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Quadropole excitation of 8Li in a microscopic three-cluster model
Physics Letters B 292 ( 1992) 235-238 North-Holland
PHYSICS LETTERS B
Quadrupole excitation of gLi in a microscopic three-cluster model P. D e s c o u v e m o n t 1 and D. Baye Physique Nucldaire Thdorique et Physique Math~matique, UniversitbLibre de Bruxelles, Campus Plaine, C.P. 229, B-I050 Brussels, Belgium Received 3 July 1992; revised manuscript received 11 August 1992
The generator coordinate method is used to investigate the SLi nucleus in a multiconfiguration (it+t) +n and (a+n) +t threecluster model. Without any parameter fit, it succeeds in reproducing several spectroscopic properties of 8Li, but contradicts the huge experimental value for the E2 transition probability between the ground and first excited states.
Recent experiments [ 1,2 ] on SLi scattering with an heavy target suggest a very large quadrupole excitation probability o f the SLi nucleus: B ( E 2 t ) = (30 + 15) e2 fm4 in SLi+ 12C data [ 1 ], and ( 5 5 + 15) e2fm 4 in SLi + natNi data [ 2 ]. In these experiments, involving a radioactive SLi beam (T~/2= 838 ms), the inelastic S L i ( 2 ÷ ) ( X , X)SLi(1 +) cross sections are measured, and fitted through a D W B A analysis. The B(E2~ ) values deduced from these analyses (28 + 14 Wu, 52 + 14 W u ) are much larger than the Weisskopf unit, and than a simple shell-model calculation ( B ( E 2 ) ~ 1 Wu, without any effective charge). This strong enhancement is still more surprising if one considers the well-established quadrupole m o m e n t o f the ground state Q ( 2 ÷ ) = (2.4 + 0.2) e fm 2 [ 3 ]. This rather small value is an indirect indication for a weak spatial extension o f the SLi ground-state wave function, and therefore cannot be interpreted consistently with the B ( E 2 ) value. We try to understand this discrepancy by using the generator coordinate method ( G C M ) [ 4 ]. The SLi wave functions are described in a three-cluster model, involving the alpha and triton particles, and a neutron. Two coupling modes are considered (see fig. 1 ): 7Li + n, with the lowest 3 - , ½_, 7_ and s - states of 7Li, and SHe + t, with the lowest 3_ and ½- states of 5He. The total wave function o f SLi with spin J and parity rt reads:
Chercheur qualifi~ FNRS.
t t
t3
R1
C)
R, Q Ct
c(
Fig. 1. Three-cluster configurations of SLi.
f ~u( R . ,
,R.~)O~. (R., , R.,)
,
otll n I n2
(1) where coefficients f ~ ' b ( R n , , Rn2) represent the generator function and where ~ t JM~ i ( R ~ , R,2) are projected Slater determinants [ 4 ]; in ( 1 ), a denotes the configuration, I the relative angular m o m e n t u m and I the channel spin, resulting from the coupling o f the 7Li and neutron spins. The 7Li nucleus is described by an u + t structure, with internal generator coordinate Rn2. For computer-time reasons, we select a single value, equal to 3.7 fm. This choice minimizes the 7Li binding energy for the V2 nucleon-nucleon interaction [ 5 ] with an oscillator parameter b = 1.50 fm. For the particle-unbound SHe nucleus, we also take a single generator coordinate R~2 equal to 1.0 fm. The generator coordinates Rn, associated to the 7 L i + n and ~ H e + t relative motion are selected from 2.4 fm to 8.4 fm with a step o f 1.2 fro. A simplified calculation (i.e. with 7Li ( ~ - ) + n and 7Li ( ½- ) + n configu-
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All fights reserved.
