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Energy dependence of the (p, np̃) reaction
Volume 40B, number 4
PHYSICS LETTERS
ENERGY
DEPENDENCE
24 July 1972
O F T H E (p, n~) R E A C T I O N *
G.W. HOFFMANN**, W.H. DUNLOP, G.J. IGO, J.G. KULLECK, C.A. WHITTEN Jr. Department of Physics, University of California, Los A ngeles, California 90024, USA and W.R. COKER Center Jor Nuclear Studies, University of Texas, Austin, Texas 78712, USA Received 9 June 1972 Excitation functions for the quasi-elastic (p, n~) reaction are presented for 91Zr, 119Sn' 2ospb and 2°9Bi from threshold to 45 MeV. Attempts to fit the data with conventional macroscopic DWBA calculations, using tim Becchetti-Greenlees (BG) optical model parameters, show that for all four nuclei considered an extremely unrealistic but consistent energy dependence is required for the strength of the complex symmetry potential. It is well established that in the local optical potential for nucleons, V0 + t" T V1/A, V0 is (at least linearly) energy dependent. The energy dependence of the nuclear symmetry potential, t" T V1/A, is far less clear, whether one looks at global optical model analyses of nucleon elastic scattering, or at charge-exchange reaction data directly sensitive to the symmetry potential. Since in the distorted wave Born approximation (DWBA), the (p, n) quasi-elastic cross section is directly proportional to t V 1 [2, such (p, n) data could provide a sensitive test of the energy dependence of V l when a macroscopic DWBA calculation is done. Some analyses of quasi-elastic (p, n) data have suggested a linear energy dependence of the symmetry potential [1, 21. For many nuclei with A ~> 60, the isobaric analog of the target ground state (IAS) populated in the quasi-elastic (p, n) reaction is proton unbound. The subsequent proton decay (~3) of this IAS is presumably isotropic. Hence the measured (p, n~) cross section at a given angle (or angles) can be used directly to infer the total (p, n) cross section to the IAS in question, i.e. oy(p, n)iAS = 4rr [da@)/d~2] avRp- 1,
(1)
where [do(g)/d~2] av is the (p, n]~) cross section at one angle (or the average for several angles) and Rp is the probability of proton decay of the IAS. One can take * Supported in part by the U.S. Atomic Energy Commission. ** Present address: Center for Nuclear Studies, University of Texas, Austin, Texas 78712, USA.
advantage of the evident fact that oT(p, n)lAS and do(~)/dg2 have the same shape as a function of energy in order to study the energy dependence of V 1 by calculating aT(P, n)IAS and comparing to do(~)/dg2. Excitation functions for the (p, nO) reaction on 91Zr, ll9Sn, 208pb and 209Bi between 20 and 45 MeV were obtained at the UCLA Cyclotron Laboratory, and are shown in the figure. Also shown for the nuclei 91Zr and ll9Sn are the data of Miller and Garvey [3] from threshold to 18 MeV. Each excitation function shows a sharp rise at threshold, a broad maximum centered about 5 MeV above threshold, and a rather rapid, nearly linear decrease in cross section with increasing bombarding energy. Calculations were made using DWBA in an attempt to fit the shapes of these excitation functions. The Becchetti-Greenlees (BG) optical model parameters [4] were used for the calculations. We used the complex V 1 t" T / A interaction of 96 (Nv/(~-Z)/2A MeV, real volume, and 48 x / ( N - Z ) / 2 A MeV, surface imaginary, as implied by the results of the BG analysis. These potentials are known to give a good overall description of (p, n)IAS angular distributions over a wide range of nuclei [1, 2, 5]. The results of these calculations using the DWBA program DWUCK [6] are shown for each o f the four nuclei as curves labeled 'a' in the figure. Plotted are the predictions for oT(P, n)IAS which is related to o(p, n~) through eq. (1). Since Rp ~< 1, the DWBA predictions represent a theoretical upper limit for the (p, n~) cross sections. These calculations do not repro453
