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Energy dependence of the surface absorption and Ni+16O scattering
Volume 58B, number 4
PHYSICS LETTERS
29 September 1975
ENERGY DEPENDENCE OF THE SURFACE ABSORPTION A N D Ni + 160 S C A T T E R I N G G.R. SATCHLER
Oak RidgeNational Laboratory, Oak Ridge, Tennessee37830, USA Received 7 July 1975 Data for Ni + 160 elastic scattering are analyzed to determine the real and imaginary potentials near the strong absorption radius as a function of bombarding energy. A strong energy dependence is found for the ratio of the imaginary part to the real part at this radius.
Recent work [1 ] on transfer reactions between "light" heavy ions (A ~< 20) and intermediate mass targets (A ~<60) has suggested that the corresponding optical potentials U have weak imaginary parts (lm U/Re U ~ 1) in the surface region, near the strong absorption radius (which is typically "" 1.5 (A~/3 +A 1/3) fro). On the other hand, analysis of scattering data [2] have led to potentials with comparable real and imaginary parts in this region. Since the elastic measurements were taken with higher bombarding energies than the transfer measurements, we have studied the optical potential as a function of energy in order to see whether the apparent discrepancy is due to an energy dependence of the absorptive potential. The results indicate a strong energy dependence of lm U/Re U in the surface. The system 58Ni + 160 was chosen as representative, w~th data being available [3] for energies from 42 to 81 MeV. In addition we used the 60Ni + 160 data [2] at 142 MeV. The results reported here were obtained by taking a 4-parameter Woods-Saxon potential with real depth V = 40 MeV. At each energy the radius (ro) and diffuseness (a) parameters were adjusted to give a minimum X2 for each of a sequence of imaginary depths I4/. In this way, we map out the X2 as a function of W/V as well as obtaining the minimum [2]. At several energies other values of V were talken also and it was verified, as expected, that the results did not depend upon this choice; the data essentially Research sponsored by the Energy Research and Development Administration under contract with the Union Carbide Corporation.
only determine the values of Re U and Im U close to the strong absorption radius. (This is defined as the distance of closest approach, D1/2, of the Rutherford orbit which has the same angular momentum L1/2 as that which gives transmission coefficient TL = 1/2. We find that r =D1/2 is also, to within 0.1 fm, the radius at which the radial kinetic energy, E c M - U CRe U - U 4. 1 / 2 , vanishes. Here UC is the Coulomb and UL the centrifugal potential). The results are summarized in fig. 1, where we see that I41/Vdecreases smoothly as the energy decreases. (The dots give the values for X2 , while the "error bars" represent a subjective estimate of the range which gives acceptable fits. For the latter, particular attention was paid to the angles where do/dw oscillates about the Rutherford cross section [2]; a difference here between experiment and optical model of more than about 5% was judged unacceptable). The corresponding optimum ro and a values also vary smoothly f r o m r o ~- 1.15 fm,a ~ 0.71 fm at 142 MeV to ro ~ 1.35 fm, a ~ 0.4 fm at 45 MeV. These variations presumably reflect inadequacies in the WoodsSaxon shape, in particular our insistence upon the same shape for the real and imaginary parts. The L1/2 values increase from about 10 at 42 MeV to about 68 at 142 MeV; the corresponding D1/2, and Re U at r = D1/2, are also shown in fig. 1. Rather similar results are obtained with a folded shape for the real potential [21. Although the decrease in Im U as the energy falls appears to be very dramatic, the variation in the mean free path A near r =DI/2 is not so great because the local kinetic energy T = E c M - U c - R e U, and hence the local velocity, is also decreasing; A(r) ~ 1.3 T(r)l/2l lm U(r)
408
Volume 58B, number 4
i 2
PHYSICS LETTERS
!
