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Dynamical fusion and direct reaction polarization potentials for heavy-ion scattering and fusion at sub-Coulomb-barrier energies
Physics Letters B 273 North-Holland
( I99 1 ) 37-4 I
PHYSICS
LETTERS
B
Dynamical fusion and direct reaction polarization potentials for heavy-ion scattering and fusion at sub-Coulomb-barrier energies B.T. Kim
and T. Udagawa
Received
22 August
199
I; revised manuscript
received
30 September
I99
1
Calculations are made of direct reactlon and fusion parts of the dynamical polarization potential obtained by reducing a large scale coupled-channels problem to that of a single-channel optical potential model. Aspects of the physical nature of the resultant potential are investigated, such as the nonlocality of the potential, and the region where absorption due to direct reactions and fusion mainly occurs.
In recent publications [ I-31, we have reported on simultaneous X’-analyses of elastic scattering, fusion and total direct reaction (DR) cross section data for a few heavy-ion systems at sub- and near-Coulombbarrier energies. These analyses are based on the extended optical model, in which the imaginary part of the optical potential is assumed to consist of a volume-type fusion part (fusion potential) and a surface-derivative-type DR part. In ref. [3], we have further considered real potentials that are expected to appear from a general dispersion relation [4,5] holding between the real and imaginary parts of the potential. .L\n important result obtained from the analyses is that the fusion potential, assumed to have a WoodsSaxon shape with a radius RF=rF(.4 I” +.-IA/’ ) and a sharp diffuseness of a F~ 0.25 fm, should have a large rFz 1.4 fm. placing R,: beyond the peak in the Coulomb barrier. The value is much larger than that usually assumed in barrier penetration model [6] or coupled-channels (CC) calculations [ 7,8] for fusion. Satchler et al. [ 91 then proposed that the large I’~-values have their origin in the known importance of couplings to DR channels; in a CC approach, the 0370-2693/9
I /$ 03.50 0 I99 1 Elsevicr Science Publishers
B.V.
so-called “bare” imaginary fusion potential E’,,F( r) may still be confined inside the Coulomb barrier and have a small radius, IoF= I .O fm. lfone projects. however, this CC problem onto the elastic channel, there appears an additional term, A I+;, in the resultant fusion potential. This is of course a part of the so-called dynamical polarization potential (DPP) AC! which appears when such a projection is done. The proposal made by Satchler et al. [ 91 is that rF becomes large due to AU; in AL:. The aim of the present paper is to study whether AM;(r) makes the r,-value of the resultant fusion potential ( Mb,+AM’F) larger than Y~)~.For this purpose. we do calculations of ALL particularly of its imaginary part Am’. The term Au/ is further decomposed into fusion (A M;) and DR (A LV”) parts. Since the necessary formulation of the calculation has already been given, for instance, in ref. [8], we shall give here only a very brief summary. In a partial wave form. the CC equations may be given as (E--/l/)%, .(1.) = 1 I
I’,/
(I.)w,, (1.) ,
(1)
37
Volume 273, number
1,2
PHYSICS
(El. -111. )oJc(r) = vclzl(r) ,
(2)
where E (Ej.) and zz(r) (oJ/. ( r ) ) are the projectile energy and radial partial wave function in the elastic (inelastic) channel, respectively, and vw is the coupling potential. The hamiltonian H/is expressed as f i 2 ( d2
t t / = - - 9"~ ~
l ( / + 1 ) ) + /~,o(,'),
r-
(3)
(4)
where AU/(r, r' ) is the DPP due to channel coupling and is given as AU/(r, r ' ) =
~ t~/l (r)G/(r, r' )~'c/(r' ) . /.
(5)
In eq. ( 5 ) G/(r, r' ) is the partial wave optical model Green's function in the excited channel/'. Because of the presence of the Green's function, 2xU¢(r, r' ) given by eq. (5) is nonlocal, as indicated. It is also/-dependent in general. The DPP AU/(r, r ' ) is complex and thus can be written as a sum of real and imaginary parts, AU/(r, r ' ) = AI'/(r, r' ) + i A W / ( r , r ' ) .
(6)
We further decompose the imaginary part AWl(r, r' ) into fusion and DR parts, A WF:/(r, r' ) and AWD:/(r, r' ). respectively;
AH](r,r')=AWv:/(r,r')+Al4"D:/(r,r').