Volume 40B, number 4
PttYSICS LETTERS
(o)
2.0 1.0
~
A
'Zr
E i-
c
4
b
1 2 , ' I~' 2'o' 2'4' 2'a' ~z' ~6'Zo' I.o "- •
(b)
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"
..,
,
5
.c
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2 I
~'6 ' io'
I.O~-
j4
' ~e ' gz'
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20 2.0 1.0 16 12
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40
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(d)
f
4
duce the shape or magnitude o f the data. The rise at threshold is much too slow, and from about 10 MeV above threshold to 45 MeV the predictions continue to rise very slowly while the data is rapidly falling. It has been suggested [ 1] that potential strengths given by (120 - E ) MeV and (60 - 0.75 E) MeV (with E in MeV) for the real volume and surface imaginary symmetry potentia',s, respectively, will reproduce the shape of the 208pb excitation function data from 30 to 45 MeV. The results o f calculations using this energy dependent symmetry potential are shown as the curves labeled 'b' in the figure. Although these predictions do not show a rapid rise at threshold, they do turn over and start to decrease at about 10 MeV above threshold for each nucleus. F r o m this point to 45 MeV these predictions do in fact have the same shape as the data. It should be mentioned that the predictions for the neutron angular distributions using the energy dependent symmetry potential fit the experimental shapes of the angular distribution data on 208pb and 209Bi, as well as, if not slightly better than, the nonenergy-dependent potential [1, 2]. Since the (p, n)IAS total cross section c o m p u t e d with DWBA is proportional to IV 1 [2, a plot of X/Oexp/ODWBA will give a potential scaling factor w h i c h when included in the DWBA calculations, will reproduce the experimentally observed excitation functions. Since the energy dependent potentials discussed do give the right shape for all four excitation functions measured from 30 to 45 MeV, a calculation of X/~exp/ODWBA was performed with the DWBA predictions using these potentials. The results are shown as the points on curves 'c' in the figure. Fits to these plots (curves 'c' in the figure) could be obtained for all four nuclei considered, resulting in complete symmetry potential strengths, as a function of energy given by
-c ZOgBi
~
d
I -
2'o
2~,
28
3'2
36
Ep ( M e V )
454
4~8
8
v
b
24
24 July 1972
4'o
i
44
-
b ,
i
48
. . . . t'lg. 1.. The. (p, np) excitation functions for 9 1 Zr, 1 1 9 S n 208Pb and 209B].. Data shown as ' X' are from ref. [3]. ~ae curves labeled 'a' are the macroscopic DWBA predictions for aT(p, n)iA S using the complex symmetry potential extracted from the Becchetti-Greenlees optical model parameters. The curves labeled 'b' are the macroscopic DWBA predictions for oT(p, n)lA S using the linearly energy dependent symmetry potentials. The curves labeled 'c' represent the symmetry potential scaling factors of eq. (2) which scale the linearly energy dependent potentials to give the macroscopic DWBA predictions for oT(p,n~) shown as curves 'd'.
Volume 40B, number 4
PHYSICS LETTERS
A {1 + (1.0+0.2) e x p [ - ( 0 . 2 5 + 0 . 0 5 ) l / f - E 0 [ ] } ( 1 2 0 - E ) MeV real volume (E in MeV), A {l+(1.0-+0.2)exp[ (0.25-+0.05)IE-E01] } ( 6 0 - 0 . 7 5 E ) MeV imaginary surface (E in MeV).
