On the nature of low-lying electric dipole excitations in light and heavy deformed nuclei

Nuclear Physics A501 (1989) 95-107 North-Holland, Amsterdam

ON THE NATURE

OF LOW-LYING

IN LIGHT T. GUHR’,

K.-D.

AND HUMMEL,

HEAVY

ELECTRIC

DIPOLE

DEFORMED

G. KILGUS,

D. BOHLE*

EXCITATIONS

NUCLEI* and

A. RICHTER

Institut fiir Kernphysik der Technischen Hochschule, Darmstadi, 6100 Darmstadt, Fed. Rep. Germany C.W.

DE JAGER,

NIKHEF-K,

H. DE VRIES

and

P.K.A.

DE WITT

HUBERTS

P.O. Box 4395, 1009 AJ Amsterdam, The Netherlands Received

1 February

1989

Form factors from high-resolution inelastic electron scattering on 4RTi, Ih4Dy, 23ZTh and 23xU display the fact that the El transition to the lowest lying J” = l- states is excited by the same mode in light and heavy nuclei. A description of the form factor in terms of surface octupole vibrations of the nucleus around a quadrupole deformed shape is shown to work quite well up to its second maximum. The El strength found at the photon point is explained satisfactorily by taking into account the mixing with the electric giant dipole resonance.

Abstract:

NUCLEAR REACTIONS 48Ti(e, e’), ‘64Dy(e, e’), “‘Th(e, e’), 238U(e, e’), E,, = 20-220 MeV; 13= 85”, 90”, 95”, 117”, 120”, 141”, 154” and 165”. 48Ti, ‘64Dy, “‘Th, 238U deduced levels, J, T, B(El)t, transition form factors.

E

1. Introduction In heavy deformed nuclei fast El transitions have been observed [see e.g. refs. lm5) and references therein] between low-lying states. On the assumption that isospin can be nuclei. charge fast El

regarded as a local symmetry, Iachello “) proposed a cluster description of In this case the center-of-mass will generally not coincide with the center-ofthus generating a large dipole moment, which in turn induces the observed transitions. The fact that the light actinides are known to exhibit enhanced

a-decay probabilities has led to the identification 6*7)of these transitions as connected with a-clustering configurations. In these nuclei the a-clustering configuration may not constitute the ground state but might still be found at excitation energies below E, = 4 MeV [refs. ‘*“)I. Octupole models, however, introduced a long time ago by Bohr and Mottelson 93’o) and refined by Donner and Greiner ‘I), offer an alternative explanation for the fast El transitions in the light actinides. The interest in octupole models arose again l

Work supported

’ Present address: ’ Present

address:

in part by the Deutsche Forschungsgemeinschaft. MPI fiir Kernphysik, 6900 Heidelberg, Fed. Rep. Germany. DFVLR, 5000 Kiiln, Fed. Rep. Germany.

0375-9474/89/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

T Guhr

96

when

it was found

state also explain

deformed

dipole excitations

12-‘4) that static octupole the experimental

state will not be octupole mechanism

et al. 1 Electric

of octupole

deformed modes

deformations

observations.

Again,

of the nuclear

in stable nuclei

ground

the ground

so that in this case we can view the excitation

as an octupole

vibration

around

a quadrupole

nucleus.

In our search

for low-lying

J” = It states in deformed

nuclei

we discovered

very

strong El transitions at low momentum transfer in all of those nuclei which we all vanish studied at sufficiently low excitation energies 15-“). These El transitions towards the lowest momentum transfer. Thus they exhibit a behaviour very similar to that of the isoscalar AT = 0 electric dipole transitions in self-conjugate nuclei “,“). At the photon point these transitions are forbidden by the isospin selection rule. In the neighbourhood