(7)
In order to achieve this decomposition, wc utilize the identity Im G/=G?- ( I m U/)G/=£2/+ ( l m g/),Q/,
(8)
whcre g/ is the free partial wave Green's function, while £2/is the M611er wave operator that generates 38
B
12 D e c e m b e r
1991
the distorted wave function upon operating on the plane wave function; ,Q~= 1 + U/Gj. Substituting eq. (8) into the imaginary part ofeq. (5), we find Al~](r, r ' ) [ = I r a AU/(r, r ' ) l
= ~ vwGF IffoFGI,~'I'I /
+ ~/' vw£2c~ (Ira g/)-Qt Vcl.
where /~'~o(r) is the " b a r e " potential, which consists of the Coulomb potential ~ , (r), an energy independent real Hartree-Fock (folding) potential ~o and an imaginary absorptive fusion potential I4\~F: Uoo(r) = l/c(r) + ~ ( r ) + i W o v ( r ) . The bare potential Uoo and the coupling potential rH (assumed to be real) essentially define the CC problem. Note that the indices l and I' used in eqs. ( 1 ) - ( 3 ) denote a set of quantum numbers specifying the channel considered, including orbital angular momenta. Eliminating oJ/(r) in eqs. ( 1 ) and (2), we obtain the reduced optical model Schr~3dinger equation,
( E - H / ) z / ( r ) = f AU/(r, r' )z/(r' )dr' ,
LETTERS
(9)
The first term on the right-hand side of the above equation clearly describes absorption due to fusion occurring in the various inelastic channels, while the second term represents absorption due to a loss of flux caused by the inelastic scattering. Therefore we may set AWv:/(r, r' ) = ~ ~'w G?: 14ovGc t'rt,
(10)
1'
A Wt):/(r, r' ) = ~ vH.E2f ( I m gc )-Qc Vet.
( 11 )
l'
Note that in obtaining the above results, use is made of the fact that L'/c is real. We have carried out numerical calculations of AI4"F:/(r, r' ), AI4"D:/(r, r'), and AV/(r, r') for the ~60 + =°SPb system at Elab = 80 MeV. Couplings of the ground state to the 2.61 McV 3 - , 3.20 MeV 5 - , and 4.07 MeV 2 + states in =°aPb, and to the 6.13 MeV 3 state in '60 were considered. The values of the parameters involved in the calculations, i.e., those for the potentials I o ( r ) and Wov(r), and also the deformation parameters, are all taken to be the same as those used in ref. [ 8 ]. The calculations were done for various/-values. The results show that AWv:l(r, r' ), AWD:/(r, r' ), and 2xl](r, r' ) are all highly nonlocal and/-dependent. The characteristic features of nonlocality are, however, found to depend weakly on l. The calculated AWl(r, r' ), kl4"F:l(r, r' ), and Al~'tT,:/(r, r' ) are displayed in figs. l a - l c , respectively, for l= 10 (in units of h) as an example. A g / ( r , r' ), kWv:/(r, r' ), and AIUD:/(r, r' ) all have a very complicated nonlocal structure. It is worth noting that the overall structure of AW/(r, r' ) is similar to that of AI4"v:/(r, r' ) in the region of radial distances smaller than the strong absorption radius (about 12.4 fm in this case) while it resembles AWD:/(r, r ' ) in the outer region. Qualitatively, we thus conclude that the region where fusion occurs is localized in the inner region of the strong absorption
Volume 273, number 1,2
PHYSICS LETTERS B
12 December 1991
•vA/
0.10
18
i
0.08 0.06 0.04 E n,,-
12
<~
10
0.02 0.00 -0.02 -0.04
8
-0.06 -0.08 -0.10 0.10
(b)
0.08 0.06 10 ¢-
o,o4 0.02
8
8
10
12
14
16
18
R(fm)
8
10
12
14
16
18 0.00
R (fm)
-0.02
Fig. I. The imaginary part (a) of/= 10f* nonlocal DPP for the J60+2°spb system at l:]~b=80 MeV. The imaginary part is further decomposed into fusion (b) and direct reaction (c) parts. The coupling scheme and the ingredients are in text. The solid contours are the 0.0 MeV level, otherwise indicated as the i0.0 MeV one. The dotted ones are the 0.01 MeV level. radius, while DR occurs in the outer region. More quantitative information on the area where fusion and D R take place may be obtained by analyzing the absorption densities A~(r) ( i = F and D ) defined as
Ai(r) =
A Wi,z(r, r' ) Re[pl(r, r' ) ] d r ' 0
(#/(r, r' ) =-Z'f(r)x/(r'
)) .
