the size of the fusion potential

Nuclear Physics A491 (1989) 492-508 North-Holland, Amsterdam

THE SIZE OF THE FUSION POTENTIAL

S.-W. HONG, T. UDAGAWA and T. TAMURA+

Department of Physics, University of Texas, Austin, Texas 78712,

Received 6 October 1987

(Revised 21 September 1988)

USA

Abstract: Recently, Satchler ef al. claimed that fusion data can be fitted with an optical potential, in

which the fusion part of the imaginary potential had a radius parameter r, as small as 1 fm. This

potential, however, does not give elastic cross sections that fit data. It will be shown that, if the fit

to experimental elastic cross sections is demanded, as it should be, there is no way to avoid making

rF as large as 1.40 fm.

1. Introduction

In our earlier publications I*‘), we proposed a simple way to calculate the fusion

cross section uF. It is to take the inner portion of the imaginary part of the optical

potential W as the fusion potential W,. uF is then obtained as the portion due to

W, of the total reaction cross section uR. Since W itself remains unmodified, (TV

can be calculated in consistency with the elastic scattering and other direct reaction

calculations.

In the actual calculations, we chose W, such that W, = W for r < R, and W, = 0

for r > RF and treated R, (or rF defined by RF= rF(Ai”+Ai’3)) as adjustable. It

was found ‘) that a large body of experimental data taken both below and above

the Coulomb barrier were fitted very well, by choosing rF= 1.45 fm. This rF value

is certainly rather large. It is in particular much larger than the rF value (1.0 fm)

normally used in the barrier penetration model (BPM) calculations ‘). We discussed

in ref. ‘) reasons why it is by no means unreasonable to have such a large rF.

The above conclusion was, however, questioned recently by Satchler et al. 435)

who argued that a much smaller rF could be used, if the energy dependence ‘-‘) of

the optical potential due to the coupling to direct reaction channels is taken into

account. Thus, in ref. ‘), optical-model potentials were searched so that they could

reproduce the elastic-scattering cross section obtained in a large-scale coupled-

channels (CC) calculation ‘). It was shown ‘) that this optical model potential, which

we shall henceforth call an equivalent elastic potential (EEP), did have a rather

short-ranged fusion potential, with rF= 1.0 fm. It was further shown that, with this

EEP, it was possible to fit the experimental fusion cross section aFp (and also to

reproduce the calculated CC total reaction cross section (T:’ and the CC fusion

cross section ~7:“).

f Deceased on 10 October 1988.

0375.9474/89/$03.50 @ Elsevier Science Publishers B.V.

(North-Holland Physics Publishing Division)

S-W. Hong et al. / Fusion patential 493

It is our opinion that the conclusion made in refs. 4,5), that a rather small rF was

enough, is premature. The reason for this opinion is that the elastic-scattering cross

section aEEP obtained with EEP does not agree with the experimental cross section

a$‘. (The elastic-scattering cross section obtained in the large scale CC calculation ‘)

did not agree sufficiently well with r’E”Lp, and as remarked above EEP was chosen

so that aEFP reproduced the former.) It was emphasized in ref.‘) that the lit to c’E;ll

was improved dramatically when EEP was used, although the fit was very poor

when the elastic part of the CC potential was used in the one-channel calculation.

In spite of the improvement thus achieved, however, it does not seem appropriate

to dismiss completely the discrepancy still remaining between a::” and r$‘.

In this paper, we address ourselves to this remaining discrepancy, and intend to

see how the EEP of ref. 5, is to be modified if this discrepancy is to be removed.

We shall then see how this modification of EEP affects the behavior of the calculated

*F.

We present in sect. 2 the EEP of ref. ‘) and discuss its characteristic features. In

sect. 3, step-by-step modifications of the EEP are made and their influence on cEL

as well as rF is examined. In sect. 4, we discuss the energy dependence of EEP

thus modified. This paper is concluded in sect. 5.

