Collective correlations for continuum states of 12C in the region of the giant dipole resonance

I.D.2: [ 2.C

]

Nuclear Physics A205 (1973) 581 --592; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

C O L L E C T I V E C O R R E L A T I O N S F O R C O N T I N U U M STATES OF lZC I N T H E R E G I O N OF T H E G I A N T D I P O L E R E S O N A N C E E. D. MSHELIA lnstitut fiir Theoretische Physik der Universitiit Frankfurt, Frankfurt am Main, Germany t

and R. F. BARRETT School of Physics, University of Melbourne, Parkville, Vic. 3052, Australia

Received 30 November 1972 (Revised 7 February 1973) Abstract: The theory of collective correlations in nuclei is extended to the nuclear continuum treating the lp-lh dipole continuum states in the framework of the eigenchannel theory of nuclear reactions. The basis states for the diagonalization of the total Hamiltonian are obtained by coupling the lp-lh continuum states to surface quadrupole vibrations. Numerical calculations based on two coupling schemes are performed for the giant dipole resonance in lzC, taking into account the low-lying excited states of the residual nuclei by core-hole coupling.

1. Introduction

The one-particle-one-hole ( l p - l h ) shell model of nuclear reactions has been successful in explaining the gross features (doorway structure) in the giant dipole resonance ( G D R ) of light and medium heavy nuclei. Higher-resolution experimental investigations have, however, revealed more structure in the giant dipole resonances of most nuclei than is explicable by this model. This is the intermediate structure, whereby the G D R , due to its interaction with other collective degrees of freedom splits into several resonances. The observed structure cannot be explained in terms of the l p - l h model. Several authors 1 - 3 ) h a v e attempted to explain this additional intermediate structure by extending the l p - l h model to consider the interaction of the nucleus with other degrees of freedom. One such attempt was made by Drechsel et aL 4), who, in their theory of collective correlations, have coupled the l p - l h G D R states to the quadrupole surface vibrations. The interaction was taken from the dynamic collective model ( D C M ) and translated semi-microscopically. For the nuclei ~2C, 2sSi and 6°Ni they obtained encouraging results. In the past few years the ordinary l p - l h shell model has also been extended in another direction; namely, the unrealistic infinite harmonic-oscillator potential well used for the description of the single-particle levels has been replaced by the more * This work was supported by the Deutsche Forschungsgemeinschaft and by the Bundesministerium fiir Bildung und Wissenschaft. 581

582

E.D. MSHELIA AND R. F. BARRETT

realistic Woods-Saxon potential. This enables one to include the effect of the particle continuum in the calculation. Two methods have been extensively used for numerical calculations. They are the eigenchannel (EC) method and the coupled-channel method. These are discussed fully in refs. s, 6), respectively. In the present paper we present the results of a calculation in which these two extensions of the l p - l h bound-state shell model have been combined; i.e., we have performed a calculation based on the collective correlation model in which the continuum nature of the particle states is properly treated. This model is described more fully in sect. 2. In sect. 3 we present the result of the calculation and a summary of the conclusion is given in sect. 4.

