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Effect of deformation on the angular correlation of sequentially emitted light particles after fusion near the Coulomb barrier
Volume 127B, number 1,2
PHYSICS LETTERS
21 July 1983
EFFECT OF DEFORMATION ON THE ANGULAR CORRELATION OF SEQUENTIALLY EMH'TED LIGHT PARTICLES AFTER FUSION NEAR THE COULOMB BARRIER N. ALAMANOS, C. LEVI, C. Le METAYER, W. MITTIG 1 and L. PAPINEAU
DPh-N,CENSaclay,91191 Gif-sur-Yvette Cedex,France Received 7 February 1983
Angular correlations of light particles are calculated for sequential statistical decay following fusion. It is shown that these correlations are sensitive to the deformation, and the calculation is compared to experimental data for the reaction 28Si(12C 2a)32S_o It is not possible to reproduce the angular correlation governed by the angular momenta of the first step transition 4 0 ~ , ~ 36At, using standard transmission coefficients. Good quantitative agreement is obtained introducing a strong deformation (#2 ~ 1.1) compatible with the deformation of fusion states as predicted from the TDHF calculations, but much higher than expected from the rotating liquid drop model.
By the measurement of angular correlations o f sequentially emitted light particles from a highly excited compound nucleus one can obtain information about the angular momenta involved in the transitions [ 1 - 3 ]. This possibility motivated us to make detailed statistical model calculations. We will especially consider here sequential a-decay because this reaction selects angular momenta peaked at l~. We have studied the reaction 12C + 2~Si leading to the low lying states of 32S by emission o f 8Be or of two sequential a-particles using a 28Si beam with energies from 60 MeV to 90 MeV and different detection techniques and geometries. In this letter, we focus the attention on angular correlation measurements between the two sequentially emitted alpha particles. The angular correlation measurements correspond to angles in the neighbourhood of 0cm = 180 ° where direct processes are expected to be very small. The beam o f 28Si was delivered by the Saclay Tandem Van de Graaff. The angular correlation measurements have been done at 70 MeV incident energy and the two alpha particles were detected in coincidence between a standard surface barrier A E - E 1 Present address: GANIL, B.P. 5027, 14021 Caen Cedex, France.
telescope fixed at zero degree and a movable position sensitive detector. The telescope (AtE = 100/2m, E = 3000/2m) subtended a solid angle of 1.5 msr and the movable detector (E = 500/2m, S = 1 X 5 cm 2) covered an angular spread of 30 ° . For the position sensitive detector particle identification is accomplished due to the limit of energy loss of the protons. A 14/~m Ta foil was used in fron of the counters in order to absorb the scattered 28Si ions. An energy resolution of 300 keV in the energy sum spectra Eal + Ea2 was obtained (insert of fig. 1) and the ground state of 32S was clearly separated from other states. It was verified that the statistical model reproduced the relative intensities of the different final states of groups of final states well. The experimental error on absolute values of the cross section is estimated to be +50% and good agreement between experimental and calculated cross sections was obtained. Thus, the use of the statistical model is justified. The cross section for statistical sequential decay, for spinless particles, final spin zero and special geometries can be written, [1,3,4] d4a(go 'El 'g2 ,~1 ' ~ 2 ) dE1 dE2 d~21 d ~ 2 =
1
~ja(Eo'E1 'E2'J) (47r)~ P)(g2).(1) 23
0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
Volume 127B, n u m b e r 1,2
PHYSICS LETTERS
21 July 1983
2.o" '-' 3000 ,m
i
2000 t
loooI 30
4.0 50 E~.E2
10
I
10
15
20
25
30
35
/.0 ELab
Fig. 1. Projection of the c~-~ coincident events leading to 32Sg s obtained with a split detector centered at 0 ° and compared to a statistical model calculation (line) for a 1 = ~ 2 = 0°- The arrow indicates an instrumental cut-off. The insert shows the s u m spect r u m (E~I + E~2 ) corrected for energy loss in the foils.
