HEAVY ION POTENTIALS : A CRITICAL REVIEW OF VARIOUS APPROACHES

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HEAVY ION POTENTIALS : A CRITICAL REVIEW

OF VARIOUS APPROACHES

D. Brink

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C5., supplément au n°\\, Tome 3 7 , Novembre 1976., page C5-47

HEAVY ION POTENTIALS: A CRITICAL REVIEW OF VARIOUS APPROACHES

D.M. Brink

Department of Theoretical Physics, 12 Parks Road, Oxford 0X1 3PQ

Résume.- Des méthodes semi classiques démontrent explicitement que l'amplitude de la dif-fusion élastique de particules alpha peut avoir des contributions significatives des on-des pénétrant à l'intérieur du noyau cible. Au contraire,la diffusion élastique d'ions lourds est principalement un phénomène de surface. Des méthodes de convolution donnent ur-ne bonur-ne description de la forme et même de la grandeur de la partie réelle du potentiel optique. Ces méthodes n'incluent pas les effets d'échanges et de polarisation et des cal-culs microscopiques indiquent que les- corrections dues à de tels effets peuvent être inpor-tantes.

Abstract: Semi-classical methods demonstrate explicitly that the elastic amplitude for a-particle scattering can have significant contributions from waves which penetrate the target nucleus. In contrast, heavy-ion elastic scattering is mainly a surface pheno-menon. Folding procedures give a good description of the shape and even the magnitude of the real part of the optical potential. These procedures do not include some

exchange and polarisation effects and microscopic calculations indicate that corrections from such effects can be large.

INTRODUCTION

Before discussing different approaches for calculating heavy-ion potentials, it is useful to understand the characteristics of a potential which are relevant for an accurate description

o f e x p e r i m e n t a l d a t a . For this reason in the first part of my talk I will discuss the qualitative relations between optical potentials, partial wave amplitudes and elastic scattering cross-sections. This question has been examined by many authors and Satchler [1] in particular has made an empirical investigation of families of optical potentials which fit experimental data and has noted their common features. From these studies he concludes that the data determine the strength of the real and imaginary potentials at the strong absorption radius. This question will be approached from a different point of view in this lecture, the aim being to use semi-classical ideas developed by Malfliet et al. [2], Knoll and Schaeffer [3] and others to see directly which properties of a

potential influence scattering amplitudes calculat-ed from it. I will also compare and contrast a-scattering with heavy-ion scattering.

The second part of this talk is concerned with different methods for calculating optical potentials for heavy-ion collisions. Folding models and theories based on energy density

forma-lism or on Hartree-Fock have many features in common, but give rather different results. Folding models tend to overestimate the heavy-ion real potential near the strong absorption radius unless there is a free parameter which can be adjusted to give the correct result. Energy density approaches, on the other hand, tend to underestimate the po-tential at this radius. It is interesting to try to understand the reasons for these differences. Another approach leads to the idea of proximity forces which have been discussed in a number of recent works l"_4]. An interaction potential in the proximity theory is a product of a simple geo-metrical factor and a universal function of

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C5-48 D.M. BRINK separation of t h e surfaces of t h e i n t e r a c t i n g nu-

c l e i . The theory is s o general -chat it should include folding approaches and energy density approaches a s s p e c i a l cases.

The i n t e r a c t i o n energy in an energy density formalism i s e s s e n t i a l l y t h e energy surface of t h e generator coordinate approach t o a microscopic theory of heavy-ion interactions. This f o m l i s m leads t o a shallow energy-dep,endent o p t i c a l po- t e n t j a l [ 5 , 6 1 . In t h e resonalzing -up approach exchange e f f e c t s lead t o a highly non-local energy- dependent p o t e n t i a l , but t h e r e are indications C71 t h a t t h i s non-local i n t e r a c t i o n can be replaced by a deep almost l o c a l p o t e n t i a l . Pauli exchange e f f e c t s a r e taken i n t o account by requiring t h a t t h e wave function of r e l a t i v e motion of t h e two nuclei should be orthogonal t o c e r t a i n forbidden s t a t e s [71 of r e l a t i v e motion.