2. Description of the model The collective correlation model was developed by Drechsel et al. 4) to include the effect of coupling the low-lying vibrational states of nuclei to the lp-1 h states required for a description of the GDR. It thus represents a combination of the DCM 7) with the l p - l h model of nuclear structure. This union of inherently different models is achieved by replacing the collective dipole term in the Hamiltonian of the DCM by the corresponding l p - l h expression. The collective correlation model has been described in detail elsewhere 4) and we will not discuss it further here. Although the calculations of Drechsel et al. predict the observed splitting of the G D R , they suffer from the disadvantages of all bound-state shell-model calculations; namely, they give no idea of the widths of the calculated resonances, or of the branching ratios to excited states of the residual nuclei. Preliminary results of the present calculations have been presented in an earlier letter 8). To treat the particle continuum correctly we have chosen to use the eigenchannel (EC) method developed by Danos and Greiner 9). This method has been discussed at length recently in a review article 5), and we will not elaborate on it further here. We have chosen the EC method in including continuum effects in the calculation in preference to the alternative coupled-channel method because of the ease with which the nuclear model may be extended past the simple l p - l h model in the EC method. This is because the EC method is basically a diagonalization technique. In our calculations we have employed two different angular momentum coupling schemes. In the first (coupling scheme A) we coupled the l p - l h states to an intermediate angular momentum, I, which is then coupled with the phonon angular momentum to give the final angular momentum of the compound nucleus. This is the coupling scheme used by Drechsel et al. ~). They found that only states where I = 1 contribute appreciably, and we have also used this approximation. The reduction in computation time achieved by use of this approximation enabled us to investigate the effect of two-phonon excitation on our results. The phonon energies (ho~) required for the calculation were obtained phenomenologically from the low-lying vibrational states of 12C (e.g. the first 2 + state at 4.43 MeV in 12C was interpreted as a one-

COLLECTIVE CORRELATIONS

583

phonon vibrational state). The form o f the matrix elements of the Hamiltonian in this coupling scheme has been given in ref. 4). The strength of the coupling between lp-lh and collective modes is derived from the B(E2; 0 ÷ ~ 2 ÷) value between the ground state and the first 2 ÷ state in ~2C. A recent B(E2) value obtained from a Darmstadt experiment lo) was used here, from which the coupling strength flo was obtained to be 0.4. Because of the harmonic nature of the collective potential well assumed in this model, the first four excited states in the residual nuclei are predicted to be degenerate. This is unrealistic as may be seen from an examination of the experimental energy spectra for ~IB and 1~C. In order to allow for this degeneracy, and to enable the calculation of cross sections to the excited states of I~B and ~ C , a different angular momentum coupling scheme s) (coupling scheme B) was usea in a separate calculation. TABLE 1 The 22 channels considered for the diagonalization of the energy matrix in taking into account the low-lying energy states of ~~B and ~IC (coupling scheme B) e(MeV)

Configurations

li B

HC

0.00

0.00

[ [ p - t xO+I{.- Xs.,l_]l[[p.t_-1 ×O+l 1-- x d } l 1-

[[0.t.-1 × 0 + l,l. - X d.,].]~-

2.14

2.00

[[p~.-i x2+],~- x s t . ] l -

[[p~-i x2+]½- xd~]l-

4.46

4.32

[[p.t -~ x2+]-'l-- xd.,l.] l -

[[p4_-~ x2+]½ - x d ~ ] ~-

5.03

4.81

[[pi - ~ x 2 + ] 4 - - x s ½ ] -1 [[p½-1 x 2 + l ~ - x d i ] l -

[ [ p l . - l x 2 + ] ~ -xd,1.] t -

6.76

6.49

[lpt. -1 × 2 + ] ~ - x d t ] t -

In this scheme, we have coupled the hole to the phonon to give the angular momentum, !, of the residual nucleus. This was then coupled with the particle to give the total angular momentum of the compound nucleus. In this coupling scheme the angular momentum 7 of the residual nucleus is a good quantum number and so reactions leaving the nucleus in an excited state could be described. The excited-state cross sections and angular distributions were obtained in an analogous way to the method outlined for the lp-lh model by Wahsweiler et al. tl). The formulae for the matrix elements of the Hamiltonian in coupling scheme B were obtained by the usual methods of angular momentum recoupling from those of coupling scheme A. In this case the restriction to states where i = 1- was not made (in contrast to scheme A) since this represents the angular momenta of 11B and ~tC. This meant that limitations of computer time restricted us to the consideration of onephonon excitation only. In order to reduce computation time somewhat, configurations involving the deep-lying ls¢ hole were neglected. This approximation has been