(12)
The above expression can be derived from the divergence of the current density. In figs. 2a and 2b we present the calculated A F ( r ) and AD(r) as functions ofr. The DR absorption density AD(r) is appreciable in the region o f r > 12 fm. It has an oscillatory long range tail. The period o f oscillation reflects that of the distorted wave function z / ( r ) , and the long range tail comes from C o u l o m b excitation. The nuclear excitation contribution is confined in the peak region o f 1 2 < r < 14 fm. The center o f the peak is located at r ~ 13 fm. This region is simply the so-called grazing region where D R is supposed to occur. The fusion absorption density A F ( r ) has also an
"<
-0.04 -0.06 -0.08 I
I
12'0
1~'0
-0.10 6{0
~ I0
10'0
1~]0
R (fm) Fig. 2. (a) The fusion absorption density .4v(r) and (b) direct reaction absorption densib .4D(r) defined by eq. (19). oscillatory structure, The period of oscillation reflects that o f the nonlocal density R e [ p ( r , r' ) ] in eq. (12). The envelope of oscillation, peaked at r ~ 9.7 fm, is roughly proportional to z,~/,(r) H])F(r)Iz/(r) F-~, the limit o l A F ( r ) obtained when the G r e e n ' s funclion is assumed to be a d-function ( a d i a b a t i c ass u m p t i o n ) . The region wherc ,4F(r) takes an appreciable value is about 8 < / " < 12 fm, where z,//(r) and HoF(r ) strongly overlap. (Note that the peak of z,# (r) appears at R o = 9 . 8 fm o f I o ( r ) , while g])v(r) has RoF=8.4 fm,) Particularly, the region where Av(r) takes a large positive value (absorption region) is limited "to r~< 10.2 fm. This absorptive region is somewhat shifted outward as c o m p a r e d with the " b a r e " fusion region defined by I~oFIZI] 2, but still stays inside the barrier (the top o f the C o u l o m b barrier ~ 11.4 fm). It is much inside the fusion region predicted in our model [3] with r v = l . 4 fm (R~_= 1 1,8 fm). Note that, next to this absorptive region, there appears a creative region ( 1 0 < r < 12 fro) where ,'Iv(r) becomes negative. The existence o f such 39
Volume 273, number 1,2
PHYSICS LETTERS B
a creative region (observed also by other authors [ 10] ) may effectively reduce the overall effects from the channel-couplings on fusion. The result obtained above is consistent with that o f a CC study made by Satchler et al. [ 8 ]. They studied a projection problem similar to the one studied in this paper. Instead of calculating D P P explicitly, however, they tried to find "effective local polarization potentials" that can reproduce the calculated CC elastic, fusion and total DR cross sections. They obtained such potentials by performing z2-analyses using the calculated CC cross sections as data. They were able to find a few sets of effective potentials in which AWv is set to be zero. Namely, there was no need to introduce any additional fusion term in their polarization potential in order to reproduce the calculated CC cross sections. A large enhancement was obtained in the CC calculation, but that was interpreted as due to a large polarization term appeared in the real part of the potential. As shown in fig. la, AH](r, r' ) is highly nonlocal. This is also the case for the real Al~](r, r' ). For practical purposes, and also for the purpose of establishing a connection between the calculated D P P and the potentials obtained by phenomenological analyses, it is useful to localize the DPPs. The most popular method is the so-called equivalent effective potential (EEP) method [11], in which a local EEP AU~H'(r) is constructed as A Cf/EEP ( r ) =
1 _
x/(r)
Z~ I f E E l ' ( F )
-i- i A 14"~E,'(r)
The above A W ) ( r ) has no physical significance as an imaginary potential; it does not represent any absorption due to real nuclear reactions. Therefore, it should not give any contribution to the absorption cross section. This is indeed the case and is seen from the fact that the expectation value of A W ) ( r ) identically vanishes, i.e.,
f xt(r)AW) (r)x/(r)dr= 0. A similar argument can also be applied to the real potential ( A t ' ) ( r ) ) coming from the imaginary part of AUt(r, r' ). It is important to note here, however, that All') (r) and AV) (r) plays an essential role in reproducing the correct elastic distorted wave function when one generates it by solving eq. (14). As is clear, the origin of such terms in the nonlocal character of Ut(r, r' ), and in this sense we may call these terms the "genuine" nonlocal terms. In fact, the terms become bigger when nonlocality of the original potential is larger, and vice versa. For instance, consider the nonlocal potential introduced by Perey and Buck [ 12 ], where nonlocality is represented by the nonlocality range parameterfl. We investigated the genuine nonlocal terms of this potential, finding that their magnitudes increase with the increasing ft. For a reasonable value offl, however, the magnitudes are still not very large. This is not the case for the present AU/(r, r' ). To illustrate this, we present, in fig. 3 using a dotted line,
r
JZxC,(,',r')x,(,")d,".
(E-HI)xI(r) = AU~VP(r)xI(r) .
0.10
~
~
J
~
0.06
c
o.o4
=) ,.4 •"
0.00
/:..........
0.02 -
•
(14)
One disturbing feature in the EEP method is that AU~EP(r) contains a imaginary (real) term that originates from the real (imaginary) parts of the original AUt(r, r' ). Let us take, as an example, the imaginary part and denote it as A W ) ( r ) . Explicitly, AW) (r) can be given as
AW)(r)=lmCj@2~fA~,](r,r')xl(r')dr'
008
(13)
In fact, AU~EP(r) yields the same distorted wave function as the nonlocal one, as a solution of the following local Schr6dinger equation:
40
12 December 1991
) .
(15)
-0.04
•
-0.06
- !
-0.08
•
:/
-0.10 6.0
I 8.0
i 10.0
I 12.0
{
14.0
16.0
R (f~) Fig. 3. The absorption densities as a function of r. The full line represents A O(r) ofeq. 931 ). The dotted line denotes the genuine nonlocal absorption density A ~ ( r ) defined by eq. (30). The broken curve is the total sum A(r) of those.
