- Email: [email protected]
Virtual breakup to spin-singlet states in elastic scattering of deuterons from nuclei
__ __ l!!B
*H J-L
ELSEVIER
20 July 1995 PHYSICS
LETTERS
B
Physics Letters B 354 (1995) 183-188
Virtual breakup to spin-singlet states in elastic scattering of deuterons Tom nuclei Y. Iseria, M. Tanifuji b a Department of Physics,Chiba-Keizai College, Chiba 263, Japan b Department of Physics, Hosei University, Tokyo 102, Japan
Received 13 February 1995; revised manuscript received 30 May 1995 Editor: C. Mahaux
Abstract Contributions to cross sections and analyzing powers from virtual breakup of deuterons to spin-singlet states are quantitatively investigated by the CDCC method in elastic scattering of deuterons at low and intermediate energies. Their effects are selectively remarkable in the tensor analyzing power A,. The invariant-amplitude method shows that the singlet-state breakup contribution behaves like TL-type tensor interactions in the scattering amplitude and the observables. Keywords: Deuteron elastic scattering; Deuteron breakup; Singlet-state breakup; CDCC method; Invariant amplitude method; Effective
tensor interaction
Elastic scattering of deuterons from nuclei provides fundamental information on mechanisms of reactions by composite projectiles. From such view points, attention has been called to deuteron virtual breakup processes; that is, spin-triplet states in the p-n continuum are taken into account as the components which are coupled to the ground state by the coupled-discretized-continuum-channels (CDCC) method and their effects on the elastic-scattering observables have been extensively studied [ l-31. These studies have clarified the indispensable role of the virtual breakup process in analyses of experimental data. Recently, a calculation which uses the adiabatic approximation within the two-step model has found important effects of the breakup to spin-singlet states on the tensor analyzing power A,, in intermediateenergy scattering [ 41. This inspires sophisticated calculations which include the singlet-state breakup and the triplet-state one on an equal-footing, for more
quantitative examination of the effect. The present paper reports the result of the CDCC calculation on the deuteron elastic scattering, which takes account of the deuteron breakup to spin-singlet continuum states as well as that to spin-triplet ones in a multi-step way. The calculation is performed in a wide energy range, Td = 700,400, 200 and 80 MeV, where the target nucleus is fixed to 58Ni to see the energy dependence of the effect of the singlet-state breakup. The deuteron-nucleus interaction is obtained by folding proton-nucleus and neutron-nucleus optical potentials. The central parts of the nucleon potentials produce the central and tensor interactions of the deuteron, which excite the deuteron to the spin-triplet states [ 1,2]. The spin-orbit parts induce the transitions to both the spin-triplet and spin-singlet states as will be discussed below. The change of the variables from rP and r,, the nucleon-nucleus relative coordinates, to R and p, the
0370-2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDlO370-2693(95)00679-6
Y. ken’. hf. Tanifui/ Physics L&ten B 354 (1995) 183-188
184
deuteron center of mass and p-n relative coordinates, transforms the nucleon-nucleus spin-orbit potentials into
veff
L =
h(R)
x{-p+fL.S++([[email protected]+@Sil*)}~ (4)
Uso(r pA
P ) e P .u
= $J+ +$J_
P
+ Uso(r nA
(L+e).s+
+
n ) e,.o +U_
(L+e).s_+fu+
”
(RxV,
with
+
ipxV,&S+
h.(R) =
(RxV,+$pxV,&S_, (1)
&
s
(~s_O’~‘ppo)~ dp,
and the U+R x V, . S- interaction produces which includes the TR-type tensor potential
L&r R
where v,ff~=b(R) x { $R2 + (VW which are expanded
U+(R,p)=4?rc
into the multipoles
vR(R)
@ G(P)loo>
and S& = ;
is+ @ S+lz) },
(5)
with
(-)” A &iX
x uS;O’*)(R,p) [r,(R)
Rl2.
