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CCBA fit to anomalous angular distribution of 40Ca(13C, 14N) 39K(32+) reaction
Volume 67B, number 1
PHYSICS LETTERS
14 March 1977
CCBA F I T T O A N O M A L O U S A N G U L A R D I S T R I B U T I O N O F 4 ° C a ( 1 3 C , 14N) 3 9 K ( 3 / 2 +) R E A C T I O N K.S. LOW t Ddpartment de Physique Nucl4aire, CEN Saclay, BP2, 91190 Gifisur. Yvette, France and
T. TAMURA and T. UDAGAWA Department o f Physics 2, University o f Texas, Austin, Texas 78712, USA Received 26 May 1976 Revised version received 20 January 1977 It is shown that the anomalous angular distribution in the 4°Ca(13C, 14N)39K (3/2 +) reaction can be fitted very well by an EFR-CCBAcalculation. Reasons why CCBAworks are discussed in some detail. Recently Bond et al. [ 1] reported on a rather surprising observation, namely, that while the angular distribution of the 40Ca(13C, 12C)41 Ca ( 7 / 2 - ) reaction could be fitted nicely by DWBA, that of the 40Ca(13C, 14N)39K(3/2 +) reaction was completely out of phase with the predicted DWBA angular distribution. The purpose of the present article is to show that these two apparently quite different reactions can be explained on an equal footing, if CCBA (coupled-channel Born approximation) rather than DWBA calculations are made. Actually Bond et al. [1] already suggested such a possibility, through the suggestion was accompanied by a suspicion that it might be difficult to explain why the higher order processes so much affect one reaction without affecting the other. After showing first that the CCBA does fit both sets of data, we shall present several reasons why the higher order processes are so important in the (13C, 14N) reaction, but not in the (13C, 12C) reaction. In carrying out exact-finite-range (EFR) CCBA calculations [2, 3], we took into account for both (13C, 12C) and (13C, 14N) reactions the inelastic excitation of the 3.73 MeV 3 - state of 40Ca. For the (13C, 14N) reaction, the calculations also included the inelastic excitation of the 4.48 MeV 5 - state as 1 Present address: Jabatan Fizik, Universiti Malaya, Kuala Lumpur, Malaysia. 2 Work supported in part by the U.S.E.R.D.A.
well. Because of the limitation of the core available in our computer, the calculations were made by coupling the 3 - and 5 - states separately to the ground 0 + state. This procedure introduces some error in the transition amplitudes, but the error only occurs in orders higher than the third and is expected to be small. The deformation parameters used in the coupledchannels calculation were/3 3 = 0.37 and ~/5 = 0.23, which are averaged of respective values determined by various experimental studies [4]. The spectroscopic amplitudes were generated from the RPA wave functions for 40Ca [5], and from pure shell model wave functions for 39K and 41Ca. Both the 3 - and 5wave functions are dominated by an (f7/2d~/12) configuration, and we ignored the other components in the calculation. The spectroscopic amplitudes pertaining to various transitions evaluated by using the above wave functions are summarized in table 1. This table also gives the binding energies of the transferred particle and the orbital angular momenta, l, transferred in the transition. As for the 13C-14C and 13C-12C systems, the spectroscopic factors 0.69 and 0.71, respectively [6], were used. In the DWBA analyses reported in ref. [1], use was made of optical potential parameters that were confirmed to fit the elastic scattering data of C on Ca and N on K. We do not consider the coupled-channel effect in the exit channel; thus the optical parameters of ref. [1] can be used there without modification. In the incident (13C) channel, we have to modify the
Volume 67B, number 1
PHYSICS LETTERS
14 March 1977
Table 1 Parameters involved in the 4°Ca(13C, 14N)39K and 4°Ca(13C, 12C)41Ca reactions via the various states of 4°Ca. Exit channel
