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The Coulomb correction term in proton absorbing optical potentials
Volume 92B, number 3,4
PHYSICS LETTERS
19 May 1980
THE COULOMB CORRECTION TERM IN PROTON ABSORBING OPTICAL POTENTIALS ¢r
J. RAPAPORT Physics Department, Ohio University, Athens, OH 45701, USA Received 23 Janaury 1980
The results of phenomenological analyses of nucleon elastic scattering by 4°Ca are analyzed, and the Coulomb correction term of the absorbing part of the optical model potential is empirically extracted. The results show general agreement with the theoretical calculations of Jeukenne, Lejeune and Mahaux.
The phenomenological local optical model potential (OMP) may be expressed as a sum of terms:
-U(r, E) = UN(r, E ) - Vc(r) - Uso(r, E ) , where Vc(r ) is the Coulomb potential due to a uniformly charged sphere and Uso(r, E) is the spin-orbit potential assumed to be of the Thomas form. U(r, E ) denotes in general a complex quantity U = V + iW. The nuclear part, UN(r , E), on which we will concentrate the analysis may be written in a consistent Lane [1] form as:
UN(r,E)=Uo(r,E)+-eUt(r,E)+ /\Uc(r,E),
(1)
where Uo(r , E ) and U 1 (r, E ) represent the isoscalar and isovector parts of the OMP, e = (N - Z)/A is the asymmetry coefficient and the + ( - ) sign is for protons (neutrons). The last term, AUc(r , E ) = A Vc(r ' E) + iA Wc(r , E), arises from the Coulomb field in the presence of an energy dependence of the OMP and is denoted as the Coulomb correction term [2]. The Coulomb field decreases the local kinetic energy of the incident proton and thus changes the effective local potential because of its energy dependence. Satchler [3] has expressed the Coulomb correction term as AUc(r, E) = Vc(Rc) ~UN(r, E ) / 0 E , where ffc(Rc) is an average of the Coulomb interaction potential and it is assumed that DUN(r , E)/DE has ap¢' Supported in part by the National Science Foundation.
proximately the same radial shape as UN(r, E); thus, the above expression reduces itself to a correction to the well depth of the OMP. With a linear approximation for the energy dependence of V, V(E) = V(O) - a.E, the real part of the Coulomb correction term, in MeV, is usually expressed [3] as AVc(r, E ) = 0.4(Z/A 1/3) X f ( r ) ; in this expression values ~ = 0.3 MeV -1 and R c = 1.3A 1/3 (fin) have been assumed. A Coulomb correction term in the absorbing part of the OMP, A Wc (r, E), should also be included in proton OMP analyses. An estimation of its value requires the knowledge of the energy dependence of W(r, E). It is well known that the radial dependence of W is energy dependent; with nucleon energies up to 2 0 - 2 5 MeV, it is a pure surface (Woods-Saxon derivative), changing to a mixture of surface and volume up to about 50 MeV and finally to a pure volume (Woods-Saxon) for higher energies. With such changes in the radial form of the phenomenological values o f W(r, E), and each one with a typical energy dependence [4], it is not clear how to extract an expression for OW(r, E)/OE. In the absence of any reliable estimates for AWc(r , E), it has been neglected (assumed to be zero) in previous proton phenomenological OMP analyses. This may have contributed to wrong evaluations of W1, the imaginary term of the isovector part of the OMP and to wrong predictions for the quasi-elastic (p, n) cross sections when a Laneconsistent OMP has been used. For T = 0 nuclei, eq. (1) may be written for protons and neutrons: 233
Volume 92B, number 3,4
PHYSICS LETTERS
Up(r, E) = Uo(r, E) + A Uc(r, E) ,
Un(r, E )
energy for 0 <~ E n <~ 25 MeV. This behavior is physically expected because as the incident nucleon energy increases, more inelastic channels are opened which results in increased absorption. For higher nucleon energies the energy dependence is almost negligible. A recent global OMP for 8 0 - 2 0 0 MeV protons [8] indicates a constant value of W. The Jw/A values obtained from OMP analyses of nucleon elastic scattering by 40Ca are presented versus incident nucleon energy in fig. I. The Jw/A values for neutrons are obtained from refs. [5,7] while those for protons from refs. [9,10]. An uncertainty of about 10% is estimated in each point. The lines represent least square fits with the functional dependence given in table 1. The following main features, in 40Ca, about the contribution to Jw/A due to the Coulomb correction term, may be deduced from fig. 1. (a) Ja wc/A is energy dependent, decreasing to almost zero at 40 MeV. (b) JAWc/A has the opposite sign of JA Ve/A , in agreement with the microscopic predictions for these terms done by Jeukenne et al. [ 11 ]. The magnitude and energy dependence ofJarec/A
-- t o ( r , E ) .