(3)
The orbital angular momenta L and 1 correspond to the coordinates R and p, respectively. The quantity S+ is the usual deuteron spin operator and S_ induces the transitions between the spin-triplet states and the spin-singlet ones. To save computer time, we will adopt the following simplifications. The term of U+px VR.S_ is neglected because its contributions to observables are estimated to be small[ 51. In the U_ (L + 1) . S_ term, f2 is neglected, because effective values of L are mostly larger than e and then the e-term will give small contributions. The contribution of the U+R x V, . S- term is taken into account in the form of an effective potential by the prescription based on the adiabatic closure approximation, which is discussed below. After such simplifications, we take only the U-L. S- term as the coupling interaction between the triplet states and the singlet ones. In the adiabatic closure approximations with the two-step model, the virtual breakup due to the interactions which include S- in Eq. (1) produces effective potentials [ 51. TheU-L.S- interaction produces V& L which includes the TL-type tensor potential
=
&+;(“)~)2&.x
In the derivations of (4) and (5)) ‘PI and ‘So waves in the continuum are assumed, respectively, for the intermediate states and the 3S1 wave is assumed for the ground state, the wave function of which is denoted by rpa. By reference to the investigations in Refs. 14,6], it will be plausible to assume the dominant contribution of the singlet-state breakup to be the T, tensor potential type. Thus one can treat the U+R x V, .S_ term in an approximate way in the sense of the correction term. In the present calculation, we will include the effective potential V& R, as the contribution of the U+R x V, . S- term, with the further assumption that the two-step model describes the essence of the contribution when E is chosen properly. Then E is determined carefully as follows. First, we perform the CDCC calculation which includes the breakup to ‘PI states due to the U_ L . S_ coupling interaction, neglecting the breakup to triplet states. Secondly, we carry out the calculation which includes the effective potential V& L by Eq. (4) as the ‘PI breakup effect, instead of the coupled-channel calculation by the use of the U_ L. S- interaction. As will be seen later, the effect of the breakup to singlet states appears remarkably in the analyzing power A,.Then E is determined so that the second calculation simulate A, obtained by the first calculation. Then the final CDCC calculations contain the singlet-state breakup effects in two ways; (i) the effective potential V, R by (5) with E determined above, as the contribution of the U+R x V,. S-
Y. Iseri, M. Tanifuji/Physics
term, and (ii) the coupling between the triplet states and the singlet ones due to the U-L .S- interaction. In the following, U,“: is assumed to have the same strength and geometry as Uzz and then the A components of U+ (U- ) vanish for odd (even) A’s because of the property of Y, (13) ; that is, effective triplet (singlet) states in the continuum are limited to even( odd)-parity states for the U+S+ (U-S-) interactions. For example, the U_ L .S_ term does not excite the deuteron to the ‘SO states. Because of the limited ability of the computer, the breakup states to be coupled in the calculation are restricted to the 3St and 3D~ (J =1,2,3) waves as the triplet states and the ‘Pi wave as the singlet one. The wave functions of these breakup states and that of the ground state which contains the D-wave component are supplied by the Reid soft-core potential. The maximum p-n relative momentum of the continuum, kmax, is taken as k,,, = 1.O fm-’ and the continuum is discretized into two bins at Td = 700, 400 and 200 MeV and four bins at Td = 80 MeV. Further details of the discretization are referred to in Ref. [ 11. The nucleon-nucleus optical potential is assumed to be the effective Schrodinger form of the so-called Dirac phenomenology for Td = 700 and 400 MeV and the standard Woods-Saxon shape potential for Td = 200 and 80 MeV The parameters of these potentials are given in Ref. [ 11. The Rutherford ratio of the cross section U/CR, the vector analyzing power A,, the tensor analyzing power A,, and the X2 parameter (for 80 MeV) obtained by the present calculation are displayed in Figs. 1 and 2 for Td = 400 MeV (intermediate energy) and 80 MeV (low energy), respectively. The solid lines include the virtual breakup to both of