2. Equivalent elastic potential

The EEP determined in ref. ‘) is the starting point of the present study. Therefore,

we shall recapitulate it here. The origin of EEP traces back to the experimental data

taken for the “0 + z”xPb system at an incident energy E,,, = 80 MeV I*). This energy

is close to the height of the Coulomb barrier, where (YF’ is considerably larger than

is predicted by BPM ‘).

As remarked in the Introduction, the EEP was determined by using the results

of the CC calculation “). In the CC calculation almost all the important inelastic

and reaction channels (except those Ieading to “C, with effects to be discussed

later) were coupled. Therefore, it was unnecessary to include any absorptive poten-

tials other than a short-ranged fusion potential W,. We shall call the elastic part

of the potential used in the CC calculation the CC elastic potential and denote it

by UcCE. It consists of the real potential V,, which is of Woods-Saxon form, and

a short-ranged fusion potential W,. Specifically, UCC‘E’ is written as

where

u CC‘E

(‘c‘t- v =

wc‘c” _

= _ v’ C’E _ i w“c-E’ )

V,(r) = V

l+exp(X,)’

W,(P) = WF

1 + exp (X,)

(1)

(2)

(3)

494

with

S.-W. Hong et al. / Fusion potential

Xi=(r-&)/a,, (4)

R, = ri(A;‘3+A;‘3) (i=O or F). (5)

The values of the parameters in UccE as used in the CC calculation ‘) are listed in

table 1 (first column). Note that the radius parameter of W,(r) is rF = 1.0 fm, which

is the value normally assumed in the BPM “).

TABLE 1

The optical potentials used in the present study, along with the resulting Xz values, the fusion cross

section (TV and the total reaction cross section (+e. (The parameters fixed during the search are underlined)

Potential: CCE EEP Ml M2 M3 M4 MS M6

V (MeV) 60.5 m 156.7 173.9 165.7 133.6 55.0 w

r0 (fm) 1.179 1.179 1.179 1.179 1.179 1.179 1.179 m

a, (fm) 0.658 w 0.584 0.584 0.584 0.584 0.638 M

W,, (MeV) b 0 a404 0.0775 0.0775 0.1144 0.7702 4.926 0.4570

ru (fm) u u u I_s 1.426 u 1.497

a, (fm) 0.786 0.786 0.786 0.786 0.434 0.0765 0.280

tVF ( MeV) 10.0 10.0 10.0 10.0 10.0 10.0 10.0 8.9

r, (fmt j.J _lxJ j.cJ U! j.J jJJ j.J 1.401

aF (fm) 0.25 0.25 0.25 0.25 0.25 0.22 0.25 0.13

CR 82.3 78.6 113.1 127.2 153.4 128.3 126.7 120.1

Cl- 37.6 36.4 34.6 48.2 39.5 12.3 0.19 35.7

X2 26.6 11.8 4.29 2.81 0.86 0.63 0.42 “)

“) Fusion and reaction cross sections are included in this Xz value. The Xz from the fit to the elastic

cross sections only is 0.45.

Using UCCE as given above [see ref. ‘) for other details of the calculation], the

CC calculation was performed, and it predicted ogc = 37.6 mb and (T$~ = 82.3 mb.

(These values are also listed in table 1.) The experimental fusion and total reaction

cross sections, however, were given as cr?* = 36 mb and &Jp = 113 mb (see sect. 3

for the discussion of this value of a:“), respectively r”*“). Thus the calculated ozc

is in good agreement with ap”. On the other hand, aF is too small by about 30 mb

compared with uFp. This is, however, understandable, because as remarked above,

the CC calculation did not include the coupling to the “C channels. In fact, the

cross section of the reactions to the 12C channels has been measured ‘oSr’), and it

was about 20 mb. The above discrepancy will thus be removed, if the CC calculation

is redone with the “C channel coupling included. We shall come back to this matter

in sect. 3.

