fragmentation of two-phonon γ vibration strength in deformed nuclei

a . __ . -_ lYi!iB EISEWER

8 August 1996

>hysics Letters B 382 (1996) 214-219

PHYSICS LETTERS B

Fragmentation of two-phonon y vibration strength in deformed nuclei

C.Y. Wu, D. Cline Nuclear Structure Research Laboratory, University of Rochester; Rochestel; NY 14627, USA

Received 10 April 1996 Editor: R.H. Siemssen

Abstract

The intrinsic E2 matrix elements between AK = 2 bands for 186~188*190Y1920s have been deduced from previously measured interband E2 transition matrix elements after correcting for the coupling between the rotational and intrinsic motion. The ratio of the intrinsic E2 matrix element between the K = 4 and 2 bands to that between the'K = 2 and 0 bands, together with the ratio of their excitation energies, provide strong evidence for the existence’ of rather pure harmonic double-phonon y-vibrational excitation. A similar result is obtained for 232Th whereas significant fragmentation of the two-phonon strength , is deduced for 156Gd and laDy.

PACS: 21.1O.K~; 21.1O.Re; 23.2O.J~; 25.70.De Keywords: Two-phonon y vibration

Rotational and vibrational modes of collective motion are very useful in classifying the low-lying ex-

cited states in deformed nuclei. The rotational mode

of collective motion is characterized by rotational

bands having correlated level energies and strongly- enhanced E2 matrix elements. The lowest intrinsic excitation with I, K" = 2,2+ in even-even deformed nuclei, typically occurring at ml MeV, is classified as a one-phonon y-vibration state [ 11. In a pure

harmonic vibration limit, the expected two-phonon y- vibration states with I, K" = 0, 0 + and 4, 4+ should have excitation energies at twice that of the I, KS = 2,2+ excitation, i.e. ~2 MeV, which usually is above the pairing gap leading to possible mixing with two- quasiparticle configurations. Therefore, the question of the localization of two-phonon y-vibration strength has been raised [ 21 because mixing may lead to fragmentation of the two-phonon strength over a range of

excitation energy. For several well-deformed nuclei, an assignment of

I, K" = 4,4$ states as being two-phonon vibrational

excitations has been suggested based on the excita-

tion energies and the predominant y-ray decay to the I, Km = 2,2+ state [3]. However, absolute B(E2) values connecting the presumed two- and one-phonon states are the only unambiguous measure of double phonon excitation. Such B (E2) data are available for

156Gd [4], t60Dy [5], 16*Er [6], 232Th [7,8], and 186,188,190,1920s [9-111. Except for i60Dy, the measured B(E2) values range from 2-3 Weisskopf units in ls6Gd to lo-20 Weisskopf units in osmium nuclei; enhancement that is consistent with collective modes of motion.

Statements regarding the phonon character that are based on a comparison of the measured E2 matrix elements, are ambiguous because the interband E2 matrix

0370-2693/96/$12&l Copyright 0 1996 Elsevier Science B.V. All rights reserved. PII SO370-2693(96)00658-Z

C.Y. Wu, D. Cline/Physics Letters B 382 (1996) 214-219 215

elements depend on the mixing of the AK = 2 bands, due to coupling of the rotational and intrinsic motion, as well as the intrinsic E2 matrix element between the

bands. The intrinsic E2 matrix elements are required for an unambiguous measure of the multiphonon structure. In this letter, these intrinsic E2 matrix elements are extracted from the measured E2 matrix elements

by assuming a first order angular momentum depen- dence of the coupling between the rotation and intrinsic motion [ 121. The interband E2 matrix elements

are assumed to be correlated by the following equa- tion (Eq. (4-210) in Ref. [ 11):

B(E2, ZKt -+ ZK) = (ZK,K’2 - 2)ZKK)

x [~l-~z(~~(~K+1)-~K’(zK’+1))1 5. (1)

where Mr and M2 are the fitting parameters, K’ = K + 2 and 5 is equal to 2/2 if K = 0 and equal to 1

otherwise. This relation underlies use of the Mikhailov plot [ 121. The applicability of Fq. (1) is based on the assumption that both bands are rotational bands having

the same intrinsic deformation. The intrinsic matrix elements between bands are related to the MI and Mz matrix elements by (Eq. (4-211) in Ref. [l]),

(K’IE21K) = Ml +4(K+ 1)Mz. (2)

The matrix elements coupling the AK = 2 bands can be deduced from the the level-energy spacing and the wavefunction mixing amplitude derived from Mz (p.

