- Email: [email protected]
Intermediate resonance of inelastic 12C + 12C scattering
Nuclear Physics A334 (1980) 177-188 ; © North-Holbnd Pwbliahlrt~ Co ., At>tsterdatn Not to be reproduced by photoprint or microfilm without written permission from the publisher
INTERMEDIATE RESONANCE OF INELASTIC '=C+ t= C SCATTERING OSAMU TANIMURA
Institute Jôr Nuclear Study" , Unluersity of Tokyo, Tanashi, Tokyo 188, Japan
Received l4 May (979 (Revised 24 July 1979) Abstract : The intermediate resonances observed in the inelastic ` = C+'~C cross sections to the single and mutual 2;(4 .43 MeV) excitations and the single 3~(9 .64 MeV) excitation are studied by the coupledchannel method with the use of the coupling interaction derived by the folding procedure between 'zC and "C . It is shown that the model is successful in reproducing the gross structures of the inelastic cross sections and espaially the correlated resonance energies of the inelastic channels . The inelastic resonanes are shown to be due to the molecular resonances in an adiabatic potential between two ' Z C, which reproduces correctly the coupled channel resonanes.
1. Introduction The oscillatory structures observed in the total inelastic cross sections of the single and mutual excitations of the 2i (4.43 MeV) state of t zC in the t ZC + t ZC scattering') have been discussed from the viewpoint of the molecular resonance formation Z~ s) and the non-resonant diffraction model 4), Similar gross structure has been found in the excitation functions of the elastic scattering at 90° [ref. S)] and the fusion cross section in the ' ZC+ t ZC system 6). In this paper we discuss on the oscillatory structure of the t ZC+ t ZC cross section confïning ourselves to the total inelastic cross sections in the ' zC+ 12C(2; ), 1 ZC(2;)+' ZC(2i ) and t 2C+' ZC(3 - ) channels, which are abbreviated in the following as (2+ , 0+ ), (2 +, 2 +) and (3 -, 0 +), respectively . From the viewpoint of the molecular resonance formation, the present author has shown that the maxima in the total inelastic cross sections to the single and mutual 2i excitations are respectively attributed to the resonances in the lowest and the second lowest adiabatic potentials between two nuclei Z). However, the fits to the data were poor in the magnitude of the cross sections and the correlation of the resonance energies between the single and mutual 2+ cross sections. This may be due to the strong coupling interaction used there. Kondo et al. a) have showed that the observed oscillatory features can be understood and reproduced by the band crossing model'), which is the extension of the Nogami-Imanishi model 8). In their calculation, however, many adjustable parameters were introduced, i,e., the potential depths of the elastic, (2+, 0+), (2+, 2 +) and (3- , 0+) channels. Nevertheless, the energies ofthe maxima ofthe resonances ofthe three inelastic cross sections are different, namely, the deviations between (2+, 0+) and (2+, 2 +), and between (2+ , 0+ ) and 177
178
O . TANIMURA
(3 - , 0+) are about 2 and 4 MeV, respectively . The elastic, single 2 +, mutual 2+ and single 3 - channels all seem to be correlated in the observed maxima ofthe excitation functions 1 9) at energies above 20 c.m . MeV. From the viewpoint of the non-resonant diffraction model, i.e., the Austern-Blair model' °), Phillips et al . a) have discussed that maxima in the total inelastic cross sections of the single and mutual 2; excitations can be associated with a particularly favorable kinematic matching between pairs of entrance- and exit-channel grazing partial waves. Since the maximum phase-shift parameter never exceeds 60°, they concluded that the maxima in the cross sections are non-resonant . However, even if the phase shift does not exceed 90°, the resonance which is represented with the Breit-Wigner formula can arise as is seen in the coupled channel calculation 2's) . The Austern-Blair model'°) is a very phenomenological model, in which the phase shift is parametrized with the smooth cut-off model. Therefore, it is necessary to investigate the applicability of the model and to ascertain whether the enhancement of the inelastic cross section is non-resonant or not by means of the coupled channel calculation with the use of the realistic interaction. In this paper we study the correlated structures of the enhancements of the total inelastic cross sections to the (2+, 0+ ), (2 + , 2+), and (3 - , 0+) excitations. We make coupled channel calculations taking into account the elastic, (2 +, 0 +), (2 +, 2+) and (3 - , 0+ ) channels with the use of an improved coupling interaction 2). We show that this model reproduces the correlated structures simply without introducing the different potential depths among the channels and that the enhancements are interpreted as resonances ofthe adiabatic potential between two nuclei. Another purpose of the present paper is to compare this model with the band crossing model') and with the Austern-Blair model'°). In sect . 2 we give the coupled equations based on the folding interaction and the equation of the modified adiabatic potential. In sect. 3 we discuss the numerical results for the total inelastic cross sections and discuss the relation between our model and the above mentioned models . A summary is given in sect. 4. "
2. Coupled equation based on the folding interaction In the analysis of elastic and inelastic heavy ion scattering, the interaction derived by the folding procedure of the nuclear force between two nuclei has often been used. The use of the realistic nunclear force reproducing the low-energy nucleon-nucleon phase shift leads to too deep a diagonal potential to reproduce the elastic cross section ' 1), whereas the off-diagonal interaction is successful in reproducing the inelastic cross section z) and has a similar behavior to that given by the microscopic calculation' 2). The reason is that, in the folding model, the antisymmetrization between nuclei is neglected, which has strong effect on the diagonal potential and has little effect on the off-diagonal potential. Therefore, we adopt a modilïed folding model in which the diagonal part is replaced by a phenomenological potential.
' 2 C+' 3 C
179
The matrix element of the nucleus-nucleus interaction U,~(r) is assumed as follows : Ut~(r) _ (1-8,~) I d~dP~r,'M~*(p, ~) ~i~aPi(ri)Ra(rx~{Ir+rl - rzl) J
X ~~.rx)(p~ ~)+~r)Si~+iW(r~,~,
(1)
with Here suffix i denotes the quantum number specifying the sub-channel ((I112)1, L; J). In eq. (1) p,(r,) (i = 1, 2) is the density of 12C, and is taken from the charge distribution data ") : (3) p(r) = po(1-0 .026866r Z)/[1 +exp ((r-R(9'))/0 .5224)], with R(B') = 2355(1+ßsYzo(e~)+ß~y4o(~)+ ~ ~S(b3~ +( -y'b ;-u)YsN(p)).
