STRENGTH FUNCTION OF GIANT VIBRATIONS

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STRENGTH FUNCTION OF GIANT VIBRATIONS

P. Bortignon, R. Broglia, Xia Ke-Ding

To cite this version:

P. Bortignon, R. Broglia, Xia Ke-Ding. STRENGTH FUNCTION OF GIANT VIBRATIONS. Journal

de Physique Colloques, 1984, 45 (C4), pp.C4-209-C4-220. �10.1051/jphyscol:1984416�. �jpa-00224081�

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JOURNAL DE PHYSIQUE

Colloque C4, supplCment a u n03, Tome 45, mars 1984 page C4-209

STRENGTH FUNCTION OF GIANT VIBRATIONS

P .F. Bortignon, R.A. ~ r o g l i a * and Xia ~e-ding**+

Dipartimento d i Fisica GaliZeo G a l i l e i , Universitd d i Padova, Padova and I . N . F. N., Laboratori Nazionali d i Legnaro, I t a l y

*'The NieZs Bohr I n s t i t u t e , University of Copenhagen, DK-2100 -Copenhagen Q, Denmark

**I. N . F. N., Laboratori Nazionali d i Legnaro, I t a l y

R6sum6.

-

On rgsume l a s i t u a t i o n a c t u e l l e d e s c a l c u l s c o n c e r n a n t 1 ' 6 -

nergie p r o p r e d e s 6 t a t s de p a r t i c u l e e t d e s r 6 s o n a n c e s g 6 a n t e s en m e t t a n t en gvidence l e s probl2mes non r e s o l u s .

A b s t r a c t .

-

The p r e s e n t s t a t u s of t h e c a l c u l a t i o n of t h e s e l f - e n e r g y of s i n g l e - p a r t i c l e s t a t e s and g i a n t r e s o n a n c e s i s b r i e f l y reviewed w i t h emphasis on t h e open problems.

1 .

-

I n t r o d u c t i o n .

The s t u d y of t h e n u c l e a r r e s p o n s e h a s w i t n e s s e d a remarkable pro- g r e s s d u r i n g t h e l a s t few y e a r s ( c f . e . g . ( 1 ) and r e f s . t h e r e i n ) . New modes of e x c i t a t i o n have been i d e n t i f l e d and a s i m p l e d e s c r i p t i o n ex- i s t s . I t i s based on t h e random phase approximation w i t h t h e s e l f - -energy c o r r e c t i o n s a r i s i n g from t h e i n e l a s t i c c o l l i s i o n s of p a r t i c l e s w i t h t h e n u c l e a r s u r f a c e ( . 2 ) . I n t h e p r e s e n t c o n t r i b u t i o n t h e s t a n d a r d model i s reviewed.

2 .

-

S t r e n g t h f u n c t i o n .

A c o n v e n i e n t framework f o r t h e d i s c u s s i o n of t h e r e s p o n s e func- t i o n i s p r o v i d e d by t h e s t r e n g t h f u n c t i o n . I t g i v e s t h e p r o b a b i l i t y t o f i n d a s t a t e 5 p e r u n i t energy and i s w r i t t e n a s ( 3 )

The energy s h i f t AEa(E) i s d e f i n e d a s

The q u a n t i t y T a ( E ) r e a d s

The "doorway" s t a t e s h a v e energy E and c o u p l e t o t h e s i m p l e s t a t e a t h r o u g h t h e m a t r i x element V

.

The p a r a m e t e r C1 A r e p r e s e n t s t h e e n e r -

-

gy i n t e r v a l around E o v e r whicgathe a v e r a g e s a r e c a r r i e d o u t . I t t a k e s i n t o a c c o u n t , i n some a v e r a g e way, t h e c o u p l i n g of t h e doorway s t a t e s t o more c o m p l i c a t e d c o n f i q u r a t i o n s .

