IL NUOVO CIMENTO VOL. 83 A, N. 2 21 Settembre 1984

Quasi-Free Experiments as a Tool for the Study of eli

Cluster Structure.

M. LATTUADA~ 1 ~. I~IGGI~ C. SPITAL:ERI and D. VINCIGLrERRA

Ist i tut i di l~isica dell' U~iversit& - Catania, I tal ia Istituto Nazionale di Fis ica ~Yucleare - Sezione di Catania Laboratorio .~azionale del Sud - Catania, I tal ia

(ricevuto 1'11 Aprile 1984)

Summary. - - The value of the g-d clustering probability in eLi deduced from quasi-free experiments may be influenced by the choice of the inter- cluster wave function. We have examined several functional forms usually taken to describe the relative motion of the two clusters. The effect of the choice of the intercluster wave function on the information deduced by analysing quasi-free data in the plane-wave impulse approximation was investigated.

PACS. 21.60. - Nuclear-structure models and methods.

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

When a light nucleus is described in the cluster model, one is usually

concerned with the knowledge of the extent of the overlap of the actual

nuclear wave function with u part icular cluster configuration. For instance

in case of 6Li much work has been done both experimental ly (1-82) and thee-

(t) C. RUHLA, ~[. RIOU, M. GUSAKOW, J.C. JACMART, M. LIN and L. VALENTIN: Phys. Lett., 6, 282 (1963). (e) D.W. Divvies, B.L. SCOTT and H. H. FORST]~a: Rev. Mod. Phys . , 37, 396 (1965). (3) H. DAvIEs, H. MUIRn]~AD and J. N. WOULI)s: iVucl. Phys . , 78, 663 (1966). (4) H.B. l~ J. W. WATSON, D. A. GOLDBERG, :P. G. ROOS, D. I . BONBRIGttT and

151

152 ~. LATTUADA, F. RIGGI, C. SPITALERI and D. VINCIGUERRA

r e t i c a l l y (an.,5) in o r d e r to o b t a i n i n f o r m a t i o n on t h e f a c t o r

(1) P = [(,Li]~ + ~) [2 .