(4)
Here B' is the polar angle in the body-fixed system andb3~,(b3 _ ~ is the phonon annihilation (creation) operator. The deformation parameters are assumed to be ßZ = - 0.45 and ßa = 0.12, which are taken from the analysis'°) of the electron scattering. The strength parameter S is assumed to be 0.5, which roughly reproduces the magnitude of the inelastic excitation function of the (3- , 0+) channel at B,,b = 25° [ref. 9)]. In eq. (2) cpr(~) represents the intrinsic wave function of 12C, which is assumed to be given by the product of the rotational and vibrational wave functions : ~Pr~~) _ [fil
8n D~i~o(~), Xr 2iu=(h)~iet+
(5)
with = Io) or 6SNIO) . The nucleon-nucleon interaction v(r) is assumed to be
(6)
Xrar(n)
o(r) _ -16.625 exp ~-
~1 .5523)Z~
- 2.4057 exp ~-
~1 .8506~ 2] ,
(7)
which is the spin-isospin independent part of the nuclear force fitted to the nucleonnucleon phase shift 1 s). The second term of eq. (1) represents the phenomenological r~l potential, the form of which is assumed to be : where Vc° , is the Coulomb potential of uniform charge distribution with radius 5.72 fm. This potential is shallower than that obtained by the empirical analysis ~
180
O. TANIMURA
because the coupled channel effect gives an additional term to t~(r). The modified adiabatic potential defined below corresponds to the empirical optical potential . The third term of eq . (1) represents the imaginary potential which is assumed to
1+exp((E, - E~.m .)/dE)1+exp((r - R,)/a,)'
where E~ = 0.09 J(J+1)+7, dE = 0.5 MeV, R, = 6 .18 fm and a, = 0 .35 fm. The parameters E,, and dE were chosen to reproduce the width of the resonance enhancement of the total inelastic cross section and R, and a, are taken from ref. 9). The coupled channel equations for the relative motion in the elastic and inelastic scattering of 1 ZC on ' ZC are given by the coupled channel method : ~z L(L+1} fiZ d2 +V(r)+iW(r) ~ u!'~(r) _ ~ Ur~(r)uj'~(r) . (10) E~ .m .- Er- -- + Zit 2~t dr2 r2 n ~ r+ Here er is the excitation energy of the 12C+' ZC system in the channel i and u;'+ (r) is the relative wave function . According to ref. 2), the total inelastic cross section (integrated cross section) for the transition of c = (1 1 12) into c' _ (I~IZ) is given by Qi~o~ -
(' 1
fz)Zkc f .~L'JZ(1
+S,~t~Xl
+S,~t~)Is~ Î'L'.cIL~
2
and the absorption cross section in the channel c is given by n
~
JZ(1 +(S,itz)2(att'cL'-`~cÎ~L' .ctL), (12) 1 1' 2)Zkc JtLf'L where ~i'L' .~,L represents the S-matrix elanent. We estimate the fusion cross section by subtracting the sum of the total inelastic cross sections from the absorption cross section: Qabe
(
QCus = Qab~ - ~ ~Inal'
(13)
In order to investigate the structure of the resonance enhancement, we examine the modified adiabatic potential 1 '), which is defined by the diagonalization of the "interaction matrix" at a fixed distance r [ref.'ri)] : ~s(r)
_ ~ Aar(r) -1 ~r~r)A~(r), r.1
(14)
with ~ ~ 2 L1 (L, + 1)~ ~( l~ r~(r) = Er+ ari+ Ur r) . 2 r2
(15)
Here AQ,(r) is the diagonalization matrix of ~1~r) and suffix a denotes the number designated in order from the lowest in energy ls) .
3. Resnlta and discu:~ioos The folding interaction given by eq. (1) is numerically calculated by expanding in terms with the spherical harmonics Y o(F) and by making the Fourier transformation of the force air- rt +r2p. Taking into account the elastic, (2+, 0+ ), (2+ , 2+) and (3 - , 0+) channels, we obtain the matrix elements among 13 subchannels, which are designated in table 1 . Fig. l displays part of the Ui~{r) for J = 10. The oRdiagonal p(r)
T~BL~ 1
The channel representation of the' 2 C+' =C system for J ~ 4 i
((I, Is)I, L ; J)
i
I 2 3 4
((oo)o, J ; J) ((20)2, J-2 ; J) ((20)2, J ; !) ((20)2, J+2 ; J)
s ((u~, J ; J) 6 7 8
((I, I,)I, L ; J) ((22)2, J-2 ; J) ((~)2, J ; J) ((22~, J-4 ; J)
i 9 10 11 12
((I, h)1, L ; J) ((~>4, ((22~, ((~)3, ((30)3,
J-2 : J) J ; J) J-3 ; J) J-1 ; J)
10
O
Fig. 1 . Real part of the interactions of the' = C+' Z C system in the case of J = 10 . The suffix is designated in table I . Only the interaction related to the aligned coupling subchannels are shown .
interaction is fairly different from the phenomenological interaction given by the r-derivative of the radius with the surface diffuseness'). It is noted that the element between the elastic and (2 +, 0+) channels, e.g., Us, t, is very small, so that the large cross section of the (2+ , 2+ ) channel is brought about through the (2+, 0 +) channel as an intermodiate step, i.e, through the interactions of U8. Z and U2, t . 3 .1 . THE TOTAL INELASTIC CROSS SECTION
We estimate the total inelastic cross sections for the (2 +, 0 +), (2 + , .2 +) and (3', 0+ )
l82
O. TANIMURA (mb)
100
TOTAL
INELASTIC CROSS SECTIONS
t
..
:Y .
I
I 10
I 18
II
I 20
2S 30 36 40 Ee.m.(INW) Fig. 2 . Energy dependence of the total inelastic cross sections and the deviation of the fusion cross sxtion from the Glas and Mosel model 's) . The single 2 *, mutual 2 * and single 3 - cross sections are shown in order from the top . Solid curves show the results by the present model and solid points show the data points from ref. ') .
