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Coupled-channel approach to deuteron elastic scattering at intermediate energy
Volume 182, number
2
PHYSICS
LETTERS
B
18 December
1986
COUPLED-CHANNEL APPROACH TO DEUTERON ELASTIC SCATTERING AT INTERMEDIATE ENERGY M. YAHIRO Shimonoseki University ofFisheries, Shimonoseki 759-65, Japan
H. KAMEYAMA,
Y. ISERI, M. KAMIMURA
and M. KAWAI
Department ofPhysics. Kyushu University, Fukuoka 812, Japan Received
8 July 1986
Deuteron elastic scattering at 400 MeV is analyzed with the coupled-channel method to treat the deuteron breakup process. Substantial improvement over the single-folding optical model is obtained in the fit to the experimental cross section. Wine-bottle shaped nucleon optical potentials are superior to Woods-Saxon ones for the deuteron scattering, while both types of potentials reproduce proton-scattering data equally well.
experimental data on 40Ca(G, c)40Ca at 200 MeV [ 5,6], as shown in fig. 2, extrapolated with respect to mass number from 40 to 58 in the manner described in I. The parameters of the potentials thus obtained are listed in tables 1, 2. The nucleon optical potentials are slightly refined from those in I because the experimental data now include the spin rotation parameter in 4oCa(i;, jj)40Ca at 200 MeV [6]. The refinement does not affect the conclusion of I: U,(WB) is superior to U,(WS) for the deuteron scattering, while the two types of nucleon optical potentials reproduce the proton-scattering data equally well. Thus, the deuteron seemed to be, in a sense, a better probe of the nucleon optical potential than the nucleon itself. The success of the folding model, however, is limited because it under-predicts the cross section at angles 8> 15”, reaching a factor of 4 at 30 O, as one sees in fig. 1. This clearly indicates that some mechanism of the process which is important in this angular region is missed by the model. It was suggested in I that it might be the virtual deuteron breakup process, because it could significantly affect the very small cross section in this angular region, the value at 30” being of the order of lop6 of the value at 5”, though the process may be small in absolute terms at this energy. The purpose of the present paper is to
The elastic scattering of deuterons is of particular interest in nuclear reaction studies as the scattering of the simplest kind of composite particle. Experiments have been performed and analysed at low energies on various target nuclei [ 11. It has been recognized with the method of coupled discretized continuum channels (CDCC) [ 21 that the virtual breakup of the deuteron (d) plays an important role at those energies, and that its effect decreases as energy increases. The question is, then, what is the situation at higher energies? Recently, elastic scattering of polarized deuterons was measured at intermediate energies of T, = 200, 400, and 700 MeV [ 31. In an analysis of the data at 400 MeV for the target nucleus 5*Ni [4], hereafter referred to as I, we found that the single folding optical model works well with wine-bottle shaped nucleon optical potentials from the Dirac phenomenology in its Schrijdinger equation form, V,( WB). It is definitely better than with standard Woods-Saxon type potentials, U,,(WS), which reproduce the proton-nucleus scattering data as well as V,( WB). We illustrate these results in figs. 1 and 2. Fig. 1 shows the comparison of the single folding optical model with U,( WB) and V,(+WS) together with the experimental data on 58Ni(d, d)5sNi at 400 MeV. The proton optical potentials are those which fit the 0370-2693/86/$ (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division)
B.V.
135
Volume 182, number
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LETTERS
B
18 December
1986
be more sensitive to small changes in the scattering amplitudes than the cross section. We assume a three-body model hamiltonian H=T,+T,+&(r,)+U”(r”)+V”,(P)
30” cm ed
0”
IO”
20”
30” cm ed
Fig. 1. Cross section, vector and tensor analyzing powers for 58Ni(& d)‘*Ni at 400 MeV. Solid and dashed lines are results of the folding-model calculations with (/,(WB) and II,( respectively. The dotted lines are the result of the coupled-channel calculation which adds the deuteron breakup process onto the folding model with U,( WB). Experimental data are from ref. [ 31.