R . A . J . RIDDLE: Phys. Rev. Lett., 22, 408 (1969). (5) K. BAHR, T. :B~.cKER, 0. M. BILANIUK and R. JAHHR: Phys. Rev., 178, 1706 (1969). (s) V . K . DOLINOV, Yu. V. MELIKOV, A. F. TULINOV and 0. V. BORMOT: Nq~cl. Phys. A, 129, 577 (1969); V. K. DOLINOV, D. V. MEBONIYA and A. F. TLrLINOV: Nucl. Phys. A, 129, 597 (1969). (7) J . R . PIZZI, M. GAILLARD, P. GAILLARD, A. GUICHARD, M. GUSAKOW, G. REBOULET. and C. RUHLA: Nucl. Phys. A, 136, 496 {1969); P. GAILLARD, M. CHEV'ALLIER, J. Y. GROSSIORD, A. GUICHARD, M. GUSAKOW, M. GUSAKOV, J. R. :PIzzI and J. P. MAILLARD: Phys. Rev. Lett., 25, 593 (1970). (a) M. JAIN, P . G . Roos, H .G . PUGH and H . D . HOLMGREN: Phys. Rev. A, 153, 49 (1970). (9) J . M . LAMBERT, R . J . KANE, P . A . TREADO, L . A . BEACH, E . L . PETERSEN and R . B . THEUS: Phys. Rev. C, 4, 2010 (1971). (lo) j . W . WATSON, H. G. PUGH, P. G. Roos, D .A . GOLDBERG, R. A. J. RIDDLE and D . I . BONERIGHT: ~Vucl. Phys. A, 172, 513 (1971). (11) I . A . MACKENZIE, S . K . MARK and TS~H Y. LI: Nucl. Phys. A, 178, 225 (1971). (12) D. BACHELIER: P h . D . Thesis, Facult4 de Sciences, 0 rsay (1971), unpublished. (13) j . C . ALDER, W. DOLLI.IOPF, W. ]~OSSLER, C. 1 ~. PERDRISAT, W . K . ROBERTS, P. KITCHINO, G .A. MOSS, W. C. 0LSEN and J. R. PRIEST: Phys. Rev. C, 6, 18 (1972). (14) R. HAGELBERG, E . L . HAASE and Y. SAKAMOTO: Nucl. Phys. A, 207, 366 (1973). (15) D. MILJANIC, T. ZABEL, R . B . LIEBERT, G.C. PHILLIPS and V. VALKOVIC: Nucl. Phys. A, 215, 221 (1973). (16) M . F . WERBY, M. B. GREENFIELD, K. W. KEMPER, D. L. MCSHAN and S. EDWARDS: Phys. Rev. C, 8, 106 (1973). (17) R . B . LIEBERT, K. H. PURSER and R. L. BURMAN: Nucl. Phys. A, 216, 335 (1973). (18) A . K . JAIN, J . Y . GROSSIORD, M. CHEVALLIER, P. GAILLARD, A. GUICI.IARD, M. GUSAKOW and J. R. PIZZI: Nucl. Phys. A, 216, 519 (1973). (19) D. MILJANIC, J. HUDOMALJ, G.S. MUTCHLER, E. ANDRkDE and G.C. PHILLIPS: Phys. •ett. B, 50, 330 (1974). (20) j . p . GENIN, J. JULIEN, M. RAMBAUT, C. SAMOUR, A. :PALMERI and D. VINCI- GUERRA: Phys. Left. B, 52, 46 (1974). (21) W. DOLLHOPF, C .F . P:ERDRISAT, 1 ~. KITCI.IING and W.C . OLSEN: Phys. Lett. B, 58, 425 (1975). (22) p. KITCHING, W. C. 0LSEN, H. S. SHERIF, W. DOLLHOPF, C. LUNKE, C. :F. I~ERDRISAT, J. R. PRIEST and W . K . ROBERTS: Phys. Rev. C, 11, 420 (1975). (23) J . p . GENIN, J. JULIEN, M. RAMBAUT, C. SAMOUR, X. PALMERI and D. VINCIGUERRA: Lett. Nuovo Cimento, 13, 693 (1975). (24) D. VINCIGUERRA, E. MODICA, A. PALMERI, J. JULIEN, C. SAMOUR and J. P. GENIN: Lett. Nuovo Cimento, 14, 333 (1975). (2~) p . G . Roos, D . A . GODBERG, N .S . CHANT and R. WOODY: Nucl. Phys. A, 257, 317 (1976). (26) p . G . Roos, N. S. CHANT, A. A. COWLEY, D . A . GOLDBERG, H . D . HOLMGREN and R. WooDY: Phys. Rev. C, 15, 69 (1977). (27) SUN HAN-CHENG, YAO JIN-ZHANG, SUN TSU-XUN, WEN KE-LING, LC HuI-JIN, DAI NENG-XIOXG, JIN RONG-HuA and YXN CHHEN: Chin. J. Nucl. Phys. , l , 1 (1979). (28) W . E . DOLLHOPF, C. F. PERDRISAT, P. KITCHING and W. C. OLSEN: Nucl. Phys. A,

QI)-ASI-FREE E X F E R I M E N T S ETC. 153

Hero a + d refers to an ~-d cluster description of 8Li. One of the most useful ways of determining experimentally P, which is sometimes referred to as clustering probability, is the study of quasi-free (QF) scattering or reac- tions. In fact, within the impulse approximation (4s), the cross-section for the coincident detection of particles 1 and 2 produced in the reaction N(0, 12)S can be written

(2) d.Q~ d~Q~ dE, ~ ~r '

where Ek is a phase-space factor (~7), and (da/dQ)~ r is the cross-section for the virtual two-body reaction T(0, 1)2, once the nucleus 5~ is considered as made up by the spectator S and the interacting duster T. The distribution G:(ps) of the spectator momentum F s is given in the distorted-wave impulse approximation (DWIA) by