The index 0, 1,2, refers to the entrance channel, the first and second step emissions respectively. The directions of the emitted particles are given by ~21 and ~22, E are the excitation energies of the nuclei involved. This form of the cross section is valid in three cases: (1) If E21 = ~22, J is the angular momentum of the initial compound nucleus. (2) If f22 = 0 , J is the angular momentum of the first transition. (3) If ~ 1 = 0, J is the angular momentum of the intermediate nucleus. The explicit form of a(E0, E l , E2, J ) has been given in ref. [5], It depends, apart from geometrical coefficients, only on the statistical partial branching ratios b ( E i , J i , li+ l ,Ji+ l , Ei+I)To obtain the branching ratios needed, the code CASCADE [6] was modified. The level densities were calculated using standard parameters [7] and included experimental discrete states for low excitation energies. With these parameters, good general agreement with experimental evaporation-residue cross sections in this mass region had been obtained within a factor two [7]. We applied this factor to the calculated cross sections. The formation cross section of the compound nucleus is calculated using crl = rr ~,2(21 + 1)T/,
(2)
where Tf = [1 + exp (t - L o ) / d f ] 24
-1
The formation cross sections and branching ratios were introduced in a code (CORANG) written to calculate the angular correlations using the formula given above. In order to compare with experimental data, the calculations were done in the laboratory system. The calculated energy spectrum (fig. 1) shows that the first step (¢Xl) and second step (¢x2) alpha particles can be selected by energy conditions. This energy separation in the statistical model calculation is due to the fact that relatively low excitation energies in 36Ar are favoured since we impose decay to the ground state of 32S. The experimental data (fig. 1) agree with the statistical model calculation. The absence of the central dip is due to the presence in the geometry of this measurement of the alpha-alpha events of 8Begs decay, but the angular correlations discussed below are out of the kinematical domain of this reaction. The subsequent discussion of the angular correlation concerns the position of the first oscillation and not the absolute values. It is the position of the oscillation that reflects the angular momenta involved in the transitions concerned in each special geometry. The population of the initial nucleus is determined by L 0 and df [eq. (2)] and is related to the angular correlation for the geometry ~ 1 = ~ 2 " L 0 has been chosen to reproduce the total fusion cross section [7 ] of = ?_,lOl . This value is L 0 = 13 for E = 70 MeV for the system 12C + 28Si, and ho significant change can be made without distroying the agreement with the data of ref. [7].
Volume 127B, number 1,2
PHYSICS LETTERS
21 July 1983 10 __ 13=0 df=l.0 _____ 13=1.1 df=l.0
100t', 12C(28Sic~ cx)32Sg.s. \~ EL(28Si)=70MeV
E~b=31.5 HeY N
~,~A ,_ x
__
df=lO
D.I=O
N
tab E= =19.5 HeV .0-2=0
1
1~)
210
30
+\+
Otab Fig. 2. Energy integrated angular correlations for s21 = fZ2 . The statistical model calculation is done with df = 1.0 (full line) and df = 1.5 (dashed line) and standard T1 (# = 0). °-1 Lt0
The diffuseness d f was chosen to be 1 in ref. [7]. Due to the fact the T : for fusion must be always smaller than the optical model T! o f ref. [8] the estimation for an upper limit of d f is 1.5. In fig. 2 are reported, together with the experimental angular correlation f21 = ~22, the calculations for d f = 1 and d f = 1.5. From the position of the experimental maximum there is good evidence that d f has not to be taken greater than d f = 1. The experimental fusion cross section and the measured angular correlation o f fig. 2 permit a maximum variation o f L 0 by +1. Anyhow the variation o f d r and L 0 within the above limits has no significant influence on the angular correlations o f fig. 3. The experimental angular correlation selecting aparticles detected at 0 ° around 19.5 MeV (lower part o f fig. 3) corresponds to ~ 2 = 0 and is determined by the mean angular m o m e n t u m o f the first step transition. Selecting 31.5 MeV in the 0 ° detector (upper part o f fig. 3) corresponds to ~ 1 = 0 and depends on the angular momenta involved in the second step. The full line curves o f fig. 3 correspond to the calculation with d f = 1 [eq. (2)] and standard transmission coefficients. The agreement in absolute value is satisfactory for a statistical model calculation. We will focus attention now on the position o f the maximum o f the first oscillation. For ~ 1 = 0 (upper part in fig. 3), the
J 1~)
20
J 0tab
I
30
Fig. 3. Angular correlation for two energies o f a-particles detected at zero degree with 31.5 +- 1.5 MeV for the upper part governed by the a6Ar* --* 32 S~. transition angular m o m e n t a
and 19.5 + 1.5 MeV governed~y the 4°Ca + 36Ar* transition angular momenta for the lower part. The full line corresponds to the statistical model calculation with #2 = 0, the dashed line to #2 = 1.1.
