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Measurement of isospin mixing between the 1+ doublet in 12C using pion inelastic scattering
Volume 99B, number 5
PHYSICS LETTERS
5 March 1981
MEASUREMENTS OF ISOSPIN MIXING BETWEEN THE 1+ DOUBLET IN 12C USING PION INELASTIC SCATTERING C.U MORRIS, R.L. BOUDRIE and J. PIFFARETTI x Los Alamos Seientifi'e Laboratory, Los Alamos, NM 87545, USA W.B. COTTINGAME 2, W.J. BRAITHWAITE, S.J. GREENE 2 C.J. HARVEY, D.B. HOLTKAMP 3 and C.Fred MOORE University of Texas, Austin, TX 78712, USA S.J. SEESTROM-MORRIS 4 University of Minnesota, Minneapolis, MN 55455, USA Received 12 November 1980
Pion inelastic scattering has been used to extract the off-diagonal charge dependent matrix element, Hob between the jTr = 1+ doublet of states at 12.71 (T = 0) and 15.11 (T = 1) MeV in 12C. The reported measurements yield a value, Hol = 148 +- 29 keV, in good agreement with recent electromagnetic measurements.
One method of detecting charge dependent components of the strong interaction is to measure the size of the off-diagonal charge dependent matrix elements, HO1 , between analog-antianalog pairs o f states in nuclei [1 ]. The size of these matrix elements can be measured by determining the mixing between states o f otherwise good isospin, T. If a systematic study of isospin mixing can show this mixing to be different than expected from the Coulomb force alone, charge dependence in the strong interaction will have been established. Such studies have been inconclusive because of both experimental and theoretical problems. Experimentally there has been no generally reliable method o f measuring isospin mixing. Measurements made with hadronic ,1 Present address: Schweizerisches Institut ftir Nuklearforschung, CH-5234 Villigen, Switzerland. ,5 Present address: New Mexico State University, Los Cruces, NM 88003, USA. ,3 Present address: University of Minnesota, Minneapolis, MN 55455, USA. ,4 Present address: Los Alamos Scientific Laboratory, Los Alamos, NM 87545, USA.
probes, including both stripping and pickup reactions and isospin forbidden reactions, have been shown to suffer from difficulties in trying to separate isospinviolating parts of the reaction mechanism from isospin impurities in the final nuclear states [2]. Measurements with electromagnetic probes rely on model dependent wave function calculations, in order to separate contributions to the T = 0 cross section due to the T = 1 wave function admixtures from those due to convection currents. The 1+ doublet in 12C provides an ideal testing ground for the various methods o f measuring isospin mixing, because of its relatively pure shell model configuration and because good shell model calculations are available. Adelberger et al. [2] have reviewed the measurements of H01 between these states and, in view of the fact that the wave functions are well known, they have concluded that the most reliable measurements are those using electromagnetic probes. Indeed, although measurements made using hadronic probes are generally inconsistent [ 1,3,4], those made with electromagnetic probes are all in good agreement [2,5,6]. The recent measurements using back angle 387
Volume 99B, number 5
PflYSICS LETTERS
electron scattering by Flanz et al. [6] provide the most accurate determination to date of H01 = 140 +- 35 keV. All of the above mentioned measurements have yielded values for H01 which are larger than those expected from the Coulomb force alone [3,7], if differences between the proton and neutron radial wave functions are ignored. However, when such differences are taken into account (as was shown to be necessary in 8Be by Dalton and Robson [8]) values of H01 between 60 and 240 keV due only to the Coulomb force can be generated [ 9 - 1 1 ] . In this letter we report on data obtained for pion inelastic scattering to the 1+ doublet in 12C. Simple impulse model arguments are used to predict the 7r+/Tr- cross section ratios for both states and also the cross section ratios between the states. These are seen to be in agreement with the data and a value of 148 -+ 29 keV is extracted for H01. Evidence for the validity of a simple one-stepdirect reaction mechanism to describe pion inelastic scattering can be seen in recent studies of pion inelastic scattering to states in 12C and 13C [12,13]. In b o t h cases rather dramatic differences between the excitation functions for natural and unnatural parity transitions are well described by this assumption. The expected n e