of the photon

point the electron-scattering

form factor in plane-

wave Born approximation (PWBA) can be expressed by expanding the Bessel function ‘O). Since the transition is forbidden in the long wavelength limit the power series in qr starts with a term proportional to (qr)3 instead of the one proportional to qr. Therefore AT = 0 isoscalar El transitions to J” = l- states in self-conjugate nuclei have form factors that drop rapidly with decreasing (low) momentum transfer. In heavy nuclei proton and neutron numbers are no longer equal so that the isovector El transitions to low-lying J” = l- states of the same isospin as the ground state should not be hindered. Therefore we were very surprised that the form factors of the El transitions in heavy deformed and in self-conjugate nuclei displayed a striking similarity at small momentum transfer. Because of this similarity we will call the El transitions to J” = l- states of the same isospin as the ground-state isoscalar transitions by virtue of their form factor behaviour. We are thus led to believe that most of the El strength at the photon point is indeed concentrated in the giant resonance even when no isospin selection rule forbids transitions to low-lying states. Finite El transition strengths to low excitation energies at the photon point can be explained by an admixture of the giant resonance ‘I). Since an electron-scattering form factor has not yet been calculated in the framework of the a-clustering model we confine our discussion to the much terms of octupole vibrations of the nuclear surface.

simpler

analysis

in

The experimental method is explained in sect. 2. It is only briefly reviewed here since detailed reports are given elsewhere *l,**). In sect. 3 the transition density is derived in the octupole vibrational model and the results obtained using this density are then discussed in sect. 4. The conclusions are summarized in sect. 5.

2. Experiment The low-q electron-scattering experiment has been performed at the electron linear accelerator which is described in detail elsewhere *I). data have been taken at the Amsterdam accelerator discussed in ref. *‘). used in the present experiment were self-supporting foils of about

Darmstadt The high-q The targets 10 mg/cm*

T. Guhr et al. / Electric dipole excitations

thickness. and

99.3%

Darmstadt

The isotopic

were 99.1% (48Ti), 98.4% (164Dy), 100% (232Th)

(238U). A 169” double for the detection

QDD magnetic Fig. 1 shows The upper

abundances

97

focussing

magnetic

of inelastically

scattered

spectrometer

served the same purpose.

two electron

scattering

spectra

is taken in the first maximum

spectrometer electrons.

on 48Ti, measured

was used in

In Amsterdam

a

in Amsterdam.

of the form factor of the J” = l- state at

E, = 3.702 MeV, the lower in the second maximum. The level of interest is indicated by arrows. Close to a well-known J” = 2+ state at E, = 3.374 MeV a good candidate is found for the J” = 3- state of the Km = O- band at E, = 3.354 MeV. In fig. 2 two to the electron-scattering spectra on ‘64Dy are compared. The peak corresponding El excitation of the J” = l- state at E, = 1.675 MeV is marked by arrows. The level is seen to be strongly

excited

in both spectra.

Again,

the upper

spectrum

is taken

in the first maximum of the El form factor and the lower, measured in Amsterdam, in the second maximum. At E, = 1.757 MeV the transition to the J” = 3- state that is the member of the Km = O- band of which the J” = l- state is the band head can be found

easily

in both

spectra.

Under the kinematical conditions of the spectrum in fig. 2, also spectra on 232Th and 238U displayed in fig. 3 have been The excitation energy of the J” = l-, K * = O- band head which is marked by in fig. 3 is considerably lower in the actinides than in the rare-earth nucleus I

I

I

I

2.0

arrows 164Dy.

I

48Ti

(e,e’)

9

=1540

E,=

upper taken.

94MeV

f z

1.0

‘4

4

4

C 0

4

4 44

5 u E ;

0 JR=Z*

0.08

1

0’ 0

I

E,=171MeV

0.04

3.2

Excitation

3.4

3.6

Energy

3.8

(MeV)

Fig. 1. Inelastic electron scattering spectra taken on 48Ti in the first and second maximum of the isoscalar El-transition form factor. The excitation of the low-lying J” = l- state is marked by arrows.