present the result of a CDCC analysis of this effect, in order to verify this premise, and to see if the conclusion of I is not affected by the inclusion of the breakup effect. This is important because in general the analyzing powers, depending on differences in the cross sections for different spin orientations, could 136
(1)
where the U are the nucleon-nucleus interaction potentials, V,, is the interaction potential between proton (p) and neutron (n), the r are the displacements of the nucleons from the target nucleus, p=r,-r,,, R=(r,+r,)/2,and T,and T,arethecorresponding kinetic-energy operators. Correction for the relativistic kinematics [ 71 is applied to T,. We take for the U the same wine-bottle shaped potentials as the ones used in fig. 1. The Coulomb part of UP is approximated by the value at r,=R. The hamiltonian is solved with the CDCC method of ref. [ 21. We restrict the state of the n-p system to 3S,, 3D,, 3D2, and ‘D3 states, neglecting the spin-singlet states because only spin-orbit terms of the U can cause transitions to them from the spin-triplet states. We restrict k, the relative momentum of n and p in the breakup state, to 0~ k< km,,= 1.0 fm-‘, and the kcontinuum is discretized into two bins of the size Ak= 0.5 fm- ‘. The values of k,,,,, and Ak are sufficient for the present CDCC analysis, since the calculated cross section and analyzing powers converge at those values as Ak decreases and as km,, increases. The deuteron ground state wave function is constructed with the Reid soft core potential [ 81. The breakup state wave fuctions, however, are calculated with gaussian potentials, V,,(p) = V,exp[ - (p/po)2], one for each of the states mentioned which fits the low-energy data on that state. The parameters of the gaussian potentials are ( VO(MeV), p,,( fm)) =( -72.15, 1.4840) for 3S,, (30.0, 2.357) for 3DI, (-40.0, 2.209) for 3D2, and (-38.0, 1.240) for 3D3. Therefore, the breakup 3D, state wave function is not strictly orthogonal to the ground state wave function. But the overlap between them turns out to be less than 0.02. The result of the CDCC calculation is shown in fig. 1 by dotted lines. The breakup effect is seen to enhance the cross section for 8 2 15”, and much improves the agreement with the observed Rutherford ratio of the cross section. At 30”, for example, the discrepancy is now a factor of 2, which we consider to be within the limit of the present model. The breakup effect does not much affect the cross section
Volume 182, number
PHYSICS
2
LETTERS
18 December
B
1986
irn&) IO4
IO2
1
IO2
p-n p
40"
0"
80"
AY
0.5
0
-0.5
-1.0 0"
40”
Table 1 The proton optical wine-bottle
potential
0cf-n p
Fig. 2. Results of a search of the proton optical potentials for 40Ca( 5, G)%a at 200 MeV. Solid and dashed lines are results of the best-tit wine-bottle type and Woods-Saxon type potentials, U,(WB) and U,(WS), respectively. Experimental data are from refs. [ 561.
80"
used in the calculation.
,
C~(WB,r)=U~+(mlE,)(I,-(1/2E,)(U~-~:)-(1/2E,)V,(V,+2~~)+Lio,,,,-(1/2E,Br)(dBldr)(o~L)+V,
E, is the proton total energy in the CM system, lJ,= Vaf(xp1) +i W&x:), and Us(r) = Vf(xR) +i WJ(x;) with x; = (rV, is the CouV, is the Coulomb potential, Up.,,, is the so-called Darwin term, and B= (E,+ m+ Us- U,- V,)l(E,+m). lomb potential. The neutron optical potential is assumed to be the same as the proton potential with the charge of the particle set to zero.
where
r;A”‘3)/a;,
344.6
1.016
0.6680
- 82.56
1.053
0.6010
-466.3
1.004
0.6949
65.60
1.075
0.5201
137
Volume 182, number
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2
Table 2 The proton optical Woods-Saxon
potential
LETTERS
B
I8 December
I986
used in the calculation.
ci,(WS,~)=-[Vf(x~)+iWf((~,)]+~,+(2/r)(dldr)[V,~(f(xff”)+iW,~(x;”)](a.L) V, is the Coulomb potential. particle set to zero.