u t r o n - p r o t o n selectivity of 7r-/Tr + scattering can also be seen in the 13C work [14] where the 7r-/~r + cross section ratios are in good agreement with shell model predictions [15]. The most convincing evidence is the observation o f a pure neutron transition to a state in 13C at 9.5 MeV, where the observed cross section ratio of 10 +- 2 is in good agreement with the expected ratio of 9:1 for such a transition. These data were obtained using the EPICS spectrometer at LAMPF, at angles of 25 ° and 35 ° in the laboratory, and at energies between 100 and 180 MeV. Targets of natural carbon with thicknesses of 110 mg/cm 2 for 7r+ and 227 mg/cm 2 for v - were used. This resulted in somewhat better energy resolution for the 7r+ scattering. The 12.71 (T = 0 ) a n d 15.11 ( T = 1) MeV levels could be clearly identified at energies below 130 MeV, but the 15.11 MeV level was obscured at higher pion bombarding energies by a broad state (F' 2 MeV) located at 15.4 MeV, and so these data were not used in the present analysis. The cross section ratios for v + and 7r- inelastic scattering to an isospin mixed doublet have been given 388
5 March 1981
in ref. [16]. There it was assumed that only the resonant [3/2, 3/2] p i o n - n u c l e o n amplitude contributed to the scattering, and consequently the o(~r+ + p)/o0r + + n) ratio was 9 / l . In the present case, where spin flip must be involved in the interaction, this assumption can be relaxed by defining the L = 1 spin flip amplitude ratio as:
X(E) = [cr(Tr+ + p)/o0r + + n)] 1/2 = (13a31 - 3a33]/12all - 2a13 +a31 - a331),
(1)
where a 2 j ' 2 T are given in terms of the v-nucleon phase shifts, 6 [17], and the pion momentum, k, by:
a2j, 2 T = k - 1 exp(i6 2J, 2 T) sin(6 2J, 2 T) •
(2)
Then, assuming two state mixing, one can write the wave functions for the isospin mixed states in terms of states of pure isospin as: (Al=c~(01+/3(11,
(Bl=c~(11-13(01,
(3)
where a 2 + 132 = 1 and both are real for states whose separation is large compared to their widths. The parameter 13 is related to the charge dependent matrix element by: o~13(EB - E A ) = (0 [HcDI 1 ) = H0j . Here E A and E B are the perturbed energies of state A and B respectively, HCD is the charge 'dependent part of the hamiltonian, and H01 is its off diagonal matrix element. The T = 0 and T = 1 wave functions ((01 and
1)1312/16
@, = o 0 [ ( X + 1)c~- ( X -
1)1312/16,
+
oB = o0[(X-
1 ) ~ - ( X + 1)t312/16,
crB = cr0 [ ( X - 1)e~ + (X + 1)/312/16. The experimental spectra have been fit using these relationships to constrain the yields for ~r+ and 7rscattering to the states at 12.71 and 15.1 1 Mev. This results in a two parameter fit of o 0 and HOl to the four measured yields. Cross sections to all other states in this energy region, as well as backgrounds were assumed to be equal in rr+ and rr- scattering. The result-
Volume 99B, number 5
PHYSICS LETTERS
5 March 1981
Table 1 Er r (MeV)
0lab (deg)
q (fm-1)
Oo a) (#b/sr)
Ho I (keV)
X2 (reduced)
100 116 130 130
35 25 25 35
0.59 0.47 0.51 0.70
133(6) 99(4) 101(4) 97(4)
118(64) 156(46) 124(60) 170(62)
0.89 0.67 0.88 1.07
a) The error bars for the cross sections do not include an estimated _+15% absolute normalization uncertainty. ing values for o 0 and H01 are presented in table 1 and a typical fit and spectrum are shown in fig. i. Although the reduced X2 for all of these fits were approximately 1, as a check we allowed the four cross section yields to vary individually. Good agreement between these methods was obtained for both the cross section ratios and for the parameter/3. These simple impulse model results have been checked for normal parity transitions using the DWlA code DWPI [18]. The ratios o+/o were seen to vary from unity by up to 10% at the lower energy due to Coulomb distortions, when [3/2, 3/2] dominance was assumed and no isospin mixing was included. However, the double ratio OAOB/OAO + + B was found to be within 0.1% of the expected ratio of 1.0. In order to
a)
~Zc ('rr,rr') JaC~ Err = 116MeV OLAB:25° REDUCED XZ :0.6'7 Hm:156 ± 46 keV
,~
i
i
12.71
I .0
i
i
i
i
I5.Z
lEO
(I ÷)
h
IL8
i
13.6
E
14.4
i
I
16.8
ENERGY (MeV)
Fig. l. A typical spectrum and fit to the excitation energy spectrum near the 12.71, 15.11 MeV region. The dashed line is the fitted quadratic background; the solid line is a sum of the background and the fitted peaks. The peaks were fit by folding experimental line shapes with lorenzian line shapes using compiled [20] energy levels and widths for the states marked.
