Seediscussions,stats,andauthorprofilesforthispublicationat:https://www.researchgate.net/publication/252859816

Thenucleus-nucleuspotential

ArticleinNuclearPhysicsA·March1976

DOI:10.1016/0375-9474(76)90501-7

CITATIONS

4

READS

4

2authors,including:

HenryGiacaman

BirzeitUniversity

29PUBLICATIONS788CITATIONS

SEEPROFILE

Availablefrom:HenryGiacaman

Retrievedon:11August2016

2.N I NuclearPhystcs A259 (1976) 157--172, (~) North-HollandPublishing Co., Amsterdam

Not to be reproduced by photoprmt or microfilm without v, ratten permission from the pubhsher

T H E N U C L E U S - N U C L E U S P O T E N T I A L )

H R. J A Q A M A N and A Z M E K J I A N

Department of Phystcs, Rutgers University, New Brunswtck, New Jersey 08903

Received 10 N o v e m b e r 1975

Abstract: Using the genera tor coordinate m e t h o d to invest igate the vahd~ty o f the use o f ,t potent ia l to describe heavy ~on elastic scattering, it is f ound that , under certain condi t ions , art effective potentxal exists These condi t ions are generally satisfied by heavy ~ons, especmlly for the po ten tml ' s tazl T he po ten tml ob ta ined includes a kinetic energy cont r ibu t ion The poten tml ~s fur ther invest igated us ing a dens i ty-dependent delta interact ion

1. Introduction

The generator coordinate method (GCM) 1,2) offers one of the most practical frameworks for the description of nuclear reactions between heavy ~ons. There have been numerous investigations of thts method m recent years [see e g., refs. 3-6,8), and references c~ted in ref. 2)]. In this case one starts f rom the Griffin-Hill-Wheeler ( G H W ) equation and solves for wave functions m the continuum. These solutions have to be carried out numerically [see e.g refs. 3,4)] since the G H W equation is an integral equation; th~s prevents one f rom gaming insight into the nature of the solu- tion, especially since one is accustomed to solving a Schrbdlnger-type differential equation. One would also hke to find a connection between the solutions to the G H W equahon and the phenomenological optical potentials obtained by fitting the data m heavy ion scattering. Apar t f rom these theoretical interests, the numerical lnte- grat~on and solution of the G H W equation meets with some techmcal dlfficulhes 4). Furthermore, most calculations usmg the G C M have so far been hmited to very light ions (mostly e-s), and although probably not out of the question, ~ts apphcat~on to heavier ions would entail a huge amount of calculation that has been the mare ob- stacle so far. In this paper we will at tempt to transform the G H W equation into a dlfferentml (Schr6dmger-type) equation that will allow us to get a better insight into the processes taking place and to obtain the solutions more easdy In particular, th~s method gives us an effectwe potentml for the interaction between two heavy ions that can be compared with phenomenolog~cal ophcal potentmls.

In sect. 2 we review the G C M as it has been apphed to heavy ~on reachons, and then we use ~t in sect. 3 to get the differential equation for the wave funchon of relative motion and consequently an expression for the nucleus-nucleus potentml and the condition of ~ts vahdity Thts potentml IS then further invest:gated m sect 4, where

t Suppor ted m part by the Na t iona l Science F o u n d a t t o n (grant no N S F GP39126X)

157

158 H. R. JAQAMAN AND A Z MEKJIAN

the e-~ 60 potential is calculated using a density-dependent force, and then used to calculate cluster states m 2°Ne The ~-~60 example is chosen partly because it is easy to get analytic expressions for it and partly because of the avallablhty of other calculations In general, however, these calculations have to be done numeri- cally, and the expressions given m subsect 4 2 pay special at tenhon to this The 1 6 0 - 1 6 0 potentml is also discussed in subsect 4 2

2. Review of GC~VI for reactions

In the G C M we start w~th a wave function of the form (.Y/is the ant isymmetnzmg operator):

A1 + A2 qb(x. ~) = ~¢ [ H ¢,(r,, ~t~)] = det {¢,(r,, ~)}, (1)

z=l

that has A~ + A 2 nucleons distributed into a number ). of potential wells (here two). each well characterized by a set =; of parameters The final many-body wave function ~s the hnear combination

~(x) = J- ~(x, a)f(~t)d~ (2)

Applying the Ritz varmtional principle to ~b. we get the generator coordinate (GHW) wave equation

f[H(~, ~')-El(a, ~t')]f(~t')d~' - 0. (3a)

where

II( , { 11 a')J = Uj

~')~ (3b)

These involve mtegratlons over the A I +A2 smgle-partlcle coordinates {r,} m the system Although the use of the Ritz varlatlonal principle is acceptable for bound state solutlons, ~t does not necessarily follow that it can be used for continuum solu- tions, However, the use of the G H W equatlon can be justified for these soluhons w~th or wlthout the use of some varlattonal prmclple 4, s). It is also known that the G C M breaks down for reactions between two unequal fragments 4,6); therefore we w:ll limit ourselves to two equal fragments. In the case of two unequal fragments, the wave function has a center-of-mass (c .m) spunoslty m it (because the total c m motion does not drop out) but its effect would be smaller for heavy ions than for light ions, so that our results are expected to also apply falrly well to two unequal heavy ~ons