(2)
The quantity b~O is given by/~) ~ Eth + 1.5 MeV, where tffl~ is the (p, n) charge-exchange threshold energy. Predictions using DWBA and these symmetry potential strengths are shown as curves 'd' in the figure. All four excitation functions are fit quite well. We submit that these unrealistic but consistent energy dependent strengths are required for the nuclear symmetry potential if the (p, nl~) excitation function data is to be fit within the framework of the DWBA using the BG optical model parameters. It is to be emphasized that in our macroscopic DWBA calculations the BG global optical model parameters are used for generating the distorted waves in the incoming and outgoing channels, as has been the customary procedure in all recent surveys in the literature of quasi-elastic (p, n) scattering on a wide range of nuclei [5, 7]. Although the BG proton optical parameters may be very reliable, the neutron parameters were derived from data substantially inferior to the proton data. Further, at neutron energies below 10 MeV important compound elastic corrections of considerable ambiguity had to be included in the BG
24 July 1972
analysis [4]. The observation that the curves 'c' in the figure all have the same shape as a function of neutron energy (E n ~ Ep + Q), and are particularly ill-behaved from E n = 0 to 10MeV, suggests that the poor fit may be due to the neutron optical potential in this energy region. An important result of the present work is that (p, n) studies to determine the strength of V 1 by fitting quasi-elastic angular distributions at a given energy with the usual global optical potentials and DWBA can obtain symmetry potentials differing greatly from those obtained via a similar analysis at a different energy. Thus recent surveys [5, 7] of quasi-elastic scattering on a wide range of nuclei may yield doubtful values for V1 unless care is taken in the theoretical analysis of the data.
References [ 1 ] T.J. Woods, G.J. lgo and C.A. Whitten, Phys. Lett. 39B
(1972) 193. [2] G.W. Hoffmann et al., in press. [3] P.S. Miller, thesis, Princeton University (1968); G.T. Garvey and P.S. Miller, Phys. Lett. 28B (1968) 244. [4] F.D. Becchetti and G.W. Greenlees, Phys. Rev. 182 (1969) 1190.
[5] C. Wong et al., Phys. Rev. C5 (1972) 158. [6] P.D. Kunz, University of Colorado, Boulder, Colorado (private communication). [7] R.F. Bently et al., Phys. Rev. Lett. 27 (1971) 1081.
455
PHYSICS LETTERS
ENERGY
DEPENDENCE
24 July 1972
O F T H E (p, n~) R E A C T I O N *
G.W. HOFFMANN**, W.H. DUNLOP, G.J. IGO, J.G. KULLECK, C.A. WHITTEN Jr. Department of Physics, University of California, Los A ngeles, California 90024, USA and W.R. COKER Center Jor Nuclear Studies, University of Texas, Austin, Texas 78712, USA Received 9 June 1972 Excitation functions for the quasi-elastic (p, n~) reaction are presented for 91Zr, 119Sn' 2ospb and 2°9Bi from threshold to 45 MeV. Attempts to fit the data with conventional macroscopic DWBA calculations, using tim Becchetti-Greenlees (BG) optical model parameters, show that for all four nuclei considered an extremely unrealistic but consistent energy dependence is required for the strength of the complex symmetry potential. It is well established that in the local optical potential for nucleons, V0 + t" T V1/A, V0 is (at least linearly) energy dependent. The energy dependence of the nuclear symmetry potential, t" T V1/A, is far less clear, whether one looks at global optical model analyses of nucleon elastic scattering, or at charge-exchange reaction data directly sensitive to the symmetry potential. Since in the distorted wave Born approximation (DWBA), the (p, n) quasi-elastic cross section is directly proportional to t V 1 [2, such (p, n) data could provide a sensitive test of the energy dependence of V l when a macroscopic DWBA calculation is done. Some analyses of quasi-elastic (p, n) data have suggested a linear energy dependence of the symmetry potential [1, 21. For many nuclei with A ~> 60, the isobaric analog of the target ground state (IAS) populated in the quasi-elastic (p, n) reaction is proton unbound. The subsequent proton decay (~3) of this IAS is presumably isotropic. Hence the measured (p, n~) cross section at a given angle (or angles) can be used directly to infer the total (p, n) cross section to the IAS in question, i.e. oy(p, n)iAS = 4rr [da@)/d~2] avRp- 1,
(1)
where [do(g)/d~2] av is the (p, n]~) cross section at one angle (or the average for several angles) and Rp is the probability of proton decay of the IAS. One can take * Supported in part by the U.S. Atomic Energy Commission. ** Present address: Center for Nuclear Studies, University of Texas, Austin, Texas 78712, USA.