insure the present results are independent of such distortion effects the data were fit with the relative ~r+ to rr normalizations changed by 10%. No significant effect was seen in the resultant isospin mixing parameters. The m o m e n t u m transfer, q, spanned by these measurements was 0.47 to 0.70 f m - 1 and the incident pion energy, f , was varied from 100 to 130 MeV. From table 1 one can see there are no statistically significant trends seen in H01 as either a function of E or q. The weighted average value of H01 obtained from all of these measurements is 148 -+ 29 keV, in good agreement with the most recent electromagnetic measurements. Recently Siciliano and Weiss [19] have studied asymmetries in rr- versus rr+ scattering resulting from continuum effects. Their study of the u n b o u n d 4 states in 12C, which predicts longer tails in the proton wave functions due to the lower proton binding energy, shows that such effects result in an energy depem dent enhancement of ~r- scattering to both T = 0 and T = 1 states. Isospin mixing, on the other hand, enhances 7r+ scattering to one state and rr- scattering to the other. Within the errors of the present data, no evidence for wave function tail effects is seen for the nearly bound 1+ doublet in 12C. In conclusion, pion inelastic scattering provides a method for measuring isospin mixing between nuclear levels which is not sensitive to the details of the nuclear wave function. Even in the case of the l + doublet in 12C, where the mixing is small, pion inelastic scattering provides an accurate determination of the isospin mixing with a model hldependent method. This work was supported in part by the Department of Energy, the Robert A. Welch Foundation, and the Swiss Institute of Nuclear Physics. 389
Volume 99B, number 5
PHYSICS LETTERS
References [1] W.J. Braithwaite, J.E. Bussoletti, F.E. Cecil and G.T. Garvey, Phys. Rev. Lett. 29 (1972) 376. [2] F.G. Adelberger, R.C. Marrs, K.A. Snorer and J. Bussoletti, Phys. Rev. C15 (1977) 484. [3] F.D. Reisman, P.I. Connors and J.B. Marion, Nucl. Phys. A153 (1970) 244. [4] J.M. Lind, G.T. Garvey and R.E. Tribble, Nucl. Phys. A276 (1977) 25. [5] F.E. Cecil, L.W. Fagg, W.L. Bendel and E.C. Jones Jr., Phys. Rev. C9 (1974) 798. [6] J.B. Flanz et al., Phys. Rev. Lett. 43 (1979) 1922. [7] F.C. Barker, Nucl. Phys. 83 (1966) 418. [81 B.J. Dalton and D. Robson, Phys. Lett. 20 (1966) 405. [9] H. Sato and L. Zamick, Phys. Lett. 70B (1977) 285. [10] R.D. Lawson, Phys. Lett. 78B (1978) 371. [11] F.C. Barker, Aust. J. Phys. 32 (1978) 27. [12] W.B. Cottingame et al., Bull. Amer. Phys. Soc. 24 (1979) 821.
390
5 March 1981
[13] S.J. Tripp et al., in: Workshop on nuclear structure with intermediate-energy probes, ed. H.A. Thiessen, LASL Report no. LA-8303-C (1980) p. 328. [14] D. Dehnhard et al., Phys. Rev. Lett. 43 (1979) 1091. [15] T.-S.H. Lee, in: Workshop on nuclear structure with intermediate-energy probes, ed. H.A. Thiessen, LASL Report no. LA-8303-C (1980) p. 2. [16] C.L. Morris et al., Phys. Lett. 86B (1979) 31. [17] G. Rowe, M. Salomon and R.H. Landau, Phys. Rev. C18 (1978) 584. [18] R.A. Eisenstein and R.A. Miller, Comput. Phys. Commun. 11 (1976) 950. [19] E.R. Siciliano and D.L. Weiss, in: Workshop on nuclear structure with intermediate-energy probes, ed. H.A. Thiessen, LASL Report no. LA-8303-C (1980) p. 278. [20] F. Ajzenberg-Selove and C.L. Busch, Nucl. Phys. A336 (1980) 1.