good agreement shows that the angular momenta involved in the second step are correctly described. However, for ~22 = 0 (lower part in fig. 3), the calculation using standard transmission coefficients (full line) is completely out of phase with respect to the experiment. This disagreement signifies that the mean /-value o f the first transitions is greater than predicted. Because of the good agreement for the angular momenta involved in the entrance channel and in the second step, this discrepancy can be due only to inadequate transmission coefficients for the first transition. The T l normally used in statistical calculations are obtained from an optical model fit to elastic scattering from the ground state o f the corresponding nucleus. This assumption is not valid for the first transition taking place between the fusion state o f 4°Ca and highly excited states o f the intermediate nucleus 36Ar and it is reasonable to expect deviation from the 25
Volume 127B, number 1,2
PHYSICS LETTERS
standard T l [6]. We considered here the modification that could be caused by deformation, using the methods of ref. [9]. Optical model transmission coefficients were calculated introducing an effective radius r~ and a Coulomb barrier V~ that depend on the deformation. The deformation was introduced for protons and a particles and not for the neutrons which represent only a very small fraction of the cross section. The T l for second step emission were not modified. A search on ~2 yielded good agreement of the calculated position of the oscillation with the data for /32 = 1.1 as seen in fig. 3 (lower part) the estimated uncertainty for/32 is +3. For/32 = 1.1, we obtained rl.1/r 0 = 1.55 and VI.1/V 0 = 0.77 for Ar + a. As expected, the agreement for ~1 = 0 (upper part of fig. 3) is not destroyed for ~2 = 1.1 within the experimental uncertainty of the position of the maximum since only the parameters of the first transition have been changed. The analysis done for successive energy bins of a particles detected at zero degrees gives the same results. From 34.5 MeV to 22.5 MeV (o~1 contribution, see fig. 1) the experimental angular correlations are in phase with the calculation. From 22.5 MeV to 13.5 MeV (a 2 contribution) the experiment is out of phase with the calculation for/32 = 0 and in phase with the ~2 = 1.1 calculation. In conclusion, a good fit to the data was obtained using a big deformation in the initial stage o f the reaction 28Si(12C, 2ot)32Sg s. This evidence has been obtained by a statistical model calculation of the angular correlation of sequentially emitted a-particles, to our knowledge the first complete calculation of this kind. Such a big deformation is compatible with fusion configurations obtained in TDHF calculations [10,
26
21 July 1983
11 ], but is much higher than expected from the rotating liquid model that, for the moderate angular momenta involved here (J0 = 14,J1 ~ 7), predicts a deformation ~ ~ 0.05 (ref. [12]). Therefore the effect discussed here is different from the modification of the energy spectra for a-particles due to the deformation induced by high angular momentum predicted in ref. [13]. We have shown that the deformation can be obtained from the statistical model analysis of angular correlations of sequentially emitted particles. Particularly appropriate for such studies are reactions leading to final spin zero, in the vicinity of 0cm = 180 ° where all direct contributions are small. We think that the method exposed here is a powerful tool for the study of the dynamics of fusion states.
References [1] E. Da Silveira, Th~se d'Etat (Orsay, 1977), unpublished. [2] D. Guerreau and R. Babinet, J. Phys. (Paris) 41, Coll. C10 (1980) 217. [3] Tai Kuang-Hsi, T. Dossing, C. Gaarde and J.S. Larsen, Nucl. Phys. A316 (1979) 189. [4] L.C. Biederharn, in: Nuclear spectroscopy, part B, ed. F. Ajzenberg-Selove (Academic Press, New York, 1960). [5] N. Alamanos, C. Le M&ayer, C. L6vi, W. Mittig and L. Papineau, Intern. Winter Meeting on Nuclear physics (Bormio, 1982) (University of Milan, Milan). [6] F. PiJlhofer, Nucl. Phys. A280 (1977) 267. [7] S. Gary and C. Volant, Phys. Rev. C25 (1982) 1877. [8] C.M. Cheng et al., Phys. Rev. C20 (1979) 1042. [9] M. Beckerman and M. Blann, Phys. Rev. Lett. 42 (1979) 156. [10] H. Flocard, S.E. Koonin and M.S. Weiss, Phys. Rev. C17 (1978) 1682. [11] P. Bonche and B. Grammaticos, Phys. Lett. 59B (1980) 198. [12] S. Cohen, F. Plasil and W.J. Swiatecki, Ann. Phys. (NY) 82 (1974) 557. [13] M. Blann, Phys. Rev. C21 (1980) 1770.