235
Volume 292, number 3,4
PHYSICS LETTERS B
rations only) has been presented in ref. [6], but no investigation of E2 transitions was performed. In the present letter, we show results obtained with different central nucleon-nucleon forces (Volkov V1, V2, V3 and V4) in order to evaluate the sensitivity of the SLi wave functions with respect to the interaction. The Coulomb potential is treated exactly, and a spin-orbit force [7] [with So=53.07 M e V f m 5, in order to reproduce the experimental excitation energy of the 7Li ( ½- ) state ] is introduced. Several properties of the SLi nucleus are experimentally well known [ 3 ]. The magnetic moment/t, the quadrupole moment Q and the B (M 1, 1 ÷ ~ 2 ÷ ) transition probability can therefore be used as a test of the model. In addition, the proton width Fp of the first excited state in the mirror 8B nucleus is also well known. Before presenting results obtained with the total wave functions ( 1 ), we use a simplified model, where we calculate the binding energy of the 2 ÷ and 1 + 8Li states as a function of the generator coordinates R~ (between 7Li and n) and R2 (between a and t). This preliminary investigation, where a single Slater determinant is employed in ( 1 ), is expected to provide a qualitative information concerning the 8Li structure. The 5He + t configuration, which is negligible for the 2 + and 1 + states, has been omitted here. In this first approach, the nucleon-nucleon interaction is the Volkov force V2 with the standard Majorana parameter M = 0.6. From results presented in fig. 2, it can be seen that the spatial structures of the 2 + and 1+ states are, in the GCM, very similar. The energy difj=
2+
=1
+
,3
4
1
0
~
2
.3
4
5
0
6 R1
1
2
5
6
(fro)
Fig. 2. Binding energy of the 8Li 2 + and 1 ÷ states as a function of the 7Li+ n generator coordinate R~ and c~+ t generator coordinate R2. The nucleon-nucleon interaction is V2 with M = 0.6. The contour lines are plotted by steps of 1 MeV.
236
15 October 1992
ference at the minimum is about 1 MeV, in agreement with experiment (0.98 MeV). The precise location of the minimum is (R~=2.67 fm, R2=3.26 fm) for the 2 + ground state, and (R1=2.83 fm, R2= 3.37 fm) for the 1 ÷ state. We therefore conclude that, in the 8Li bound states, the best R2 value should be close to 3.3 fm. This is slightly lower than our choice (3.7 fm) which is intended to optimize the 7Li binding energy. The additional neutron therefore brings closer the et particle and the triton. Consequently, we may expect that matrix elements involving only the spatial dependence of the wave function, like the quadrupole moment, will be slightly overestimated in our model. Fig. 2 also suggests that the SLi nucleus cannot be considered as a 7Li core surrounded by a neutron. At the minimum energy, R1, is significantly lower than R2. Even if this picture is schematic only, it indicates that, in 8Li, the additional neutron is rather inside the 7Li core than orbiting far from it. After this qualitative study of SLi, we come now to the full model, corresponding to the mixing of generator coordinates presented in ( 1 ). For each Volkov interaction, the Majorana parameter M i s adjusted to the SLi binding energy with respect to the 7Li ( 3 _ ) + n threshold. We obtain M = 0.6234, 0.6300, 0.6265 and 0.6242 for V1, V2, V3 and V4 respectively. These values are very close to each other, and to the "standard" value M = 0.6. When the nucleon-nucleon force is determined, the model does not contain any free parameter. We give in table 1 some spectroscopic properties obtained with the four nucleon-nucleon interactions. Let us recall that there is no parameter fit. It appears that the GCM results are almost insensitive to the choice of the force. As expected from the choice of the R2 generator coordinate in 7Li, the GCM quadrupole moment is slightly too large, if one compares with experiment. However, the order of magnitude is quite good, and a part of the remaining discrepancy should be cancelled by a smaller value of R2. A striking result is the large difference between the 7Li and SLi quadrupole moments. The experimental [3] value in 7Li is ( - 3 . 4 + 0 . 6 ) e f m 2, and the GCM gives [6] Q(7Li) = - 4 . 0 e f m 2. In order to understand this strong difference with 8Li, we have used a simplified local model, involving a point-like 7Li ( ~ - ) nucleus and a neutron. Neglecting antisym-
Volume 292, number 3,4
PHYSICS LETTERSB
150~ober 1992
Table 1 Spectroscopicproperties of SLi.
