Mixed-Symmetry Interpretation of Some Low-Lying Bands in Deformed

7 Mixed-Symmetry Interpretation of Some Low-Lying Bands in Deformed Nuclei and the Distribution of Collective Magnetic Multipole Strength 1

2

2

3

O. Scholten, K. Heyde, P. Van Isacker, and T. Otsuka 1

Michigan State University, East Lansing, MI 48824 Institute for Nuclear Physics, Proeftuinstraat 86, B-9000 Gent, Belgium Japan Atomic Energy Research Institute, Tokai, Ibaraki-ken, Japan

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The effect of neutron-proton symmetry breaking on the distri bution of M1 strength in the SU(3) limit of the Interacting Boson Model (IBA-2) is studied. A possible alternative choice for the Majorana force is investigated, with a structure that resembles more closely that which is calculated in microscopic theories. It is found that the specific choice for the Majorana interaction has important consequences for the magnetic strength distribution function. In addition it allows for an alternative interpretation of the second excited Κ=0 band in rare earth nuclei, as a mixed -symmetry state. π

Experimental

+

e v i d e n c e i s a c c u m u l a t i n g [Boh84]

on t h e e x i s t e n c e and

p r o p e r t i e s o f s t a t e s i n deformed n u c l e i t h a t a r e n o t f u l l y s y m m e t r i c

i n the

n e u t r o n and p r o t o n degrees o f freedom.

mixed-

symmetry modes have been p r e d i c t e d Suz77]

The o c c u r r e n c e o f t h e s e

i n various

geometrical

models

[LoI79,

a n d i n t h e v e r s i o n o f t h e I n t e r a c t i n g Boson Model where n e u t r o n a n d

p r o t o n degrees o f freedom are e x p l i c i t l y

taken i n t o account (IBA-2) [ D i e 8 3 ,

Ari83] . In

t h e IBA-2 model [ A r i 8 3 ] , t h e s t r u c t u r e o f t h e c o l l e c t i v e

even-even

nuclei

i scalculated

by c o n s i d e r i n g

n e u t r o n and p r o t o n s a n d d b o s o n s .

states i n

a system o f i n t e r a c t i n g

We w i l l f o c u s a t t e n t i o n on t h e M a j o r a n a

force, M

= ξ (3 d ν 2

VÏÏ

- d s ) ν π

ÏÏ

-(s d - d s ) ν π ν π

-2

)

^

ξ. ( d d ) k ν ττ

«(d d ) ν ττ

,

which s e r v e s t o r a i s e t h e e x c i t a t i o n energy o f t h e s t a t e s t h a t a r e n o t f u l l y symmetric i n t h e n e u t r o n and p r o t o n degrees o f freedom. contains

three

parameters, £

a p p l i c a t i o n s , a s f o r example these

l f

ξ , and ξ . 2

given i n refs.

3

This Majorana

force

I n most p h e n o r n e n o l o g i c a l

[Van84 ,Sch85 ] , f o r s i m p l i c i t y ,

p a r a m e t e r s have been chosen equal, i n which case t h e Majorana 0097-6156/ 86/ 0324-0047S06.00/ 0 © 1986 American Chemical Society

Meyer and Brenner; Nuclei Off the Line of Stability ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

force

NUCLEI OFF T H E LINE OF STABILITY

48

p u s h e s up

t h e mixed-symmetry s t a t e s by an e q u a l amount, i . e . t h i s f o r c e i s a

t r u e Majorana f o r c e . While

for

the

ε,

κ

and

γ

parameters

extensive

microscopic

V , IT

calculations

exist

[ O t s 7 8 ,Sch83] , t h e m i c r o s c o p i c o r i g i n o f t h e

f o r c e i s at best only p a r t i a l l y there ξ

3

i s very l i t t l e ,

should

be

o r no,

chosen equal

parameters of

the

[Sch83»Dru85] .

understood

evidence [Dru85].

Microscopic

the

l

3

made a p h e n o m e n o l o g i c a l

force,

be

taken

ξ = 0 . i s more r e a l i s t i c .

3

by

The

two

d i f f e r e n c e between the i n the SU(3)

f o c u s our

of

the

observables

that

I ) , and

one

microscopic choices

using

some

detail

the

1 5 6

nucleus

Gd.