The neutron
optical
potential
is assumed
to be the same as the proton
potential
with the charge of the
V
rR
4
W
rI
4
VW
rF
ag
WSO
r;
a;
11.71
1.354
0.6477
24.90
0.9657
0.8294
3.177
0.9881
0.7265
-2.126
0.9986
0.5787
at forward angles, 135 15 ‘, as the optical model amplitude predominates there. It is important to note that the analyzing powers are not much affected, either, at any angle shown in the figure. Hence, the conclusion of I is well preserved. In summary, the folding potential constructed from a wine-bottle shaped nucleon-nucleus potential from a Dirac optical model reproduces ~$1 the major part of the experimental data on 58Ni (d, d) “Ni at 400 MeV. Much of the discrepancy with the experimental data on the cross section at large angles, 19Z 15 ‘, can be attributed to the effect of the virtual breakup of the deuteron. Hence, the present analysis reinforces the conclusion of I that nucleon optical potentials which are equivalent in reproducing the nucleon scattering data can be distinguished by an analysis of deuteron scattering data at twice the proton incident energy; the present analysis shows that the wine-bottle shaped potential of the Dirac optical model is definitely superior to the standard Woods-Saxon type one for the nucleon optical potential at 200 MeV. In this work, the Dirac optical model is used for deriving the wine-bottle type potential. There are, however, other methods of doing this, such as microscopic calculations based on the Brueckner theory [ 91, multiple-scattering theory including the Lorentz-Lorenz effect [ lo], etc. Proton optical potentials obtained with such methods, even if they are equally good for proton scattering, may be distinguished by an analysis of deuteron elastic scattering with the method of the present work. It is a great pleasure of the authors to thank Dr. T. Hasegawa for very helpful discussions about the experimental data of ref. [ 31. They also wish to thank Dr. Stephenson for his supplying them with the numerical data of ref. [ 61. The numerical calcula-
138
tions were done at the Computing Center of Kyushu University using a FACOM M382 computer (financial aid from the Research Center for Nuclear Physics) and at the Institute of Nuclear Study, University of Tokyo using a FACOM M380 computer. References [ 1] C.M. Perey and F.G. Perey, At. Data Nucl. Data Tables I7 (1976) 1; W.W. Daehnick, J.D. Childs and Z. Vrcelj, Phys. Rev. C 2 1 (1980) 2253. [2] M. Yahiro, M. Nakano, Y. Iseri and M. Kamimura, Prog. Theor. Phys. 67 (1982) 1467, and references therein; M. Yahiro, Y. Iseri, M. Kamimura, M. Kawai, M. Nakano and M. Tanifuji, Proc. Tsukuba Intern. Workshop on Deuteron involving reactions and polarization phenomena (Tsukuba, 1985)(World Scientific, Singapore, 1986) pp.45-66. [3 81N. van Sen, J. Arvieux, Ye. Yanlin, G. Gaillard, B. Bonin, A. Boudard, G. Bruge, J.C. Lugol, R. Babinet, T. Hasegawa, F. Soga, J.M. Cameron, G.G. Neilson and D.M. Sheppard, Phys. Lett B 156 (1985) 185; and private communications. [4 .] M. Yahiro, H. Kameyama, Y. Iseri, M. Kamimura and M. Kawai, Proc. 6th Intern. Symp. on Polarization phenomena in nuclear physics (Osaka, 1985), J. Phys. Sot. Jpn. 55 (1986) Suppl. 322, pp. 322-326. [ 51 B.C. Clark, S. Hama, S.G. Kaelbermann, E.D. Cooper and R.L. Mercer, Phys. Rev. C 31 (1985) 694, 1975. [6] E.J. Stephenson, Proc. 6th Intern. Symp. on Polarization phenomena in nuclear physics (Osaka, 1985)) J. Phys. Sot. Jpn. 55 (1986) Suppl. 316, pp. 316-321; and private communication. [ 71 R.G. Newton, Scattering theory of waves and particles (Springer, Berlin, 1966); W.R. Coker, L. Ray, and G.W. Hoffmann, Phys. Lett. B 64 (1976) 403. [8]R.V.Reid,Ann.Phys.(NY)50(1968)411. [ 91 L. Rikus and H.V. von Geramb, Nucl. Phys. A 426 (1984) 496. [lo] M. Thies, Phys. Lett. B 162 (1985) 255.