584

E.D. MSHELIA AND R. F. BARRETT

shown elsewhere to have little effect on the results in the G D R region of excitation. In scheme B, the diagonal energy matrix elements were adjusted to correctly reproduce the thresholds o f reactions to excited state of the residual nuclei. In effect this meant that we phenomenologically introduced a hole-phonon interaction in the residual nuclei t 1B and 11C which removed the degeneracy of the excited states of it B and 11C caused by the harmonic approximation for the collective potential. This is similar to the model used by Clegg 12) to describe the low-lying states of 11B and 11C. Alternatively, one could say that we have partially included the effects of anharmonicities in a phenomenological way. A total of 22 coupled channels were obtained when using coupling scheme B. These are presented in table 1. To obtain the single-particle and single-hole wave functions for use in the construction of the l p - l h part of the basis, we proceeded in an exactly analogous way to that used in previous EC calculations. The experimental single-particle energy scheme of 12C was reproduced as closely as possible by a Woods-Saxon potential well including spin-orbit and Coulomb terms. The actual potential parameters used were those already published by Antony-Spies 13) for an earlier 12C EC calculation. The residual two-body force was assumed to be a zero-range force of the form

V(rl, r2) = Vo6(rx--r2)[ao+ao~r1" 0"2]. The parameters Vo, ao and ao were varied somewhat to see their effect on the results obtained. The strength V0 controls the position obtained for the GDR. The expression for the energy matrix elements for a zero-range force in the l p - l h approximation already exists in the literature [see, e.g., Wahsweiler et al. 11)]. To calculate the dipole absorption cross section in the EC theory we have treated the coupling of the electromagnetic field to the nucleus by perturbation theory using the long-wave approximation in a manner analogous to Wahsweiler et al. 11). This expression has a non-zero contribution only from the basis states where no phonon has been excited. This reflects the one-body nature of the dipole operator and indicates that here one cannot directly excite a vibrational (np-nh) state by an incoming 1 - photon. The extra structure obtained in the dipole absorption cross section in this calculation is thus caused by the coupling of the vibrational states to the l p - l h states via the dipole-phonon interaction. In sect. 3 we present the results of the calculation. 3. Discussion of results

3.1. COUPLING SCHEME A The total partial cross sections in coupling scheme A are presented in figs. la and b calculated with the set o f parameters Vo = - 650 MeV. fm 3, ao = 0.865, a, = 0.135. In fig. la only the first 2 + phonon state at 4.43 MeV has been considered. Comparing this with the result of the pure lp-lh model (fig. lc), we can recognize two new resonances. The smaller one at 19.2 MeV has been identified as the vibrational satellite

COLLECTIVE CORRELATIONS

585

o f the [p~ 1 x s½] resonance at 17.5 MeV, and the resonance at 22.6 MeV as the vibrational satellite o f the giant resonance at 21.4 MeV, which is mostly composed of [p~ t x d~] configurations. The n u m b e r o f channels considered here was 12 compared with only 6 in the l p - l h model.

] 4B C~

,i 32

16

I!' !,;i \ ' * \-%.. //I

b) 40 32

£ b 16

80Ilp_lh ~ 6oI

C)

I/

+t //=:,= E [MeV] Fig. 1. Theoretical photoabsorption cross sections with collective correlations (coupling scheme A): (a) including the 2 + one-phonon state at 4.43 MeV; (b) including the 2 + one-phonon state at 4.43

MeV and the 0 + two-phonon state at 7.66 MeV; (c) the result of the lp-lh model without collective correlations. A residual force strength of Vo = --650 MeV. fm3 and exchange mixture of the Meshkov-Soper type were used in all three cases.

I n the h a r m o n i c approximation the t w o - p h o n o n state is three times degenerate. The experimental spectrum on the other hand, shows a splitting o f this state in three levels. The 0 + state lies at 7.66 MeV, the 2 + state at about 10.3 MeV and the 4 + state

586

E.D.