98

T. Guhr

ef al. / Electric

I

dipole

excitations

c

I

I

1.8

2.0

164

Dy (e,e’)

1.6

Excitation Fig. 2. Inelastic electron scattering isoscalar El-transition form factor.

spectra taken The excitation

In case of 238U the strongly

excited

Energy

2. 2

(MeV)

on ‘64Dy in the first and second maximum of the of the low-lying J” = l- state is marked by arrows.

level not much

higher

in excitation

energy

is

again the J” = 3- state that is the member of the K ?i= O- rotational band. At E, ^- 1 MeV the transitions to the J” = 2+ members of the p- and y-rotational bands are found. In 7132Ththese levels lie close to the J” = 3- state of the K n = Op band at about 0.78 MeV and are not resolved.

3. Transition charges in the octupole vibrational model Octupole vibrations are treated ‘I) as phonons which couple strongly to the deformed quadrupole spheroid. Dipole transitions of the octupole state occur because of the admixture of the giant dipole resonance. In this section we want to show how to describe such a transition by the transition charge density. The transition charge density of the transition from the J” = O+ ground state to the J” = lP state, pr(r) = pi(r)+

a”&(r),

(3.1)

T. Guhr et al. / Electric dipole excitations

r

I

I

232Th

(e,e’)

238U

(e,e’)

+/

99

I

I

7

Eo= 40MeV

I

I

0.6

0.8

Excitation

I

I

1.0

1.2

Energy

(MeV)

Fig. 3. Comparison of forward-angle electron scattering spectra on “*Th and “* U. Both spectra are taken in the first maximum of the form factor of the isoscalar El excitations that are marked by arrows.

is a sum of an isoscalar

and an isovector

part. The isoscalar

density

pi(~) is caused

by the pure octupole vibration while the isovector density a’pr(r) is due to the admixture of the giant dipole resonance with an admixture coefficient a”. Note that the isovector part photon point.

is solely

responsible

for a non-zero

transition

strength

at the

In order to determine the isoscalar transition density pS( r), we follow a procedure described by Hirsch et al. 23-25) using the rotational model of the nucleus. If the intrinsic excitation is interpreted as an axial symmetric octupole vibration of small amplitude around the deformed equilibrium density p,,(r), the corresponding transition density is obtained by a Taylor expansion in terms of the vibration amplitude. Including center-of-mass corrections the EL transition density can be written as

pdr) = Y where

(I

ape,(r) dr

y and 1) are constants

Ym(~Q) Kc,(a) da+ common

T

ape&r) az

YLO(~) da

to all states in the band.

>

,

(3.2)

It is easy to show

100

T. Guhr

the importance cos s(a/ar) m

of the

would

orthogonality octupole

the

of the

equilibrium

integrand

contains

YLD. If the equilibrium

vanish

density

for L # 3 and the second

of the spherical

transitions

Electric dipole excitations

deformation

- (sin s/r)(a/as),

cos 6 and

integral

et al. /

without

harmonics. deformation

density: the

were radial integral

Therefore

Since

product

(a/az) = of

symmetric,

for L # 1 by virtue

it is impossible

of the equilibrium

Y10 = the first of the

to describe

the

density.

Here we use eq. (3.2) to calculate the isoscalar part p;(r) of eq. (3.1). In this case one can evaluate n by forcing the dipole transition strength to vanish, which is equivalent

to the condition 00 pi(r)r3dr=0.

