and scattered 58Ni-ions, with energies from 100 to 126 MeV by gating on the measured particlc energy. Two Ge-detectors, placed at 0 = 1.5 ° and 60 ° with respect to the direction of the incoming beam, were used to detect deexcitation y-rays in coincidence with the events in the particle detectors. The 1.5 ° detector was placed at a distance of 4.4 cm from the target and the 60 ° detector was placed at a distance of 7.0 cm from the target. The Ge-detectors were covered by Cd- and Cu-absorbers in order to minimize the count rate from X-rays from the Pb-stopper near the target. The detectors were efficiency calibrated using radioactive sources. Doppler-corrected y-ray spectra are shown in Figs. 1 and 2.

2.3. 2°spb experiments In the 2°sPb experiments the b e a m was obtained from the S u p e r H I L A C at the Lawrence Berkeley laboratory with a b e a m energy of 878 + ~0 MeV, which is safe at -15 all scattering angles. The targets were a 1.06 m g / c m 2 self-supporting ~°6Pd-foil enriched to 98.48%, and a 0.96 m g / c m 2 self-supporting l°8pd-foil enriched to 98.88%. The scattered Pb- and recoiling Pd-nuclei were detected in kinematic coincidence by a pair of position-sensitive parallel-plate proportional counters. They were square, 120 × 120 m m 2, the center of the left detector was 27.9 cm from the target at 0 = 26 °, and the center of the right detector 23.5 cm from the target at 0 = 32.5 °. In the off-line analysis each detector was divided into two regions, thus giving four different particle gates, the limits of which are listed in Table 1. In these experiments the recoiling Pd-nuclei were detected, and the scattering angles of the 2°sPb-projectiles given in Table 1 were determined from the recoil angle. Two Ge-detectors, placed at 150 ° to the right and to the left of the incoming beam, respectively, were used to detect y-rays coincident with events in the particle

550

L.E. St,,ensson et al. / Nuclear Physics A584 (1995) 547-572

-1-~ C'I

10 `4

T

rj3

+~

+~

0

%~ I

+w

T

Z

T Cq

T

-r~ ~10 3

+~

,.~10 2

2;

,oJ 10 3 >

+

T

'

";"'

0"2

c~

+~

T

-}-e~

:

+~_ ¢.D

~1°1 oz; lO

I 5oo

600

I

700

800

900

1000

11 O0

Gamma Ray Energy (keV) Fig. 1. Ge-spectra of 1°6pd from the t60 experiment (top), 58Ni experiment (middle) and 2°spb experiment (bottom). The top spectrum is obtained with a gate on the 2~- ~ 0~- transition energy in the NaI-spectrum. The 2~- ~ 0~- transition itself is seen, since the energy gate in the NaI-spectrum is broad and includes other lines, as well as part of the Compton distribution from higher-energy transitions feeding the 2]~ state.

L.E. Svensson et al. / Nuclear Physics .4584 (1995) 547-572

'1

~ 10 4

0 r~

-

~+.n:

-

~

o~

I[ ~

551

j

+~

+~

~

+~

i

~0 .,,,.j

~S

',

lu

¢3 +m ~10

t~ ~ 3

+=

C
T

© q-~

o

10

+~

t

t

~

~.o

Vo L

4

Z m

J

77

.

.

.

.

o lO I

L

500

I

600

I

700

800

900

1000

11 O0

Gamma Ray Energy (keV) Fig. 2. Ge-spectra of msPd (see caption to Fig. 1).

detectors. The Ge-detectors were placed at distances of 83 and 81 m m from the target, respectively. Both detectors were covered by Cd- and Cu-absorbers. Doppler-corrected y-ray spectra are shown in Figs. 1 and 2. For experimental details, see [15].

L.E. Svensson et al./Nuclear Physics A584 (1995) 547-572

552

3. Analysis

3.1. Level schemes of l°°pd and mspd The low-spin level structures of l°6pd and l°8pd are well established from (n,y)-reactions [16,17], from other light-ion reactions, and from radioactive decay (see references quoted in [1-3]). The yrast states of l°6pd are known up to I ~ = 16 + from a heavy-ion reaction study [18], and the low-spin states have been Coulomb excited [4-9] and recently studied in electron-scattering experiments [19]. The states observed in the present experiment are shown in Figs. 3 and 4. All the states observed in l°6pd were known, but in l°8pd six new y-rays, with energies 576, 635, 778, 798, 908 and 1027 keV, were observed. Based on some of them we tentatively propose three new states with spins and parities (4~), (6~-) and (8~-), respectively. Below follow some comments on these new y-rays. A relatively strong, previously unobserved, 778 keV line is seen in the 2°spb experiment. We assign it as the 8 ~ 6~ transition. This assignment is supported by the observed relative yield (see Fig. 5 below), by the systematics, and by the fact that this transition has not been observed previously in radioactive decay or in n-capture experiments [2,3], nor in Coulomb excitation with low-Z projectiles. A y-ray transition with energy 694 keV was assigned as the 4~-~ 2~ transition in [8]. This transition is observed in the present experiment as well as a weaker transition with an energy of 576 keV, which we suggest is the 4~-~ 4~ transition. A state at 1956 keV was observed in a (t,p)-experiment [20] and assigned spin 4 +. We propose that the observed 908 keV transition decays from this 4~ state to the 4~- state.

Table 1 Particle-detector geometry and projectile scattering ranges in the 58Ni and 2°8pb experiments. In the 2°spb experiment the recoils were detected, and the given projectile ranges are calculated from the recoil angles Projectile

5SNi

2°8pb

Detector

A~ B C D b D c E F G H I J

0min

0max

q~min

~bmax

0rneanlab

0mea nc.m. (deg)

(deg)

(deg)

(deg)

(deg)

(deg)

l°6pd

l°8Pd

163 116 56 43 43 101 71 14 27 19 34

175 124 64 47 47 109 79 27 39 34 49

0 281 281 253 253 251 251 241 247 64 65

360 289 289 257 257 259 259 299 293 116 115

169 120 60 45 62 105 75 28 30 30 28

175 148 88 68 91 137 107 141 109 131 95

175 148 67 90 136 106 143 105 135 93

a A n n u l a r detector. b This entry for detector D corresponds to the projectiles being detected (see the text). c This entry for detector D corresponds to the recoils being detected (see the text), and the corresponding projectile center-of-mass scattering angles are therefore 91 ° and 90 °, respectively.

L.E. Suensson et al. / Nuclear Physics A584 (1995) 547-572

553

2963 2077 1229 512

6+1

1932

4_-~2

l 4+! .... 1128--1---2+2

1562 1134] i

2 +3

1707

O+s

0+2

tJ

2% ....... 0+1 ......

Fig. 3. Observed levels in l°6pd.