The

we

as

determined

can

(calculation

be

from

taken

II).

bandheads u s i n g

Even though

as

In

the

position

a = 0 . 1 5 MeV

F i g . 1 the

these

i n both

such

symmetry

two

of

the

1

(calculation

+

level I)

calculation 3 and

the Κ -1

( I ) the

choices

π

Κ «1

the

s t r e n g t h of

band l i e s n e a r Ε = 3 MeV,

+

χ

two +

calculations

π

+

calculation

and

discuss

the

Ε ^3

MeV

b = 0.3

MeV

near as

IBA-2

Majorana χ

e n e r g i e s of

the

force

spectrum of

the Majorana

1 5 6

are Gd.

force i s

the spectrum of mixed-

is totally

different.

In

band i s t h e l o w e s t mixed-symmetry band, w h i l e i n near

the

a l l o t h e r mixed-symmetry s t a t e s o c c u r

bandhead o r h i g h e r . (II) results

will

f o r the Majorana

( I I ) t h e l o w e s t mixed-symmetry b a n d l i e s Ύ band

of the Κ = 1

K^-ot

that

π

or

calculated excitation

different

calculations

states i n the

calculation

the

2

therefore

strength of the

c o m p a r e d w i t h some known b a n d h e a d s i n t h e e x p e r i m e n t a l

chosen

ξ =0.0

f o r c e i s most will

parameters of

h a m i l t o n i a n have been t a k e n f r o m r e f . [ S c h 8 0 ] . The

[Boh84],

Majorana

attention.

more

force,

the

[ S c h 8 5 ,Dru85] .

f o r the Majorana which

we

depend

and

l

calculations

l i m i t o f t h e IBA m o d e l o n

but

compare

b=ξ =ξ3

As an example o f a t y p i c a l d e f o r m e d r a r e e a r t h n u c l e u s we in

the

equal,

u s i n g the standard c h o i c e f o r the

(calculation

2

( c a l c u l a t i o n I I ) as suggested

intriguing

for

For t h i s reason

2

investigation

c a l c u l a t i o n s , one

ά=ξ =ξ =ξ 1

and

2

calculations

s t r o n g l y on t h e s t r u c t u r e o f t h e M a j o r a n a f o r c e . I n t h i s work we r e s u l t s o f two

ξ ,

ρ

I B A - 2 h a m i l t o n i a n , as r e p o r t e d i n r e f . [Dru85], i n d i c a t e

E, = e, =b a n d

choice

Majorana

particular

that the three parameters ξ

t h a t the parameters ξ i n the Majorana f o r c e s h o u l d not that

In

The

lowest

mixed-symmetry

position

at the π

Κ =0

+

band

i n e n e r g y v e r y n e a r an e x p e r i m e n t a l l y w e l l

band.


of

position in

known

7.

Mixed-Symmetry Interpretation

SCHOLTEN ET AL.

49 M SU-85-115

Ps 2

0

S

l

M

0

M

2

0 1

°S S 2

°M M !

3 MeV

2

•—

—

_—

—

—

—!

1

l 5 6

Gd

.

0

F i g . 1 A c o m p a r i s o n between c a l c u l a t e d bandhead e n e r g i e s and t h o s e o b s e r v e d in Gd. I n t h e c a l c u l a t i o n p r e s e n t e d on t h e l e f t - h a n d s i d e (calculation I ) t h e M a j o r a n a f o r c e i s p a r a m e t e r i z e d ( s e e t e x t ) by a=0.15 MeV w h i l e i n t h e c a l c u l a t i o n o n t h e r i g h t - h a n d s i d e ( I I ) b=0.3 MeV i s used. A b o v e an e x c i t a t i o n e n e r g y o f 2.5 MeV, t h e symmetry a s s i g n m e n t may n o t be a c c u r a t e due t o t h e F - s p i n n o n s c a l a r n a t u r e o f the h a m i l t o n i a n . Reproduced with permission from [Sch85]. Copyright 1985 American Physical Society. 1 5 6

In

this

section

symmetry K^^Og calculated

some o f t h e E2 decay p r o p e r t i e s o f t h e l o w - l y i n g

band a r e d i s c u s s e d .

E 2

- e Q IT

(2) Tf

and

Q

(2) are ν

the

phénoménologie c a l c u l a t i o n s boson

are

u s i n g the o p e r a t o r T

where Q

mixed

I n t h e IBA-2 m o d e l , E2 t r a n s i t i o n s

effective

charges

( 2 )

• e

π

p r o t o n and

one are

usually

Q υ ν

( 2 )

neutron quadrupole

operators.

assumes t h a t t h e n e u t r o n a n d

equal,

e =e

.

