LETTJ~RE AL NIYOVO CIMENTO VOL. 23, N. 10 4 N o v e m b r e 1978

Configuration Mixing in the 12C(6Li, d)~SO Reaction.

A. CUNSOLO, A. FOT1, G. IM~'~}, G. PAPPALARDO a n d G. RACITI

I s t i t u t o N a z i o n a l e d i F i s i e a N u c l e a t e - S e z i o n e d i Ca t an i a

Centre S i c i l i a n o d i F i s i c a _Nucleare e d i S t r u t t u r a della M a t e r i a - C a t a n i a

I s t i t u t o di F i s i c a d e l P U n i v e r s i t d - Corse I t a l i a , 57, Catan ia , I t a l i a

N. SAUNIER

D d p a r t m e n t de P h y s i q u e Nuclda ire , C E N Sae lay , B P 2 - 91190 Gi] - sur -Yve t t e , F r a n c e

( r i c e v u t o il 28 L u g l i o 1978)

A n i n t e r e s t i n g p o s s i b i l i t y o p e n e d b y h e a v y - i o n (1) i n d u c e d n u c l e a r r e a c t i o n s is t h e s t u d y of t h e m a n y - p a r t i c l e - m a n y - h o l e c o m p o n e n t s of t h e n u c l e a r w a v e f u n c t i o n s .

I m p o r t a n t d e v e l o p m e n t s b o t h t h e o r e t i c a l l y a n d e x p e r i m e n t a l l y h a v e t a k e n p l a c e a l o n g t h e s e l i n e s d u r i n g t h e l a s t f ew y e a r s (2). I n p a r t i c u l a r (eLi, d) a n d (TLi, t ) r e ac -

t i o n s s e e m to b e e x t r e m e l y s u i t a b l e fo r s u c h t y p e of s t u d i e s l a r g e l y d u o to t h e f a c t t h a t t h e s e r e a c t i o n s s e e m to p r o c e e d m a i n l y t h r o u g h a s i m p l e d i r e c t a - t r a n s f e r a l so if a

c o m p o u n d - n u c l e u s m e c h a n i s m c a n c o n t r i b u t e (3). R e c e n t l y , s y s t e m a t i c s t u d i e s of (eLi, d) r e a c t i o n s on s d - s h e l l ( 4 , s ) a n d f p - s h e l l ( e )

n u c l e i h a v e b e e n c a r r i e d o u t a n d a t t e m p t s (4) h a v e b e e n m a d e to e x p l a i n t h e e x p e r i -

m e n t a l u - s p e c t r o s c o p i c s t r e n g t h s in t e r m s of t h e S U a m o d e l o f E l l i o t .

T h e e x p e r i m e n t a l ~ - s p e c t r o s e o p i c s t r e n g t h s w e r e a l w a y s d e d u c e d i n t h e s i m p l i f y i n g h y p o t h e s i s t h a t t h e s t a t e s e x c i t e d in t h e r e a c t i o n a r e we l l d e s c r i b e d as p u r e m a n y -

