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Pockets in heavy-ion potentials and nucleon transfer
Volume 158B, number 3
PHYSICS LETTERS
15 August 1985
P O C K E T S IN HEAVY-ION P O T E N T I A L S AND N U C L E O N T R A N S F E R ~ D.P. RUSSELL, W.T. P I N K S T O N and V.E. O B E R A C K E R Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 3723.5, USA Received 18 April 1985
A schematic model is described for taking into account the effect of potential pockets (motivated by positron experiments at GSI) on sub-Coulomb nucleon transfer cross sections. Results for single-neutron transfer in 23Su-23SU collisions bear a remarkable resemblance to data for some bombarding energies.
Discrete lines in the spectra of positrons [1,2] have been interpreted as resulting from "spontaneous decay of the vacuum" [3,4]. For this interpretation to work [3], it seems that the colliding nuclei must stick together for times of order 1 0 - 1 9 - 1 0 - 2 0 s - long compared to the time scale of collisions assuming Rutherford trajectories. A possible mechanism for producing such time delays is a minimum or "pocket" in the heavy-ion potential corresponding to surface contact [5]. The notion of a pocket is supported to some extent by theoretical models [6,7] ; however its primary motivation has been the need for a mechanism for producing the positron lines. This idea will be completely convincing only if evidence can be found for the pockets in nuclear physics data. The purpose of this letter is to report on some calculations of the effect of using heavy-ion potentials with pockets on nucleon transfer reactions. The energy range of interest is in the vicinity of and below the Coulomb barrier. Nucleon transfer at such energies is a very weak process with a characteristic angular distribution. It seems inevitable that any modification of the scattering potential which causes the nuclei to spend more time close together would strongly influence transfer. (There is also an effect on quasielastic scattering, but it would be more difficult to observe, as will be seen.) Radioehemical data exist [8,9] for nucleon transfer in 238U-238U collisions. A Supported in part by NSF grant PHY-84-4525 and DOE contract grant DE-A505-84ER40137.
0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
differential cross section for single-neutron transfer is shown in fig. 1. Also shown is the characteristic angular distribution expected in sub-Coulomb transfer. The data are fiat at back angles, then show a peak before falling off. It is well known [10] from collisions involving much lighter ions that, as the energy approaches the Coulomb barrier, a dip or an oscillatory behavior at back angles develops, resulting from the •
.
i
i
|
|
!
i
i
16 14
-~
lO
b "o 6 4
2 80
IO0
120
140
160
180
e (d.o,..,) Fig. 1. Single-neutron transfer cross sections at CM energy of 673 MeV. The bar graph gives the experimental results [8]. The smooth curve shows the characteristic shape expected in sub-Couiomb transfer (arbitrarily normalized to the experimental total cross section).
201
Volume 158B, number 3
PHYSICS LETTERS
15 August 1985
onset of absorption. However, such a mechanism would not provide an adequate explanation of the radiochemical data; the flatness in the cross section at back angles persists [9] down to energies as low as 82% of the Coulomb barrier (600 MeV). Transfer between such heavy ions, both deformed, is extremely difficult to treat theoretically, and no attempt has been made to construct an accurate quantitative theory. Instead, a schematic, two-state model has been studied to help understand the kind of behavior to expect if the interaction potentials of very heavy ions exhibit pockets. In this model, a neutron bound in a 4s state on one (spherical) Woods-Saxon core (radins = (1.25 fm)A 1/3, diffuseness = 0.85 fm, BE = 6 MeV) is transferred to an identical state on the other. The reaction Q is zero, which along with the l = 0 nature of the transfer greatly simplifies things. Recoil and non-orthogonality corrections are neglected. For each angular momentum L of the relative motion, a pair of coupled radial equations for elastic and transfer channels (i = 1,2) must be solved. [rL + v(r) + e - elR
i , / : 1, 2,
sLl(elastic ) = [exp(2iS+) + exp(2i8 _)]/2,
(2a)
SL12(transfer) = [exp(2iS+) -- exp(2iS_)]/2.