In ref. “) the elastic-scattering cross section ucc EL obtained from the above CC

calculations was used as “data” (the comparison of o:E with the experimental data

will be discussed later), and the usual optical-mode1 x2 analysis of the “data” was

carried out to obtain the EEP, which we denote by UEEP. The following form was

S.-W. Hong et al. / Fusion potential 495

assumed for UEEP;

where

with

u EEP = _ V”E’_ iWEEP f (6)

V EEP= V,(r), (7)

WEEP= W,(r)-l- W”(r) (8)

ev (XDt w~(r)=4wD(1+exp(X,,))‘. (9)

The real part VEEP is given by the same formal expression as for V,(r) in UC“E.

However, the values of the parameters, particularly the strength (V) and diffuseness

( aO) parameters, were readjusted significantly. The imaginary part WEEP now consists

of two parts: W,(r) and W,(r). W,(r) is exactly the same as that used in UcCE,

while W,,(r) is a new term. It was introduced ‘) in order to take care of the absorption

due to the direct reactions. Thus in ref. ‘), the fusion cross section uF and the total

direct reaction cross section unR are calculated as

giving the total reaction cross section as

trR = flF t 0-n ff. (12)

In both (IO) and (ll), u is the incident velocity and xc+) is the distorted wave

calculated by using the (full) optical potential UEEp of (6), which contains both

W,(r) and W,(r).

In ref. ‘), four different sets of EEP parameters were obtained. In the present

study, we choose the set 2. This choice is made, since the fusion cross section

calcufated with this set agrees best with experiment. We present the parameters

of this set 2 EEP (which we shall henceforth call simply the EEP) in column 2 of

table 1.

An important feature of the ULt” IS that the strength of the real potential VEEP is much greater than that in Vcc” . This is due to the fact that the couplings effectively

increase the attraction between the two colliding ions in the entrance channel ‘,‘.“).

It was stressed that this increased attraction played the most decisive role in

enhancing the fusion at the near- and sub-barrier energies “).

Another important feature of UEEP . IS that the absorption in the barrier region is

very weak. In order to show this, we present in fig. 1 WEEP as a function of r for

the radial range of 8 < r < 1.5 fm. For comparison, we also present the imaginary

part WoM of the usual optical-model potential U”“, which was determined from

496 S-W. Hong et al. / Fusion potential

Fig. I. W”.’ and W”” as functions of I.

the optical-model analysis of the experimental elastic-scattering cross section I”).

As seen, W EEP is much smaller than WclM for lo< r < 13 fm, i.e., in the barrier

region. We shall see shortly that this weakness of WEEP under the barrier causes

the EEP to lead to a poor fit to the elastic-scattering cross section.

The optical-model calculation with this EEP predicts that pEEP= 36.4 mb and

CF:” = 78.6 mb, both agreeing with u:c and afdc, respectively. These values are

listed in table 1. As already mentioned above, aze agrees with experiment. Thus,

uEEp also agrees with u~‘~. azC is, however, about 30 mb smaller than cryp, and so

is aEEP. The calculated elastic-scattering cross section, ~r~~~~, naturally agrees with

a$. However, it does not agree with the experimental cross section &‘jJ’, as

emphasized in the Introduction. A comparison of CF::’ (a full line) with a”E”:’ is

given in fig. 2. The discrepancy between CT:“,” and CFF? is clearly seen; aEEP starts

to fall off at an angle about 10” larger than the angle where vY’L” starts to fall off.

This indicates that the absorption in EEP is too weak.

In the next section, we carry out a number of ,$ analyses, where x2 is defined as


Fig. 2. Angular distributions of elastic-scattering cross sections (ratios to Rutherford) of “0.t ““Pb at

E,,, = 80 MeV. The solid, dashed and dash-dotted curves are theoretical cross sections with the EEP,

M2 and M3 potential, respectively. The circles are experimental cross sections.

In eq. (13), R( Sj) and Rexp( 8:) are, respectively, the theoretical and experimental

ratios of the elastic-scattering cross section to the Rutherford cross section. (We

took ten of them.) The 6, stands for the scattering angle. For the weight function

AR(6,), we use AR(6i) = 0.0182, which is taken from ref. I”). With the above

definition, we obtain x2 = 26.6 for the EEP, which is quite large.