S h e l l - m o d e l , and

In

proton recent

Tf ν p h e n o m e n o l o g i c a l [ V e r 8 5 ] c a l c u l a t i o n s t e n d t o f a v o r a v a l u e o f 1.5e Se^ £2.0e . F o r t h i s r e a s o n we p r e s e n t i n t a b l e 1 some B ( E 2 ) v a l u e s f o r t h e ν mixed-symmetry s t a t e s , c a l c u l a t e d f o r two d i f f e r e n t c h o i c e s f o r t h e b o s o n

effective

charges,

e =2e and e =e . E v e n t h o u g h t h e n e u t r o n and p r o t o n π ν IT ν c h a r g e s d i f f e r by a f a c t o r 2, t h e t r a n s i t i o n s f r o m t h e Κ =0 π

boson e f f e c t i v e

3

+

band t o t h e o t h e r l o w - l y i n g

+

bands i s o n l y a f r a c t i o n o f t h e 2 + 0 1

and o f t h e o r d e r o f a few s.p.u.

The

transitions within

b u t somewhat weaker t h a n t r a n s i t i o n s w i t h i n a r e v e r y s i m i l a r t o what one

+

expects

t h e g.s.

transition

1

t h e band a r e s t r o n g

band. T h e s e

i n t h e t r a d i t i o n a l 2 qp

predictions

interpretation


50

NUCLEI OFF THE LINE OF STABILITY

T a b l e 1 . C a l c u l a t e d B ( E 2 ^ v a l u e s f o r v a r i o u s t r a n s i t i o n s i n v o l v i n g members o f t h e mixed-symmetry Κ = 0 band f o r t w o d i f f e r e n t c h o i c e s f o r t h e b o s o n e f f e c t i v e charges. π

Transition

2

1

+ 0

1

e »e π ν

100.

100. 0.86

0.36

MS

y

«

V

ο* °ι

y

0.85

1.11

2.13

3.38

0.005

0.12 0.05

0.10

y > 2

e =e /2 ν IT

70.86

78.15

MS* °MS

Source: Reproduced with permission from [Sch85]. Copyright 1985 American Physical Society. o f t h e s e bands.

boson e f f e c t i v e

charges the

t r a n s i t i o n s l e a d i n g t o t h e mixed-symmetry s t a t e s do n o t v a n i s h ,

indicating a

considerable neutron rupole

Even f o r t h e c a s e o f e q u a l

amount o f F - s p i n

and proton

bosons

breaking.

Since

l 5 6

in

(N =7, Ν =5) i s n o t e q u a l , π ν

Gd

t h e number o f

t h e quadrupole-quad-

i n t e r a c t i o n acting only between n e u t r o n s and protons

strong

F-spin

breaking

i n t e r a c t i o n matrix

term

i n the hamiltonian.

introduces

I n t h e present

e l e m e n t between t h e p u r e 0* a n d 0 *

s

a

case t h e

i s of the order of

200 k e V . Another important interaction

d i f f e r e n c e between t h e two c h o i c e s

s h o w s up i n t h e c a l c u l a t e d M1 s t r e n g t h d i s t r i b u t i o n .

IBA-2 m o d e l , t h e M1 t r a n s i t i o n o p e r a t o r

τ

Μ1

= / 3Ô75Î ( g

where g a n d g Tf ν of

f o r t h e Majorana

f

(d d

f

+ g (d d

Tf IT IT V a r e t h e boson g - f a c t o r s .

V

) V

'

On t h e b a s i s o f t h e c o l l e c t i v i t y

t h e b o s o n s one c a n a r g u e t h a t t h e s p i n c o n t r i b u t i o n s t o g

tially

c a n c e l and t h a t

therefore

In the

i s w r i t t e n as

the g-factors are equal

and g

essen-

to the orbital

nucléon g - f a c t o r s , g - 1.0 μ and g = 0.0 μ„ [Sam84]. π Ν ν Ν Μ

+

I n t a b l e 2 t h e Μ1 t r a n s i t i o n s t r e n g t h l e a d i n g t o t h e l o w e s t is

given.

level,

In calculation

I a l l strength

i s concentrated

as i s expected i n t h e pure S U ( 3 ) l i m i t

[Sch85] .

1

states

i n the f i r s t

1

+

In calculation I I

the strength i s d i s t r i b u t e d over s e v e r a l l e v e l s , r e s u l t i n g i n a considerably l o w e r s t r e n g t h f o r t h e 1* l e v e l . T h i s c o u l d

be p a r t

of the explanation of

t h e o b s e r v e d s p l i t t i n g o f t h e M1 s t r e n g t h i n e x p e r i m e n t [Boh8M].

I t i s also


7.

SCHOLTEN ET AL.