(3.3)

I0 Integration

by parts yields 7) =

d3r

_~~P’“(‘)Y,o(4)Y,o(9)

5p,,(r) d3r

(3.4)

*

This equation displays the physical meaning of n in our special constant n is the normalized overlap of the equilibrium density

isoscalar case: The and the product of

the spherical harmonics Y,. and YJo. Obviously n would be equal to zero without deformation of the nucleus. In other words, n measures the influence of the deformation to the transition. The constant y is determined by adjusting the form factor calculated with the transition density pi(r) to the experimental form factor points of the E3 excitation to the J” = 3- state of the K” = Op band. A note on the equilibrium density used is in order. the low excitation

energy

the equilibrium

density

p,,(r)

We assume

of the excited

very much different from the ground-state density p,.,.(r). evaluate pi(r) in eq. (3.2) we used a deformed ground-state

p,.,.(r) =

that because

of

state is not

Therefore, in order to density of the form

Pg5.0

1+ex~{(r-c(l+~2Y~~(~)+~~y~~(~))l/~}

(3.5)

We took the deformation parameters pZ and p4 from the data tables. Further we assumed that the thickness t is the same as in the known spherical ground-state density. The parameter c was determined by the condition that the normalization of p,.,.(r) is the same as the normalization of the spherical ground-state density. Having fixed all parameters in the isoscalar density p:(r) we are left with the isovector part a’py(r) which is due to the admixture of the giant resonance. To calculate p;(r) we used the Tassie model 26). The admixture coefficient u” was determined by adjusting it to the experimental form factor points in the following way: Since u” is the only remaining free parameter in the theoretical form factor calculated with the total transition density (3.1) a” can be evaluated from the best

7-1Guhr et al. / Electric dipole excitations

fit to the data. Then the transition

strength

101

is given by

B(EI)T=Ilo’loSp:(~~~‘d~l*. In fig. 4 the calculated compared

isoscalar

with the isovector

Due to the condition case the corresponding

transition

El transition

(3.6)

density density

for 232Th as an example

taken

from the Tassie

is

model.

that the El transition strength has to vanish in the isoscalar transition density has a node in the interior of the nucleus.

I a-

I

I

I

I 5

I 10

15

4C

0

s

0.

f!

e

l-

-4

0

Radial Fig. 4. Isoscalar

transition

density

One can also evaluate above. In first-order given by ‘I)

I

Distance

(fm)

for 232Th (eq. (3.2)) and isovector model.

the Coulomb

perturbation

matrix element

theory

transition

matrix element

from the Tassie

(Hc) with the model described

the wave function

of the J” = l- state is

Izy.GDR>, where (Hc) is the Coulomb

density

(3.7)

and 1?Po,,)and E,,, are the wave function

and energy of the pure octupole vibrational state while IPGDR) and EGDR denote the wave function and energy of the K = 0 component of the giant dipole resonance. The transition

strength

from the J” = 0’ ground

state I ‘Pg.,.) is

B(EI)~=I(~l10(EI)II~~.,.)12, where O(E1) is the El transition operator. out above we find from eq. (3.7)

Since (!P,,,jl O(E1) )I!Pg.,.) = 0 as pointed

(&)=I&~-&DR~

B,BdR:lE);jf 7 J

with B(El)T

given by eq. (3.6).

(3.8)

(3.9)

102

T. Guhr et al. / Eiectric dipole excitations

4. Results and discussion The comparison between theory and experiment is shown in figs. 5-7. The data on 232Th and 238U that have been measured at high bombarding energies are the results of previous experiments at the Bates linear accelerator 23-25).The theoretical form factors are computed in distorted-wave Born approximation (DWBA) using the transition densities explained above. The curves for 48Ti and ‘64Dy are calculated for a scattering angle of @= 154”, those for 232Th and 238U for @= 117”. The data that have been measured at various scattering angles are recalculated to the one for which the theoretical form factor is shown in figs. 5-7. This is a standard procedure in electron scattering. The following assumptions enter in the application of the correction factor. First, we exclude electric transverse contributions to the form factor which should be small for the low excitation energies of the excited J” = lstates. Second, we point out again that in recalculating the experimental form-factor points with the help of a given theory one assumes that this theory is a reasonable approximation to reality. Correction factors that vary between 0.9 and 5.0 had to be applied to the data. This way of presenting the data has the striking advantage, however, that all measured data can be compared with a single curve, whereas without this procedure it would be neccessary to compare theory and experiment for each scattering angle separately. As a consequence of the model dependence, a smooth form factor as shown in the figs. 5-7 as a function of effective momentum