Incident channel
Transferred ] 1a
Binding energy (MeV)
Spectroscopic amplitude
Alls
39K G.S. 3/2 +
4°Ca G.S. 0 + 3.73 34.48 5-
d3/2 f7/2 f7/2
2,1 4,3 4,3
8.33 1.09 b 1.09 b
1.81 0.52 0.71
41Ca G.S. 7 / 2 -
4°Ca G.S. 0 +
f7/2 d3/2
4,3 2,1
8.36 15.6 c
0.9 0.52
3.73 3a Only underlined l values were used in the actual calculations. bThis is the binding energy difference between 41Ca and 4°Ca. c This is the binding energy difference between 39Ca and 4°Ca. parameters of ref. [1 ] in such a way that the elastic cross section is reproduced when the 3 - state is coupled. It was found that this can be achieved by changing the value o f r v = 1.27 fm of ref. [1] into r v = 1.30 fm, other parameters being unchanged. Bound state parameters were taken the same as in ref. [1 ], except that a spin orbit interaction with Vso = 8 MeV, rso = 1.25 fm and aso = 0.65 was added to the central potential of ref. [1 ]. The theoretical cross sections plotted in fig. 1, together with experimental data [ 1 ], were obtained by first taking into account only the direct transition (o0), then adding the indirect transition via the 3 state (o03), and adding finally the transition via the 5 - states (0035). It should be noted that o 0 is almost the same as the DWBA cross section, i.e. o 0 -~ o (DwBA). For the (13C, 12C) reaction we first note in fig. 1 that o 0 "" o03 , i.e. o (CCBA)~ o (DWBA) (although 003 does agree better with experiment than does o0). The fact that o (CCBA) -~ o (DwBA) means that the effect of the multi-step process is extremely small in this reaction, which can also be inferred from the fact that the cross section,obtained by considering a pure indirect transition is about three orders of magnitude smaller than that which is obtained with a pure direct transition. Since DWBA fits the data [1], and CCBA is close to DWBA, the CCBA result also fits the data. This conclusion is not altered even when the CCBA cross section is obtained by coupling also the 5 state [7]. The situation is very different in the (13C, 14N) reaction, where the indirect transitions are competitive with the direct transition. As is seen in fig. 1, o03,
including only the 3 - coupling, already differs markedly from o 0 and thus from o(DWBA), although the difference is not large enough to reproduce the observed cross section. When the 5 - coupling is also added, however, we obtain o035 which has an angular distribution completely out-of-phase from that of o 0, thus agreeing very nicely with experiment. The mysterious feature of the (13C, 14N) reaction data has thus been well accounted for. We shall now present several reasons for why the two-step process is so strong in the (13C, 14N) reaction, but is weak in the (13C, 12C) reaction. (i) The first is the difference in the single particle binding energies. In the two-step mode of the (13C, 14N) reaction, a proton is picked up from the f7/2 orbit, after being excited from the d3/2 orbit to form the 3 - state at 3.73 MeV (or 5 - state at 4.48 MeV). With the usual separation energy procedure, this f7/2 proton is less bound by 3.73 MeV (or by 4.48 MeV) compared with the d3/2 proton that is picked up in the one-step process; a fact which greatly favors the two-step processes. A more careful treatment based on the formalism of Pinkston and Satchler [8] shows, however, that the energy to be used in our case is not the separation energy, but just the single-particle (SP) energy that one has before the RPA calculation is made. As is seen in table 1, the SP energy of the f7/3 proton is as small as 1.1 MeV. The corresponding wave function thus has a large amplitude in the external region, and the two-step process can be very strong. The same argument applies to the stripping of a neutron into the d3/2 hole in the 3 - state, when the (13C, 12C) reaction proceeds via the two-step mode.