2
PHYSICS
LETTERS
B
18 December
1986
COUPLED-CHANNEL APPROACH TO DEUTERON ELASTIC SCATTERING AT INTERMEDIATE ENERGY M. YAHIRO Shimonoseki University ofFisheries, Shimonoseki 759-65, Japan
H. KAMEYAMA,
Y. ISERI, M. KAMIMURA
and M. KAWAI
Department ofPhysics. Kyushu University, Fukuoka 812, Japan Received
8 July 1986
Deuteron elastic scattering at 400 MeV is analyzed with the coupled-channel method to treat the deuteron breakup process. Substantial improvement over the single-folding optical model is obtained in the fit to the experimental cross section. Wine-bottle shaped nucleon optical potentials are superior to Woods-Saxon ones for the deuteron scattering, while both types of potentials reproduce proton-scattering data equally well.
experimental data on 40Ca(G, c)40Ca at 200 MeV [ 5,6], as shown in fig. 2, extrapolated with respect to mass number from 40 to 58 in the manner described in I. The parameters of the potentials thus obtained are listed in tables 1, 2. The nucleon optical potentials are slightly refined from those in I because the experimental data now include the spin rotation parameter in 4oCa(i;, jj)40Ca at 200 MeV [6]. The refinement does not affect the conclusion of I: U,(WB) is superior to U,(WS) for the deuteron scattering, while the two types of nucleon optical potentials reproduce the proton-scattering data equally well. Thus, the deuteron seemed to be, in a sense, a better probe of the nucleon optical potential than the nucleon itself. The success of the folding model, however, is limited because it under-predicts the cross section at angles 8> 15”, reaching a factor of 4 at 30 O, as one sees in fig. 1. This clearly indicates that some mechanism of the process which is important in this angular region is missed by the model. It was suggested in I that it might be the virtual deuteron breakup process, because it could significantly affect the very small cross section in this angular region, the value at 30” being of the order of lop6 of the value at 5”, though the process may be small in absolute terms at this energy. The purpose of the present paper is to
The elastic scattering of deuterons is of particular interest in nuclear reaction studies as the scattering of the simplest kind of composite particle. Experiments have been performed and analysed at low energies on various target nuclei [ 11. It has been recognized with the method of coupled discretized continuum channels (CDCC) [ 21 that the virtual breakup of the deuteron (d) plays an important role at those energies, and that its effect decreases as energy increases. The question is, then, what is the situation at higher energies? Recently, elastic scattering of polarized deuterons was measured at intermediate energies of T, = 200, 400, and 700 MeV [ 31. In an analysis of the data at 400 MeV for the target nucleus 5*Ni [4], hereafter referred to as I, we found that the single folding optical model works well with wine-bottle shaped nucleon optical potentials from the Dirac phenomenology in its Schrijdinger equation form, V,( WB). It is definitely better than with standard Woods-Saxon type potentials, U,,(WS), which reproduce the proton-nucleus scattering data as well as V,( WB). We illustrate these results in figs. 1 and 2. Fig. 1 shows the comparison of the single folding optical model with U,( WB) and V,(+WS) together with the experimental data on 58Ni(d, d)5sNi at 400 MeV. The proton optical potentials are those which fit the 0370-2693/86/$ (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division)
B.V.
135
Volume 182, number
2
PHYSICS
LETTERS
B
18 December
1986
be more sensitive to small changes in the scattering amplitudes than the cross section. We assume a three-body model hamiltonian H=T,+T,+&(r,)+U”(r”)+V”,(P)
30” cm ed
0”
IO”
20”
30” cm ed
Fig. 1. Cross section, vector and tensor analyzing powers for 58Ni(& d)‘*Ni at 400 MeV. Solid and dashed lines are results of the folding-model calculations with (/,(WB) and II,( respectively. The dotted lines are the result of the coupled-channel calculation which adds the deuteron breakup process onto the folding model with U,( WB). Experimental data are from ref. [ 31.