No line corresponding to the deexcitation of the known 4~ state at 1990 keV was observed. However, we observe a relatively strong line at 635 keV. It was observed also in the (n,y)-experiment [17], but not placed in their level scheme. We tentatively suggest that it corresponds to the deexcitation of a (4~-) state at 2076 keV. We also observe, both in the 5SNi and 2°sPb experiments, a new line with a y-ray energy of 1027 keV (see Fig. 2). It cannot be the 1026 keV 4 ~ 2~~

(8÷1) (24-22)

(6~'~) (2076)

1771

/-54

1957

4+

Z+5

1514

_0%

1624 I

!048

(4+5)

4+

2

1441

931 --

1

---T .

.

.

.

.

_

_

_

_

_

_

08pd Fig. 4. Observed levels in i°spd.

3

554

L.E. Svensson et al. / Nuclear Physics A584 (1995) 547-572

transition nor the 1028 keV 4 ~ 4~ transition, since both the 4 ~ 4~" and the 4~---, 2~- transitions are observed in the 2°spb experiment but not in the 5SNi experiment. We leave the 1027 keV line unassigned. In some of the spectra from the 2°8pb-excitation a weak 798 keV y-ray is seen, which possibly is the 62+ ~ 42+ transition, and was used in the analysis as such. The measured yield is compatible with this assignment. 3.2. Fitting o f matrix elements

The coupled-channels least-squares search code G O S I A [21] was used to deduce the reduced E2 and M1 matrix elements from the observed y-ray yields. G O S I A treats the Coulomb excitation process semi-classically. The input to G O S I A contains spins, parities and excitation energies of the states involved, starting values for matrix elements connecting these states, experimentally determined y-ray yields, geometry of y-ray and particle detectors, projectile kinetic energy and energy loss in the target, internal conversion coefficients, and spectroscopic data known from other experiments. In the calculation of the deexcitation y-ray yields, integrations over particle-detector solid angle and projectile energy are performed. The starting values of the matrix elements are used to calculate for each state its Coulomb excitation cross section, from which the deexcitation y-ray yields are determined. The matrix elements are then varied using the steepest-descent method to obtain a best possible fit of calculated deexcitation y-ray yields to the experimentally observed yields. The goodness of the fit is measured by a minimum xZ-criterion. The dependence of the deexcitation y-ray yields on the reduced matrix elements is measured as a ratio d log(yield)/d log M,., quantities which are obtained by GOSIA, and which can be used to judge which matrix elements are sensitive to the experimental data. The Coulomb excitation cross-section for each state depends on products of all reduced electromagnetic matrix elements connecting it with the ground state via a number of intermediate states, some of which are weakly excited. Because of these virtual-excitation effects it is necessary in the analysis of m6pd and l°spd to include most of the previously known low-spin ( I ~ < 6 +) states (see Fig. 6 below), plus several additional states of higher spin with energies estimated from systematics (see [15]). These states were coupled to more than 125 non-diagonal and diagonal E2 and M1 matrix elements in each nucleus. The calculated y-ray yields were fitted to 202 and 190 experimental y-ray yields, respectively, from several different combinations of y-detectors and particle energy and scattering ranges. In addition, the previously known 2 ~ 0~- and 2 [ ~ 2 7 E2 matrix elements, as well as E 2 / M 1 mixing ratios and branching ratios known from other experiments, and listed in Tables 2 and 3, were used as input data in the fitting procedure. In all, 247 and 206 data were used in the analysis of m6pd and l°8pd, respectively. Multiple Coulomb excitation is sensitive to both the magnitude and relative signs of the EA matrix elements. The wave function of each state has an arbitrary phase which was selected by fixing the sign of one E2 matrix element for each state, marked with an asterix in Table 4. This ensured a consistent phase conven-

L.E. &,ensson et al. / Nuclear Physics A584 (1995) 547-572

555

Table 2 Previously known T-ray branching ratios used in the analysis of m6Pd [1] and m s p d [2,3,17] Transition

1116Pd

ms Pd

03+ - , 2 ~ / 0 f ~ 2 7 04+ ---, 23+/04 ~ ~ 2 + (14+ ---' 2+1/u4/ll+~ 2+= 0 ; -" 2 3 / 0 5 - ---' 27 11~ --' 2~/()~- --+ 27

0.151(11) 0.029(5) 0,0075(13) 0,292(13) 0.089(6)

0.210(33)

22 --+ 2 ~ / 2 ~ ~ 11~

1.84(5) 0.434(13) 0.125(13) 9.59(20) 0.40(5) 11.029(10)

2.96(40) 0.59(27)

-9 + -3

__~

+ , ~ + ~ Or [12/,~

2 3÷ ~ 2 j / 2 ~ ~ 117 -3 '~ + ~" 2 1 / 2 ~ - ~ O( ~+ -4 ---' 0 ~ / 2 ~ - ---, 2 7 24+ ---, 2 + / 2 ~ ~ 21+ 24+ --, O f / 2 ] - ~ 27 "~+ ---' -3+/'~ -4 +l/~4 ~ 2( 2~ --, 2 3+/ 2 8 + ~ 3~ 2~. ---, 0+/9:= _> - , 3~+ ")+ + -)~ -. - ' 2 2 / - s ~ 3 f -)+ - s ~ 2 ] / 2 + ---' 3 ( 2 s --, 0 1 / 2 ~ ~ 3~+ + , "~+ ~ 31 /2~, ~ 2i + "~+ ---, 2 ,=+ / 2 ~ -+ 2~ _~, 2~ ~ 1 / 7 / 2 , + --+ 2 ( 0 . 0 8 1 1 ( 1 3 I' ~ 2 ~ / 3 i + ~ 2V 3~+ ~ 4 ( / 3 ( --, 2 + +

+

+

4+ -+ 3 E / 4 : ---, 41 4 + ~ 2 + / 4 + ~ 41+ 4 ; ---, 2 7 / 4 j -~ 4~+ 4 + -+ 2 ~ + / 4 3 ÷___+ 21+ 4 +~ ---, 4 1 / 4 1 --' 2t + 4q+ - ' 2 I / 4 4 + ---' 4 1 4~ ~ ,~/~+/4=4 -~ 4+t 44+ ---' " 1+/ /443 + ~ 4 +l 4+4 -~'~+-z / 4 a + - - + 4 ( 4 ; -~ 4 ; / 4 + ~ 3 ~" 4+ ~ 4 +/A4I/'s -+ 31+ 4 s+ --, :"~ / 4 , + ~ 3 ~

4.57(54) (I.47(9) 1/.33(7) 0.167(36) 0.133(29)

1.8(3) 1.10(10) 2.34(19) 0.42(3) 0.369(21) 0.039(8) 0.526(15) 1 ) 2.25(13) 1/.1187(5) 0.059(8) 2.79(15) 1/./108(4) 0.40(8) 3.3(1.1) 0.1/42191

0.96(8)

11.16(4) 0.37(7) 0.27(4) (I.20(6)

0.063(12) 0.098(12)