G(ps) = (2~)-tf~*yJ*z(r)~odr, (3) J

316, 350 (1979). (29) ZtIANG PEI-HuA, Su~" Zu-XuN, WE~ ~ KE-LI.~G, WANG FANa-LI.w and WANG SnlJ-MAO: Chin. J. ~u Phys., 2, 219 (1980). (31) U. SE~'.\~nAUSER, H.-J. PF~IFF~R, H. K. WALTER, F. W. SCHLEI'UTZ, H. S. PRUYS, R. ENGFER, R. HARTMANN, E. A. HERMES, P. HEUSI and H. P. ISAAK: Nucl. Phys. A, 386, 429 (1982), and references therein. (3~) D. GOLA, W. BRETFELD, W. BU'RGMER, H. EICHNER, CII. IIEI~,'RICH, H. J. HELTEN, II. KRETZER, K. PRESCHER, ]I. OSWALD, W. SCHNORRENBERG and H. PAETZ gen. SCmECK : Phys. Rev. C, 27, 1394 (1983). (32) G. CALVI, ~r LATTUADA, F. RIGGI, C. SPITALERI, D. VINCIGUERRA. and D. M~LJA- .~IC: Left. ~ruovo Cimento, 37, 279 (1983). (a3) K. WILDERMUTtI and W. McCLuRE: Springer Tracts in Modern Physics, Vol. 41 (1966). (3a) S. SAITO, J. HI~:RA and H. TASAKA: Prog. Theor. Phys., 39, 635 (1968). (as) Yu. A. KUDEYAROV, V.G. NEUDATCHIN, S.G. SEREBRYAKOV and y~r. F. S.~IR- .~OV: Soy. J. Nucl. Phys., 6, 876 (1968). (3~) V-. G. NiEUDATCHIN and Yv. F. S-~IRNOV: Prog. Nucl. Phys., 10, 273 (1969). (3~) A . K . JAIN, N. SARMA and B. BANERJEE: IVuovo Cimento B, 62, 219 (1969). (as) G .L . PAY~'E and P. L. vo~- BEr[REN: Phys. Rev. C, 5, 1955 (1972). ('~g) T . K . LIM: Phys. Lett. B, 47, 397 (1973). (40) A . K . JAIl" and N. SAR)fA: Nucl. Phys. A, 233, 145 (1974). (41) j . V . NOBLE: Phys. Lett. B, 55, 433 (1975). (42) N .S . CIIA~'T and P . G . Roos: Phys. Rev. C, 15, 57 (1977). (4a) D . R . CHAKrtABARTY and M.A. ESWXRA~': Phys. Rev. C, 25, 1933 (1982). (44) R. KRIVEC and M. V. MHIAILOVlC: J. Phys. G, 8, 821 (1982). (4~) K. KUMAR and A. K. J A I l : J. Phys. G, 8, 827 (1982). (4s) G . F . CHEW: Phys. Rev., 80, 196 (1950); G. F. CHEW and G. C. WICK: Phys. Roy., 85, 636 (1952). (47) p.G.F.4.LLICA, F. RIGGI, C. SI'ITXLEaI and M.C. SUTERA: Lett. Nuovo Cimento, 22, 547 (1978).

11 - ll Nuovo Cimenlo A.

154 ~. LATTUADA, F. RIGGI, C. SPITALERI and D. VINCIGUERRA

where the ~v2s are the distorted wave functions of the incident and outgoing particles, while z(r) is the intereluster wave function. The normalization conditions are

f lz(r ) l~dr = X, (4)

(5) f iG(p )l = 1.

In the plane-wave impulse approximation (PWIA), G(ps) becomes the Fourier t ransform of x(r):

(6) (ps) = (2 )- fVcxp d r .

I t has already been pointed out (lo) tha t the reported values of /~ obtained from quasi-free experiments vary from a few percent to unity, in the case of the g-d configuration of 6Li. Also a connection has been found (lo) between the FWItlV[ of the measured momentum distribution and the value of /~. In fact, when the momentum transfer is higher than about 300 MeV/c, the experiments give a momentum distribution with a maximum at Ps = 0 (which is expected since the g-d motion is mainly in a s-state) and with (As) a FWHN[ of about 70 NIeV/c. When the energy or rather the transferred momentum involved in the reaction is low, the measured cross-section becomes smaller, implying a low value of P, while the momentum distribution becomes narrower. Both phenomena seem to be correlated with absorption effects for sm~ll intercluster distances (lO,4S,49), as will be discussed later in greater detail, and have justified the introduction (lO) of a cut-off radius Ro in the radial part of the wave function z(r) in the P W I A approach. Hopefully, the t r ea tment of these effects through a DWI2~ or a 1)WIA with cut-off radius should imply the derivation of a unique value of / ' .

Sometimes, FWHMs definitely larger than 70 MeV/c have been reported in the li terature (12,2~.~2,2S). However, this is connected with an experimental broadening (3), or with an improper data t rea tment (ds).