3. Fit to elastic scattering cross section and its influence on fusion cross section

We now introduce several modifications of the EEP presented in the previous

section. We shall consider six different modifications and will call the resultant

potentials the Ml through M6 potentiais. The values of the parameters of these

potentials are listed in table 1. There we list also the xz values, the calculated fusion

and total reaction cross sections. In what follows, we shall explain what these

498 S.-W. Hong ef al. / Fusion potential

modifications are and discuss the effects of these modifications upon the elastic

scattering and fusion cross sections.

In discussing the properties of the various potentials, we frequently compare the

predicted on with ayp, and we feel it is desirable to comment here on what value

is to be taken as a:“. In the text of ref. lo), a value of 100 * 10 mb was given as a

measured uyp, and this value plus 5 mb, which was the expected upper limit of the

evaporation residue, was quoted in ref. ‘). However, as remarked above, the authors

of ref. lo) also performed optical-model calculations and presented the thus calcu-

lated value 113 mb as acceptable for uR. The same value of oR = 113 mb was

predicted by optical-model calculations in a more recent experimental paper ‘). Also

the comparison of fig. 17 in ref. lo) with fig. 14 in ref. I’), in which cross sections

resulting in various channels were given, seems to indicate that on = 100 mb given

in ref. lo) was a slight underestimate. From these considerations we decided to take

in the following that (T;~ = 113 + 10 mb. On the other hand, we take 36 mb in ref. lo)

as it stands for u’;“.

As remarked in the previous section, the couplings to the 12C channels were

neglected in the CC calculation, and it was suggested ‘) that this was the reason

why the calculated (T:’ (and thus cF’) were smaller than rTp. We now examine

the effects of these neglected couplings, which we do by modifying the parameters

in V,(r) and W,,(r), particularly by increasing the strength parameter W,,.

Ml potential. We first readjusted the value of W, so as to reproduce ar”

(= 113 mb). The EEP with this modified W,, value is called the Ml potential. The

required value is 0.0775 MeV, which may be compared with the original value

(0.0404 MeV). We then examined the influence of this modification on the elastic

scattering cross section. It was found that the x2 value was reduced by a factor of

2 to 11.8. The fit to vyf nevertheless remained rather poor.

M2 potential. As the next step, we allowed V, the magnitude of V,(r) to vary,

and a x2 analysis based on eq. (13) was done, keeping, however, all the other

parameters the same as in the Ml potential. The potential with the thus modified

V-value is called the M2 potential. The V-value obtained is V= 173.9 MeV, which

is about 20 MeV larger than that of the EEP. This amount of increment may be

regarded as reasonable as a correction originating from the coupling to the 12C

channels.

The elastic-scattering cross section obtained with the M2 potential is shown in

fig. 2 by the dashed line. The fit to the data has been considerably improved as

compared with that obtained by the EEP (the full line). However, the fit is still

unsatisfactory. The x2 value with the M2 potential is 4.29, while that with the

optical-model (OM) potential given in ref. lo) is 1.85. This means that the fit with

the M2 potential is 2.5 times worse than that obtained with the OM potential.

M3 potential. A x2 analysis was then performed by varying V and W, simul-

taneously. The resulting potential is the M3 potential, and the elastic-scattering

cross section thus calculated is presented by the dash-dotted line in fig. 2. The x2


value is now 2.81. In the sense of x2 the fit has thus been further improved, but as

seen in fig. 2, a noticeable discrepancy between the calculated and experimental

cross sections has appeared in the shoulder region (the 100” < 6 < 120” range). In

this angular range, RexP(rYi) remains almost unity, while the calculated R( aj) is

considerably lower than that. This shows that the absorption in the tail of W,, is

now too strong. (The W, value of the M3 potential is 0.1144 MeV, which is 2.8

times that of the EEP.) As also seen in fig. 2, however, the M3 potential gives a

fairly good fit to data at larger angles (6 > 120”), showing that a stronger absorption

needed at smaller radii (the barrier region) has now been provided adequately.