The e l a s t i c cross-section f o r heavy-ion s c a t t w - i n g is determined by phase s h i f t s which can i n t u r n be calculated from an i n t e r a c t i o n potential. An a l t e r n a t i v e approach t o parameterising a p o t e n t i a l

i s t o parameterise phase s h i f t s d i r e c t l y . An ad- vantage of a p o t e n t i a l description i s t h a t a poten- t i a l determined from e l a s t i c s c a t t e r i n g c+n be used i n calculations of reaction cross-sections using DWBk o r coupled-channelmethods. There a r e , however, appmximate formulations of reaction theories i n which t h e e l a s t i c s c a t t e r i n g p o t e n t i a l appears only through t h e e l a s t i c phase s h i f t s L83. If these f o m ~ ~ ~ l a t i o n s a r e successful one can ask i f t h e po- t e n t i a l description of interactions between nuclei i s useful o r necessary.

POTENTIALS AND SCATTERING AMPLITUDES

This section w i l l be devoted t o a discussion of q u a l i t a t i v e r e l a t i o n s between o p t i c a l potentials, p a r t i a l wave amplitudes and e l a s t i c s c a t t e r i n g C-S-

sections. The semi-classical methods applied t o nu- clean s c a t t e r i n g problems by Malfliet e% al.C21, Knoll and Schaeffer C31 and others C91 give express- ions which l o c a l i s e t h e regions of t h e o p t i c a l p o - t e n t i a l which detewnine various c h a r a c t e r i s t i c s of s c a t t e r i n g amplitudes. These r e l a t i o n s can be understood b e s t

b;r

studying some examples. I have chosen examples from a-scattering and heavy-ion s c a t t e r i n g C9lwhich i l l u s t r a t e a range of d i f f e r e n t e f f e c t s .

( a , 1 6 ~ ) s c a t t e r i n g , El& = 104 EleV: The angular d i s t r i b u t i o n calculated with Saxon-Woods p o t e n t i a l parameters Vo = 8 2 MeV, Wo = 29.7 MeV, Ro = 3.53 fm and a = 0.72 fm i s shown i n f i g . 1 .

Opticalpawmeters were taken from Harakeh e t a l . [ I 0 3 and give a good f i t t o t h e experimental data. The angular d i s t r i b u t i o n shows d i f f r a c t i o n

s t r u c t u r e a t forward angles and has a pronounced bump f o r Q

-

50' which i s c h a r a c t e r i s t i c of t h e nuclear rainbow s c a t t e r i n g discussed by Goldbe~g and Smith C 11

I .

I n t h i s example, t h e grazing o r strong absarp- t i o n angular momentum, defined as t h e value of R f o r which t h e transmission coefficient T _R=

4

is R = 18 and f i g . 2 shows t h a t t h e real p a r t of t h e

g

t o t a l p o t e n t i a l V(r) which Is t h e sum of t h e nuclear Vn(r)

,

Coulomb Vc(r) and c e n t r i f u g a l poten- t i a l s Vn(r) has no b a r r i e r f o r angular momenta near

R This i s a case where t h e simplest version of g'

!&B can be used t o c a l c u l a t e t h e nuclear s c a t t e r i n g phase s h i f t

I n e q . ( l )

m

and rc a r e t h e nuclear and Coulomb turning points, i . e . they are zeros o f t h e inte- grands of t h e f i r s t and second i n t e g r a l s in eq.Cl), and R i s any point outside t h e range o f t h e nuclear i n t e r a c t i o n V n ( r ) . The nuclear turning point

m

i s complex because t h e o p t i c a l p o t e n t i a l V (r) i s

n

Fig.1. Cross-section 5 for(a-160) s c a t t e r i n g

(El& = 104 NeV) from a quanta1 o p t i c a l model calculation. oa is calculated from t h e WKB

amplitude e q . ( l ) .

complex. The variation of r with R i s shown i n n

f i g . 3 . Also shown a r e t r a j e c t o r i e s of other zeros of E-V(r) which a r e too f a r from t h e real r-axis -to make any s i g n i f i c a n t contribution t o t h e scatter-

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HEAVY ION POTENTIALS C5-49

16

Fig. 2 . ReV(r) f o r (a- 0 ) scattering f o r three angulm momntum values. Centre-of-mass energy i s E = 83 MeV.

the

WKB

cross-section ad. It is very

similar

t o the quanta1 cross-section og obtained from a

numerical solution of the Schddinger equation us-

ing a standard optical model code. Fig. L i shows

the semi-classical deflection function

2

e t a )

=

2 d (ReG)/dR and lqR1 where qR=exp(2ib(R)) i s the scattering amplitude. The graph of

lnR1

indicates t h a t the absorption increases smoothly a s 1 decreases. It i s due t o the increasing effect of the absorptive potential I m V n ( r ) as the distance of closest approach decreases. The nuclear rainbow angle i s Bnr : - 6 0 ~ . P a r t i a l waves with 7hRG 17

s c a t t e r into t h e angular region 20' -+ 60° and

produce the rainbow peak i n the cross-section. The absorption is strong for these angular mmenta but

i s weak enough t o allow a peak i n t h e cross-section.

