- Email: [email protected]
A test of a model t-matrix for direct reaction inelastic scattering
Volume 86B, number 2
PHYSICS LETTERS
24 September 1979
A TEST OF A MODEL t-MATRIX FOR DIRECT REACTION INELASTIC SCATTERING ~
M. FOX and I. MORRISON School of Physics, University o f Melbourne, Parkville, Victoria 3052, Australia and K. AMOS 1 and D. WEISS Physics Department, Indiana University, Bloomington, IN 47401, USA Received 4 December 1978
Differential cross sections and analyzing powers from the inelastic scattering of 65 MeV protons leading to the 1+ T = 0 (12.71 MeV) and 1÷ T = 1 (15.11 MeV) states in 12C have been analysed as tests of a model two-nucleon t-matrix.
Inelastic proton scattering from nuclei can be a most useful probe of nuclear structure and one which complements the test provided by electromagnetic excitation of, or transitions between, the same nuclear states [1 ]. The latter processes, such as inelastic electron scattering, provide information primarily about proton distributions and transitions in nuclei, although in particular cases new measurements of the transverse form factors may be quite sensitive to magnetization currents and hence neutron distributions [2]. Proton scattering analyses, on the other hand, are most influenced by the neutron distribution and transitions in accord with the properties of the inter-nucleon force. To be a useful probe of nuclear structure, however, all pertinent details of the reaction mechanism responsible for inelastic proton scattering must be understood and specified for use in any calculation. A number of studies [1,3,4] have shown that, predicated upon a distorted wave approximation for direct reaction inelastic proton scattering, a two-nucleon t-matrix (usually represented by an effective two-nucleon interaction) requires corrections for core polarization effects [5] (especially for natural parity transitions) Research supported by a grant from the Australian Research Grants Committee. 1 On leave from the School of Physics, University of Melbourne.
and for virtual excitation of giant resonances [3]. The latter higher-order processes are very strongly (projectile) energy dependent whence careful selection of data for analysis ( > 40 MeV protons for most nuclei) abrogates any need to allow for such resonance effects. Likewise a careful choice of reactions to be studied minimises the role to be played by core polarization corrections. Of course, with the appropriate nuclear structure calculations, even the so called "collective" transitions can be analyzed without any obvious need for core polarization corrections [6]. Even so, one must then evaluate numerous amplitudes * x, so numer. ous usually that sensitivity to details of the chosen nuclear t-matrix is lost. This should not be the case with unnatural parity transitions. For such reactions, relatively few spectroscopic amplitudes (transition density matrix elements) are involved [3,7] and the usual core polarization corrections [5] are not required. Indeed, studies of inelastic proton scattering to unnatural parity states have ascertained that the appropriate twonucleon t-matrix should have central, tensor and twobody spin--orbit force components and should also be complex [4]. A recent study [8] of elastic and inelastic scattering suggests that the t-matrix should be energy and momentum dependent as well. To date, however, tests o f model t-matrices, even ~1 This is one definition of collectivity [1,6]. 121
Volume 86B, number 2
PHYSICS.LETTERS
with reactions for which the spectroscopy is well defined [4,7,8] have been based upon analyses of differential cross-section data. Recently, data from the inelastic scattering from 12C of 65 MeV polarized protons was obtained [9], and these data included the analyzing powers as well as the differential cross sections for the excitation of the T = 0 (12.71 MeV) and T = 1 (15.11 MeV) 1+ states. As the spectroscopy of these unnatural parity transitions is well established [3] and involves only a few significant transition density matrix elements and as analyzing powers, by being related to differences between (incoherent) scattering probabilities [7], are sensitive tests of details in any reaction theory, we present herein an analysis of those data seeking new information about the two-nucleon t-matrix for inelastic proton scattering. As starting values we have chosen the effective twonucleon interaction that has been used with some success in analyses of other inelastic scattering data, most of which, however, were obtained from experiments with smaller projectile energies [1,3,4,7]. This effective interaction consists of a central [10], tensor and two-body spin-orbit forces [11 ] (the latter having been used infrequently) and which are VcentraI = - 2 5 exp(-0.272r~2)SS08 T1 -- 47 exp(--O.337r22)SS18TO,