(2b)
The phase shifts were calculated in WKB approximation. Both scattering amplitudes can be written in the form
202
+ 1)sL/pL(cos 0)/2ik.
LOMB
700
'~
660.
.2°t R(fm)
o
(1)
The radial kinetic energy operator, including the centrifugal potential, is T L , and e is the single-particle binding energy. The transfer form factor G is computed in the usual way as the overlap of a binding potential between 4s wave functions centered on different cores. Systems of equations such as this have been studied in connection with fusion reactions [ 11]. The~, are easily decoupled by the transformation, I¢~ = R~ -+R 2 ; these new radial functions correspond to motion in potentials, lz_+ G, respectively. The S-matrix elements describing elastic scattering and transfer are easily obtained from the phase shifts resulting from motion in the potentials, V -+ G,
fi(O) = ~ ( 2 L L
~
a
I
:
i :/:/.
740
(3)
"0
o 80
100
120
140
160
180
0 (d.0.,..) Fig. 2. (a) Potential with a shoulder (used in calculations) compared to the Coulomb potential and to a potential with a pocket. The bombarding energy, E = 673 MeV, is indicated. (b) Elasticcross section (unsymmetrized) for 23Su-23Su using the shoulder potential.The circlesshow the effect of crier. gy and angle averaging.
Calculations were made using the potential with a "shoulder" (a more conservative approach than using a pocket) in fig. 2a. A potential with a pocket is also shown; however it turns out that nothing as drastic as a pocket is needed to cause significant changes from the standard sub-Coulomb behavior. The shoulder suffices. The Coulomb potential for 238U-238U is also shown for comparison. Results for elastic scattering are shown in fig. 2b. The deviations of the cross see-
Volume 158B, number 3
PHYSICS LETTERS
tion from Rutherford are characteristic of a quantum mechanical rainbow. The oscillations in angle are too rapid to be observable; also the effective beam energy in the experiments has a spread of order 10 MeV because of target straggling. The circles in fig. 2b show the effect of averaging over energy and angle. Because it occurs at back angles, which are difficult to observe, and because the cross section must be syrrmaetrized
a 0.24 .O
o
a.
0.16 E
0 w.-
0.08
COULOMB
0
40
80
"~
120
160
2C)0
L, Ang. Morn. Quant. No.
70
50 .,D
E
•~ b
30
15 August 1985
around 0CM = 90 °, a small effect like that shown is essentially unobservable. More interesting are the transfer results shown in fig. 3. The probability of transfer is shown in fig. 3a, along with the probability computed using the Coulomb potential. There is a dramatic increase in the transfer probability for an Lvalue corresponding to the classical impact parameter for orbiting. The cross section is shown in fig. 3b; again the circles show the effect of energy and angle averaging. A flattening, with a peak at smaller angles, similar to that exhibited by the data in fig. 1, is predicted. The aim of this research has been to develop a schematic model - as a fftrst step toward a more realistic theory - which permits a treatment of the influence of potential pockets on transfer reactions. We believe the model is qualitatively correct, but its quantitative predictions are not yet reliable. Nuclear deformations, which have been neglected, may have several important consequences. For instance, Coulomb excitation, a strong process in 238U, is left out of the present treatment. In addition, existing theories [6,7] suggest that potential modifications should be quite dependent on orientation, so that any effects on cross sections should be multiplied by a factor less than one, giving the probability that the nuclei collide in relative orientations favorable for "sticking". The predictions of the schematic model are very similar to the published data [8] at 673 MeV; however it fails to reproduce the lower energy data [9]. It predicts, as one would expect and contrary to what is observed, t h a t the cross sections have the canonical sub-Coulomb angular distributions when the energy is lower than the "shoulder". Nevertheless, the results are encouraging and support transfer reaction studies as a suitable testing ground for recent speculations on the existence of pockets in heavy-ion potentials.
References 10
80
J 100
120
140
160
180
0 Fig. 3. (a) Transfer probability as a function of L for motion in the shoulder and Coulomb potentials. (b) Transfer cross section using the shoulder potential. Circles show the effect of energy and angle averaging.
