Polarization potentials in heavy ion scattering and fusion

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PHYSICS

POLARIZATION

POTENTIALS

LETTERS

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1990

IN HEAVY ION SCATTERING AND FUSION

B.T. KIM a,b, M. NAITO a and T. UDAGAWA a a Department ofPhysics, University of Texas, Austin, TX 78712, USA b Department Received

ofphysics,

29 September

Sung Kyun Kwan University, &won 1989; revised manuscript

440-746, Korea

received 2 I December

1989

A search is made for a polarization potential fully satisfying the dispersion relation, by carrying out simultaneous X2-analyses of elastic scattering and fusion data for the I60 + 208Pb system near the Coulomb barrier energy. The imaginary part, W, of the potential is assumed to consist of a volume-type fusion and a surface-derivative-type direct reaction potential, while the real part has the corresponding polarization parts along with an energy-independent bare potential. Characteristic features of the potential thus obtained are discussed.

In our recent publications [ 1,2 1, we have reported simultaneous X2-analyses of elastic scattering and fusion data [ 3-61 for the I60 + 208Pb and 32S+ 58,64Ni systems at sub- and near-Coulomb-barrier energies. These analyses were done within the optical model, assuming an imaginary potential W that consists of a volume-type fusion part, IV,, and a surface-derivative-type direct reaction (DR) part, W,,. The fusion and DR cross sections, OF and fir, respectively, were calculated by using the expression [ 7 ] cr,= f
lV,(r)]~(+‘)

(i=ForD).

(1)

In eq. (1 ), x (+) is the usual distorted wave function that satisfies the Schrodinger equation with the full optical model potential U. The total reaction cross section oR is then given as crR= a,+ an. The primary purpose of the analyses was to deduce information on the radius parameter rF of WF. The results showed that rF should be around 1.4 fm. This value is much larger than that assumed in the usual barrier-penetration model [ 81, and also in coupledchannel (CC) studies [9-l 21 made for sub- and nearbarrier scattering and fusion. The analyses have revealed other interesting features of wr and WD. For the I60 + 208Pb system, WF at the strong absorption radius RA was found to exhibit a so-called threshold anomaly [ 131, i.e., a sudden decrease of WF when the incident energy E is be0370-2693/90/$03.50

0 Elsevier Science Publishers

low the Coulomb barrier energy. For this system, meanwhile, WD stayed rather constant, and did not show the threshold anomaly observed in WF. In the 32S+ 58,64Ni systems, however, the situation was different: both W, and WD at r= R, showed the anomalous behavior. It was found that for the 32S+64Ni system, wr% f WD, while for the 32S+ 58Ni system, WF z 0 at r= RA. Further, the value of WD for the 3’S+ 64Ni system is larger by a factor of 1.5 than WD for the 3’S+58Ni system. It is remarkable that with these values of W, and W,, along with the real potential V(r) that we determined, we were able to explain all the features of the observed data for c+ and CD, such as the large sub-barrier enhancement in or, the characteristic target-dependence of OF nd a,, and the correlation between the magnitude of or and op. It is also remarkable that the real and imaginary potentials satisfied very well the required dispersion relation [ 14,15 ] at r= R,. The potentials determined in refs. [ 1,2 1, however, do not satisfy the dispersion relation at very radial point r. The purpose of the present study is to try to remove this unsatisfactory feature and to obtain potentials that satisfy the dispersion relation over all space [ 16 1. This provides also an opportunity to extract information on the polarization part of the potential, as separated from the so-called bare (folding) potential part. We do this by considering, as an example, the 160+ “‘Pb system. This work is moti-

B.V. (North-Holland

)

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vated by recent work by Mahaux et al. [ 17- 19 ] who have successfully obtained such optical potentials satisfying the dispersion relation everywhere, for neutron scattering from 40Ca and ‘08Pb. The optical model potential U to be used in the present study is given as U=lic(r)-[[o(r)+V(r)+iW(r)],

(2)

where U,-(r) is the Coulomb potential and V,(r) is the bare potential [ 9,111 that corresponds to the Hartree-Fock potential in refs. [ 17-191. Note that Vo( r) may have an E-dependence coming from the non-locality of the exchange contribution. We ignore such effects, however, in the present study, as they are expected to be small for heavy-ion scattering [ 201. The quantities I/and Win eq. (2) are, respectively, the real and imaginary parts of the so-called polarization potential [ 2 1 ] that originates from couplings to reaction channels. They are assumed to have a volume-type fusion and surface-derivative-type DR parts. Explicitly, Vo( r), V(r) and W( r ) are given, respectively, as

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8 March

1990

rF = 1.40 fm,

‘+ =0.25 fm,

(7)

rp = 1.50 fm,

up ~0.45 fm.