161 in Ref. [ 11). The above method has been used to extract in-

trinsic E2 matrix elements for 156Gd, 16’Dy, 168Er, 186~188~1go~1g20s and 232Th. These are nuclei for which

the absolute E2 data are available from either lifetime measurements or Coulomb excitation measurements that are sufficiently complete to give unambigu-

ous matrix elements. The justification for the use of Eq. (2) to correlate the interband E2 matrix elements between K = 0 and 2 bands and between K = 2 and 4 bands is demonstrated by analyses using the rotational- invariant technique [ 10,151 applied to cases such as 168Er [ 181 and 186,188*1go,1g20s [9-l 11. These studies show almost constant centroids of both the magnitude and asymmetry for the quadrupole deformation of the K = 0 and 2 bands in these nuclei, as well as an ad- ditional K = 4 band in ‘the osmium nuclei. This is consistent with the interpretation that, in each nucleus,

-30 -20 -10 0 10 20 30 IK=,(I,,fl) - 1,&,=,+1)

Fig. 1. Mikhailov plot for the interband E2 transitions between

K = 2 and 0 bands in osmium nuclei together with the best fit

(solid line) using Eq. (1) and parameters shown. Spin is labelled

for transitions with AI = 0.

these bands are rotational bands with similar intrinsic deformation.

Osmium nuclei. For the osmium nuclei, the excitation energy for the I, quasi-Km = 2,2+ state ranges

from 489 keV in lg20s to 767 keV in lg60s. Possible two-phonon y-vibrational I, quasi-K” = 4,4+ states exist at about twice the one-phonon energies and these states decay predominantly to members of the one- phonon quasi-K = 2 band [ 13,141. Coulomb exci-

tation work on 186~188~190~39’%s has determined nearly

complete sets of E2 matrix elements and these can be described approximately by a y-soft type of collective model [9-l 11.

The interband E2 matrix elements between K = 2 and 0 bands are correlated well using Eq. (1) for ‘e6~‘s80s and moderately well for ‘goOs, as shown in Fig. 1. The matrix elements, Ml and M2, determined from a least-square fit of Eq. ( 1) to the data, are given in Fig. 1. From these matrix elements, the intrinsic E2 matrix element between K = 2 and 0 bands can be deduced from Eq. (2) and are listed in Table 1. The

216 C.Y. Wu, D. Cline / Physics Letters B 382 (1996) 214-219

Table 1 Extracted intrinsic E2 moments (eb) between bands with AK = 2.

Nucleus (K = 21 E2 IK = 0) (K = 41 E2 ]K = 2) Ratio a

tS6Gd lecDy 168Er

1860s 1% OS ‘~0s 1920s 232Th

0.238f0.004 0.157-f0.017 0.66f0.07 0.253f0.006 0.045*0.007 0.18f0.03 0.243ztO.008 0.16 < < 0.2gb 0.7 < < 1.1

0.21 < < 0.47” 0.9 < < 1.9 0.417f0.011 0.67f0.29 1.60f0.70 0.401f0.017 0.55f0.19 1.3710.48 0.396ztO.037 0.53ztO.06 1.3410.20

0.45 d 0.50-0.55d 1.11-1.22 0.31e 0.49&O. I 1 1.58f0.35

aTheratioof (K=4] E2 ]K=2) to (K=21 E2 ]K=O). b From the decay of the 4+ state. c From the decay of the 5+ state. d From three-band-mixing (K = 0, 2, and 4) calculation. e From three-band-mixing (K = 0, 2, and 0’) calculation.

