- Email: [email protected]
On the nature of “three-phonon” excitations in 112Cd
17 October 1996
PHYSICS
LElTERS
B
Physics Letters B 387 (1996) 259-265
On the nature of “three-phonon” excitations in ‘12Cd H.
Lehmann a~1,P.E. Garrett b, J. Jolie a, C.A. McGrath b, Minfang
Yeh b, S.W. Yates b
a lnstifur de Physique, Universire’ de Fribourg. Pe’rolles, CH-1700 Fribourg, Switzerland b University of Kenfucky,Lexington, KY 40506-0055, USA Received 4 June 1996; revised manuscript received 15 July 1996 Editor: R.H. Siemssen
Abstract The lifetimes of proposed 3-phonon states in *‘*Cd have been measured using the Doppler shift attenuation method (DSAM) following the (n,n’y) reaction. The experimental results allow the determination of relevant B( E2) values which are compared to different theoretical descriptions. Collective enhancements favouring the interpretation of collectivity in the phonon model are observed for 3+ and 4+ members of the 3-phonon quintuplet. However, for the proposed 2+ member, this simple model fails most likely due to the subtle interplay between intruder, mixed-symmetry,and phonon states, and
the decay pattern is strongly perturbed. The best overall agreement is obtained with 1BM-2calculations. PACS: 21.10R,T; 21.60E.F; 23.20 Keywords: 3-phonon states; DSAM; Lifetimes; B(E2)
values; Phonon model; IBM
Collective states in nuclei have been the subject of intense study since Bohr and Mottleson first proposed their collective hamiltonian [ 1I. An alternative way to describe these states is demonstrated by the Interacting Boson Model (IBM) [ 23 in which bosons, representing coupled valence-nucleon pairs, are used to describe the collective properties of nuclei. Often, this approach leads to similar predictions as the phonon model, although the underlying mechanism is different. While one would think that most questions regarding the nature of collective excitations would have already been answered, this is, in fact, not the case. The main difficulty encountered is the complex structure of excited nuclei. Thus, it is not clear, for example, how high in excitation energy, and hence into the region where single-particle degrees of freedom can become important, the collective states survive.
Recently, the level schemes of vibrational even-even nuclei near mass 100 have been developed into regions of increasing complexity (e.g. [ 3-S]) . Particularly, the combination of multiple Coulomb excitation, neutron capture and (a,xn) reactions to the Cd isotopes has been pivotal in the investigation of multiphonon states and shape coexistence due to intruding multi particle-hole excitations. Because both phenomena appear in the same nucleus at similar energies, their interference complicates the interpretation. For example, for 3-phonon states in ‘14Cd different interpretations have been invoked [ 3,6,7]. As illustrated in a recent survey [ 81, most proposed 3-phonon states are based on energy and spin systematics or are assigned on the basis of relative B(E2) values (essentially branching ratios). In order to assess their true nature, however, some measure of the collective character must be obtained. Often, only the
’ E-mail: [email protected]. 0370-2693/%/$12.00
Copyright 8 1996 Elsevier Science B.V. All rights reserved.
PII SO370-2693(96)01038-6
260
H. Lehmunn et al./Physics
determination of absolute transition probabilities between excited states allows one to obtain the wave function sensitive information needed. Especially in a complicated region, where interference effects can obscure the collective “fingerprints” [ 61, level interpretations and multiphonon assignments have to rely to a large part on absolute B(E2) values. The aim of the present work is to improve the understanding of the proposed [4,8] 3-phonon states in ‘l*Cd. Although the experimental knowledge of the level scheme of this nucleus is rather extensive [ 45 3, only a limited amount of experimental data exists for the lifetimes of excited states, and no lifetimes were available for states above 1.5 MeV excitation energy. Therefore, the absolute B(E2) values for transitions depopulating the 3-phonon quintuplet [4,8] have remained completely unknown. To remedy this situation, in measurements performed at the University of Kentucky Van de Graaff facility, the lifetimes of the proposed 3-phonon states have been measured using the Doppler shift attenuation method (DSAM) following inelastic neutron scattering (INS). Nearly monoenergetic neutrons with an energy of 2.5 MeV were produced with the 3H(p,n)3He reaction. This neutron energy was selected for several reasons: higher-lying levels which could possibly feed the 3-phonon states were not significantly populated, the population cross sections of the 3-phonon states were near their maximum values, and the resulting y-ray spectra were still relatively simple, thus allowing accurate fits of the peaks of interest. The scattering sample consisted of 50 g of “*Cd0 (98.17% isotopically enriched) contained in a cylindrical polyethylene container. By measuring the energy of the y rays as a function of angle, the experimental attenuation factor F(r)exp can be found from the relation E(6) = #??,(I + /3F(7)expCOSe>
(1)
where E(B) is the energy of the y ray measured at an angle B with respect to the direction of the incident neutron, E, is the unshifted y-ray energy, and /? = u/c represents the recoil velocity of the nucleus. The lifetime of the state is deduced by comparing the measured F( 7)exp values to those calculated theoretically using the Winterbon formalism [9]. In order to get a precise value for F( r)ext,, two experiments were performed with spectra recorded at 10 angles. One an-
Letters B 387 (1996) 259-265
150
250
1600
1700
I
350
450
1800
1900
,
550
650
750
.I
,
2000
2100
2200
EY (kev)
Fig. 1. y-spectrum at an angle of 90”. The prominent peaks from ‘12Cd are labelled with their energies.