(8)

These values are chosen as the best possible values, taking into consideration results of studies reported in refs. [ 1,9 1. Note, however, that these values are still not unique, and this non-uniqueness might introduce a certain ambiguity in the present study. Once the geometrical parameters are fixed in this way, the dispersion relation is reduced to one for the strength parameters I’, and W, ( i = F and D). The relation now reads [ 14,15 ] Y:(E) = K:(E) +E-Es

YP

m

s

0

dE’

W,(E’ ) (E’-E,)(E’-E)

’

(9)

wheref(X,)=[l+exp(X,)]-’ with Xi=(r-R,)/u, (i = 0, F and D) is the usual Woods-Saxon function. The optical potential given above involves eleven parameters, excluding the Coulomb radius parameter in .!J,-,which we fix as r,= 1.25 fm. It is impossible to determine all the eleven parameters uniquely from a X2-analysis; we thus fix a priori some of the parameters by taking values from the literature. Following refs. [ 9,111, we fixed the parameters of the bare potential Vo( r) as

where P stands for the principal value and vi( Es) is the value of the potential at a reference energy E= Es. Using the above relation, it is possible to evaluate values of I’F and L’p, once the values of WE(E) and WD (E) are known. In an attempt to obtain the values of W,(E) and WD (E), we first carried out a X2-analysis, treating all the four strength parameters VF, vp, W, and WD as adjustable parameters. We took into account all the data [3,4] available for incident energies between Elab= 78 MeV and 104 MeV. We included the total DR cross sections, when available, in theX2-analysis. The values of the parameters determined in this way showed, as expected, a considerable fluctuation; i.e., they changed appreciably from one energy point to another, excepting those of WD. For this exceptional case, the values were determined as a fairly smooth function of E. To illustrate this and also the fluctuation mentioned above, we show in fig. 1 the values of VD and W,. As seen, the VD values fluctuate fairly strongly, while those of WD behave rather smoothly as a function of E. The later WD values can be represented well by the following function of E (in units of MeV):

vo=60.4MeV,

W,(E)=0

V,(r) = Vof(Xo), V(r) = V,(r)+

(3)

I/,(r)

=V,J”(X,)+4V,&~,

df(Xp) D

W(r)= W,(r) + WD(r) = w,f(&)+4W,,~,-

df( xD) mD

ro=1.176fm,

’

ao=0.658fm.

(6)

With these values, Vo(r) has values very similar to those of the folding potential at rxR,. Further, we fix all the geometrical parameters in the polarization potential V(r)+iW(r) as 20

for EG 70, =O.O185(E-70)

for 70
=0.37

for 90 GE.

The solid line shown in fig. lb is WD predicted

(10)

by the

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The solid curve shown in fig. 1a shows the V, values thus calculated. The V,(E,) value used is V,(E,) = 0.5 MeV at Es = 90 MeV. In what follows, we shall use thus calculated IJb (E) as our final V,,(E). The second x2 analysis was then carried out by using these V, and W,, and by treating VF and W, as adjustable parameters. Fig. 2 shows the values of V, and W, values thus determined. As seen, both are now determined as fairly smooth functions of E. The W, values can now be well represented (in units of MeV) as W,(E)=0

80

90

100

110

EIob(MeVl

Fig. 1. The strength parameters (a) V,(E) and (b) W,(E) of the direct reaction potential as functions of E. The dots are those extracted from the X2-analyses, while the full lines drawn for I+‘, and V,(E) are the predictionsofeq. (IO) andeq. (9) using (IO), respectively.