exp Vl V2 Va V4
#(2 + )
Q(2 + )
E~(I + )
/'p (SB, 1 + )
B ( M 1 , 1 +--.2 + )
B(E2, 2+-. 1+ )
(/iN)
(efm2)
(MeV)
(keY)
(#~ fm 2)
(e2fm4)
1.65 1.41 1.39 1.43 1.45
2.4+0.2 3.10 3.10 3.08 3.07
0.98 0.54 0.51 0.59 0.61
37+5 54 55 51 50
5.0_+ 1.6 7.0 7.0 6.8 6.8
30_+15, 55-+15 2.1 2.1 2.1 2.1
metrization between them yields for the quadrupole moment of 8Li a ( s L i ) = ½Q(7Li) + ~--~e(p 2) ,
(2)
where ( p 2 ) is the square of the mean distance between 7Li and neutron. According to the quadrupole moments of 7Li and 8Li, eq. (2) should require quite unphysical values for ( p 2 ) to be verified. This result supports the conclusion drawn from fig. 2, saying that the 8Li structure is very different from a 7Li core and an external neutron. Our theoretical Q(8Li) value shows that the GCM is able to explain this difference between the structures of 7Li and 8Li. The magnetic moment is improved with respect to our earlier study [ 6 ] (/z = 1.18/tN), which takes account of the 7Li ( 3 -, ½- ) "Jrn configurations only. The excitation energy of the 1 + state is also improved (we obtained 0.30 MeV in ref. [ 6 ] ); it is possible that the tensor force, which is missing in the present calculation, is partly responsible for the experimental energy difference. On the other hand, examination of fig. 2 suggests that a smaller R2 value might increase this excitation energy. In the mirror 8B nucleus, the 1 ÷ first excited state is located above the proton threshold. We have tuned the Majorana parameter in order to reproduce its experimental energy (0.64 MeV), and calculated the proton width. As for the quadrupole moment, the difference between the GCM value (see table 1 ) and experiment is most likely due to our 7Be description. At this point, we have shown that all spectroscopic properties concerning a single state are reasonably well reproduced by the present microscopic model, without adjusting any parameter. In addition, table 1 indicates that the M 1 reduced transition probability, involving the 2 ÷ and 1 + states simultaneously, is also in satisfactory agreement with experiment. This
means that the overlap between the 2 + and 1+ wave functions is realistic. If, in addition, one remembers that the good description of the GCM 2 ÷ wave function is supported by the quadrupole moment, one would expect an E2 reduced transition probability consistent with experiment. However, we obtain B ( E 2 ) = 2 . 1 e2fm 4 for each interaction, which is considerably lower than experiment. Of course, an important, but yet unknown, effect might be missing in the present three-cluster model, explaining the very large experimental B(E2~) value. However, this hypothesis is in contradiction with the good description of all other experimentally known quantities. A large B (E2t) would simultaneously require a broad spatial extension of the involved states and a strong overlap between them. The GCM description of the spatial dependence is supported by the 2 ÷ quadrupole moment and by the proton width of the 1+ resonance in 8B. Furthermore, the M1 transition probability is a good test of the theoretical overlap. Let us also point out that the microscopic model is able to reproduce several other excited states, such as the 3 ÷ (Ex=2.255 MeV) and 4 ÷ (Ex=6.55 MeV) states on which we do not focus in the present letter. The large discrepancy between experiment and a microscopic model which realistically reproduces several properties of low-lying states, certainly deserves new experimental and theoretical investigations. We would like to thank the Services de Programmation de la Politique Scientifique for a supercomputer grant on the CRAY-YMP computer of the U L B - V U B computer centre.
237
Volume 292, number 3,4
PHYSICS LETTERS B
References [ 1 ] R.J. Smith, J.J. Kolata, K. I.amkin, A. Morsad, F.D. Becchetti, J.A. Brown, W.Z. Liu, J.W. J~inecke, D.A. Roberts and R.E. Warner, Phys. Rev. C 43 ( 1991 ) 2346. [2] J.A. Brown, F.D. Becchetti, J.W. J~.necke, K. Ashktorab, D.A. Roberts, J.J. Kolata, R.J. Smith, K. I.amkin and R.E. Warner, Phys. Rev. Lett. 66 ( 1991 ) 2452.
238
15 October 1992
[ 3 ] F. Ajzenberg-Selove, Nucl. Phys. A 490 (1988 ) 1. [4] D. Baye and P. Descouvemont, Proe. 5th Intern. Conf. on Clustering aspects in nuclear and subnuelear systems (Kyoto, Japan, 1988), J. Phys. Soc. Jpn. 58 (1989), Suppl., p. 103. [5] A.B. Volkov, Nucl. Phys. A 74 (1965) 33. [6] P. Deseouvemont and D. Baye, Nucl. Phys. A 487 (1988) 420. [7] D. Baye and N. Pecher, Bull. C1. So. Acad. R. Belg. 67 ( 1981 ) 835.