4 ; --, 2 , + / 4 ; - , 3 ( 4 + 4<,+ + 4 e/4<, ~ 31+

1.20(81 0.35(5) 0.022(9)

4~ ---' 4 + I / ~a +, - + 3 ~

0.057(8)

tion for the experimental and theoretical results for both nuclei. The signs of the remaining matrix elements are physical observables once the phase of the wave function has been defined. The fitting procedure was very time-consuming due to the large system of strongly correlated matrix elements. The most efficient way of working with the code was to fit different subsets of matrix elements and the complete set of matrix elements alternately. Examples comparing the agreement between the Coulomb

L.E. Suensson et al. / Nuclear Physics A584 (1995) 547-572

556

Table 3 Previously known E 2 / M 1 mixing ratios used in the analysis of l°6pd [1] and m8pd [22] Transition

l°6pd

l°8pd

2 ~- --+ 2 ~23 ---,2~2~ ---,2~26~ --+ 2~2g --+ 2~-

- 9.4(2.0) +0.24(1) + 1.5(3) - 0.06(12) + 0.25(2)

- 3.1(4)

3~- --+ 2~3 ~- ---,2 ~

-7.9(8)

4~- ~ 4~ 4~- --+ 3~ 46~ --+ 3?

- 2.30(2) - 7.5(1.5) + 1.0(8)

- 3.8(4)

/.

10->

/

+~ o

T 2

/ //I ~-I

4t ~ V

/

./ J

t

10-2- __,~

I

6+

/ /

/ f

>. 10-310-'-

/

TE// +// (82) -~ 6

80 I 0 0

60

80 i 0 0 120 140 160 180

100 120 140

16 0 S8Ni 2oSpb Scattering angle (degrees) in C.M. system Fig. 5. Experimental and fitted y-ray yields relative to the 2~- ~ 0~ yield.

L.E. Svensson et al. / Nuclear Physics A584 (1995) 547-572

557

excitation predictions, using the fitted matrix elements, and the data are shown in Fig. 5. Note that the sizes of the experimental and calculated yield ratios vary over several orders of magnitude. This reflects the large differences between the interaction strengths in the various experiments. Similar fits were obtained for all observed y-ray yields, some of which are shown in [15]. The weakest observed transitions had a yield of about 10 - 3 relative to the 2~-~ 0~- yield. The overall

Table 4 E x p e r i m e n t a l t r a n s i t i o n a l E 2 m a t r i x e l e m e n t s a n d B ( E 2 ) v a l u e s in l ° 6 p d a n d m s p d . T h e p h a s e s of the m a t r i x e l e m e n t s s e l e c t e d to fix the arbitrary p h a s e of the wave functions are m a r k e d by an asterix. A + sign in f r o n t o f a m a t r i x e l e m e n t indicates that the p h a s e w a s undetermined Transition

2~- - o 0 [ 0~- + 2 [ 0~- ~ 2 ~ 2~- ~ 0 [

l°6pd

m8pd

Present

Present

Previous

( f lIE2 IIi)

B(E2)

[e.b]

[W.u.l

+ f~ "1(1+ 0.04 * . . . . . 0.04 + c~ a~+o.02 • . . . . . 0.02 +004 + 0.24 _ 0 1 0 2 - 0 114 +0.0o6 "

-0.006

42+4 -4 43+6 -9 19+7

Present

Present

Previous b

B(E2)

( f lIE2 IIi)

B(E2)

B(E2)

[W.u.l

[e.b]

[W.u.]

[W.u.]

a

+3

+006*

46 _ 3 +7 30-7

+ 0 . 8 7 _ o.04 ~_ n An+o.02. r

0.02

a_ N ,:IQ+o.o2

- 3

T U.Jo

0 87 +°"1° "

L,..*u

-0.09

x'z"u-

0.10

_ 0.05

n no• +o.oo5

-- 'J.v"o

- 0.005

51 1 +13

50 _ 5 +6 52 6 47+5

1.3

•

4 4 _+ 1 01(~

- 1l

0 63 +0.07 "

-0.07

0.82

+ O.(~

2~ ~ 2 ~ -

D -/~ + 0.04 *

-v..v

0.04

39-4

4 5 _+4

- - u . vRR u l ) +0"04 * 0.04 -

51-+5

782~

47~21 ~

+ 1 -~+o.o7. . . . . . 0.07

71-+7

73+1o - - lo

± 1 A~+o.o7* 7- •.-,~ _ o.o7

+5 74 _ 8 1 2 +14 • -11

92 + 12

N 1/+0.30

4[~2f

.....

t~"/+7.2

0.03

+0006

0 ~ --~ 2~

+ 0.085 -o~oo6

o; --, 2~

--

nv . ~,~n +o.o2 * u _ 0.02

2 ; ~ O?

+0 03 + 0.045 _ olo3 + 0.76+o°.o~ * _.

2 f ---, 2~

- " . - " - 0.03

2+~4?3

+ 0.28+°'~

n an+0.04

-

--~

4f ~ 2f

+0.97

,) A+0.4 ~'~-0.3

< 0.06

< 1

< 0.24

< 19

13_+3 0 14 +0.02 • - o.oz

•

0.004

+007* 0105

-0.07

2 +2.2

• - 1.5 •

3 +2.5 -1.4

0 007 +0.006 • -o.oo3 +5 35-4

+0 002 + 0 . 0 3 8 _ 01003 -~-o~.,~o~+o.o6 * 0.05 _

0 52 + ° 1 °

5

.

.....

+0-18-o113

10

n nH+o.o05

4~- ~ 2 ?

+009

. . . . 0.3

39+~_

n n~+o.oo8

--~'~v-O.O06

2 3+ ~ 2 ~ -

+4

1 ,~n+O.lO

+7

--

"

-0.04

- 0 . . . . o104 1 ~.+0.10. x.o~_0.14

45 +11 -7

< 0.09

< 0.3

..L . . . . 1 - ) a~+ -0 . 007. 0*6

- 0.91 +0.07 -0.08

RO+10 u~_ 13

~-9 /)~+0.11 * ....... 0.I1

4+ - ~ 32 ~

+~

4f ~4~-

+f~ /)Q +0.14 u'uu - 0.20

8? ~67 a F r o m [1]• b F r o m [3].