(1) See for example the Proceedings of the International Conference on Nuclear Reactions Induced by Heavy Ions, Heidelberg, 1969, edited by R. BeCK and W. R. ItER1NG (Amsterdam, 1970). (*) See for example the Proceedings of the Symposium on Heavy Ions Reactions and J lany Particle Excitations (Saclay, 1971); J. Physique, 32, C-6 (1971), and the following review articles: M. C. LE- MAIRE: Phys. Rcp., 7 C, 281 (1973); K. S. LOW: J. Phys. C, 37, 5 (1976); S. KXHANX and A. J. BALTZ: _4dv. Nucl. Phys., 9, 1 (1977); H. T. FORTUNE: Proceedings of the International Conference on Nuclear Structure (Tokyo, 1977); Journ. Phys. Soc. Japan, Suppl. , 44, 99 (1978). (s) A. A. OGLOBLIN: Proceedings o/ the International Conference on Nuclear Reactions Induced by Heavy Ions, Heidelberg, 1969, edited by R. B e c k and W. R. HERING (Amsterdam, 1970); K. BETHG.E: Ann. Rev. Nuel. ScL, 20, 225 {1970); G. BASSANI: note CEA-N-1474 (1971). (4) N. ANANTARAMAN, C. L. BENNETT, J. P. DRAAYER, H. V~ r. FULBRIGHT, H. E. GOVE and J. TOKE: Phys. Rev. Lett., 35, 1131 (1975); G. D. GUNN, l=t. N. BOYD, N. ANANTARA.~,IAN, D. SHAPIRA, J. TOKE and H. E. GOVE: Nucl. Phys., 275A, 524 (1977). (a) ~r E. COBERN, D. J. PISANO and P. D. PARKER: Phys. Rev. C, 14, 491 (1976). (~) H . W . FULBRIGHT, C. L. BENNETT, R. A. LINDGREEN, R. G. MARKAM, S. C. Mc GUIRE, G. C. MOR- RISEN, U. STROHBUSCI:I and J. TSKE: Nucl. Phys., 284A, 324 (1977).

371

372 A. CUNSOLO, A. FOTI, G. I~[ME, G. PAPPALAI{DO, G. RACITI a n d N. SAUNIER

par t i c l e -many-ho le configurations, even if the necessi ty to include configurat ion ad- mix tu res has been claimed (4).

In th is let ter we present the rcsults obta ined in the analysis of the ~2C(~Li, d):~0 reac t ion made taking into account the ~60 level configurat ion mixing suggested by BROWN and G R ~ - ( 7 ) .

In a previous inves t iga t ion (s), the lcC(6Li, d ) 6 0 react ion was s tudied in the (20--40) MeV incident -energy range. In reL (s) we have shown tha t the essential Features of the exper imenta l cross-sections (da/d/2)~, can be explained under the hypothes is tha t the contr ibut ions from the s ta t i s t ica l compound-nucleus mechanism (da/d.(2)~ F and from direct ~-transfer mechanism (do-/d.Q)EFm.W~ ~, add incoherent ly , t ha t is

(1)

where the normal izat ion cons tan t S (~-spectroscopic s trength) contains all the spectro- scopic informat ion. In the expression (1), (da/d.O)n 7 was es t imated by per forming calculat ions wi th the Hause r -Feshbach formalism (9) and one-step (do"/d~)EFR.DWBX was ca lcula ted using the S A T U R N - M A R S code of TAMVnA and L o w ( ~ assunfing tha t the ~60 states arc described, in a cluster approximat ion , as ~ (~) ,2(~ in a re la t ive single stz, te w i t h quan tum numbers N L .

In order to invest igate the sensibil i ty of such analysis on a more realistic descrip- t ion of the 160 states, we have assumed for the 1~0 ground state, the 6.92 MeV (2 +) s ta te and the 10.35 MeV (4 +) state, the par t ic le-hole admixtures as calculated by B~OWN and G R ~ N (7). In this case the direct cont r ibut ion to the cross-section cannot be factor ized into a spectroscopic factor and a D W B A cress-section, but becomes (u)

do" t CONF'MIX N (2) \~Q/EFR-DWnA OC IE A.,'L BNL,~S(O)[~ ,

where A~-L is the spectroscopic ampl i tude for the ~60 nucleus in a definite s ta te dis- sociat ing into a 12Cg~. nucleus and an ~.part iele wi th re la t ive mot ion specified by N L. The fac tor BNL,~s(O) is the usual D W B A transi t ion ampl i tude .

In order to perform the coherent summat ion over N, in eq. (2) we u s c d ' a modi- fied (~2) version of S A T U R N - M A R S , E F R - D W B A code.