PHYSICS LETTERS B
Quadrupole excitation of gLi in a microscopic three-cluster model P. D e s c o u v e m o n t 1 and D. Baye Physique Nucldaire Thdorique et Physique Math~matique, UniversitbLibre de Bruxelles, Campus Plaine, C.P. 229, B-I050 Brussels, Belgium Received 3 July 1992; revised manuscript received 11 August 1992
The generator coordinate method is used to investigate the SLi nucleus in a multiconfiguration (it+t) +n and (a+n) +t threecluster model. Without any parameter fit, it succeeds in reproducing several spectroscopic properties of 8Li, but contradicts the huge experimental value for the E2 transition probability between the ground and first excited states.
Recent experiments [ 1,2 ] on SLi scattering with an heavy target suggest a very large quadrupole excitation probability o f the SLi nucleus: B ( E 2 t ) = (30 + 15) e2 fm4 in SLi+ 12C data [ 1 ], and ( 5 5 + 15) e2fm 4 in SLi + natNi data [ 2 ]. In these experiments, involving a radioactive SLi beam (T~/2= 838 ms), the inelastic S L i ( 2 ÷ ) ( X , X)SLi(1 +) cross sections are measured, and fitted through a D W B A analysis. The B(E2~ ) values deduced from these analyses (28 + 14 Wu, 52 + 14 W u ) are much larger than the Weisskopf unit, and than a simple shell-model calculation ( B ( E 2 ) ~ 1 Wu, without any effective charge). This strong enhancement is still more surprising if one considers the well-established quadrupole m o m e n t o f the ground state Q ( 2 ÷ ) = (2.4 + 0.2) e fm 2 [ 3 ]. This rather small value is an indirect indication for a weak spatial extension o f the SLi ground-state wave function, and therefore cannot be interpreted consistently with the B ( E 2 ) value. We try to understand this discrepancy by using the generator coordinate method ( G C M ) [ 4 ]. The SLi wave functions are described in a three-cluster model, involving the alpha and triton particles, and a neutron. Two coupling modes are considered (see fig. 1 ): 7Li + n, with the lowest 3 - , ½_, 7_ and s - states of 7Li, and SHe + t, with the lowest 3_ and ½- states of 5He. The total wave function o f SLi with spin J and parity rt reads:
Chercheur qualifi~ FNRS.
t t
t3
R1
C)
R, Q Ct
c(
Fig. 1. Three-cluster configurations of SLi.
f ~u( R . ,
,R.~)O~. (R., , R.,)
,
otll n I n2
(1) where coefficients f ~ ' b ( R n , , Rn2) represent the generator function and where ~ t JM~ i ( R ~ , R,2) are projected Slater determinants [ 4 ]; in ( 1 ), a denotes the configuration, I the relative angular m o m e n t u m and I the channel spin, resulting from the coupling o f the 7Li and neutron spins. The 7Li nucleus is described by an u + t structure, with internal generator coordinate Rn2. For computer-time reasons, we select a single value, equal to 3.7 fm. This choice minimizes the 7Li binding energy for the V2 nucleon-nucleon interaction [ 5 ] with an oscillator parameter b = 1.50 fm. For the particle-unbound SHe nucleus, we also take a single generator coordinate R~2 equal to 1.0 fm. The generator coordinates Rn, associated to the 7 L i + n and ~ H e + t relative motion are selected from 2.4 fm to 8.4 fm with a step o f 1.2 fro. A simplified calculation (i.e. with 7Li ( ~ - ) + n and 7Li ( ½- ) + n configu-
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All fights reserved.