Volume 67B, number 1 f
PHYSICS LETTERS
i
i
i
4°Ca(ISC,)2C)4'Ca(g
----
i
i
(o)
s r/z )
00
io.o
14 March 1977
Q-matched. In the case of the (13C, 12C) reaction the optimum Q value is zero, while the actual Q values o f the direct and indirect transitions via the 3 - and 5 states are, respectively, 3.41, 7.14 and 7.89 MeV. Therefore, in this case the transition via the excited states are rather strongly Q-mismatched. (iii) The third reason is the difference in the coherence of intermediate contributions. Somewhat symbolically, one may denote the two-step process as lal~ l a ~ lb, which means that we have in mind a process in which a partial wave l a in the incident channel makes a transition into l a m the excited ( 3 - or 5 - ) channel, and then finally into Ib in the exit channel. In these two steps, angular momenta l' and 1" are transferred (l' being fixed at 3 or 5 in our case). Let us take all the amplitudes with a fixed set of values of Ia, lb and l', and sum them over l a and l". A surprising fact we found was that this sum was constructive for the (13C, 14N) reaction, but was destructive for the (13C, 12C) r e a c t i o n . This favors the importance o f the two-step processes for the (13C, 14N) reaction, but suppresses them in the (13C, 12C) reaction. Which of the above three factors is the most important? In order to give a possible answer to this question, let us first note that, in the process of obtaining the a03 cross sections of fig. 1, we found that R ~ 10, where R is defined as the absolute ratio of the amplitude of the two-step process in the (13C, 14N) reaction divided by that in the (13C, 12C) reaction. We may then carry out a new calculation for the (13C, 12C) reaction by modifying somewhat artifically parameters so that the two-step process in this reaction becomes as favorable as that in the (13C, 14N) reaction, as far as the above two points (i) and (ii) are concerned. We achieved this by first making the SP binding energy of the d3/2 neutron equal to 1.1 MeV (rather than 15.6 MeV; cf. table 1), and then assigning a (negative) binding energy to the 3 - state so that the 3 - ~ 7 / 2 transition proceeds with an optimum Q-value; Q = 0. The result of this new calculation showed that the new value of R is R ~ x,/]O. This means that the above reasons (i) and (ii) together account for about half the difference in the importance of the two-step processes in the two reactions. The remaining half must be accounted for by the coherence (iii), which is thus the most important, It is seen that we now have a rather clear understanding o f what is taking place in the normal and anomalous t
IO
-_-
(b)
4 ° C 0 ( ' 3C '4N)39K (g.s ' 3/;~ )
u~
m
.
t
br4 -----
O- 0
io.o 1
,:7",
~~
1.0
i
5
i
I0
i
15
i
20
2~5
3
;
ec,m,
Fig. 1. Comparison o f EFR-CCBA calculations with experiment with Elab(4°Ca) = 68 MeV. Three types o f theoretical cross sections oo, ao3 and ao3 s are explained in the text. Note that o o ~ o(DWBA). The normalization factors used for (13C, 12C) and (13C, 14N) reactions were, respectively, 1.65 and 1.45.
The SP energy of this neutron is as large as 15.6 MeV, strongly inhibiting the two-step process. (ii) The second reason is the difference in the Q-matching condition. The optimum Q value for the (13C, 14N) reaction is found to be about 5.7 MeV, while the actual Q value for the 0 + ~ 3/2 + transition is - 0 . 7 8 MeV, making the one-step transition rather strongly Q-mismatched. The transitions via the 3 and 5 - states, on the other hand, have Q = 2.95 and 3.70 MeV, respectively, and thus are much better
Volume 67B, number 1
PHYSICS LETTERS
processes of ref. [ 11. Why the nature of the coherence differs so drastically in the two reactions is not so easy to find out, however. Obviously very complicated dynamics are involved. It is nevertheless clear that the coherence does not depend strongly on the mass number. The experiment of ref. [I] showed that the 3/Z+ cross sections in A Ca(13C, 14N) reactions with A f 40 also behaved anomalously. We believe that the technique used here for A = 40 applies to these cases too. It is known that the angular distribution of the 4oCa (L3C, 14N)39K(1/2+) reaction is also anomalous [l], but we have performed a CCBA calculation for this case too and found that the anomaly is very well accounted for. As is seen in ref. ]l ] , the anomaly in this case is somewhat weaker than it is in the 3/2+ final state. This is reasonable because in the l/2’ case the step that proceeds via 5- state is not possible: The .5- state cannot include an (f7,2s&) configuration. It is a pleasure to acknowledge fruitful discussion with Drs. M.C. Mermaz and J.C. Peng. We thank the
14 March 1977
authors of ref. [I 1, in particular Dr. P.D. Bond for making available to us the details of the data, and for his encouraging discussions.
References [I] P.D. Bond et al., Phys. Rev. Lett. 36 (1976) 300. [2] T. Tamura, Phys. Reports 14C (1974) 59. [3] T. Tamura, KS. Low and T. Udagawa, Phys. Lett. 51B (1974) 116. [4] C.R. Gruhn et al., Phys. Rev. C6 (1972) 915; R.A. Eisenstein et al., Phys. Rev. 188 (1069) 1815. [5] W.J. Gerace and A.M. Green, Nucl. Phys. All3 (1968) 641. [6] S. Cohen and D. Kurath, Nucl. Phys. Al01 (1967) 1. f7] K.S. Low, T. Tamura and T. Udagawa, unpublished; K.S. Low, to be published in: Proc. of European Conf. on Nuclear physics with heavy ions, Caen, France, September, 1976. [S] W.T. Pinkston, R.J. Philpott and G.R. Satchler, Nucl. Phys. Al25 (1969) 176.