The height of the peak is very sensitive t o

Wo.

A s

expected, the nuclear rainbow angle corresponds t o a c l a s s i c a l turning p i n t with ral R0(cf.Fig.3)

and i s due t o the inflection in the nuclear potential at r = Ro.

~ a , ~ O c a ) scattering, El& = 29 MeV: Figs.5-8 i l l u s t r a t e some r e s u l t s calculated with Saxon- Woods parameters Vo = 183.7 MeV, Wo= 16.6 MeV, Ro=

4.925 fm, = 0.564 f ~ n . This i s an adaptation of a potential given i n r e f .

C

1 2 3 which f i t s the q r i -

iP

mental elastic. scattering a t forward and backward angles, bat not f o r intermediate angles. The po- t e n t i a l gives a barrier i n V ( r ) (fig.5) f o r angular

IIm

a -

160 ( 10L MeV)

Fig.4. The semi-classical deflection function $ (R! and the reflection coefficientlnR12 f o r (a-100 ) scattering.

Fig. 3. Complex turning points f o r (w.160) scattering.

-

mmenta near the grazing momentum R

-

13, and it

22

can be shown t h a t the scattering amplitude is a sum of two parts C93

The f i r s t part fg(€l) corresponds t o a wave reflected a t t h e potential b a r r i e r and has its main contribution from p a r t i a l waves with R

>

R The second

g'

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C5-50 D.M. BRINK

contribution f o r low p a r t i a l waves R L R The semi- g'

classical derivation of eq. ( 2 1 i n r e f .

C

91 contains some b a r r i e r penetration corrections which a r e omitted i n ref.[3]. The pattern of complex turning p i n t s f o r the (a-40Ca) scattering i s shown in f i g .

6. The b a r r i e r contribution fB(8) t o t h e scattering amplitude comes from a reflection a t the external turning point rl while f I @ ) cornsponds t o a reflection from the internal turning point r3.There i s no reflection f m the second turning point r2

i n the upper half of the complex r p l a n e ; but t h i s turning point does affect the b a r r i e r penetration corrections. I f r2 i s near rl then the b a r r i e r penetration corrections are iniportant.

Fig. 7 gives the contributions oB(0) =

1

fB(0)

1

2

and a (0)=lfI(O)I t o the t o t a l cross-section ~ ( 0 ) . I

Fig. 5. Real part of V ( r ) f o r ( a , 'Oca) with

R = 1 5 .

The b a r r i e r amplitude fB(0) mkes the m j o r contribution f o r small scattering angles while fI(8) gives the dominant contribution a t backward angles. The cross-section f o r intermediate angles i s a

L o ~ a [ a , a ) ' O c a 129.0 MeV) c

--.--.

--. optical potentlal

real potentlal 0,

a

'

Fig.6. Trajectories for the turning points rl, r 2 ,

r3 f o r the ( a

-

4 0 ~ a ) potential. The dashed curves are trajectories f o r the r e a l potential only. Numbers indicated angul-ar momentum values.

r e s u l t of complicated interference effects between fB(8) and fI(8). The magnitude of t h e p a r t i a l wave amplitude f o r (a- 'Oca) scattering i s shown i n fig.8. Both

nB

and qI vary s m t h l y with R. The s t r u c t w e i n l q l =

InB+nI(

f o r R ; 1 0

is due t o interference between

nB

and rll.

0.10~

'OCa ( a , a ) ' O C ~ (29.0 MeV)

Fig. 7 . Cross-section f o r (a

-

'Oca) scattering f o r the potential parameters in the t e x t . The contributions 5 ( 0 ) from the barrier wave and aI(0) from the internal wave are also indicated.