Thus a simple comparison of neutron and proton OMP analyses should yield the values for AUc(r, E). An empirical determination of the strength of AVc has been previously presented [5 ], with a value in MeV: AVe(r ) = (0.46 -+ O.07)(Z/A 1/3)f(r). An empirical determination of AWe(r, E ) is attempted in the present paper based on nucleon OMP analyses of 4°Ca. Because of the different energy dependences of the various forms of the radial functions, it is convenient to express the results in terms of volume integrals per nucleon, Jw/A. Several papers have been published on neutron elastic scattering OMP analyses with energies up to 26 MeV [5,6] ; recently OMP analyses of 30 and 40 MeV neutron elastic scattering by 40Ca have also been reported [7]. Similarly, several papers [ 8 - I 0 ] discuss OMP analyses of proton elastic scattering by 40Ca up to 200 MeV. The results indicate an increase of the absorptive part of the OMP with increasing nucleon
• Co ( n , n ) 120
T
o Ca (p,p)
~,~
g-
o ~
o
E ¢,
19 May 1980
~ - . - - o ........~
80
o (Ow/A) n- (dwlA)p -- - (dnwe/A)
T
40
oo
0 I0
20
r ~ 30
40
50
E (MeV)
Fig. 1. Values of the volume integral per nucleon of the imaginary part of the nucleon OMP as a function of the incident lab energy for 4°Ca. More than one value at each energy represent different OMP geometrical parameters. The square data points represent the negative of the contribution to JW/A, due to the Coulomb correction term. The Jw/A values for protons at around 21 MeV are quite scattered, probably due to the high (p, n) Q-value (Q = -15.1 MeV) (see refs. [5,9]). 234
Volume 92B, number 3,4
PHYSICS LETTERS
Table 1 Nucleon Jw/A values in MeV fm 3 for 4°Ca. Assumed energy dependence: Jw/A = [Jw(E = 0)//t ] (1 + awE). Energy (MeV)
Nucleon
< 30
protons neutrons protons
>30
Jw(E = O) a W A 63 ± 7 87 _+9 110 +- 12
(MeV) -1 0.023 +- 0.006 0.017 -+ 0.007 -0.0015 -+ 0.0012
have been theoretically estimated in ref. [11 ] using a microscopic OMP obtained from realistic n u c l e o n nucleon interactions for the proton elastic scattering of 208pb. The present empirical results in 40Ca show general agreement with the above calculations. The square data points in fig. 1 were obtained by subtracting from the empirical neutron line the experimental Jw/A values obtained for protons. The line represents a linear least square fit o f the form:
JzxWc/A = - ( 4 3 + 7) [1 - (0.028 + 0.005)E] (MeV fm 3) . It has a value JzxWc/A = 0 for a proton energy E = 36 -+ 7 MeV, and, because the slope of Jw/A for nucleon energies above 30 MeV is almost zero, the value of JaWc/A for E >~ 40 MeV should be negligible. Assuming that g(r), the radial dependence of AWc(r), is the same as the surface (derivative W o o d s - S a x o n ) radial dependence of W, the above expression may be written in a general form as:
AWe(r, E ) = - ( l - 0.028E)(0.5 +- 0.1) × (Z/A I/3)g(r). The above expression applied for 208pb agrees rather well both in magnitude and energy dependence with the theoretical values presented in ref. [11]. Arnold and Clark [12] have developed a relativistic OMP based on the use o f the Dirac equation and found that a number o f features of the empirical OMP may be explained by the use o f relativistic wave equations. One of these features is the Coulomb correction term. Their recent results for the relativistic OMP analysis of 4°Ca show
19 May 1980
(a) that the real Coulomb correction term is in reasonable agreement with empirical results [5] and with the results o f microscopic non-relativistic calculations of the OMP [11 ] and (b) that the imaginary Coulomb correction term has the opposite sign o f AVe, and is energy dependent. The present empirical results show agreement with these relativistic OMP calculations. In summary, it is proposed that the imaginary term of the empirical proton OMP may be written as:
W(r,E) = Wv(E)f(r ) + [WD(E ) + eWI(E ) + AWc(E)lg(r ) where f(r) is a W o o d s - S a x o n radial form factor and g(r) its radial derivative. The value o f AWe(E ) in MeV is AWc(E ) = - ( 1 - 0.028E) 0.SZ/A 1/3, 15<~E<~36, and is always negative or zero. [AWc(E ) = 0, E ~ 36 MeV.]