(1)
2 (8 T0 [_105.25 exp(_ 1.086r212 ) Vtensor -_ r12S12 1.948 exp(-0.2417r~2)] + 6 T1 [17.918 exp(--0.7612r22 )
(2)
- 2.3085 exp(-0.5228r~2 )
effective interactions [4] and consequently in most test cases give very similar predictions of inelastic scattering differential cross sections. In the main, however, the data used to ascertain the effects of components in the effective interactions were taken at "low" energies for which the issue was often clouded by such high-order effects as virtual resonance excitations. For higher-energy data, and with the sensitive analyzing powers fine tuning the analyses, we expect that it will be necessary to vary from this basic form for the effective interaction. Indeed, some scaling at least was needed in a previous analysis of 136 MeV data [7]. In these studies we have allowed a scaling of the strengths of each component in the effective interaction and overall phase variations of the tensor and L ' S forces with respect to each other and with the cen~ tral interaction. The resultant t-matrix model is a scaled, complex version of that used previously, and we define it as t(12) = aVcentra 1 + be i° Vtensor + ce i¢ Vspin_orbi t. (4) This effective interaction has been used in antisymmetrized distorted wave approximation calculations [3] for the isoscalar and isovector 1÷ transitions. In these calculations optical model potential parameters as given in ref. [9] were used, as were harmonic oscillator wavefunctions (hw = 17.9 MeV) for the single nucleon bound states and the spectroscopic amplitudes were obtained by a shell model calculation made using the Cohen and Kurath matrix elements [ 12]. These spectroscopic amplitudes are given in table 1. The results are presented in three figures with the differential cross-section data from the inelastic scattering of 65 MeV polarized protons [9] being cornTable 1 Spectroscopic amplitudes, S(jlj 2 ; J i J f ; I), for the 1÷ excitations in 12C (the upper sign is for protons; the lower for neutrons).
+ 0.381 exp(-0.2004r~2)l }, and
Jl~]2
Vspin_orbit = L "S {8 7"0 [-213.91 e x p ( - 1.792r122) ] + 8Tl [--282.41 exp(--1.750r122)
S(jl]2;01;1) T = 0 (12.71 MeV)
T= 1 (15.11 MeV)
+ 0.8802
+ 0.8452
(3) Opa ~ Opl
-- 5.1793 exp(--0.959r122) ] }. These functional forms have Fourier transforms in the important 1 fm -1 to 2 fm -1 region and volume integrals that compare closely with those of other "realistic" 122
24 September 1979
Opl --->OP3
++0.4341
_+0.4157
Opl -~ Opl
-0.0531
+ 0.0712
Op3 ~ Opa
~ 0.0169
_+0.0932
Volume 86B, number 2
PHYSICS LETTERS
pared in fig. 1 with our calculations in which no twob o d y s p i n - o r b i t contributions were allowed. The predicted analyzing powers associated with each of the three results given in fig. 1 are then compared with the data in fig. 2. To complete the study the results obtained when the s p i n - o r b i t interaction is included are compared with the data in fig. 3. In figs. 1 and 2, three results are shown; those d e picted by the dashed curves were obtained using an 100
. ~ \
100
24 September 1979 ~1\\
0.5
d~( mb/sr )
A{O)%
I
-0.5
-12c(~, 6)
65 MeV I* T=I _ (15.11 MeV)
0.5
T: 0
~~11*
(12.71 MeV ) (P,P) 1*T=1 (15.11MeV}
"4_~ I
I
I*T=O (12.71MeV)
~'~.. i
-0.5
,'.,.\
~X-t
lg
I
o'o do
30
60
90
Oc.m.(deg.}
Fig. 3. The differential cross sections and analyzing power resuits for the inelastic scattering of 65 MeV protons displaying the effects of a two-body spin-orbit force. I
30
I
60
I
I
t
30 90 0 c.m. (deg~
60
90
Fig. 1. The differential cross section results for the inelastic scattering of 65 MeV polarized protons leading to the 1+ states in 12C.
6S MeV
12c }
1÷ T=I (15.11 MeV)
( p. p/) I*T=O (12.71MeV)
.5
.. Ale) 0
....' ",}{ {t '1% /
"--.