+

Permanent a d d r e s s : I n s t i t u t e of Nuclear Research, Shanghai, China.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984416

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C4-210 JOURNAL DE PHYSIQUE

I n t h e e n e r g y i n t e r v a l EiSESEf t h e c e n t r o i d of t h e s t r e n g t h f u n c - t i o n p ( E ) i s d e f i n e d a s

a 4)

i Ei

The damping w i d t h :I of t h e s t a t e 5 i s p r o p o r t i o n a l t o t h e s t a n d a r d d g v i a t i o n

For a Gaussian d i s t r i b u t i o n ~:-2.40.

The s t r e n g t h f u n c t i o n 1 ) c a n d i s p l a y a marked s t r u c t u r e . I t r e - f l e c t s t h e s t a t e - d e p e n d e n c e of t h e c o u p l i n g m a t r i x e l e m e n t s and t h e f a c t t h a t T a ( E ) c a n become v e r y s m a l l due t o s m a l l v a l u e s o f t h e den- s i t y of s t a t e s .

I n t h e Hartree-Fock t h e o r y of t h e n u c l e u s t h e s t r e n g t h f u n c t i o n a s s o c i a t e d w i t h a s i n g l e - p a r t i c l e w i l l d i s p l a y a s h a r p peak. When t h e r e s i d u a l i n t e r a c t i o n i s s w i t c h e d on t h e s i n g l e - p a r t i c l e s t r e n g t h becomes d i s t r i b u t e d o v e r a wide e n e r g y i n t e r v a l and t h e a s s o c i a t e d cen- t r o i d moves away from t h e o r i g i n a l u n p e r t u r b e d p o s i t i o n ( c f . f i g . 1 ) .

F i g . 1 .

-

S i n g l e - p a r t i c l e s t r e n g t h f u n c t i o n P ( E ) i n t h e Hartree-Fock t h e o r y ( u p p e r p a r t ) and t a k i n g i n t o a c c o u n t t h e r e s i d u a l i n - t e r a c t i o n ( l o w e r p a r t ) . The q u a n t i t y Ea d e n o t e s t h e c e n t r o i d e n e r g y measured from t h e Fermi e n e r g y EF.

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The c o u p l i n g t o c o l l e c t i v e d e g r e e s of freedom l i k e s u r f a c e v i b r a t i o n s ( c f . f i g . 2 a ) d e t e r m i n e s t h e p r o p e r t i e s o f t h e main p e a k . The c o u p l i n g t o n o n - c o l l e c t i v e d e g r e e s of freedom ( c f . f i g . 2b) h a s two main e f - f e c t s , t o s h i f t t h e c e n t r o i d and t o p r o v i d e t h e s t r e n g t h f u n c t i o n w i t h a l o n g t a i l .

F i g . 2.

-

B a s i c c o u p l i n g s between s i n g l e - p a r t i c l e ( - h o l e ) s t a t e s and c o l l e c t i v e v i b r a t i o n s which c o n t r i b u t e t o t h e s e l f - e n e r g i e s of s i n g l e - p a r t i c l e s ( - h o l e s ) and o f g i a n t v i b r a t i o n s . A l i n e marked by a n a r r o w p o i n t i n g u p r e p r e s e n t s a p a r t i c l e , w h i l e one p o i n t i n g down r e p r e s e n t s a h o l e . A wavy l i n e s t a n d s f o r a v i b r a t i o n , w h i l e a d a s h e d l i n e r e p r e s e n t s t h e two-body i n - t e r a c t i o n .

I n f i g . 3 w e show r e s u l t s o f c a l c u l a t i o n s f o r d e e p - h o l e s t a t e s i n ' 0 8 p b . The c o u p l i n g s o f t y p e 2a t o s u r f a c e v i b r a t i o n s h a v e b e e n t a k e n i n t o a c c o u n t ( 4 ) . B e c a u s e b o t h c o l l e c t i v e a n d n o n - c o l l e c t i v e s o l u t i o n s o f t h e random p h a s e a p p r o x i m a t i o n were i n c l u d e d , most of t h e c o n t r i b u - t i o n s o f t h e s p i n - and i s o s p i n - i n d e p e n d e n t components of t h e r e s i d u a l i n t e r a c t i o n a r i s i n g from t h e p r o c e s s 2b a r e a l s o - t a k e n i n t o a c c o u n t .