The sets of parameters ~z characterize the positions of the wells plus any other propertles such as their deformation In the present work we are only interested m the posmons of the wells and so we need two parameters =h and ~z to describe these positions that would coinclde with the expectatlon values of the c.m of the two ions

NUCLEUS-NUCLEUS POTENTIAL 159

Since the total c.m mot ion drops out, we actually need one pa ramete r • = ~ x - at2- The G H W generat ing funct ion for a system of two identical clusters as then

• (x; me) = m,,

A~ (1') ~ x ( x ~ - e a ) = d [ H ~ , ( r , - ~ ? ) ] = det {~b,(r,-ea)},

2 = 1,2.

The wave funct ion (1 ') can be writ ten

cb(x, . , , me) = q~ .i (R~ m )'x~CG(r--m)Oi"tO'z"t], (4)

= I ~ 1 , I[/2 are the internal wave func- where G(r- ~t) as a func tmn peaked at r ~, and ,,t ,,t Uons for the two nuclei and depend only on internal coordinates. I t can be easdy

shown that

9(r) = J- dm G(r- ~)f(m), (5)

m a y be regarded as the wave funct ion of relative mo t ion between the two ions by compar ing (2) with the resonat ing group wave functaon using the fo rm (4) for q~.

One of the first techn,cal dffficultaes m the G C M is tha t m a n y functions 9 ( 0 can- not be represented by eq (5) with a well-behaved f u n c t i o n f ( e ) ; f ( e ) does not always exist a l though 9(r) does 4). I t would be desirable therefore to t r ans fo rm the p rob lem into an equat ion governing 9(r) with no ment ion of f (m) .

All calculations done so far have utilized ha rmonic oscillator (h.o.) wave functions for the ~b,( r , -~x) in (1) or (1 ') above. Fo r an oscillator potent ia l with a single- nucleon pa rame te r b, the c m. mo t ion of a nucleus factors out o f a completely antl- s y m m e m z e d shell-model wave func tmn as a Gauss lan 7),

~, m. = ~b'"' exp [-a(xz-mz)2/b2], (6)

provided all shells are closed, except at mos t one. I f the tv, o nuclea have the same b, as ~s the case for identical nuclei, this leads to

G(r- m) = (2u/re) ~ exp [ - u(r- m)2], (7)

A1A 2 1 ( _ A forAl=A2=lA) u = 2 ( A a + A 2 ) bZ 8b z

where

Even in Investigations of the G C M where no explicit use of b o. wave functions was made, the expression (7) for G had to be assumed for any progress to be achLeved This is actually a very plausible assumpt ion since the c.m. mo t ion is expected to fac tor out. Eqs. (6) and (7) reflect the uncer ta inty in the posi t ion of the ions, and they are the price we have to pay for t rying to localize the c.m. We note that as the nucleus

160 H. R JAQAMAN AND A Z. MEKJIAN

becomes heawer ~t wdl oscillate less about ~ts mean c.m. and so Jt becomes more approprmte to assume that the c.m fluctuation factors out as a Gaussian that gets sharper as A increases and improvements due to GCM become less important 9)

The fact that G(r -a t ) depends on r - a t and as peaked at r - a t = 0 has been utflazed by Yukawa s) to obtain another form for the G H W equatmn. Yukawa was able to Isolate the klnetac energy of relative motmn operator m the equation, which then becomes [see Wong z)]

I lJ J ~ - V~- E l(at, e ' ) f ( K ) d e ' + I/(at, at')f(at')dat' = 0, (8a)

where

v ( , , = at)l Vl l (x, at')), (8b)

_ X',~l v(r,-rj) wath v the nucleon-nucleon EJs the energy of relative motmn, and Vt2 - z..j~2 force Thls alternative form for the GHW equatmn wall be more useful for the present work

3. Differential equation for g(at)

Using the expression (7) for G we can write the kernels (3b) and (8b) as (v,e use K to denote L H or V)

K(at, at') = f exp [ - u(r- ~)2]k(r) exp [ - u(r- ~')2]dr, (9)

where k(r), apart from a multlpllcatlve constant factor, is the resonating group method kernel, and ~t does not depend on either at or at'. Thas as to be compared with

K(at, at) = f exp [ - 2u (r - at)z] k(r)dr (10)

Noting that

( , . -a t ' ) ' - = ( r - a t + a t - a t ' ) z = ( r - a t ) z + ( a t - = ' ) Z + Z ( v - a t ) . (a t -a t ' ) ,

we can write

K(at, at') = exp [-.(at-at')z]fexp [-2u(r-at) z] exp [ - 2 u ( r - a t ) . (a t -~ ' ) ]k(r)dr

01) Because of the presence of exp [ - u(at - at,)z ] and exp [ - 2u(r- at)2 ], the contnbutton of exp [ - 2 u ( r - a t ) . (at-at')] is mainly m the region ( r - ~ ) " (a t -~ ' ) ~ 0 so that we can write (up to second order)

exp [ - 2 u ( v - a t ) . (at-at'J] ~ 1 - 2 u ( r - a t ) . (a t -a t ' )+Zua[(r-a t ) • (at_at,)]2.