channels by eq . (11). Fig. 2 displays the comparison between the calculated and observed inelastic cross sections, where the data are taken from ref.'). It also displays the deviation function of the fusion cross section 6) from the prediction of the Glas and Mosel model t~ . The calculations reproduce well the inelastic data both in magnitude and in phase of oscillation in the energy region from 20 to 40 c.m . MeV. The observed excitation function for the (3 - , 0 +) channel at B,,b = 25° [ref. 9)] shows the oscillatory structures whose maxima are almost the same in energy as those of the present calculation of the total inelastic (3 -, 0+ ) cross section. The calculated enhancematts of the inelastic cross sections are all caused by the single J resonance, i :e., the enhancements at about 14.5, 19.5, 24 .5 and 30 MeV are due to the resonanes with J = 10,12, 14 and 16, respectively. These are equal to the
spins specified by observation 1). The above results are the same as those obtained by Tanimura Z) and by Kondo et al . 3). The present calculation gives almost the same resonance energies of the same spin for the three channels, whereas the latter two calculations gave dif%rent resonance energies for the respective channels. In ref. Z ), the deviation of the resonance energy between (2 +, 0+) and (2+ , 2 +) is over 2 MeV for J Z 12. In ref. 3), the deviation is about 2 MeV between (2+, 0+ ) and (2+, 2+) and about 4 MéV between (2+ , 0+) and (3- , 0+ ) for J Z 12. The observed maxima of the three channels for the same J lie within two MeV and the resonances have large overlap one another, i.e., are correlated . This model does not reproduce the (2 +, 0 +) cross section at energies between 10 and 15 MeV . The reason may be considered that in this energy range the reaction channels such as a+ Z°Ne and eBe+ 160 are enhanced strongly as reported in ref. Z°), which we did not take into account. The present model also reproduces the oscillatory structure of the observed fusion cross section 6), which seems to be correlated with the inelastic cross sections as seen from fig. 2. It was shown in refs. Z " 21) that the maxima of the fusion cross section are caused by the single J resonances in the elastic channel. It is concluded that the resonance enhancements of the elastic, fusion and inelastic cross sections of (2+, 0+ ), (2 +, 2+) and (3- , 0 +) may be correlated. This can be interpreted as an example of the double resonance mechanism proposed by Greiner et al. a2) . 3 .2 . THE MODIFIED ADIABATIC POTENTIAL AND THE BAND CROSSING MODEL
We investigate the modified adiabatic potential (MAP) defined by eq. (14) . Fig. 3 displays the lowest MAP ~1(r) and the coupled channel resonance level for each J. The level is equal to the zero point oscillation energy in the pocket of ~1(r) within f0.2 MeV, and therefore it can be regarded as a potential resonance of ~ 1 (r). The coupled channel resonances cause the simultaneous enhancements of the calculated total inelastic cross sections for the three channels as seen in fig. 2. Thus the lowest resonance sequences form the "n = 0 molecular band" which was first suggested by Arima et al. 23) and independently Fink et at. za) . Fig. 4 displays the comparison between the resonances obtained from the MAP and CC method. At energies below 9.64 MeV, it is diflïcult to perform the CC calculation because the (3 - , 0+) channel becomes closed. Apparently, MAP reproduce very well the CC resonances . This means that MAP includes the essential part of the channel coupling eßect. We next compare the prediction by MAP with the one by the band crossing model (BCM) proposed by Matsuse et al.') . In fig. 4, the resonance trajectories predicted by BCM is also shown, which are taken from ref.'). The trajectories are closer than the resonances obtained by MAP or by CC. For J = 10, for example, four resonance trajectories predicted by BCM lie between the first and second resonances by MAP. Therefore, it is pointed out that BCM is not useful under the present coupling inter-
184
O. TANIMURA
4
3
6
7
8 r(tm)
Fig . 3 . The lowest modified adiabatic potentials (MAP) and the lowest coupled-channel resonance levels.
024 6
8
10
12
14
16
18
Fig . 4 . Comparison of the resonance energies predicted by the modified adiabatic potential, the coupled
channel model and the band-crossing model. The cross axis is shown in the scale of J(J+ 1) . Solid points and circles show resonances obtained by the modified adiabatic potential and the coupled channel model, respectively . Solid, dot-dashed, dotted and dashed lines show the resonance trajxtories of the aligned coupling subchannels of the elastic, (2+, 0+), (2+, 2 +) and (3 - , 0+) channels, respectively, in the bandcrossing model.
action, which is rather stronger than the phenomenological coupling interaction s .') . On the other hand, with the use of theweak coupling interaction, BCM was successful in predicting the CC resonance as well as MAP. 3.3 . COMPARISON W1TH THE AUSTERN-BLA1R MODEL
In this subsection, we investigate the energy dependence of the S-matrix element and compare it with the results calculated from the Austern-Blair model t°) by Phillips et a1 . 4). Fig. 5 displays the energy dependence of the real phase shift St , t and the absolute values, ISt, t I, ISz, t h ISa, t I and Ist t, t I ~ Also displayed are the absolute values of the S-matrix in the Austern-Blair model, i .e ., I SiB I and the contribution to (2 + , 0 + ) from J = L-2, which we represent ISii-zl~ From this figure we can $IJ
e
c")
40
.
J " 10
J
12
I
O -~40
" 16
i
-80 I
Oß
J " 19
r
v
:i
J 14
`
_.
-~" .
.
`
`~
~~
v ", J"12
~, J"10
J" 10
`. J"12I
20
J" 18
v
J" 18
`~
`
`.
1 J"14 P~ ~ .v`
J 16
1`v~ _r 25
Ee.mtINW)
J .18
v
v
r
0
o.a
I
` J"14
v
IS
.,I
v~v ~ 30
3S
40
Fig. 5. Energy behaviors of the real phase shift 6, . , and the absolute values of the S-matrix I S, , , I, I S2 . , h I Ss . ~ I end I S, , , , I~ The suffix is designated in table I . Solid and dot-dashed curves show the results of the present model. Dashed curves show the results of the Austere-Blair model 9).