All the opt ical-model and bound-s ta te parameters needed to calculate the t ransi- t ion ampl i tude B,vL,~s(O) have been taken to be the same as in the pure conf igurat ion case (see ref. (s) for details). The adopted spectroscopic ampl i tudes A~vL are d isplayed in the last colunm of table I. They were obta ined f rom the Brown-Green ampl i tudes and the theore t ica l spectroscopic ampl i tudes calculated by ICHI.~U~A et al. (La) in the f ramework of the S U3 model.

(~) G. BROWN and A. GREEN: N u c l . Phys . , 75, 401 (1966). (s) A. ('VNf4OI,O, A. FOTI, G. PAPPAI,ARDO, G. I~ACITI an d N. SAUNIER: Phys . Rev. , to be pl lbl ished (1978). (9) t I . FESHBA('II: ill Nuclear Spectroscopy, ed i t ed by F. AJZENBERG-SELOVE (New Y o r k , N . Y . , 1960). p. 625. (1o) T . TA.~IURX and K. S. L o w : Comp. Phys . Comm. , 8, 3i9 (1974). (u) N. AUSTERN: Direct Nuclear Reaction Theories (New York , N . Y . , 1970); T. T,X~URA: P]~ys. Rep . . 14, 59 (1974). (1,) J . (~. PENG: p r i va t e c o m m u n i c a t i o n . (.a) M. I('tII3IURA, A. ARIMA, E. C. HALBERT a n d T. TI,:RXSAWA: Nuc l . Phys . , 204 A. 225 (1973).

CONFIGURATION MIXING IN THJ~ ~2C(eLi, d):60 REACTION 373

TABLE I. -- Spectroscopic a,~plitudes for mixed con]iguration EFR-D~VBA calculations.

Levels Config- Mixing Spectroscopic A~-c E* (McV) J~ ura t ion ampl i tudes (7) ampl i tudes (~3)

g.s. 0 + Op-Oh + 0.874 - - 0.544 - - 0.475(A2s )

2p.2h + 0.469 + 0.422 - - 0.198(A3s )

4p.4h - - 0.130 + 0.494 - - 0.064(Aas )

6.92 2 + 2p-2h + 0.377 - - 0.343 - - 0.129(A.,D)

4p-4h - - 0.923 - - 0.483 - - 0.446(A3D )

10.35 4 + 2p-2h + 0.260 + 0.129 - - 0.034(A1G )

4p-4h - - 0.970 + 0.455 --0.441(A2o )

In fig. 1 the deu te ron angular d i s t r ibu t ions f rom the ~C(6Li, d):~O react ion, measured at 28 MeV, are shown toge ther wi th the theore t ica l calculat ions. No te t ha t t he mixing of configurat ions do not change the shapes of the theore t ica l curves bu t only the absolute values.

The re la t ive cross-sect ions are r epo r t ed in table II in t e rms of the normal iza t ion pa ramete r R, defined by the relat ion

(3) \dD].xp \dsg]nF + \~/EFa-DWBX '

TABLE II. -- Comparison betwee~t pure-conJiguration a.ad mixed-con]iguratiou results.

Levels J~ Pure configurat ions Mixed con fgu ra t i ons

E* (MeV) Config- R R/R~.8. Configurations R R/R~.8. ura t ion

0.00 0 + Op.Oh 0.537 1 0.874(Op-Oh)+O.469(2p.2h)-- 0.211 1 - - 0.130(4p-4h)

6.92 2 + 4p-4h 0.154 0.29 0.377(2p.2h)--O.923(4p-4h) 0.123 0.58

10.35 4 + 4p-4h 0.304 0.57 0.260(2p-2h)--O.970(4p-4h) 0.271 1.28

In the ease of pure configuration, express ion (1) becomes

(4)

We expect t h a t th i s normal iza t ion fac tor is cons tan t for different t rans i t ions . Table ! I r epor t s the ra t io R/Rg.~. for the << pure ,> conf igurat ion and the <, mixed ,>

.~onfiguration cases. We note t ha t a b e t t e r agreement wi th the expe r imen t is ob ta ined for the mixed configurat ion case; in fact the R/Rg.~. values show smaller f luctuat ions a round uni ty .