the triplet and singlet states, the dashed ones the breakup to the triplet state only and the dotted lines neglect both breakup effects. At Td = 400 MeV, the breakup to the triplet states produces remarkable changes in the cross section at 0 2 15’ but relatively small ones in the analyzing powers, while the breakup to the singlet states gives large contributions to the tensor analyzing power A, but very small ones to the cross section and the vector analyzing power A,. This nature of the singletstate contribution is very similar to that observed in the adiabatic two-step model calculations [ 41. However, the magnitude of the contribution is somewhat overestimated in Ref. [ 41, as has been pointed out in
Letters B 354 (1995) 183-188
185
Ref. [71. In the present calculation, the contribution of the U+ R x V, . S_ term turns out to be quite small and thus the dominant singlet-state effect arises from the U-L .S_ term. The characteristic feature of the breakup contributions at Td = 80 MeV is qualitatively similar to the one at Td = 400 MeV, that is, the breakup to the triplet states gives a large contribution to the cross section, while the one to the singlet states produces remarkable effects on the analyzing power A,. The present calculations are compared with the experimental data [8,9] in these figures. In the cross section, the contribution of the singlet-state breakup has a tendency to improve the theoretical fit to the data, particularly at 80 MeV, although the magnitude of the contribution is very small as was discussed already. To get better fits to the data, one should examine other nucleon-nucleus optical potentials as input, since such examinations were successful in the triplet-state breakup calculation at Td = 56 MeV [ 21. Further there are ambiguities in the choice between the Woods-Saxon type and Dirac-phenomenology one. These problems are left for future studies. Contrary to the minor effects in the cross section, the singletstate breakup improves the agreement with the data of A,,,, remarkably. The improvements are clearly seen around 8 = 25” at Td = 400 MeV and at most angles larger than 45’ at Td = 80 MeV. In the latter case, the singlet-state contribution overcomes the triplet-state one with the opposite sign, improving the fit to the data quite successfully. The agreement with the data of the vector analyzing power A, is also improved by the singlet-state breakup at both incident energies though the magnitude of the contribution is small. In order to understand the contribution of the virtual breakup in terms of the effective interactions, the scattering amplitudes are decomposed into the spin-spacetensor amplitudes classified according to the tensorial character in the spin space [ 61, by the invariantamplitude method [ lo]. The scalar amplitude denoted by U, describes the scattering by the central interaction, the vector amplitude S denotes the scattering by the spin-orbit interaction, and the tensor quantities T, and Tp denote the scattering by the TR-type and TLtype tensor interactions, respectively, in the sense of the effective interactions. Fig. 3 shows the magnitudes of these amplitudes for Td = 400 and 80 MeV. At Td = 400 MeV, the triplet-state breakup modifies considerably all amplitudes IUI, ISI, (T,I and ITpI, while the
Y. Iseri, M. Tanifuji/ Physics Letters B 354 (1995) 183-188
186
3 :
1.0
j 0.5
: 0.0
10-l
\\ 10-2
i
[
1o-3()
-------folding ----CDCC(trip.) -CDCC(trip.+sing.) I
10
I
206,,
.0.5
“I__.\ _ ‘sIIy _0.0 I.
I
I
(degr:;
0
10
I
I
I
206cm (&gE)
Fig. I. Rutherford ratio of (a) cross section U/Q and (b) vector (Ay) and tensor (Ayy) analyzing powers in 58Ni(d, d)58Ni at Td = 400 MeV. The dotted lines describe the simple folding-model calculation without the breakup effects. The dashed and solid lines include the virtual breakup to the triplet states and that to both the triplet and singlet states, respectively. The experimental data are taken from Ref. [ 8 1,
58Ni(&d)58Ni
Fig. 2. Rutherford ratio of (a) cross section (T/UR and (b) vector (Ay) and tensor (Ayy) analyzing powers in 58Ni(d: d)5sNi 80 MeV. The definitions of the lines are referred to in the caption to Fig. I. The experimental data am taken from Ref. [ 91.
at Td =
Y. Iseri, h4. Tanifuii/ Physics Letters B 354 (1995) 183-188
Fig. 3. Magnitudes of spin-space tensor amplitudes, ICJI, ISI, ITnI and ITpI in ‘*Ni(z, MeV. The values of 1111and /ToI are multiplied by a factor of 10 and 0.1, respectively. caption to Fig. 1.