O . TAN1MU1tA
186
point out in the following. (a) In the elastic component, i.e., 5 1, 1 and ISI, th the clear resonance behavior is seen for J = 10 and 12, but is not appreciable for the other J. (b) In the components of the aligned coupling channels, i.e., ISZ, t h ISe, t l and Ist t, t h a similar energy behavior is seen for all J; though width of the resonance spreads with increasing J. (c) The maxima of ISZ, t h Isa,1I and Ist 1,11 are almost the same in energy for each J, resulting in the cross-section maxima at the same energies as seen from fig. 2. (d) Though the behaviors of the ISI, tl and ISibl are fairly different, those of IS Z .1 1 and IS~i_ZI are similar, especially for J = 12. This indicates that the energy behavior of the (2 + , 0 + ) cross section is similar in both calculations. The Austern-Blair model is a very phenomenological one in which SL is parametrized with the smooth cut-off model and the amplitude is described by the one-step process with respect to the coupling interaction. Since the present model is based on the CC calculation with a realistic interaction, it can be said that the present model gives a foundation of the Austern-Blair model. Phillips et àl. a) concluded that the enhancements in the (2 + , 0 + ) cross section are not due to resonance because the maximum phase shift never exceeds 60° in their calculation. In fact, in the present calculation, 8 1, 1 does not exceed 60° for J ? 12. However, we can regard these enhancements as resonances in the sense that these are described with the Breit-Wigner resonance formula; Qinel
~
r~lrln~l ~ (~, . Eree)Z +2rZ
(16)
where rc,, r,~c, and r are the partial widths of the elastic and inelastic channels and the total width. We try to calculate the r,nc, by assuming the S-matrix as follows near the resonance energy : ~o'rL'd " orL ~o> `Sc'r'L', cIL ti Jc'r'L' . crL + ~ _ +. E Eres 2lr
(17)
where r~rL denotes the width of the subchannel i = (c = (I1IZ), 1, L; J), S~.°,?L ., ~,L the S-matrix of the non-resonant background and e~ the phase factor. We roughly estimate r from the width of the total (2 + , 0 + ) cross section. Then we can approximately calculate rc, from the behavior of ISl,tl bY as~8 e9~ (1~ . If we roughly Teste 2 The elastic and inelastic widths of the' 2C+' 2C system at the lowest resonance energy J 10 12 14 16
E,~~ (MeV)
r (MeV)
re, (MeV)
13 .9 18 .9 24 .9 31 .0
0.5 1 1 .5 2
0 .1 0 .3 0 .2 0 .1
(2 + , 0 + )
(2 + , 2 + )
(3 - , 0 + )
0.2 0.3 0.3 0.4
0 .0 0 .1 0 .2 0.2
0 .0 0 .0 0 .1 0 .5
estimate the absolute values of the second term of the r.h.s. of eq . (17) from the behaviors of I SZ . , h I sa, t I and 1 St t, t h we can obtain r,oe, for the (2 + , 0+), (2 +, 2+ ) and (3 - , 0+) channels. The result is listed in table 2. From this table we can see that the enhancement of the total inelastic cross section is caused by the single resonance with a broad inelastic width I'mc, . 4. Summary
We have studied the intermediate resonance observed in the tZC+' ZC inelastic cross section on the basis of the folding interaction by taking into account the single 2+ , mutual 2+ , and single 3 - channels. We have shown that the present model is successful in reproducing the gross structures of the total inelastic cross sections for the single and mutual 2 + excitations. Particularly, the model reproduces correctly the correlated energies of the maxima of the inelastic cross sections, which are attributed to the single J resonance . It is emphasized that the modified adiabatic potential reproduces the coupled channel. resonance correctly and is very useful for studying the strongly coupled system . It is concluded that the molecular resonance of the' ZC+ t ZC system is formed as a potential resonance in the modified adiabatic potential and exlübits itself in the inelastic channels as well as the elastic channel. We have shown that the Austern-Blair model is an approximate method to the coupled channel one in the calculation of the gross structures of the total inelastic cross sections. The author wishes to express his thanks to Professor T. Tazawa and Professor B. Imanishi for valuable discussions, He would like to acknowledge Professor T. Marumori for encouraganent. The numerical calculations were performed with the FACOM M180IIAD at INS. References 1) T. M. Cormier, J.'Applegate, G. M. Berkowitz, P. Braun-Munzinger, P. M. Cortnier, J. W. Harris, C. M. Jachcinski, L. L. Lee, Jr ., J. Barrette and H. E. Wegner, Phys . Rev . Lett . 38 (1977) 940; T. M. Cormier, C. M. lachcinski, G. M . Berkowitz, P. Braun-Munzinger, P. M. Corrnier, M . Gai, J. W. Hams, J. Barrette and H. E. Wegner, Phys. Rev . Lett . 40 (1978) 924 2) O. Tauimam, Nucl. Phys. A309 (1978) 233 3) Y. Kondo, Y. Abe and T. Maisase, Phys . Rev. C19 (1979) 1356 4) R. L. Phillips, K. A. Erb and D. A. Bromley, Phys . Rev. Lett . 42 (1979) 566 5) H. Emling, R. Nowotny, D. Pelte and G. Schrieder, Nucl. Phys . A211 (1973) 600 6) P. Sperr, T. H. Braid, Y. Eisen, D. G. Kovar, F. M . Prosser, Jr ., J. P, Schiffer, S. L. Tabor and S. Vigdor, Phys. Rev . Lett . 37 (1976) 321 7) Y. Kondo, T. Maisase and Y. Abe. Proc . of the INS-1PCR Symposium on Cluster Stn~cture of Nuclei and Transfer Reactions Induced by Heavy-Ions, Tokyo, 1975, p. 2$0; T. Maisase, Y. Kondo and Y. Abe, Prog . Theor, Phys . 59 (1978) 1009 8) B. Imanishi, Phys. Lett. 27B (1968) 267; Nucl . Phys. A215 (1969) 33 9) W. Rrilly, R. Wieland, A. Gobbi, M. W. Sechs, J. Maher, R. H. Siemssen, D. Mingay and D. A. Bromley, Nuovo Cim. A13 (1973) 913
188 10) i l) 12) 13) 14) 1~ 16) 17) 18) 19) 20) 21) 22) 23) 24) 25)
O. TANIMURA N. Austern and J. S. Blair, Ann. of Phys. 33 (1965) 15 G. R. Satchler, Nucl. Phys . A279 (1977) 493 Y. Goto and H. Horiuchi, Prog. Theor. Phys. 62 (1979), No. 3 C. W. De Jage~, H. De Vries and C. De Vries, Atomic Data and Nuclear Data Tables 14 (1974) 479 Y. Abgrall, B. Morand and E. Caurier, Nucl . Phys . A192 (1972) 372 I. Reichsfein and Y. C. Tang, Nucl . Phys. A139 (1969) 144 O. Tanimura and T. Tazawa, Phys. Lett . 83B (1979) 22 ; O. Tanimura and T. Tazawa, Phys . Rev. C20 (1979) 183 M. Nogami, Proc . of the INS-IPCR Symposium on Cluster Structure of Nuclei and Transfer Reactions Induced by Heavy Ions, Tokyo, 1975, p. 401 ; T. Tazawa and M. Nogami, Prog . Theor. Phys. 60 (1978) 1739 O. Tanimura and B. Imasishi, Phys . Lett . 80B (1979) 340 D. Glas and U. Mosel, Phys . Rev. C10 (1974) 2620 H. Voit, W. Galatat, W. Treu, H. Fröhlich and P. Drück, Phys . Left . 67B (1977) 399 O. Tanimura, Nuovo Cim. Left . 23 (1978) 70 W. Scheid, W. Greinet and R. Lemmer, Phys. Rev. Left . 25 (1970) 176 A. Arima, G. Scharff-Goldberger and K. W. McVoy, Phys. Left . 40B (1972) 7 H. J . Fink, W. Scheid and W. Greiser, Nucl . Phys . A188 (1972) 259 R. E. Malurin, J. W. Harris and P. Paul, Phys . Rev. C18 (1978) 163
INTERMEDIATE RESONANCE OF INELASTIC '=C+ t= C SCATTERING OSAMU TANIMURA
Institute Jôr Nuclear Study" , Unluersity of Tokyo, Tanashi, Tokyo 188, Japan
Received l4 May (979 (Revised 24 July 1979) Abstract : The intermediate resonances observed in the inelastic ` = C+'~C cross sections to the single and mutual 2;(4 .43 MeV) excitations and the single 3~(9 .64 MeV) excitation are studied by the coupledchannel method with the use of the coupling interaction derived by the folding procedure between 'zC and "C . It is shown that the model is successful in reproducing the gross structures of the inelastic cross sections and espaially the correlated resonance energies of the inelastic channels . The inelastic resonanes are shown to be due to the molecular resonances in an adiabatic potential between two ' Z C, which reproduces correctly the coupled channel resonanes.