1 9 ~ 1 ( 4 0 ~ a ( a , a ) ~ Q ~ a ( 2 9 . 0 MeV)

I

cu W = 16 6 MeV

0 - - 0 - - 0 W = 26.6 MeV

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HEAVY I O N POTENTIALS

How do these r e s u l t s change f o r scattering processes involving heavy nuclei? There a r e several m j o r changes, all of which r e f l e c t the diminished importance of t h e i n t e r i o r region of t h e optical potential for e l a s t i c heavy-ion scattering,

The f i r s t change concerns the nuclear rainbow effect which was a p d e n t feature of t h e

(a, 160) example discussed earlier in this section and has been seen i n many other e l a s t i c a-scattering experiments. There a r e , however, no cases of the nuclear rainbow effect known when heavier projectiles are scattered e l a s t i c a l l y . A possible reason i s t h a t the imaginary part of the optical potential i s so large in heavy-ion potentials t h a t only t h e extreme t a i l of the optical potential contributes t o e l a s t i c scattering. Alternatively, Knoll and Schaeffex C33 have pointed out t h a t the nuclear r a b b m can occur a t s m l l e r angles ;than the Coulomb one and be Mdden by the Rutherford scattering.

Although the nuclear rainbow effect seems not t o appear i n e l a s t i c heavy-ion scattering, there

are indications t h a t it may show up

in

deep in- e l a s t i c scattering o r quasi-fission a s a focusing of the reaticnpmducts i n t o a certain angular region, f o r example i n the reactions i n i t i a t e d by K r + B i a t 525 MeV L133.

I n cases where the r e a l part of the nuclear potential i s stmng enough t o give a b a r r i e r

in

the t o t a l potential, the scattering amplitude can be

divided a s i n eq. (3). The amplritude due t o the internal wave fI(0) can make significant contributions in e l a s t i c a-scattering, but i n heavy-ion systems it i s important only f o r l i g h t ions a t low energies when resonance effects occur. Normally, the optical model absorption i s so s t m g t h a t the internal wave which penetrates o r passes over the b a r r i e r i s completely &sorbed so that fICO)z 0

and only the b a r r i e r region contributes t o the scattering, f ( e ) = f (8).

The example of F 6 0 , l 6 0 ) scaYtering a t 50 MeV studied by Knoll and Schaeffer [ 3 1 i s a typical example f o r light-ions a t intermediate energies o r heavier-ions at low energies. They use a Saxon-

Woods potential with parameters Vo = 50 MeV,

Wo

=

20 MeV, Ro = 6.05 fin and a = 0.6 fm. Tne r e a l part of the nuclear potential i s strong enough t o give a b a r r i e r in the t o p 1 potential and the turning point pattern resembles f i g . 6. However, t h e absorption i s strong and the wave f I ( 8 ) reflected a t the internal turning point r3 which was important

Fig. 9 . Cross-section f o r ( 1 6 0 ,160) e l a s t i c scattering without synme.Wization. The barrier penetration correction factor N i s included in oB and neglected i n u BO.

in the ( a , 4 0 ~ a ) example i s t o t a l l y absorbed in t h i s case. The semi-classical theory gives the p a r t i a l wave amplitude a s C93

rl

=

exp (2 i ~ ~ ) / N ' ( i ~j (3) where6,is t o be calculated from eq. ( 1 with rn a s the most external turning p o h t and N(is> i s a

correction f o r b a r r i e r penetration effects. The quantity E i s reiated t o the action integral

between the turning points r2 and r on each side

1

of the barrier. The barrier penetration correction factor N i s significantly different from unity only i f

1

E

1

,< 0.5. A simple estimate gives

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C5-52 D.M. BRINK where rg i s the b a r r i e r position and wB=,&yttB/p)

i s the o s c i l l a t o r frequency of a p a r t i c l e ~ tm s s h k~ eq-1 t o -the reduced mass of the two nuclei nwving in a potential obtained by inverting the barrier. In the present example min

I

E

I

"

0.25 and the b a r r i e r penetration correction makes some difference t o the an- distribution only a t

large angles Cfig.9).

Even when the r e a l potential is strong enough t o give a , b a r r i e r in the t o t a l potential, the absorption in the barrier region can be so large t h a t most of the incident flux i s absorbed by the

imaginary part of V ( r ) outside h e barrier. This

s i tuation occurs when min

I

E

/

>

0.5. Then b a r r i e r penetration effects are not important, the e k t i c scattering is determined by the potential outside the b a r r i e r and the b a r r i e r height i t s e l f is not relevant. A deep r e a l potential with strong absorption in the barrier region is indistinguishable from a shallow r e a l potential with no barrier provided the -two potentials are similar in the strong absorption region.

I n t h i s surface absorption l i m i t t h e nuclear phase c m be calculated with t h e simple one-turning point WKB forrrolla (1 1

.