singlet-state breakup produces large effects on ITal but not on It& ISI and IT,/ in most angles. From such a behavior of the amplitudes, one will see that the main contribution from the singlet-state breakup is of a nature similar to that of the TL-type tensor interaction and thus the derivation of the effective potential (4) for the U-L . S_ interaction is qualitatively justified. This also implies that, for practical purposes, the effect of the singlet-state breakup on the elastic scattering can approximately be described by the TL-type local potential with a suitable depth. Similar features of the breakup effects are observed at Td = 80 MeV, as illustrated in Fig. 3b. With the help of the spin-space-tensor amplitudes, one can understand the characteristics of the contributions of the virtual breakup to the observables. Since IU] is the largest among the amplitudes, it will be reasonable to keep the spin-dependent amplitudes in their first order in the observables for qualitative examinations. Then we get [6] (+ P ]U]2/9, A? N 2fiIm(
(6) US*> / 9~7,
(7)
187
d)‘ENi at (a) Td = 400 MeV and (b) Td = 80 The definitions of the lines are referred to in the
-8Re(U$)
2Jz
+ aRe(UT,*) >/
9~
(8) As was discussed above, the breakup to the singlet states mainly contributes to ITpI. In Eqs. (6)-( 8)) Tp is included in A, but not in (+ and A, and therefore the dominant contribution of the singlet-state breakup appears selectively in A,,. By contrast, the triplet-state breakup contributes to IUI, ISI, IT,/ and ITpI and thus does affect all of the observables u, Ay and A,,. The effects of the breakup to the singlet states at Td = 700 and 200 MeV have similar natures to those at Td = 400 MeV but the magnitude of the contributions to the observables are considerably smaller. The reason will be speculated as follows. The dominant contributions of the singlet-state breakup are almost proportional to L2 because of its TL-type nature. This makes their effect larger for higher energies. On the other hand, experience in the studies of the triplet-state breakup [ 1,2] suggests that the virtual-breakup effect itself is larger at lower energies. These two elements with opposite characters will compete with each other and, as a result, produce the rather complicated energy
188
Y.Iseri, M. Tanifuji / Physics Letters B 354 (1995) 183-l 88
dependence of the singlet-state breakup effect. Further details of the results at Td = 700 and 200 MeV will be given elsewhere. The present work is the first attempt on a coupledchannel calculation of deuteron elastic scattering which includes the spin-singlet states as well as the spin-triplet ones in the p-n continuum, as the coupled component. The contributions of the virtual breakup to the spin-singlet states are shown to be important at incident energies both of 400 MeV and 80 MeV. They are quite similar to those of the TL-type tensor interaction, as has been suggested in Refs. [4,6], and affect selectively the tensor analyzing power A,. These features of the singlet-state breakup contributions are reasonably understood by the invariant-amplitude method. In the calculation the singlet P wave in the continuum is explicitly coupled to the ground state which contains the D-state admixture. The contribution of the singlet S wave is taken into account in the form of the effective potential, instead of including in the channel coupling. At present, the difference in shape and magnitude between l_J$ and lJ:z are neglected. However, when these are taken into account in the calculation, they will be another source of breakup to the singlet S states. Effects of such a breakup have been discussed for (d, p) reactions [ 11 I. From Eqs. (2) and (4)) their effects on the elastic scattering will be similar to those of the TL-type tensor potential. In an earlier calculation [ 21 which includes the breakup to 3St and 3D~ (J = 1,2,3) waves, the contribution of G waves has been examined and found to be small.
This suggests the effects of other singlet states of higher angular momenta to be small. However, it is desirable to examine these effects numerically, for example in the case of the F wave. Such an investigation will be attempted in the future.
References [ 1 ] M. Yahiro, Y. Iseri, H. Kameyama, M. Kamimura and M. Kawai, Prog. Theor. Phys. Suppl. 89 (1986) 32; N. Austem, Y. Iseri, M. Kamimura, M. Kawai, G. Rawitscher and M. Yahiro, Phys. Rep. 154 (1987) 125. 121 Y. Iseri, H. Kameyama, M. Kamimura, M. Yahiro and M. Tanifuji, Nucl. Phys. A 490 (1988) 383. [3] Y. Iseri, M. Tanifuji, Y. Aoki and M. Kawai, Phys. Len. B 265 (1991) 207. [4] J.S. Al-Khalili, J.A. Tostevin and R.C. Johnson, Phys. Rev. C 41 (1990) R806; Nucl. Phys. A 514 (1990) 649. [5] M. Tanifuji and Y. Iseri, Prog. Theor. Phys. 87 ( 1992) 247. 161 Y. Iseri, M. Tanifuji, H. Kameyama, M. Kamimura and M. Yahiro, NucI. Phys. A 533 (1991) 574. [7] J.S. Al-Khalili and R.C. Johnson, Nucl. Phys. A 546 (1992) 622. [8] N. van Sen, J. Arvieux, Ye. Yanlin, G. Gaillard, B. Bonin, A. Boudard, G. Bmge, J.C. Lugol, R. Babinet, T. Hasegawa, E Soga, J.M. Cameron, G.C. Nailson and D.M. Sheppard, Phys. I&t. B 156 (1985) 185; T. Hasegawa, private communications. 191 E.J. Stephenson, J.C. Collins, C.C. Foster, D.L. Friesel, W.W. Jacobs, W.P Jones, M.D. Kaitchuck and I? Schwandt, Phys. Rev. C 28 (1983) 134; E.J. Stephenson, private communication. [lOI M. Tanifuji and K. Yazaki, Prog. Theor. Phys. 40 (1968) 1023. 1111 J.D. Harvey and R.C. Johnson, J. Phys. A 7 (1974) 2017.