1. Introduction The oscillatory structures observed in the total inelastic cross sections of the single and mutual excitations of the 2i (4.43 MeV) state of t zC in the t ZC + t ZC scattering') have been discussed from the viewpoint of the molecular resonance formation Z~ s) and the non-resonant diffraction model 4), Similar gross structure has been found in the excitation functions of the elastic scattering at 90° [ref. S)] and the fusion cross section in the ' ZC+ t ZC system 6). In this paper we discuss on the oscillatory structure of the t ZC+ t ZC cross section confïning ourselves to the total inelastic cross sections in the ' zC+ 12C(2; ), 1 ZC(2;)+' ZC(2i ) and t 2C+' ZC(3 - ) channels, which are abbreviated in the following as (2+ , 0+ ), (2 +, 2 +) and (3 -, 0 +), respectively . From the viewpoint of the molecular resonance formation, the present author has shown that the maxima in the total inelastic cross sections to the single and mutual 2i excitations are respectively attributed to the resonances in the lowest and the second lowest adiabatic potentials between two nuclei Z). However, the fits to the data were poor in the magnitude of the cross sections and the correlation of the resonance energies between the single and mutual 2+ cross sections. This may be due to the strong coupling interaction used there. Kondo et al. a) have showed that the observed oscillatory features can be understood and reproduced by the band crossing model'), which is the extension of the Nogami-Imanishi model 8). In their calculation, however, many adjustable parameters were introduced, i,e., the potential depths of the elastic, (2+, 0+), (2+, 2 +) and (3- , 0+) channels. Nevertheless, the energies ofthe maxima ofthe resonances ofthe three inelastic cross sections are different, namely, the deviations between (2+, 0+) and (2+, 2 +), and between (2+ , 0+ ) and 177
178
O . TANIMURA
(3 - , 0+) are about 2 and 4 MeV, respectively . The elastic, single 2 +, mutual 2+ and single 3 - channels all seem to be correlated in the observed maxima ofthe excitation functions 1 9) at energies above 20 c.m . MeV. From the viewpoint of the non-resonant diffraction model, i.e., the Austern-Blair model' °), Phillips et al . a) have discussed that maxima in the total inelastic cross sections of the single and mutual 2; excitations can be associated with a particularly favorable kinematic matching between pairs of entrance- and exit-channel grazing partial waves. Since the maximum phase-shift parameter never exceeds 60°, they concluded that the maxima in the cross sections are non-resonant . However, even if the phase shift does not exceed 90°, the resonance which is represented with the Breit-Wigner formula can arise as is seen in the coupled channel calculation 2's) . The Austern-Blair model'°) is a very phenomenological model, in which the phase shift is parametrized with the smooth cut-off model. Therefore, it is necessary to investigate the applicability of the model and to ascertain whether the enhancement of the inelastic cross section is non-resonant or not by means of the coupled channel calculation with the use of the realistic interaction. In this paper we study the correlated structures of the enhancements of the total inelastic cross sections to the (2+, 0+ ), (2 + , 2+), and (3 - , 0+) excitations. We make coupled channel calculations taking into account the elastic, (2 +, 0 +), (2 +, 2+) and (3 - , 0+ ) channels with the use of an improved coupling interaction 2). We show that this model reproduces the correlated structures simply without introducing the different potential depths among the channels and that the enhancements are interpreted as resonances ofthe adiabatic potential between two nuclei. Another purpose of the present paper is to compare this model with the band crossing model') and with the Austern-Blair model'°). In sect . 2 we give the coupled equations based on the folding interaction and the equation of the modified adiabatic potential. In sect. 3 we discuss the numerical results for the total inelastic cross sections and discuss the relation between our model and the above mentioned models . A summary is given in sect. 4. "
2. Coupled equation based on the folding interaction In the analysis of elastic and inelastic heavy ion scattering, the interaction derived by the folding procedure of the nuclear force between two nuclei has often been used. The use of the realistic nunclear force reproducing the low-energy nucleon-nucleon phase shift leads to too deep a diagonal potential to reproduce the elastic cross section ' 1), whereas the off-diagonal interaction is successful in reproducing the inelastic cross section z) and has a similar behavior to that given by the microscopic calculation' 2). The reason is that, in the folding model, the antisymmetrization between nuclei is neglected, which has strong effect on the diagonal potential and has little effect on the off-diagonal potential. Therefore, we adopt a modilïed folding model in which the diagonal part is replaced by a phenomenological potential.
' 2 C+' 3 C
179
The matrix element of the nucleus-nucleus interaction U,~(r) is assumed as follows : Ut~(r) _ (1-8,~) I d~dP~r,'M~*(p, ~) ~i~aPi(ri)Ra(rx~{Ir+rl - rzl) J
X ~~.rx)(p~ ~)+~r)Si~+iW(r~,~,
(1)
with Here suffix i denotes the quantum number specifying the sub-channel ((I112)1, L; J). In eq. (1) p,(r,) (i = 1, 2) is the density of 12C, and is taken from the charge distribution data ") : (3) p(r) = po(1-0 .026866r Z)/[1 +exp ((r-R(9'))/0 .5224)], with R(B') = 2355(1+ßsYzo(e~)+ß~y4o(~)+ ~ ~S(b3~ +( -y'b ;-u)YsN(p)).