present the result of a CDCC analysis of this effect, in order to verify this premise, and to see if the conclusion of I is not affected by the inclusion of the breakup effect. This is important because in general the analyzing powers, depending on differences in the cross sections for different spin orientations, could 136
(1)
where the U are the nucleon-nucleus interaction potentials, V,, is the interaction potential between proton (p) and neutron (n), the r are the displacements of the nucleons from the target nucleus, p=r,-r,,, R=(r,+r,)/2,and T,and T,arethecorresponding kinetic-energy operators. Correction for the relativistic kinematics [ 71 is applied to T,. We take for the U the same wine-bottle shaped potentials as the ones used in fig. 1. The Coulomb part of UP is approximated by the value at r,=R. The hamiltonian is solved with the CDCC method of ref. [ 21. We restrict the state of the n-p system to 3S,, 3D,, 3D2, and ‘D3 states, neglecting the spin-singlet states because only spin-orbit terms of the U can cause transitions to them from the spin-triplet states. We restrict k, the relative momentum of n and p in the breakup state, to 0~ k< km,,= 1.0 fm-‘, and the kcontinuum is discretized into two bins of the size Ak= 0.5 fm- ‘. The values of k,,,,, and Ak are sufficient for the present CDCC analysis, since the calculated cross section and analyzing powers converge at those values as Ak decreases and as km,, increases. The deuteron ground state wave function is constructed with the Reid soft core potential [ 81. The breakup state wave fuctions, however, are calculated with gaussian potentials, V,,(p) = V,exp[ - (p/po)2], one for each of the states mentioned which fits the low-energy data on that state. The parameters of the gaussian potentials are ( VO(MeV), p,,( fm)) =( -72.15, 1.4840) for 3S,, (30.0, 2.357) for 3DI, (-40.0, 2.209) for 3D2, and (-38.0, 1.240) for 3D3. Therefore, the breakup 3D, state wave function is not strictly orthogonal to the ground state wave function. But the overlap between them turns out to be less than 0.02. The result of the CDCC calculation is shown in fig. 1 by dotted lines. The breakup effect is seen to enhance the cross section for 8 2 15”, and much improves the agreement with the observed Rutherford ratio of the cross section. At 30”, for example, the discrepancy is now a factor of 2, which we consider to be within the limit of the present model. The breakup effect does not much affect the cross section
Volume 182, number
PHYSICS
2
LETTERS
18 December
B
1986
irn&) IO4
IO2
1
IO2
p-n p
40"
0"
80"
AY
0.5
0
-0.5
-1.0 0"
40”
Table 1 The proton optical wine-bottle
potential
0cf-n p
Fig. 2. Results of a search of the proton optical potentials for 40Ca( 5, G)%a at 200 MeV. Solid and dashed lines are results of the best-tit wine-bottle type and Woods-Saxon type potentials, U,(WB) and U,(WS), respectively. Experimental data are from refs. [ 561.
80"
used in the calculation.
,
C~(WB,r)=U~+(mlE,)(I,-(1/2E,)(U~-~:)-(1/2E,)V,(V,+2~~)+Lio,,,,-(1/2E,Br)(dBldr)(o~L)+V,
E, is the proton total energy in the CM system, lJ,= Vaf(xp1) +i W&x:), and Us(r) = Vf(xR) +i WJ(x;) with x; = (rV, is the CouV, is the Coulomb potential, Up.,,, is the so-called Darwin term, and B= (E,+ m+ Us- U,- V,)l(E,+m). lomb potential. The neutron optical potential is assumed to be the same as the proton potential with the charge of the particle set to zero.
where
r;A”‘3)/a;,
344.6
1.016
0.6680
- 82.56
1.053
0.6010
-466.3
1.004
0.6949
65.60
1.075
0.5201
137
Volume 182, number
PHYSICS
2
Table 2 The proton optical Woods-Saxon
potential
LETTERS
B
I8 December
I986
used in the calculation.
ci,(WS,~)=-[Vf(x~)+iWf((~,)]+~,+(2/r)(dldr)[V,~(f(xff”)+iW,~(x;”)](a.L) V, is the Coulomb potential. particle set to zero.