235
Volume 292, number 3,4
PHYSICS LETTERS B
rations only) has been presented in ref. [6], but no investigation of E2 transitions was performed. In the present letter, we show results obtained with different central nucleon-nucleon forces (Volkov V1, V2, V3 and V4) in order to evaluate the sensitivity of the SLi wave functions with respect to the interaction. The Coulomb potential is treated exactly, and a spin-orbit force [7] [with So=53.07 M e V f m 5, in order to reproduce the experimental excitation energy of the 7Li ( ½- ) state ] is introduced. Several properties of the SLi nucleus are experimentally well known [ 3 ]. The magnetic moment/t, the quadrupole moment Q and the B (M 1, 1 ÷ ~ 2 ÷ ) transition probability can therefore be used as a test of the model. In addition, the proton width Fp of the first excited state in the mirror 8B nucleus is also well known. Before presenting results obtained with the total wave functions ( 1 ), we use a simplified model, where we calculate the binding energy of the 2 ÷ and 1 + 8Li states as a function of the generator coordinates R~ (between 7Li and n) and R2 (between a and t). This preliminary investigation, where a single Slater determinant is employed in ( 1 ), is expected to provide a qualitative information concerning the 8Li structure. The 5He + t configuration, which is negligible for the 2 + and 1 + states, has been omitted here. In this first approach, the nucleon-nucleon interaction is the Volkov force V2 with the standard Majorana parameter M = 0.6. From results presented in fig. 2, it can be seen that the spatial structures of the 2 + and 1+ states are, in the GCM, very similar. The energy difj=
2+
=1
+
,3
4
1
0
~
2
.3
4
5
0
6 R1
1
2
5
6
(fro)
Fig. 2. Binding energy of the 8Li 2 + and 1 ÷ states as a function of the 7Li+ n generator coordinate R~ and c~+ t generator coordinate R2. The nucleon-nucleon interaction is V2 with M = 0.6. The contour lines are plotted by steps of 1 MeV.
236
15 October 1992
ference at the minimum is about 1 MeV, in agreement with experiment (0.98 MeV). The precise location of the minimum is (R~=2.67 fm, R2=3.26 fm) for the 2 + ground state, and (R1=2.83 fm, R2= 3.37 fm) for the 1 ÷ state. We therefore conclude that, in the 8Li bound states, the best R2 value should be close to 3.3 fm. This is slightly lower than our choice (3.7 fm) which is intended to optimize the 7Li binding energy. The additional neutron therefore brings closer the et particle and the triton. Consequently, we may expect that matrix elements involving only the spatial dependence of the wave function, like the quadrupole moment, will be slightly overestimated in our model. Fig. 2 also suggests that the SLi nucleus cannot be considered as a 7Li core surrounded by a neutron. At the minimum energy, R1, is significantly lower than R2. Even if this picture is schematic only, it indicates that, in 8Li, the additional neutron is rather inside the 7Li core than orbiting far from it. After this qualitative study of SLi, we come now to the full model, corresponding to the mixing of generator coordinates presented in ( 1 ). For each Volkov interaction, the Majorana parameter M i s adjusted to the SLi binding energy with respect to the 7Li ( 3 _ ) + n threshold. We obtain M = 0.6234, 0.6300, 0.6265 and 0.6242 for V1, V2, V3 and V4 respectively. These values are very close to each other, and to the "standard" value M = 0.6. When the nucleon-nucleon force is determined, the model does not contain any free parameter. We give in table 1 some spectroscopic properties obtained with the four nucleon-nucleon interactions. Let us recall that there is no parameter fit. It appears that the GCM results are almost insensitive to the choice of the force. As expected from the choice of the R2 generator coordinate in 7Li, the GCM quadrupole moment is slightly too large, if one compares with experiment. However, the order of magnitude is quite good, and a part of the remaining discrepancy should be cancelled by a smaller value of R2. A striking result is the large difference between the 7Li and SLi quadrupole moments. The experimental [3] value in 7Li is ( - 3 . 4 + 0 . 6 ) e f m 2, and the GCM gives [6] Q(7Li) = - 4 . 0 e f m 2. In order to understand this strong difference with 8Li, we have used a simplified local model, involving a point-like 7Li ( ~ - ) nucleus and a neutron. Neglecting antisym-
Volume 292, number 3,4
PHYSICS LETTERSB
150~ober 1992
Table 1 Spectroscopicproperties of SLi.