1
"
~_~I0
-
results are plotted in this way in fig. 2; again the dots are for the X2 minima and the lines indicate the ranges of values corresponding to the "error bars" shown in fig. 1. The points scatter too much and cover too limited a distance ( ~ 0.6 fm) to determine the slope precisely, but the trend is for a ~ 0.6 to 0.7 fm if we represent Re U by exp (-r/a) in this region. These data are also compatible with the form and slope suggested previously [5], based upon a folding model,
i
,,1" ! J ' - t
29 September 1975
~.~
Re U = -St 2 e x p ( ( R V - r ) 9 #.0
withR v = 1.14 (A~/3 +A 1/3) anda v = 0.59 fm, but the strength needed, S ~- 0.9 MeV, is less than a third that deduced by these authors [5]. Also shown in fig. 2 are plots of some double-folded potentials [2], derived using shell model densities for the two nuclei and effective nucleon-nucleon interactions; RTG is a Gaussian which fits low-energy nucleonnucleon scattering [6], 2Y is a sum of Yukawas which is equivalent to the long-range parts of the even-plusodd-state Hamada-Johnston potentials [7], while 1 FG
I
Ni +~60 0.8 0.6
/
/
/
.1
f
/
~'
i '~''
m
,/ 0.4
0
40
/
60
80
•
58
•
60
t20 I00 EL(MeV)
~40
lay),
~60
Fig. 1. Results from 4-parameter Woods-Saxon potential fits to Ni + 1~O scattering. The lines are drawn to guide the eye.
in fm if T and U are in MeV. In fact, A -~ 10 fm for energies ~> 70 MeV, corresponding to quite weak absorption in this region, and increases for lower energies to about 45 +- 20 fm at 45 MeV, (Of course, A is much smaller in the interior, "" 0.5 fm with these potentials, but the scattering itself does not determine this quantity.) Demanding A ~. 10 fm at r =D1/2 for all energies would require W[ V ~. 0.4 at the lowest energies, appreciably larger than the values found. However, in addition to this effect of the relative speed of the ions, we also expect that direct reactions, which contribute a large part of this peripheral absorption, will become less probable as the energy decreases [1,4,], and this will make W decrease even faster. If we really determine Re U at r = D1/2 by fitting the elastic data, then fits at different energies allow us to map out Re U as a function or r, yielding an effective local, energy-independent, ion-ion potential. Our
2
~
,
Ni +160
1
,xl~ ~
~..~
-
%~-,,~
&
~.~ 0.5 •
58
•
60
'~2y '~%% G
"IFG 0.2
0.1
9.5
10 • (fro}
t0.5
Fig. 2. Values of the real potential determined from the elastic data. The lines correspond to double-folded potentials described in the text.
409
Volume 58B, number 4
PHYSICS LETTERS
is simply a Gaussian of range 1 fm and strength 100.6 MeV [8]. Each has been adjusted in strength to give Re U(r = 9.75) = 1.32 MeV; this required renormalizing the 2Y strength by a factor 0.60 and the RTG strength by 0.53, close to the renormalization factors which have been found necessary before [2]. Both RTG and 1FG shapes seem to be satisfactory, while the 2Y (which has no OPEP component) seems to have too long a range. It has been suggested previously [1 ] that a surface transparent potential (STP) could be constructed by taking an imaginary potential with a deep Woods-Saxon term having a relatively sharp edge plus the derivative of a weak Woods-Saxon term outside it to represent, at least in part, the peripheral absorption due to direct reactions. We have attempted to find such potentials to fit the present data sets, but were unsuccessful except for the lowest two energies. At 1.42 MeV, an STP introduces rapid oscillations into the angular distribution, even at small angles where do/do R is close to unity, whereas the data show none. The intermediate energy data ( E ~ 60 to 81 MeV) can often be fitted down to where do/daR ~ 0.01, say, but then the predicted cross sections flatten off start to rise, while the experimental ones continue to fall. This is due to reflection of the low partial waves from the deep absorptive core. When the parameters of the STP are varied in order to improve the fits, one is driven back to potential with values of Im U[Re U at r .~ D1/2 similar to those described above (fig. 1). We believe that an STP will not adequately represent the elastic scattering at all angles even at the energis where it has been invoked in order to improve fits to transfer measurements [ 1]. Consequently, if the STP are to be used in DWBA calculations, the usual rigid requirements of reproducing elastic data have to be relaxed at least partially.