gular distribution measurement employed a 57% efficient HPGe detector located N 1.1 m from the scattering sample, while the other used a 52% HPGe detector N 1.3 m from the sample. Each detector was placed inside a BGO anti-Compton shield, and time-of-flight gating was employed in order to reduce extraneous background events. The energy calibration of the detectors was continuously monitored through the use of radioactive source spectra collected concurrently with the in-beam spectra. Fig. 1 shows a typical spectrum, while in Fig. 2 the measured y-ray energy is plotted as a function of angle for selected transitions from each of the three states studied (see Fig. 3), as well as the 798 keV 4: --+ 2: transition. The half life determined for the 4; level from the data shown in Fig. 2 is 760 f 310 fs, in excellent agreement with the previously tabulated value of 900 zt 80 fs [ lo]. Consistent results for all well-resolved and moderately strong y rays were observed in the two angular distribution measurements. It should be noted that the quite small shifts are caused by the lifetimes of the states and not by the low recoil velocity. Table 1 lists the obtained lifetimes of the states. Further detaiIs of the experimental arrangement and the standard analysis procedure can be found elsewhere [ 11 I, and a complete analysis of all results will be presented in a forthcoming paper [ 121. In addition to the lifetimes, several hitherto unobserved or unplaced transitions from the 2+ state at 2 12 1 keV have been found (see Table 1) .
H. Lehmann et al./ Physics Lerrers B 387 (1996) 259-265
261
Table 1 Measuted lifetimes, relative intensities ( Ircr) and mixing coefficients (8) Jr
&lkevl
3f
2064.6
I
4f3
2081.8
2+ 4
2121.6
7 [fs]4
.Glkevl 648.9 752.2
0.133(2) 0.456(2)
1447.0 612.9 666.2 769.4 1464.0 688.2 809.1 897.1 1504.0 2121.7
0.411( 1) 0.081(3) 0.515(2) 0.368(2) 0.037 ( I ) 0.118(3) 0.024( 2) 0.096(2) 0.741(3) 0.020( 1)
680( 190)
500( 150)
740( 200)
-1 ’ I”.4 (-0.7M.1 -0.2 -0.6) -2 69+0.24
’ -0.16 _, ,58+0.10 -0.12 (-0 36+u.o4 -0.03 ) -0.4o+uo4 -0.03
d
1.36:$,
(0.24+0.03 ) -0.04
* Average of all observed deexciting transitions. b The branching ratios am determined from excitation function data recorded at 125’. c The first solution has the smaller x2. * A reliable value of S could not be determined.
In order to calculate the B( E2) values, the branching ratios and the S values have also been deduced from the data [ 121 and are given in Table 1. In many aspects, predictions of the harmonic vibrator phonon model and the IBM are very similar. One way that they differ is in the B(n) values predicted for transitions between multiphonon states. In the pure U( 5) limit of the IBM, the B (E2)‘s are scaled relative to those of the phonon model due to the finite N effect (since the number of bosons, N, is a fixed, finite quantity in a given nucleus while the number of phonons Nt,t, is not fixed): B(m;nd =
--f
N-nd+l
N
nd
-
111~~
B(m; Nph--) Nph- 1)phonon
(2)
where nd is the number of d-bosons in the IBM. In Table 2, the experimental B (E2) values for transitions between the proposed multiphonon states [4,8] are compared to model predictions, whereby each successive model is of increasing sophistication. For the sake of completeness the 0; (the subscript refers to the ordering of the O+ levels as shown in Fig. 3) as well as 2; intruder state are also included. It can be seen from the experimental values that all multiphonon states, except the first two excited Of states but including the proposed 3:: and 4: 3-phonon states, decay to the next lower-lying multiplet by very collective transitions,
as predicted for a harmonic vibrator. Therefore, they have preserved to a large degree their collective character, despite the presence of nearby intruder states. In contrast, because of the strong mixing between the normal 2-phonon O+ and the ground state of the intruder configuration, the decay of the 0; state, previously labelled as a 2-phonon state based on the energy [4,8], shows an extreme deviation from its expected decay. Since the 2: state, which has been shown to be of almost pure intruder origin [ 51, decays strongly to both the 0; and 0; levels, this indicates that these two Of levels “loose their original character and thus cease to belong to a given configuration” [ 131. For the proposed 3+ and 4+ 3-phonon states [4,8], the data favour the surface-vibration interpretation since most transitions of interest turn out to be very collective; the experimental B( E2) values are significantly greater than the U(5) IBM prediction, and thus there is no clear-cut evidence for a finite N effect. However, it is clear that there is a major problem with the proposed 2’ member of the quintuplet (2:) which shows strong deviations in its expected decay pattern. The transitions to the 2+ and 4+ 2-phonon states do not exhibit significant collective enhancements; only the 2: --) 0; transition has a large B(E2) value (26 W.U.), followed by that of the 2: + 0; transition (5.7 W.U.). This decay pattern to the first two excited O+ states is consistent with their strong mutual mixing.