I

I

I

I

l-

48fi MeV

E, = 3.702

0 Dormstodt A Amsterdam 1

0

0.5

I

I

1.0

1.5

I

2.0

L 2.5

qc,,(fm-‘1 Fig. 5. Form factor of the El transition to the J” = l- state at E, = 3.702 MeV in ‘sTi. The experimental data agree well with the octupole vibrational form factor. The data have been normalized to 0 = 154”.

T. Guhr et al. / Electric dipole excitations I

I

103

I

I

164

Dy

10

+

.H

-4

(e,e')

E, :I.675

MeV

2

h

c -5 _

-5

IO

0

u . h

c w

-6

10

. :

0 Dormstadt

V

10

A Amsterdam

-7

Y

I

I 0.8

I ‘0.4

I 1.6

1.2

qeff (fm-‘1 Fig. 6. Form factor of the isoscalar El transition to the J” = I- state at .E, = 1.675 MeV in lh4Dy. The first two maxima of the form factor are well described by the octupole vibrational form factor. The data have been normalized to 0 = 154”.

I

I

I

I

I

238”

232Th E,=

I

0.718MeV

Ex=0.680MeV

0

Dormstadt

Ll MIT

I

0

0.5

1.0

1.5 ciefr

0

0.5

I 1.0

I

1.5

(f m-l 1

Fig. 7. Comparison of the electron scattering form factor to the lowest lying J” = l- state in the actinide nuclei 232Th and ‘%. The octupole vibration theory yields for Z3ZTh a satisfying description in the first two maxima, in the case of ?J the agreement is not so good. The data have been normalized to 0 = 117”.

T. Guhr et al. / Electric dipole excitations

104

transfer

is only

the

physical

picture

It is clear from figs. 5-7 that the octupole

vibration

model yields a good description

assumptions

obtained

is essentially

of the first two maxima four nuclei.

whenever

the model

right. of the form factors

of the excitation

If a Tassie model from factor is used instaed,

can not be described

underlying

of J” = l- states in all

already

the first maximum

correctly.

As noted in the introduction the form factor of a pure isoscalar El transition already explains the measured form factors rather well. We have improved on this description by allowing for a small isovector El admixture in the transition density that has been determined shown in figs. 5-7 include The upper

with the procedure described these very small admixtures.

limits for El strengths

present approach small and depend The upper limits our data with the which gives zero

and Coulomb

matrix

above.

The form factors

elements

deduced

in the

are summarized in table 1. The El strengths obtained are all rather strongly on the explicit form assumed for the transition density. of table 1 are derived in the following way: When we compared form factor calculated with the isoscalar transition density alone, B(El)t values, the deviation from the best description possible

was already small. In the case of 232Th for example the 2 was about 2. Allowing for an isovector admixture improved the adjustment a little. Increasing the admixture beyond that corresponding to the B(El)f value listed in the table led to a description that grew rapidly worse. For example the B(El)T value for 232Th listed in the table corresponds to a 2 of about 2.5. TABLE 1 Upper limits of the B(El)f

and an estimate

Nucleus

4”Ti ‘64Dy z3q.h 238”

values and Coulomb matrix elements (H,) for upper limits of the spreading widths r deduced from the present experiment

B(El)T [ ez fm’] <2x

10-J

(25 x 10-3
<353 1258 115 < 170

<53 ~32 13 <14

Because of the procedure explained above, it is clear that our results are really upper limits and exclude transition strengths that are larger than those listed in the table. The same is true for the Coulomb matrix elements calculated with eq. (3.9). Following ref. 27) however, we can show that the upper limits for the Coulomb matrix elements given in table 1 are not too far from what one should expect: in the statistical analysis of ref. *‘) it is argued that the spreading width, l-=2rrH;/D,