51

Mixed-Symmetry Interpretation 2

i n t e r e s t i n g t o note t h a t close

the experimentally

observed s t r e n g t h

o f 1.3 u

N

i s

to the prediction of calculation I I .

T a b l e 2. Some c a l c u l a t e d B(M1+) v a l u e s , i n u n i t s o f μ* , u s i n g 6 "lu™ a n d g =0. The d i f f e r e n c e between t h e two c a l c u l a t i o n s i s e x p l a i n e d i n \ h e ^ t e x t . II

I Ε

B(M1 )

E

2.43 0.00 0.00 0.00

3.0 3.8 4.1 4.3

X 3.1 4.2 4.3 4.4

In

this

contribution

B(M1 )

x

1 .69 0.55 0.05 0.00

we h a v e i n v e s t i g a t e d

t h e e f f e c t , on c a l c u l a t e d

o b s e r v a b l e s i n t h e IBA-2 m o d e l , o f a n a l t e r n a t i v e c h o i c e force, the

as suggested

peculiar feature

o f producing i n t h e SU(3) π

t h e r e a p p e a r s a mixed-symmetry Κ = 0 as

that

o f t h e β and Ύ bandheads.

appreciable be

tested,

f o rthe Majorana

by m i c r o s c o p i c c a l c u l a t i o n s [Dru85].

+

limit

T h i s c h o i c e has

a spectrum

i n which

band a t a p p r o x i m a t e l y t h e same e n e r g y In addition

this

s p r e a d i n g o f t h e c o l l e c t i v e M1 s t r e n g t h , f o r example i n i n e l a s t i c e l e c t r o n

gives

rise

t o an

a p r e d i c t i o n which can

scattering.

Acknowledgment s We w o u l d

l i k e t o thank Drs.

for helpful discussions. Science

A.E.L. D i e p e r i n k ,

S. P i t t e l ,

T h i s w o r k was s u p p o r t e d

and D. Warner

by t h e U.S.

National

F o u n d a t i o n u n d e r g r a n t n o . P H Y - 8 4 - 0 7 8 5 8 a n d by NATO g r a n t n o .

RG85/0036.

[Ari83] [Boh84]

[Die83] [Dru85] [LoI79] [Ots78]

References A. Arima and F. Iachello, Adv. in Nucl. Phys. 13, 139 (1983). D. Bohle et al., Phys. Lett. 137B, 27 (1984). D. Bohle et al., Phys. Lett. 148B, 260 (1984). C.W. de Jager, in Interacting Boson-Boson and Boson-Fermion Sys tems, p.225, ed. O. Scholten, (World Scientific, Singapore 1984) A.E.L. Dieperink, Prog. Part. Nucl. Phys. 9, 121 (1983). C.H. Druce, S. Pittel, B.R. Barrett and P.D. Duval, Phys. Lett. 157B, 115 (1985). N. Lo Iudice and F. Palumbo, Nucl. Phys. A326, 193 (1979). T. Otsuka, A. Arima and F. Iachello, Nucl. Phys. A309, 1 (1978). T. Otsuka in Interacting Boson-Boson and Boson-Fermion Systems, p. 3, ed. O. Scholten, (World Scientific, Singapore 1984)


NUCLEI OFF T H E LINE OF STABILITY

52

[Sam84] [Sch80] [Sch83] [Sch85] [Suz77] [Van84]

[Ver85] [Sch85]

M. Sambataro, O. Scholten, A.E.L. Dieperink and G. Piccitto, Nucl. Phys. A423, 333 (1984). O. Scholten, Ph.D. Thesis, Univ. of Groningen (1980) O. Scholten, Phys. Rev. C28, 1783 (1983). O. Scholten, K. Heyde, P. Van Isacker, J. Jolie, J. Moreau, M. Waroquier and J. Sau, Nucl. Phys. A438, 41 (1985). T. Suzuki and D.J. Rowe, Nucl. Phys. A289, 461 (1977). P. Van Isacker, K. Heyde, J. Jolie, J. Moreau, M. Waroquier and O. Scholten, Phys. Lett. 144B, 1 (1984). P.D. Duval, S. Pittel, B.R. Barrett and C.H. Druce, Phys. Lett. 129B, 289 (1983). J. Vervier, Contr. "XXIII International Winter meeting on Nuclear Physics", Bormio, Italy, 1985 O. Scholten, K. Heyde, P. Van Isacker, and T. Otsuka, Phys. Rev. C30, 1730, 1731 (1985).

RECEIVED May 19, 1986


Mixed-Symmetry Interpretation of Some Low-Lying Bands in Deformed

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