fulltext

Dimers and polymers in extremely deformed neutron-rich

light nuclei (*)

W.VONOERTZEN

Hahn-Meitner-Institut, Berlin and Fachbereich Physik, Freie Universität Berlin Glienicker Straße 100, D-14109 Berlin, Germany

(ricevuto il 18 Luglio 1997; approvato il 15 Ottobre 1997)

Summary. — The structure of nuclear dimers and their rotational bands based on the a-a potential and covalent valence neutrons with up to 4 valence nucleons are discussed using published transfer reaction data for Be and boron isotopes. Based on the 01

2 state in12C, which is assumed to be an 3 a-particle chain at an excitation

energy of 7.65 MeV and adding the covalent nucleons, chain states in the system

12_{C* 1x neutrons are constructed. The energy position of the lowest chain states are}

estimated and ways for their population in reactions on 9_{Be and using radioactive}

beams are proposed. It is expected that these states are metastable and could have appreciable branches for g-decay. Extrapolations to longer chain states, polymers, in neutron-rich light isotopes are made.

PACS 21.10 – Properties of nuclei; nuclear energy levels. PACS 21.60.Gx – Cluster models.

PACS 01.30.Cc – Conference proceedings.

1. – Introduction

It has been known since more than 15 years that ground states and excited states in the isotopes of beryllium 9_{Be and} 10_{Be can be described as two-center molecules}

(dimers), where two a-particles are bound by covalent neutrons. A survey of the theoretical and experimental situation has recently been made by the author [1], where the experimental evidence from the literature has been compiled and states have been grouped into rotational bands in such a way that moments of inertia can be extracted.

There has also been a renewed interest in chain states consisting of a-particles in 4N nuclei both experimentally and theoretically [2]. After the well-established structure of8_{Be, which shows a rotational band based on a two-a-particle structure, the}

second O21 at 7.65 MeV in 12C has been proposed to be a chain of three a-particles.

(*) Paper presented at the 174. WE-Heraeus-Seminar “New Ideas on Clustering in Nuclear and Atomic Physics”, Rauischholzhausen (Germany), 9-13 June 1997.

896

Various approaches based on a cranked cluster model have confirmed that such structures exist [3] not only in 12C but also in other nuclei. Experimental evidence exists for example for16_{O [4] and recently also for}24_{Mg [5]. The origin of the existence}

of such states is the structure of the a-a potential, which (in its local form) shows an attractive part and a repulsive core [6], which describes well the 01_{and 2}1_{states in}8_Be

at 0.092 MeV and 3.04 MeV excitation energy.

With the establishment of detailed knowledge on the structure of dimers larger structures, chain states, polymers, can be predicted for carbon (13_C-18_{C) and oxygen}

(19_O-24_O).

I mention here two older references which contain similar discussions: i) Work by M. Seya, M. Kohno and S. Nagata [7] on the molecular orbital approach to the structure of9–11Be. ii) The paper by D. H. Wilkinson on a-neutron rings and chains [8]. The work of Seya et al. contains a detailed discussion of the deformations related to the molecular structure of the isotopes of Be and boron. In the rather speculative paper by Wilkinson the ground state of 10_{Be, with a binding energy for two neutrons of 8.5 MeV (which}

does not have the covalent bond structure) is used as a building block for more extended structures like rings.

2. – Dimers

2.1. Beryllium isotopes. – In order to understand the structure of the beryllium isotopes we have to discuss the properties of the two-center molecular orbitals, as they

Fig. 1. – Correlation diagram for nucleons in molecular orbitals in a two-center system with identical cores. The minimum of the a 1a-potential is located at a distance of ca. 3.5 fm (adapted from ref. [9]).

appear in a correlation diagram. In fig. 1 we show the relevant diagram [9] for the first single-particle orbits (S_{1 O2}, P_{3 O2}, P_{1 O2}, D_{5 O2}) of the separated nuclei. On the left side the well-known diagram for deformed nuclei can be identified; these orbitals merge into the two-center orbitals, which are obtained by the splitting of the separated center configurations. Quantum numbers which are necessary to classify these orbitals are indicated; the projection of the spin on the molecular axis (K); the reflection symmetry for the exchange of identical cores

(

)

has to be considered at the minimum in the a-a potential which occurs at a distance of ca. 3–4 fm. At smaller distance the a-a potential (in its local form) becomes strongly repulsive.

Fig. 2. – Energy diagrams for dimers of the beryllium isotopes. The excitation energies (and thresholds) are shown relative to the energy of the (a 1a1xn ) threshold. Only the first possible isomeric two-center states are indicated (without rotational excitations) in boxes.