OQ + 0.05 *

,,.,.o 0.04

+020

+ 0.23_oi17 1o • + 1.49 +o -0124

6f ~ 4f + '~ "ltZ + 0 . 1 2 z..JJ --0.33

*

107 _+ 1 23 6

0.7

a-n R'~ +0.10 --~'~-o.o7

23-2

4f ~ 42

~'"

12+6 -- 4

~'J~-0.()5 +3

~ ,7+02

+0.11

n .~.,+O.lO "70+004

59568

- 0.42_ 0.07

4~ -, 2~ 6?~47

0 16 +°°1

n hoe +0.010 ...... o.o15

~- 9 "/~ + 0.17 * -- ~.-u-O.14

+7 55 _ 5

3 7 +2.7 "

- 1.I

30+5 -5 +12 11

107_

+1.1

2.9 0.8 0

"

2 +L5 -0.2

+4.8

1.9_1.8 56+8 - i7 +19

149_ 15

-12

o.(~

558

L.E. Svensson et al. / Nuclear Physics A584 (1995) 547-572

quality of the fitted yields was reasonable although the final normalized X 2 were about 2 for both nuclei. The errors of the calculated matrix elements are complicated to estimate due to cross-correlation effects. The method used is described in [21,23] and normally gives asymmetric errors. After the final minimum was obtained, the results of the error calculation and the calculated sensitivity were used to select the unambiguously determined matrix elements. In all, 25 E2 matrix elements were unambiguously determined in l°6pd while 31 E2 matrix elements were determined for l°8pd. Considering the number of matrix elements that are sensitive to the data, the system was well overdetermined. 3.3. S y s t e m a t i c errors

Quantum-mechanical corrections, E4 excitations competing with E2 excitations, odd-spin state excitations, the dipole polarization effect, and uncertainty in the beam energy were discussed in [11,12,15] and found to have a small or negligible influence on the extracted matrix elements. For sufficiently long-lived states the deorientation effect changes the angular distribution of the deexcitation y-rays. The two-state model of Brenn et al. [24] is used in G O S I A to handle this effect. In a detailed investigation on Os-nuclei [25], it was found that a 20% change in the magnitude of the deorientation effect produced a less than 2% change of the E2 matrix elements. The state with the longest lifetime is the 2~- state. Since the lifetimes of these 2~- states are much shorter in l°6A°spd than in the Os-nuclei it is expected that the uncertainty induced by the deorientation effect can be neglected. After the final minimum was reached, the influence of the experimentally unobserved states was checked. In this calculation all experimentally unobserved states were excluded, apart from one 'high-spin' state in each band. This resulted in a slightly worse fit of the observed yields as a whole, even though no individual transition was severely changed. Possible contributions of the systematic errors mentioned here have been included by adding a weighted error of 5% to the correlated errors obtained from GOSIA.

4. Results

The experimentally determined transitional E2 matrix elements, both magnitude and relative phase, are presented in Table 4, and the diagonal E2 matrix elements in Table 5. All E2 matrix elements below and within the 'two-phonon' triplet were determined, as well as most matrix elements that couple to the 'three-phonon' states, and some that couple to the 'four-phonon' states. Most values agree reasonably well with the previously known values, except the B(E2; 2~-~2~-) and B(E2; 4~---,2~-) values in l°sPd, which are lower in the present experiment. The previous value of the B(E2; 2~-~ 2~-) was astonishingly large compared to the present result and corresponding values in l°6pd and ll°Pd.

L . E . Svensson et al. / N u c l e a r Physics A 5 8 4 (1995) 5 4 7 - 5 7 2

559

Table 5 E x p e r i m e n t a l d i a g o n a l E 2 m a t r i x e l e m e n t s in ~°6pd a n d l ° S p d Transition

l°6pd

l°Spd

Present ( f lIE2 IIi)

Previous a ( f lIE2 IIi)

Present ( f lIE2 IIi)

Previous h ( f lIE2 IIi)

[e'b]

[e'b]

(e-b]

[e.b]

2 { ~ 2~-

_a

4~- ---, 4~-

+0.07 -1"02-O .ll +0.23 - 1.47 o.13

6 ? --' 6 ?

,7-)+0.06 u.~ 0.07

_ f l '7"7 + 0 . ( / 5 u... 0.1)5

+°11 - 0 -78 -OlO O -16. + O ' l s - v . . . . . oAa - 1.1_o4

~ - a ~O +0"06

4~+ -* 4 ~ .

- .... o.os _ a af~ +0.18 -.~ 0.o5

2 + ~ 2~

_ n L')+0.09

3

0 " 6 u~+0"07 --0.07

+0.2

8 ~ --* 8 ? 2 + ~ 2 2~

--

- - 0 . 8 1 -+0 0. .00 94

....

0.24

+ 0 . 7 3 +0./19 0.07 +0.08 -0.02-013 --0.57

+0.16 (I. 1 2

F r o m [1]. b F r o m [3]. a

Coulomb excitation of unnatural-parity states, such as the 3 ( state, is small because destructive interference occurs between the major excitation pathways. Population of such states occurs mainly through y-ray feeding. The 31+ _ _ . ~ 22+ transition was weakly observed in some spectra in the 2°8pb experiment on l°spd. The corresponding yields were included in the fitting procedure and were reasonably well reproduced. Together with the known branching ratio of the 3~---* 2 + and 3 ~ 2~ transitions, it is implied that the B(E2; 3 ( ~ 2 7) is about 25 W.u.. In addition, several other E2 matrix elements were partly sensitive to the data, but their errors were large and they are not included in Table 4. It is worth mentioning, however, that if the assignment of the 4~- state at 2076 keV in l°sPd is correct, it is connected to the 2~- state by a large E2 matrix element of the order of 150 W.u.. For transitions between states of equal spins, both M1 and E2 matrix elements were included in the fitting procedure. In some cases the M1 matrix elements were determined [15].

5. Discussion

The low-spin structure of the Pd-isotopes has previously been discussed in terms of vibrations and rotations in the geometrical model [26,27], and by means of the boson expansion technique (BET) [28], the interacting boson model (IBA) [29-32], and the fermion dynamical-symmetry model [33], with varying success. The many E2 matrix elements measured in this work provide a stringent test of these collective models. However, in this paper we have chosen to limit the discussion to elucidate the extent to which the measured quadrupole properties, primarily of the 'two-' and 'three-phonon' states, agree with the predictions of the simple har-

L.E. Svensson et aL / Nuclear Physics A584 (1995) 547-572

560

8+1

(8*,)

I

/

0+4 - - Z .

2+5

4+~

2+4

--- 4+3 / /

/

,/ // /

2+5 ---, 64"1 4+2

0+.~ -

\\ 3 2+

/

/

',, + ~\4 4

//

/

4+2

/

/

/

/

2+4

4+1 /

0+3

/ . 2+.~ //

"4 phonon"

Z/

iI

~ 2+2

/"

/ 3f~ ~ /

-. 3+i

0+2

(5+,,)? /, f .....

(6+2)

6+1 //--

"3 phonon"

/I

.

~,.,

(Y,)

/

/

0+2

4+1

~ 2+2 .

"2 phonon"

2+~ 2+t

0+1

106pd

"1 phonon"

0+1

108pd

Fig. 6. Partial level schemes of l°6pd and t°spd. monic-vibrator model, and to extract a measure of the triaxiality. A systematic comparison is made with experimental data in neighbouring nuclei.