MSHELIA

AND

R . F. B A R R E T T

at 14.08 MeV. Including the second 0 + state into the calculation increased the number of coupled channels to 18 and the result is shown in fig. lb. Surprisingly the form of the cross section compared to that of fig. la has hardly changed. A possible explanation might be that the energy of this state is so high that it can only couple weakly in the region of the GDR. Its influence might be felt above the energy region considered here. Further calculations including the 2 + level at 10.3 MeV gave the same picture.

C~2(T't°tG )

//,¢~*, / \

20

c~)

15 10

¢{

0

48

°~ , ° +,+

it'*

!

o-<,,,o,o,/---

:-

/A

[I ~

-o'-(~,Qround state)

,til I

"

oT_ 3210

uJ

i

u~ 161 u')

°(Texcited

states)~(/, '

/ /

482

c)'

k

C12 (~',total)

32. i

16--~i O h

i

16

18

i

i

i

20 22 24 ENERGY [MeV]

~

l

26

~,

i

28

F i g . 2. Total absorption cross sections. (a) The experiment of Shevchenko and Yudin 1 s). T h e full curve represents the result of the lp-lh model multiplied by ._a,. (b) The result with collective correlations, including the low-lying states of 11B and ~1C (coupling scheme B) calculated with the parameter set Vo = - - 7 5 0 M e V . f m a, a o - 0.9, a a = 0.1. F u l l line: total cross section; dotdashed line: ground state cross section; dotted line: contributions of excited states, (c) Total cross section as under (b) but calculated with the parameter set Vo ~ - - 8 5 0 M e V • f m a, ao = 1, a~ = 0:

In the calculations just discussed, the calculated energy of the giant resonance (21.4 MeV) is lower than the experimental value (22.2 MeV) because the value of the residual interaction strength /Io was too low. Nevertheless the theoretical peaks at 19.2 MeV and 22.6 MeV can be identified with the experimental ones at 19.5 and 25.5 MeV of fig. 2a.

COLLECTIVE CORRELATIONS

587

3.2. COUPLING SCHEME B 3.2.1. Parametric behaviour o f the cross sections. In coupling scheme B, we have investigated the influence of the strength Vo, exchange mixture ao or a~ of the twobody residual interaction, and deformation parameter flo on the cross sections. The calculated results are, however, not critically dependent on the parametrization. By varying the strength Vo, it was found that, with increasing Vo, a few states are shifted to higher energies and carry most of the dipole strengths, as in the case of the schematic model. A variation of the exchange mixture a o or a~, shifted the dipole strength with increasing ao (or decreasing a~, since their sum must be l) to lower energies. Similar results were obtained by Buck and Hill 14) for 160. The influence of the surface parameter flo consists in shifting the energies and dipole strengths to higher energies. This is to be expected, since the dipole-phonon interaction has the nature of an additional two-body force between the p-h states. 3.2.2. Total absorption cross sections. "[he results of the calculation including the excited states of the residual nuclei are presented in figs. 2-5 and table 2, c~dculated with parameters Vo = - 7 5 0 MeV. fm 3, ao = 0.9, a~ = 0.1. Fig. 2 shows the total absorption cross section. The peaks at 17.7, 19.5 and 20.9 MeV of fig. 2b are in good agreement with the observed peaks of Shevchenko et al. 15) (fig. 2a). The splitting of the strength in the giant resonance is also reproduced. The vibrational satellite is however about 1 MeV lower than the experimental one at 25.5 MeV, in agreement with the bound-state calculation of Drechsel et al. This might be due to the neglect of anharmonic terms in the quadrupole Hamiltonian. The integrated total cross section up to 27.15 MeV amounts to about 160 MeV. mb in the lp-lh model as well as in the collective correlation model. Experiments performed by Wycoff et al. 16) give about 1 l0 MeV • rob. The fact that the result of the present model is no better in this respect than those of the lp-lh model is not surprising. As mentioned earlier, the states with coupled surface phonons do not contain any dipole strengths in themselves. They only have the effect of modulating the dipole strengths already contained in the lp-lh states. Including short-range correlations might help improve the result because quasi-deuteron effects, which take place at higher energies, help rob the giant resonance of dipole strength. As an example of the variation of parameters, we show in fig. 2c a total cross section with Vo = - 850 MeV. fm a and ao = 1, a~ = 0. With this set of parameter, the giant resonance still lies at about 22.2 MeV, but because of the larger ao value the vibrational satellite has less dipole strength than that of fig. 2b. Moreover the small peak at 20.9 MeV of fig. 2b has been shifted, obviously due to the larger Vo value, to higher energy at 22.9 MeV. 3.2.3. Partial absorption cross sections. In fig. 3 the partial (~, Po) (fig. 3b) and (~, p) (fig. 3c) cross sections are shown and compared with the experimental result of the inverse reaction liB(p, ~o)12C of Allas et al. 17) (fig. 3a). Besides the peaks in the giant resonance already discussed, the peaks at 17.7, 19.5 and 20.9 MeV are also seen