H. Lehmann et al./ Physics Letters B 387 (1996) 259-26.5
262 798.10
0.031
P(T) 1415.6
Table 2 Comparison of experimental and theoretical B( E2) values
t 0.019
4.d617.5
2’
Transition
1
1
L
798.00 -1.00 1447.10
1447.00
-0.50
0.00
0.50
1.00
[
: q
;; 2t 4; 0, 2: 2; 2+ 3 2:
7;; -+ 2i --) 2: -+2 -+ 0: --+ 2 +O
:
*
03
Exp. [ Wu.]
3$q6? 56(25)’ 61(S)” 0.012(l)” 0.3(1)0 0.3(2)’ 59(16)O 40(20)”
H. V.
Theory [Wu.] U(5) U(5)-O(6)
30.6 61.2 61.2 61.2 -
30.6 53.5 53.5 53.5 -
IBM-2
30.6 39.4 54.4 54.4 16.9 0.0 0.3 83 50
36.4 30.6 40.1 49.1 0.13 0.01 0.8 51 22
Proposed 3-phonon levels
0.083 1 769.10 - 1.00
F(T)
3 504.15
I
-0.50
-
2121.6
1504.05
1503.95 ’ -1.00
2081.8
0.046
0.00
t
t 0.035
4.d1415.6
1.00
0.50
T2’
1
:il..;l -0.50
0.50
0.00
-
0.003
0.422
3; ;; 4, 4; 4; 4; ;;
I;: 4; --+ -+2 ---t 2Jr ---f 4: - 2, I:$
24(9)b 62( 17) 0.28(S) 69121) 28(9) 48(14) “;!$;b)
26.2 65.6 48.1 43.7 :
49.2 19.7 36.1 32.8 :
20.6 52.2 0.003 38.2 34.4 3 0.0004c 0.03c
47.1 17.1 0.12gd 39.2 23.1d 23.4” 0.06 0.2
2t 2ft 2t 2:
--+ 04 12F * 4T + ot_
5.7(15) < 2.4 n.0. 26(7)
17.5 31.5 42.8
13.1 23.6 32.1
3.2c 9.2c 17.2“ 34.3c
6.34 2.73 0.13 45.9
” From Ref. [4]. b Calculated using the first value for S given in Table I. c Assumed to correspond to the fifth theoretical level. d Assumed to correspond to the second theoretical 4+state predicted at 1981.4 keV.
is concluded that there is no evidence for a pure 2+ 3-phonon state in “*Cd. In order to explain both the nature of the 2: level and the vibrational “purity” of the other 3-phonon states, the effect of the presence of the intruder configuration on the 3-phonon states must be considered. Since the phonon model cannot account for the intruder configuration, the experimental B (E2) values are compared with predictions obtained in a mixed IBM-l calculation [ 141 using the parameters of the U(5)-O(6) model [ 131 (see Table 2). This model allows for a description of normal states (by the U( 5) dynamical symmetry) interacting with the intruding configuration (described by the 0( 6) limit). For most states, the inclusion of the intruder states in the calculation has little effect on the B(E2) values. Only it
cos
-
4’
0.014
2’+617.5
1.7(5)b
1.00
0
Fig. 2. Measured y-ray energy as a function of cos6 for selected transitions. Noted is the F(T) value determined from a linear fit to the data.
Furthermore, the other decay branches suggest a large degree of fragmentation of the 2+ 3-phonon strength. The next possible candidate for the 2+ 3-phonon state, the 2f level at 2156.2 keV, shows an intense transition to the 2; intruder state [ 121 indicating that the 2f level has a sizable intruder component. Therefore,
H. Lehmann et al. / Physics Letters B 387 (19%) 259-265
the first two excited O+ states and the 2: state show a considerably different decay pattern. This selective behavior is a consequence of the influence of the O( 5) symmetry on the interaction between normal and intruder states, which explains why many states can preserve their multiphonon character despite the nearby intruder states [ 151. To visualize this, consider the schematic level scheme given in Fig. 4. Because of the O(5) symmetry, strong mixing can only take place between states having the same 0( 5) quantum number v and a small energy difference. For the 2-phonon states, this allows very strong mixing between the first and second excited O+ states, and their electromagnetic decays thus change appreciably (Table 2). The other 2-phonon states remain unaffected by close-lying intruder states due to the selective mixing and the separation in excitation energy. The same mechanism explains the purity of the 4; and 3: states. For the 2: level, the situation remains unclear. The 2: state is identified as corresponding to the fifth theoretical 2+ state in the U( 5) -O(6) calculation. There are two main reasons for this assignment: the small difference in excitation energy of the 2: and 2: state, both experimentally and theoretically, and the absence of a strong transition between the 25 and the 2: intruder state in the calculation. With this assignment the U(5)-O(6) model describes the fragmentation of the B( E2) strength to the two excited O+ states, but is unable to describe the decay to the other 2-phonon states. This seems to be a major failure of this description, as it is precisely these states that stay rather pure due to the selective mixing (see Fig. 4). However, it might also indicate that the 2: level contains a major component not considered in the model space. To investigate this, the last column of Table 2 lists the predictions of a configuration mixing IBM-2 calculation [ 161. The parameters of this calculation were obtained from a microscopic calculation or from fits to low-lying levels. The IBM-2 calculation improves the description of the decay of the 0; state and the 24’ state while retaining reasonable values for all other transitions. To learn why the IBM-2 provides a better description of the latter decay than the IBM- 1, the complicated wave function of this state obtained in the IBM-2 calculation must be considered. It consists approximately of three parts:
124’)x 17%]2$,) + 33%]2$
263
+ 50%(2;,,)