Volume 273, number 1,2
PHYSICS LETTERS B
the "'genuine nonlocal absorption density" A ~(r) defined in terms o f A W ] (r) as
A~(r) =ZT'(r)AW)(r)z/(r) .
(16)
The "true absorption density" A ° ( r ) which is the sum of Av(r) and AD(r) defined earlier in eq. (12) may now be given in terms of A W ° ( r ) = A W ~ E V ( r ) AW) (r) as
A°(r) =zT(r)AW°(r)zt(r)
.
(17)
I n fig. 3, we display A o(r) by the full line and the sum .4 (r) = A o(r) +,4 ~(r) by the broken curve. The absolute magnitude of the genuine nonlocal absorption density A 1(r) is essentially the same as that of the true absorption density A ° ( r ) , but the sign is opposite over most of the range of r. Therefore the shape of the resultant total.4 (r) becomes completely different from that of A ° ( r ) . This shows how important it is to subtract A ~(r) from A ( r ) in order to obtain the correct absorption density A ° ( r ) . The above result suggests that it is improbable that a highly nonlocal DPP, like the one in the present work, can be localized in a meaningful m a n n e r [ 13 ]. As remarked, the DPP is also/-dependent. In ref. [ 13 ] a very elaborate effort was made to obtain localized and /-averaged DPPs. Unfortunately, the potentials obtained are unable to reproduce consistently the calculated CC elastic scattering cross sections. In contrast to this, we have always been successful in finding a local, /-independent optical potential that can reproduce the experimental elastic scattering, fusion and total DR cross section data very well. Apparently, it has not been possible so far to derive such a local, /-independent potential such as is obtained from phenomenological analyses, directly from CC calculations. One possible origin of this failure may be ascribed to the insufficient n u m b e r of channels included in the CC calculations and the use of a too simplified form of the coupling scheme (neglecting the coupling between inelastic channels, etc.). One might generally expect [ 14] that if one includes more channels and uses a more realistic coupling scheme, complicated nonlocality may be washed away. The Green's functions in the different inelastic channels have different oscillations in r-space and the resulting potential becomes more local because of the random-phase nature of the contributions from a large n u m b e r of terms. T h e / - d e p e n d e n c e may also be averaged out. It would be interesting and important if
12 December 1991
such calculations could be done. In summary, we performed calculations of the direct reaction and fusion parts of the dynamical polarization potential obtained by reducing a large scale coupled-channels problem to that of a single-channel optical potential model. It is found that these terms are highly nonlocal and cannot be localized in a meaningful manner. We also studied the CC effects on the region where fusion occurs. The results indicate that the region is somewhat shifted outward as compared with the original bare fusion region, but still remains inside the peak of the C o u l o m b barrier. The region is further inside the fusion region predicted in our earlier phenomenological analyses. We thank Professor W.R. Coker and Professor S.-W. Hong for a careful reading of the manuscript. This work was supported in part by the Ministry of Education, through the basic Science Institute Program, 1991, and in part by the US Department of Energy, G r a n t No. DE-FG05-84ER40145.
References
[ 1] T. Udagawa, B.T. Kim and T. Tamura, Phys. Rev. C 32 (1985) 124. [2] T. Udagawa, T. Tamura and B.T. Kim, Phys. Rev. C 39 (1989) 1840. [3] B.T. Kim, M. Naito and T. Uclagawa, Phys. Left. B 237 (1990) 19. [4] C. Mahaux, H. Ngo and G.R. Satchler, Nucl. Phys. A 449 (1986) 354;A 456 (1986) 134. [5] M.A. Nagarajan, C. Mahaux and G.R. Satchler, Phys. Rev. Leu. 54(1985) 1136. [6] J.R. Birkelund and J.R. Huizenga, Annu. Rev. Nucl. Part. Sci. 33 (1983) 265. [ 7 ] S.G. Steadman and M.J. Rhoades-Brown,Annu. Rev. Nucl. Part. Sci. 36 (1986) 649; M.J. Rhoades-Brownand P. Braun-Munzinger,Phys. Left. B 136 (1984) 19: S. Landowneand S.C. Pieper, Phys. Rev. C 29 (1984) 1352. [8]G.R. Satchler, M.A. Nagarajan, J.S. Lilley and 1.J. Thompson, Ann. Phys. 178 (1987) 110. [9]G.R. Satchler, M.A. Nagarajan, J.S. Lilley and l.J. Thompson, Phys. Rcv. C41 (1990) 1869. [10] A.A. loannides and R.S. Mackinlosh, Phys. Len. B 161 (1990) 43. [ 11 ] M.E. Franey and P.J. Ellis, Phys. Rev. C 23 ( 1981 ) 787. [12] F. Percy and B. Buck, Nucl. Phys. 32 (1962) 353. [13]I.J. Thompson, M.A. Nagarajan, J.S. Lilley and M.J. Smithson, Nucl. Phys. A 505 (1989) 84. [14]L. Canton, Y. Hahn and G. Cauapan, Phys. Rev. C 43 (1991)2441. 41
( I99 1 ) 37-4 I
PHYSICS
LETTERS
B
Dynamical fusion and direct reaction polarization potentials for heavy-ion scattering and fusion at sub-Coulomb-barrier energies B.T. Kim
and T. Udagawa
Received
22 August
199
I; revised manuscript
received
30 September
I99
1
Calculations are made of direct reactlon and fusion parts of the dynamical polarization potential obtained by reducing a large scale coupled-channels problem to that of a single-channel optical potential model. Aspects of the physical nature of the resultant potential are investigated, such as the nonlocality of the potential, and the region where absorption due to direct reactions and fusion mainly occurs.