NUCLEUS-NUCLEUS POTENTIAL 161

Tiffs gives

1 [v~ K(~, ~)]. v~+ 1 r(~, ~)v~

_ [o , - 1 ]-V2K(a,~t)]+ 1_~ X K(*t, 0t g(a), (12) + 16u 32u 2,,J=x,y,: a~,Sc~j

where differential operators contained withm square brackets operate only on the functions inside those brackets. Substituting (12) back into the GHW eq (Sa)we get

j/--h2 V _E)(Z ¼1(,,,)+ __ [V,/(ot,,)], V , + - / ( , , , ) V 2 + I 1 1 IV21(,, ,)] ) ~I 2/2 4u 8u

1 V(~,a)V~ + 1 [V2V(ot,,)])}gOt)=O, + (~v(',')+ l [v~ v(" ~)] "v~+ ~

where we have neglected the last term in (12) above Such a term would introduce an anisotropy into the effective mass. The above equation can be reduced to an equa- tion in Ig (arguments are dropped from now on):

}{ V2 1 [ ~ I ] } I g = - {~ +~ul [ _~] . . . . 1 [VVl [Vl] J-h---~v~-e i + v -v 2,u 14u 28u 7u I I

1 V V 2 _ 1 V [V2I ] I V [_~]2 1 [V~V]} (13) + , + - - - + Ig.

14u I 14u I I 7u I 28u

This can be put m the form

V 2 - q - - - - I - E VZ+V, rr+[VU] • V - E 19 = O, ~k2/2 14u 2/2 I 4/2 I

(14) where

[ (V h 2 [ - 7 ] ) 1 1 [V2V] E [vZI] v o . = - v + 1__ v 2 + + , I 14u 4# 28u I 28u I

1 V h 2 IV2/1 U - +

7u I 28#u I

It is observed that (14) is a fourth-order differential equation or, in other words, it includes a velocity-dependent effective mass in addition to its radial dependence On the other hand, the ordinary Schr6dlnger equation is of second order, and if we want a description of the scattering by a potential to be acceptable, we should try to find a second-order differential equation

- - - V2 + V I - E Ig = 0 , (15) 2/2

162 H R. JAQAMAN AND A Z MEKJIAN

whose solutmns satisfy (14). The po ten tml V 1 is to be found by subst i tut ing (15)

in (14) and requir ing self-consistency [1.e., we should get (15) back] . The choice

VI = (V/I )+(h2/4#)([V2I][I) is found to satisfy the greater pa r t o f this con&t Ion :

it exact ly cancels out the veloci ty and radia l dependence o f the effective mass and the

veloci ty dependent po ten tml U, and would app rox ima te ly satisfy the th i rd cond l tmn

tha t arises" V 1 = Vef f - [V2V1]/14/.t Our V 1 satisfies ins tead

14u = V I + I 2--8-u 4## [V2I] 28u /

Therefore we can conclude tha t (15) exists, at least to a g o o d approx ima t ion , ~f

28-u V2I - << V + - - V2I . (16) 28u 4/2

Numer ica l calculat ions were pe r fo rmed to check the extent to wtuch eq. (16) is

satisfied for the case of e -e scat ter ing using the Volkov force no. 2 [ref. 25)] Table 1

shows the values o f d such tha t the lef t -hand side o f (16) is < 10 ~ and < 2 0 ~ of the

TABLE l The values of d as explained m the text

E (MeV) 0 50 I00 150 200 250

e-~ 10 ~ 2 9 2 85 2.65 4 4 4.75 4.95 20 ~ 2 75 2.55 2.60 2 7 4 15 4 5

2 nuclei of 5 ~ 2.6 4 4 4 55 4 6 4 65 4.7 mass 10 ~ 1 7 1.2 4.45 4.55 4 6 4 6 160 20 ~o 1 2 1 0 0 85 4.45 4 5 4 55

The results for mass 100 are true for separations r ~ d except r ~ 4 2 fm where the right-hand sxde of eq. (16) vanishes However, tins Js dxrectly related to tile use of the ~-ct analytic expressxon and not relevant to the two A = 100 nuclei. For both cases only separations _--< 5 fm were in- vestigated and there are always subintervals m r < d where the 5 ~o, 10 ~o and/or 20 ~ reqmrements are satisfied also

r igh t -hand side for separa t ion distances > d The lef t -hand side IS not less than 5 ~o

o f the r igh t -hand side for a lmos t all separa t ion distances. I t is thus obvious tha t (16)

is no t well satisfied for the a - e system especial ly since cc-a scat ter ing is sensitive to the

inner region of the potent ia l . A l so shown in table I are s imilar results ob ta ined f rom

the same analyt ic expression used for the a - e case bu t with larger values o f # and u

app rop r i a t e for a system of two nuclei o f mass 100. I t is appa ren t tha t up to an energy

,.~ 100 MeV eq (16) is much bet ter satisfied for this case, especial ly in the tai l region

o f the potent ia l ; this region is a l l - impor tan t in heavy ion scattering. This agrees with

wha t is expected f rom examining eq. (16) i tself where, for larger/~ and u, all the terms

would become smal ler c o m p a r e d to V.