410
29 September 1975
In summary, we have found that the ratio of imaginary part to real part of the optical potential near the strong absorption radius is strongly energy dependent, although we point out that this does not necessarily require a large increase in the mean free path at this radius as the energy is reduced. In any case, even Im U = Re U a t 142 MeV corresponds to a mean free path of nearly 10 fm, and consequently weak absorption, in this region. We are indebted to the authors of ref. [3] for their elastic data.
References [1] A.J. Batz, et al., Phys. Rev. C, to be published; E.H. Auerbach, et al., Phys. Rev. Lett. 30 (1973) 1078; C. Chasman, et al., Phys. Rev. Lett. 31 (1973) 1074; J.B. Ball, et al., Phys. Lett. 49B (1974) 348. [2] G.R. Satchler, Phys. Lett. 55B (1975) 167; J.B. Ball, et al,, Nucl. Phys., to be published; M.L. Halbert, et al., to be published. [3] P.R. Christensen, et al., Nucl. Phys. A207 (1973) 433; M.S. Zisman, et al., Phys. Rev. Cll (1975) 809; L. West, K.W. Kemper and N.R. Fletcher, Phys. Rev. Cll (1975) 859; P.D. Bond, et al., private communication. [4] W. yon Oertzen, et al., in Reactions between complex nuclei, ed. R.L. Robinson, et al. (North-Holland, Amsterdam, 1974), p. 83; W. Henrting, et al., to be published. [5] R.A. Broglia and A. Winter, Phys. Rep. 4C (1972) 153. A strength S ~ 3 MeV is given in eq. (3.44) of this paper, but the numerical values of the constants that the authors quote leads to S ~ 5.25 MeV. [6] I. Reichstein and Y.C. Tang, Nucl. Phys. A139 (1969) 144. [7] W.G. Love, private communication. [8] C.B. Dover, P.J. Moffa and LP. Vary, to be published.
PHYSICS LETTERS
29 September 1975
ENERGY DEPENDENCE OF THE SURFACE ABSORPTION A N D Ni + 160 S C A T T E R I N G G.R. SATCHLER
Oak RidgeNational Laboratory, Oak Ridge, Tennessee37830, USA Received 7 July 1975 Data for Ni + 160 elastic scattering are analyzed to determine the real and imaginary potentials near the strong absorption radius as a function of bombarding energy. A strong energy dependence is found for the ratio of the imaginary part to the real part at this radius.
Recent work [1 ] on transfer reactions between "light" heavy ions (A ~< 20) and intermediate mass targets (A ~<60) has suggested that the corresponding optical potentials U have weak imaginary parts (lm U/Re U ~ 1) in the surface region, near the strong absorption radius (which is typically "" 1.5 (A~/3 +A 1/3) fro). On the other hand, analysis of scattering data [2] have led to potentials with comparable real and imaginary parts in this region. Since the elastic measurements were taken with higher bombarding energies than the transfer measurements, we have studied the optical potential as a function of energy in order to see whether the apparent discrepancy is due to an energy dependence of the absorptive potential. The results indicate a strong energy dependence of lm U/Re U in the surface. The system 58Ni + 160 was chosen as representative, w~th data being available [3] for energies from 42 to 81 MeV. In addition we used the 60Ni + 160 data [2] at 142 MeV. The results reported here were obtained by taking a 4-parameter Woods-Saxon potential with real depth V = 40 MeV. At each energy the radius (ro) and diffuseness (a) parameters were adjusted to give a minimum X2 for each of a sequence of imaginary depths I4/. In this way, we map out the X2 as a function of W/V as well as obtaining the minimum [2]. At several energies other values of V were talken also and it was verified, as expected, that the results did not depend upon this choice; the data essentially Research sponsored by the Energy Research and Development Administration under contract with the Union Carbide Corporation.