[1] J. Schweppe, A. Gruppe, K. Bethge, H. Bokemeyer, T. Cowan, H. Folger, J.S. Greenberg, H. Grein, S. Ito, R. Schule', D. Schwalm, K.E. Stiebing, N. Trautmarm, P. Vincent and M. Waldsehmidt, Phys. Rev. Lett. 51
(1983) 2261. [2] M. Clemente, E. Berdermann, P. Kienle, H. Tsertos, W. Wagner, C. Kozhuharov, F. Bosch and W. Koenig, Phys. Lett. 137B (1984) 41. [3] J. Reinhardt, B. Muller and W. Greiner, Z. Phys. A303 (1981) 173; 203
Volume 158B, number 3
[4] [5] [6] [7]
204
PHYSICS LETTERS
U. Muller, G. Soft, T. de Reus, J. Reinhardt, B. Muller and W. Greiner, Z. Phys. A313 (1983) 263. C. Bottcher and M.R. Strayer, Phys. Rev. Lett., to be published. U. Heinz, U. Muller, J. Reinhardt, B. Muller, W. Greiner and W.T. Pinkston, Z. Phys. A316 (1984) 341. M.J. Rhoades-Brown, V.E. Oberacker, M. Seiwert and W. Greiner, Z. Phys. A310 (1983) 2878. M. Seiwert, W. Greiner and W.T. Pinkston, J. Phys. G l l (1985) L21.
15 August 1985
[8] G. Wirth, W. Bruchle, Wo Fan, H. Gaeggeler, K. Schlitt, K. Summerer, J.V. K?atz, M. Lerch and N. Trautmann, GSI Scientific Report (1982) p. 13, unpublished. [9] G. Wirth and J.V. Kxatz, private communication. [10] L.J.B. Goldfarb, in: Proc. Intern. Conf. on Reactions between complex nuclei (Nashville, TN), eds. R.L. Robinson, F.K. McGowan and J.B. Ball (North-Holland, Amsterdam, 1974). [11] C.H. Dasso, S. Landowne and A. Winther, Nucl. Phys. A405 (1983) 381; R. Lindsay and N. Rowley, J. Phys. G10 (1984) 805.
PHYSICS LETTERS
15 August 1985
P O C K E T S IN HEAVY-ION P O T E N T I A L S AND N U C L E O N T R A N S F E R ~ D.P. RUSSELL, W.T. P I N K S T O N and V.E. O B E R A C K E R Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 3723.5, USA Received 18 April 1985
A schematic model is described for taking into account the effect of potential pockets (motivated by positron experiments at GSI) on sub-Coulomb nucleon transfer cross sections. Results for single-neutron transfer in 23Su-23SU collisions bear a remarkable resemblance to data for some bombarding energies.
Discrete lines in the spectra of positrons [1,2] have been interpreted as resulting from "spontaneous decay of the vacuum" [3,4]. For this interpretation to work [3], it seems that the colliding nuclei must stick together for times of order 1 0 - 1 9 - 1 0 - 2 0 s - long compared to the time scale of collisions assuming Rutherford trajectories. A possible mechanism for producing such time delays is a minimum or "pocket" in the heavy-ion potential corresponding to surface contact [5]. The notion of a pocket is supported to some extent by theoretical models [6,7] ; however its primary motivation has been the need for a mechanism for producing the positron lines. This idea will be completely convincing only if evidence can be found for the pockets in nuclear physics data. The purpose of this letter is to report on some calculations of the effect of using heavy-ion potentials with pockets on nucleon transfer reactions. The energy range of interest is in the vicinity of and below the Coulomb barrier. Nucleon transfer at such energies is a very weak process with a characteristic angular distribution. It seems inevitable that any modification of the scattering potential which causes the nuclei to spend more time close together would strongly influence transfer. (There is also an effect on quasielastic scattering, but it would be more difficult to observe, as will be seen.) Radioehemical data exist [8,9] for nucleon transfer in 238U-238U collisions. A Supported in part by NSF grant PHY-84-4525 and DOE contract grant DE-A505-84ER40137.