~\
\ -.5
30
60
90
60
90
0 c.m. (degrees)
Fig. 2. The analyzing power results for the inelastic scattering of 65 MeV polarized protons leading to the 1+ states in 12C. The individual results coincide with those digplayed in fig. 1, and are as specified in the text.
unsealed interaction (a = b = 1, c = ~ = 0 = 0); those depicted b y the solid curve being obtained with values a = b = 1, O = 90 °, c = @ = 0; while those shown b y the dotted curve were obtained b y using a = 1, b = 0.8, 0 = 90 ° and c = 0 = 0. It is quite evident that the isovector excitation differential cross section is quite well reproduced b y all calculations while the isoscalar excitation is not. Overall the continuous curve results (90 ° phase and full tensor weight) give the best simultaneous fit. But these calculations less adequately reproduce the measured analyzing powers. From fig. 2 it is again evident that all three model t-matrices reasonably reproduce the observed structure o f the isovector excitation data but again, and more obviously in this case, none yield predictions that adequately resemble the measured isoscalar excitation data. Nevertheless, the effect o f a complex nature o f the t-matrix is quite dramatic and suggests that such analyzing power data do test this character of the t-matrix. In this regard, therefore, the results presented in fig. 3 are rather disappointing. Shown therein are the results o f using the complete t-matrix model o f eq. (4) in the DWA calculations and allowing the values of b, c, 0 and ¢ to vary 123
Volume 86B, number 2
PHYSICS LETTERS
and to obtain the best simultaneous fit to all of the data. These results, depicted by the solid curves, required parameter values o f a = 1, b = 1.1,c = 1.3, 0 = 180 ° and q~= 0. For comparison, and depicted by the dashed curves, we present the predictions of the unscaled t-matrix (a = b = c = 1 ; 0 = ~b= 0). Clearly the isovector transition data have been very well reproduced. But the isoscalar transition predictions are totally inadequate. The effect of this model t-matrix L "S component has been quite dramatic as comparison between the dashed curve predictions of figs. 1 and 2 with those in fig. 3 reveal, but the effects worsen overall agreement with the data. It seems clear therefore that this model t-matrix is inadequate when detailed aspects of data including spin dependent data are considered. It yields, however, transition strengths of appropriate magnitudes (as in previous analyses of other data) and, in some circumstances, quite good predictions of differential crosssection data. Nevertheless, from these analyses, an improved t-matrix will have to have complex strengths, with the imaginary parts influencing the predictions of analyzing powers with some significance.
References [ 1] K. Amos, A. Faessler, I. Morrison, R. Smith and H. Muether, Nucl. Phys. A304 (1978) 191.
124
24 September 1979
[2] G.A. Peterson, private communication. [3] H.V. Geramb et al., Phys. Rev. C12 (1975) 1697. [4] G. Bertsch, J. Borysowicz, H. McManusand W.G. Love, Nucl. Phys. A284 (1977) 399; R. Smith and K. Amos, Aust. J. Phys., to be published. A. Picldesimer and G.E. Walker, Phys. Rev. C (1978); H.V. Geramb and P.E. Hodgson, Nucl. Phys. A246 (1975) 173; L. Rikus, R. Smith, I. Morrison and K. Amos, Nucl. Phys. A286 (1977) 494. [5] W.G. Love and G.R. Satchler, Nucl. Phys. A172 (1971) 449; V.R. Brown and V.A. Madsen, Phys. Rev. C l l (1975) 1298. [6] P. Nesci and K. Amos, Nucl. Phys. A284 (1977) 239. [7] R. Smith, J. Morton, I. Morrison and K. Amos, Aust. J. Phys. 31 (1978) 1; R. Smith and K. Amos, Phys. Lett. 55B (1975) 162; H.V. Geramb, R. Sprickmann and G.L. Strobel, Nucl. Phys. A199 (1973) 545. [8] F.A. Brieva, H.V. Geramb and R.J. Rook, Phys. Lett. 79B (1978) 177. [9] K. Hosono et al., Phys. Rev. Lett. 41 (1978) 621. [10] C.W. Wong and C.Y. Wong, Nucl. Phys. A91 (1967) 433. [ 11 ] H. Eikemeier and H.H. Hackenbroich, Nucl. Phys. A 169 (1971) 407. [12] S. Cohen and D. Kurath, Nucl. Phys. AI01 (1967) 1.