Ekpmding eq. (11 t o f i r s t order in V , b ) gives the classical p e m b a - t i o n r e s u l t andteven t h i s i s quite accurate

C3l

In eqs.(6) and (7) d i s the distance of closest approach f o r a Rutherford o r b i t with angular mo-

mentum X = 2

+

4,

vo i s the asymptotic r e l a t i v e velocity, k = pv0/+, is the wave number and h = z1,z2 g2/*;v0 the sonnnerfeld parameter. They a r e related by (?,/dl2 = k2

-

2k n/d. For an

exponential potential

Vn(r) =

- CVo

+

i Wo) exp

-

Cr-Ro)/a eq.(7) gives

The strong absdrption radius dl C11 i s the value 2

of d f o r which the transmission coefficient TR= $. This corresponds t o Im(26) = 0.346. Then eq.C8)

can be used t o estimate the d, i f the imaginary 2

p-wt of Vn(d) i s known. The e l a s t i c scattering of by '08pb a t 96 MeV i s an example where

1 2

Fig. 1 0 . Cross-section f o r ( C , 2 0 S ~ b ) e l a s t i c scattering a t 96 MeV C141

surface absorption i s strong. The angulw distribution i s fit-bed well [I41 by a Saxon-Woods potential with Vo = 40 MeV, Wo = 25 MeV, RO=10.31

fm, a = 0.56 fm. Solving eq.C8) gives d& = 12.2 f m

which is the sane a s the value obtained by calcula- t i n g from a f u l l opticalnmdel calculation.

Fig.10 gives a sketch of 0 (e)/aRCe) f o r t h i s

example. The grazing angular mmentum i s R = 52

,

i?

arg q = 2651

-

32O f o r R = R and the transmission g'

coefficient f a l l s from 0.9 t o 0.1 in a range

AR F 12,

ANGUIAR DISTIiIBUTIONS

There are several different approaches t o t h e problem of calculating angular distributions by semi-classical methods. Their camon s t a r t i n g point is the Poisson s m t i o n . formula which gives C15,161

with :1 (0) =

6

JX

exp CiX(zei 2rrm)lqCA-&)dX (10) This is a useful r e s u l t because generally only a few terms i n the sumrration (9) a r e important and the different terms :1 (0) have a p i c t o r i a l association with classical trajectories. Some commn combinations a r e i l l u s t r a t e d i n fig.11 taken

16 1fjO) from r e f . C151. The cross-sectknsfor ( 0 , scattering a t 50 MeV Cfig.9) f o r (3 < 20° and (I2C, 2 0 ' ~ b ) scattering a t 96 MeV (fig.10) f o r

0 < 45O have conWibutions m i n l y from

I-

(fig.11). 0

This term has a d o r h a n t Coulomb contribution

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HEAVY I O N POTENTIALS

Fig.11. Classical t r a j e c t o r i e s corresponding -to (a) forward angle o s c i l l a t i o n s , (b) intermediate angle oscillations and ( c ) backward angle oscillations.

d i f f r a c t i v e and refractive effects of t h e nuclear surface. These contributions i n t e r f e r e t o give a Fresnel tylZe diffraction pattern. For 0 > 30°,

1: and 1: contribute. Both terms vary smoothly with angle and interfere t o give the diffraction structure shown in fig.9. Backward angle oscilla-

+

tions are pmduced by interference of I. and

I

;

.

Different s d - c l a s s i c a l approaches vary i n t h e methods used t o calculate t h e integrals I:(8) and i n the interpretation of results; Methods are discussed i n many references. I n t h i s lecture I w i l l make a few remrks about interpretation. The

gross features of e l a s t i c angular distributions are

determined by two parameters, the Coulomb parameter n and the grazing angular moment R The

8'

r a t i o of t h e e l a s t i c cross-section t o the Ruther-. ford cross-section resembles a Fresnel diffraction pattern when n >> 1 with o/oR =

I,

f o r the classi- c a l grazing angle, When n

5

1 t h e angular distribu- t i o n i s l i k e a Frauenhofer diffraction pattern. The refractive effects of the nuclear potential m e subtle but important modifications t o these simple patterns.

There are two interpretations of angular distributions in t h e l i m i t n >>I (eg. ( 1 6 ~ , 2 0 8 ~ b ) i n f i g . 10). I n one view C16lelastic and i n e l a s t i c heavy-ion collisions are dominated by d i f f r a c t i v e scattering of t h e Fresnel type. There are modifications of t h e pattern because the cut-off i n t h e scattering amplitude near t h e grazing angulax mmentum i s smooth rather than sharp, and because there is a r e a l nuclear phase. These modifications produce a perturbation of the basic Fresnel pattern. I n t h i s view t h e scattering i s mainly influenced by

t h e imaginary p m t of t h e o p t i c a l potential. This approach i s associated with a methcd f o r calculating t h e scattering amplitude i n which the leading Fresnel amplitude i s m d i f i e d by corrections depending on the detailed angular momentum dependence of m he p a r t i a l wave amplitudes.