The t h e o r e t i c a l s t r e n g t h f u n c t i o n s d i s p l a y t h e main o b s e r v e d p r o - p e r t i e s . The c e n t r o i d i s however p r e d i c t e d a p p r o x i m a t e l y 1 MeV t o o h i g h . I t w i l l b e i m p o r t a n t t o s t u d y w h e t h e r c o u p l i n g s of t h e t y p e 2b t o t h e t e n s o r and t h e s p i n - and i s o s p i n - d e p e n d e n t p a r t s o f t h e n u c l e a r f o r c e c a n remove t h i s d i s c r e p a n c y w i t h o u t p r o d u c i n g m a j o r c h a n g e s i n t h e s t r u c t u r e of t h e main p e a k .

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J O U R N A L DE PHYSIQUE

ENERGY I MeV) ENERGY (MeV)

Fig. 3 .

-

Adapted from ref. (5). On the left side the normalized experimental spectrum (smooth full drawn curve) extracted (5) from the analysis of the reaction 2 0 8 ~ b ( 3 ~ e , a ) 2 0 7 ~ b at 283 MeV incident energy and at the angle 8=2O is compared with theoretical predictions. The full drawn curve strongly peak- ed at r10 MeV is the result of the calculation discussed in the text. The solid, dotted and dashed bars show the contributions of the hole states lh11/2, 19712 and lg9/2 respectively, as calculated in ref. ( 6 ) . The energy scale for the theoretical predictions have been shifted downward by 1.2 MeV for the full line and 2.0 MeV for the bars. On the right hand- -side, the partial contributions to the theoretical spectrum discussed in the text are displayed. The main contributions arise from the Ihll/2 ( - - - ) , 1g7/2 ( - . - . - . I ! ^1g9/2( . - . . . ) and 2f5/2 (xxxx) hole states. We thank E. Magllone for help in the preparation of the figure.

4.

-

Giant resonances.

A giant resonance is a coherent superposition of particle-hole excitations. The damping of these modes arise from the coherent decay of the particle and of the hole according to the processes shown in figs 2c

-

2f.

Calculations making use of the coupling 2c and 2d have been carried out for a variety of giant resonances and nuclei ( 4 , 7 ) . Both col lective and non-collective roots of the random phase approximation have been included in the intermediate states and a discussion similar to that carried out in the previous section applies. Results are displayed in fig. 4 in comparison with the experimental data (8). Strong cancellations among the contribution 2c and 2d to Va, are observed (9).

Examples are shown in fig. 5. It is noted that the largest contributions are connected with the presence of the low-lying octupole vibration in the intermediate states. The experiments on the gamma-decay of the giant resonances, as discussed also at this Conference (101, will test this feature of the theory.

The width of the peaks are well reproduced. The centroid is however predicted for the density vibrations somewhat too high and in general the percentage of the EWSR observed is smaller than calculated. Also strength is observed at energies few MeV below the energy at which the calculated strength function becomes essentially zero.

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Fig. 4.

-

Comparison of the experimental data ( 8 ) with the theoreti

-

cal results ( 4 , 5 ) for the centroid energy, the spreading width and the fraction of sum rule exhausted by the Giant Quadrupole Resonance, the Giant Octupole Resonance and the Gamow-Teller Resonance of some closed-shell nuclei.

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J O U R N A L DE PHYSIQUE

F i g . 5.

-

Examples o f c a n c e l l a t i o n between t h e c o n t r i b u t i o n s shown i n f i g , 2c and 2d t o t h e m a t r i x ,,V a s s o c i a t e d w i t h t h e coup- l i n g of t h e GQR, GDR and GTR of 2 0 8 ~ b t o 2p-2h s t a t e s con- t a i n i n g a c o l l e c t i v e s u r f a c e v i b r a t i o n ( 9 )

.