I t is obvious tha t a different choice of g(r) may affect the overall features of G2(Ps), namely its l e W H ~ and its top value G2(0). In particular, once a cut-off radius larger than about 4 fm is introduced, all wave functions should give appreciably the same shape of G2(ps), and hence the same FWHN[, since z(r) is in its asymptot ic region. The absolute value of G:(0), and hence the value of P, depends however on the absolute va lue of g(r), which in tu rn depends on its overall shape.

(~s) S. BXR~ARINO, M. LAT~UADA, F. ]~IGGI, C. SPITAL:ERI and D. VINCIGU~ERRA: Phys. t{ev. C, 21, 1104 (1980). (49) ~V~. LATTUADA, F. RIGGI, C. SPITALERI, D. VINCIGUERRA, C.M. SUTERA and A. PANTAL~O: NUOVO Cimento A, 71, 429 (1982).

Q~AS~-~'R~E ~XP~RIM~NTS ~Te. 155

TILe aim of the present work is then

i) to determine the influence of the choice of x(r) on the deduced value of / ) ;

ii) to discuss the limits of val idi ty of the P W I A with a cut-off radius and revisit the existing da ta in eLi quasi-free reactions within the same approach, to see whether an unique value of P can be obtained.

2. - T h e eLi w a v e f u n c t i o n .

To determine the influence of the choice of the wave funct ion z(r) on the value of P which can be deduced from the experiment , we calculated first the mom e n tum distr ibution in the PWIA. By considering only a s-wave motion between the two clusters, and neglecting ant isymmetr izat ion, the intercluster wave funct ion is writ ten

(7) z ( r ) - - R(r)Yoo(O, q~)

and the m o m e n t u m distr ibution is simply given by

(s)

where

(9)

G(ps) --= (4~)-~G(ps),

co

G(ps) -= (2z)-14~fio(psr/h)R(r) r 2 d r .

0

I t is well known tha t a small admixture of d-wave is necessary to reproduce the quadrupolo moment of the ground s ta te of SLi. Any relat ive motion with 1 ~-- 0 introduces a broad and shallow contr ibut ion in the momen tum distri- bution, with a zero at Ps ~ 0. We have calculated the two contributions for the square-well wave funct ion of subscct. 2"3. The result is shown in fig. 1 by assuming equal weight for s- and d-wave. F ro m the figure it is clear t ha t even a large admixture of a d-wave in g(r) would change only slightly the F W H M of the momen tum distribution, leaving its top value unchanged. Incidental ly this means t ha t unless one measures the m o m en tu m distr ibution

for large values of Ps and is able to in terpret it properly (for large Ps values the P W I A is cer tainly not adequate), no information can be obtained on the d-state and P refers only to the s-state.

We have ment ioned t ha t cq. (7) does not take into account the wave function ant isymmctr izat ion. This approximat ion is justified by the limited influence ant isymmotr iza t ion has in the calculations (33,,5).

Within the assumption (7), we have then calculated the momen tum distri-

156 M. LATTIYADA, F. RIGGI, C. SPITALERI a n d D. VINCIGUERRA

10 I I

10 2 60

3-

E

i0 -I

2 G 2

/ /

/ /

/ /

/ /

10-21 ,/ I I ~ I I 0 50 100 150 200 250

PS (MeV/c )

Fig. 1. - Momentum distribution of ~- and d-clusters of 6Li calculated for a square- well intercluster potential, for two l imiting cases: only s-wave (continuous curve) and only d-wave in the actual 6Li wave function does not affect the momentum distribu- tion for p.~< 100 MeV/c.

but ion G*(ps), and in particular its FWHM and its top value O2(0), through eq. (9), for the fo l lowing f u n c t i o n a l dependences of R(r):

a) Hankel,

b) harmonic osci l lator,

c) square well,

d) Eckart,

e) Woods-Saxon,

]) Woods-Saxon with hard core.

QUASI-FREE EXPERIMENTS ETC. I~7

I n t h e c l u s t e r m o d e l , t h e = - a n d d - c l u s t e r s a r e c o n s i d e r e d t o b e i n a 2s

s t a t e , t o s ~ t i s f y t h e P a u l i p r i n c i p l e . T h i s w a s k e p t i n t o c o n s i d e r a t i o n i n a l l

c a s e s e x c e p t f o r t h e H a n k e l f u n c t i o n a n d t h e W o o d s - S ~ x o n f u n c t i o n w i t h

h a r d co re .