PHYSICS LETTERS
5 March 1981
MEASUREMENTS OF ISOSPIN MIXING BETWEEN THE 1+ DOUBLET IN 12C USING PION INELASTIC SCATTERING C.U MORRIS, R.L. BOUDRIE and J. PIFFARETTI x Los Alamos Seientifi'e Laboratory, Los Alamos, NM 87545, USA W.B. COTTINGAME 2, W.J. BRAITHWAITE, S.J. GREENE 2 C.J. HARVEY, D.B. HOLTKAMP 3 and C.Fred MOORE University of Texas, Austin, TX 78712, USA S.J. SEESTROM-MORRIS 4 University of Minnesota, Minneapolis, MN 55455, USA Received 12 November 1980
Pion inelastic scattering has been used to extract the off-diagonal charge dependent matrix element, Hob between the jTr = 1+ doublet of states at 12.71 (T = 0) and 15.11 (T = 1) MeV in 12C. The reported measurements yield a value, Hol = 148 +- 29 keV, in good agreement with recent electromagnetic measurements.
One method of detecting charge dependent components of the strong interaction is to measure the size of the off-diagonal charge dependent matrix elements, HO1 , between analog-antianalog pairs o f states in nuclei [1 ]. The size of these matrix elements can be measured by determining the mixing between states o f otherwise good isospin, T. If a systematic study of isospin mixing can show this mixing to be different than expected from the Coulomb force alone, charge dependence in the strong interaction will have been established. Such studies have been inconclusive because of both experimental and theoretical problems. Experimentally there has been no generally reliable method o f measuring isospin mixing. Measurements made with hadronic ,1 Present address: Schweizerisches Institut ftir Nuklearforschung, CH-5234 Villigen, Switzerland. ,5 Present address: New Mexico State University, Los Cruces, NM 88003, USA. ,3 Present address: University of Minnesota, Minneapolis, MN 55455, USA. ,4 Present address: Los Alamos Scientific Laboratory, Los Alamos, NM 87545, USA.
probes, including both stripping and pickup reactions and isospin forbidden reactions, have been shown to suffer from difficulties in trying to separate isospinviolating parts of the reaction mechanism from isospin impurities in the final nuclear states [2]. Measurements with electromagnetic probes rely on model dependent wave function calculations, in order to separate contributions to the T = 0 cross section due to the T = 1 wave function admixtures from those due to convection currents. The 1+ doublet in 12C provides an ideal testing ground for the various methods o f measuring isospin mixing, because of its relatively pure shell model configuration and because good shell model calculations are available. Adelberger et al. [2] have reviewed the measurements of H01 between these states and, in view of the fact that the wave functions are well known, they have concluded that the most reliable measurements are those using electromagnetic probes. Indeed, although measurements made using hadronic probes are generally inconsistent [ 1,3,4], those made with electromagnetic probes are all in good agreement [2,5,6]. The recent measurements using back angle 387
Volume 99B, number 5
PflYSICS LETTERS
electron scattering by Flanz et al. [6] provide the most accurate determination to date of H01 = 140 +- 35 keV. All of the above mentioned measurements have yielded values for H01 which are larger than those expected from the Coulomb force alone [3,7], if differences between the proton and neutron radial wave functions are ignored. However, when such differences are taken into account (as was shown to be necessary in 8Be by Dalton and Robson [8]) values of H01 between 60 and 240 keV due only to the Coulomb force can be generated [ 9 - 1 1 ] . In this letter we report on data obtained for pion inelastic scattering to the 1+ doublet in 12C. Simple impulse model arguments are used to predict the 7r+/Tr- cross section ratios for both states and also the cross section ratios between the states. These are seen to be in agreement with the data and a value of 148 -+ 29 keV is extracted for H01. Evidence for the validity of a simple one-stepdirect reaction mechanism to describe pion inelastic scattering can be seen in recent studies of pion inelastic scattering to states in 12C and 13C [12,13]. In b o t h cases rather dramatic differences between the excitation functions for natural and unnatural parity transitions are well described by this assumption. The expected n e u t r o n - p r o t o n selectivity of 7r-/Tr + scattering can also be seen in the 