An alternative interpretation i s t h a t the fornard angle oscillations in a case l i k e fig.10 are pro- duced by interference between two branches of a c l a s s i c a l deflection function near t h e Coulomb rainbow angle C17,183. The integral (TO) i s calculated by the stationary phase method C181 o r t h e saddle point method C33 and t h e interfering c l a s s i c a l paths correspond t o points of stationary phase o r to saddles i n one of these methods. Advo- cates of t h i s method argue C191 t h a t t h e angular distribution f o r e l a s t i c scattering i s governed by t h e Coulomb rainbow angle in t h e c l a s s i c a l deflection function and t h a t t h e angular distribution i s mainly expected t o depend on t h e real of t h e nuclear potential with perturbations due t o t h e imaginary part.

Empirical investigations by Satchler C11 and collaborators C141 seem t o indicate t h a t t h e long range parts of both t h e r e a l and t h e imaginary parts of t h e optical potential are important f o r f i t t i n g e l a s t i c scattering data. I n examples l i k e

(c,

2 0 8 ~ b ) t h e strong absorption radius i s mst sensitive t o t h e iTFagk.ary part of the potential, but t h e r e a l part of the optical potential a t t h e strong absorption radius is also r a t h e r well d e t e ~ mined. This second condition on the r e a l potential could be almost equivalent t o a reqdrement'that t h e r e a l potential should give a correct value t o t h e Coulomb rainbow angle C191.

CALCULATING HEAVY-ION POTENTIALS

The discussion i n t h e preceding sections shows t h a t t h e e l a s t i c scattering of heavy-ions i s sensitive only t o the extreme t a i l of an internu- c l e a r interaction potential, except possibly i n some cases involving l i g h t e r ions a t low energies. A number of approaches have been used f o r calculating the r e a l p a of the potential. Usually the imaginary part is treated phenomenologically, a l - though it seems t o be a t l e a s t as important as t h e r e a l p a r t of t h e potential C 201.

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C5-54 D .M. BRINK

include exchange and polarization correctims. In the single folding model one takes an o p t i c a l p o t - e n t i a l U _N(r) f o r scattering of a nucleon from me nucleus and folds t h i s w i t h the density distribution of the other nucleus. The p a r m t e r s of U N ( r )

are chosen empirically f m f i t s t o nucleon-nucleus scattering data. In most calculatims t h i s m d e l overestim3.tes the heavy-ion potential near the s t r m g absorption. radius by a factor of two o r m r e . Hcwever, a recent investigation by Rickert- sen and Satchler L211 shms t h a t the simple folding m e 1 can give b e t t e r results i f a shape, which differs from Saxon-Woods

,

i s used f o r U N ( r ) .

The double folding &el represents the heavy- ion potential in the form

3 3

U ( r ) =

I/

pl(rl) p2(r2) v(r12) d rld r2 (11)

where r12 = r

+

r2

-

r and v i s an effective nuc- 1

lea-nucleon interaction. This &el gives a good account of e l a s t i c scattering data provided the effective i n t e r a c t i m v has a reasonably short range and i t s strength i s taken as a f r e e para- m t e r . Effective interactions v with a range which i s too big

are

excluded because they give heavy-ion potentials which disagree with experi- rrental data. For example, an effective interac- t i o n derived from the long range part of the Hamada-Johnston interaction does not give a s a t i s - factory potential i n the double folding model C141.

There, the f a u l t l i e s with the long range one-pion exchange component of the interaction. I f the effective interaction has a reasonably short range, the U ( r ) can be expanded as

3 2 3

U =

A

[ f p _{1 2}p d r

-

* < r > / Vpl.Vp2d rli.. ( 1 2 )

where A i s the volwre integral of the effective

2

interaction and <r > i s i t s man square radius. 2

This formula shms t h a t <r > i s a convenient rneasure of the range of an effective interaction f o r the purposes of the double folding mdel. A

2 ' 2

value <r > = 11.8 fm appropriate f o r a one-pion exchange potential seems t o be too large t o f i t the heavy-ion data. An effective i n t e r a c t i m derived from r e a l i s t i c g-matrix e l e m t s by Love and

2 2

Satchler r221 has <r > = 3.11 fm and even f i t s the magnitude of U ( r ) t o within 10%.