exp Vl V2 Va V4
#(2 + )
Q(2 + )
E~(I + )
/'p (SB, 1 + )
B ( M 1 , 1 +--.2 + )
B(E2, 2+-. 1+ )
(/iN)
(efm2)
(MeV)
(keY)
(#~ fm 2)
(e2fm4)
1.65 1.41 1.39 1.43 1.45
2.4+0.2 3.10 3.10 3.08 3.07
0.98 0.54 0.51 0.59 0.61
37+5 54 55 51 50
5.0_+ 1.6 7.0 7.0 6.8 6.8
30_+15, 55-+15 2.1 2.1 2.1 2.1
metrization between them yields for the quadrupole moment of 8Li a ( s L i ) = ½Q(7Li) + ~--~e(p 2) ,
(2)
where ( p 2 ) is the square of the mean distance between 7Li and neutron. According to the quadrupole moments of 7Li and 8Li, eq. (2) should require quite unphysical values for ( p 2 ) to be verified. This result supports the conclusion drawn from fig. 2, saying that the 8Li structure is very different from a 7Li core and an external neutron. Our theoretical Q(8Li) value shows that the GCM is able to explain this difference between the structures of 7Li and 8Li. The magnetic moment is improved with respect to our earlier study [ 6 ] (/z = 1.18/tN), which takes account of the 7Li ( 3 -, ½- ) "Jrn configurations only. The excitation energy of the 1 + state is also improved (we obtained 0.30 MeV in ref. [ 6 ] ); it is possible that the tensor force, which is missing in the present calculation, is partly responsible for the experimental energy difference. On the other hand, examination of fig. 2 suggests that a smaller R2 value might increase this excitation energy. In the mirror 8B nucleus, the 1 ÷ first excited state is located above the proton threshold. We have tuned the Majorana parameter in order to reproduce its experimental energy (0.64 MeV), and calculated the proton width. As for the quadrupole moment, the difference between the GCM value (see table 1 ) and experiment is most likely due to our 7Be description. At this point, we have shown that all spectroscopic properties concerning a single state are reasonably well reproduced by the present microscopic model, without adjusting any parameter. In addition, table 1 indicates that the M 1 reduced transition probability, involving the 2 ÷ and 1 + states simultaneously, is also in satisfactory agreement with experiment. This
means that the overlap between the 2 + and 1+ wave functions is realistic. If, in addition, one remembers that the good description of the GCM 2 ÷ wave function is supported by the quadrupole moment, one would expect an E2 reduced transition probability consistent with experiment. However, we obtain B ( E 2 ) = 2 . 1 e2fm 4 for each interaction, which is considerably lower than experiment. Of course, an important, but yet unknown, effect might be missing in the present three-cluster model, explaining the very large experimental B(E2~) value. However, this hypothesis is in contradiction with the good description of all other experimentally known quantities. A large B (E2t) would simultaneously require a broad spatial extension of the involved states and a strong overlap between them. The GCM description of the spatial dependence is supported by the 2 ÷ quadrupole moment and by the proton width of the 1+ resonance in 8B. Furthermore, the M1 transition probability is a good test of the theoretical overlap. Let us also point out that the microscopic model is able to reproduce several other excited states, such as the 3 ÷ (Ex=2.255 MeV) and 4 ÷ (Ex=6.55 MeV) states on which we do not focus in the present letter. The large discrepancy between experiment and a microscopic model which realistically reproduces several properties of low-lying states, certainly deserves new experimental and theoretical investigations. We would like to thank the Services de Programmation de la Politique Scientifique for a supercomputer grant on the CRAY-YMP computer of the U L B - V U B computer centre.
237
Volume 292, number 3,4
PHYSICS LETTERS B
References [ 1 ] R.J. Smith, J.J. Kolata, K. I.amkin, A. Morsad, F.D. Becchetti, J.A. Brown, W.Z. Liu, J.W. J~inecke, D.A. Roberts and R.E. Warner, Phys. Rev. C 43 ( 1991 ) 2346. [2] J.A. Brown, F.D. Becchetti, J.W. J~.necke, K. Ashktorab, D.A. Roberts, J.J. Kolata, R.J. Smith, K. I.amkin and R.E. Warner, Phys. Rev. Lett. 66 ( 1991 ) 2452.
238
15 October 1992
[ 3 ] F. Ajzenberg-Selove, Nucl. Phys. A 490 (1988 ) 1. [4] D. Baye and P. Descouvemont, Proe. 5th Intern. Conf. on Clustering aspects in nuclear and subnuelear systems (Kyoto, Japan, 1988), J. Phys. Soc. Jpn. 58 (1989), Suppl., p. 103. [5] A.B. Volkov, Nucl. Phys. A 74 (1965) 33. [6] P. Deseouvemont and D. Baye, Nucl. Phys. A 487 (1988) 420. [7] D. Baye and N. Pecher, Bull. C1. So. Acad. R. Belg. 67 ( 1981 ) 835.