898

The 3O22_{ground state of}9

Be corresponds here to the K 43O2, p2_{-binding orbital}

(p_{3 O2}2, g), whereas the next excited state, the 1 O21state, corresponds to the K 41O2,

s-ungerade-binding orbital (s_{1 O2}1, u), and the 1 O22 state to the K 41O2,

s-gerade-anti-binding orbital (p_{1 O2}2, g). Using the Pauli principle for two neutrons in these orbits (p_{3 O2}2) and (s_{1 O2}1) (or neutron and proton in a T 41 state) we obtain the well-established four states at excitation energies of ca. 6–7 MeV in 10_{Be with J}p

4 01_{; 2}1_{; 1}2 _{and 2}2_{, which form rotational bands as shown in fig. 3. The ground}

state of10_{Be and its first-excited state}

₍

₎

rotational band with a smaller deformation parameter, whereas the dimers show rotational excitations with a much larger moment of inertia

(

₎

see fig. 3 and sect. 3.

Figure 2 shows the relevant information on binding energies including the heavier isotopes (11_{Be and}12_{Be). The excitation energies are shifted so as to have a common line}

Fig. 3. – Excitation energies of states forming rotational bands of dimers based on 2 a 1 x-nucleons for9Be,10_Be,10_{B and}11_{Be as a function of J (J 11).}

Fig. 4. – Single-neutron stripping reactions on 9_{Be populating strong single-particle states}

in10_{Be, in particular the rotational bands of the dimers, the}9_{Be(a ,}3_{He ) reaction (in comparison}

with10_B).

for the relevant thresholds ( 2 a 1xn ) for all isotopes. The binding energy introduced by the addition of valence nucleons can thus be read from this line. Also the thresholds for the decay into other channels (with normal shapes) are indicated.

We repeat here the evidence for rotational bands in9Be and10Be. This evidence [10] is based mostly on inelastic a-scattering, (a ,3_{He) reactions as well as (d, p) reactions on} 9_{Be. In fig. 3 the different rotational bands are summarised and their properties are}

discussed in sect. 3. For the rotational bands in 10_{Be, fig. 4 shows as an example the}

population of10_{Be states in} 9_{Be(d, p) and}9_{Be(a ,}3_{He) reactions. The (a ,}3_{He) reaction is}

particularly selective at higher excitation energy; the very selective population of the

K 412_{band is very conspicuous (the Q-values are very negative!).}

The evidence for states in11_{Be corresponding to dimers can be obtained by looking}

into the population of states in two neutron transfer reactions like the9_{Be(t, p) reaction}

(fig. 5). Whereas these reactions should strongly populate such states, they should not

be observed in the one neutron stripping on 10Be, because the ground state does not have the appropriate structure. If there is a rotational band structure based on the

11

Be, 1 O21_{ground state, we must expect a similar drastic Coriolis coupling effect as for}

900

Fig. 5. – Spectra of two neutron stripping reactions on9_{Be populating strongly the rotational band}

of the dimer in11_{Be by the (t , p) reaction. The dashed line at 5.25 MeV indicates a not separated}

contaminant.

from the population of the 11_{Be states in the} 9_{Be(t, p) reaction we can suggest the}

following sequence of states for the K 41O21 _{band: ( 1 O2}1_{, 0.0 MeV; 5 O2}1_{, 1.78 MeV;}

3 O21_{, 2.69 MeV; 9 O2}1_{, 3.88 MeV and 7 O2}1 _{at 6.510 MeV); they are shown in fig. 3}

together with the K 43O22 _{band starting at 3.96 MeV. The binding energy of three}

covalent valence neutrons for the K 43O2 band amounts to approximately 5 MeV and the other states align perfectly in a rotational band as shown in fig. 3. These states are narrow and will possibly show g-transitions, which have not yet been studied. As discussed in sect. 3, the neutron pair is coupled to spin 01_{in the (t, p) reaction and most}

likely populates states with the (s_{1 O2}1u )2 configuration. These states show a strong relation to the most strongly populated states in the9Be(d, p) 10Be reaction, which are the negative-parity states with Jp

4 22, 32_{, 4}2_{at 6.26, 7.37 and 9.27 MeV.}

For 12Be there is little information available so far. If we fill 4 neutrons into the molecular orbitals (p_{3 O2}2) and (s_{1 O2}1) we expect a rather compact shape for12Be similar to10_Be

01,g.s. due to the (s1 O21)23 (p_{3 O2}2)2 configuration. In the two-center correlation diagram the s_{1 O2}1 (see fig. 1) configuration has a slope with an increasing binding energy for larger deformation (or two-center distance). Thus the fact that the moment of inertia is larger for the 12Beg.s.-band

(

01-band can be understood. Starting with values of the n-binding energies obtained for11_{Be, and assuming that the fourth neutron can contribute approximately}

1.5 MeV (like in9_{Be) the dimers in}12_{Be should be observed from excitation energies of}

approximately 4.5 MeV. These could possibly be observed in inelastic scattering on12_Be

using a radioactive beam of 12Be or in a 9Be(9Li ,6Li )12Be reaction and other three neutron stripping reactions on9_Be.