(3)
where the normal component - composed by the 3phonon and the first excited mixed-symmetry (ms) state (ms referring to the proton-neutron degree of freedom) - and the intruder component have the same probability. This strong mixing in the calculation arises from the fact that all components have Y = 1, and because the normal and the intruder states are separated by only 150 keV. This small energy separation is clearly a major deficiency of the calculation, since experimentally the energy difference is much greater (M 650 keV) . As the ]2:,,) component decays mainly to the 0’ states, this is the origin of the overestimation of the B( E2) values towards these states. The poor energy correspondence as well as the still remaining discrepancies (although smaller than in the other calculations) in the description of the B (E2) values in the IBM-2 may be an indication that other effects outside the model space of the IBM-2 (i.e., single particle excitations) contribute to the wave function of the 2: state, or that some fine tuning of model parameters may be necessary. However, the spirit of the present work was to test predictions of various model approaches, rather than just fit the data. Nevertheless, it is concluded that - although some problems remain - the IBM-2 gives the best description of the perturbed decay pattern and this is mainly due to the inclusion of a mixed symmetry component in the wave function of the 2: state. In conclusion, DSAM following the (n,n’r) reaction has been used to measure lifetimes of states in “‘Cd. The nature of vibrational excitations in l12Cd, considered as a typical spherical vibrator, has been investigated by examining the absolute E2 transition rates from 3-phonon states. From the measured transition probabilities, it is observed that collectivity remains intact at an excitation energy where quasipartitle degrees of freedom start to become important. There is no need to invoke large anharmonicities to explain the data [3,6] if the presence of the wellestablished intruder states is taken into account. The B( E2) values from the 3+ and 4+ members of the 3-phonon quintuplet show no evidence for finite N ffects, and the data favour the interpretation in the harmonic vibrator model. For the proposed 2+ 3-phonon member, however, the phonon picture fails, and only an IBM-2 calculation can qualitatively reproduce its
264
Ff. Lehmann et al./ Physics Letters 6 387 (1996) 259-265 2121.7 1504.0 897
1
1464.0 769 4
2’ 3 phonon
intruder
2 phonon
2’
1 phonm
0’
-
618
1
v=l
0
u=o
Fig. 3. Partial level scheme showing the 3-phonon states and their decays studied in the present work. The widths of the arrows correspond to the branching intensities. The 0( 5) quantum number Y for each level is also given.
U(5)
O(6)
W
O-
o+
Rg. 4. The possible mixing of normal and intruder states is shown. The thickness of the anwws indicates the strength of the interaction between the states.
strongly perturbed decay pattern. The improvement in the IBM-2 description compared to the other models can be related to a major component of the lowest mixed-symmetry state in the 2: wave function.
Quantitative discrepancies, however, remain and the description has still to be improved. At the excitation energy of the 3-phonon quintuplet, mixed symmetry states are excepted to be present, and thus a description in the IBM-I is not feasible. As the purity of the 3: and 4; states, as well as the mixing of the 22 level in the IBM-2 case, follow the U(5)-O(6) selection rules [ 151, it is concluded that these rules, derived in an IBM-l framework, appear to maintain their validity in the IBM-2. All this demonstrates the subtle interplay between multiphonon, mixed-symmetry and intruder states present in the Cd isotopes. This work was supported by the Swiss National Science Foundation, and by the U.S. National Science Foundation under grant PHY-9300077. H.L. wants to thank the group at the University of Kentucky for their hospitality during his stay. We also want to thank Tam& Belgya for advice and the use of his analysis codes. References [ I] A. Bohr and B. Mot&on,
K. Danske Vidensk. S&k. Mat-
H. Lehmann et al./ Physics Letters B 387 119961259-265
Fys. Medd. 27 No. 16 (1953). [2] F. Iachello and A. Arima, The Interacting Boson Model (Cambridge University Press, 1987). [3] C. Fahlander et al., Nucl. Phys. A 485 (1988) 327. [4] M. D&se et al., Nucl. Phys. A 554 (1993) 1. [5] R. Hertenbetger et al., Nucl. Phys. A 574 (1994) 414. [6] R.F. Casten et al., Phys. Lett. B 297 (1992) 19. [ 71 K. Heyde et al., Nucl. Phys. A 586 ( 1995) 1. [8] J. Kern et al., Nucl. Phys. A 593 (1995) 21. [9] K.B. Winterbon, Nucl. Phys. A 246 (1975) 293.
265
[lo] D. De Frenne et al., Nucl. Data Sheets 57 (1989) 443. [ 111 T. Belgya et al., in Pmt. of the 8th. Int. Symp. on Capture Gamma-Ray Spectroscopy and related topics, ed. J. Kern (World Scientific, 1994) 878. [ 121 PE. Garrett et al., to be published. [ 131 H. Lehmann and J. Jolie, Nucl. Phys. A 588 ( 1995) 623. [ 141 D. Kusnezov, The computer code Octupole ( 1987). [ 151 J. Jolie and H. Lehmann, Phys. Lett. B 342 (1995) 1. [ 161 M. Del&e et al., Nucl. Phys. A 551 (1993) 269.