NUCLEUS-NUCLEUS POTENTIAL 163

We can now make the following further observations: (a) The above equations [(13), (14) and (15)] are for I(c¢, ~)g(~) and not for

y(~), but since I(~, ~) --, 1 for large 0c (the only region where g(~) has to be known) we can regard I(~, x)#(~) as our wave function This ambiguity m the wave function at short distances is a reflection of the fact that the kinetic energy operator IS not unique at short distances (with a complementary ambiguity in the effectwe potential) and as a consequence of antlsymmetrizatlon [see ref. 2) p. 337]. I t is also intimately connected with the removal of velocaty-dependent potentials 13) so that I(~, c¢) here plays a role similar to that of F m ref 13) Tlus can be seen f rom (15) which, when changed to an equation for 9(~) alone, would get a new term added to the potential plus a velocity-dependent term

(b) The potential V 1 can be looked upon as equivalent to the real part of an op- tical potential and can be compared with phenomenologlcal optical potentials It consists of two parts V 1 = (V/l)+(h2/41.t)(V2I/I), the first part is the potential that one would get m a Heatler-London approximation 14) and the second part has a form that suggests it is a kinetic energy contribution to the potential for relative motion between the two nuclei. There has been some discussion 15) by Zlnt and Mosel of such a contrabutaon, but whereas these authors predict a repulsive contribu- tion (based on the Born-Oppenheimer approximation), the term (h2/4~t)(V2I)/I is not repulsave everywhere In fact at is attractive for part of the tad region, although it :s neghgible there

(c) There have been some objections l o) against the interpretation of V a (or s,irnlar terms) as being equivalent to the real part of an opt,cal potentml These objections depend mainly on model calculations that show a strong varmtlon of ~lCfr m the anteractlon region 11). We have a few observations about this point" 0) The calculations in ref. 11) do not include the velocity dependence of the effectave mass This dependence [as those authors observe 11)] as important for heavy ion ~cattering and neglecting at would amount in eq (14) above to omitting the V2V 2 term which in turn would make it amposslble to get to eq (15) where the effective mass IS a constant (11) Those calculations ~ere done using the cranking formula xx hose validity ~s questionable for heavy aon reactions. Thls formula applies to adi- abatic processes where at as safe to assume that the velocity of the collective motion (m the present case this would be the relative motion between the two heavy Ions) is small compared with the Fermi motion 12) Th~s is why the cranking formula has a perturbatlon-theoretacal character (m) There is no reason why effectave masses ob- tained in different models or by dafferent processes should be the same, since these masses have no physical significance by themselves, and are not necessardy defined In the same way in all models. The effectwe masses obtained in thas work do not show any of the strong r-dependence or the order of magnitude devmt~on from the reduced mass # that was found mre f . 11). Even if we Include the h~gher-order terms that were omitted an getting to (12) and (13) no major changes in the above conclu- sions are expected, since the important terms are all in (13) already However, ~t

t64 H R JAQAMAN AND A Z MEKJIAN

could stdl be the case that the objections raised mre f . ~ o) are s:,Jl valid where a po- tential description is not possible [Le, where (16) is not satisfied].

4. The real part of the optical model potential

4.1 GENERAL DISCUSSION

In sect. 3 It was remarked that V I can be regarded as the real part of an optical potential for the elastic scattering of heavy ions and that the term VII m VI corre- sponds to the Heitler-London approximation. This approximation amounts te putting A~ nucleons m one well to form one nucleus and A2 nucleons m another well, at a chstance at from the first, and then calculate the lnteractmn between the two nuclei This is to be contrasted with the two-center shell model (which is eqmvalent to the molecular-orbital method) m which each nucleon is s:~pposed to be m the field of both wells

Denoting the nucleons m nucleus 1 by the numbers 1 -~ A 1, and m nucleus 2 by

A 1 + 1 -* A1 +A2, we define

s, , 2,, 2s = 1, 2;

<Ol vii, z> = j ¢ * ( r , - at~,)¢~(,2 - at~,)v(at + , 2 - r , ) ~ k ( , i - a tO¢, (r2 - a t 0 d ' i dr2 ,

g < AI , J > AI , at = atz,-at~s,

and define D . , kt as the determinant formed from [Sp~] by omitting the ith and j th rows and the kth and lth columns, mult:plied by the pan ty of the transformation that brings the ith and j th rows to the posmons of the first and second rows, and the kth and lth columns to the posmons of the first and second columns. The two vertical hnes in IS,~l denote the determinant of the matrix S w. The notatmn here ~s s~mllar to that of Slater 14) but not identical In what follows t ____ AI, A1 < j <_ A I + A 2 m all expressmns, With these definmons we have i4)