References [1] A.M. Lane, Phys. Rev. Lett. 8 (1962) 171;Nucl. Phys. 35 (1962) 676. [2] A.M. Lane, Rev. Mod. Phys. 29 (1957) 191. [3] G.R. Satchler, Isospin in nuclear physics, ed. D.H. Wilkinson (North-Holland, Amsterdam, 1969) Ch. 9. [4] J. Rapaport, V. Kulkarni and R.W. Finaly, Nucl. Phys. A330 (1979) 15. [5] J. Rapaport, J.D. Carlson, D. Bainum, T,S. Cheema and R.W. Finlay, Nucl. Phys. A286 (1977) 232. [6] J. Rapaport and R.W. Finlay, IEEE Trans. Nucl. Sci. NS-26 (1979) 1196. [7] R.P. DeVito, S.M. Austin, U.E.P. Berg, W.A. Sterrenburg and L.E. Young, Bull. Am. Phys. Soc. 24 (1979) 830. [8] A. Nadasen et al., Phys. Rev. C, to be published. [9] J.F. Dicello, G. Igo and W.T. Leland, Phys. Rev. C4 (1971) 1130. [10] W.T.H. Van Oers, Phys. Rev. C3 (1971) 1971; N.E. Davison, private communication (1977). [11] J.-P. Jeukenne, A. Lejeune and C. Mahaux, Phys. Rev. C15 (1977) 10; C16 (1977) 80. [12] L. G. Arnold and B.C. Clark, Phys. Lett. 84B (1979) 46. [13] B.C. Clark, L.G. Arnold and R.L. Mercer, Bull. Am. Phys. Soc. 24 (1979) 839; and private communication.
235
PHYSICS LETTERS
19 May 1980
THE COULOMB CORRECTION TERM IN PROTON ABSORBING OPTICAL POTENTIALS ¢r
J. RAPAPORT Physics Department, Ohio University, Athens, OH 45701, USA Received 23 Janaury 1980
The results of phenomenological analyses of nucleon elastic scattering by 4°Ca are analyzed, and the Coulomb correction term of the absorbing part of the optical model potential is empirically extracted. The results show general agreement with the theoretical calculations of Jeukenne, Lejeune and Mahaux.
The phenomenological local optical model potential (OMP) may be expressed as a sum of terms:
-U(r, E) = UN(r, E ) - Vc(r) - Uso(r, E ) , where Vc(r ) is the Coulomb potential due to a uniformly charged sphere and Uso(r, E) is the spin-orbit potential assumed to be of the Thomas form. U(r, E ) denotes in general a complex quantity U = V + iW. The nuclear part, UN(r , E), on which we will concentrate the analysis may be written in a consistent Lane [1] form as:
UN(r,E)=Uo(r,E)+-eUt(r,E)+ /\Uc(r,E),
(1)
where Uo(r , E ) and U 1 (r, E ) represent the isoscalar and isovector parts of the OMP, e = (N - Z)/A is the asymmetry coefficient and the + ( - ) sign is for protons (neutrons). The last term, AUc(r , E ) = A Vc(r ' E) + iA Wc(r , E), arises from the Coulomb field in the presence of an energy dependence of the OMP and is denoted as the Coulomb correction term [2]. The Coulomb field decreases the local kinetic energy of the incident proton and thus changes the effective local potential because of its energy dependence. Satchler [3] has expressed the Coulomb correction term as AUc(r, E) = Vc(Rc) ~UN(r, E ) / 0 E , where ffc(Rc) is an average of the Coulomb interaction potential and it is assumed that DUN(r , E)/DE has ap¢' Supported in part by the National Science Foundation.