However, crR is now too large.

M4 potential. The undesirable feature of the M3 potential, i.e., the too strong

absorption in the tail region of W,(r), may be remedied by modifying the geometrical

parameters, in particular, ao in W,,(r). Thus a x2 analysis was done by starting

from the M3 potential and letting ur, and rD vary. With only a, and rD allowed to

vary, however, we were unable to reduce the x2 value and we decided to add W, and V as additional variables. The resultant potential is the M4 potential, and as

seen in table 1, the x2 value is now as small as 0.86. uR = 128 mb is still somewhat

too large but not too much. A disturbing fact encountered is that 12.3 mb obtained

for err is less than half that of ayp = 36 mb. In any case, characteristics of the M4

potential are that a, has been reduced, as expected, by about a factor of 1.7 from

that in EEP. rD did not change very much, but W,, has increased significantly. Also

V in the M4 potential is smaller than that in the M3 potential, even smaller than

that in EEP. The fit to the elastic cross section achieved by the M4 potential is

presented in fig. 3.

M5 Fofe~tiul. By starting from the M3 potential, another x2 analysis was done.

In this case rO = 1.5 fm was retained, but a, in V(r) was allowed to vary along with

a,, W, and V. The resultant potential is the M5 potential, and it gives a ,$ value

which is as small as 0.63. As also seen in fig. 4, the fit to the elastic cross section is

very good. CT~ = 127 mb is also close to the experimental value (clyp = 113 i 10 mb).

The M5 potential has, nevertheless, a few disturbing features. First of all, it predicts

an extremely small or (= 0.19 mb), because W,, is very large and V is very small.

Such a small value of V ( =L. 55 MeV) also contradicts the spirit of refs. “-8), which

advocated a rather large V. Somewhat too small a,, together with too large W, (when rD= 1.5 fm) produces an unusual shape for a potential and this is also a

source of concern.

M6 poi~~tial. We may summarize the essence of what we have done so far in the

following way. Since it has been more or less obvious (see, e.g., fig. 1) that EEP

had a too weak absorption, we have increased the absorption gradually, by modifying

the parameters in W,,(r), although some of the parameters in V(r) were also

modified, because otherwise we were unable to improve the fit to elastic-scattering

cross section c??(G). By the time we arrived at either the M4 or the M5 potential,

the needed improvement in fitting c$’ has in fact been achieved satisfactorily, and

500 S.-W. Hong et al. / Fusion potential

Fig. 3. The same as in fig. 2, except that the solid line represents the theoretical cross section with the

M4 potential.

as expected this was done by an increased W,(r). (The volume integral of W,(r),

e.g., in the M5 potential is about ten times that of the W,(r) in EEP.) A consequence

of the increased W,,(r), however, is that it is very difficult to keep the amplitude

of the relative motion sufficiently large in the region in which the fusion potential

W,(r) is appreciable. This resulted in a too small or. It should be emphasized that

so far we have left W,(r) of EEP completely untouched.

From what we have experienced, as summarized in the preceding paragraph, it

may not be unreasonable to conclude that if we want to fit simultaneously ~‘E1p and

cre,Xp, it is inevitable that we start to modify the W,(r) of EEP. We thus carried out

a final x2 analysis, this time by varying freely all the six parameters in W,(r) and

W,,(r) (but keeping V(r) the same as it was in EEP.) In carrying this x2 analysis, CT:?’ and uFxp were also included as part of the data to be fitted. The potential

obtained in this way is the M6 potential, and fig. 5 shows that the calculated

elastic-scattering cross section fits rather well cG!‘. The x2 value obtained (for the

elastic-scattering cross section only) is 0.45, which is smaller than any x2 value we

S.-W. Hong et al. / Fusion potenfial 501


M5 potential.

have obtained above. Also uR = 120.3 mb and wF= 35.7 mb obtained with the M6

potential fit the respective data very well.