(ij]vlkl)D,j,u v(at, at) = < ~ " ~ ( ~ , at)lVx21~(x, at)> = ,j~, I(at, at) ( ~(a)(x, ae)l~(x, at)) ISpq I

Some terms vanish, namely those corresponding to exchanges between nucleons be- longing to the same nucleus, and we get

V _ 1 ~ {[( iJ lv l i j ) - ( iJIv l j i )]D. , .+ ~ (ijlvlkl)D.,kt}, (17) I IS,~l ,,J k~

for at least one of k, l ~ i, j . In the important tall region, where there IS no substantial overlap between the two

ions, the above expressmn s~mphfies considerably since all the exchange terms be-

H R. JAQAMAN AND A Z MEKJIAN 165

come negligible, Spq ~ ~Spq, ISp~l ~ 1, D,j, ,~ ~ 1 and

V, --, V ~ E <,Jlvl,j> = ~P,(r l )p2(r2)v(~+r2-rl)dr , dr2 , (18) I ,j d

i e , we end up with a folding model for the potential that has been previously in- troduced, more or less, phenomenologically t6,tT)

It is also worthwhile to study (17) assulmng it apphes for all separation distances between two identical ions. As ct ~ 0, ISpql -* 0 and, if we use as we should, a nu- cleon-nucleon force v that allows for nuclear saturation [a density-dependent force ,7, is) for instance], this will lead to a steeply repulsive core which prevents the two ions f rom completely overlapping, as expected by the Pauh principle. This repulsion can be related to non-physical states excluded by the Pauh pnnclple, the so-called null states 2.s)

For the sake of comparison, we define also an adiabatic hmlt for V/I, considering (17) as discussed above to be some sort of a non-adiabatic or sudden limit for the potential. Actually, the GCM, on which (17) is based, is intimately connected to a sudden-approximation picture of the interaction between the two nuclei: freeze the internal wave function of each nucleus, calculate the kernels and then allow the nucleons to interact. In the adiabatic limit we assume that the two nuclei are at rest with respect to each other at a certain distance, and allow their internal wave functions to readjust as required by the exclusion principle, the wave functions of the nucleons in one nucleus can be expanded in terms of only the unoccupied eJgenstates of the Hamdtoman of the other nucleus. Then S,j = 6,~, I = 1, and

V,(,) = ~ [<ijlvliJ>- <ijlvlJt>] *l

= f p,(r,)pz(r2)v(= + r.,-r,)dr, dr 2 - ~ ' <iJIvljt> (19) tJ

For ~t = 0 and a zero-range force the exchange contnbut lon is ¼ the direct contrlbu- uon to VI [since < q l v l J * > = <Ol@J> f o r ~ = 0 and i , j having same spin and lso- ~pm] and we get

1,'1(0) = ¼ | p,(r,)p2(r2)v(r 2 - r,)dr, dr 2 . (20) d

As ~ increases the exchange integrals become less and less important until they be- come negligible in the tall region so that the folding expression (18) emerges again Two points are worth noting about (19), 0) it does not include the effect of the null states and this has to be kept in mind when using the expression (19) (see subsect 4 2), and 0i) pt and P2 are not free-nucleus densities, except in the tall region of the potential where there is very little overlap, but are the Instantaneous densities that include the deformation that the nuclei induce in each other. These are obviously difficult to calculate and are similar to the two-center shell-model densities As a

166 H R JAQAMAN AND A Z. MEKJIAN

first approxamatton, however, they can be taken as the free-nucleus densatles ("frozen" density approxlmat~on) Unhke (17), (19) does not have a steeply repulsave core, af any.

For v, the nucleon-nucleon force, ~t ~s customary to use one of the effectave rater- actions, such as the Skyrme force 19), used an nuclear structure calculataons. How- evcr, for scattering calculations we have to add to thas force a non-hermatean term that accounts for compound nucleus formation 2o). Such a term would mainly con- trabute to the imaginary part of the potentml, is difficult to calculate so that a param- crazed form is used instead, and would be neglected here as far as the real part of the potentml is concerned. Furthermore, all the precedang dlscussaon has been earned out m a one-channel pacture, whereas in practice a number of channels are usually open These can be taken into account by a coupled-channels calculation 2a) which adds another term to the elastac channel effective operator, its real part will also be considered neghgable The neglect of all the above terms as justified in the surface region by the success of such phenomenological potentials as the foldang model.

The above dtscusslon throws some hght on possible sources of an energy depen- dence m V 1. Among these we identify:

(1) The energy dependence of the effectave mteractaon itself in addition to the energy dependent terms added to at m order to account for compound nucleus formation and open channels (including both the non-locahty of these terms and the increase an the number of open channels wath energy)

(li) The Pauh principle whose role changes with energy. (lIl) The energy dependence that is expllcltly found in the expressaon for V~ff m

eq. (14), which would be felt ff eq. (16) is not well satisfied Thus at as difficult to pre&ct the energy dependence of the potentml unless at is

the case that one of the above sources dominates all the others.