The neutron
optical
potential
is assumed
to be the same as the proton
potential
with the charge of the
V
rR
4
W
rI
4
VW
rF
ag
WSO
r;
a;
11.71
1.354
0.6477
24.90
0.9657
0.8294
3.177
0.9881
0.7265
-2.126
0.9986
0.5787
at forward angles, 135 15 ‘, as the optical model amplitude predominates there. It is important to note that the analyzing powers are not much affected, either, at any angle shown in the figure. Hence, the conclusion of I is well preserved. In summary, the folding potential constructed from a wine-bottle shaped nucleon-nucleus potential from a Dirac optical model reproduces ~$1 the major part of the experimental data on 58Ni (d, d) “Ni at 400 MeV. Much of the discrepancy with the experimental data on the cross section at large angles, 19Z 15 ‘, can be attributed to the effect of the virtual breakup of the deuteron. Hence, the present analysis reinforces the conclusion of I that nucleon optical potentials which are equivalent in reproducing the nucleon scattering data can be distinguished by an analysis of deuteron scattering data at twice the proton incident energy; the present analysis shows that the wine-bottle shaped potential of the Dirac optical model is definitely superior to the standard Woods-Saxon type one for the nucleon optical potential at 200 MeV. In this work, the Dirac optical model is used for deriving the wine-bottle type potential. There are, however, other methods of doing this, such as microscopic calculations based on the Brueckner theory [ 91, multiple-scattering theory including the Lorentz-Lorenz effect [ lo], etc. Proton optical potentials obtained with such methods, even if they are equally good for proton scattering, may be distinguished by an analysis of deuteron elastic scattering with the method of the present work. It is a great pleasure of the authors to thank Dr. T. Hasegawa for very helpful discussions about the experimental data of ref. [ 31. They also wish to thank Dr. Stephenson for his supplying them with the numerical data of ref. [ 61. The numerical calcula-
138
tions were done at the Computing Center of Kyushu University using a FACOM M382 computer (financial aid from the Research Center for Nuclear Physics) and at the Institute of Nuclear Study, University of Tokyo using a FACOM M380 computer. References [ 1] C.M. Perey and F.G. Perey, At. Data Nucl. Data Tables I7 (1976) 1; W.W. Daehnick, J.D. Childs and Z. Vrcelj, Phys. Rev. C 2 1 (1980) 2253. [2] M. Yahiro, M. Nakano, Y. Iseri and M. Kamimura, Prog. Theor. Phys. 67 (1982) 1467, and references therein; M. Yahiro, Y. Iseri, M. Kamimura, M. Kawai, M. Nakano and M. Tanifuji, Proc. Tsukuba Intern. Workshop on Deuteron involving reactions and polarization phenomena (Tsukuba, 1985)(World Scientific, Singapore, 1986) pp.45-66. [3 81N. van Sen, J. Arvieux, Ye. Yanlin, G. Gaillard, B. Bonin, A. Boudard, G. Bruge, J.C. Lugol, R. Babinet, T. Hasegawa, F. Soga, J.M. Cameron, G.G. Neilson and D.M. Sheppard, Phys. Lett B 156 (1985) 185; and private communications. [4 .] M. Yahiro, H. Kameyama, Y. Iseri, M. Kamimura and M. Kawai, Proc. 6th Intern. Symp. on Polarization phenomena in nuclear physics (Osaka, 1985), J. Phys. Sot. Jpn. 55 (1986) Suppl. 322, pp. 322-326. [ 51 B.C. Clark, S. Hama, S.G. Kaelbermann, E.D. Cooper and R.L. Mercer, Phys. Rev. C 31 (1985) 694, 1975. [6] E.J. Stephenson, Proc. 6th Intern. Symp. on Polarization phenomena in nuclear physics (Osaka, 1985)) J. Phys. Sot. Jpn. 55 (1986) Suppl. 316, pp. 316-321; and private communication. [ 71 R.G. Newton, Scattering theory of waves and particles (Springer, Berlin, 1966); W.R. Coker, L. Ray, and G.W. Hoffmann, Phys. Lett. B 64 (1976) 403. [8]R.V.Reid,Ann.Phys.(NY)50(1968)411. [ 91 L. Rikus and H.V. von Geramb, Nucl. Phys. A 426 (1984) 496. [lo] M. Thies, Phys. Lett. B 162 (1985) 255.