In recent publications [ I-31, we have reported on simultaneous X’-analyses of elastic scattering, fusion and total direct reaction (DR) cross section data for a few heavy-ion systems at sub- and near-Coulombbarrier energies. These analyses are based on the extended optical model, in which the imaginary part of the optical potential is assumed to consist of a volume-type fusion part (fusion potential) and a surface-derivative-type DR part. In ref. [3], we have further considered real potentials that are expected to appear from a general dispersion relation [4,5] holding between the real and imaginary parts of the potential. .L\n important result obtained from the analyses is that the fusion potential, assumed to have a WoodsSaxon shape with a radius RF=rF(.4 I” +.-IA/’ ) and a sharp diffuseness of a F~ 0.25 fm, should have a large rFz 1.4 fm. placing R,: beyond the peak in the Coulomb barrier. The value is much larger than that usually assumed in barrier penetration model [6] or coupled-channels (CC) calculations [ 7,8] for fusion. Satchler et al. [ 91 then proposed that the large I’~-values have their origin in the known importance of couplings to DR channels; in a CC approach, the 0370-2693/9
I /$ 03.50 0 I99 1 Elsevicr Science Publishers
B.V.
so-called “bare” imaginary fusion potential E’,,F( r) may still be confined inside the Coulomb barrier and have a small radius, IoF= I .O fm. lfone projects. however, this CC problem onto the elastic channel, there appears an additional term, A I+;, in the resultant fusion potential. This is of course a part of the so-called dynamical polarization potential (DPP) AC! which appears when such a projection is done. The proposal made by Satchler et al. [ 91 is that rF becomes large due to AU; in AL:. The aim of the present paper is to study whether AM;(r) makes the r,-value of the resultant fusion potential ( Mb,+AM’F) larger than Y~)~.For this purpose. we do calculations of ALL particularly of its imaginary part Am’. The term Au/ is further decomposed into fusion (A M;) and DR (A LV”) parts. Since the necessary formulation of the calculation has already been given, for instance, in ref. [8], we shall give here only a very brief summary. In a partial wave form. the CC equations may be given as (E--/l/)%, .(1.) = 1 I
I’,/
(I.)w,, (1.) ,
(1)
37
Volume 273, number
1,2
PHYSICS
(El. -111. )oJc(r) = vclzl(r) ,
(2)
where E (Ej.) and zz(r) (oJ/. ( r ) ) are the projectile energy and radial partial wave function in the elastic (inelastic) channel, respectively, and vw is the coupling potential. The hamiltonian H/is expressed as f i 2 ( d2
t t / = - - 9"~ ~
l ( / + 1 ) ) + /~,o(,'),
r-
(3)
(4)
where AU/(r, r' ) is the DPP due to channel coupling and is given as AU/(r, r ' ) =
~ t~/l (r)G/(r, r' )~'c/(r' ) . /.
(5)
In eq. ( 5 ) G/(r, r' ) is the partial wave optical model Green's function in the excited channel/'. Because of the presence of the Green's function, 2xU¢(r, r' ) given by eq. (5) is nonlocal, as indicated. It is also/-dependent in general. The DPP AU/(r, r ' ) is complex and thus can be written as a sum of real and imaginary parts, AU/(r, r ' ) = AI'/(r, r' ) + i A W / ( r , r ' ) .
(6)
We further decompose the imaginary part AWl(r, r' ) into fusion and DR parts, A WF:/(r, r' ) and AWD:/(r, r' ). respectively;
AH](r,r')=AWv:/(r,r')+Al4"D:/(r,r').