a) ~

B l l (P,7~) C l z

o~

°%o,

too

,~°

°~

¢

50

- L---

i

~_

i

I

b)

I

12

ZE

11

O u m i--

u')

20

~ lO rr

0

•,"

i

I

I

I

I

c) E 20

~

0

Cu(T'P) Bll

L

18

20 22 24 ENERGY EMeV]

26

28

Fig. 3. Photoproton cross sections. (a) experimental result of the inverse reaction l lB[p, To)t2C of Alias et aL tT); (b) and (c) (7, Po) and (7, P) cross sections respectively with collective correlations in coupling scheme B.

_. •!"°

z+

.~

¢,. ~

.1

2 E 0

a)

. ,2

# j,,

•

I"

r~.. •

~ ,-.,~.-.."'~" ", " ~ ' ; ' ~

b

•

~"i

. . . . -t

I/Yt! /ill

12

i

4

/

~

'

22

-",

" ' f "\-

" "

/'t

20

b)

.

24

"

26

EEMeV3 Fig. 4. Cross-section comparison of (a) experimental (7, no) data 18) with (b) theoretical (7, n) (dot-dashed curve) and (7, no ) (full curve) predictions. The lowest curves in (a) and (b) represent contributions to the cross sections of neutron emission and transition to the excited states.

COLLECTIVE CORRELATIONS

589

experimentally. The latter two are not present in a normal lp-lh model calculation and have, by an examination of the wave function, been identified as one-phonon satellites of the [p~l x st] peak at 17.7 MeV. The (y, n) and (y, no) cross sections are shown in fig. 4. The experimental points correspond to the (~,, no) data of Fultz et al. is), and the dashed curve to that of Verbinski et al. t 9). The lowest curves in figs. 4a and b represent the contributions of the excited states. In comparing, it should be remembered that in our model only negativeparity states have been considered. From the calculated areas under the curves of fig. 4b up to 27.15 MeV we conclude that 10 ~ of all neutrons take part in transitions to the excited states while 90 Y/ogo to the ground state. The corresponding values of Fultz et al. up to 28 MeV excitation energy are 17 ~o and 83 ~. TAeI~ 2 Comparison of the theoretical integrated differential cross sections at 90 ° in coupling scheme B with the measured results of Medicus e t al. zl) f~o (~)dE

f~° (d~2) dE (MeV " mb/sr):

(MeV" mb/sr): x2C(7 , p)lIB

Energy of level (MeV)

12C(7 ,

theory Eo = 27.15 MeV

exp. Eo = 27 MeV

Energy of level (MeV)

0.11 0.91 0.22 0.86

0 0.02 0.06 0.05 0.05 0. l 0 0.05 0.30

8.10 7.50 6.90 6.35 6.49 4.81 4.32 2.00

Total excited states

2.10

0.6(5)

Ground state

8.20

6.20

8.57 7.99 7.30 6.81 6.76 5.03 4.46 2.14

Total excited states Ground state

n)11C

theory exp. Eo = 27.15 MeV Eo = 27 MeV

0.01 0.03 0.05 0.38

0 0 0.03 0.04 0.03 0.10 0.03 0.1 (5)