The single and double folding models neglect certain exchange effects. They also neglect contributions t o the internuclear potential caused by polarization of the nuclei by the nuclear and Coulomb fields. Some idea of the magnitudes of these effects can be f o m d by studying mdels

ii

I

fnl

Fig.12. (160,160) potential from ref.C231. The

various curves are explained in the text. with simple effective interactions but with effects of exchange and polarizatim included.

An

example of such a calculaticn i s the study of the (160 ,160) i n t e r a c t i m using Skyrrre' s effective interaction and i t s self-consistent extensim

C231. In this model the t o t a l energy of the

1 6 16

( 0, 0) system a t a fixed separation R i s identified with the heavy-icn potential. The t o t a l energy i s computed using completely anti- symcstrized wave-functions s o t h a t e f f e c t s of exchange and the Pauli exclusicn principle are included exactly. Tne p m t e r s of the Skyrm

interaction were f i t t e d t o nuclear binding energies so the model should give a reasonable estimate of the shape and strength of the internuclear interaction. The results i l l u s t r a t e d in

f i g . 1 2 are taken from a review t a l k by Mosel C231.

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HEAVY ION PI

the unperturbed wave functions of the isolated 160 nuclei neglecting anti-synnretry, while the curve FAS includes so= part of the exchange corrections. The difference between the curves FAS and PES i s mainly a k i n e t i c energy e f f e c t . When the two nuc- l e i overlap the Paul? principle forces sorre nuc- leons i n t o higher orbits increasing the mutual k i n e t i c energy and making the mutual attraction l e s s strong. The curve SCPES i s the r e s u l t of a self-consistent calculation. The interacting nuclei were allcwed t o change t h e i r wave-functions t o minimize the t o t a l energy. Such a self-cmsist- ent procedure i s a way of calculating s t a t i c p o l e ization effects. Tnis f i n a l curve i s i n reasonable agreerrent with the empirical curve. This calculation shows t h a t both exchange and polariza- t i o n effects can be large; each can change the interaction strength by a f a c t o r of two. In the

(160,160) example they almost cancel, but there i s

no guarantee t h a t t h i s w i l l happen f o r other pairs of nuclei. Should polarization e f f e c t s be included 1 ' the heavy-ion potential? The answer t o this

question depends on the t i m required f o r polarization compared with the collision t i m e .

A r a t h e r different a p p m c h t o heavy-im potentials i s based on t h e idea of proximity forces developed by the Berkeley group C41 which generalizes som e a r l i e r ideas of Wilczynski C261. In t h i s theory t h e interaction energy of two nuclei

given by a formula

U = & e ( D ) d a

The i n t e g r a l i s taken over a surface &J between the interacting nuclei, D i s the separation distance between the smfaces of the two nuclei measured perpendicular t o the s u r f a c e g , a d e(D) i s an interaction energy per unit area of two para- l l e l nuclear surfaces separated by a distance D.

Approximate calculation of t h i s i n t e g r a l f o r i n t e r - acting spherical nuclei with r a d i i R1 and R gives

2 U(s) = 2 IT

R

5; e(D) d (D) (13) where s = r

-

(R1

+

R2) and the average radius

= R ~ R ~ @1+~2 )

.

This corresponds t o a force

F(s) = - 2 IT

R

e (s). (14)

A very nice feature of eq. (13) is t h a t t h e i n t e r - action energy U ( S ) of two nuclei with surfaces separated by a distance s i s given in t e r m of a universal function e ( s ) . I f e ( s ) i s kncwn o r can

be calculated f o r one p a i r of nuclei then one i m d i a t e l y has information about other pairs of nuclei. The idea of t h e proximity formula i s based

on the liquid drop model, but the formula i s very general and applies a l s o t o other theories. For example, suppose the interaction energy i s

represented by the folding f o m l a

which corresponds t o a double folding model with a &-function effective interaction v. I f the densities pl and p:, have Saxon-Woods shapes

then the integral (15) can be evaluated approxi- mately using the methods i n t h e appendix of r e f . C271 t o give

2 - s d s

U(s0)" 2nAoO

R

J s O ( w a )

where s o = ( r

- R1

-

R 2 ) and r i s t h e separation between the centres of the two nuclei. This has exactly t h e proximity form with a particular expression f o r e ( s ) . Eq.(16) has been derived assuming t h a t s o and a are both much smaller than R1 and R2.