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5.

-

Open problems.

The isoscalar giant quadrupole vibration of 208??b has been studied through electron and hadron inelastic scattering and, recen- tly, also in coincidence experiments. In what follows, we comment shortly on the various results to give an indication of some of the open problems.

In the high resolution electron scattering experiment of ref.

( I ? ) , due to the low momentum transfer selected by the experimental set-up, it is expected that no other multipolarity than L=2 be excit- ed. In the energy interval between 8 and 12 MeV 54 peaks have been observed. The associated width of each of them is $ 5 0 keV, which is the energy resolution (cf. fig. 6).

8 9 io i I 12 - E (MeV)

E ( MeV)

Fig. 6.

-

In the upper part of the figure the quadrupole strength distribution for '"pb obtained in the (e,e') (11) and (a,a'n)

(12) experiments discussed in the text are displayed. In the lower part, the theoretical strength function calculated in ref. (4) is shown. Note the difference between the energy scales of the two parts of the figure.

The theoretical strength function in fig. 6 seems to display many of the observed structures when use is made of a very small averaging parameter A=50 kev (4). It is noted that the spectrum is in average predicted = I MeV too high. Also the amount of observed strength is only half of that theoretically expected.

The bum observed in the excitation spectrum of the

208Pb(a.a1)2g8Pb ( G Q ) reaction at E =10.2 MeV ( 1 3 ) can be well fitted with a Gaussian function displaying a width of 2 2 . 4 MeV and exhausting about 75% of the EWSR. The lack of structure can hardly be ascribed

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C4-2 16 JOURNAL DE PHYSIQUE

t o t h e e x p e r i m e n t a l r e s o l u t i o n which i s 1 5 0 keV.

I n t h i s c a s e s t a t e s w i t h m u l t i p o l a r i t i e s d i f f e r e n t from L = 2 can be e x c i t e d . I n f a c t t h e spectrum shows two broad bumps ( c f . f i g . 7 ) . One which i s a s s o c i a t e d w i t h t h e g i a n t q u a d r u p o l e v i b r a t i o n . A second which c o r r e s p o n d s t o t h e g i a n t monopole v i b r a t i o n . The t h e o r e t i c a l spectrum ( 1 4 ) i n t h e e x c i t a t i o n r e g i o n between 9 and 1 6 MeV c a l c u l a t e d w i t h t h e a v e r a g i n g p a r a m e t e r A = 1 MeV a p p a r e n t l y reproduce b o t h t h e shape and w i t h i n 2 0 % t h e a b s o l u t e magnitude o f t h e experdental one a f t e r background s u b t r a c t i o n . The observed c e n t r o i d i s i n t h i s c a s e -1 MeV h i g h e r t h a n p r e d i c t e d . However, t h e c a l c u l a t e d s t r e n g t h f u n c t i o n con- t a i n s c o n t r i b u t i o n s a r i s i n g from t h e p r e s e n c e of L = 4 and L = 6 v i b r a - t i o n s ( c f . f i g . 7 ) . I f t h e s e c o n t r i b u t i o n s a r e removed t h e a b s o l u t e magnitude of t h e r e s u l t i n g spectrum which c o n t a i n s ~ 7 0 % of t h e qua- d r u p o l e EWSR and 9 0 % of t h e monopole EWSR i s a b o u t a f a c t o r of 2 s m a l l e r t h a n t h a t o b s e r v e d .

E (MeV)

--

F i g . 7 .

-

I n t h e upper p a r t t h e e x p e r i m e n t a l ( 1 3 ) and t h e o r e t i c a l ( 1 4 ) s p e c t r a a t 8=11° f o r 152 MeV a - s c a t t e r i n g from '''~b a r e d i s p l a y e d . I n t h e lower p a r t , t h e p r e d i c t e d c o n t r i b u t i o n s of t h e L = 2 ( - - - ) , L = 4 ( - . - . - ) , L=6 (-.