PHYSICS LETTERS
21 July 1983
EFFECT OF DEFORMATION ON THE ANGULAR CORRELATION OF SEQUENTIALLY EMH'TED LIGHT PARTICLES AFTER FUSION NEAR THE COULOMB BARRIER N. ALAMANOS, C. LEVI, C. Le METAYER, W. MITTIG 1 and L. PAPINEAU
DPh-N,CENSaclay,91191 Gif-sur-Yvette Cedex,France Received 7 February 1983
Angular correlations of light particles are calculated for sequential statistical decay following fusion. It is shown that these correlations are sensitive to the deformation, and the calculation is compared to experimental data for the reaction 28Si(12C 2a)32S_o It is not possible to reproduce the angular correlation governed by the angular momenta of the first step transition 4 0 ~ , ~ 36At, using standard transmission coefficients. Good quantitative agreement is obtained introducing a strong deformation (#2 ~ 1.1) compatible with the deformation of fusion states as predicted from the TDHF calculations, but much higher than expected from the rotating liquid drop model.
By the measurement of angular correlations o f sequentially emitted light particles from a highly excited compound nucleus one can obtain information about the angular momenta involved in the transitions [ 1 - 3 ]. This possibility motivated us to make detailed statistical model calculations. We will especially consider here sequential a-decay because this reaction selects angular momenta peaked at l~. We have studied the reaction 12C + 2~Si leading to the low lying states of 32S by emission o f 8Be or of two sequential a-particles using a 28Si beam with energies from 60 MeV to 90 MeV and different detection techniques and geometries. In this letter, we focus the attention on angular correlation measurements between the two sequentially emitted alpha particles. The angular correlation measurements correspond to angles in the neighbourhood of 0cm = 180 ° where direct processes are expected to be very small. The beam o f 28Si was delivered by the Saclay Tandem Van de Graaff. The angular correlation measurements have been done at 70 MeV incident energy and the two alpha particles were detected in coincidence between a standard surface barrier A E - E 1 Present address: GANIL, B.P. 5027, 14021 Caen Cedex, France.
telescope fixed at zero degree and a movable position sensitive detector. The telescope (AtE = 100/2m, E = 3000/2m) subtended a solid angle of 1.5 msr and the movable detector (E = 500/2m, S = 1 X 5 cm 2) covered an angular spread of 30 ° . For the position sensitive detector particle identification is accomplished due to the limit of energy loss of the protons. A 14/~m Ta foil was used in fron of the counters in order to absorb the scattered 28Si ions. An energy resolution of 300 keV in the energy sum spectra Eal + Ea2 was obtained (insert of fig. 1) and the ground state of 32S was clearly separated from other states. It was verified that the statistical model reproduced the relative intensities of the different final states of groups of final states well. The experimental error on absolute values of the cross section is estimated to be +50% and good agreement between experimental and calculated cross sections was obtained. Thus, the use of the statistical model is justified. The cross section for statistical sequential decay, for spinless particles, final spin zero and special geometries can be written, [1,3,4] d4a(go 'El 'g2 ,~1 ' ~ 2 ) dE1 dE2 d~21 d ~ 2 =
1
~ja(Eo'E1 'E2'J) (47r)~ P)(g2).(1) 23
0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
Volume 127B, n u m b e r 1,2
PHYSICS LETTERS
21 July 1983
2.o" '-' 3000 ,m
i
2000 t
loooI 30
4.0 50 E~.E2
10
I
10
15
20
25
30
35
/.0 ELab
Fig. 1. Projection of the c~-~ coincident events leading to 32Sg s obtained with a split detector centered at 0 ° and compared to a statistical model calculation (line) for a 1 = ~ 2 = 0°- The arrow indicates an instrumental cut-off. The insert shows the s u m spect r u m (E~I + E~2 ) corrected for energy loss in the foils.