PHYSICS LETTERS
14 March 1977
CCBA F I T T O A N O M A L O U S A N G U L A R D I S T R I B U T I O N O F 4 ° C a ( 1 3 C , 14N) 3 9 K ( 3 / 2 +) R E A C T I O N K.S. LOW t Ddpartment de Physique Nucl4aire, CEN Saclay, BP2, 91190 Gifisur. Yvette, France and
T. TAMURA and T. UDAGAWA Department o f Physics 2, University o f Texas, Austin, Texas 78712, USA Received 26 May 1976 Revised version received 20 January 1977 It is shown that the anomalous angular distribution in the 4°Ca(13C, 14N)39K (3/2 +) reaction can be fitted very well by an EFR-CCBAcalculation. Reasons why CCBAworks are discussed in some detail. Recently Bond et al. [ 1] reported on a rather surprising observation, namely, that while the angular distribution of the 40Ca(13C, 12C)41 Ca ( 7 / 2 - ) reaction could be fitted nicely by DWBA, that of the 40Ca(13C, 14N)39K(3/2 +) reaction was completely out of phase with the predicted DWBA angular distribution. The purpose of the present article is to show that these two apparently quite different reactions can be explained on an equal footing, if CCBA (coupled-channel Born approximation) rather than DWBA calculations are made. Actually Bond et al. [1] already suggested such a possibility, through the suggestion was accompanied by a suspicion that it might be difficult to explain why the higher order processes so much affect one reaction without affecting the other. After showing first that the CCBA does fit both sets of data, we shall present several reasons why the higher order processes are so important in the (13C, 14N) reaction, but not in the (13C, 12C) reaction. In carrying out exact-finite-range (EFR) CCBA calculations [2, 3], we took into account for both (13C, 12C) and (13C, 14N) reactions the inelastic excitation of the 3.73 MeV 3 - state of 40Ca. For the (13C, 14N) reaction, the calculations also included the inelastic excitation of the 4.48 MeV 5 - state as 1 Present address: Jabatan Fizik, Universiti Malaya, Kuala Lumpur, Malaysia. 2 Work supported in part by the U.S.E.R.D.A.
well. Because of the limitation of the core available in our computer, the calculations were made by coupling the 3 - and 5 - states separately to the ground 0 + state. This procedure introduces some error in the transition amplitudes, but the error only occurs in orders higher than the third and is expected to be small. The deformation parameters used in the coupledchannels calculation were/3 3 = 0.37 and ~/5 = 0.23, which are averaged of respective values determined by various experimental studies [4]. The spectroscopic amplitudes were generated from the RPA wave functions for 40Ca [5], and from pure shell model wave functions for 39K and 41Ca. Both the 3 - and 5wave functions are dominated by an (f7/2d~/12) configuration, and we ignored the other components in the calculation. The spectroscopic amplitudes pertaining to various transitions evaluated by using the above wave functions are summarized in table 1. This table also gives the binding energies of the transferred particle and the orbital angular momenta, l, transferred in the transition. As for the 13C-14C and 13C-12C systems, the spectroscopic factors 0.69 and 0.71, respectively [6], were used. In the DWBA analyses reported in ref. [1], use was made of optical potential parameters that were confirmed to fit the elastic scattering data of C on Ca and N on K. We do not consider the coupled-channel effect in the exit channel; thus the optical parameters of ref. [1] can be used there without modification. In the incident (13C) channel, we have to modify the
Volume 67B, number 1
PHYSICS LETTERS
14 March 1977
Table 1 Parameters involved in the 4°Ca(13C, 14N)39K and 4°Ca(13C, 12C)41Ca reactions via the various states of 4°Ca. Exit channel
Incident channel
Transferred ] 1a
Binding energy (MeV)
Spectroscopic amplitude
Alls
39K G.S. 3/2 +
4°Ca G.S. 0 + 3.73 34.48 5-
d3/2 f7/2 f7/2
2,1 4,3 4,3
8.33 1.09 b 1.09 b
1.81 0.52 0.71
41Ca G.S. 7 / 2 -
4°Ca G.S. 0 +
f7/2 d3/2
4,3 2,1
8.36 15.6 c
0.9 0.52
3.73 3a Only underlined l values were used in the actual calculations. bThis is the binding energy difference between 41Ca and 4°Ca. c This is the binding energy difference between 39Ca and 4°Ca. parameters of ref. [1 ] in such a way that the elastic cross section is reproduced when the 3 - state is coupled. It was found that this can be achieved by changing the value o f r v = 1.27 fm of ref. [1] into r v = 1.30 fm, other parameters being unchanged. Bound state parameters were taken the same as in ref. [1 ], except that a spin orbit interaction with Vso = 8 MeV, rso = 1.25 fm and aso = 0.65 was added to the central potential of ref. [1 ]. The theoretical cross sections plotted in fig. 1, together with experimental data [ 1 ], were obtained by first taking into account only the direct transition (o0), then adding the indirect transition via the 3 state (o03), and adding finally the transition via the 5 - states (0035). It should be noted that o 0 is almost the same as the DWBA cross section, i.e. o 0 -~ o (DwBA). For the (13C, 12C) reaction we first note in fig. 1 that o 0 "" o03 , i.e. o (CCBA)~ o (DWBA) (although 003 does agree better with experiment than does o0). The fact that o (CCBA) -~ o (DwBA) means that the effect of the multi-step process is extremely small in this reaction, which can also be inferred from the fact that the cross section,obtained by considering a pure indirect transition is about three orders of magnitude smaller than that which is obtained with a pure direct transition. Since DWBA fits the data [1], and CCBA is close to DWBA, the CCBA result also fits the data. This conclusion is not altered even when the CCBA cross section is obtained by coupling also the 5 state [7]. The situation is very different in the (13C, 14N) reaction, where the indirect transitions are competitive with the direct transition. As is seen in fig. 1, o03,