Calculations based on t h e energy density approach C261 o r on Skyme's interaction with t h e Thomas-Fee approximat ion C 27 I are generally consistent with t h e predictions of t h e proximity model, though there a r e some small deviations. For example, the outer part of the potentials i n r e f . C271 were f i t t e d by Saxon-Woods potentials. Accord- ing t o the proximity m d e l the surface diffuseness should be constant. The r e s u l t s i n r e f . C271 give values between a = 0.56 f o r ( 1 6 0 , 1 6 0 ) t o a = 0.66 f o r (Pb, Pb). Also, the miuumforce between two nuclei should be given by the formula C41.

F- =-27

E

e(0) = TI

Ey

where y i s t h e surface energy coefficient f o r 2

nuclear matter ( y l 1 MeV/frn )

.

Values of y ex- tracted from heavy-ion potentials i n ref.C271 vary

2

between 0.70 MeV/fm f o r (160,160) t o 0.90 P4ev/fm2 f o r (Pb,Pb) with m s t values concentrated near 0.85

2 MeV/fm

.

The proximity formula w i l l certainly be a very useful guide f o r the study of heavy-ion potentials because of i t s simplicity and generality. There are, however, a number of reasons why it should be used with some caution. It should be accurate i n cases where s << R and R,. A t the strong absorp-

L

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C5-56 D.M. BRINK

polarization effects should follow the proximity f oysrmla.

Bibliography

]

G.R. Satchler, Pmcs. Conf. on Reactions between Complex Nuclei, Nashville (North-Holland 1974) Vol. 2, 174; Symposium on Macroscopic Features of Heavy-Ion Collisions, Argonne, Illinois, April 1976.

[2] R.A. Malfliet et al. Phys. Lett.

*

(1973)238.

T.

Koeling and R.A. Malfliet, F'hys. Reports 22C (1975) 181.

13

1 J. Knoll and R. Schaeffer

,

Annals of Phys

.

97

(1976) 307.

14

]

J. Blocki et al. Laurence Berkeley Report LBL-5014 (1976).

[5J D.H. Brink: and F1. Stancu, NucI. Phys. A243 (1975) 175.

r61 P.G. Zint and U. Mosel, Phys. Lett. - 56B (1975) 424.

173 S. Saito, Prog. Theor. Phys.

5

(1969) 705

B.

Buck et dl., preprint, Oxford 11/76 (1976). [8] S. Landowme et dl., Nucl. (1976) 99. [9

]

D.M. Brink and N. Takigawa, preprint

,

Oxford

(1976).

[ j 0 M.N. Harakeh et al, preprint Groningen KV-63

(1976).

[ll] D.A, Goldberg et dl., Phys. Rev. (1974) 1362.

[l21 G. Gaul et dl., Nucl. Phys.

A137

(1969) 177. r13

-

1

H.H. Deubler and K. Dietrich, F'hys. Lett.

56B

(1975) 241.

1141 J.B. Ball et dl., Nucl. Phys. A252 (1975) 208. [lcl N. RowLey and C. Marty, preprint Orsay

PNO/TH 76-3.

1161 W.F. Frahn, Heavy-Ion High-spin states and Nuclear Structure, Vol.1

(IAEA,

Vienna,1975) p.157.

W.E. Frahn and

D.H.E.

Gross, to be published. W.E. Frahn and K.E. Rehm, Confer- ence on Nucl.Phys.with Heavy-Ions, Caen(1976) 1171 R.A. Malfliet et dl., F'hys. Lett.

*

(1973)238. 1183 R. da Silvera, Phys. Lett.

9

(1973) 211. [191 P.R. Christensen and A. Winther, preprint.

LZOJ

P.J. Moffa et dl.,

Phys. Lett. - 35 (1975) 992. kl] L.D. Rickertsen and G.R. Satchler

,

Conference on Nucl. Phys. with Heavy-Ions, Caen (1976) 5. [22] W .G. Love and G .R. Satchler

,

ConFerence an Nucl

.

Phys. with Heavy-Ions, Caen (1976) 5.

H. Flocard, Phys. Lett.

(1974) 129. U. Mosel, Symposium on Macroscopic Froperties of Heavy-Ion Collisions, Argonne (1976). [24] J. Wilczynski, Phys. Lett.

9

(1973) 484. [2g D.M. Brink and N. Rowley

,

Nucl. Phys

.

A219

(1974) 79.

1261 C. Ngo et dl., Nucl. Phys. A= (1975) 237.