advantage of the evident fact that oT(p, n)lAS and do(~)/dg2 have the same shape as a function of energy in order to study the energy dependence of V 1 by calculating aT(P, n)IAS and comparing to do(~)/dg2. Excitation functions for the (p, nO) reaction on 91Zr, ll9Sn, 208pb and 209Bi between 20 and 45 MeV were obtained at the UCLA Cyclotron Laboratory, and are shown in the figure. Also shown for the nuclei 91Zr and ll9Sn are the data of Miller and Garvey [3] from threshold to 18 MeV. Each excitation function shows a sharp rise at threshold, a broad maximum centered about 5 MeV above threshold, and a rather rapid, nearly linear decrease in cross section with increasing bombarding energy. Calculations were made using DWBA in an attempt to fit the shapes of these excitation functions. The Becchetti-Greenlees (BG) optical model parameters [4] were used for the calculations. We used the complex V 1 t" T / A interaction of 96 (Nv/(~-Z)/2A MeV, real volume, and 48 x / ( N - Z ) / 2 A MeV, surface imaginary, as implied by the results of the BG analysis. These potentials are known to give a good overall description of (p, n)IAS angular distributions over a wide range of nuclei [1, 2, 5]. The results of these calculations using the DWBA program DWUCK [6] are shown for each o f the four nuclei as curves labeled 'a' in the figure. Plotted are the predictions for oT(P, n)IAS which is related to o(p, n~) through eq. (1). Since Rp ~< 1, the DWBA predictions represent a theoretical upper limit for the (p, n~) cross sections. These calculations do not repro453
Volume 40B, number 4
PttYSICS LETTERS
(o)
2.0 1.0
~
A
'Zr
E i-
c
4
b
1 2 , ' I~' 2'o' 2'4' 2'a' ~z' ~6'Zo' I.o "- •
(b)
o.o/-
"
..,
,
5
.c
/ c ~
.tO
E
3
b
to. i-
b
2 I
~'6 ' io'
I.O~-
j4
' ~e ' gz'
|
?;s
'4b
' 4'4 '
2°ePb
12
v I-
b
8 4
d b
20 2.0 1.0 16 12
E
v
28
32
36
40
4'4
I-
(d)
f
4
duce the shape or magnitude o f the data. The rise at threshold is much too slow, and from about 10 MeV above threshold to 45 MeV the predictions continue to rise very slowly while the data is rapidly falling. It has been suggested [ 1] that potential strengths given by (120 - E ) MeV and (60 - 0.75 E) MeV (with E in MeV) for the real volume and surface imaginary symmetry potentia',s, respectively, will reproduce the shape of the 208pb excitation function data from 30 to 45 MeV. The results o f calculations using this energy dependent symmetry potential are shown as the curves labeled 'b' in the figure. Although these predictions do not show a rapid rise at threshold, they do turn over and start to decrease at about 10 MeV above threshold for each nucleus. F r o m this point to 45 MeV these predictions do in fact have the same shape as the data. It should be mentioned that the predictions for the neutron angular distributions using the energy dependent symmetry potential fit the experimental shapes of the angular distribution data on 208pb and 209Bi, as well as, if not slightly better than, the nonenergy-dependent potential [1, 2]. Since the (p, n)IAS total cross section c o m p u t e d with DWBA is proportional to IV 1 [2, a plot of X/Oexp/ODWBA will give a potential scaling factor w h i c h when included in the DWBA calculations, will reproduce the experimentally observed excitation functions. Since the energy dependent potentials discussed do give the right shape for all four excitation functions measured from 30 to 45 MeV, a calculation of X/~exp/ODWBA was performed with the DWBA predictions using these potentials. The results are shown as the points on curves 'c' in the figure. Fits to these plots (curves 'c' in the figure) could be obtained for all four nuclei considered, resulting in complete symmetry potential strengths, as a function of energy given by
-c ZOgBi
~
d
I -
2'o
2~,
28
3'2
36
Ep ( M e V )
454
4~8
8
v
b
24
24 July 1972
4'o
i
44
-
b ,
i
48
. . . . t'lg. 1.. The. (p, np) excitation functions for 9 1 Zr, 1 1 9 S n 208Pb and 209B].. Data shown as ' X' are from ref. [3]. ~ae curves labeled 'a' are the macroscopic DWBA predictions for aT(p, n)iA S using the complex symmetry potential extracted from the Becchetti-Greenlees optical model parameters. The curves labeled 'b' are the macroscopic DWBA predictions for oT(p, n)lA S using the linearly energy dependent symmetry potentials. The curves labeled 'c' represent the symmetry potential scaling factors of eq. (2) which scale the linearly energy dependent potentials to give the macroscopic DWBA predictions for oT(p,n~) shown as curves 'd'.