A few interesting features of the M6 potential may be remarked. First of all, r D- - 1.497 fm is very close to its starting value of 1.5 fm, in spite of the fact that it

was permitted to vary freely in the x2 analysis. On the other hand, rF has changed

drastically (from 1 fm to 1.4 fm), making the absorption under the barrier stronger,

as we expected it would do. This value 1.4 fm is rather close to the value of rF = 1.45 fm

we imposed in our fusion calculations I,*). In refs. I,*), W,(r) was cut off sharply

at r = rF(Ai” + A:“). On the other hand, the W,(r) in the M6 potential has a finite

diffuseness. However, its value (uF = 0.13 fm) is rather small, so that this W,(r) also

is sharply cut off. The fusion part of the M6 potential is thus surprisingly close to

the W,(r) that we used in our earlier work I**). It is noticeable that the resultant

total imaginary potential W(r)( = W,(r) + W,,(r)) has now a reasonably large

value. Indeed, W(r) at the strong absorption radius of r = 12.4 fm is 0.49 MeV,

which is in good agreement with what was observed in ref. “). (See fig. 2 of ref. “)).

502 S-W. Hong et al. / Fusion potential

M6

nl d

N

d

O(c. M. )


M6 potential.

The EEP, on the other hand, gives only 0.04 MeV at the same radius, which is too

small.

We may also remark that the M6 potential predicts the average spin of the

compound nucleus to be (I”)= 142, which is significantly larger than the value 104

obtained in ref. ‘). Although our value 142 is not as large as the experimental value

= 180 obtained by Murakami et al. 13), it is larger than any theoretical value quoted

in ref. r3), including the value obtained with our earlier approach.

~7~~te~~~ff~. Finally, we remark that the fit to the elastic scattering can further

be improved beyond what was obtained in fig. 5, if we do not include the total

reaction cross section data in the x2 analysis. An example of such a fit is presented

in fig. 6. As seen, the fit to the data is now almost perfect, the small yet clearly

visible discrepancy observed in fig. 5 at 6 = 90”-110” now being removed. The optical

potential used in obtaining this result, and may be called the M7 potential consists

of a modified EEP and an additional surface-derivative imaginary potential Wb( r)

with Wb=3.72 MeV, rb= 1.418 fm, and ai,= 0.192 fm. The modified EEP used

S.-W Hong et al. / Fusion potential 503


M7 potential delined at the end of sect. 3.

differs from the original EEP only in the values of W, and rF. The modified values

are W, = 5.0 MeV and rF= 1.3 fm. With this new potential, we get CT~= 130 mb,

which is only 7 mb larger than the upper limit of the experimental value app =

113 * 10 mb. However, to obtain gyp requires one to measure every possible reaction

cross sections and thus the current value of ay” may in fact be an underestimate.

4. Energy dependence of the M6 potential

In this section we examine whether the M6 potential, a potential with a large rF, can explain data at other energies beyond that (E,,, = 80 MeV) considered in the

previous section. For this purpose, we perform x2 analyses of ueErp, aFnp and uFp

taken from refs. “J’ ) by using the M6 potential as a starting potential. We treat V,,

W, and C+ as free parameters to be determined from the analysis. The vaiue of rF


is kept the same as that of the M6 potential. We shall keep calling the resultant

(energy dependent) potential the energy-dependent M6 (EDM6) potential.

The fit to age and u’;‘” thus obtained is presented in figs. 7 and 8, respectively.

As seen, the fit is very good in both GEL and cF. A similar fit has also been obtained

in (TV. This shows that the EDM6 potential is able to reproduce the data at other

energies as well.

In fig. 9, the values of the real and imaginary parts, V and W, respectively, of

the EDM6 potential at the strong absorption radius of RA = 12.4 fm are presented

(by the filled circles) as functions of Elab. Presented also (by the crosses) are

contributions from W, to the total W. As seen, both V and W show the characteristic

energy dependence now known as “threshold anomaly” 6), i.e., the values of V and

W increase as Elab approaches approximately the Coulomb-barrier energy.