2.2. Boron isotopes. – The concept of isospin tells us immediately that the dimers in the neutron-rich isotopes have isospin analogues in the neighbouring nuclei (see [1]).

For10_{B we start with the basic structure of}9_{Be. By adding a proton we can form in} 10

B the T 41 analogue states (TD) of the four states in

10_{Be*, the residual interaction}

states have been identified in 10_{B in the excitation energy region of 6.56–7.5 MeV.}

Adding the proton to form the T_{E(T 40) states we will have again four states as in the}

10_{Be* case, and using the generalised Pauli principle the corresponding states, which}

will form rotational bands are Jp

4 11, 31 _{for (p}

3 /22)2 and 12, 22 for the (p3 /22) 3 (s1 /21) configurations, respectively. Now the nucleon-nucleon interaction is slightly more attractive and we expect that the corresponding states are lowered by ca. 2 MeV relative to the T_{Dstates. The (a , t) reaction populates T 41 and T40 states; with its} strong momentum mismatch it shows a strong preference of high angular momentum states (see fig. 4).

For the 11_{B nucleus there are detailed studies of (}3_{He, p) and (a , d) reactions on} 9_{Be, which could strongly populate states in} 11_{B with T}

D4 3 /2 and TE4 1 /2 . The TD

states are analogues of the states in 11_{Be already discussed. The state (T 43/2) at} 5.24 MeV of 11_{Be will be located at approximately 17.50 MeV in} 11_{B. In fact strong}

transitions with narrow width have been observed in the (3_{He, p) reaction at high}

excitation energy. The strongest transitions are in direct correspondence to the strong transitions in the9_{Be(t, p) reaction (see fig. 5).}

3. – Moments of inertia and assignment of configurations to states in the isotopes10Be and11Be

The moments of inertia, u , of the dimers are of course the most relevant points of information for the discussion of their shapes (in particular also the g-ray transition probabilities between these states if they are observed). In fig. 3 the excitation energies of states in the beryllium and boron isotopes are plotted in diagrams representing rotational bands. I use the simplest expression for the rotational energy for a nucleus with moment of inertia, u , as a function of angular momentum J : K D 1 O2: E(J) 4ˇ2

O2 u[ J(J 1 1 ) 2 K(K 1 1 ) ]. For K 4 1 O2, we use, incorporating the effect of Coriolis decoupling [11]: E(J) 4ˇ2

/2 u[ J(J 11)1 (2)J 11/2_{a(J 11/2) ]; here,}

a, is the Coriolis decoupling parameter. The energy in this case can also be written as E(J) 4b[ (J1 (11sa/2) ]2

, where s is the signature, (2)J 11/2_{, defining the reflection}

symmetry of the intrinsic state. For the9

Be* (K 41/21_{) band I obtain the values b(4ˇ}2

/2 u) 40.35–0.46 [MeV] and

a ` 2.0; the same a-value is obtained for the K 41/2 band of11Be shown in fig. 3 (b is smaller A02–0.3 MeV) (a`2.0). This value

(

if only one orbital is considered) must be interpreted as the fact that the original spin of the orbit, whose K-quantum number is considered, is j 43/2. The Coriolis decoupling parameter in the two cases gives thus a strong support for the two-center correlation diagram interpretation of these states.

For the moments of inertia the following results are obtained:

a) For the9

Be K 43/22_and10_Be

g.s.bands (and8Be) similar values for ˇ2/2 u in the

range of 0.5–0.6 MeV. If we use a simple formula for the moments of inertia, (mR2_{), we}

have for m 42.0 and m42.22 and R4r0

(

)

with r0B 1.3 fm ( 2 Q A1 /3). These states still have a rather compact shape. As suggested

by Seya et al. [7], the ground state of10_{Be ( 0}1_{) can be assumed to be the (p}

3 /22, g)2₀1, configuration. In the (p3 /2)201configuration a strong pairing effect gives a shape which

corresponds to the smallest possible distance of the two a particles.

b) For the rotational bands of the excited dimer configurations in 10Be*, 11Be*,

902

distance between the two a-particles in the two-center diagram must be much larger.

Using this information from the rotational bands I can identify the structure of these states in terms of the diagram in fig. 1. The two lowest states at an internuclear distance of R B3.5 fm for the two centers are the (p3 /22, g) and (s1 /21, u) configurations, they correspond to the two lowest states of 9_{Be. The (s}

1 /21, u) configuration crosses the (p3 /22, g) at ca. R B4 fm; it has a decreasing slope (increasing binding energy) for R D4 fm, whereas the (p3 /22, g) configuration shows an increase of the binding energy by approaching smaller distances (R E4 fm) with a possible minimum at R B3 fm. We may thus give the following interpretation of the valence neutron structure of the two 01_{states in}10_{Be (see also ref. [1]).}

N10Be01 (g.s.)b4(2.19)1O2(p3O22g)₀211(0.01)1O2(s_1O21u)2₀1; N10Be*01 ( 6 .26 MeV )b 4 a0(s1 O21u)2₀1.