5. I. Energies The experimental excitation energies of l°6j°spd are shown in Fig. 6. Almost all known low-spin states below an excitation energy of about 2.4 MeV can be accounted for in the vibrational model if one includes the 'four-phonon' states. The 'two-phonon' states are slightly split in energy, which may be understood by means of a small anharmonic term in the vibration. The presumed 'three-phonon' states, however, have a much larger energy splitting, with the highest and lowest states within a multiplet several hundreds of keV apart. It is difficult to envisage such a large energy splitting as caused by anharmonicities alone. However, in this energy regime one expects single-particle excitations to mix with the vibrations. The proton single-particle levels of ~°8pd, calculated using a Woods-Saxon potential [35], are shown in Fig. 7 as a function of quadrupole deformation. The vds/2 and vg7/2 orbitals, as well as the 7rg9/2 orbital, are expected to play a dominant role in the positive-parity low-spin structure of these nuclei. For example, the v(g7/2 )2 configuration gives rise to a 0 +, 2 ÷, 4 ÷ and 6 + multiplet of states, and similar multiplets are obtained by the v(ds/2) 2 and l'r(g9/2 )2 configurations. The total routhian surfaces (TRS) of the lowest-lying positive-parity configuration of l°2-1mPd and of H4Cd were calculated using a deformed Woods-Saxon potential including the monopole pairing interaction as described in [36]. They are shown for l°2'l°6"t°8'll°pd in Fig. 8 at a rotational frequency of ha, = 0.25 MeV. Two minima are observed, one along the prolate collective axis and the other along the

L.E. St~ensson et aL / Nuclear Physics A584 (1995) 547-572 -2

.

~ > . _ _ , -

.

,-

I

.

---..~-~.,

56l

,

,

:.-\\ _

\\~"

\

-

~-6

~-9 t~ -10

-0.3

-0.2

-0. l

0.0

132

0.1

0.2

0.3

0.4

Fig. 7. Proton single-particle levels for l°spd. For qualitative arguments the diagram can be used also for the neighboring nuclei as well as for neutrons.

prolate non-collective axis. The calculated ground-state deformation fie of the six nuclei are 0.12, 0.15, 0.16, 0.18, 0.21 and 0.14, respectively. The deformation increases significantly from l°2pd to t°4pd. For l°6'l°8'u°pd the minimum becomes more stable in the/32-direction and more unstable in the y-direction. This picture is qualitatively confirmed from the energy ratio E(4~)/E(2-~), which can be used as a measure of the deformation. It is 2 for a vibrator, and 3.33 and 2.5 for a stable deformed and y-unstable nucleus, respectively. The experimental ratios of the six nuclei above are 2.29, 2.40, 2.40, 2.42, 2.46 and 2.30, respectively. The increase between l°2pd and l°4pd is rather large considering that the energies of their 2~ states are the same. There is no increase between t°4pd and l°6pd, which may be understood from the fact that an increased deformation is compensated by an increasing y-instability. It is now interesting to study the proton single-particle levels in Fig. 7 once again. At Z = 46 and 132 --- 0.15 the rrpt/2 orbital cuts through the Fermi surface. Two particles or two holes in the 7rp~/2 orbital can only give spin 0 + or 1 +, which gives us an extra degree of freedom to create 0 + states. They will be lowered relative to states with higher angular momentum. In l°2pd the ~'Pt/2 orbital lies far away from the Fermi surface, but when the deformation increases in m4pd it comes closer to the Fermi surface, and this 0 + state may eventually become low enough to enter into the energy regime of the 'three-phonon' states. Another complication is the proton two-particle-two-hole excitations across the Z = 50 shell, which involve the strongly deformation-driving g7/2 proton orbital [37,38]. They are known, particularly in the Sn [39] and possibly also in the Cd-nuclei [40], to intrude into the two-phonon vibrational region. These states have larger deformation and are associated with rotational band structure [39]. Thus, one must keep in mind that several different degrees of freedom other than vibrational are expected in the low-lying positive-parity states in the Pd-nuclei.

L.E. Svensson et al. /Nuclear Physics A584 (1995) 547-572

562

5.2. Transition probabilities The B(E2; 2~--~ 0~) values of ~°2-11°pd and H4Cd are plotted as a function of neutron number in Fig. 9. In all the following figures the point at N = 66 will be represented by 114Cd. The collectivity increases from 33 W.u. in mzPd to 54 W.u. in ~°Pd. The collectivity decreases to 31 W.u. in ll4cd, although it has the same number of valence particles as 11°pd. However, H4Cd has two proton holes less, which indicates the importance of the proton-neutron interaction to create collectivity. All measured B(E2) values of t°6pd and t°spd are given in Table 4. Their transitions between the 'one-', 'two-' and 'three-phonon' states are compared with 1°2'1°4'11°pd and lt4Cd in Table 6. To facilitate the comparison, the B(E2) values

g" 0.20

i! ~~

t-Z/,

- .-.-.-

-::..c.w.-.-c:,.c:...-;,

v_ "10 ~K3 _ - 0.00 r,b,xx

-0.10

.,.

.~

;;~g//2///]t//~

I~ I\1 -0.20 0.00

0.10

0.20

0.30

0.00

X=~2cos(7+30 °)

0.30

0.30

0.20

0.20

6_, ~e~ 0.10

0.10

0.20

0.30

0.10

0.20

0.30

X=lB2cos(7+30 °)

0.10 " ] ~

.! 0.00 ~ -0.10

-0.20 0.00

-0.10 !

0.10

0.20

X=13~cos(7+30°)

0.30

-0.20 0.00

X=[~2cos(7+30°)

Fig. 8. Total routhian surface calculation for the lowest-lying (~-, a ) = ( + , 0) configuration, the vacuum configuration, at hw = 0.25 MeV. Top-left figure l°2pd: /32 = 0.12 and 3' = - 1 . 9 °. Top-right figure I°6Pd:/32 = 0.16 and 3' = - 4 . 2 °. Bottom-left figure l°8pd:/32 = 0.18 and 3' = - 4 . 1 °. Bottom-right figure lmpd: /32 = 0.21 and 3' = - 16.5 °. T h e r e is 100 k e V b e t w e e n the contour lines.