(4.1)

T Guhr et al. / Electric dipole excitations

lies between mean

level

10 and spacing

100 keV in all nuclei and

i?$

for which

is the average

Although

our analysis

involving

resonance

is obviously

not statistical,

data are available.

squared

only the octupole

105

Coulomb

vibration

we try to estimate

matrix

D is the element.

state and the giant

the spreading

width

by

taking

(Hc)’ for 3 and (E,,, - EGDRl instead of the mean level spacing D. The results for r are also given in table 1, they are quite reasonable, only the spreading

width for 232Th seems to be too small. The values, which have been derived

in this work, also agree with those of other

experiments 28-30). In an early (‘y, 7’) experiment where no spin assignment was possible for the E, = 3.702 MeV state in 48Ti a width was measured that corresponds to B(El)T=l.l x 10m3 e* fm’. In ref. 29) a value for the El strength to the J” = lstate at E, = 680 keV in 238U is given as B(E1)?=(0.75-2.0) x 10m4 e* fm* which is much smaller than the upper limit that we are able to give. Those smaller transition strengths lead according to eq. (3.9) also to smaller Coulomb elements, i.e. (&) q 262 keV for 48Ti and to (Hc) = 17-29 keV for 238U.

B(El)T matrix

Furthermore, the upper limits for the El strength found is in good agreement with what has been expected on theoretical grounds in the framework of the octupole vibration model ‘l) where El strengths of some low3 e* fm* are easily explained as due to the admixture of the giant El resonance. For a more detailed comparison more experimental

data have to be accumulated.

5. Summary High-resolution

inelastic-electron-scattering

form

low-lying K, J” = 0, l- states have been measured mental data are all easily explained in a simple strength

to the low-lying

J” = l- state is treated

factors

of El

transitions

to

with high accuracy. The experitwo-state model where the El

as due to the admixture

of the El

giant resonance in the wave function of the low-lying state. The resulting values for the El strengths and Coulomb matrix elements are upper limits. In a systematic study of excitation energies and transition strengths of low-lying negative-parity states in rare-earth and actinide nuclei Cottle and Bromley 3’) point out that all presently available data can be explained in the framework of octupole vibrational models. We have shown here that this model also yields a good description of the form factors Of,,, nucleus

of “isoscalar” El transitions 48Ti, the rare-earth nucleus

to low-lying

J” = l- states in the light nuclei 232Th and

‘64Dy and the actinide

238U, thus encompassing nuclei from three different mass regions. Lastly, the upper limit on the El strength found in the present experiment corresponds only to a very small fraction of the molecular sum rule 32), therefore, even disregarding the strong evidence for the octupole vibrational character of the El excitation of the lowest J” = l- states in deformed nuclei El strength connected with cluster states has to be looked for at higher excitation energies. This fact has also been stated recently by Iachello 6, for heavy nuclei and by Assenbaum et al. 33)

T. Guhr et at. / Electric dipole exciiarions

106

for light nuclei where the cluster states are expected Experimental

verification,

We would

like to thank

on 164Dy and

H. J. Daley,

however,

is presently

K. Alrutz-Ziemssen F. Iachello

One of us (A.R.) had an interesting

around

still lacking. for his help with the measurements

and P. Manakos

conversation

4 or 14 MeV, respectively.

for valuable

with W. Greiner

discussions.

who focussed

his

attention onto the model of ref. I’). Part of this work belongs to the research program of the National Instituut voor Kernfysica en Hoge Energie Fysica (NIKHEF, Sectie K), which is financially supported by the Stichting voor Fundamentel Onderzoek der Materie

(FOM)

Onderzoek

(NWO).

and the Nederlandse

Organisatie

voor Zuiver Wetenschappelijk

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