Table 2 Coupling matrix elements( keV) extracted from either the method outlined in p. 161 of Ref. [l] or three-band-mixing calculation.

Nucleus (K=2] hz ]K=O) (K = 41 hz]K = 2)

ls6Gd -1.1 -1.2 160Dy -0.8 -0.3 168Er -0.6 -0.6 < < -0.4a

-0.8 < < -0.3 b 186-11900~ -4 to -2 -3c

1920s -3c -1.5 to -2c 232,j.h -Id -0.6

a From the decay of the 4+ state. b From the decay of the 5+ state. c From three-band-mixing (K = 0, 2, and 4) calculation. d From three-band-mixing (K = 0, 2, and 0’) calculation.

same analysis has been applied for the decay of I, K” = 4,4+ state to members of K = 2 band. The data are well correlated by Eq. (1) for 186~188~1900s as shown in Fig. 2. The extracted matrix elements, iI41 and M2, and deduced intrinsic E2 matrix element between K = 4 and 2 bands were listed in Fig. 2 and Table 1, respectively.

The matrix elements coupling the intrinsic K = 2 and 0 bands range from -2 keV for 1860s to -4 keV for t9’Os as listed in Table 2. The coupling matrix elements between K = 4 and 2 bands cannot be estimated reliably via the method outlined in p. 161 of Ref. [ l] due to the larger coupling and have been estimated to be -3 keV from a three-band-mixing (K = 0, 2,

Fig. 2. Mikhailov plot for the interband E2 transitions between I, Kn = 4,4+ state and band members of K = 2 in osmium nuclei together with the best fit (solid line) using Eq. ( 1) and parameters shown.

and 4 bands) calculation. The estimated intrinsic E2 matrix elements between the K = 2 and 0 bands and between the K = 4 and 2 bands, from the three-band- mixing calculation, are consistent with those obtained

from Mikhailov plot analyses. For the case of 1920s, a three-band-mixing (K = 0,

2, and 4 bands) calculation was performed to estimate the intrinsic E2 matrix elements and coupling matrix elements. Good agreement can be achieved with the band-mixing calculation. The estimated intrinsic E2 matrix elements and coupling matrix elements, listed in Tables 1 and 2 respectively, have the same magnitude as those in other osmium nuclei.

To illustrate the extent to which the multiphonon y- vibration scheme can be applied to I, Kn = 2,2+ and 4,4+ states in osmium nuclei, the ratio of the intrinsic E2 matrix element between K = 4 and 2 bands to that between the K = 2 and 0 bands versus the ratio of experimental excitation energies is plotted in Fig. 3. Note that a pure harmonic vibrator has an intrinsic E2 matrix element ratio equal to A. Both the excitation


I mlaoDy

I I

E,+IK=,) / Ez+(K=z)

Fig. 3. The ratio of intrinsic E2 moments vs. the ratio of experimental excitation energies between two- and one-phonon vibration for all cases studied here.

energy and intrinsic moment ratios are consistent with the interpretation of I, K" = 4, 4+ states in osmium nuclei being predominantly harmonic two-phonon ‘y- vibration states. The slightly lower ratio of intrinsic E2 matrix elements and increased excitation-energy

ratio for 1920s probably is a reflection of anharmonic

vibrational motion [ 16,171. A possible two-phonon vibrational I, Kg = 0, Of

state exists in all the osmium nuclei studied here, in that the energy and decay properties are consistent with the characteristic of two-phonon excitations. However, the intrinsic E2 matrix element, derived from the matrix element, (0, K = 0' 11 E2 11 2, K = 2), accounts only for about 70% of the harmonic vibration limit implying significant mixing with other

low-lying K = 0 states, such as the one-phonon P-vibration excitation. Although there is some in- formation regarding branching ratios, no absolute

B (E2) values are known for possible two-phonon O+ states in other deformed nuclei. Thus the following

discussion is limited to two-phonon 4+ states. *56Gd. The band built on I, KT = 2,2+ at 1.154