only determine the values of Re U and Im U close to the strong absorption radius. (This is defined as the distance of closest approach, D1/2, of the Rutherford orbit which has the same angular momentum L1/2 as that which gives transmission coefficient TL = 1/2. We find that r =D1/2 is also, to within 0.1 fm, the radius at which the radial kinetic energy, E c M - U CRe U - U 4. 1 / 2 , vanishes. Here UC is the Coulomb and UL the centrifugal potential). The results are summarized in fig. 1, where we see that I41/Vdecreases smoothly as the energy decreases. (The dots give the values for X2 , while the "error bars" represent a subjective estimate of the range which gives acceptable fits. For the latter, particular attention was paid to the angles where do/dw oscillates about the Rutherford cross section [2]; a difference here between experiment and optical model of more than about 5% was judged unacceptable). The corresponding optimum ro and a values also vary smoothly f r o m r o ~- 1.15 fm,a ~ 0.71 fm at 142 MeV to ro ~ 1.35 fm, a ~ 0.4 fm at 45 MeV. These variations presumably reflect inadequacies in the WoodsSaxon shape, in particular our insistence upon the same shape for the real and imaginary parts. The L1/2 values increase from about 10 at 42 MeV to about 68 at 142 MeV; the corresponding D1/2, and Re U at r = D1/2, are also shown in fig. 1. Rather similar results are obtained with a folded shape for the real potential [21. Although the decrease in Im U as the energy falls appears to be very dramatic, the variation in the mean free path A near r =DI/2 is not so great because the local kinetic energy T = E c M - U c - R e U, and hence the local velocity, is also decreasing; A(r) ~ 1.3 T(r)l/2l lm U(r)
408
Volume 58B, number 4
i 2
PHYSICS LETTERS
!
1
"
~_~I0
-
results are plotted in this way in fig. 2; again the dots are for the X2 minima and the lines indicate the ranges of values corresponding to the "error bars" shown in fig. 1. The points scatter too much and cover too limited a distance ( ~ 0.6 fm) to determine the slope precisely, but the trend is for a ~ 0.6 to 0.7 fm if we represent Re U by exp (-r/a) in this region. These data are also compatible with the form and slope suggested previously [5], based upon a folding model,
i
,,1" ! J ' - t
29 September 1975
~.~
Re U = -St 2 e x p ( ( R V - r ) 9 #.0
withR v = 1.14 (A~/3 +A 1/3) anda v = 0.59 fm, but the strength needed, S ~- 0.9 MeV, is less than a third that deduced by these authors [5]. Also shown in fig. 2 are plots of some double-folded potentials [2], derived using shell model densities for the two nuclei and effective nucleon-nucleon interactions; RTG is a Gaussian which fits low-energy nucleonnucleon scattering [6], 2Y is a sum of Yukawas which is equivalent to the long-range parts of the even-plusodd-state Hamada-Johnston potentials [7], while 1 FG
I
Ni +~60 0.8 0.6
/
/
/
.1
f
/
~'
i '~''
m
,/ 0.4
0
40
/
60
80
•
58
•
60
t20 I00 EL(MeV)
~40
lay),
~60
Fig. 1. Results from 4-parameter Woods-Saxon potential fits to Ni + 1~O scattering. The lines are drawn to guide the eye.
in fm if T and U are in MeV. In fact, A -~ 10 fm for energies ~> 70 MeV, corresponding to quite weak absorption in this region, and increases for lower energies to about 45 +- 20 fm at 45 MeV, (Of course, A is much smaller in the interior, "" 0.5 fm with these potentials, but the scattering itself does not determine this quantity.) Demanding A ~. 10 fm at r =D1/2 for all energies would require W[ V ~. 0.4 at the lowest energies, appreciably larger than the values found. However, in addition to this effect of the relative speed of the ions, we also expect that direct reactions, which contribute a large part of this peripheral absorption, will become less probable as the energy decreases [1,4,], and this will make W decrease even faster. If we really determine Re U at r = D1/2 by fitting the elastic data, then fits at different energies allow us to map out Re U as a function or r, yielding an effective local, energy-independent, ion-ion potential. Our
2
~
,
Ni +160
1
,xl~ ~
~..~
-
%~-,,~
&
~.~ 0.5 •
58
•
60
'~2y '~%% G
"IFG 0.2
0.1
9.5
10 • (fro}
t0.5
Fig. 2. Values of the real potential determined from the elastic data. The lines correspond to double-folded potentials described in the text.