13C work [14] where the 7r-/~r + cross section ratios are in good agreement with shell model predictions [15]. The most convincing evidence is the observation o f a pure neutron transition to a state in 13C at 9.5 MeV, where the observed cross section ratio of 10 +- 2 is in good agreement with the expected ratio of 9:1 for such a transition. These data were obtained using the EPICS spectrometer at LAMPF, at angles of 25 ° and 35 ° in the laboratory, and at energies between 100 and 180 MeV. Targets of natural carbon with thicknesses of 110 mg/cm 2 for 7r+ and 227 mg/cm 2 for v - were used. This resulted in somewhat better energy resolution for the 7r+ scattering. The 12.71 (T = 0 ) a n d 15.11 ( T = 1) MeV levels could be clearly identified at energies below 130 MeV, but the 15.11 MeV level was obscured at higher pion bombarding energies by a broad state (F' 2 MeV) located at 15.4 MeV, and so these data were not used in the present analysis. The cross section ratios for v + and 7r- inelastic scattering to an isospin mixed doublet have been given 388
5 March 1981
in ref. [16]. There it was assumed that only the resonant [3/2, 3/2] p i o n - n u c l e o n amplitude contributed to the scattering, and consequently the o(~r+ + p)/o0r + + n) ratio was 9 / l . In the present case, where spin flip must be involved in the interaction, this assumption can be relaxed by defining the L = 1 spin flip amplitude ratio as:
X(E) = [cr(Tr+ + p)/o0r + + n)] 1/2 = (13a31 - 3a33]/12all - 2a13 +a31 - a331),
(1)
where a 2 j ' 2 T are given in terms of the v-nucleon phase shifts, 6 [17], and the pion momentum, k, by:
a2j, 2 T = k - 1 exp(i6 2J, 2 T) sin(6 2J, 2 T) •
(2)
Then, assuming two state mixing, one can write the wave functions for the isospin mixed states in terms of states of pure isospin as: (Al=c~(01+/3(11,
(Bl=c~(11-13(01,
(3)
where a 2 + 132 = 1 and both are real for states whose separation is large compared to their widths. The parameter 13 is related to the charge dependent matrix element by: o~13(EB - E A ) = (0 [HcDI 1 ) = H0j . Here E A and E B are the perturbed energies of state A and B respectively, HCD is the charge 'dependent part of the hamiltonian, and H01 is its off diagonal matrix element. The T = 0 and T = 1 wave functions ((01 and
1)1312/16
@, = o 0 [ ( X + 1)c~- ( X -
1)1312/16,
+
oB = o0[(X-
1 ) ~ - ( X + 1)t312/16,
crB = cr0 [ ( X - 1)e~ + (X + 1)/312/16. The experimental spectra have been fit using these relationships to constrain the yields for ~r+ and 7rscattering to the states at 12.71 and 15.1 1 Mev. This results in a two parameter fit of o 0 and HOl to the four measured yields. Cross sections to all other states in this energy region, as well as backgrounds were assumed to be equal in rr+ and rr- scattering. The result-
Volume 99B, number 5
PHYSICS LETTERS
5 March 1981
Table 1 Er r (MeV)
0lab (deg)
q (fm-1)
Oo a) (#b/sr)
Ho I (keV)
X2 (reduced)
100 116 130 130
35 25 25 35
0.59 0.47 0.51 0.70
133(6) 99(4) 101(4) 97(4)
118(64) 156(46) 124(60) 170(62)
0.89 0.67 0.88 1.07
a) The error bars for the cross sections do not include an estimated _+15% absolute normalization uncertainty. ing values for o 0 and H01 are presented in table 1 and a typical fit and spectrum are shown in fig. i. Although the reduced X2 for all of these fits were approximately 1, as a check we allowed the four cross section yields to vary individually. Good agreement between these methods was obtained for both the cross section ratios and for the parameter/3. These simple impulse model results have been checked for normal parity transitions using the DWlA code DWPI [18]. The ratios o+/o were seen to vary from unity by up to 10% at the lower energy due to Coulomb distortions, when [3/2, 3/2] dominance was assumed and no isospin mixing was included. However, the double ratio OAOB/OAO + + B was found to be within 0.1% of the expected ratio of 1.0. In order to
a)
~Zc ('rr,rr') JaC~ Err = 116MeV OLAB:25° REDUCED XZ :0.6'7 Hm:156 ± 46 keV
,~
i
i
12.71
I .0
i
i
i
i
I5.Z
lEO
(I ÷)
h
IL8
i
13.6
E
14.4
i
I
16.8
ENERGY (MeV)
Fig. l. A typical spectrum and fit to the excitation energy spectrum near the 12.71, 15.11 MeV region. The dashed line is the fitted quadratic background; the solid line is a sum of the background and the fitted peaks. The peaks were fit by folding experimental line shapes with lorenzian line shapes using compiled [20] energy levels and widths for the states marked.