0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
differential cross section for single-neutron transfer is shown in fig. 1. Also shown is the characteristic angular distribution expected in sub-Coulomb transfer. The data are fiat at back angles, then show a peak before falling off. It is well known [10] from collisions involving much lighter ions that, as the energy approaches the Coulomb barrier, a dip or an oscillatory behavior at back angles develops, resulting from the •
.
i
i
|
|
!
i
i
16 14
-~
lO
b "o 6 4
2 80
IO0
120
140
160
180
e (d.o,..,) Fig. 1. Single-neutron transfer cross sections at CM energy of 673 MeV. The bar graph gives the experimental results [8]. The smooth curve shows the characteristic shape expected in sub-Couiomb transfer (arbitrarily normalized to the experimental total cross section).
201
Volume 158B, number 3
PHYSICS LETTERS
15 August 1985
onset of absorption. However, such a mechanism would not provide an adequate explanation of the radiochemical data; the flatness in the cross section at back angles persists [9] down to energies as low as 82% of the Coulomb barrier (600 MeV). Transfer between such heavy ions, both deformed, is extremely difficult to treat theoretically, and no attempt has been made to construct an accurate quantitative theory. Instead, a schematic, two-state model has been studied to help understand the kind of behavior to expect if the interaction potentials of very heavy ions exhibit pockets. In this model, a neutron bound in a 4s state on one (spherical) Woods-Saxon core (radins = (1.25 fm)A 1/3, diffuseness = 0.85 fm, BE = 6 MeV) is transferred to an identical state on the other. The reaction Q is zero, which along with the l = 0 nature of the transfer greatly simplifies things. Recoil and non-orthogonality corrections are neglected. For each angular momentum L of the relative motion, a pair of coupled radial equations for elastic and transfer channels (i = 1,2) must be solved. [rL + v(r) + e - elR
i , / : 1, 2,
sLl(elastic ) = [exp(2iS+) + exp(2i8 _)]/2,
(2a)
SL12(transfer) = [exp(2iS+) -- exp(2iS_)]/2.
(2b)
The phase shifts were calculated in WKB approximation. Both scattering amplitudes can be written in the form
202
+ 1)sL/pL(cos 0)/2ik.
LOMB
700
'~
660.
.2°t R(fm)
o
(1)
The radial kinetic energy operator, including the centrifugal potential, is T L , and e is the single-particle binding energy. The transfer form factor G is computed in the usual way as the overlap of a binding potential between 4s wave functions centered on different cores. Systems of equations such as this have been studied in connection with fusion reactions [ 11]. The~, are easily decoupled by the transformation, I¢~ = R~ -+R 2 ; these new radial functions correspond to motion in potentials, lz_+ G, respectively. The S-matrix elements describing elastic scattering and transfer are easily obtained from the phase shifts resulting from motion in the potentials, V -+ G,
fi(O) = ~ ( 2 L L
~
a
I
:
i :/:/.
740
(3)
"0
o 80
100
120
140
160
180
0 (d.0.,..) Fig. 2. (a) Potential with a shoulder (used in calculations) compared to the Coulomb potential and to a potential with a pocket. The bombarding energy, E = 673 MeV, is indicated. (b) Elasticcross section (unsymmetrized) for 23Su-23Su using the shoulder potential.The circlesshow the effect of crier. gy and angle averaging.
Calculations were made using the potential with a "shoulder" (a more conservative approach than using a pocket) in fig. 2a. A potential with a pocket is also shown; however it turns out that nothing as drastic as a pocket is needed to cause significant changes from the standard sub-Coulomb behavior. The shoulder suffices. The Coulomb potential for 238U-238U is also shown for comparison. Results for elastic scattering are shown in fig. 2b. The deviations of the cross see-
Volume 158B, number 3
PHYSICS LETTERS
tion from Rutherford are characteristic of a quantum mechanical rainbow. The oscillations in angle are too rapid to be observable; also the effective beam energy in the experiments has a spread of order 10 MeV because of target straggling. The circles in fig. 2b show the effect of averaging over energy and angle. Because it occurs at back angles, which are difficult to observe, and because the cross section must be syrrmaetrized
a 0.24 .O
o
a.