proximately the same radial shape as UN(r, E); thus, the above expression reduces itself to a correction to the well depth of the OMP. With a linear approximation for the energy dependence of V, V(E) = V(O) - a.E, the real part of the Coulomb correction term, in MeV, is usually expressed [3] as AVc(r, E ) = 0.4(Z/A 1/3) X f ( r ) ; in this expression values ~ = 0.3 MeV -1 and R c = 1.3A 1/3 (fin) have been assumed. A Coulomb correction term in the absorbing part of the OMP, A Wc (r, E), should also be included in proton OMP analyses. An estimation of its value requires the knowledge of the energy dependence of W(r, E). It is well known that the radial dependence of W is energy dependent; with nucleon energies up to 2 0 - 2 5 MeV, it is a pure surface (Woods-Saxon derivative), changing to a mixture of surface and volume up to about 50 MeV and finally to a pure volume (Woods-Saxon) for higher energies. With such changes in the radial form of the phenomenological values o f W(r, E), and each one with a typical energy dependence [4], it is not clear how to extract an expression for OW(r, E)/OE. In the absence of any reliable estimates for AWc(r , E), it has been neglected (assumed to be zero) in previous proton phenomenological OMP analyses. This may have contributed to wrong evaluations of W1, the imaginary term of the isovector part of the OMP and to wrong predictions for the quasi-elastic (p, n) cross sections when a Laneconsistent OMP has been used. For T = 0 nuclei, eq. (1) may be written for protons and neutrons: 233
Volume 92B, number 3,4
PHYSICS LETTERS
Up(r, E) = Uo(r, E) + A Uc(r, E) ,
Un(r, E )
energy for 0 <~ E n <~ 25 MeV. This behavior is physically expected because as the incident nucleon energy increases, more inelastic channels are opened which results in increased absorption. For higher nucleon energies the energy dependence is almost negligible. A recent global OMP for 8 0 - 2 0 0 MeV protons [8] indicates a constant value of W. The Jw/A values obtained from OMP analyses of nucleon elastic scattering by 40Ca are presented versus incident nucleon energy in fig. I. The Jw/A values for neutrons are obtained from refs. [5,7] while those for protons from refs. [9,10]. An uncertainty of about 10% is estimated in each point. The lines represent least square fits with the functional dependence given in table 1. The following main features, in 40Ca, about the contribution to Jw/A due to the Coulomb correction term, may be deduced from fig. 1. (a) Ja wc/A is energy dependent, decreasing to almost zero at 40 MeV. (b) JAWc/A has the opposite sign of JA Ve/A , in agreement with the microscopic predictions for these terms done by Jeukenne et al. [ 11 ]. The magnitude and energy dependence ofJarec/A
-- t o ( r , E ) .
Thus a simple comparison of neutron and proton OMP analyses should yield the values for AUc(r, E). An empirical determination of the strength of AVc has been previously presented [5 ], with a value in MeV: AVe(r ) = (0.46 -+ O.07)(Z/A 1/3)f(r). An empirical determination of AWe(r, E ) is attempted in the present paper based on nucleon OMP analyses of 4°Ca. Because of the different energy dependences of the various forms of the radial functions, it is convenient to express the results in terms of volume integrals per nucleon, Jw/A. Several papers have been published on neutron elastic scattering OMP analyses with energies up to 26 MeV [5,6] ; recently OMP analyses of 30 and 40 MeV neutron elastic scattering by 40Ca have also been reported [7]. Similarly, several papers [ 8 - I 0 ] discuss OMP analyses of proton elastic scattering by 40Ca up to 200 MeV. The results indicate an increase of the absorptive part of the OMP with increasing nucleon
• Co ( n , n ) 120
T
o Ca (p,p)
~,~
g-
o ~
o
E ¢,
19 May 1980
~ - . - - o ........~
80
o (Ow/A) n- (dwlA)p -- - (dnwe/A)
T
40
oo
0 I0
20
r ~ 30
40
50
E (MeV)