including only the 3 - coupling, already differs markedly from o 0 and thus from o(DWBA), although the difference is not large enough to reproduce the observed cross section. When the 5 - coupling is also added, however, we obtain o035 which has an angular distribution completely out-of-phase from that of o 0, thus agreeing very nicely with experiment. The mysterious feature of the (13C, 14N) reaction data has thus been well accounted for. We shall now present several reasons for why the two-step process is so strong in the (13C, 14N) reaction, but is weak in the (13C, 12C) reaction. (i) The first is the difference in the single particle binding energies. In the two-step mode of the (13C, 14N) reaction, a proton is picked up from the f7/2 orbit, after being excited from the d3/2 orbit to form the 3 - state at 3.73 MeV (or 5 - state at 4.48 MeV). With the usual separation energy procedure, this f7/2 proton is less bound by 3.73 MeV (or by 4.48 MeV) compared with the d3/2 proton that is picked up in the one-step process; a fact which greatly favors the two-step processes. A more careful treatment based on the formalism of Pinkston and Satchler [8] shows, however, that the energy to be used in our case is not the separation energy, but just the single-particle (SP) energy that one has before the RPA calculation is made. As is seen in table 1, the SP energy of the f7/3 proton is as small as 1.1 MeV. The corresponding wave function thus has a large amplitude in the external region, and the two-step process can be very strong. The same argument applies to the stripping of a neutron into the d3/2 hole in the 3 - state, when the (13C, 12C) reaction proceeds via the two-step mode.
Volume 67B, number 1 f
PHYSICS LETTERS
i
i
i
4°Ca(ISC,)2C)4'Ca(g
----
i
i
(o)
s r/z )
00
io.o
14 March 1977
Q-matched. In the case of the (13C, 12C) reaction the optimum Q value is zero, while the actual Q values o f the direct and indirect transitions via the 3 - and 5 states are, respectively, 3.41, 7.14 and 7.89 MeV. Therefore, in this case the transition via the excited states are rather strongly Q-mismatched. (iii) The third reason is the difference in the coherence of intermediate contributions. Somewhat symbolically, one may denote the two-step process as lal~ l a ~ lb, which means that we have in mind a process in which a partial wave l a in the incident channel makes a transition into l a m the excited ( 3 - or 5 - ) channel, and then finally into Ib in the exit channel. In these two steps, angular momenta l' and 1" are transferred (l' being fixed at 3 or 5 in our case). Let us take all the amplitudes with a fixed set of values of Ia, lb and l', and sum them over l a and l". A surprising fact we found was that this sum was constructive for the (13C, 14N) reaction, but was destructive for the (13C, 12C) r e a c t i o n . This favors the importance o f the two-step processes for the (13C, 14N) reaction, but suppresses them in the (13C, 12C) reaction. Which of the above three factors is the most important? In order to give a possible answer to this question, let us first note that, in the process of obtaining the a03 cross sections of fig. 1, we found that R ~ 10, where R is defined as the absolute ratio of the amplitude of the two-step process in the (13C, 14N) reaction divided by that in the (13C, 12C) reaction. We may then carry out a new calculation for the (13C, 12C) reaction by modifying somewhat artifically parameters so that the two-step process in this reaction becomes as favorable as that in the (13C, 14N) reaction, as far as the above two points (i) and (ii) are concerned. We achieved this by first making the SP binding energy of the d3/2 neutron equal to 1.1 MeV (rather than 15.6 MeV; cf. table 1), and then assigning a (negative) binding energy to the 3 - state so that the 3 - ~ 7 / 2 transition proceeds with an optimum Q-value; Q = 0. The result of this new calculation showed that the new value of R is R ~ x,/]O. This means that the above reasons (i) and (ii) together account for about half the difference in the importance of the two-step processes in the two reactions. The remaining half must be accounted for by the coherence (iii), which is thus the most important, It is seen that we now have a rather clear understanding o f what is taking place in the normal and anomalous t
IO
-_-
(b)
4 ° C 0 ( ' 3C '4N)39K (g.s ' 3/;~ )
u~
m
.