L.E. Svensson et al. / Nuclear Physics A584 (1995) 547-572 7

8O

I

I

[

I

I

56.3

I

6o T 40

.~qr 20

I

56

I

58

I

60

I

62

I

64

I

66

Neutron Number Fig. 9. The B(E2;2~-~0[) values in W.u. as a function of neutron number. The filled symbols represent the Pd-isotopes, and the open symbol represents 1~4Cd.

are all normalized to the B(E2; 2~ ~ 0~-) values of the respective nuclei. They are plotted as a function of neutron number in Figs. 10 and 11 for the A n = 1 transitions ( A n is the difference in our assigned phonon number between the states involved in the transition). Some interesting general trends are observed. All A n = 0, 2, 3 transitions are smaller than 1 W.u., with some exceptions discussed below, and all A n = 1 transitions are larger than 10 W.u., with a few exceptions (see Table 6). The B(E2) values of the transitions between the 'one-', 'two-' and 'three-phonon' states are thus qualitatively in agreement with the harmonic-vibrator model. Quantitatively, however, there are large differences. In the Pd-nuclei most A n = 1 transitions are smaller than predicted, in many cases by as much as a factor of two or more. In ll4cd, on the other hand, many transitions fit into the vibrational picture up to the three-phonon states, which was already pointed out in [12], and later in [42]. For example, a large value of B(E2) consistent with the vibrator model is observed between the 0~ state and the 2~- state in 114Cd. This is true also in l°2pd, but in the other Pd-nuclei the Of--+ 2~ transitions are weak, and are difficult to fit into the vibrational picture. Also the generally accepted two-phonon vibrational states show large deviations from the vibrational predictions. The 'two-' to 'one-phonon' 0~---* 2~ and 22~-~ 2~ transitions are in some cases smaller by as much as a factor of five as compared to the expected vibrator strength. Even in ll4Cd they are much smaller than predicted, about a factor of two, although 114Cd has several higher-lying states that seem to fit into the vibrational picture. One may therefore question the vibrational nature of these states, particularly for the 03 and 2~- states, since it is difficult to envisage a mechanism, such as mixing, that in all nuclei would reduce the expected vibrational transition strengths by large factors. The non-vibrational character of the 0~- and 2~- states is also indicated by the strong A n = 0 0~-~ 2~ transitions in some of the Pd-nuclei (see Fig. 10). Fig. 10 also shows the relative B(E2) values predicted by the rigid rotor for the yrast 4 ] ~ 2~, 6 ~ - ~ 4 ~ and 8 ~ - ~ 6 ( transitions. The experimental values fall between the harmonic-vibrator and rigid-rotor values, lending support to the picture of the Pd-isotopes as/3- and y-soft quadrupole-deformed nuclei. Apprecia-

L.E. Svensson et al. / Nuclear Physics A584 (1995) 547-572

564

Table 6 Experimental B(E2)values in l ° 2 - 1 1 ° p d a n d l l 4 c d . The B(E2; 2~ ~ 0 ~ ) v a l u e s for the six nuclei are 33, 35, 42, 50, 54 and 31 Weisskopf units, respectively, but they have been normalized to 100 to facilitate the comparison with the spherical harmonic-vibrator model (HV) Transition 102pd c logpd c 106pd d 108pd d llOpd e ll4cd f HV An 2~- ~ 0~-

100

100

O f ---' 21~

< 0.001 a

Of ~ 2f

291+121-121a

2~- ~ 0~-

5 •2 +3.4 --3.4

2 f ---' 21~

48+3-3

4~- ~ 21~ 4~- ~ 2 f 03 ~ 2~03 ~ 2f

39+9 -9 a

23 ~ 0~-

av. 21+°"06 -0.06 52 -TM 34

4 0 + 66

aJ ' wA+o.3 0.3

100

100

102 -21 + 14

12 .1/)A+ . . . 12

4t;17j_7 +

94+1°-22

21+o.2 " -0.2

13 +°'2 • --0.2

1.1 +°"1 --0.1

"~ 9+1.2 -'--1.2

6~ -~ 4~-

0

16 +°"1 " 0.I

0

2

102+1°-1o

89+4-4

90+6

200

1

163+217 +815

200+13-13

200

1

< 80

5 •7 +1° -O.7

< 0.23

< 2

0 "33 -0.05 +°'°5

<38 nv. 19+o.o2 -0.03 118+16 -12

9,:t ~ -+1 10 0

<3.1

12+0.2 •-0.2

1 7 +0.9 •-0.9 74+37

"1A+0.4 . . . . 1.4

-37

24 +12 8 on+22 ~ 14

n nl"7+0.015 ..... 0.007 83+12

-I0

4 f ~ 2~4 f ~4~-

1

0

148 +10_ 16 "~ . . .A+2.8 . 2.2

23~ --* 4 (

~-27

200

5 •8+2"9-1.3

93+1°-1o

24_+54 13+6 -3 9 1 +1"8 •-1.8 1~,1+27

1

169--17+17 17+17.5 •-0.7

2f

4~-~2~-

8 7 -+3 6

113+59-4

+6 13-2

0

0

a. . ."7+0.4 0.009_+3 0 . 0.6 93 +19 :t~2+26 -9 JJ -26 300 ~ 7 -0.6 +0.4 1.1 -0.1 +o.1 0 J"

2

1 1 1 _+4 20

5 2 _+77

17 +38 --9

23~2~-

4f~2~-

5 2 +62

100

66 +66

23 ---' 03 23

100

*~vlA~ +14-14

31 +7 -5

2 3+ ~ Of

100

55-+7 212 +24 -31

<0.6 +14 110-10 74+5.4 • -2.2

0"7+0.4 ~-~ 0.4

26 -4 +8 78+11 -11

b

139

1

21 --9 +9

0

0

0

2

58

1

~ 1~+0.3 . . . . . 0.7

100+73 42+11 -4 b

0 3 0 +0"11 1 6 +0.22 • -O.ll •-0.22 +6 1 +39

61 6 12+10 -7

60+IO -10

59+6 -6

214 +24 -22

196 -7 +17

1 3

103

1

0

2

03-6 +42 - 26

158

1

0

0

+11 41-6 384 +126 - 107

142

1

300

1

370

a It was suggested in [10] that the Of state was the three-phonon state and the 0 3 state was the two-phonon state. b It was pointed out in [12] that the B(E2) values of the 2 3 ~ 0 f and 2 3 ~ 4~- transitions could be wrong since they did not reproduce the measured branching ratio from an (n, y) experiment [41], and alternative values of 197 +25 - 2 5 and 139 +39 - 1 3 , respectively, were proposed. The branching ratio was re-measured in [42] confirming the value from [41]. We therefore adopt the values of 197_25 +25 and 139 +39 --13 for these two transitions. c From [10]. d From present work. e From [11]. f From [12].

ble average

triaxiality can be inferred

from

the existence

of low-lying pseudo-y

bands in these nuclei. One may attempt to extract the centroid of the triaxiality by c o m p a r i n g t h e E 2 d a t a w i t h t h e p r e d i c t i o n s o f t h e t r i a x i a l - r o t o r m o d e l [34] s i n c e many of these E2 data are insensitive to the y-softness degree of freedom. For yrast transitions those predictions differ little from the rigid-rotor value (Fig. irrespective of the average y-deformation, w h e r e a s t h e 2~---* 0~- a n d 2 ~ - ~ transitions are sensitive to the centroid of the y-deformation as shown in Fig.

the 10) 2~12.

L.E. Svensson et al. / Nuclear Physics A584 (1995) 547-572 250 2O0 ~r3 ~-~

I

i

t

I

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500

I

_Harm..ohie_V..ih_rator.. . . . . . . . .