Fig. 1. Values of the volume integral per nucleon of the imaginary part of the nucleon OMP as a function of the incident lab energy for 4°Ca. More than one value at each energy represent different OMP geometrical parameters. The square data points represent the negative of the contribution to JW/A, due to the Coulomb correction term. The Jw/A values for protons at around 21 MeV are quite scattered, probably due to the high (p, n) Q-value (Q = -15.1 MeV) (see refs. [5,9]). 234
Volume 92B, number 3,4
PHYSICS LETTERS
Table 1 Nucleon Jw/A values in MeV fm 3 for 4°Ca. Assumed energy dependence: Jw/A = [Jw(E = 0)//t ] (1 + awE). Energy (MeV)
Nucleon
< 30
protons neutrons protons
>30
Jw(E = O) a W A 63 ± 7 87 _+9 110 +- 12
(MeV) -1 0.023 +- 0.006 0.017 -+ 0.007 -0.0015 -+ 0.0012
have been theoretically estimated in ref. [11 ] using a microscopic OMP obtained from realistic n u c l e o n nucleon interactions for the proton elastic scattering of 208pb. The present empirical results in 40Ca show general agreement with the above calculations. The square data points in fig. 1 were obtained by subtracting from the empirical neutron line the experimental Jw/A values obtained for protons. The line represents a linear least square fit o f the form:
JzxWc/A = - ( 4 3 + 7) [1 - (0.028 + 0.005)E] (MeV fm 3) . It has a value JzxWc/A = 0 for a proton energy E = 36 -+ 7 MeV, and, because the slope of Jw/A for nucleon energies above 30 MeV is almost zero, the value of JaWc/A for E >~ 40 MeV should be negligible. Assuming that g(r), the radial dependence of AWc(r), is the same as the surface (derivative W o o d s - S a x o n ) radial dependence of W, the above expression may be written in a general form as:
AWe(r, E ) = - ( l - 0.028E)(0.5 +- 0.1) × (Z/A I/3)g(r). The above expression applied for 208pb agrees rather well both in magnitude and energy dependence with the theoretical values presented in ref. [11]. Arnold and Clark [12] have developed a relativistic OMP based on the use o f the Dirac equation and found that a number o f features of the empirical OMP may be explained by the use o f relativistic wave equations. One of these features is the Coulomb correction term. Their recent results for the relativistic OMP analysis of 4°Ca show
19 May 1980
(a) that the real Coulomb correction term is in reasonable agreement with empirical results [5] and with the results o f microscopic non-relativistic calculations of the OMP [11 ] and (b) that the imaginary Coulomb correction term has the opposite sign o f AVe, and is energy dependent. The present empirical results show agreement with these relativistic OMP calculations. In summary, it is proposed that the imaginary term of the empirical proton OMP may be written as:
W(r,E) = Wv(E)f(r ) + [WD(E ) + eWI(E ) + AWc(E)lg(r ) where f(r) is a W o o d s - S a x o n radial form factor and g(r) its radial derivative. The value o f AWe(E ) in MeV is AWc(E ) = - ( 1 - 0.028E) 0.SZ/A 1/3, 15<~E<~36, and is always negative or zero. [AWc(E ) = 0, E ~ 36 MeV.]
References [1] A.M. Lane, Phys. Rev. Lett. 8 (1962) 171;Nucl. Phys. 35 (1962) 676. [2] A.M. Lane, Rev. Mod. Phys. 29 (1957) 191. [3] G.R. Satchler, Isospin in nuclear physics, ed. D.H. Wilkinson (North-Holland, Amsterdam, 1969) Ch. 9. [4] J. Rapaport, V. Kulkarni and R.W. Finaly, Nucl. Phys. A330 (1979) 15. [5] J. Rapaport, J.D. Carlson, D. Bainum, T,S. Cheema and R.W. Finlay, Nucl. Phys. A286 (1977) 232. [6] J. Rapaport and R.W. Finlay, IEEE Trans. Nucl. Sci. NS-26 (1979) 1196. [7] R.P. DeVito, S.M. Austin, U.E.P. Berg, W.A. Sterrenburg and L.E. Young, Bull. Am. Phys. Soc. 24 (1979) 830. [8] A. Nadasen et al., Phys. Rev. C, to be published. [9] J.F. Dicello, G. Igo and W.T. Leland, Phys. Rev. C4 (1971) 1130. [10] W.T.H. Van Oers, Phys. Rev. C3 (1971) 1971; N.E. Davison, private communication (1977). [11] J.-P. Jeukenne, A. Lejeune and C. Mahaux, Phys. Rev. C15 (1977) 10; C16 (1977) 80. [12] L. G. Arnold and B.C. Clark, Phys. Lett. 84B (1979) 46. [13] B.C. Clark, L.G. Arnold and R.L. Mercer, Bull. Am. Phys. Soc. 24 (1979) 839; and private communication.
235