*H J-L
ELSEVIER
20 July 1995 PHYSICS
LETTERS
B
Physics Letters B 354 (1995) 183-188
Virtual breakup to spin-singlet states in elastic scattering of deuterons Tom nuclei Y. Iseria, M. Tanifuji b a Department of Physics,Chiba-Keizai College, Chiba 263, Japan b Department of Physics, Hosei University, Tokyo 102, Japan
Received 13 February 1995; revised manuscript received 30 May 1995 Editor: C. Mahaux
Abstract Contributions to cross sections and analyzing powers from virtual breakup of deuterons to spin-singlet states are quantitatively investigated by the CDCC method in elastic scattering of deuterons at low and intermediate energies. Their effects are selectively remarkable in the tensor analyzing power A,. The invariant-amplitude method shows that the singlet-state breakup contribution behaves like TL-type tensor interactions in the scattering amplitude and the observables. Keywords: Deuteron elastic scattering; Deuteron breakup; Singlet-state breakup; CDCC method; Invariant amplitude method; Effective
tensor interaction
Elastic scattering of deuterons from nuclei provides fundamental information on mechanisms of reactions by composite projectiles. From such view points, attention has been called to deuteron virtual breakup processes; that is, spin-triplet states in the p-n continuum are taken into account as the components which are coupled to the ground state by the coupled-discretized-continuum-channels (CDCC) method and their effects on the elastic-scattering observables have been extensively studied [ l-31. These studies have clarified the indispensable role of the virtual breakup process in analyses of experimental data. Recently, a calculation which uses the adiabatic approximation within the two-step model has found important effects of the breakup to spin-singlet states on the tensor analyzing power A,, in intermediateenergy scattering [ 41. This inspires sophisticated calculations which include the singlet-state breakup and the triplet-state one on an equal-footing, for more
quantitative examination of the effect. The present paper reports the result of the CDCC calculation on the deuteron elastic scattering, which takes account of the deuteron breakup to spin-singlet continuum states as well as that to spin-triplet ones in a multi-step way. The calculation is performed in a wide energy range, Td = 700,400, 200 and 80 MeV, where the target nucleus is fixed to 58Ni to see the energy dependence of the effect of the singlet-state breakup. The deuteron-nucleus interaction is obtained by folding proton-nucleus and neutron-nucleus optical potentials. The central parts of the nucleon potentials produce the central and tensor interactions of the deuteron, which excite the deuteron to the spin-triplet states [ 1,2]. The spin-orbit parts induce the transitions to both the spin-triplet and spin-singlet states as will be discussed below. The change of the variables from rP and r,, the nucleon-nucleus relative coordinates, to R and p, the
0370-2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDlO370-2693(95)00679-6
Y. ken’. hf. Tanifui/ Physics L&ten B 354 (1995) 183-188
184
deuteron center of mass and p-n relative coordinates, transforms the nucleon-nucleus spin-orbit potentials into
veff
L =
h(R)
x{-p+fL.S++([[email protected]+@Sil*)}~ (4)
Uso(r pA
P ) e P .u
= $J+ +$J_
P
+ Uso(r nA
(L+e).s+
+
n ) e,.o +U_
(L+e).s_+fu+
”
(RxV,
with
+
ipxV,&S+
h.(R) =
(RxV,+$pxV,&S_, (1)
&
s
(~s_O’~‘ppo)~ dp,
and the U+R x V, . S- interaction produces which includes the TR-type tensor potential
L&r R
where v,ff~=b(R) x { $R2 + (VW which are expanded
U+(R,p)=4?rc
into the multipoles
vR(R)
@ G(P)loo>
and S& = ;
is+ @ S+lz) },
(5)
with
(-)” A &iX
x uS;O’*)(R,p) [r,(R)
Rl2.