These values approximately satisfy the following dispersion relation ‘),

v= V, { “-Esp W(E’) 77 (E'-E,)(E'-E)'

El,, = 83 MeV

‘“0 + “*Pb

I_-_

bU I I ,

60 100

0,,(deg)

(14)

Fig. 7. Angular d~strjbutions of elastic-scattering cross sections (ratios to Rutherford) of ‘hO+21)XPb at

various incident energies. The solid curves are theoretical cross sections with the modified M6 potential, while the circles are experimental cross sections “~“).

S.-W. Hong et al. / Fusion potential

‘60 + *08Pb

Fig. 8. The calculated and experimental fusion cross sections as a function of the incident energy.

In order to demonstrate this, we presented in fig. 9 the values of V evaluated from

(14) by the full line shown there. In the calculation, use was made of the following

expression for W(E),

80 100 E ,ob(MeV)

505

W(E)= w” 1 +ee’“-t”‘/‘3 ’ (15)

with E,,= 83.5 MeV, WC,= 1.6 MeV and A = 1.8 MeV. Use was also made of v, =

1.8 MeV and E, = 90 MeV. The line drawn for W in fig. 9 is the W-value predicted

from (15) with the parameters given above. As seen, W thus predicted fits with the

empirical W-value very well. Now V calculated from eq. (14) explains the empirical V-values, particularly the energy dependence, fairly well. This shows that the V-

and W-values determined from the x2 analysis indeed satisfies the dispersion

relation, which may be considered to support the validity of the EDM6 potential.

It is important to remark here that even at r = RA (= 12.4 fm), the contribution

from W, to W plays an important role, and that the energy dependence of W

essentially comes from that of W,. In fact, the values of W,, = W - W, is essentially

a constant in the energy range considered in fig. 9. This may be understood as


I h I I I

2.5 - ‘60 + 208Pb

7 r” 2.0 - 7

1.5 -

80 90 100

E ,J MeV )

Fig. 9. The values of the real and imaginary potentials at r = 12.4 fm as functions of the incident energies.

The circles shown for W is the full W-value, while the crosses are the contribution from W,. The curve

drawn for V is the prediction from the dispersion relation using W given by the curve drawn

for it.

arising from the fact that agp is a much more smoothly varying function of Elab

than is ~$2~.

5. Summary and conclusions

By starting with the equivalent elastic potential (EEP) of ref. 5), we searched for

better parameters so that good fits to the actual experimental elastic-scattering cross

section were obtained, rather than fitting to the theoretical elastic-scattering cross

section predicted by coupled-channels calculations ‘) as was done in ref. ‘). The

search was done step by step, resulting in potentials called Ml through M6, which

are summarized in table 1. The EEP of ref. ‘) contains two imaginary potentials;

the surface type W,(r) and the volume type WF(r), and they were used in such a

way as to be responsible for direct reactions and fusion, respectively. We have

followed this prescription throughout.

As we went from EEP through to M4, the fit to the elastic-scattering data was

improved gradually, but not yet satisfactorily. A good fit to the elastic scattering

data was obtained, however, with M5, in which the magnitude of W,,(r) was

S.-W. Hong et al. / Fusion potenfial 507

increased substantially. This indicates that EEP has to be modified by increasing

the absorption at a larger radius. (Note that W,(r) has a much larger radius than

W,(r) does.) Trouble was encountered with M4 and M5, however, because the

fusion cross section gradually decreased as x2 decreased (table l), and it became

as small as 0.19 mb for the M5 potential.

We thus went back to the original EEP form of the potential, but now allowed

essentially all the parameters in the imaginary potentials to vary freely. In particular

the radius of W, which had been kept at 1.0 fm in all potentials up to M5 was

permitted to vary for the first time. The resulting potential, called M6, achieved

good fits to all the data: elastic scattering, total reaction and fusion cross sections.

Because of the good simultaneous fits to all the data, we may consider that the

M6 potential has a real physical significance. A notable fact is that the radius

parameter of W, in the M6 potential is as large as 1.4 fm, being very close to the

value of rF (= 1.45 fm) we needed to use in our fusion cross section calculations I,‘).