The index 0, 1,2, refers to the entrance channel, the first and second step emissions respectively. The directions of the emitted particles are given by ~21 and ~22, E are the excitation energies of the nuclei involved. This form of the cross section is valid in three cases: (1) If E21 = ~22, J is the angular momentum of the initial compound nucleus. (2) If f22 = 0 , J is the angular momentum of the first transition. (3) If ~ 1 = 0, J is the angular momentum of the intermediate nucleus. The explicit form of a(E0, E l , E2, J ) has been given in ref. [5], It depends, apart from geometrical coefficients, only on the statistical partial branching ratios b ( E i , J i , li+ l ,Ji+ l , Ei+I)To obtain the branching ratios needed, the code CASCADE [6] was modified. The level densities were calculated using standard parameters [7] and included experimental discrete states for low excitation energies. With these parameters, good general agreement with experimental evaporation-residue cross sections in this mass region had been obtained within a factor two [7]. We applied this factor to the calculated cross sections. The formation cross section of the compound nucleus is calculated using crl = rr ~,2(21 + 1)T/,
(2)
where Tf = [1 + exp (t - L o ) / d f ] 24
-1
The formation cross sections and branching ratios were introduced in a code (CORANG) written to calculate the angular correlations using the formula given above. In order to compare with experimental data, the calculations were done in the laboratory system. The calculated energy spectrum (fig. 1) shows that the first step (¢Xl) and second step (¢x2) alpha particles can be selected by energy conditions. This energy separation in the statistical model calculation is due to the fact that relatively low excitation energies in 36Ar are favoured since we impose decay to the ground state of 32S. The experimental data (fig. 1) agree with the statistical model calculation. The absence of the central dip is due to the presence in the geometry of this measurement of the alpha-alpha events of 8Begs decay, but the angular correlations discussed below are out of the kinematical domain of this reaction. The subsequent discussion of the angular correlation concerns the position of the first oscillation and not the absolute values. It is the position of the oscillation that reflects the angular momenta involved in the transitions concerned in each special geometry. The population of the initial nucleus is determined by L 0 and df [eq. (2)] and is related to the angular correlation for the geometry ~ 1 = ~ 2 " L 0 has been chosen to reproduce the total fusion cross section [7 ] of = ?_,lOl . This value is L 0 = 13 for E = 70 MeV for the system 12C + 28Si, and ho significant change can be made without distroying the agreement with the data of ref. [7].
Volume 127B, number 1,2
PHYSICS LETTERS
21 July 1983 10 __ 13=0 df=l.0 _____ 13=1.1 df=l.0
100t', 12C(28Sic~ cx)32Sg.s. \~ EL(28Si)=70MeV
E~b=31.5 HeY N
~,~A ,_ x
__
df=lO
D.I=O
N
tab E= =19.5 HeV .0-2=0
1
1~)
210
30
+\+
Otab Fig. 2. Energy integrated angular correlations for s21 = fZ2 . The statistical model calculation is done with df = 1.0 (full line) and df = 1.5 (dashed line) and standard T1 (# = 0). °-1 Lt0
The diffuseness d f was chosen to be 1 in ref. [7]. Due to the fact the T : for fusion must be always smaller than the optical model T! o f ref. [8] the estimation for an upper limit of d f is 1.5. In fig. 2 are reported, together with the experimental angular correlation f21 = ~22, the calculations for d f = 1 and d f = 1.5. From the position of the experimental maximum there is good evidence that d f has not to be taken greater than d f = 1. The experimental fusion cross section and the measured angular correlation o f fig. 2 permit a maximum variation o f L 0 by +1. Anyhow the variation o f d r and L 0 within the above limits has no significant influence on the angular correlations o f fig. 3. The experimental angular correlation selecting aparticles detected at 0 ° around 19.5 MeV (lower part o f fig. 3) corresponds to ~ 2 = 0 and is determined by the mean angular m o m e n t u m o f the first step transition. Selecting 31.5 MeV in the 0 ° detector (upper part o f fig. 3) corresponds to ~ 1 = 0 and depends on the angular momenta involved in the second step. The full line curves o f fig. 3 correspond to the calculation with d f = 1 [eq. (2)] and standard transmission coefficients. The agreement in absolute value is satisfactory for a statistical model calculation. We will focus attention now on the position o f the maximum o f the first oscillation. For ~ 1 = 0 (upper part in fig. 3), the
J 1~)
20
J 0tab
I
30
Fig. 3. Angular correlation for two energies o f a-particles detected at zero degree with 31.5 +- 1.5 MeV for the upper part governed by the a6Ar* --* 32 S~. transition angular m o m e n t a
and 19.5 + 1.5 MeV governed~y the 4°Ca + 36Ar* transition angular momenta for the lower part. The full line corresponds to the statistical model calculation with #2 = 0, the dashed line to #2 = 1.1.