above eq. ( 10). As seen, the line fits the empirical values rather well. The fact that W, was able to be fixed as a smooth function of E indicates that these values are reliable. There is also a reason that W, would be determined rather unambiguously and hence appear as a smooth function of E. The reason lies in the fact that W, is the dominant absorptive term in the peripheral region. Therefore, the elastic scattering cross section is quite sensitive to the value of W,. This is not the case for Vi,; even in the peripheral region, VD is generally smaller than the bare potential Vo( r), making it difficult to determine the value unambiguously. The fluctuation seen in fig. la may be understood to arise from this difficulty. Since a reliable value of W, is now available, one can generate better values of P’b by using the dispersion relation (9). In doing this, we need to know one more parameter, i.e., the value of Vi, at E=E,. We may fix this VD(Es) by fitting the average of the resultant Vi, to that of the empirically determined I’,,.

for EG 77.5,

=0.52(E-77.5)

for 77.5
=3.4

for 84 GE.

(11)

The solid curve shown in fig. 2b is the prediction of eq. ( 11). The curve fits the empirical values reasonably well. We then calculated values of V, by using eq. (9) and ( 11). The constant value involved was chosen as VF(Es) = 1.8 MeV at Es= 84 MeV. The solid line shown in fig. 2a is the prediction. As seen, the

80

,

I

I

I

I

I

a

>a 5

4.0-

. .

.

Y 200

I 70

00

90

100

HO

E,,b(MeV) Fig. 2. The strength parameters (a) V,(E) and (b) W,(E) of the fusion potential as functions of E. The dots are those extracted from the X*-analyses, while the full lines drawn for Wo and V,(E) are the predictions ofeq. ( I1 ) and eq. (9) using ( 1 1), respectively. 21

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predicted V,-values again agree reasonably well with the empirically determined values, except at the lowest energy point (E= 78 MeV). The discrepancy seen there, however, should not be taken too seriously, since the data are rather insensitive to the vr-values at such a low energy. Excepting this point, the empirically determined VF and W, are seen to satisfy the dispersion relation fairly well. This result then suggests that we may use, as our final potential, the one with W, and W,, given, respectively, by eqs. ( 11) and ( lo), and also VF and L’b generated from the dispersion relation as done above. This potential fully satisfies the dispersion relation. It can also reproduce the data well. This is seen in figs. 3 and 4, where we have presented R= oEL/ is the Rutherford cross secGRLJTH (where ORUTH tion), CJ~and cub, calculated by using our final potential. The results are also compared with the experimental data. As seen, all experimental R, oF and aD results are well reproduced by the calculations. We now wish to give a few remarks on the polarization potential we obtained. The first point is that

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1990

I

I60 + “*Pb

Fig. 4. Comparison of the calculated direct reaction, fusion, and total reaction cross sections using our final optical potential for the ‘60+208Pb system, with the experimental data. The data are taken from refs. [ 3,4].

_I

Wd 40

‘23

160

8 c m!dfd

Fig. 3. Comparison of the calculated ratio R of the elastic scattering cross sections to Rutherford cross sections, using our final optical potential for the 160+ 208Pb system, with the experimental data. The broken line shown for I&,,,= 88 MeV is R calculated by neglecting Vo( r). The data are taken from refs. [ 3,4].

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W, depends on E much more slowly than does W,, around the Coulomb barrier energy. This is seen by comparing the slopes in eqs. ( 10 ) and ( 11). The slope (0.0185) of W, is much smaller than that (0.52) of W,. This seems to reflect the fact that a, changes much more slowly with E than does a,. The resultant energy-dependence of W, and W, seems thus physically plausible. The second remark is that, at the strong absorption radius RA, the values of the real polarization potential ( V=O.4-0.7 MeV) are generally smaller than those of the bare potential (V,= 1.4 MeV). This result is quite different from that obtained in the CC studies [ 9,12 1, in which much larger polarization potentials, often larger than the bare potentials, have been obtained. The origin of such a large polarization potential obtained may be ascribed to the fact that the calculations assumed a small rF value of rF= 1.O fm as already remarked above. Use of such a small rF value makes the absorption in the barrier region ex-