4 2. CALCULATIONS WITH A DENSITY-DEPENDENT INTERACTION

In a prevaous work a 7), the present authors used a densaty-dependent zero-range nuclear force hnear in the local density.

v ( r a - r2) = - ~o 6 ( r l - r2) + flP(½(rl + r 2 ) ) 6 ( r l - r 2 ) . (21)

Such a force can be regarded as a very sampllfied form of the Skyrme mteractaon (which atself can be used if desired), and ~s chosen for the ease at provides in carrying out calculations and obtaining analytic results and ~ts abdtty to reproduce important nuclear effects, especaally saturation. A more sophastacated force is clearly needed for accurate calculations. I t was shown a 7) that by choosing ~o = 956 MeV • fm 3 and flo = 2175 MeV. fm 6, (21) gaves a reasonable value for the tall of the potential obtained m a foldang procedure (eq. 08)) . This does not provide a sensltave test of the effect of the density-dependent part of (21) because of the httle overlap between the densities of the two nuclei m th~s case. A better test would revolve the inner region of the potentaal where there is an appreciable overlap between the nuclez Such a

NUCLEUS-NUCLEUS POTENTIAL 167

situation is prowded by cluster-model energy levels m nucleL For th~s case ~t Is rea- sonable to use the admbatlc hmlt expression (19)

Vary and Dover ~ 6) used this expression for the calculation of energy levels of ~- clusters on ~60 in Z°Ne with a density-independent delta interaction, but they had to introduce a considerable renormahzatlon of the strength of the lnteractaon (as compared to the one used for scattering calculations) m order to obtain a qmte suc- cessful reproduction of the bound and resonant energy levels m ZONe. The use of density-dependent forces explains the major part of the renormahzat~on needed Substituting (21) in (19) we get

i { f ( ) ( ) f ( ) ( )Pl°¢darl} ( ) VI(~) = - S o Pl rx P2 r l - a t drl+-~flo P~ r l P2 r l - - g , 22

where P~oc as the local density at the s~te of interaction, the factor ¼ s~mulates the effect of the Pauh principle, and the ~2 factor ~s due to the rearrangement effects 22) that result f rom using a density-dependent force. Both factors are exact at the origin and are a fairly good approximation m the region near the ongm that determines the depth of the potential. They go to 1 as the separation increases.

The first problem that arises is what to use for p~o¢. A first guess might be the sum of the densities Px and P2 at the site of the mteractton, however, thas ~s not the case. Even where the sudden approximation is apphcable, Paoc = Pl +pz+mter fe rence terms, since we should really add the wave functions and not the dens~tles In addi- tion to the interference terms, Pi and Pz m~ght be d~storted from the free state dens~taes On the other hand, we have to keep m mind that the two clusters (a and x 60) are part of the nucleus Z°Ne which, as a first approximation, can be thought of as a uniform sphere of density Po- Neglecting edge effects, (22) reduces to

3

where, now, cq = ~ o - ~ z f l o P o . Putting Po ~ 0.16 fm -3, ct o = 956 MeV- fm 3, flo = 2175 MeV • fm 6, we get. cq ~ 434 MeV • fm 3 to be compared with the value used by Vary and Dover x 6)

, 4 2rrh z 4 2n(197) 2 ~1 - f - 1.237 = 4 2 7 M e V fm 3,

3 M 3 940

while the value they used for scattering calculations is -,~ s o So whde the very close agreement between ~ and ~1' could be accidental, xve can at least conclude that we are able to account for the magmtude of the normahzmg "strength f a c t o r " f that they determine phenomenologlcally. The null states are taken into consideration by making sure that the calculated wave functions have the correct number of nodes, e .g, the 0 + state m the ground state band should have four nodes, etc. On the other hand, if we use (17) we do not do this, the afore-mentioned 0 + state will have no nodes

168 H. R J A Q A M A N A N D A. Z. M E K J I A N

The repulswe core suppresses the tuner oscillations of the wave function [see ref. 23) for a smular discussion about 160-160 scattering].

We now consider a classical-geometrical model in whtch the two overlapping nuclei remain segments of uniformly dense spheres whose radn are functions of the distance between the two centers of the nuclei under the assumptmn that the enclosed volume is a constant (lncompressiblhty) This g~ves the equatmn (see fig. 1):

3 3 1 3 2 2 3 R 1 + R z - ~ r +¼r(R 1 + R 2 ) + 8rr (R2-R2)2 = 2(R°al +R°a2)"

Exammatmn of th~s equation m&cates that R2 + Rt + {R~I +Ro*2} ~ as r -+ 0, i.e., the radii of the spheres become larger. For two identical spheres

R 2~ - 1 r 1 1 r 2 1 1 r s - - ~ - - - + + + . . . . Ro 4 R 0 2 ~ 16 R 2 96 2 ~'Ro 3

For Rol ~> Ro2, the smaller sphere undergoes a more drastic change than the larger one.