~7~ A. CUNSOLO, A. FOTI, G. 1MM]~, G. PAPPALAI~DO, G. RACITI and .W. SAUNIER

1021 ~ ~ - . . . . . . .

103

.Q

.~ 102

10 2

10

• .92 (2+)

~ , ~ . �9

g's'(O+)

' , ,~, \~ , , ?--.-

30 60 90 ~c.rn.

Fig . 1. - D e u t e r o n a n g u l a r d i s t r i b u t i o n s for t he lzC(~Li, d)160 r e a c t i o n a t 28 MeV. T h e d a s h e d c u r v e s r e p r e s e n t H F ca lcu la t ions , t h e ful l ones r e p r e s e n t the s u m of H F a n d E F R - D ' W B A w i t h m i x e d -

c o n f i g u r a t i o n con t r ibu t ions . ESL i ~ 28 MeV.

This can be understood in terms of the overlap function (la).

N

( ~ being the radial wave function for a given ~-particle final-state configuration) displayed in fig. 2 for the two cases.

The improved description of the leO final-state products in the /r function tail near the nuclear surface, a more pronounced enhancement for the g.s. than for the 2 + state (the 4 + state being left essentially unchanged), thus producing a different increase in the calculated cross-sections and hence a different decrease in the corre- sponding R values.

CONFIGURATION M I X I N G IN THE rzO(eLi, d)~60 R~ACT[ON 375

10 o I I ' I I I

lo-'

10 -2

10-3

w

~" 10-2 ,.Q

10 .3

lo-4

10 -5

10 -6

a) I

b)

' \

"i',-,,, ':,,,

\

f . ~

i f

i I , I 3 6 9 12 15

r'~,_ 12c(f m)

' I i

Fig . 2. - O v e r l a p f u n c t i o n s for t h e g r o u n d s t a t e ( a ) ) a n d t h e 6 .92 M e V 2 § s t a t e (b) ) of t h e '~O n u c l e u s , v s . t h e a-l~C r e l a t i v e d i s t a n c e . T h e d a s h e d a n d t h e fu l l c u r v e s r e f e r to t h e m i x e d a n d p u r e c o n f i g u r a t i o n s c a s e s r e s p e c t i v e l y ( s ee t e x t ) .

The main d rawback of present analysis is t ha t i t refers only to three transit ions. Moreover one of t h e m leads to an unbound s ta te ; fur thermore we recognize tha t some approximat ions are made in the theore t ica l calculat ion of the spectroscopic ampl i tudes (as e.g. the use of equal harmonic-osci l la tor size paramete r for pa ren t and daughter nuclei) and there are some inconsistencies in the use of these quant i t ies ( the A~L are calculated using harmonic oscillator w a v e funct ion, while in the B~,L,2s(O) the a-cluster wave funct ions are generated in a Saxon-Woods potent ia l ) , Never theless we th ink

3 7 6 A. CUiNSOLO, A. FOTI , G. IMM]~, G. PAPPALARDO, G. ]~ACITI and N. SAUNIER

that our analysis points out that the many-particle-many-hole configuration mixing in the description of the ~sO nucleus plays an important role in a quantitative analysis of the ~C(eLi, d)-'eO reaction. Further work is in progress in order to test more sophisticated 380 wave functions (14).

* * *

The authors would like to thank Dr. J. C. P ~ 6 for his kind help in partially mod- ifying the EFR-DWBA code, Prof. M. ICHIMU~A for enlightening discussions and sug- gestions, Prof. R. ]31~AGLIA for interesting comments and Dr. F. Rlzzo for his help in numerical calculations.

(~) Y. SUZUKI: Prog. Theor. Phys., 56, 111 (1976); 55, 1751 (1976); a n d p r i v a t e c o m m u n i c a t i o n .