(3)
The orbital angular momenta L and 1 correspond to the coordinates R and p, respectively. The quantity S+ is the usual deuteron spin operator and S_ induces the transitions between the spin-triplet states and the spin-singlet ones. To save computer time, we will adopt the following simplifications. The term of U+px VR.S_ is neglected because its contributions to observables are estimated to be small[ 51. In the U_ (L + 1) . S_ term, f2 is neglected, because effective values of L are mostly larger than e and then the e-term will give small contributions. The contribution of the U+R x V, . S- term is taken into account in the form of an effective potential by the prescription based on the adiabatic closure approximation, which is discussed below. After such simplifications, we take only the U-L. S- term as the coupling interaction between the triplet states and the singlet ones. In the adiabatic closure approximations with the two-step model, the virtual breakup due to the interactions which include S- in Eq. (1) produces effective potentials [ 51. TheU-L.S- interaction produces V& L which includes the TL-type tensor potential
=
&+;(“)~)2&.x
In the derivations of (4) and (5)) ‘PI and ‘So waves in the continuum are assumed, respectively, for the intermediate states and the 3S1 wave is assumed for the ground state, the wave function of which is denoted by rpa. By reference to the investigations in Refs. 14,6], it will be plausible to assume the dominant contribution of the singlet-state breakup to be the T, tensor potential type. Thus one can treat the U+R x V, .S_ term in an approximate way in the sense of the correction term. In the present calculation, we will include the effective potential V& R, as the contribution of the U+R x V, . S- term, with the further assumption that the two-step model describes the essence of the contribution when E is chosen properly. Then E is determined carefully as follows. First, we perform the CDCC calculation which includes the breakup to ‘PI states due to the U_ L . S_ coupling interaction, neglecting the breakup to triplet states. Secondly, we carry out the calculation which includes the effective potential V& L by Eq. (4) as the ‘PI breakup effect, instead of the coupled-channel calculation by the use of the U_ L. S- interaction. As will be seen later, the effect of the breakup to singlet states appears remarkably in the analyzing power A,.Then E is determined so that the second calculation simulate A, obtained by the first calculation. Then the final CDCC calculations contain the singlet-state breakup effects in two ways; (i) the effective potential V, R by (5) with E determined above, as the contribution of the U+R x V,. S-
Y. Iseri, M. Tanifuji/Physics
term, and (ii) the coupling between the triplet states and the singlet ones due to the U-L .S- interaction. In the following, U,“: is assumed to have the same strength and geometry as Uzz and then the A components of U+ (U- ) vanish for odd (even) A’s because of the property of Y, (13) ; that is, effective triplet (singlet) states in the continuum are limited to even( odd)-parity states for the U+S+ (U-S-) interactions. For example, the U_ L .S_ term does not excite the deuteron to the ‘SO states. Because of the limited ability of the computer, the breakup states to be coupled in the calculation are restricted to the 3St and 3D~ (J =1,2,3) waves as the triplet states and the ‘Pi wave as the singlet one. The wave functions of these breakup states and that of the ground state which contains the D-wave component are supplied by the Reid soft-core potential. The maximum p-n relative momentum of the continuum, kmax, is taken as k,,, = 1.O fm-’ and the continuum is discretized into two bins at Td = 700, 400 and 200 MeV and four bins at Td = 80 MeV. Further details of the discretization are referred to in Ref. [ 11. The nucleon-nucleus optical potential is assumed to be the effective Schrodinger form of the so-called Dirac phenomenology for Td = 700 and 400 MeV and the standard Woods-Saxon shape potential for Td = 200 and 80 MeV The parameters of these potentials are given in Ref. [ 11. The Rutherford ratio of the cross section U/CR, the vector analyzing power A,, the tensor analyzing power A,, and the X2 parameter (for 80 MeV) obtained by the present calculation are displayed in Figs. 1 and 2 for Td = 400 MeV (intermediate energy) and 80 MeV (low energy), respectively. The solid lines include the virtual breakup to both of