(4)
Here B' is the polar angle in the body-fixed system andb3~,(b3 _ ~ is the phonon annihilation (creation) operator. The deformation parameters are assumed to be ßZ = - 0.45 and ßa = 0.12, which are taken from the analysis'°) of the electron scattering. The strength parameter S is assumed to be 0.5, which roughly reproduces the magnitude of the inelastic excitation function of the (3- , 0+) channel at B,,b = 25° [ref. 9)]. In eq. (2) cpr(~) represents the intrinsic wave function of 12C, which is assumed to be given by the product of the rotational and vibrational wave functions : ~Pr~~) _ [fil
8n D~i~o(~), Xr 2iu=(h)~iet+
(5)
with = Io) or 6SNIO) . The nucleon-nucleon interaction v(r) is assumed to be
(6)
Xrar(n)
o(r) _ -16.625 exp ~-
~1 .5523)Z~
- 2.4057 exp ~-
~1 .8506~ 2] ,
(7)
which is the spin-isospin independent part of the nuclear force fitted to the nucleonnucleon phase shift 1 s). The second term of eq. (1) represents the phenomenological r~l potential, the form of which is assumed to be : where Vc° , is the Coulomb potential of uniform charge distribution with radius 5.72 fm. This potential is shallower than that obtained by the empirical analysis ~
180
O. TANIMURA
because the coupled channel effect gives an additional term to t~(r). The modified adiabatic potential defined below corresponds to the empirical optical potential . The third term of eq . (1) represents the imaginary potential which is assumed to
1+exp((E, - E~.m .)/dE)1+exp((r - R,)/a,)'
where E~ = 0.09 J(J+1)+7, dE = 0.5 MeV, R, = 6 .18 fm and a, = 0 .35 fm. The parameters E,, and dE were chosen to reproduce the width of the resonance enhancement of the total inelastic cross section and R, and a, are taken from ref. 9). The coupled channel equations for the relative motion in the elastic and inelastic scattering of 1 ZC on ' ZC are given by the coupled channel method : ~z L(L+1} fiZ d2 +V(r)+iW(r) ~ u!'~(r) _ ~ Ur~(r)uj'~(r) . (10) E~ .m .- Er- -- + Zit 2~t dr2 r2 n ~ r+ Here er is the excitation energy of the 12C+' ZC system in the channel i and u;'+ (r) is the relative wave function . According to ref. 2), the total inelastic cross section (integrated cross section) for the transition of c = (1 1 12) into c' _ (I~IZ) is given by Qi~o~ -
(' 1
fz)Zkc f .~L'JZ(1
+S,~t~Xl
+S,~t~)Is~ Î'L'.cIL~
2
and the absorption cross section in the channel c is given by n
~
JZ(1 +(S,itz)2(att'cL'-`~cÎ~L' .ctL), (12) 1 1' 2)Zkc JtLf'L where ~i'L' .~,L represents the S-matrix elanent. We estimate the fusion cross section by subtracting the sum of the total inelastic cross sections from the absorption cross section: Qabe
(
QCus = Qab~ - ~ ~Inal'
(13)
In order to investigate the structure of the resonance enhancement, we examine the modified adiabatic potential 1 '), which is defined by the diagonalization of the "interaction matrix" at a fixed distance r [ref.'ri)] : ~s(r)
_ ~ Aar(r) -1 ~r~r)A~(r), r.1
(14)
with ~ ~ 2 L1 (L, + 1)~ ~( l~ r~(r) = Er+ ari+ Ur r) . 2 r2
(15)
Here AQ,(r) is the diagonalization matrix of ~1~r) and suffix a denotes the number designated in order from the lowest in energy ls) .
3. Resnlta and discu:~ioos The folding interaction given by eq. (1) is numerically calculated by expanding in terms with the spherical harmonics Y o(F) and by making the Fourier transformation of the force air- rt +r2p. Taking into account the elastic, (2+, 0+ ), (2+ , 2+) and (3 - , 0+) channels, we obtain the matrix elements among 13 subchannels, which are designated in table 1 . Fig. l displays part of the Ui~{r) for J = 10. The oRdiagonal p(r)
T~BL~ 1
The channel representation of the' 2 C+' =C system for J ~ 4 i
((I, Is)I, L ; J)
i
I 2 3 4
((oo)o, J ; J) ((20)2, J-2 ; J) ((20)2, J ; !) ((20)2, J+2 ; J)
s ((u~, J ; J) 6 7 8
((I, I,)I, L ; J) ((22)2, J-2 ; J) ((~)2, J ; J) ((22~, J-4 ; J)
i 9 10 11 12
((I, h)1, L ; J) ((~>4, ((22~, ((~)3, ((30)3,
J-2 : J) J ; J) J-3 ; J) J-1 ; J)
10
O
Fig. 1 . Real part of the interactions of the' = C+' Z C system in the case of J = 10 . The suffix is designated in table I . Only the interaction related to the aligned coupling subchannels are shown .
interaction is fairly different from the phenomenological interaction given by the r-derivative of the radius with the surface diffuseness'). It is noted that the element between the elastic and (2 +, 0+) channels, e.g., Us, t, is very small, so that the large cross section of the (2+ , 2+ ) channel is brought about through the (2+, 0 +) channel as an intermodiate step, i.e, through the interactions of U8. Z and U2, t . 3 .1 . THE TOTAL INELASTIC CROSS SECTION
We estimate the total inelastic cross sections for the (2 +, 0 +), (2 + , .2 +) and (3', 0+ )
l82
O. TANIMURA (mb)
100
TOTAL
INELASTIC CROSS SECTIONS
t
..
:Y .
I
I 10
I 18
II
I 20
2S 30 36 40 Ee.m.(INW) Fig. 2 . Energy dependence of the total inelastic cross sections and the deviation of the fusion cross sxtion from the Glas and Mosel model 's) . The single 2 *, mutual 2 * and single 3 - cross sections are shown in order from the top . Solid curves show the results by the present model and solid points show the data points from ref. ') .