t
br4 -----
O- 0
io.o 1
,:7",
~~
1.0
i
5
i
I0
i
15
i
20
2~5
3
;
ec,m,
Fig. 1. Comparison o f EFR-CCBA calculations with experiment with Elab(4°Ca) = 68 MeV. Three types o f theoretical cross sections oo, ao3 and ao3 s are explained in the text. Note that o o ~ o(DWBA). The normalization factors used for (13C, 12C) and (13C, 14N) reactions were, respectively, 1.65 and 1.45.
The SP energy of this neutron is as large as 15.6 MeV, strongly inhibiting the two-step process. (ii) The second reason is the difference in the Q-matching condition. The optimum Q value for the (13C, 14N) reaction is found to be about 5.7 MeV, while the actual Q value for the 0 + ~ 3/2 + transition is - 0 . 7 8 MeV, making the one-step transition rather strongly Q-mismatched. The transitions via the 3 and 5 - states, on the other hand, have Q = 2.95 and 3.70 MeV, respectively, and thus are much better
Volume 67B, number 1
PHYSICS LETTERS
processes of ref. [ 11. Why the nature of the coherence differs so drastically in the two reactions is not so easy to find out, however. Obviously very complicated dynamics are involved. It is nevertheless clear that the coherence does not depend strongly on the mass number. The experiment of ref. [I] showed that the 3/Z+ cross sections in A Ca(13C, 14N) reactions with A f 40 also behaved anomalously. We believe that the technique used here for A = 40 applies to these cases too. It is known that the angular distribution of the 4oCa (L3C, 14N)39K(1/2+) reaction is also anomalous [l], but we have performed a CCBA calculation for this case too and found that the anomaly is very well accounted for. As is seen in ref. ]l ] , the anomaly in this case is somewhat weaker than it is in the 3/2+ final state. This is reasonable because in the l/2’ case the step that proceeds via 5- state is not possible: The .5- state cannot include an (f7,2s&) configuration. It is a pleasure to acknowledge fruitful discussion with Drs. M.C. Mermaz and J.C. Peng. We thank the
14 March 1977
authors of ref. [I 1, in particular Dr. P.D. Bond for making available to us the details of the data, and for his encouraging discussions.
References [I] P.D. Bond et al., Phys. Rev. Lett. 36 (1976) 300. [2] T. Tamura, Phys. Reports 14C (1974) 59. [3] T. Tamura, KS. Low and T. Udagawa, Phys. Lett. 51B (1974) 116. [4] C.R. Gruhn et al., Phys. Rev. C6 (1972) 915; R.A. Eisenstein et al., Phys. Rev. 188 (1069) 1815. [5] W.J. Gerace and A.M. Green, Nucl. Phys. All3 (1968) 641. [6] S. Cohen and D. Kurath, Nucl. Phys. Al01 (1967) 1. f7] K.S. Low, T. Tamura and T. Udagawa, unpublished; K.S. Low, to be published in: Proc. of European Conf. on Nuclear physics with heavy ions, Caen, France, September, 1976. [S] W.T. Pinkston, R.J. Philpott and G.R. Satchler, Nucl. Phys. Al25 (1969) 176.