4

20C

I

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58

i

I

60

i

i

I

62

200

i

I

66

2O0

R.oIor

. . . . . . . . . . . .

i

I

H a2n2onicVibrat2r_ . . . . . . . . .

I

I

I

I

58

60

62

64

66

I

I

I

l

I

I

H _aAv~_°nic_Vib_rator . . . . . . . . .

200

2 ; --, 2+

150

_Rigid_R..olPr...

I

56 250

100 . . . . . . . . . . .

6+

250

_Rigid

I

64

300

"4~

I

lOG

56

400

I

_H_atn2o_nic 7i brat_or.. . . . . . . . . .

2~-

50

500

I

6 + --, 4+

3O0

100 4~- ~

I

400

......

150

I

565

i

50

I

I

I

I

I

I

56

58

60

62

64

66

i

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[

I

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56

58

60

62

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66

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i

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i

300 250

Ijarm_o_ni~ V_ib_ra,_or_ . . . . . . . . .

2O0

fa3

2+

150

[

150 100

l O0

t

÷

, !50

~C~

i

50

__Ha2m_onic_Vib_rator . . . . . . . . .

10

I

I

56

58

I

60

I

62

I

64

I

I

66

56

10.

N = 66,

Relative which

B(E2) represents

values

as a function

of neutron

58

I

60

I

62

•

I

64

I

66

Neutron Number

Neutron Number Fig.

I

number

for

the

Pd-isotopes,

except

the

point

at

ll4cd.

Using this simple model, the 2 3--+ 0 r transition infer y-values ranging between 25 ° and 28 °, whereas the 2~--+ 2~- transition implies about 25 °. The differences in triaxiality extracted from these two transitions probably reflects the y-softness and other degrees of freedom not included in the extreme rigid triaxial-rotor model. 5.3. Quadrupole moments The diagonal E2 matrix elements of the 2~- and 22~ states are plotted as a function of neutron number in Fig. 13. The magnitude of the <2 ~- IIE2 II2 ~-> matrix element increases with neutron number for the Pd-nuclei, which may reflect an increase in deformation as predicted by the TRS calculations. However, the diagonal E2 matrix elements are primarily sensitive to triaxial deformation. They are plotted as a function of angular momentum for l°6pd and l°8pd in Fig. 14, where they are compared to the predictions of the triaxial-rotor model [34] for various values of y using a value of/3 derived from the transition E2 strength. The

566

L.E. Svensson et at/Nuclear Physics A584 (1995) 547-572 500

I

i

i

i

i

300

t

I

I

I

I

I

I

250 400 200 _ H__a~rn2n~c..V_ibr_at-°r. . . . . . . . 150 ".~

200

300

,

,

r

I

,

,

0 300

56

58

60

62

64

66

t

i

I

I

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i

25O

250

~ , ~ 200 c'~

200

150

150

-~ loo

I

_H_a2"~_onic..Vib_rato r_ . . . . . . . . .

300

.

.

.

.

.

.

.

.

t

I 56

t 58

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I 64

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I

i

i

I

l

I

2~ ~ 4+

I

100 50

5O 0

.

t

100 50

0

Ita_rn2onic_Vib_rat_°r_

,

T

I

'

, 56

58

60

62

I 64

I 66

i

i

I

I

i

I

4+ -~ 2~

25O

0 300

I

I

T

I

l

56

58

60

62

64

I 66

I

I

I

I

i

I

I 64

I 66

4~ -+ 4~

250

2"0 200

20{3 Itafn2onic.Vibrator . . . . . . .

~2

_nL~2o_"ic_v_ib2a'_°L . . . . . . . . .

150 100 5O I 56

I 58

I 60

I 62

I 64

Neutron N u m b e r

I 66

0

I 56

I 58

I 60

I 62

Neutron Number

Fig. 11. See caption of Fig. 10. data are consistent with an average triaxiality of 3' of about 20 ° for the ground and pseudo-gamma bands. Note the opposite sign of the static moment of the 2 f state as is expected for a band head using the rotor model. These model-dependent triaxiality values, derived from the static moments, are consistent with those derived from the E2 transition matrix elements. Note that the TRS calculations do not predict the y-deformation correctly, but considering the shallow minimum in the y-direction, one may expect that small variations in the input parameters to the TRS calculations may change the calculated y-deformation. 5.4. Intrinsic shape p a r a m e t e r s

A model-independent description of the quadrupole collectivity of the nuclear states can be obtained by application of the Cline and Flaum rotational invariants [45,46]. Two parameters, Q and 6, are defined, analogous to the Bohr parameters 3 and y, the expectation values of which can be evaluated in the intrinsic frame of

L.E. St, ensson et al. / Nuclear Physics A584 (1995) 547-572 4

I

I

+~

I

.....

I

..''.

. .....'

I

567

I

...,"-.. .. ". 20 °

2

T

........

+~

....

....

• . 15o

....... i

...

"'" 250

~,,ft ................ , ...... ;... • .........o ; ; 0

I

i

56

58

I

I

I

I

60

62

64

66

Neutron Number 80

I

I

I

I .."

I

I

. .-".

....... .......... i) ...... illlli"...... }....... ~.';~ii-.

+~

...... ............ ....,

1. 4 0

+eq c,q ~:q~ 2 0

..

..

. ...--

,. . ...- 28 ° -. ". 27o "§ 25 °

½ ",.

•

"-., 20 °

ca

I

I

I

I

I

I

56

58

60

62

64

66

15 °

Neutron Number Fig. 12. The B(E2; 22" --+ 0~- ) and B(E2; 22~ --+ 2~- ) values in W.u. as a function of neutron number. The filled symbols represent the Pd-isotopes, and the open symbol represents u4Cd.

reference by summing various products of matrix elements along closed paths starting and ending in the state (for details see [45,46,11]). The simplest rotational invariant is used to obtain the expectation value of Q2 for a state I:

v~Q

~

~(lllg2lIJ)(J j I

~.~

0

~

v -

-1

E21II)

I

2 I

I

2 I

0 J

'

I

Harmonic Vibrator

..................

t

v

~

21~

'

'

}

I

I

I

I

I

I

56

58

60

62

64

66

Neugron N u m b e r Fig. 13. The 2{- and 2 5 diagonal matrix elements as a function of neutron number. The filled symbols represent the Pd-isotopes, and the open symbol represents ll4Cd. The data for 1°2'1°4pdis from [43]; for u°Pd from [11]; for ll4cd from [12].