409
Volume 58B, number 4
PHYSICS LETTERS
is simply a Gaussian of range 1 fm and strength 100.6 MeV [8]. Each has been adjusted in strength to give Re U(r = 9.75) = 1.32 MeV; this required renormalizing the 2Y strength by a factor 0.60 and the RTG strength by 0.53, close to the renormalization factors which have been found necessary before [2]. Both RTG and 1FG shapes seem to be satisfactory, while the 2Y (which has no OPEP component) seems to have too long a range. It has been suggested previously [1 ] that a surface transparent potential (STP) could be constructed by taking an imaginary potential with a deep Woods-Saxon term having a relatively sharp edge plus the derivative of a weak Woods-Saxon term outside it to represent, at least in part, the peripheral absorption due to direct reactions. We have attempted to find such potentials to fit the present data sets, but were unsuccessful except for the lowest two energies. At 1.42 MeV, an STP introduces rapid oscillations into the angular distribution, even at small angles where do/do R is close to unity, whereas the data show none. The intermediate energy data ( E ~ 60 to 81 MeV) can often be fitted down to where do/daR ~ 0.01, say, but then the predicted cross sections flatten off start to rise, while the experimental ones continue to fall. This is due to reflection of the low partial waves from the deep absorptive core. When the parameters of the STP are varied in order to improve the fits, one is driven back to potential with values of Im U[Re U at r .~ D1/2 similar to those described above (fig. 1). We believe that an STP will not adequately represent the elastic scattering at all angles even at the energis where it has been invoked in order to improve fits to transfer measurements [ 1]. Consequently, if the STP are to be used in DWBA calculations, the usual rigid requirements of reproducing elastic data have to be relaxed at least partially.
410
29 September 1975
In summary, we have found that the ratio of imaginary part to real part of the optical potential near the strong absorption radius is strongly energy dependent, although we point out that this does not necessarily require a large increase in the mean free path at this radius as the energy is reduced. In any case, even Im U = Re U a t 142 MeV corresponds to a mean free path of nearly 10 fm, and consequently weak absorption, in this region. We are indebted to the authors of ref. [3] for their elastic data.
References [1] A.J. Batz, et al., Phys. Rev. C, to be published; E.H. Auerbach, et al., Phys. Rev. Lett. 30 (1973) 1078; C. Chasman, et al., Phys. Rev. Lett. 31 (1973) 1074; J.B. Ball, et al., Phys. Lett. 49B (1974) 348. [2] G.R. Satchler, Phys. Lett. 55B (1975) 167; J.B. Ball, et al,, Nucl. Phys., to be published; M.L. Halbert, et al., to be published. [3] P.R. Christensen, et al., Nucl. Phys. A207 (1973) 433; M.S. Zisman, et al., Phys. Rev. Cll (1975) 809; L. West, K.W. Kemper and N.R. Fletcher, Phys. Rev. Cll (1975) 859; P.D. Bond, et al., private communication. [4] W. yon Oertzen, et al., in Reactions between complex nuclei, ed. R.L. Robinson, et al. (North-Holland, Amsterdam, 1974), p. 83; W. Henrting, et al., to be published. [5] R.A. Broglia and A. Winter, Phys. Rep. 4C (1972) 153. A strength S ~ 3 MeV is given in eq. (3.44) of this paper, but the numerical values of the constants that the authors quote leads to S ~ 5.25 MeV. [6] I. Reichstein and Y.C. Tang, Nucl. Phys. A139 (1969) 144. [7] W.G. Love, private communication. [8] C.B. Dover, P.J. Moffa and LP. Vary, to be published.