With our approach ‘*2), the fusion part of the imaginary potential was cut off sharply

at a radius r,(A:” +AA:‘3). It is then interesting to note further that the W,(r) in

the M6 potential has a rather small diffuseness. This means that the M6 W,(r) has

copied rather faithfully even the sharp cutoff nature of the fusion potential of refs. I,‘).

We then applied the M6-potential to other energies by varying the values of V,,

WI, and aF as functions of the incident energy. The values of these parameters were

fixed from x2 analyses. The resultant fit was found to be very good, and the values

of the real and imaginary potentials at the strong absorption radius were shown to

satisfy the dispersion relation. This further supports the validity of the M6 potential.

We may conclude our summary of the results of the present analysis as follows.

(i) In order to fit the experimental elastic-scattering data, it is unavoidable to have

an imaginary potential that has a significant strength at a larger radius. In other

words, we must have a potential that causes a substantial absorption under the

Coulomb barrier. (ii) However, if the increased absorption under the barrier is

achieved solely by the increased W,,(r) (keeping the fusion potential short ranged

as in EEP), the resultant fusion cross section becomes too small (as seen in the case

of the M4 potential). (iii) In order to achieve a simultaneous fit to the scattering

and fusion data, it becomes inevitable to increase the radius of W,(r). This implies

that the identity of the incident system is lost at a fairly large distance rF and that

once the flux reaches the fusion potential radius rF, it never comes back to simple

direct reaction channels.

Our conclusion is that an automatic parameter search performed by starting with

the EEP of ref. ‘) resulted in a potential that resembles very closely the potential

we used earlier I,‘). The key concept used here is a simultaneous fit to the scattering

and fusion data, a concept which we have used all along.

We wish to thank Professor W.R. Coker for his careful reading of the manuscript.

The present work was supported in part by the US Department of Energy.


References

1) T. Udagawa and T. Tamura, Phys. Rev. C29 (1984) 1922

2) T. Udagawa, B.T. Kim and T. Tamura, Phys. Rev. C32 (1985) 124

3) J.R. Birkehmd and J.R. Huizenga, Ann. Rev. Nucl. Part. Sci. 33 (1983) 265

4) M.A. Nagarajan and G.R. Satchler, Phys. Lett. B173 (1986) 29

5) G.R. Satchler, M.A. Nagarajan, J.S. Lilley and 1.J. Thompson, Ann. of Phys. 178 (1987) 110

6) J.S. Lilley, B.R. Fulton, M.A. Nagarajan, I.J. Thompson and D.W. Banes, Phys. Lett. BlSl(1985) 181

7) M.A. Nagarajan, C. Mahaux and G.R. Satchler, Phys. Rev. Lett. 54 (1985) 1136 8) C. Mahaux, H. Ngo and G.R. Satchler, Nucl. Phys. A449 (3986) 354; Nuci. Phys. A456 (1986) 134

9) 1.J. Thompson, M.A. Nagarajan, J.S. Lilley and B.R. F&on, Phys. Lett. B157 (1985) 250

10) F. Videbaek et al, Phys. Rev. Cl5 (1977) 954

11) E. Vulgaris, L. Grodzins, SC. Steadman and R. Ledoux, Phys. Rev. C33 (1986) 2017; Phys. Rev.

C34 (1986) 1495; and optical parameters are in Phys. Rev. C34 (1986) 1495

12) C.H. Dasso, S. Landowne and A. Winther, Nucl. Phys. A405 (1983) 381;

M.J. Rhoades-Brown and P. Braun-Munzinger, Phys. Lett. B136 (1984) 19;

S. Landowne and SC. Pieper, Phys. Rev. C29 (1984) 1352. 13) T. Murakami, C.-C. Sahm, R. Vandenbosch, D.D. Leach, A. Ray and M.J. Murphy, Phys. Rev. 04

(19%) 1353

the size of the fusion potential

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