the triplet and singlet states, the dashed ones the breakup to the triplet state only and the dotted lines neglect both breakup effects. At Td = 400 MeV, the breakup to the triplet states produces remarkable changes in the cross section at 0 2 15’ but relatively small ones in the analyzing powers, while the breakup to the singlet states gives large contributions to the tensor analyzing power A, but very small ones to the cross section and the vector analyzing power A,. This nature of the singletstate contribution is very similar to that observed in the adiabatic two-step model calculations [ 41. However, the magnitude of the contribution is somewhat overestimated in Ref. [ 41, as has been pointed out in
Letters B 354 (1995) 183-188
185
Ref. [71. In the present calculation, the contribution of the U+ R x V, . S_ term turns out to be quite small and thus the dominant singlet-state effect arises from the U-L .S_ term. The characteristic feature of the breakup contributions at Td = 80 MeV is qualitatively similar to the one at Td = 400 MeV, that is, the breakup to the triplet states gives a large contribution to the cross section, while the one to the singlet states produces remarkable effects on the analyzing power A,. The present calculations are compared with the experimental data [8,9] in these figures. In the cross section, the contribution of the singlet-state breakup has a tendency to improve the theoretical fit to the data, particularly at 80 MeV, although the magnitude of the contribution is very small as was discussed already. To get better fits to the data, one should examine other nucleon-nucleus optical potentials as input, since such examinations were successful in the triplet-state breakup calculation at Td = 56 MeV [ 21. Further there are ambiguities in the choice between the Woods-Saxon type and Dirac-phenomenology one. These problems are left for future studies. Contrary to the minor effects in the cross section, the singletstate breakup improves the agreement with the data of A,,,, remarkably. The improvements are clearly seen around 8 = 25” at Td = 400 MeV and at most angles larger than 45’ at Td = 80 MeV. In the latter case, the singlet-state contribution overcomes the triplet-state one with the opposite sign, improving the fit to the data quite successfully. The agreement with the data of the vector analyzing power A, is also improved by the singlet-state breakup at both incident energies though the magnitude of the contribution is small. In order to understand the contribution of the virtual breakup in terms of the effective interactions, the scattering amplitudes are decomposed into the spin-spacetensor amplitudes classified according to the tensorial character in the spin space [ 61, by the invariantamplitude method [ lo]. The scalar amplitude denoted by U, describes the scattering by the central interaction, the vector amplitude S denotes the scattering by the spin-orbit interaction, and the tensor quantities T, and Tp denote the scattering by the TR-type and TLtype tensor interactions, respectively, in the sense of the effective interactions. Fig. 3 shows the magnitudes of these amplitudes for Td = 400 and 80 MeV. At Td = 400 MeV, the triplet-state breakup modifies considerably all amplitudes IUI, ISI, (T,I and ITpI, while the
Y. Iseri, M. Tanifuji/ Physics Letters B 354 (1995) 183-188
186
3 :
1.0
j 0.5
: 0.0
10-l
\\ 10-2
i
[
1o-3()
-------folding ----CDCC(trip.) -CDCC(trip.+sing.) I
10
I
206,,
.0.5
“I__.\ _ ‘sIIy _0.0 I.
I
I
(degr:;
0
10
I
I
I
206cm (&gE)
Fig. I. Rutherford ratio of (a) cross section U/Q and (b) vector (Ay) and tensor (Ayy) analyzing powers in 58Ni(d, d)58Ni at Td = 400 MeV. The dotted lines describe the simple folding-model calculation without the breakup effects. The dashed and solid lines include the virtual breakup to the triplet states and that to both the triplet and singlet states, respectively. The experimental data are taken from Ref. [ 8 1,
58Ni(&d)58Ni
Fig. 2. Rutherford ratio of (a) cross section (T/UR and (b) vector (Ay) and tensor (Ayy) analyzing powers in 58Ni(d: d)5sNi 80 MeV. The definitions of the lines are referred to in the caption to Fig. I. The experimental data am taken from Ref. [ 91.
at Td =
Y. Iseri, h4. Tanifuii/ Physics Letters B 354 (1995) 183-188
Fig. 3. Magnitudes of spin-space tensor amplitudes, ICJI, ISI, ITnI and ITpI in ‘*Ni(z, MeV. The values of 1111and /ToI are multiplied by a factor of 10 and 0.1, respectively. caption to Fig. 1.