PHYSICS LETTERS
24 September 1979
A TEST OF A MODEL t-MATRIX FOR DIRECT REACTION INELASTIC SCATTERING ~
M. FOX and I. MORRISON School of Physics, University o f Melbourne, Parkville, Victoria 3052, Australia and K. AMOS 1 and D. WEISS Physics Department, Indiana University, Bloomington, IN 47401, USA Received 4 December 1978
Differential cross sections and analyzing powers from the inelastic scattering of 65 MeV protons leading to the 1+ T = 0 (12.71 MeV) and 1÷ T = 1 (15.11 MeV) states in 12C have been analysed as tests of a model two-nucleon t-matrix.
Inelastic proton scattering from nuclei can be a most useful probe of nuclear structure and one which complements the test provided by electromagnetic excitation of, or transitions between, the same nuclear states [1 ]. The latter processes, such as inelastic electron scattering, provide information primarily about proton distributions and transitions in nuclei, although in particular cases new measurements of the transverse form factors may be quite sensitive to magnetization currents and hence neutron distributions [2]. Proton scattering analyses, on the other hand, are most influenced by the neutron distribution and transitions in accord with the properties of the inter-nucleon force. To be a useful probe of nuclear structure, however, all pertinent details of the reaction mechanism responsible for inelastic proton scattering must be understood and specified for use in any calculation. A number of studies [1,3,4] have shown that, predicated upon a distorted wave approximation for direct reaction inelastic proton scattering, a two-nucleon t-matrix (usually represented by an effective two-nucleon interaction) requires corrections for core polarization effects [5] (especially for natural parity transitions) Research supported by a grant from the Australian Research Grants Committee. 1 On leave from the School of Physics, University of Melbourne.
and for virtual excitation of giant resonances [3]. The latter higher-order processes are very strongly (projectile) energy dependent whence careful selection of data for analysis ( > 40 MeV protons for most nuclei) abrogates any need to allow for such resonance effects. Likewise a careful choice of reactions to be studied minimises the role to be played by core polarization corrections. Of course, with the appropriate nuclear structure calculations, even the so called "collective" transitions can be analyzed without any obvious need for core polarization corrections [6]. Even so, one must then evaluate numerous amplitudes * x, so numer. ous usually that sensitivity to details of the chosen nuclear t-matrix is lost. This should not be the case with unnatural parity transitions. For such reactions, relatively few spectroscopic amplitudes (transition density matrix elements) are involved [3,7] and the usual core polarization corrections [5] are not required. Indeed, studies of inelastic proton scattering to unnatural parity states have ascertained that the appropriate twonucleon t-matrix should have central, tensor and twobody spin--orbit force components and should also be complex [4]. A recent study [8] of elastic and inelastic scattering suggests that the t-matrix should be energy and momentum dependent as well. To date, however, tests o f model t-matrices, even ~1 This is one definition of collectivity [1,6]. 121
Volume 86B, number 2
PHYSICS.LETTERS
with reactions for which the spectroscopy is well defined [4,7,8] have been based upon analyses of differential cross-section data. Recently, data from the inelastic scattering from 12C of 65 MeV polarized protons was obtained [9], and these data included the analyzing powers as well as the differential cross sections for the excitation of the T = 0 (12.71 MeV) and T = 1 (15.11 MeV) 1+ states. As the spectroscopy of these unnatural parity transitions is well established [3] and involves only a few significant transition density matrix elements and as analyzing powers, by being related to differences between (incoherent) scattering probabilities [7], are sensitive tests of details in any reaction theory, we present herein an analysis of those data seeking new information about the two-nucleon t-matrix for inelastic proton scattering. As starting values we have chosen the effective twonucleon interaction that has been used with some success in analyses of other inelastic scattering data, most of which, however, were obtained from experiments with smaller projectile energies [1,3,4,7]. This effective interaction consists of a central [10], tensor and two-body spin-orbit forces [11 ] (the latter having been used infrequently) and which are VcentraI = - 2 5 exp(-0.272r~2)SS08 T1 -- 47 exp(--O.337r22)SS18TO,