\ 0.6

0,4

\

/ \

./ \ "~ <, 0.2 I~' '"~--.. "<"-:

i E ~0

q :

t' i I " l ;

o.a 'U

o 2 L, 6 ," ( f r n )

8 t l

I0 12

Fig. 2. - R a d i a l p a r t R(r) of t he i n t e r e lu s t e r w a v e func t ion m u l t i p l i e d by r vs. r o ( . . . . . Hanke l , - - - s q u a r e well, - - - Ecka r t , - - . . . . . Woods-S ' lxon , - - - h a r d core).

158 ~ . LATTUADA, F. RIGGI, C. SPITALERI and D. VINCIGIJ-ERRA

I n all cases the roo t -moan- squa re d is tance be tween the two clusters d

~- (r~)~ was deduced a nd in some cases i t was kep t fixed a t 3.46 f m (35,3s),

in order to de te rmine t he p a r a m e t e r s of t he wave func t ion /~(r). This va lue

fa i r ly compares wi th the one deduced b y simple classical considerat ions , re la t -

ing t h e r .m.s, rad ius Re of e l i t o t he r .m.s, radi i R~ and R2 of the r162 and

d-clusters , respect ively , a nd t o d t h r o u g h

16

:By us ing R2 -~ 2.095 fm (5o), R4 ---- 1.672 fm (5o) and R8 ~- 2.55 f m (~3) the

va lue d - 3.80 is obta ined . The b e h a v i o u r of the rad ia l wave func t ion R(r)

t imes the dis tance r is r epo r t ed in fig. 2 as a func t ion of v for all t h e func t iona l

dependences discussed here.

2"1. Hanke l ] u n c t i o n . - I f the long- range Coulomb effects be tween t he

two clusters are neglected, t he in torc lus te r w a v e func t ion has an a s y m p t o t i c

b e h a v i o u r p ropor t i ona l t o t h e H a n k e l func t ion exp [ - - k r ] / r . H e r e k ~

(2/~B) t, where # a n d B are, respect ive ly , t he r educed mass and the b ind ing

TABLE I. - Calculation o/ the mean intercluster distance d, o/ the FWHM and o/ the top value o/ the intercluster momentum distribution /or the reported wave /unctions. :For the harm(role oscillator (line 2), the square well and the Eckart wave /unction, the distance d was kept /ixed to 3.46 fro.

Radial wave function Parameters d FWHM G2(0) (fm) (MeV/c) (fm a sr -1)

1) Itankel k : 0.306 fm -1 2.31 78 3.5 2) harmonic oscillator (2s) Q = 0.765 fm -1 3.46 113 2.7 3) harmonic oscillator (2s) Q = 0.427 fm -1 5.83 67 5.8 4) square well (2s) a = 2.50 fm 3.46 70 6.4 5) Eckart (2s) b = 0.92 fm 3.46 66 6.7

r o = 1.62 fm 6) Woods-Saxon (2s)(2~) 7) Woods-Saxon with hard core (4a) 8) Woods-Saxon with hard core (lo)

4.27 68 10.3 /~ ~ 0.6 fm 3.80 73 9.0 1~ c ~ 1.25 fm 4.26 71 10.9

ene rgy of t he :r sys tem. I n some cases the H a n k e l func t ion has been t a k e n

to descr ibe the a -d m o t i o n also at small distances. Fo r t he I t a n k e l func t ion

the F W I t ~ is g iven b y 2(~/2- 1)t]lk, while d : (~/~-~)-1. F o r k : 0.3063 fm

the F W H M of the m o m e n t u m d is t r ibu t ion has a value abou t 10 % h igher

t h a n the expe r imen ta l resul t (see tab le I) b u t the value of d is m u c h too

small.

(50) C. CIOFI DEGLI ATTI: Prod. Part. iYucl. Phys., 3, 163 (1980).

QUASI-FI~EE ~:XFERIMENTS XTC. 1~9

2"2. H a r m o n i c - o s c i l l a t o r wave ]unc t ion . - Historical ly, harmonic-osci l la tor

( H e ) wave functions were the first to be used in the cluster model (33,35,3s) to describe the relat ive mot ion of the two clusters. This was in par t icular the case for el i . Le t us consider a 2s t tO wave funct ion

R ( r ) - - (2~)-i3JQt[1 - - 1Q2r2] exp [ - ~Q2r2] .