(7)
In order to achieve this decomposition, wc utilize the identity Im G/=G?- ( I m U/)G/=£2/+ ( l m g/),Q/,
(8)
whcre g/ is the free partial wave Green's function, while £2/is the M611er wave operator that generates 38
B
12 D e c e m b e r
1991
the distorted wave function upon operating on the plane wave function; ,Q~= 1 + U/Gj. Substituting eq. (8) into the imaginary part ofeq. (5), we find Al~](r, r ' ) [ = I r a AU/(r, r ' ) l
= ~ vwGF IffoFGI,~'I'I /
+ ~/' vw£2c~ (Ira g/)-Qt Vcl.
where /~'~o(r) is the " b a r e " potential, which consists of the Coulomb potential ~ , (r), an energy independent real Hartree-Fock (folding) potential ~o and an imaginary absorptive fusion potential I4\~F: Uoo(r) = l/c(r) + ~ ( r ) + i W o v ( r ) . The bare potential Uoo and the coupling potential rH (assumed to be real) essentially define the CC problem. Note that the indices l and I' used in eqs. ( 1 ) - ( 3 ) denote a set of quantum numbers specifying the channel considered, including orbital angular momenta. Eliminating oJ/(r) in eqs. ( 1 ) and (2), we obtain the reduced optical model Schr~3dinger equation,
( E - H / ) z / ( r ) = f AU/(r, r' )z/(r' )dr' ,
LETTERS
(9)
The first term on the right-hand side of the above equation clearly describes absorption due to fusion occurring in the various inelastic channels, while the second term represents absorption due to a loss of flux caused by the inelastic scattering. Therefore we may set AWv:/(r, r' ) = ~ ~'w G?: 14ovGc t'rt,
(10)
1'
A Wt):/(r, r' ) = ~ vH.E2f ( I m gc )-Qc Vet.
( 11 )
l'
Note that in obtaining the above results, use is made of the fact that L'/c is real. We have carried out numerical calculations of AI4"F:/(r, r' ), AI4"D:/(r, r'), and AV/(r, r') for the ~60 + =°SPb system at Elab = 80 MeV. Couplings of the ground state to the 2.61 McV 3 - , 3.20 MeV 5 - , and 4.07 MeV 2 + states in =°aPb, and to the 6.13 MeV 3 state in '60 were considered. The values of the parameters involved in the calculations, i.e., those for the potentials I o ( r ) and Wov(r), and also the deformation parameters, are all taken to be the same as those used in ref. [ 8 ]. The calculations were done for various/-values. The results show that AWv:l(r, r' ), AWD:/(r, r' ), and 2xl](r, r' ) are all highly nonlocal and/-dependent. The characteristic features of nonlocality are, however, found to depend weakly on l. The calculated AWl(r, r' ), kl4"F:l(r, r' ), and Al~'tT,:/(r, r' ) are displayed in figs. l a - l c , respectively, for l= 10 (in units of h) as an example. A g / ( r , r' ), kWv:/(r, r' ), and AIUD:/(r, r' ) all have a very complicated nonlocal structure. It is worth noting that the overall structure of AW/(r, r' ) is similar to that of AI4"v:/(r, r' ) in the region of radial distances smaller than the strong absorption radius (about 12.4 fm in this case) while it resembles AWD:/(r, r ' ) in the outer region. Qualitatively, we thus conclude that the region where fusion occurs is localized in the inner region of the strong absorption
Volume 273, number 1,2
PHYSICS LETTERS B
12 December 1991
•vA/
0.10
18
i
0.08 0.06 0.04 E n,,-
12
<~
10
0.02 0.00 -0.02 -0.04
8
-0.06 -0.08 -0.10 0.10
(b)
0.08 0.06 10 ¢-
o,o4 0.02
8
8
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18
R(fm)
8
10
12
14
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R (fm)
-0.02
Fig. I. The imaginary part (a) of/= 10f* nonlocal DPP for the J60+2°spb system at l:]~b=80 MeV. The imaginary part is further decomposed into fusion (b) and direct reaction (c) parts. The coupling scheme and the ingredients are in text. The solid contours are the 0.0 MeV level, otherwise indicated as the i0.0 MeV one. The dotted ones are the 0.01 MeV level. radius, while DR occurs in the outer region. More quantitative information on the area where fusion and D R take place may be obtained by analyzing the absorption densities A~(r) ( i = F and D ) defined as
Ai(r) =
A Wi,z(r, r' ) Re[pl(r, r' ) ] d r ' 0
(#/(r, r' ) =-Z'f(r)x/(r'
)) .