.-.

^{. - )} ^{and L=O} ^{( .}

. . .

^.)

m u l t i p o l a r i t i e s a r e shown.

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Recent d a t a ( 1 5 ) have been p u b l i s h e d which c o n f i r m t h e t h e o r e t i c a 1 p r e d i c t i o n s ( c f . f i g . 7 and t a b l e I ) , in p a r t i c u l a r , t h e f a c t t h a t t h e s t r e n g t h distributions a r e v e r y f a r from having Gaussian s h a p e s . The experiment was c a r r i e d o u t a t somewhat h i g h e r e n e r g y t h a n t h e p r e v i o u s o n e . The main d i f f e r e n c e however was t h a t d a t a were t a k e n a t v e r y s m a l l a n g l e s , where t h e c o n t r i b u t i o n of t h e d i f f e r e n t m u l t i p o l a - r i t i e s can b e e x t r a c t e d w i t h more c o n f i d e n c e .

T A B L E

1

T a b l e 1.

-

Comparison of t h e e x p e r i m e n t a l d a t a ( 1 5 ) w i t h t h e t h e o r e t i c c a l r e s u l t s ( 4 ) f o r t h e c e n t r o i d , t h e s p r e a d i n g width and t h e f r a c t i o n of EWSR e x h a u s t e d by i s o s c a l a r g i a n t v i b r a - t i o n s i n 2 0 8 ~ b . The RPA r e s u l t s of r e f . ( 1 6 ) have been used f o r t h e i s o s c a l a r d i p o l e s t a t e s a s i n p u t of t h e s p r e a d i n g c a l c u l a t i o n s .

J"

0 ⁺ 2+

3

-

4 +

6 + 3

-

5- 1

-

The f u l l l i n e i n f i g . 6 shows t h e s t r e n t h d i s t r i b u t i o n o b t a i n e d i n t h e ( a , a l n ) c o i n c i d e n c e e x p e r i m e n t on "'Pb f o r t h e GQR e n e r g y r e g i o n ( 1 2 ) . The p a t t e r n i s r o u g h l y c o r r e l a t e d w i t h what i s observed i n e l e c t r o n s c a t t e r i n g and t h e o r e t i c a l l y e x p e c t e d . I n t h i s experiment a somewhat l a r g e r s t r e n g t h t h a n i n t h e ( e r e 1 ) r e a c t i o n h a s been o b s e r v ed

.

I t h a s been s u g g e s t e d ( R . d e Haro e t a l . , p r e p r i n t , J f i l i c h , 1982) t h a t t h e d i s c r e p a n c i e s between t h e low-energy e l e c t r o n s c a t t e r i n g ex- p e r i m e n t (qz0.25 fm-') and t h e ( q z l fm-') ( a , a l ) d a t a can be unders- t o o d i n t e r m s of t h e d i f f e r e n c e i n q-value t r a n s f e r r e d i n t h e two ex- p e r i m e n t s and t h e a s s o c i a t e d changes i n t h e r e s p o n s e f u n c t i o n . A t qz0.25 fm-' h a l f of t h e e n e r g y weighted sum r u l e i s e x p e c t e d t o b e con c e n t r a t e d i n few peaks mounted on a broad s t r u c t u r e . I t i s argued t h a t t h e magnitude of t h i s s t r u c t u r e i s s o broad t h a t it may have been con- f u s e d w i t h t h e background. A l t h o u g h t h i s e x p l a n a t i o n i s a t t r a c t i v e l y s i m p l e t h e p r e d i c t e d p e a k s a r e much wider t h a n e x p e r i m e n t a l l y obsesv- e d . Second, w h i l e a b o u t 15% o f t h e ET87SR h a s been observed i n t h e

- E (MeV)

Th

.

^Exp

.