MeV is well established. The interband E2 matrix elements between this and the ground-state bands have been analyzed previously using Eq. ( 1) by Backlin et al. [4]. The intrinsic E2 matrix element is deduced to be 0.238f0.004 eb from the moments, Ml = 0.258f0.004 eb and M2 = -0.005 1 f0.0005 eb. There are two low-lying I, K* = 4, 4+ states at 1.5 11 and 1.861 MeV, respectively. The 4+ state at 1.861 MeV decays mainly to members of the K = 4 band at 1.5 11 MeV which then decays predominantly to members

of the K = 2 band with enhanced B(E2) ‘s, characteristic of two-phonon states. The intrinsic matrix element between these K = 4 and 2 bands is deduced to

be 0.157f0.017 eb from the matrix elements, MI = 0.366f0.013 eb and MP = -0.0174f0.0009 eb, de-

termined by fitting Eq. ( 1) to the interband E2 matrix elements for the transitions of the 4+ and 5+ states of

K=4bandtomembersofK=2band[4]. 160Dy. The interband matrix elements have been

measured for the transitions between the I, K" = 2,2+

.state at 966 keV and members of the ground-state band [ 51. The intrinsic E2 matrix element between K = 2 and 0 is deduced to be 0.253f0.006 eb from

the matrix elements, Ml = 0.272f0.006 eb and A& = -0.0047fO.0008 eb. There are two low-lying I, K" = 4, 4+ states at 1.694 and 2.097 MeV, respectively. Both are candidates for being two-phonon states because they decay mainly to members of K = 2 band and the ratio of their excitation energies to the I, KT = 2,2+ state. Absolute E2 data are available only for

4+ state at 1.694 MeV. {5] The deduced intrinsic E2

matrix element, between the K = 4 and 2 bands, is 0.045f0.007 eb.

‘@Er. The interband matrix elements between

K = 2 band at 821 keV and ground-state band have been determined in Coulomb excitation work by Kotlinski et al. [ 181. The intrinsic matrix element, (K = 2jE2jK = 0) = 0.243 f 0.008 eb, is deduced from the matrix elements, MI = 0.262f0.007 eb and M2 = -0.0047f0.0008 eb. Enhanced ,B(E2) values were measured in a lifetime measurement [ 61 for the

transitions from the 4+ and 5+ states of the K = 4 band at 2.055 MeV to members of the K = 2 band. This I, KT = 4,4+ state has been identified as a possi-

ble pure two-phonon y-vibration state [ 61. Since only lower and upper limits were extracted for the individ-

ual matrix elements, the intrinsic E2 matrix element between the K = 4 and 2 bands was deduced from the E2 data of the 4+ and 5+ states separately. For the

4+ state, the intrinsic matrix element is deduced from 0.014 e2b2< B(E2;4;, + 2iz2) ~0,041 e2b2[61

and the y branching [191 Zr<4+ -+ 3 +)/ZY(4+ -+ 2+) = 0.47 assuming a pure E2 4 + -+ 3+ transition. For the 5+ state, the intrinsic matrix element is deduced from 0.017 e2b2< B(E2;5;, --+

3&:,,) co.090 e2b2[ 61 and the y branchings [ 191,

Zy(5+ --$ 4+)/1,(5+ --+ 3 +) = 0.51f0.27 and

218 C.Y. Wu, D. C&e/Physics Letters B 382 (1996) 214-219

1,<5+ --+ 5+)/1,(5 + -+ 3+) = 0.2750.17 assuming pure E2 transitions. The branching ratio [ 191 Z,(5+ -+ 7+)/1,(5+ -+ 3+) = 0.087 f 0.053 was

omitted from the least-squares fit because it is not correlated well by Eq. ( 1). However, there is little

change in the final result when this branching ratio is included.