singlet-state breakup produces large effects on ITal but not on It& ISI and IT,/ in most angles. From such a behavior of the amplitudes, one will see that the main contribution from the singlet-state breakup is of a nature similar to that of the TL-type tensor interaction and thus the derivation of the effective potential (4) for the U-L . S_ interaction is qualitatively justified. This also implies that, for practical purposes, the effect of the singlet-state breakup on the elastic scattering can approximately be described by the TL-type local potential with a suitable depth. Similar features of the breakup effects are observed at Td = 80 MeV, as illustrated in Fig. 3b. With the help of the spin-space-tensor amplitudes, one can understand the characteristics of the contributions of the virtual breakup to the observables. Since IU] is the largest among the amplitudes, it will be reasonable to keep the spin-dependent amplitudes in their first order in the observables for qualitative examinations. Then we get [6] (+ P ]U]2/9, A? N 2fiIm(
(6) US*> / 9~7,
(7)
187
d)‘ENi at (a) Td = 400 MeV and (b) Td = 80 The definitions of the lines are referred to in the
-8Re(U$)
2Jz
+ aRe(UT,*) >/
9~
(8) As was discussed above, the breakup to the singlet states mainly contributes to ITpI. In Eqs. (6)-( 8)) Tp is included in A, but not in (+ and A, and therefore the dominant contribution of the singlet-state breakup appears selectively in A,,. By contrast, the triplet-state breakup contributes to IUI, ISI, IT,/ and ITpI and thus does affect all of the observables u, Ay and A,,. The effects of the breakup to the singlet states at Td = 700 and 200 MeV have similar natures to those at Td = 400 MeV but the magnitude of the contributions to the observables are considerably smaller. The reason will be speculated as follows. The dominant contributions of the singlet-state breakup are almost proportional to L2 because of its TL-type nature. This makes their effect larger for higher energies. On the other hand, experience in the studies of the triplet-state breakup [ 1,2] suggests that the virtual-breakup effect itself is larger at lower energies. These two elements with opposite characters will compete with each other and, as a result, produce the rather complicated energy
188
Y.Iseri, M. Tanifuji / Physics Letters B 354 (1995) 183-l 88
dependence of the singlet-state breakup effect. Further details of the results at Td = 700 and 200 MeV will be given elsewhere. The present work is the first attempt on a coupledchannel calculation of deuteron elastic scattering which includes the spin-singlet states as well as the spin-triplet ones in the p-n continuum, as the coupled component. The contributions of the virtual breakup to the spin-singlet states are shown to be important at incident energies both of 400 MeV and 80 MeV. They are quite similar to those of the TL-type tensor interaction, as has been suggested in Refs. [4,6], and affect selectively the tensor analyzing power A,. These features of the singlet-state breakup contributions are reasonably understood by the invariant-amplitude method. In the calculation the singlet P wave in the continuum is explicitly coupled to the ground state which contains the D-state admixture. The contribution of the singlet S wave is taken into account in the form of the effective potential, instead of including in the channel coupling. At present, the difference in shape and magnitude between l_J$ and lJ:z are neglected. However, when these are taken into account in the calculation, they will be another source of breakup to the singlet S states. Effects of such a breakup have been discussed for (d, p) reactions [ 11 I. From Eqs. (2) and (4)) their effects on the elastic scattering will be similar to those of the TL-type tensor potential. In an earlier calculation [ 21 which includes the breakup to 3St and 3D~ (J = 1,2,3) waves, the contribution of G waves has been examined and found to be small.
This suggests the effects of other singlet states of higher angular momenta to be small. However, it is desirable to examine these effects numerically, for example in the case of the F wave. Such an investigation will be attempted in the future.
References [ 1 ] M. Yahiro, Y. Iseri, H. Kameyama, M. Kamimura and M. Kawai, Prog. Theor. Phys. Suppl. 89 (1986) 32; N. Austem, Y. Iseri, M. Kamimura, M. Kawai, G. Rawitscher and M. Yahiro, Phys. Rep. 154 (1987) 125. 121 Y. Iseri, H. Kameyama, M. Kamimura, M. Yahiro and M. Tanifuji, Nucl. Phys. A 490 (1988) 383. [3] Y. Iseri, M. Tanifuji, Y. Aoki and M. Kawai, Phys. Len. B 265 (1991) 207. [4] J.S. Al-Khalili, J.A. Tostevin and R.C. Johnson, Phys. Rev. C 41 (1990) R806; Nucl. Phys. A 514 (1990) 649. [5] M. Tanifuji and Y. Iseri, Prog. Theor. Phys. 87 ( 1992) 247. 161 Y. Iseri, M. Tanifuji, H. Kameyama, M. Kamimura and M. Yahiro, NucI. Phys. A 533 (1991) 574. [7] J.S. Al-Khalili and R.C. Johnson, Nucl. Phys. A 546 (1992) 622. [8] N. van Sen, J. Arvieux, Ye. Yanlin, G. Gaillard, B. Bonin, A. Boudard, G. Bmge, J.C. Lugol, R. Babinet, T. Hasegawa, E Soga, J.M. Cameron, G.C. Nailson and D.M. Sheppard, Phys. I&t. B 156 (1985) 185; T. Hasegawa, private communications. 191 E.J. Stephenson, J.C. Collins, C.C. Foster, D.L. Friesel, W.W. Jacobs, W.P Jones, M.D. Kaitchuck and I? Schwandt, Phys. Rev. C 28 (1983) 134; E.J. Stephenson, private communication. [lOI M. Tanifuji and K. Yazaki, Prog. Theor. Phys. 40 (1968) 1023. 1111 J.D. Harvey and R.C. Johnson, J. Phys. A 7 (1974) 2017.