PHYSICS
LElTERS
B
Physics Letters B 387 (1996) 259-265
On the nature of “three-phonon” excitations in ‘12Cd H.
Lehmann a~1,P.E. Garrett b, J. Jolie a, C.A. McGrath b, Minfang
Yeh b, S.W. Yates b
a lnstifur de Physique, Universire’ de Fribourg. Pe’rolles, CH-1700 Fribourg, Switzerland b University of Kenfucky,Lexington, KY 40506-0055, USA Received 4 June 1996; revised manuscript received 15 July 1996 Editor: R.H. Siemssen
Abstract The lifetimes of proposed 3-phonon states in *‘*Cd have been measured using the Doppler shift attenuation method (DSAM) following the (n,n’y) reaction. The experimental results allow the determination of relevant B( E2) values which are compared to different theoretical descriptions. Collective enhancements favouring the interpretation of collectivity in the phonon model are observed for 3+ and 4+ members of the 3-phonon quintuplet. However, for the proposed 2+ member, this simple model fails most likely due to the subtle interplay between intruder, mixed-symmetry,and phonon states, and
the decay pattern is strongly perturbed. The best overall agreement is obtained with 1BM-2calculations. PACS: 21.10R,T; 21.60E.F; 23.20 Keywords: 3-phonon states; DSAM; Lifetimes; B(E2)
values; Phonon model; IBM
Collective states in nuclei have been the subject of intense study since Bohr and Mottleson first proposed their collective hamiltonian [ 1I. An alternative way to describe these states is demonstrated by the Interacting Boson Model (IBM) [ 23 in which bosons, representing coupled valence-nucleon pairs, are used to describe the collective properties of nuclei. Often, this approach leads to similar predictions as the phonon model, although the underlying mechanism is different. While one would think that most questions regarding the nature of collective excitations would have already been answered, this is, in fact, not the case. The main difficulty encountered is the complex structure of excited nuclei. Thus, it is not clear, for example, how high in excitation energy, and hence into the region where single-particle degrees of freedom can become important, the collective states survive.
Recently, the level schemes of vibrational even-even nuclei near mass 100 have been developed into regions of increasing complexity (e.g. [ 3-S]) . Particularly, the combination of multiple Coulomb excitation, neutron capture and (a,xn) reactions to the Cd isotopes has been pivotal in the investigation of multiphonon states and shape coexistence due to intruding multi particle-hole excitations. Because both phenomena appear in the same nucleus at similar energies, their interference complicates the interpretation. For example, for 3-phonon states in ‘14Cd different interpretations have been invoked [ 3,6,7]. As illustrated in a recent survey [ 81, most proposed 3-phonon states are based on energy and spin systematics or are assigned on the basis of relative B(E2) values (essentially branching ratios). In order to assess their true nature, however, some measure of the collective character must be obtained. Often, only the
’ E-mail: [email protected]. 0370-2693/%/$12.00
Copyright 8 1996 Elsevier Science B.V. All rights reserved.
PII SO370-2693(96)01038-6
260
H. Lehmunn et al./Physics
determination of absolute transition probabilities between excited states allows one to obtain the wave function sensitive information needed. Especially in a complicated region, where interference effects can obscure the collective “fingerprints” [ 61, level interpretations and multiphonon assignments have to rely to a large part on absolute B(E2) values. The aim of the present work is to improve the understanding of the proposed [4,8] 3-phonon states in ‘l*Cd. Although the experimental knowledge of the level scheme of this nucleus is rather extensive [ 45 3, only a limited amount of experimental data exists for the lifetimes of excited states, and no lifetimes were available for states above 1.5 MeV excitation energy. Therefore, the absolute B(E2) values for transitions depopulating the 3-phonon quintuplet [4,8] have remained completely unknown. To remedy this situation, in measurements performed at the University of Kentucky Van de Graaff facility, the lifetimes of the proposed 3-phonon states have been measured using the Doppler shift attenuation method (DSAM) following inelastic neutron scattering (INS). Nearly monoenergetic neutrons with an energy of 2.5 MeV were produced with the 3H(p,n)3He reaction. This neutron energy was selected for several reasons: higher-lying levels which could possibly feed the 3-phonon states were not significantly populated, the population cross sections of the 3-phonon states were near their maximum values, and the resulting y-ray spectra were still relatively simple, thus allowing accurate fits of the peaks of interest. The scattering sample consisted of 50 g of “*Cd0 (98.17% isotopically enriched) contained in a cylindrical polyethylene container. By measuring the energy of the y rays as a function of angle, the experimental attenuation factor F(r)exp can be found from the relation E(6) = #??,(I + /3F(7)expCOSe>
(1)
where E(B) is the energy of the y ray measured at an angle B with respect to the direction of the incident neutron, E, is the unshifted y-ray energy, and /? = u/c represents the recoil velocity of the nucleus. The lifetime of the state is deduced by comparing the measured F( 7)exp values to those calculated theoretically using the Winterbon formalism [9]. In order to get a precise value for F( r)ext,, two experiments were performed with spectra recorded at 10 angles. One an-
Letters B 387 (1996) 259-265
150
250
1600
1700
I
350
450
1800
1900
,
550
650
750
.I
,
2000
2100
2200
EY (kev)
Fig. 1. y-spectrum at an angle of 90”. The prominent peaks from ‘12Cd are labelled with their energies.