(12)
The above expression can be derived from the divergence of the current density. In figs. 2a and 2b we present the calculated A F ( r ) and AD(r) as functions ofr. The DR absorption density AD(r) is appreciable in the region o f r > 12 fm. It has an oscillatory long range tail. The period o f oscillation reflects that of the distorted wave function z / ( r ) , and the long range tail comes from C o u l o m b excitation. The nuclear excitation contribution is confined in the peak region o f 1 2 < r < 14 fm. The center o f the peak is located at r ~ 13 fm. This region is simply the so-called grazing region where D R is supposed to occur. The fusion absorption density A F ( r ) has also an
"<
-0.04 -0.06 -0.08 I
I
12'0
1~'0
-0.10 6{0
~ I0
10'0
1~]0
R (fm) Fig. 2. (a) The fusion absorption density .4v(r) and (b) direct reaction absorption densib .4D(r) defined by eq. (19). oscillatory structure, The period of oscillation reflects that o f the nonlocal density R e [ p ( r , r' ) ] in eq. (12). The envelope of oscillation, peaked at r ~ 9.7 fm, is roughly proportional to z,~/,(r) H])F(r)Iz/(r) F-~, the limit o l A F ( r ) obtained when the G r e e n ' s funclion is assumed to be a d-function ( a d i a b a t i c ass u m p t i o n ) . The region wherc ,4F(r) takes an appreciable value is about 8 < / " < 12 fm, where z,//(r) and HoF(r ) strongly overlap. (Note that the peak of z,# (r) appears at R o = 9 . 8 fm o f I o ( r ) , while g])v(r) has RoF=8.4 fm,) Particularly, the region where Av(r) takes a large positive value (absorption region) is limited "to r~< 10.2 fm. This absorptive region is somewhat shifted outward as c o m p a r e d with the " b a r e " fusion region defined by I~oFIZI] 2, but still stays inside the barrier (the top o f the C o u l o m b barrier ~ 11.4 fm). It is much inside the fusion region predicted in our model [3] with r v = l . 4 fm (R~_= 1 1,8 fm). Note that, next to this absorptive region, there appears a creative region ( 1 0 < r < 12 fro) where ,'Iv(r) becomes negative. The existence o f such 39
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PHYSICS LETTERS B
a creative region (observed also by other authors [ 10] ) may effectively reduce the overall effects from the channel-couplings on fusion. The result obtained above is consistent with that o f a CC study made by Satchler et al. [ 8 ]. They studied a projection problem similar to the one studied in this paper. Instead of calculating D P P explicitly, however, they tried to find "effective local polarization potentials" that can reproduce the calculated CC elastic, fusion and total DR cross sections. They obtained such potentials by performing z2-analyses using the calculated CC cross sections as data. They were able to find a few sets of effective potentials in which AWv is set to be zero. Namely, there was no need to introduce any additional fusion term in their polarization potential in order to reproduce the calculated CC cross sections. A large enhancement was obtained in the CC calculation, but that was interpreted as due to a large polarization term appeared in the real part of the potential. As shown in fig. la, AH](r, r' ) is highly nonlocal. This is also the case for the real Al~](r, r' ). For practical purposes, and also for the purpose of establishing a connection between the calculated D P P and the potentials obtained by phenomenological analyses, it is useful to localize the DPPs. The most popular method is the so-called equivalent effective potential (EEP) method [11], in which a local EEP AU~H'(r) is constructed as A Cf/EEP ( r ) =
1 _
x/(r)
Z~ I f E E l ' ( F )
-i- i A 14"~E,'(r)
The above A W ) ( r ) has no physical significance as an imaginary potential; it does not represent any absorption due to real nuclear reactions. Therefore, it should not give any contribution to the absorption cross section. This is indeed the case and is seen from the fact that the expectation value of A W ) ( r ) identically vanishes, i.e.,
f xt(r)AW) (r)x/(r)dr= 0. A similar argument can also be applied to the real potential ( A t ' ) ( r ) ) coming from the imaginary part of AUt(r, r' ). It is important to note here, however, that All') (r) and AV) (r) plays an essential role in reproducing the correct elastic distorted wave function when one generates it by solving eq. (14). As is clear, the origin of such terms in the nonlocal character of Ut(r, r' ), and in this sense we may call these terms the "genuine" nonlocal terms. In fact, the terms become bigger when nonlocality of the original potential is larger, and vice versa. For instance, consider the nonlocal potential introduced by Perey and Buck [ 12 ], where nonlocality is represented by the nonlocality range parameterfl. We investigated the genuine nonlocal terms of this potential, finding that their magnitudes increase with the increasing ft. For a reasonable value offl, however, the magnitudes are still not very large. This is not the case for the present AU/(r, r' ). To illustrate this, we present, in fig. 3 using a dotted line,
r
JZxC,(,',r')x,(,")d,".
(E-HI)xI(r) = AU~VP(r)xI(r) .
0.10
~
~
J
~
0.06
c
o.o4
=) ,.4 •"
0.00
/:..........
0.02 -
•
(14)
One disturbing feature in the EEP method is that AU~EP(r) contains a imaginary (real) term that originates from the real (imaginary) parts of the original AUt(r, r' ). Let us take, as an example, the imaginary part and denote it as A W ) ( r ) . Explicitly, AW) (r) can be given as
AW)(r)=lmCj@2~fA~,](r,r')xl(r')dr'
008
(13)
In fact, AU~EP(r) yields the same distorted wave function as the nonlocal one, as a solution of the following local Schr6dinger equation:
40
12 December 1991
) .
(15)
-0.04
•
-0.06
- !
-0.08
•
:/
-0.10 6.0
I 8.0
i 10.0
I 12.0
{
14.0
16.0
R (f~) Fig. 3. The absorption densities as a function of r. The full line represents A O(r) ofeq. 931 ). The dotted line denotes the genuine nonlocal absorption density A ~ ( r ) defined by eq. (30). The broken curve is the total sum A(r) of those.