0.47

0.40

4.70

3.10

Further cross sections for transitions to the various excited states of 11B and t t C were calculated. Unfortunately at present there are no corresponding experiments with which to compare them. The differential cross section for (~, no) at 90 ° has been measured by Verbinski et al. 2o). The agreement of the theoretical cross section with experiment in this case was comparable to that shown in fig. 4 for the integrated-overangles case. Integrated differential cross sections at 90 ° for ground and excited state transitions have been compared with the experiment of Medicus et al. 21). The results are tabulated in table 2. As in the other cases already discussed, the theoretical values for the excited-state cross sections are higher than the experimental ones. Also, the theoretical

590

E . D . MSHELIA AND R. F. BARRETI"

(7, P3) value is larger than the (7, Pl) value. This might lie in the fact that because this state has the same angular momentum 7 = ½ as the ground state it can acquire some transition strength from the ground state. 3.2.4. Angular-distribution coefficients. In fig. 5 we present the coefficient a2/a o in a Legendre polynomial expansion of the photoproton angular distribution. In figs. 5a and b we compare the calculated lp-lh results (no collective correlations)

a) I

-o.s~-

I,

f

I

b)

t -o.s"

"~c)

6' 0.5}

0 ~_-----'7-~~____ -0.5

d)

0.~

E[MeV]

"~b..2'2 •

.----J

~4

_

26 ~8 ~" i

-0.5 L

Fig. 5. The a2/ao coefficient in a Legendre polynomial expansion dcr/d.Q -- ao+a2P2 (cos #) of the photoproton angular distribution for the reaction 12C(y, po)llB. (a) in the l p - l h model including the deep-lying configuration [s{ -1 ×p~_]l-; (b) in the l p - l h model without the configuration [s½- I × p{ l t - ; (c) with collective correlations; (d) experimental result of Frederick and Sherick 22).

with and without the configuration [s~ t × p~]t- respectively. The effect of this configuration on the angular distribution is small and can be seen only at higher energies (above 26 MeV where the curve of fig. 5a slightly rises with increasing energy). For comparison, the experimental results of Frederick and Sherick 22) (fig. 5d) are also shown. The inclusion of collective correlations (fig. 5c) is seen to have no effect on the predicted values of the aSa o coefficient compared with those obtained from the simple lp-lh model. 4. Summary and conclusion

In the work presented here we have investigated the giant dipole photonuclear cross section of 12C in the framework of the collective correlation model, whereby the continuum particle states are properly taken into account. In our model, calculations were performed according to two angular momentum coupling schemes. In coupling scheme A, the excited states of the residual nuclei