B y adjust ing the p a r a m e t e r Q one can obta in ~ correct mean intercluster

distance bu t with a much too large FWI:IM (see second en t ry in table I). On

the other hand, if the F W H M is correct ly reproduced, the distance is not (third

entry) . The top value of the m o m e n t u m distr ibut ion varies within a fac tor

of two between the two c~ses. This d iscrepancy is to be connected with the

well-known inadequacy of the t i e wave functions to describe proper ly a bound

sys tem for large distances. This effect is emphasized in *Li by the low binding

energy (B----1.47 MeV) which introduces a very slowly decaying a sympto t i c bohaviour in the r~dial pa r t of the interclustcr wave function. We have verified t h a t the same effect is obta ined when using a different polynomial

in the t i e wave funct ion.

2"3. Square -we l l wave ]unct ion . - The pa rame te r s of a square-well potent ia l

( V - ~ - Vo for r < a) are comple te ly de termined once the binding energy B

and the r.m.s, distance d are fixed. For a 2s wave function, and for B ~ 1.47 I~oV

�9 ~nd d ---- 3.46 fm, one obtains a ---- 2.50 fm and Vo ~- 6[.26 • This wave

funct ion gives u good value for the F W H ~ (table I). The zero-crossing distance is a t r0---- 1.62 fm.

As will be seen later , whoa calculat ing the m o m e n t u m dis t r ibut ion for different wave functions repor ted in the l i terature, the r .m.s, intercluster distance takes different values. In ordcr to afford ,~ comparison of various results under similar conditions, we have calculated (fig. 3) the values of the FWH-M: ~md of G'-(0) for the square-well wave funct ion as a funct ion of d. For completeness also the square-well radius a is reported. The l imit ing

case is for a ---- 0, when the Hanko l funct ion is obtained. ] t is clear t h a t by

increasing d (or a) t he FWtt-M: decreases while G~(0) increases in ~ more

d rama t i c way. In fig. 3 are also indicated the values of G~-(0) and of the

F W H ~ for m o m e n t u m distr ibut ions deduced f rom the wave functions discussed in the next subsections.

2"4. E c k a r t wave ]unct ion . - As an a l te rna t ive to the previous wave func- tion, an E c k a r t fo rm

R ( r ) --= co,lsl {1-- exp [-- b(r - - r o ) ] } { l - exp [-- br]} exp [-- kr]/r

can be used. By using the same value of re as for the square well and by keeping d fixed, a slightly smaller FWH.~I is obtMued, while tile top value is the same within abou t 6 ~o.

160 .~. I, ATTUADA, F. RIGGI, C. SPITALERI a n d D. VINCIGUERRA

4

10

T'- 8

~E 6 v

v

4

80

"r 4(]

20

' I ' I

1:1 0

, I , I ,

l l a I i 3 4

c/.(fm)

Fig. 3. - Top value G2(0) and FWIIM of the ~-d momentum distribution calculated for a square-well potential, as a function of the mean intercluster distance d. The square-well radius a is also reported. The symbols refer to calculations performed for various intercluster wave functions as reported in table I (m Hankel, �9 Eckavt, D Woods- Saxon, o, �9 hard core).

2"5. W o o d s - S a x o n p o t e n t i a l . - W h e n calcula t ing t he m o m e n t u m dis t r ibu-

t i on wi th t he wave func t ion ob ta ined for the W o o d s - S a x o n po ten t i a l of ref. (36),

one finds a va lue of t he F W l t M (68 HoV/c) in good a ~ e e m e n t wi th t he p rev ious

ones. H o w e v e r , t he r .m.s, d is tance is ove res t ima ted (d ---- 4.27 fm), while t he

t op va lue (G~(0)----10.3 fm 3 sr -1) is defini tely larger t h a n for o the r wave

funct ions .

QUASI-FREE E X P E R I M E N T S ETC. l ~ l

2"6. W o o d s - S a x o n poten t ia l w i th hard core. - W a v e funct ions obta ined

f rom nuclear potent ia ls wi th ha rd core have been used in the l i terature. WATSO~ et al. (lo) in t roduced a 1.25 fm hard core. A similar potent ia l , with the hard-core decreased to 0.6 fm was found to be more sa t is factory in ref. (43).

T o t e t h a t in bo th cases the wave funct ion is nominal ly in a 1s-state, bu t due to the ha rd core, its general bohaviour is similar to the 2s-state, for not too

small intercluster distances. This can be seen in fig. 2. For bo th values of the hard core the F W t t M is comparable with the ones ob ta ined for the wave

funct ion of subsect. 2"5. Moreover, the t rend toward higher values of G~(0)

is confirmed (see t ab le I and fig. 3).