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the "'genuine nonlocal absorption density" A ~(r) defined in terms o f A W ] (r) as
A~(r) =ZT'(r)AW)(r)z/(r) .
(16)
The "true absorption density" A ° ( r ) which is the sum of Av(r) and AD(r) defined earlier in eq. (12) may now be given in terms of A W ° ( r ) = A W ~ E V ( r ) AW) (r) as
A°(r) =zT(r)AW°(r)zt(r)
.
(17)
I n fig. 3, we display A o(r) by the full line and the sum .4 (r) = A o(r) +,4 ~(r) by the broken curve. The absolute magnitude of the genuine nonlocal absorption density A 1(r) is essentially the same as that of the true absorption density A ° ( r ) , but the sign is opposite over most of the range of r. Therefore the shape of the resultant total.4 (r) becomes completely different from that of A ° ( r ) . This shows how important it is to subtract A ~(r) from A ( r ) in order to obtain the correct absorption density A ° ( r ) . The above result suggests that it is improbable that a highly nonlocal DPP, like the one in the present work, can be localized in a meaningful m a n n e r [ 13 ]. As remarked, the DPP is also/-dependent. In ref. [ 13 ] a very elaborate effort was made to obtain localized and /-averaged DPPs. Unfortunately, the potentials obtained are unable to reproduce consistently the calculated CC elastic scattering cross sections. In contrast to this, we have always been successful in finding a local, /-independent optical potential that can reproduce the experimental elastic scattering, fusion and total DR cross section data very well. Apparently, it has not been possible so far to derive such a local, /-independent potential such as is obtained from phenomenological analyses, directly from CC calculations. One possible origin of this failure may be ascribed to the insufficient n u m b e r of channels included in the CC calculations and the use of a too simplified form of the coupling scheme (neglecting the coupling between inelastic channels, etc.). One might generally expect [ 14] that if one includes more channels and uses a more realistic coupling scheme, complicated nonlocality may be washed away. The Green's functions in the different inelastic channels have different oscillations in r-space and the resulting potential becomes more local because of the random-phase nature of the contributions from a large n u m b e r of terms. T h e / - d e p e n d e n c e may also be averaged out. It would be interesting and important if
12 December 1991
such calculations could be done. In summary, we performed calculations of the direct reaction and fusion parts of the dynamical polarization potential obtained by reducing a large scale coupled-channels problem to that of a single-channel optical potential model. It is found that these terms are highly nonlocal and cannot be localized in a meaningful manner. We also studied the CC effects on the region where fusion occurs. The results indicate that the region is somewhat shifted outward as compared with the original bare fusion region, but still remains inside the peak of the C o u l o m b barrier. The region is further inside the fusion region predicted in our earlier phenomenological analyses. We thank Professor W.R. Coker and Professor S.-W. Hong for a careful reading of the manuscript. This work was supported in part by the Ministry of Education, through the basic Science Institute Program, 1991, and in part by the US Department of Energy, G r a n t No. DE-FG05-84ER40145.
References
[ 1] T. Udagawa, B.T. Kim and T. Tamura, Phys. Rev. C 32 (1985) 124. [2] T. Udagawa, T. Tamura and B.T. Kim, Phys. Rev. C 39 (1989) 1840. [3] B.T. Kim, M. Naito and T. Uclagawa, Phys. Left. B 237 (1990) 19. [4] C. Mahaux, H. Ngo and G.R. Satchler, Nucl. Phys. A 449 (1986) 354;A 456 (1986) 134. [5] M.A. Nagarajan, C. Mahaux and G.R. Satchler, Phys. Rev. Leu. 54(1985) 1136. [6] J.R. Birkelund and J.R. Huizenga, Annu. Rev. Nucl. Part. Sci. 33 (1983) 265. [ 7 ] S.G. Steadman and M.J. Rhoades-Brown,Annu. Rev. Nucl. Part. Sci. 36 (1986) 649; M.J. Rhoades-Brownand P. Braun-Munzinger,Phys. Left. B 136 (1984) 19: S. Landowneand S.C. Pieper, Phys. Rev. C 29 (1984) 1352. [8]G.R. Satchler, M.A. Nagarajan, J.S. Lilley and 1.J. Thompson, Ann. Phys. 178 (1987) 110. [9]G.R. Satchler, M.A. Nagarajan, J.S. Lilley and l.J. Thompson, Phys. Rcv. C41 (1990) 1869. [10] A.A. loannides and R.S. Mackinlosh, Phys. Len. B 161 (1990) 43. [ 11 ] M.E. Franey and P.J. Ellis, Phys. Rev. C 23 ( 1981 ) 787. [12] F. Percy and B. Buck, Nucl. Phys. 32 (1962) 353. [13]I.J. Thompson, M.A. Nagarajan, J.S. Lilley and M.J. Smithson, Nucl. Phys. A 505 (1989) 84. [14]L. Canton, Y. Hahn and G. Cauapan, Phys. Rev. C 43 (1991)2441. 41