good agreement shows that the angular momenta involved in the second step are correctly described. However, for ~22 = 0 (lower part in fig. 3), the calculation using standard transmission coefficients (full line) is completely out of phase with respect to the experiment. This disagreement signifies that the mean /-value o f the first transitions is greater than predicted. Because of the good agreement for the angular momenta involved in the entrance channel and in the second step, this discrepancy can be due only to inadequate transmission coefficients for the first transition. The T l normally used in statistical calculations are obtained from an optical model fit to elastic scattering from the ground state o f the corresponding nucleus. This assumption is not valid for the first transition taking place between the fusion state o f 4°Ca and highly excited states o f the intermediate nucleus 36Ar and it is reasonable to expect deviation from the 25
Volume 127B, number 1,2
PHYSICS LETTERS
standard T l [6]. We considered here the modification that could be caused by deformation, using the methods of ref. [9]. Optical model transmission coefficients were calculated introducing an effective radius r~ and a Coulomb barrier V~ that depend on the deformation. The deformation was introduced for protons and a particles and not for the neutrons which represent only a very small fraction of the cross section. The T l for second step emission were not modified. A search on ~2 yielded good agreement of the calculated position of the oscillation with the data for /32 = 1.1 as seen in fig. 3 (lower part) the estimated uncertainty for/32 is +3. For/32 = 1.1, we obtained rl.1/r 0 = 1.55 and VI.1/V 0 = 0.77 for Ar + a. As expected, the agreement for ~1 = 0 (upper part of fig. 3) is not destroyed for ~2 = 1.1 within the experimental uncertainty of the position of the maximum since only the parameters of the first transition have been changed. The analysis done for successive energy bins of a particles detected at zero degrees gives the same results. From 34.5 MeV to 22.5 MeV (o~1 contribution, see fig. 1) the experimental angular correlations are in phase with the calculation. From 22.5 MeV to 13.5 MeV (a 2 contribution) the experiment is out of phase with the calculation for/32 = 0 and in phase with the ~2 = 1.1 calculation. In conclusion, a good fit to the data was obtained using a big deformation in the initial stage o f the reaction 28Si(12C, 2ot)32Sg s. This evidence has been obtained by a statistical model calculation of the angular correlation of sequentially emitted a-particles, to our knowledge the first complete calculation of this kind. Such a big deformation is compatible with fusion configurations obtained in TDHF calculations [10,
26
21 July 1983
11 ], but is much higher than expected from the rotating liquid model that, for the moderate angular momenta involved here (J0 = 14,J1 ~ 7), predicts a deformation ~ ~ 0.05 (ref. [12]). Therefore the effect discussed here is different from the modification of the energy spectra for a-particles due to the deformation induced by high angular momentum predicted in ref. [13]. We have shown that the deformation can be obtained from the statistical model analysis of angular correlations of sequentially emitted particles. Particularly appropriate for such studies are reactions leading to final spin zero, in the vicinity of 0cm = 180 ° where all direct contributions are small. We think that the method exposed here is a powerful tool for the study of the dynamics of fusion states.
References [1] E. Da Silveira, Th~se d'Etat (Orsay, 1977), unpublished. [2] D. Guerreau and R. Babinet, J. Phys. (Paris) 41, Coll. C10 (1980) 217. [3] Tai Kuang-Hsi, T. Dossing, C. Gaarde and J.S. Larsen, Nucl. Phys. A316 (1979) 189. [4] L.C. Biederharn, in: Nuclear spectroscopy, part B, ed. F. Ajzenberg-Selove (Academic Press, New York, 1960). [5] N. Alamanos, C. Le M&ayer, C. L6vi, W. Mittig and L. Papineau, Intern. Winter Meeting on Nuclear physics (Bormio, 1982) (University of Milan, Milan). [6] F. PiJlhofer, Nucl. Phys. A280 (1977) 267. [7] S. Gary and C. Volant, Phys. Rev. C25 (1982) 1877. [8] C.M. Cheng et al., Phys. Rev. C20 (1979) 1042. [9] M. Beckerman and M. Blann, Phys. Rev. Lett. 42 (1979) 156. [10] H. Flocard, S.E. Koonin and M.S. Weiss, Phys. Rev. C17 (1978) 1682. [11] P. Bonche and B. Grammaticos, Phys. Lett. 59B (1980) 198. [12] S. Cohen, F. Plasil and W.J. Swiatecki, Ann. Phys. (NY) 82 (1974) 557. [13] M. Blann, Phys. Rev. C21 (1980) 1770.