L.E. Svensson et aL / Nuclear Physics A584 (1995) 547-572

568

Table 7 The intrinsic shape parameters Q and State

Q2 [e2b 2] io6pd

lOSpd

uopd

I14Cd b

a

0 • 63 +0.03 -0.03

0 "77 +°"1° -0.11

0 •86 +0.02 -0.06

0 "53 +0.07 -0.08

0 76 +0.04

n Qo+o.02

~ n,~+o.o5

0 64 +0.09

4~-

tl 2 ' 7 + 0 . 0 7 ~.~0.07

1 nO+0.0~i x'u~-o.07

1 3 7 + °" 2 1 • -0.13

1 n2 +0.09 x.uu 0.06

2f

0 58 +0.07 -0.07

0 92 +0.05 -0.06

a 2"/+0.07 u'u" 0.08

0

Of

fl 2 " 7 + 0 . 0 4 u.ut 0.04

1 9"3 + 0 - 1 2 x.~ 0.14

1 5 8 +0"11 " --0.22

1 12 +°17

0~2~-

0~-

"

-0,04

u.uJ

0.59

l ' u z ~ - 0.07

"

•

-0.17

73 +°"12 -0.06

"

~[o1

~[o1

a[o]

a[ o1

20-+2

19+45

16-+I

27+2-2

--0.15

a From [11]. b From [12].

I

I

I

I

L__!_- J___l

7=25 °

7=25 ° ..."

..."" 7=20 o

A ¢',1

-1 """'"'t,.

7=I5 °

v

lO6pd -2

.................. ;r._.igaI

t

lO8pd

i

SpinI

2

4

I

6

I

8

I

i

i

i

SpinI

2

4

i~:

6

I

8

I

"i-

-i!i,

- ;5. - "..-..

,.Q ,_.%~ o A.¢,a \ ~.'..

• v.~ v

::ii@i).'...

-._'.....

~a _]

".', ".. 3,=25 ° "- '.., ". 7 = 1 5 ° • ..;....:

";...... 7 = 2 5 ° - .,.. "...3,=15 ° .... %-.:i';:.

106pd I 2

.r=2o° I 4

Spin I

I 6

108pd I 2

..

7=20 ° I 4

I 6

Spin I

Fig. 14. Diagonal matrix elements in 1°6'1°8pd as a function of angular momentum for the ground bands (top figures), and for the quasi-T bands (bottom figures).

L.E. Svensson et al. /Nuclear Physics A584 (1995) 547-572

569

where the last factor is a 6j-symbol and the index J runs over all states connected to state I through an E2 matrix element. The QZ-values of the 0~- and 2~ states, and of the three two-phonon states, are given in Table 7. One may argue that not all E2 matrix elements that couple to the respective states are known, and thus that the summation is not exhausted, but most likely all the strong matrix elements that couple to the two-phonon states are known, and further, since Q2 is obtained from squares of all non-zero matrix elements coupling to state I, missing matrix elements must always lead to an underestimation of the QZ-value. The QZ-values for the ground state and two-phonon states are plotted in Fig. 15 as a function of neutron number. Also shown in the figure are the values of Q2 obtained if the rotational invariant is applied to the predicted E2 matrix elements of the harmonic-vibrator model normalized to the ground state. It is very interesting to note that the quadrupole collectivity, as measured by the p a r a m e t e r Q2, can be accounted for in the vibrational model, at least for the 0 f and 4~- states of the heaviest isotopes. It implies that the vibrational quadrupole strength of these states is not concentrated into one transitional B(E2) value, but rather that it is fragmented and shared between several E2 matrix elements. It is also interesting to note that only about half of the vibrational quadrupole strength is obtained for the 2~- state by summing all matrix elements that couple to it. Either this state is less vibrational than the 0~- and 4~-

½ 0

I 60

2

I

I 62

I 64

I 66

Neutron Number I

I

I

",,~ 4-{

¢D

½

0

I 60

I 62

I 64

I 66

Neutron Number Fig. 15. The intrinsic shape parameter Q2 plotted as a function of neutron number for the ground state, and for the three two-phonon states. The filled symbols represent the Pd-isotopes, and the open symbol represents n4Cd. The dashed line is Q2 calculated for the harmonic-vibrator model.

570

L.E. Svensson et al. / Nuclear Physics A584 (1995) 547-572 40

i

I

i

i

30 20 10

I

60

I

62

I

64

i

66

68

Neutron Number Fig. 16. The intrinsic shape parameter 6 of the ground state plotted as a function of neutron number. The filled symbols represent the Pd-isotopes, and the open symbol represents ll4cd. For an axially symmetric prolate-deformed nucleus 6 = 0 ° is expected, and for a harmonic vibrator 6 = 30 ° is expected,

states, or we have not observed half of the vibrational E2 strength that couples to it, which is unlikely. The large value of Q2 for the 0~- states may also be interpreted as a large dynamic quadrupole deformation, and thus one may associate it with the band head of a rotational band as indicated in Figs. 3 and 4. The possibility of such rotational-band structures were also discussed in ll°pd [11] and ll4cd [12]. The triaxiality of the intrinsic E2 moments for the ground state also can be obtained from the expectation value of the centroid of cos 36. These asymmetry values, listed in Table 7, are model independent and thus are more significant than asymmetry values derived using the rigid triaxial-rotor model. A model can be used to relate the E2 asymmetry 6 to the shape-asymmetry parameter 3'; they would be similar in many models. The 6-values of the ground state are plotted as a function of neutron number in Fig. 16. They vary from 20° in l°6pd to 16 ° in ll°Pd, but is 27 ° in '14Cd. The measured value of 6 confirms the importance of triaxial deformation as was implied from the model-dependent 3' derived using the rigid-triaxial-rotor model. 6. Conclusions The quadrupole collective properties of the nuclei a°6pd and l°8pd have been measured through several Coulomb excitation experiments, resulting in a modelindependent determination of the magnitudes and relative signs of E2 matrix elements in each nucleus. Combined with earlier similar measurements on 1°2'1°4'11°pd and l l a c d these data form an extensive set of experimental data suitable for systematic comparisons with models of transitional nuclei. These data have been used to elucidate the extent to which the simple harmonic quadrupole vibrator is applicable and to extract a measure of the triaxiality of the quadrupoledeformed collectivity. It is shown that vibrational degrees of freedom are important for the description of the low-spin level structure of the Pd-nuclei, but that they cannot account for all their observed decay properties, not even for the

L.E. Svensson et al. / Nuclear Phys&s A584 (1995) 547-572

571

generally accepted two-phonon vibrational states, and that it is necessary to invoke rotational motion and triaxiality for explaining the structure of some of these states. The sum-rule method of Cline and Flaum was applied to the data, and the result implies that the vibrational quadrupole strength of the 0 ÷ and 4 ÷ two-phonon states is fragmented and shared between several E2 matrix elements. This does, however, not apply to the 2 ÷ state of the two-phonon multiplet. The y-deformation, implied by the E2 rotational invariants, and the individual static quadrupole moments and transition probabilities range between 15 ° and 27 ° for the observed states. The completeness of the present data set of matrix elements and level spectra provides a stringent test for collective models of nuclear structure in these shape-transitional nuclei.

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