The general conclusions t h a t can be drawn f rom the above analysis, not

t ak ing into account the H e wave functions, are:

1) The value of the F W t t ~ of the m o m e n t u m distr ibut ion is not par-

t icular ly sensitive to the choice of the wave funct ion (table I) . Also, changes in the wave funct ion p a r a m e t e r s which result in a var ia t ion of d have a l imited

influence on the F W H M (fig. 3).

2) The value of G2(0) depends clearly on the choice of the intercluster

wave function, even if, once a compar ison is made for the same value of d,

the var ia t ions do not exceed :[: 20 %. On the other hand, the choice of the

p a r a m e t e r values is critical since relat ivc]y small var ia t ions in d m a y corre- spond to sizeable changes in G2(0).

80

60

~_ 40 :E

h_

20

, I ~ I , [ i I , 2 4. 6 8 10

R c ( fm)

Fig. 4. - FWHM of the interclustcr nmmentum distribution for the wave functions reported in fig. 2, as a function of the cut-off radius Re.

162 M. LATTUADA, F. RIGGI, C. SPITALERI and D. VINCIGUERRA

3 . - P W I A w i t h c u t - o f f r a d i u s .

I t has been pointed out earlier tha t , to t ake into account reabsorp t ion

effects, or even to s imulate the inadequacy of the cluster model for small

intercluster distances, a cut-off radius Re can be in t roduced in the intercluster

wave function, such t h a t

R(r) = 0 for r <= _Re.

Lot us now consider thc influence of introducing a cut-off radius in the

radial wave functions discussed in the previous section. In fig. 4 is repor ted

the behav iour of the F W H M of the m o m e n t u m distr ibutiou as a funct ion of Re.

,21 I ~ I I I I

10

- = : : : _ _ _ , : . . .

C 6 ' \

% \ \

\ &, \ \

',N, N. \

. \ - \ \

~ " ' ~ . . .

o" " i s 6 8 ~o Rr

Fig. 5. - Top va lue G~(0) of t he i n t c r c l u s t e r m o m e n t u m d i s t r i b u t i o n for the wave func t i ons r epo r t ed in fig. 2, as a f unc t i on of the cut-off r ad ius R c.

QUASI-FREE EXPERIMENTS ETC. 163

The curves are for all the wave functions described above except the HO. As expected from the arguments discussed in sect. 1, all curves emerge in the same one for Re > 5 fm. The variation of G2(0) vs. Re is reported in fig. 5. I t is evident here that all curves have the same behaviour oven if the absolute values are different.

I t is interesting to note that different wave functions may yield the same value ef G2(0), provided Rr is changed accordingly, usually by a small amount. At the same time the F W H ~ (fig. 4) does not change appreciably. In other words, when working within the PWIA with a cut-off radius, there is no straightforward way of choosing between different functional forms of the intercluster wave function.

The validity of this conclusion can be extended also to the DWIA. In this case, changes in the parameters of the absorbing and distorting potentials should have the same effect as changes in the strong-absorption distance introduced in the PWIA treatment by Re. Obviously, the DWIA gives completely different predictions (2~,42) for values of Ps where the PWIA pre- dicts zeroes in G2(ps). However, for eLi, thc first zero falls well beyond 100 MeV/c, so that it has no influence on the FWH2r and on the top value of the momentum distribution (see fig. 1).

4 . - C o n c l u s i o n .

From the analysis reported in sect. 2 and 3, a few conclusions can be drawn, concerning the study of the ~-d structure of 6Li. First, there is not a simple and selective way of choosing between the various in terchster wave func- tions proposed in the literature by just looking at QF data. Actually these experiments seem to probe essentially the eli cluster structure for large inter- cluster distances. A precise choice should then be based on more information, obtainable from a different kind of experiment, as in ref. (43).

Second, the value of P that can be deduced from QF experiments through eq. (2) depends largely on the value of R c, when working within the PWIA.

A similar conclusion can be predicted for the DWIA, when considering the variation of the potential parameters.