Fig. 1. Continuous hne shows the original size of the nuclei, and the dotted hne the sizes w~th the assumptmn of no compression.

Motivated by the above picture, we can get a faarly good approximation for the local density when the two nuclei have a large overlap by giving them larger effective radii (and a smaller central density to conserve the total mass), and then putting Pl.c = Pl +P2. It is interesting that exact calculations 24) ofpl.o that include the inter- ference effects tend to agree quahtatwely with this picture, the nuclei appearing to have larger radi~, in the static hrmt that apphes when the relatwe motmn between the two nuclei can be neglected. Harmonic oscillator densities are used for 160 and ct:

p,60(r)=4(~)~(l+2vlr2)e -~1:, (24)

with v I = 0.314 and v2 = 0.538 fm -a for the free nuclei. To get larger radii, hob- ever, we have to use smaller values for vl and v 2, which should be functions of the

N U C L E U S - N U C L E U S P O T E N T I A L 169

MeV

0

- I

-p

-5

-4-

- 5

2 + _ _ . . . _ _ s I ~ " ~ x _.....~ x I

/ ]"

0 + - - t

a) b) c) d) e}

Fig 2. Energy levels for the Z°Ne ground-s ta te band. (a) Exper imenta l results. (b) Resul t s us ing eq. (23) wRh m, = 417 M e V - f m 3. (c) Resul t s us ing eqs. (22) and (24) wRh P~oo = p l -kpz and 1,1 ---- vz = 0.314. (d) Same as (c) bu t wRh vl = 0.314 and ~z ---- 0.3. (e) Same as (e) but wlthv~ = ~'z = 0.3. No te tha t it is posmbl¢ to fit each level by a shgh t r ead jus tment o f ~,~ and ~z and tha t this gives the correct dep th for the potent ial . Energies are measu red with respect to the c<-t60

threshold, and only b o u n d levels are conmdcred.

(1.)

v

Fig. 3

0

-201

- 4 0

- 6 0

-80

- I00

-120

-140

-160

J / / /

/ /

/ /

/ /

/ t

o~-t60 potenhal

i I I I ) t + t I [ I

2 + ,~ 5 e 7 + 9 to +l ~2 r ( f r o )

The real par t o f the ~-160 potent ia l m the adiabat ic hml t . 0 ) con t inuous curve cor- r esponds to (c) m fig. 2, a n d 01) dashed curve cor responds to (b) m fig 2.

170 H. R J A Q A M A N A N D A Z M E K J I A N

>

>

A, IO 9

~ 160-160 potenhal '~ \ \

l /

,oo

o I \ , , / , ,

- I 0 0 ~_10 ~ /.// / / / { / / - 200 / , / / (c)~lo) / /

/ , / / ,'t (~)/" I / , - oo / / /

- 500 / I

/lb~ /lbt~lo I

J I ~ '1 T ) I ' [ I I l l 2 5 4 5 G -( 8 9 I0 II 12

r ( f m )

Fig. 4 The 16Oo160 potentml . (a), (b), and (d) are calculated m the admbatac hml t ; (a) is vahd near the origin, (b) and (d) are vahd m the tad r egmn [the actual po ten tml curve m the mter - medmte r egmn lying between (a) and (d)] (c) cor responds to V/I (eq (17)), ' the sudden ap- p rox imat ion ' (b), (c), and (d) are also magnif ied ten t imes for r > 5 5 fm to show the t ad r egmn more clearly. (e) is the (~2/4#)(VzI/I) correc tmn tha t has to be added to (c) to give Vz (eq (15)) The ~ 6 0 + x 6 0 elastic scat ter ing is sensitive to the po ten tml for r > 6 5 fm only

separation distance. In practice, t h i s is not easy to do and constant values are chosen so that the resulting potential is probably not acceptable for the surface region Results for cluster-state calculations are shown in fig. 2. Fig. 3 gives the correspondmg ~-t 60 potentials. The t 60-a 60 potential is also calculated and shown in fig. 4 where curve (a) IS the potential obtained when eq (23) is used with ~t = 420 MeV. fm 3 (this potential is good near the origin and gives the depth of the potential), (b) is the potential obtained from eq (18) using (21) with ~o = 956 MeV • fm 3 and neglecting the density-dependent part since the resulting potential is good only in the tall region where this part is neghglble Curve (c) is the t 60-~ 60 potential according to eq. (17); It is taken f rom the work of Yukawa 8) who uses a Volkov i s ) effective force which has a finite range, and is shown here for comparison The longer range of (c) In the far tall region is due to the use of a finite-range force in the calculation (as opposed to the zero-range force used for (b)). The difference between (b) and (c) for r < 7 fm ~s due to the neglect of the density-dependent part of the delta interaction and the omlssmn of exchange effects. This is obvious f rom curve (d) which is the same as (b) except that the density-dependent part of the force is Included. Also shown in fig 4, curve (e), is the kinetic energy term (h2/4#)(V2I/I) that has to be added to VII (curve