COLLECTIVE CORRELATIONS

591

could not be suitable described. However approximations could be made by reducing the size of the configuration space and enabling two-phonon states to be considered. Coupling scheme B was introduced in order to take the excited states of residual nuclei into account. This was achieved by means of the core-hole coupling model. Although the model described in this work represents a marked improvement over the simple l p - l h model, it has some weak points: Firstly, surface vibrations were assumed to be harmonic, although the experimental spectra possess anharmonicities. A consequence of this might have been that the calculated position of the vibrational satellite was about 1 MeV lower than the experimental value. Secondly, the two-body short-range correlations were neglected and the collective dipole operator in the dipole-quadrupole interaction was approximated by the p-h dipole operator. As a result the absolute magnitudes of the calculated cross sections were larger than the experimental ones. Other continuum calculations have been performed for 12C based on models which have some degree of overlap with the one used here. Baur and Alder 23) have coupled the collective one-phonon 2 + state at 4.43 MeV in 12C to the lp-lh states in a coupledchannel method. The particle-vibration coupling interaction was however defined via the optical model potential. This model is equivalent to our coupling scheme A. However, in the results of Baur and Alder there was no indication of the vibrational satellite at 25.5 MeV. Recently Birkholtz 24) has also calculated the G D R of 12C in a coupled-channel method, whereby the ground state and the excited states of the residual mass-11 nuclei were described in a lp-shell intermediate-coupling model. There, the correct experimental position of the vibrational satellite at 25.5 MeV could be reproduced and the energy-integrated total cross section below 30 MeV has been decreased from his lp-lh model value of 164 MeV- mb to 138 MeV. mb. In order to reproduce this experimental value one must include effects (e.g. short-range correlations) which can shift the excess dipole strength in the giant resonance to higher energies. Calculations along these lines could be valuable, but have not yet been performed. We are indebted to Prof. Dr. W. Greiner who suggested this work and for his continuous interest. We gratefully acknowledge fruitful discussions with Drs. P. P. Delsanto and P. Antony-Spies. Computing facilities were provided by the Zentrales Recheninstitut, University of Frankfurt am Main. R. F. Barrett received financial support in the form of post-doctoral fellowships from the Alexander von HumboldtStiftung while in Frankfurt and from AINSE while in Melbourne. References 1) C. M. Shakin and W. L. Wang, Phys. Rcv. Lett. 26 (1971) 902 2) M. Kamimura, K. Ikcda and A. Arima, Nucl. Phys. A95 (1967) 129 3) V. Gillot, M. A. Melkanoff and J. Raynal, Nucl. Phys. /07 (1967) 631 4) D. Drechs¢l, J. B. Seaborn and W. Greiner, Phys. Rev. 162 (1967) 983

592

E . D . MSHELIA A N D R. F. BARRETT

5) R. F. Barrett, L. C. Biedenharn, M. Danos, P. P. Delsanto, W. Greiner and H. G. Wahsweiler, Rev. Mod. Phys. 45 (1973) 44 6) C. Mahaux and H. A. Weiclenmiiller, Shell-model approach to nuclear reactions (North-Holland, Amsterdam, !969) 7) J. Le Tourneux, Mat. Fys. Medd. Dan. Vid. Selsk. 34 (1965) 11; H. J. Weber, M. G. Huber and W. Greiner, Z. Phys. 192 (1966) 182, 223 8) E. D. Mshelia, R. F. Barrett and W. Greiner, Phys. Rev. Lett. 28 (1972) 847 9) M, Danos and W. Greiner, Phys. Rev. 146 (1966) 708 10) H. Theissen, Inst. fiir Techn. Kernphys. Darmstadt, private communication 11) H. G. Wahsweiler, W. Greiner and M. Danos, Phys. Rev. 170 (1968) 893 12) A. B. Clegg, Nucl. Phys. 38 (1962) 353 13) P. Antony-Spies, Nucl. Phys. A188 (1972) 641 14) B. Buck and A. D. Hill; Nucl. Phys. A95 (1967) 271 15) V. Sbevchenko and N. Yudin, At. Energy Rev. 3 (1965) 3 16) J. M. Wycoff, B. Ziegler, H. W. Koch and R. Uhlig, Phys. Rev. 137 (1964) B576 17) R. G. Alias, S. S. Hanna, L. Meyer-Schiitzmeiter and R. E. Segel, Nucl. Phys. 58 (1964) 122 18) S. C. Fultz, J. J. Caldwell, B. L. Berman, R. L. Bramblett and R. R. Harvey, Phys. Rev. 143 (1966) 790 19) V. V. Verbinski, J. C. Courtney, D. F. Herring, R. B. Walton and R. E. Sund, Bull. Am. Phys. Soc. 9 (1964) 628 20) V. V. Verbinski and J. C. Courtney, Nucl. Phys. 73 (1965) 398 21) H. A. Medicus, E. M. Bowey, D. B. Gaytber, B. H. Patrick and E. J. Winhold, Nucl. Phys. A156 (1970) 257 22) D. E. Frederick and A. D. Sherick, Phys. Rev. 176 (1968) 117 23) G. Baur and K. Alder, Heir. Phys. Acta 44 (1971) 49 24) J. Birkholz, Nucl. Phys. A189 (1972) 385