New, since changes in Rr not only influence the value of G2(0) but also

the FWttM, one may hope to obtain an unique value of P from different QF ex- periments, by adjusting R so as to fit the experimental widths of the momentum distributions. This procedure has already been adopted in a previous work (3~) concerning the measurement of the eLi(p, pd) QF reaction at 20 and 42 MeV incidcnt energy. The two experimental momentum distributions ( F W H ~ ~- = (55 =~ 3) ~eV/c and (60 • 2) ~eV/c) required R c-~ (5.2 • 0.6) fm and

Rr = (4.2 • 0.3)fm for the two cases, respectively. Appreciably the same

164 ~. L&TTUADA, F. RIGGI, C. SPITALERI and D. VINCIGUERRA

value of P was found ( P ~ 0 . 1 8 • and P ~ 0 . 1 7 ~=0.07 for the 20 and 42 MeV exper iment , respectively).

W e have then t r ied to app ly the same rout ine to other QF da ta on 5Li,

r epor ted in the l i terature. Only a few references repor t absolute values for the cross-section of for G~(ps) and could be t aken into account. The resul ts

of the analysis repor ted in table I I are ra ther discouraging a t first sight.

Actual ly the deduced values of P cover two orders of magni tude . However ,

a few words of caution mus t be said. The exper iment of ref. (19) gives a much

too large cross-section. P robab l y the sequential processes, which domina te

the react ion a t low bombard ing energy, still influence the da ta in the QF

region.

TABLE II. - Values el the parameter P (see text) obtained ]rom the analysis o/ several quasi-]tee experiments on 6Li as reported in the literature.

Reaction Energy (MeV) References FWHM (MeV/e) P

(p, 9d) 20 (as) 55 0.18 (P, 9d) 42 (32) 60 0.17 (9, pd) 19 (~:) ~ 50 0.30 --0.33 (p, pd) 156 (12) 72 0.51 (a, ad) 23.6 (5) 48 0.054 (a, ~d) 50 (3) 60 0.048 (d, 2d) 27 (13) 65 --70 0.13 (d, 2d) 7.5, 9, 10.5 (15) ~ 3 0 2.6--3.5

Heglect ing the findings of ref. (18), two clusters of P value can be identified.

The first with P = 0.13 to 0.5 is obta ined f rom (p, pd) and (d, 2d) exper iments . The second, with P values a round 0.05 comes f rom (~, ~d) exper iments . We have not found a sat isfactory explanat ion for this large discrepancy.

I n a n y case the present work indicates tha t , when affording the deduct ion

of the value of P f rom QF exper iments , great care mus t be t aken in order to

eva lua te the effects of the choice of the intercluster wave funct ion and of the

va lue of Rr (when working within the P WIA) , or of the potent ia l pa rame te r s

(when using the DWIA) .

�9 R I A S S U N T O

I1 valore della probabilit~ di clustering ~-d nel 6Li dedotto da esperimenti quasi liberi pub essere influenzato dalla scelta della funzione d'onda intereluster. Noi abbiamo esa- minato alcune forme funzionali spesso usate per descrivere il mote relative dei due cluster. ]~ state studiato l'effetto della scelta della funzione d'onda intercluster sulle informazioni dedotte analizzando dati di proeessi quasi liberi nell'ambito della appros- simazione impulsiva di onde piane.

QUASI-FREE EXPERIMENTS ETC. 16~

]~Ba311-CBO60~Hble 3KCllOpIIMeHTbl, KaK IIHCTpyMeHT ~Jl~l IICCJIe~oBaHH~ ~c~acTepnofi

crpyKryph, e l i .

Pe3mMe (*). - - BenHqHna BepOflTHOCTH 0 6 p a a o B a a ~ a-d KnacTepoa B SLi, nonyqeHHa~ H3 Kaa3H-CaO60~HblX aKcnCpHMCHTOB, MO~CT 3aaHceTl. OT ab l60pa Me~vr~acTepHo~t BOaaoao~i qbyHrttmi. MbI ~tccac~yeM ~ecroabKo ~yHr, u~onan~HbLX ~OpM, O6~lqHO ~cnonb- 3ycM~/x , R ~ o n . c a n n o t OTHOCHTe.r/sHOFO ,RBlt;~eH/4g ;~ayx K~aCTOpOB. HCCne,RyCTCI~I Bmi~tH~e ab t60pa Me>r/lyrnacTepno/i BO~HOaOi~ qbyHrtmH a a nHqbOpM~mm, rtoay~enHym H3 aHanH3a KBa3H-CBO60,IIHMX ~anm,LX a IIJIOCKOBOJ'IHOBOM HMHyflbCHOM npI46an~cenHa.

(') IZepeae()eno pec)artluef4.