gular distribution measurement employed a 57% efficient HPGe detector located N 1.1 m from the scattering sample, while the other used a 52% HPGe detector N 1.3 m from the sample. Each detector was placed inside a BGO anti-Compton shield, and time-of-flight gating was employed in order to reduce extraneous background events. The energy calibration of the detectors was continuously monitored through the use of radioactive source spectra collected concurrently with the in-beam spectra. Fig. 1 shows a typical spectrum, while in Fig. 2 the measured y-ray energy is plotted as a function of angle for selected transitions from each of the three states studied (see Fig. 3), as well as the 798 keV 4: --+ 2: transition. The half life determined for the 4; level from the data shown in Fig. 2 is 760 f 310 fs, in excellent agreement with the previously tabulated value of 900 zt 80 fs [ lo]. Consistent results for all well-resolved and moderately strong y rays were observed in the two angular distribution measurements. It should be noted that the quite small shifts are caused by the lifetimes of the states and not by the low recoil velocity. Table 1 lists the obtained lifetimes of the states. Further detaiIs of the experimental arrangement and the standard analysis procedure can be found elsewhere [ 11 I, and a complete analysis of all results will be presented in a forthcoming paper [ 121. In addition to the lifetimes, several hitherto unobserved or unplaced transitions from the 2+ state at 2 12 1 keV have been found (see Table 1) .
H. Lehmann et al./ Physics Lerrers B 387 (1996) 259-265
261
Table 1 Measuted lifetimes, relative intensities ( Ircr) and mixing coefficients (8) Jr
&lkevl
3f
2064.6
I
4f3
2081.8
2+ 4
2121.6
7 [fs]4
.Glkevl 648.9 752.2
0.133(2) 0.456(2)
1447.0 612.9 666.2 769.4 1464.0 688.2 809.1 897.1 1504.0 2121.7
0.411( 1) 0.081(3) 0.515(2) 0.368(2) 0.037 ( I ) 0.118(3) 0.024( 2) 0.096(2) 0.741(3) 0.020( 1)
680( 190)
500( 150)
740( 200)
-1 ’ I”.4 (-0.7M.1 -0.2 -0.6) -2 69+0.24
’ -0.16 _, ,58+0.10 -0.12 (-0 36+u.o4 -0.03 ) -0.4o+uo4 -0.03
d
1.36:$,
(0.24+0.03 ) -0.04
* Average of all observed deexciting transitions. b The branching ratios am determined from excitation function data recorded at 125’. c The first solution has the smaller x2. * A reliable value of S could not be determined.
In order to calculate the B( E2) values, the branching ratios and the S values have also been deduced from the data [ 121 and are given in Table 1. In many aspects, predictions of the harmonic vibrator phonon model and the IBM are very similar. One way that they differ is in the B(n) values predicted for transitions between multiphonon states. In the pure U( 5) limit of the IBM, the B (E2)‘s are scaled relative to those of the phonon model due to the finite N effect (since the number of bosons, N, is a fixed, finite quantity in a given nucleus while the number of phonons Nt,t, is not fixed): B(m;nd =
--f
N-nd+l
N
nd
-
111~~
B(m; Nph--) Nph- 1)phonon
(2)
where nd is the number of d-bosons in the IBM. In Table 2, the experimental B (E2) values for transitions between the proposed multiphonon states [4,8] are compared to model predictions, whereby each successive model is of increasing sophistication. For the sake of completeness the 0; (the subscript refers to the ordering of the O+ levels as shown in Fig. 3) as well as 2; intruder state are also included. It can be seen from the experimental values that all multiphonon states, except the first two excited Of states but including the proposed 3:: and 4: 3-phonon states, decay to the next lower-lying multiplet by very collective transitions,
as predicted for a harmonic vibrator. Therefore, they have preserved to a large degree their collective character, despite the presence of nearby intruder states. In contrast, because of the strong mixing between the normal 2-phonon O+ and the ground state of the intruder configuration, the decay of the 0; state, previously labelled as a 2-phonon state based on the energy [4,8], shows an extreme deviation from its expected decay. Since the 2: state, which has been shown to be of almost pure intruder origin [ 51, decays strongly to both the 0; and 0; levels, this indicates that these two Of levels “loose their original character and thus cease to belong to a given configuration” [ 131. For the proposed 3+ and 4+ 3-phonon states [4,8], the data favour the surface-vibration interpretation since most transitions of interest turn out to be very collective; the experimental B( E2) values are significantly greater than the U(5) IBM prediction, and thus there is no clear-cut evidence for a finite N effect. However, it is clear that there is a major problem with the proposed 2’ member of the quintuplet (2:) which shows strong deviations in its expected decay pattern. The transitions to the 2+ and 4+ 2-phonon states do not exhibit significant collective enhancements; only the 2: --) 0; transition has a large B(E2) value (26 W.U.), followed by that of the 2: + 0; transition (5.7 W.U.). This decay pattern to the first two excited O+ states is consistent with their strong mutual mixing.