insure the present results are independent of such distortion effects the data were fit with the relative ~r+ to rr normalizations changed by 10%. No significant effect was seen in the resultant isospin mixing parameters. The m o m e n t u m transfer, q, spanned by these measurements was 0.47 to 0.70 f m - 1 and the incident pion energy, f , was varied from 100 to 130 MeV. From table 1 one can see there are no statistically significant trends seen in H01 as either a function of E or q. The weighted average value of H01 obtained from all of these measurements is 148 -+ 29 keV, in good agreement with the most recent electromagnetic measurements. Recently Siciliano and Weiss [19] have studied asymmetries in rr- versus rr+ scattering resulting from continuum effects. Their study of the u n b o u n d 4 states in 12C, which predicts longer tails in the proton wave functions due to the lower proton binding energy, shows that such effects result in an energy depem dent enhancement of ~r- scattering to both T = 0 and T = 1 states. Isospin mixing, on the other hand, enhances 7r+ scattering to one state and rr- scattering to the other. Within the errors of the present data, no evidence for wave function tail effects is seen for the nearly bound 1+ doublet in 12C. In conclusion, pion inelastic scattering provides a method for measuring isospin mixing between nuclear levels which is not sensitive to the details of the nuclear wave function. Even in the case of the l + doublet in 12C, where the mixing is small, pion inelastic scattering provides an accurate determination of the isospin mixing with a model hldependent method. This work was supported in part by the Department of Energy, the Robert A. Welch Foundation, and the Swiss Institute of Nuclear Physics. 389
Volume 99B, number 5
PHYSICS LETTERS
References [1] W.J. Braithwaite, J.E. Bussoletti, F.E. Cecil and G.T. Garvey, Phys. Rev. Lett. 29 (1972) 376. [2] F.G. Adelberger, R.C. Marrs, K.A. Snorer and J. Bussoletti, Phys. Rev. C15 (1977) 484. [3] F.D. Reisman, P.I. Connors and J.B. Marion, Nucl. Phys. A153 (1970) 244. [4] J.M. Lind, G.T. Garvey and R.E. Tribble, Nucl. Phys. A276 (1977) 25. [5] F.E. Cecil, L.W. Fagg, W.L. Bendel and E.C. Jones Jr., Phys. Rev. C9 (1974) 798. [6] J.B. Flanz et al., Phys. Rev. Lett. 43 (1979) 1922. [7] F.C. Barker, Nucl. Phys. 83 (1966) 418. [81 B.J. Dalton and D. Robson, Phys. Lett. 20 (1966) 405. [9] H. Sato and L. Zamick, Phys. Lett. 70B (1977) 285. [10] R.D. Lawson, Phys. Lett. 78B (1978) 371. [11] F.C. Barker, Aust. J. Phys. 32 (1978) 27. [12] W.B. Cottingame et al., Bull. Amer. Phys. Soc. 24 (1979) 821.
390
5 March 1981
[13] S.J. Tripp et al., in: Workshop on nuclear structure with intermediate-energy probes, ed. H.A. Thiessen, LASL Report no. LA-8303-C (1980) p. 328. [14] D. Dehnhard et al., Phys. Rev. Lett. 43 (1979) 1091. [15] T.-S.H. Lee, in: Workshop on nuclear structure with intermediate-energy probes, ed. H.A. Thiessen, LASL Report no. LA-8303-C (1980) p. 2. [16] C.L. Morris et al., Phys. Lett. 86B (1979) 31. [17] G. Rowe, M. Salomon and R.H. Landau, Phys. Rev. C18 (1978) 584. [18] R.A. Eisenstein and R.A. Miller, Comput. Phys. Commun. 11 (1976) 950. [19] E.R. Siciliano and D.L. Weiss, in: Workshop on nuclear structure with intermediate-energy probes, ed. H.A. Thiessen, LASL Report no. LA-8303-C (1980) p. 278. [20] F. Ajzenberg-Selove and C.L. Busch, Nucl. Phys. A336 (1980) 1.