232Th. The lowest I, Kn = 2,2+ at 785 keV is nearly

degenerate with the 775 keV 2+ state of the first ex- cited K = 0 ’ band. Their mixing makes it impractical to use Eq. ( 1) to determine the Mt and h42 matrix el-

ements from the E2 matrix elements for the 2+ state transitions. The intrinsic E2 matrix element between the K = 2 and 0 bands is estimated to be 0.31 eb using a three-band-mixing (K = 0, 2 and 0’ bands) calculation and fitting to the available data [ 201. The

extracted coupling matrix element between the K = 2 and 0’ bands is about -0.33 keV and the intrinsic E2 matrix element between K = 0’ and 0 bands about 0.20 eb. The state with I, Kr = 4, 4+ at 1.414 MeV shows many characteristics of being a two-phonon state as reported in Refs. [ 7,8], where the lifetime and decay branching were measured. The intrinsic matrix

element between the K = 4 and 2 bands is deduced to be 0.49&O. 11 eb from the matrix elements, Ml = 0.586f0.074 eb and iPI:! = -0.0076f0.0070 eb.

The ratio of the intrinsic E2 matrix elements vs.

the ratio of experimental excitation energies between two- and one-phonon states for all the nuclei are il-

lustrated in Fig. 3. The large intrinsic E2 matrix element ratios for the osmium nuclei and 232Th are consistent with the existence of rather pure harmonic two-

phonon y-vibration states in these nuclei. A common factor for these two-phonon states is that their excitation energies are below the pairing gap. In contrast to the uniform one-phonon vibrational strengths in

156Gd, 160Dy, and 168Er (listed in Table 1) , substantial strength fragmentation for the two-phonon vibration in 15(jGd and 16’Dy is suggested by the intrinsic E2 matrix element ratios, as shown in Fig. 3. However, the appreciable two-phonon strength for the observed transitions in 156Gd at 1.511 MeV and t’j8Er at 2.055 MeV, suggests that the centroid of the two-phonon strength may not be as high as 3-4 MeV proposed in Ref. [ 21. The anharmonicity exhibited by the excitation energy ratio for 168Er is consistent with a reduced ratio of intrinsic E2 matrix elements relative to the harmonic vibration limit [ 16,171.

Many of the I, Kr = 4,4+ states, suggested as candidates for two-phonon excitation in well-deformed nuclei [ 31, may contain significant two-quasiparticle

components, as has been addressed by Burke [ 211 based on nucleon-transfer and P-decay data. This

appears true for 156Gd and 160Dy discussed in the present study. For 190*1920s, the two-phonon strength is shown to approach the harmonic vibration limit

despite the proposed [ 211 existence of a significant

hexadecapole phonon component. The magnitude of this hexadecapole component is still an open question due to the uncertainty involved in the (~,a) reaction mechanism [ 221. The static E2 matrix element, as well as the three AK = 2 E2 transition matrix ele-

ments, measured for the I, quasi-Kr = 4,4+ states, in each osmium nucleus, all are consistent with coupled rotation-vibrational models [ 9- 111. The enhancement of these four E2 matrix elements, is strong evidence that the two-phonon components dominate over two-quasiparticle components in the structure of these states. Contrary to statements made in Ref. [ 211, including the g-boson [ 231 in the interacting boson model [ 241, which is equivalent to the hexade-

capole phonon, actually underestimates the measured B( E2) values [ 91 by up to a factor of 4 for the

osmium nuclei. In summary, the intrinsic AK = 2 E2 matrix ele-

ments between the presumed double, single and zero phonon y-vibrational intrinsic states have been deduced from measured interband E2 matrix elements for 156Gd, 160~~, 168~~, %!Th, ad 186,188,190,1920s.

These intrinsic E2 matrix elements, which are a sen-

sitive measure of multiphonon structure, plus the excitation energies, show unambiguously the existence of rather pure harmonic double-phonon y-vibrational I, K” = 4,4+ intrinsic states in 232Th and the osmium nuclei. Even though there is evidence for anharmonicity, the double-phonon y-vibrational strength still is largely focussed in a single state in 168Er whereas significant fragmentation of the two-phonon strength is manifest in 156Gd and 160Dy. The localization of double-phonon intrinsic states, demonstrated here, opens the possibility of identifying other multiphonon structures in nuclei.


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fragmentation of two-phonon γ vibration strength in deformed nuclei

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