H. Lehmann et al./ Physics Letters B 387 (1996) 259-26.5
262 798.10
0.031
P(T) 1415.6
Table 2 Comparison of experimental and theoretical B( E2) values
t 0.019
4.d617.5
2’
Transition
1
1
L
798.00 -1.00 1447.10
1447.00
-0.50
0.00
0.50
1.00
[
: q
;; 2t 4; 0, 2: 2; 2+ 3 2:
7;; -+ 2i --) 2: -+2 -+ 0: --+ 2 +O
:
*
03
Exp. [ Wu.]
3$q6? 56(25)’ 61(S)” 0.012(l)” 0.3(1)0 0.3(2)’ 59(16)O 40(20)”
H. V.
Theory [Wu.] U(5) U(5)-O(6)
30.6 61.2 61.2 61.2 -
30.6 53.5 53.5 53.5 -
IBM-2
30.6 39.4 54.4 54.4 16.9 0.0 0.3 83 50
36.4 30.6 40.1 49.1 0.13 0.01 0.8 51 22
Proposed 3-phonon levels
0.083 1 769.10 - 1.00
F(T)
3 504.15
I
-0.50
-
2121.6
1504.05
1503.95 ’ -1.00
2081.8
0.046
0.00
t
t 0.035
4.d1415.6
1.00
0.50
T2’
1
:il..;l -0.50
0.50
0.00
-
0.003
0.422
3; ;; 4, 4; 4; 4; ;;
I;: 4; --+ -+2 ---t 2Jr ---f 4: - 2, I:$
24(9)b 62( 17) 0.28(S) 69121) 28(9) 48(14) “;!$;b)
26.2 65.6 48.1 43.7 :
49.2 19.7 36.1 32.8 :
20.6 52.2 0.003 38.2 34.4 3 0.0004c 0.03c
47.1 17.1 0.12gd 39.2 23.1d 23.4” 0.06 0.2
2t 2ft 2t 2:
--+ 04 12F * 4T + ot_
5.7(15) < 2.4 n.0. 26(7)
17.5 31.5 42.8
13.1 23.6 32.1
3.2c 9.2c 17.2“ 34.3c
6.34 2.73 0.13 45.9
” From Ref. [4]. b Calculated using the first value for S given in Table I. c Assumed to correspond to the fifth theoretical level. d Assumed to correspond to the second theoretical 4+state predicted at 1981.4 keV.
is concluded that there is no evidence for a pure 2+ 3-phonon state in “*Cd. In order to explain both the nature of the 2: level and the vibrational “purity” of the other 3-phonon states, the effect of the presence of the intruder configuration on the 3-phonon states must be considered. Since the phonon model cannot account for the intruder configuration, the experimental B (E2) values are compared with predictions obtained in a mixed IBM-l calculation [ 141 using the parameters of the U(5)-O(6) model [ 131 (see Table 2). This model allows for a description of normal states (by the U( 5) dynamical symmetry) interacting with the intruding configuration (described by the 0( 6) limit). For most states, the inclusion of the intruder states in the calculation has little effect on the B(E2) values. Only it
cos
-
4’
0.014
2’+617.5
1.7(5)b
1.00
0
Fig. 2. Measured y-ray energy as a function of cos6 for selected transitions. Noted is the F(T) value determined from a linear fit to the data.
Furthermore, the other decay branches suggest a large degree of fragmentation of the 2+ 3-phonon strength. The next possible candidate for the 2+ 3-phonon state, the 2f level at 2156.2 keV, shows an intense transition to the 2; intruder state [ 121 indicating that the 2f level has a sizable intruder component. Therefore,
H. Lehmann et al. / Physics Letters B 387 (19%) 259-265
the first two excited O+ states and the 2: state show a considerably different decay pattern. This selective behavior is a consequence of the influence of the O( 5) symmetry on the interaction between normal and intruder states, which explains why many states can preserve their multiphonon character despite the nearby intruder states [ 151. To visualize this, consider the schematic level scheme given in Fig. 4. Because of the O(5) symmetry, strong mixing can only take place between states having the same 0( 5) quantum number v and a small energy difference. For the 2-phonon states, this allows very strong mixing between the first and second excited O+ states, and their electromagnetic decays thus change appreciably (Table 2). The other 2-phonon states remain unaffected by close-lying intruder states due to the selective mixing and the separation in excitation energy. The same mechanism explains the purity of the 4; and 3: states. For the 2: level, the situation remains unclear. The 2: state is identified as corresponding to the fifth theoretical 2+ state in the U( 5) -O(6) calculation. There are two main reasons for this assignment: the small difference in excitation energy of the 2: and 2: state, both experimentally and theoretically, and the absence of a strong transition between the 25 and the 2: intruder state in the calculation. With this assignment the U(5)-O(6) model describes the fragmentation of the B( E2) strength to the two excited O+ states, but is unable to describe the decay to the other 2-phonon states. This seems to be a major failure of this description, as it is precisely these states that stay rather pure due to the selective mixing (see Fig. 4). However, it might also indicate that the 2: level contains a major component not considered in the model space. To investigate this, the last column of Table 2 lists the predictions of a configuration mixing IBM-2 calculation [ 161. The parameters of this calculation were obtained from a microscopic calculation or from fits to low-lying levels. The IBM-2 calculation improves the description of the decay of the 0; state and the 24’ state while retaining reasonable values for all other transitions. To learn why the IBM-2 provides a better description of the latter decay than the IBM- 1, the complicated wave function of this state obtained in the IBM-2 calculation must be considered. It consists approximately of three parts:
124’)x 17%]2$,) + 33%]2$
263
+ 50%(2;,,)