(1)
2 (8 T0 [_105.25 exp(_ 1.086r212 ) Vtensor -_ r12S12 1.948 exp(-0.2417r~2)] + 6 T1 [17.918 exp(--0.7612r22 )
(2)
- 2.3085 exp(-0.5228r~2 )
effective interactions [4] and consequently in most test cases give very similar predictions of inelastic scattering differential cross sections. In the main, however, the data used to ascertain the effects of components in the effective interactions were taken at "low" energies for which the issue was often clouded by such high-order effects as virtual resonance excitations. For higher-energy data, and with the sensitive analyzing powers fine tuning the analyses, we expect that it will be necessary to vary from this basic form for the effective interaction. Indeed, some scaling at least was needed in a previous analysis of 136 MeV data [7]. In these studies we have allowed a scaling of the strengths of each component in the effective interaction and overall phase variations of the tensor and L ' S forces with respect to each other and with the cen~ tral interaction. The resultant t-matrix model is a scaled, complex version of that used previously, and we define it as t(12) = aVcentra 1 + be i° Vtensor + ce i¢ Vspin_orbi t. (4) This effective interaction has been used in antisymmetrized distorted wave approximation calculations [3] for the isoscalar and isovector 1÷ transitions. In these calculations optical model potential parameters as given in ref. [9] were used, as were harmonic oscillator wavefunctions (hw = 17.9 MeV) for the single nucleon bound states and the spectroscopic amplitudes were obtained by a shell model calculation made using the Cohen and Kurath matrix elements [ 12]. These spectroscopic amplitudes are given in table 1. The results are presented in three figures with the differential cross-section data from the inelastic scattering of 65 MeV polarized protons [9] being cornTable 1 Spectroscopic amplitudes, S(jlj 2 ; J i J f ; I), for the 1÷ excitations in 12C (the upper sign is for protons; the lower for neutrons).
+ 0.381 exp(-0.2004r~2)l }, and
Jl~]2
Vspin_orbit = L "S {8 7"0 [-213.91 e x p ( - 1.792r122) ] + 8Tl [--282.41 exp(--1.750r122)
S(jl]2;01;1) T = 0 (12.71 MeV)
T= 1 (15.11 MeV)
+ 0.8802
+ 0.8452
(3) Opa ~ Opl
-- 5.1793 exp(--0.959r122) ] }. These functional forms have Fourier transforms in the important 1 fm -1 to 2 fm -1 region and volume integrals that compare closely with those of other "realistic" 122
24 September 1979
Opl --->OP3
++0.4341
_+0.4157
Opl -~ Opl
-0.0531
+ 0.0712
Op3 ~ Opa
~ 0.0169
_+0.0932
Volume 86B, number 2
PHYSICS LETTERS
pared in fig. 1 with our calculations in which no twob o d y s p i n - o r b i t contributions were allowed. The predicted analyzing powers associated with each of the three results given in fig. 1 are then compared with the data in fig. 2. To complete the study the results obtained when the s p i n - o r b i t interaction is included are compared with the data in fig. 3. In figs. 1 and 2, three results are shown; those d e picted by the dashed curves were obtained using an 100
. ~ \
100
24 September 1979 ~1\\
0.5
d~( mb/sr )
A{O)%
I
-0.5
-12c(~, 6)
65 MeV I* T=I _ (15.11 MeV)
0.5
T: 0
~~11*
(12.71 MeV ) (P,P) 1*T=1 (15.11MeV}
"4_~ I
I
I*T=O (12.71MeV)
~'~.. i
-0.5
,'.,.\
~X-t
lg
I
o'o do
30
60
90
Oc.m.(deg.}
Fig. 3. The differential cross sections and analyzing power resuits for the inelastic scattering of 65 MeV protons displaying the effects of a two-body spin-orbit force. I
30
I
60
I
I
t
30 90 0 c.m. (deg~
60
90
Fig. 1. The differential cross section results for the inelastic scattering of 65 MeV polarized protons leading to the 1+ states in 12C.
6S MeV
12c }
1÷ T=I (15.11 MeV)
( p. p/) I*T=O (12.71MeV)
.5
.. Ale) 0
....' ",}{ {t '1% /
"--.