Volume 40B, number 4
PHYSICS LETTERS
A {1 + (1.0+0.2) e x p [ - ( 0 . 2 5 + 0 . 0 5 ) l / f - E 0 [ ] } ( 1 2 0 - E ) MeV real volume (E in MeV), A {l+(1.0-+0.2)exp[ (0.25-+0.05)IE-E01] } ( 6 0 - 0 . 7 5 E ) MeV imaginary surface (E in MeV).
(2)
The quantity b~O is given by/~) ~ Eth + 1.5 MeV, where tffl~ is the (p, n) charge-exchange threshold energy. Predictions using DWBA and these symmetry potential strengths are shown as curves 'd' in the figure. All four excitation functions are fit quite well. We submit that these unrealistic but consistent energy dependent strengths are required for the nuclear symmetry potential if the (p, nl~) excitation function data is to be fit within the framework of the DWBA using the BG optical model parameters. It is to be emphasized that in our macroscopic DWBA calculations the BG global optical model parameters are used for generating the distorted waves in the incoming and outgoing channels, as has been the customary procedure in all recent surveys in the literature of quasi-elastic (p, n) scattering on a wide range of nuclei [5, 7]. Although the BG proton optical parameters may be very reliable, the neutron parameters were derived from data substantially inferior to the proton data. Further, at neutron energies below 10 MeV important compound elastic corrections of considerable ambiguity had to be included in the BG
24 July 1972
analysis [4]. The observation that the curves 'c' in the figure all have the same shape as a function of neutron energy (E n ~ Ep + Q), and are particularly ill-behaved from E n = 0 to 10MeV, suggests that the poor fit may be due to the neutron optical potential in this energy region. An important result of the present work is that (p, n) studies to determine the strength of V 1 by fitting quasi-elastic angular distributions at a given energy with the usual global optical potentials and DWBA can obtain symmetry potentials differing greatly from those obtained via a similar analysis at a different energy. Thus recent surveys [5, 7] of quasi-elastic scattering on a wide range of nuclei may yield doubtful values for V1 unless care is taken in the theoretical analysis of the data.
References [ 1 ] T.J. Woods, G.J. lgo and C.A. Whitten, Phys. Lett. 39B
(1972) 193. [2] G.W. Hoffmann et al., in press. [3] P.S. Miller, thesis, Princeton University (1968); G.T. Garvey and P.S. Miller, Phys. Lett. 28B (1968) 244. [4] F.D. Becchetti and G.W. Greenlees, Phys. Rev. 182 (1969) 1190.
[5] C. Wong et al., Phys. Rev. C5 (1972) 158. [6] P.D. Kunz, University of Colorado, Boulder, Colorado (private communication). [7] R.F. Bently et al., Phys. Rev. Lett. 27 (1971) 1081.
455