(3)
where the normal component - composed by the 3phonon and the first excited mixed-symmetry (ms) state (ms referring to the proton-neutron degree of freedom) - and the intruder component have the same probability. This strong mixing in the calculation arises from the fact that all components have Y = 1, and because the normal and the intruder states are separated by only 150 keV. This small energy separation is clearly a major deficiency of the calculation, since experimentally the energy difference is much greater (M 650 keV) . As the ]2:,,) component decays mainly to the 0’ states, this is the origin of the overestimation of the B( E2) values towards these states. The poor energy correspondence as well as the still remaining discrepancies (although smaller than in the other calculations) in the description of the B (E2) values in the IBM-2 may be an indication that other effects outside the model space of the IBM-2 (i.e., single particle excitations) contribute to the wave function of the 2: state, or that some fine tuning of model parameters may be necessary. However, the spirit of the present work was to test predictions of various model approaches, rather than just fit the data. Nevertheless, it is concluded that - although some problems remain - the IBM-2 gives the best description of the perturbed decay pattern and this is mainly due to the inclusion of a mixed symmetry component in the wave function of the 2: state. In conclusion, DSAM following the (n,n’r) reaction has been used to measure lifetimes of states in “‘Cd. The nature of vibrational excitations in l12Cd, considered as a typical spherical vibrator, has been investigated by examining the absolute E2 transition rates from 3-phonon states. From the measured transition probabilities, it is observed that collectivity remains intact at an excitation energy where quasipartitle degrees of freedom start to become important. There is no need to invoke large anharmonicities to explain the data [3,6] if the presence of the wellestablished intruder states is taken into account. The B( E2) values from the 3+ and 4+ members of the 3-phonon quintuplet show no evidence for finite N ffects, and the data favour the interpretation in the harmonic vibrator model. For the proposed 2+ 3-phonon member, however, the phonon picture fails, and only an IBM-2 calculation can qualitatively reproduce its
264
Ff. Lehmann et al./ Physics Letters 6 387 (1996) 259-265 2121.7 1504.0 897
1
1464.0 769 4
2’ 3 phonon
intruder
2 phonon
2’
1 phonm
0’
-
618
1
v=l
0
u=o
Fig. 3. Partial level scheme showing the 3-phonon states and their decays studied in the present work. The widths of the arrows correspond to the branching intensities. The 0( 5) quantum number Y for each level is also given.
U(5)
O(6)
W
O-
o+
Rg. 4. The possible mixing of normal and intruder states is shown. The thickness of the anwws indicates the strength of the interaction between the states.
strongly perturbed decay pattern. The improvement in the IBM-2 description compared to the other models can be related to a major component of the lowest mixed-symmetry state in the 2: wave function.
Quantitative discrepancies, however, remain and the description has still to be improved. At the excitation energy of the 3-phonon quintuplet, mixed symmetry states are excepted to be present, and thus a description in the IBM-I is not feasible. As the purity of the 3: and 4; states, as well as the mixing of the 22 level in the IBM-2 case, follow the U(5)-O(6) selection rules [ 151, it is concluded that these rules, derived in an IBM-l framework, appear to maintain their validity in the IBM-2. All this demonstrates the subtle interplay between multiphonon, mixed-symmetry and intruder states present in the Cd isotopes. This work was supported by the Swiss National Science Foundation, and by the U.S. National Science Foundation under grant PHY-9300077. H.L. wants to thank the group at the University of Kentucky for their hospitality during his stay. We also want to thank Tam& Belgya for advice and the use of his analysis codes. References [ I] A. Bohr and B. Mot&on,
K. Danske Vidensk. S&k. Mat-
H. Lehmann et al./ Physics Letters B 387 119961259-265
Fys. Medd. 27 No. 16 (1953). [2] F. Iachello and A. Arima, The Interacting Boson Model (Cambridge University Press, 1987). [3] C. Fahlander et al., Nucl. Phys. A 485 (1988) 327. [4] M. D&se et al., Nucl. Phys. A 554 (1993) 1. [5] R. Hertenbetger et al., Nucl. Phys. A 574 (1994) 414. [6] R.F. Casten et al., Phys. Lett. B 297 (1992) 19. [ 71 K. Heyde et al., Nucl. Phys. A 586 ( 1995) 1. [8] J. Kern et al., Nucl. Phys. A 593 (1995) 21. [9] K.B. Winterbon, Nucl. Phys. A 246 (1975) 293.
265
[lo] D. De Frenne et al., Nucl. Data Sheets 57 (1989) 443. [ 111 T. Belgya et al., in Pmt. of the 8th. Int. Symp. on Capture Gamma-Ray Spectroscopy and related topics, ed. J. Kern (World Scientific, 1994) 878. [ 121 PE. Garrett et al., to be published. [ 131 H. Lehmann and J. Jolie, Nucl. Phys. A 588 ( 1995) 623. [ 141 D. Kusnezov, The computer code Octupole ( 1987). [ 151 J. Jolie and H. Lehmann, Phys. Lett. B 342 (1995) 1. [ 161 M. Del&e et al., Nucl. Phys. A 551 (1993) 269.