13.6 13.8

11.2 11.4

r e g i o n GQR

11 12.2

11.2 13.7

20.1 18.7

20.8

20.5 21.3

r +

(MeV)

Th

.

^Exp

.

1 . 7 2.6

2.6 2.7

5 . 3 4.5

4.8 7.8

5.0 5.0

5.8

6 . O 5.9

EWSR ( % )

Th. Exp

.

9 2 90k2O

7 2 70t20

6 5

18 30t- 3

12 30t 3

6 3 6 0

12

5 4 9 0

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C4-218 JOURNAL DE PHYSIQUE

energy range 8 MeVcEt9.3 MeV the calculation of ref.(l5) predicts no strength below 9.3 MeV. ( * )

A mechanism has to be invoked to bring strength to this energy region without a major change in the position and shape of the peak which reproduces V e r y well the experimental data. With a loss of the order of 15% the observed EVSR will still be consistent with the data which displays such types of uncertainties.

Another example of the standing discrepancies between theory and experiment is provided by the giant quadrupole resonance of 4 0 ~ a . Through a-inelastic scattering a peak has been identified at 17.8 MeV with a width of 12.5 MeV and carrying about 40% of the EWSR (18)

.

^From

recent angular correlation measurements of a-decay to 3 6 ~ r ( 1 9 ) it is inferred that (60+:2)% of the auadrupole EWSR is found in the energy interval 10.0 PleV<E<15.5 MeV, with most of the strength concentrat- ed in the region 12.5 MeV(E(15.5 MeV. These results strongly suggest that in previous inelastic hadron experiments the contribution from the continuum for E<15 MeV has been highly overestimated.

Calculations have been carried out taking into account all the couplings shown in fig. 2 (c-f). Only collective states were included in the contributions 2c and 2d. The M3Y (20) force was used in the calculationof2e and 2f. The results are shown in fig. 8 in comparison with the experimental data. Also shown are the results of the surface model. In this calculation only the couplings shown in figs. 2c and

2d were taken into account and both collective and non-collective roots of the random phase approximation were included in the intermediate 2p-2h states.

ENERGY (MeV) ENERGY (MeV)

Fig. 8.

-

The experimental ( - ) isoscalar EWSR distribution for L=2 ob_

tained for "Ca from the data of refs. ( 1 8 ) and (19) is corn pared with the two theoretical distributions ( - - ) discussed in the text. On the left-hand side, results of the surface model (4) are reported. For the results shown on the right- -hand side, only the collective surface vibrations were used for the coupling in figs. 2c and 2d and the M3Y force for the contribution 2e and 2f.

( * ) Cf. also the 208~b(e,e'n) data reported to this Conference (17).

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The volume effects arising from the M3Y interaction seem to pro- duce some of the desired effects, although much strength is shifted not only to lower but also to higher energies.

A more stringent test of the M3Y force is provided by the calculation of the Ganow-Teller strength in 4 0 ~ a (21). In this case any strength comes from the ground state correlations induced by the interaction. A preliminar comparison with the data in fig. 9 does not seem to lead to obvious contradictions. Systematic calculations are needed before any firm conclusion can be drawn.

Fig. 9.

-

The neutron spectrum for the reaction 4 0 ~ a ( p , n ) 4 0 ~ c at ED=

=I60 MeV and at zero degree, is compared with the theoretic a1 results ( - - - ) for the L=O Gamow-Teller transitions calculated as discussed in the text. We thank C. Gaarde for help in the preparation of the figure.

6.

-

Conclusion.

The surface model provides a good description of the main peaks associated with the different giant resonances.

The need for a mechanism which brings strength away from the peak without altering its structure in a major way seems to be indi- cated by the data. Coupling to 2p-2h configurations associated with volume excitations have many of the desired effects.

To become more quantitative in the study of the response function it is necessary to resolve the contradictions existing between different experimental data and to gain a better understanding of the nature of the background.

(1) K. Goeke and J. Speth; Ann. Rev. Nucl. Part. Sci.

32

(1982) 65.

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