0.16 E
0 w.-
0.08
COULOMB
0
40
80
"~
120
160
2C)0
L, Ang. Morn. Quant. No.
70
50 .,D
E
•~ b
30
15 August 1985
around 0CM = 90 °, a small effect like that shown is essentially unobservable. More interesting are the transfer results shown in fig. 3. The probability of transfer is shown in fig. 3a, along with the probability computed using the Coulomb potential. There is a dramatic increase in the transfer probability for an Lvalue corresponding to the classical impact parameter for orbiting. The cross section is shown in fig. 3b; again the circles show the effect of energy and angle averaging. A flattening, with a peak at smaller angles, similar to that exhibited by the data in fig. 1, is predicted. The aim of this research has been to develop a schematic model - as a fftrst step toward a more realistic theory - which permits a treatment of the influence of potential pockets on transfer reactions. We believe the model is qualitatively correct, but its quantitative predictions are not yet reliable. Nuclear deformations, which have been neglected, may have several important consequences. For instance, Coulomb excitation, a strong process in 238U, is left out of the present treatment. In addition, existing theories [6,7] suggest that potential modifications should be quite dependent on orientation, so that any effects on cross sections should be multiplied by a factor less than one, giving the probability that the nuclei collide in relative orientations favorable for "sticking". The predictions of the schematic model are very similar to the published data [8] at 673 MeV; however it fails to reproduce the lower energy data [9]. It predicts, as one would expect and contrary to what is observed, t h a t the cross sections have the canonical sub-Coulomb angular distributions when the energy is lower than the "shoulder". Nevertheless, the results are encouraging and support transfer reaction studies as a suitable testing ground for recent speculations on the existence of pockets in heavy-ion potentials.
References 10
80
J 100
120
140
160
180
0 Fig. 3. (a) Transfer probability as a function of L for motion in the shoulder and Coulomb potentials. (b) Transfer cross section using the shoulder potential. Circles show the effect of energy and angle averaging.
[1] J. Schweppe, A. Gruppe, K. Bethge, H. Bokemeyer, T. Cowan, H. Folger, J.S. Greenberg, H. Grein, S. Ito, R. Schule', D. Schwalm, K.E. Stiebing, N. Trautmarm, P. Vincent and M. Waldsehmidt, Phys. Rev. Lett. 51
(1983) 2261. [2] M. Clemente, E. Berdermann, P. Kienle, H. Tsertos, W. Wagner, C. Kozhuharov, F. Bosch and W. Koenig, Phys. Lett. 137B (1984) 41. [3] J. Reinhardt, B. Muller and W. Greiner, Z. Phys. A303 (1981) 173; 203
Volume 158B, number 3
[4] [5] [6] [7]
204
PHYSICS LETTERS
U. Muller, G. Soft, T. de Reus, J. Reinhardt, B. Muller and W. Greiner, Z. Phys. A313 (1983) 263. C. Bottcher and M.R. Strayer, Phys. Rev. Lett., to be published. U. Heinz, U. Muller, J. Reinhardt, B. Muller, W. Greiner and W.T. Pinkston, Z. Phys. A316 (1984) 341. M.J. Rhoades-Brown, V.E. Oberacker, M. Seiwert and W. Greiner, Z. Phys. A310 (1983) 2878. M. Seiwert, W. Greiner and W.T. Pinkston, J. Phys. G l l (1985) L21.
15 August 1985
[8] G. Wirth, W. Bruchle, Wo Fan, H. Gaeggeler, K. Schlitt, K. Summerer, J.V. K?atz, M. Lerch and N. Trautmann, GSI Scientific Report (1982) p. 13, unpublished. [9] G. Wirth and J.V. Kxatz, private communication. [10] L.J.B. Goldfarb, in: Proc. Intern. Conf. on Reactions between complex nuclei (Nashville, TN), eds. R.L. Robinson, F.K. McGowan and J.B. Ball (North-Holland, Amsterdam, 1974). [11] C.H. Dasso, S. Landowne and A. Winther, Nucl. Phys. A405 (1983) 381; R. Lindsay and N. Rowley, J. Phys. G10 (1984) 805.