~\
\ -.5
30
60
90
60
90
0 c.m. (degrees)
Fig. 2. The analyzing power results for the inelastic scattering of 65 MeV polarized protons leading to the 1+ states in 12C. The individual results coincide with those digplayed in fig. 1, and are as specified in the text.
unsealed interaction (a = b = 1, c = ~ = 0 = 0); those depicted b y the solid curve being obtained with values a = b = 1, O = 90 °, c = @ = 0; while those shown b y the dotted curve were obtained b y using a = 1, b = 0.8, 0 = 90 ° and c = 0 = 0. It is quite evident that the isovector excitation differential cross section is quite well reproduced b y all calculations while the isoscalar excitation is not. Overall the continuous curve results (90 ° phase and full tensor weight) give the best simultaneous fit. But these calculations less adequately reproduce the measured analyzing powers. From fig. 2 it is again evident that all three model t-matrices reasonably reproduce the observed structure o f the isovector excitation data but again, and more obviously in this case, none yield predictions that adequately resemble the measured isoscalar excitation data. Nevertheless, the effect o f a complex nature o f the t-matrix is quite dramatic and suggests that such analyzing power data do test this character of the t-matrix. In this regard, therefore, the results presented in fig. 3 are rather disappointing. Shown therein are the results o f using the complete t-matrix model o f eq. (4) in the DWA calculations and allowing the values of b, c, 0 and ¢ to vary 123
Volume 86B, number 2
PHYSICS LETTERS
and to obtain the best simultaneous fit to all of the data. These results, depicted by the solid curves, required parameter values o f a = 1, b = 1.1,c = 1.3, 0 = 180 ° and q~= 0. For comparison, and depicted by the dashed curves, we present the predictions of the unscaled t-matrix (a = b = c = 1 ; 0 = ~b= 0). Clearly the isovector transition data have been very well reproduced. But the isoscalar transition predictions are totally inadequate. The effect of this model t-matrix L "S component has been quite dramatic as comparison between the dashed curve predictions of figs. 1 and 2 with those in fig. 3 reveal, but the effects worsen overall agreement with the data. It seems clear therefore that this model t-matrix is inadequate when detailed aspects of data including spin dependent data are considered. It yields, however, transition strengths of appropriate magnitudes (as in previous analyses of other data) and, in some circumstances, quite good predictions of differential crosssection data. Nevertheless, from these analyses, an improved t-matrix will have to have complex strengths, with the imaginary parts influencing the predictions of analyzing powers with some significance.
References [ 1] K. Amos, A. Faessler, I. Morrison, R. Smith and H. Muether, Nucl. Phys. A304 (1978) 191.
124
24 September 1979
[2] G.A. Peterson, private communication. [3] H.V. Geramb et al., Phys. Rev. C12 (1975) 1697. [4] G. Bertsch, J. Borysowicz, H. McManusand W.G. Love, Nucl. Phys. A284 (1977) 399; R. Smith and K. Amos, Aust. J. Phys., to be published. A. Picldesimer and G.E. Walker, Phys. Rev. C (1978); H.V. Geramb and P.E. Hodgson, Nucl. Phys. A246 (1975) 173; L. Rikus, R. Smith, I. Morrison and K. Amos, Nucl. Phys. A286 (1977) 494. [5] W.G. Love and G.R. Satchler, Nucl. Phys. A172 (1971) 449; V.R. Brown and V.A. Madsen, Phys. Rev. C l l (1975) 1298. [6] P. Nesci and K. Amos, Nucl. Phys. A284 (1977) 239. [7] R. Smith, J. Morton, I. Morrison and K. Amos, Aust. J. Phys. 31 (1978) 1; R. Smith and K. Amos, Phys. Lett. 55B (1975) 162; H.V. Geramb, R. Sprickmann and G.L. Strobel, Nucl. Phys. A199 (1973) 545. [8] F.A. Brieva, H.V. Geramb and R.J. Rook, Phys. Lett. 79B (1978) 177. [9] K. Hosono et al., Phys. Rev. Lett. 41 (1978) 621. [10] C.W. Wong and C.Y. Wong, Nucl. Phys. A91 (1967) 433. [ 11 ] H. Eikemeier and H.H. Hackenbroich, Nucl. Phys. A 169 (1971) 407. [12] S. Cohen and D. Kurath, Nucl. Phys. AI01 (1967) 1.