NUCLEUS-NUCLEUS POTENTIAL 171

(c)) to form the optical potential V x (see eq (15) and the discussion following it). It ~s apparent that this term modifies the potentml and the final result ~s much shallow-

er than without this term [cf. ref. 15)]. In the ~-160 calculations, the Coulomb potential used was that due to the inter-

action between two spheres 26) which is probably better than the sphere-point

charge potential usually used However, the energy level splitting is not sensitive to the Coulomb potential, and choosing different forms for the Coulomb potential merely shifts their position, which can be compensated for by a slight readjustment of the depth of the nuclear potential

5. Conclusion

We have shown that the GCM for heavy ion reactions is equivalent, to a good approximation, to solving a Schrodlnger equation with an effective potential whose real part has mainly a sample expression and includes a kinetic energy contribution besides the usual term that results from the interaction between the nucleons be- longing to the different nuclei. This lmphes that a potential description of heavy 1on reactions is not without a physical bas,s, and It explains the fact that phenomenolog~- cal optical potentials have been very successful. The GCM has been a valuable tool, and thxs would become more apparent when calculations for heavier nuclei are per- formed. The simple density-dependent delta force we used has proved to be quite successful m reproducing many results.In spite of all the simplifications introduced the potential obtained stdl retains most of the properties of interest, and its simplicity gives a better feel for the interaction between nuclei, at least as a first step towards a comprehensive and more accurate unified microscopic description

The authors would like to thank Profs L Zamlck and S Yoshlda for some in- teresting and helpful dlscussmns

References

1) D. J Hdl and J. A Wheeler, Phys Rev. 89 (1953) 1102, J J. Griffin and J. A Wheeler, Phys. Rev. 108 (1957) 311

2) C W. Wong, Phys. Lett 15C (1975), and references cited thereto 3) N do Takacsy, Phys Rev C5 (1972) 1883 4) R Beck, J. Borysowlcz, D M Brink and M. V Mlhallovlc, Nucl Phys A244 (1975) 45 and 58 5) B. Glraud, J. LeTourneux and E. Osnes, Ann of Phys. 89 (1975) 359 6) B Glraud and J. Le Tourneux, Nucl. Phys A240 (1975) 365 7) J. P. Elhott and T H. R Skyrme, Proc. Roy. Soc A232 (1955) 561 8) T. Yukawa, Nucl Phys A186 (1972) 127, Phys Lett 38B (1972) 1, Phys Rev. C8 (1973)

1593 9) T Fhessbach, Z Phys. 242 (1971) 287, Z. Phys. 247 (1971) 117;

D. M Brink and FI. Stancu, Nucl Phys. A243 (1975) 175 10) H Fnednch, K. Langanke, A. Welguny and R Santo, Phys Lett 55B (1975) 345

172 H. R JAQAMAN AND A. Z. MEKJIAN

11) P. Llchtner, D. Drechsel, J. Maruhn and W. Greiner, Phys. Rev. Lett. 28 (1972) 829 12) L. Zamick, invited talk, Tucson Conf. on effective interactions and operators, University of

Arizona, June 2, 1975 (preprint); Y. M. Engel et aL, Umversity of Oxford, Department of Theoretical Physics, preprmt

13) F. G. Percy, in Nuclear spectroscopy and reactmns, ed. J. Cerny (Academic Press, New York, 1974) p. 145

14) W. Heltler and F. London, Z Phys 44 (1927) 455; H. Margenau, Phys. Rev. 59 (1941) 37; J. C. Slater, Quantum theory of molecules and sohds (McGraw-Hall, New York, 1963) voL 1

15) P. G. Zmt and U. Mosel, Phys. Lett. B56 (1975) 424 16) J. P Vary and C. B. Dover, Phys. Rev. Lett. 31 (1973) 1510,

G. W Greenlees, G. J Pyle and Y. C. Tang, Phys. Rev. 171 (1968) 1115, D°

17) H 18) B. 19) T. 20) N. 21) H. 22) R. 23) A. 24) T. 25) A. 26) T

M Brink and N. Rowley, Nucl. Phys 3.219 (1974) 79 Jaqaman and A. Z. Mekjlan, Phys. Lett. 56B (1975) 237 Smha, Phys Rev. Lett. 33 (1974) 600 H. R. Skyrme, Nucl. Phys 9 (1959) 615 Austern, Direct nuclear reaction theories (Wdey, New York, 1970) p 63 Feshbach, Ann. of Phys. 5 (1958) 357; 19 (1962) 287 Sharp, Ph. D. Thesls, Rutgers Umverslty (unpubhshed) Tohsaki, F. Tanabe and R. Tamagakl, Prog. Theor. Phys. 53 (1975) 1022 Fhessbach, Z. Phys. 238 (1970) 329,242 (1971) 287 B. Volkov, Nucl. Phys 74 (1965) 33 W Donnelly, J. Dubach and J D. Walecka, Nucl. Phys. A232 (1974) 355