channels by eq . (11). Fig. 2 displays the comparison between the calculated and observed inelastic cross sections, where the data are taken from ref.'). It also displays the deviation function of the fusion cross section 6) from the prediction of the Glas and Mosel model t~ . The calculations reproduce well the inelastic data both in magnitude and in phase of oscillation in the energy region from 20 to 40 c.m . MeV. The observed excitation function for the (3 - , 0 +) channel at B,,b = 25° [ref. 9)] shows the oscillatory structures whose maxima are almost the same in energy as those of the present calculation of the total inelastic (3 -, 0+ ) cross section. The calculated enhancematts of the inelastic cross sections are all caused by the single J resonance, i :e., the enhancements at about 14.5, 19.5, 24 .5 and 30 MeV are due to the resonanes with J = 10,12, 14 and 16, respectively. These are equal to the
spins specified by observation 1). The above results are the same as those obtained by Tanimura Z) and by Kondo et al . 3). The present calculation gives almost the same resonance energies of the same spin for the three channels, whereas the latter two calculations gave dif%rent resonance energies for the respective channels. In ref. Z ), the deviation of the resonance energy between (2 +, 0+) and (2+ , 2 +) is over 2 MeV for J Z 12. In ref. 3), the deviation is about 2 MeV between (2+, 0+ ) and (2+, 2+) and about 4 MéV between (2+ , 0+) and (3- , 0+ ) for J Z 12. The observed maxima of the three channels for the same J lie within two MeV and the resonances have large overlap one another, i.e., are correlated . This model does not reproduce the (2 +, 0 +) cross section at energies between 10 and 15 MeV . The reason may be considered that in this energy range the reaction channels such as a+ Z°Ne and eBe+ 160 are enhanced strongly as reported in ref. Z°), which we did not take into account. The present model also reproduces the oscillatory structure of the observed fusion cross section 6), which seems to be correlated with the inelastic cross sections as seen from fig. 2. It was shown in refs. Z " 21) that the maxima of the fusion cross section are caused by the single J resonances in the elastic channel. It is concluded that the resonance enhancements of the elastic, fusion and inelastic cross sections of (2+, 0+ ), (2 +, 2+) and (3- , 0 +) may be correlated. This can be interpreted as an example of the double resonance mechanism proposed by Greiner et al. a2) . 3 .2 . THE MODIFIED ADIABATIC POTENTIAL AND THE BAND CROSSING MODEL
We investigate the modified adiabatic potential (MAP) defined by eq. (14) . Fig. 3 displays the lowest MAP ~1(r) and the coupled channel resonance level for each J. The level is equal to the zero point oscillation energy in the pocket of ~1(r) within f0.2 MeV, and therefore it can be regarded as a potential resonance of ~ 1 (r). The coupled channel resonances cause the simultaneous enhancements of the calculated total inelastic cross sections for the three channels as seen in fig. 2. Thus the lowest resonance sequences form the "n = 0 molecular band" which was first suggested by Arima et al. 23) and independently Fink et at. za) . Fig. 4 displays the comparison between the resonances obtained from the MAP and CC method. At energies below 9.64 MeV, it is diflïcult to perform the CC calculation because the (3 - , 0+) channel becomes closed. Apparently, MAP reproduce very well the CC resonances . This means that MAP includes the essential part of the channel coupling eßect. We next compare the prediction by MAP with the one by the band crossing model (BCM) proposed by Matsuse et al.') . In fig. 4, the resonance trajectories predicted by BCM is also shown, which are taken from ref.'). The trajectories are closer than the resonances obtained by MAP or by CC. For J = 10, for example, four resonance trajectories predicted by BCM lie between the first and second resonances by MAP. Therefore, it is pointed out that BCM is not useful under the present coupling inter-
184
O. TANIMURA
4
3
6
7
8 r(tm)
Fig . 3 . The lowest modified adiabatic potentials (MAP) and the lowest coupled-channel resonance levels.
024 6
8
10
12
14
16
18
Fig . 4 . Comparison of the resonance energies predicted by the modified adiabatic potential, the coupled
channel model and the band-crossing model. The cross axis is shown in the scale of J(J+ 1) . Solid points and circles show resonances obtained by the modified adiabatic potential and the coupled channel model, respectively . Solid, dot-dashed, dotted and dashed lines show the resonance trajxtories of the aligned coupling subchannels of the elastic, (2+, 0+), (2+, 2 +) and (3 - , 0+) channels, respectively, in the bandcrossing model.
action, which is rather stronger than the phenomenological coupling interaction s .') . On the other hand, with the use of theweak coupling interaction, BCM was successful in predicting the CC resonance as well as MAP. 3.3 . COMPARISON W1TH THE AUSTERN-BLA1R MODEL
In this subsection, we investigate the energy dependence of the S-matrix element and compare it with the results calculated from the Austern-Blair model t°) by Phillips et a1 . 4). Fig. 5 displays the energy dependence of the real phase shift St , t and the absolute values, ISt, t I, ISz, t h ISa, t I and Ist t, t I ~ Also displayed are the absolute values of the S-matrix in the Austern-Blair model, i .e ., I SiB I and the contribution to (2 + , 0 + ) from J = L-2, which we represent ISii-zl~ From this figure we can $IJ
e
c")
40
.
J " 10
J
12
I
O -~40
" 16
i
-80 I
Oß
J " 19
r
v
:i
J 14
`
_.
-~" .
.
`
`~
~~
v ", J"12
~, J"10
J" 10
`. J"12I
20
J" 18
v
J" 18
`~
`
`.
1 J"14 P~ ~ .v`
J 16
1`v~ _r 25
Ee.mtINW)
J .18
v
v
r
0
o.a
I
` J"14
v
IS
.,I
v~v ~ 30
3S
40
Fig. 5. Energy behaviors of the real phase shift 6, . , and the absolute values of the S-matrix I S, , , I, I S2 . , h I Ss . ~ I end I S, , , , I~ The suffix is designated in table I . Solid and dot-dashed curves show the results of the present model. Dashed curves show the results of the Austere-Blair model 9).