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tremely weak [ I], which in turn makes the coupling effectively very strong. This makes the resultant polarization potential extremely large. This suggests then that if CC calculations are repeated with a large rF value such as used in the present study, the resultant polarization potential may become much smaller than that obtained before, and hopefully agree with what is deduced in the present study. It is thus interesting to do such calculations. Our final comment is on V,,(r), which is a new feature in this study. This potential gives rise to an important effect on the calculated oEEL.This is shown in fig. 3, where we show, by the broken line, a calculated R for E=88 MeV, obtained by setting V,=O, in comparison with what we obtained before (solid line) and also the experimental data. As seen, the neglect of V,, clearly worsens the fit to the experimental data in the angle region between, say, 19,and &, where 0, is the angle at which R starts to fall off while 19~is the angle where the value of R becomes R - 10 -2, respectively. Note that the above angular region corresponds to partial waves that pass through the region where VD( r) has an appreciable value. Therefore, the effect of V,,(r) on aEL seen above is not surprising. We may then expect also that VD( r) will also give rise to important effects on the direct reaction cross sections. A study of the effects on such cross sections would also be interesting. In short, we have carried out simultaneous X2-analysis of elastic scattering and fusion (and also DR cross sections when available) for the ‘60+208Pb system at near- and sub-barrier energies, and determined a polarization potential that satisfies the dispersion relation over all space. This work is supported by the US Department of Energy and by the Ministry of Education, Korea,

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through the Basic Science Institute

8 March

Program,

1990

1989.

References [I ] S.-W. Hong, T. Udagawa

and T. Tamura, Nucl. Phys. A 491 (1989) 492. [2] T. Udagawa, T. Tamura and B.T. Kim, Phys. Rev. C 39 (1989) 1840. [ 31 F. Videvaek et al., Phys. Rev. C 15 ( 1977) 954. [4] E. Valgaris, L. Gradzins, S.G. Steadman and R. Ledoux, Phys. Rev. C 33 (1986) 2017. [5] A.M. Stefanini et al., Phys. Rev. Lett. 59 ( 1987)2852. [6] A.M. Stefanini et al., Nucl. Phys. A 456 (1986) 509. [ 71 T. Udagawa and T. Tamura, Phys. Rev. C 29 (1984) 1922; T. Udagawa, B.T. Kim and T. Tamura, Phys. Rev. C 32 (1985) 124. [ 81 J.R. Birkelund and J.R. Huizenga, Annu. Rev. Nucl. Part. Sci. 33 (1985) 124.

[9] G.R. Satchler, M.A. Nagarajan, J.S. Lilley and I.J. Thompson, Ann. Phys. (NY) 178 (1987) 110. [IO] C.H. Dasso, S. Landowne and A. Winther, Nucl. Phys. A 405 (1983) 381. [ 111M.J. Rhoades-Brown and M. Prakash, Phys. Rev. Lett. 53 (1984) 333; M.J. Rhoades-Brown and P. Braun-Munzinger, Phys. Lett. B 136 (1984) 19. [ 121 S. Landowne and S.C. Pieper, Phys. Rev. C 29 ( 1984) 1352; S.C. Pieper, M.J. Rhoades-Brown and S. Landowne, Phys. Lett. B 162 (1985) 43. [ 131 J.S. Lilley, B.R. F&on, M.A. Nagarajan, I.J. Thompson and D.W. Banes,Phys. Lett. B 151 (1985) 181. [ 141 C. Mahaux, H. Ngo and G.R. Satchler, Nucl. Phys. A 449 (1986) 354; A 456 (1986) 134. [ 151 M.A. Nagarajan, C. Mahaux and G.R. Satchler, Phys. Rev. Lett54 (1985) 1136. [ 161 B.T. Kim, H.C. Kim and K.E. Park, Phys. Rev. C 37 (1988) 998. [ 171 C.H. Johnson, D.J. Horen and C. Mahaux, Phys. Rev. C 36 (1987) 2252. [ 181 C.H. Johnson and C. Mahaux, Phys. Rev. C 38 (1988) 2573. [ 191 C.H. Johnson and C. Mahaux, Phys. Rev. C 38 ( 1988) 2589. [20] G.R. Satchler and W.G. Love, Phys. Rep. 55 ( 1979) 183. [ 2 1] W.G. Love, T. Terasawa and G.R. Satchler, Nucl. Phys. A 291 (1977) 183.

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