O . TAN1MU1tA
186
point out in the following. (a) In the elastic component, i.e., 5 1, 1 and ISI, th the clear resonance behavior is seen for J = 10 and 12, but is not appreciable for the other J. (b) In the components of the aligned coupling channels, i.e., ISZ, t h ISe, t l and Ist t, t h a similar energy behavior is seen for all J; though width of the resonance spreads with increasing J. (c) The maxima of ISZ, t h Isa,1I and Ist 1,11 are almost the same in energy for each J, resulting in the cross-section maxima at the same energies as seen from fig. 2. (d) Though the behaviors of the ISI, tl and ISibl are fairly different, those of IS Z .1 1 and IS~i_ZI are similar, especially for J = 12. This indicates that the energy behavior of the (2 + , 0 + ) cross section is similar in both calculations. The Austern-Blair model is a very phenomenological one in which SL is parametrized with the smooth cut-off model and the amplitude is described by the one-step process with respect to the coupling interaction. Since the present model is based on the CC calculation with a realistic interaction, it can be said that the present model gives a foundation of the Austern-Blair model. Phillips et àl. a) concluded that the enhancements in the (2 + , 0 + ) cross section are not due to resonance because the maximum phase shift never exceeds 60° in their calculation. In fact, in the present calculation, 8 1, 1 does not exceed 60° for J ? 12. However, we can regard these enhancements as resonances in the sense that these are described with the Breit-Wigner resonance formula; Qinel
~
r~lrln~l ~ (~, . Eree)Z +2rZ
(16)
where rc,, r,~c, and r are the partial widths of the elastic and inelastic channels and the total width. We try to calculate the r,nc, by assuming the S-matrix as follows near the resonance energy : ~o'rL'd " orL ~o> `Sc'r'L', cIL ti Jc'r'L' . crL + ~ _ +. E Eres 2lr
(17)
where r~rL denotes the width of the subchannel i = (c = (I1IZ), 1, L; J), S~.°,?L ., ~,L the S-matrix of the non-resonant background and e~ the phase factor. We roughly estimate r from the width of the total (2 + , 0 + ) cross section. Then we can approximately calculate rc, from the behavior of ISl,tl bY as~8 e9~ (1~ . If we roughly Teste 2 The elastic and inelastic widths of the' 2C+' 2C system at the lowest resonance energy J 10 12 14 16
E,~~ (MeV)
r (MeV)
re, (MeV)
13 .9 18 .9 24 .9 31 .0
0.5 1 1 .5 2
0 .1 0 .3 0 .2 0 .1
(2 + , 0 + )
(2 + , 2 + )
(3 - , 0 + )
0.2 0.3 0.3 0.4
0 .0 0 .1 0 .2 0.2
0 .0 0 .0 0 .1 0 .5
estimate the absolute values of the second term of the r.h.s. of eq . (17) from the behaviors of I SZ . , h I sa, t I and 1 St t, t h we can obtain r,oe, for the (2 + , 0+), (2 +, 2+ ) and (3 - , 0+) channels. The result is listed in table 2. From this table we can see that the enhancement of the total inelastic cross section is caused by the single resonance with a broad inelastic width I'mc, . 4. Summary
We have studied the intermediate resonance observed in the tZC+' ZC inelastic cross section on the basis of the folding interaction by taking into account the single 2+ , mutual 2+ , and single 3 - channels. We have shown that the present model is successful in reproducing the gross structures of the total inelastic cross sections for the single and mutual 2 + excitations. Particularly, the model reproduces correctly the correlated energies of the maxima of the inelastic cross sections, which are attributed to the single J resonance . It is emphasized that the modified adiabatic potential reproduces the coupled channel. resonance correctly and is very useful for studying the strongly coupled system . It is concluded that the molecular resonance of the' ZC+ t ZC system is formed as a potential resonance in the modified adiabatic potential and exlübits itself in the inelastic channels as well as the elastic channel. We have shown that the Austern-Blair model is an approximate method to the coupled channel one in the calculation of the gross structures of the total inelastic cross sections. The author wishes to express his thanks to Professor T. Tazawa and Professor B. Imanishi for valuable discussions, He would like to acknowledge Professor T. Marumori for encouraganent. The numerical calculations were performed with the FACOM M180IIAD at INS. References 1) T. M. Cormier, J.'Applegate, G. M. Berkowitz, P. Braun-Munzinger, P. M. Cortnier, J. W. Harris, C. M. Jachcinski, L. L. Lee, Jr ., J. Barrette and H. E. Wegner, Phys . Rev . Lett . 38 (1977) 940; T. M. Cormier, C. M. lachcinski, G. M . Berkowitz, P. Braun-Munzinger, P. M. Corrnier, M . Gai, J. W. Hams, J. Barrette and H. E. Wegner, Phys. Rev . Lett . 40 (1978) 924 2) O. Tauimam, Nucl. Phys. A309 (1978) 233 3) Y. Kondo, Y. Abe and T. Maisase, Phys . Rev. C19 (1979) 1356 4) R. L. Phillips, K. A. Erb and D. A. Bromley, Phys . Rev. Lett . 42 (1979) 566 5) H. Emling, R. Nowotny, D. Pelte and G. Schrieder, Nucl. Phys . A211 (1973) 600 6) P. Sperr, T. H. Braid, Y. Eisen, D. G. Kovar, F. M . Prosser, Jr ., J. P, Schiffer, S. L. Tabor and S. Vigdor, Phys. Rev . Lett . 37 (1976) 321 7) Y. Kondo, T. Maisase and Y. Abe. Proc . of the INS-1PCR Symposium on Cluster Stn~cture of Nuclei and Transfer Reactions Induced by Heavy-Ions, Tokyo, 1975, p. 2$0; T. Maisase, Y. Kondo and Y. Abe, Prog . Theor, Phys . 59 (1978) 1009 8) B. Imanishi, Phys. Lett. 27B (1968) 267; Nucl . Phys. A215 (1969) 33 9) W. Rrilly, R. Wieland, A. Gobbi, M. W. Sechs, J. Maher, R. H. Siemssen, D. Mingay and D. A. Bromley, Nuovo Cim. A13 (1973) 913
188 10) i l) 12) 13) 14) 1~ 16) 17) 18) 19) 20) 21) 22) 23) 24) 25)
O. TANIMURA N. Austern and J. S. Blair, Ann. of Phys. 33 (1965) 15 G. R. Satchler, Nucl. Phys . A279 (1977) 493 Y. Goto and H. Horiuchi, Prog. Theor. Phys. 62 (1979), No. 3 C. W. De Jage~, H. De Vries and C. De Vries, Atomic Data and Nuclear Data Tables 14 (1974) 479 Y. Abgrall, B. Morand and E. Caurier, Nucl . Phys . A192 (1972) 372 I. Reichsfein and Y. C. Tang, Nucl . Phys. A139 (1969) 144 O. Tanimura and T. Tazawa, Phys. Lett . 83B (1979) 22 ; O. Tanimura and T. Tazawa, Phys . Rev. C20 (1979) 183 M. Nogami, Proc . of the INS-IPCR Symposium on Cluster Structure of Nuclei and Transfer Reactions Induced by Heavy Ions, Tokyo, 1975, p. 401 ; T. Tazawa and M. Nogami, Prog . Theor. Phys. 60 (1978) 1739 O. Tanimura and B. Imasishi, Phys . Lett . 80B (1979) 340 D. Glas and U. Mosel, Phys . Rev. C10 (1974) 2620 H. Voit, W. Galatat, W. Treu, H. Fröhlich and P. Drück, Phys . Left . 67B (1977) 399 O. Tanimura, Nuovo Cim. Left . 23 (1978) 70 W. Scheid, W. Greinet and R. Lemmer, Phys. Rev. Left . 25 (1970) 176 A. Arima, G. Scharff-Goldberger and K. W. McVoy, Phys. Left . 40B (1972) 7 H. J . Fink, W. Scheid and W. Greiser, Nucl . Phys . A188 (1972) 259 R. E. Malurin, J. W. Harris and P. Paul, Phys . Rev. C18 (1978) 163