The other states at ca. 6 MeV energy in10Be must have the following configurations: N10Be01 ( 5 .958 )b 4 a1(p3 O22g )2₂1,

N10Be12( 5 .960 )b 4a2(p3 O22g ) 7 (s_{1 O2}1u ) ]12 and

N10Be22( 6 .263 )b 4a₃[ (p_{3 O2}2g ) 7 (s_{1 O2}2u ) ]₂2.

The second excited 21_{state appears much higher, because of the unfavourable overlap}

of the two (p_{3 O2}2g )-orbitals if they are coupled to spin 21. There should be in fact a further rotational band based on this state with states of 31 _{and 4}1 _{in the region of}

7–9 MeV, which is possibly covered by the other states.

Concerning the configurations in11_{Be, I can make the following assignments:}

N11Be_{3 O2}* 2; ( 3 .94 )b 4 (s_{1 O2}1u )2₀17 (p_{3 O2}2g ) and

N11Be_{1 O2}1; g.s. b 4 (p_{3 O2}2g )2₀17 (s_{1 O2}1u ) .

The latter configuration shows a less pronounced energy lowering of the (p_{3 O2}2g )2₀1 configuration than in the10_Be

01 ground state; I attribute this to the influence of the (s_{1 O2}1u ) configuration of the third neutron, which attains its minimum energy at larger distances, off-setting the effect of the (p_{3 O2}1g )2 levels. The negative parity, K 41O2, state is suggested to have the configuration: N11Be_{1 O2}2; ( 0 .32 MeV )b 4 (p_{3 O2}2g )22 (s_{1 O2}2g )1. This configuration, (s_{1 O2}2g ), appears as the next at a smaller two-center distance of ca. 3 fm, where the minimum for the 10_Be

01 ground state is expected. This state has a spectroscopic amplitude (in the deformed shell model) for the p_{1 O2}structure determined from the 10_{Be( d , p) reaction of (0.63)}1 O2 _{and is thus related to the more}

compact shape of the10Be01ground state; this state in11Be is thus much more compact in shape than the members of the K 43O22_band.

This discussion of the moments of inertia is strikingly confirmed by the calculations of Kanada-Enyo and Horiuchi (ref. [12]) using the antisymmetrised molecular

Fig. 6. – Density distributions of nucleons (neutrons and protons) in the beryllium isotopes as obtained from calculations based on antisymmetrised molecular dynamics (AMD) by Kanada-Enyo and Horiuchi (private communication). The neutron densities include those of the a-particles, still the covalent nature of the neutron bond is well seen. The possible molecular orbital configurations according to fig. 1 are indicated.

dynamics (AMD) approach. The AMD calculations reflect the properties of the two-center correlation diagram and the densities of the valence neutrons obtained correspond to those of the (p_{3 O2}) and (s_{1 O2})-orbitals. I reproduce in fig. 6 density plots as they are obtained in these calculations. They are obtained from Slater determinants after parity projection and represent the lowest states for a given parity. The covalent nature of the valence neutrons is immediately recognised. The tentative configuration assignments are indicated. We observe also a remarkable effect, the mixing of covalent orbitals of different reflection symmetry, which produces very spectacular density distributions for the case of the p 4 (2) states in10

Be. The K 43O2 band in 11_{Be with}

negative parity (which is not shown) is expected to have similar density distribution like the (2)-parity band in10_{Be with (s}

1 O21p_{3 O2}2) configurations.

4. – Isomeric chain states of the carbon isotopes and polymers

12_{C. The excited state of} 12_{C with spin parity 0}1_{at 7.66 MeV is just 288 keV above}

the threshold for the decay into 3a-particles at 7.366 MeV. This state is used as a reference point in fig. 7 for all carbon isotopes. Because of its peculiar properties it has been associated with a-particle cluster structure by Brink [3], and other authors, in particular in the form of an a-particle chain. This description is not completely uncontested, the bending modes of the pure a-particle chains, however, could be restricted once the valence-neutron bonds are added.

13_{C. The threshold in} 13_{C for the decay into (}12_C*

0211 n ) is at 12.6 MeV. Using the binding energy of the neutron in the dimer (9_{Be) we predict an isomeric state as a chain}

state with spin 3O22 _{at ca. 11.04 MeV. Another important threshold is for (}9

Be 1a), which is situated at 10.65 MeV. We can thus predict that coupling of 9_{Be with an}

a-particle should give a state at ca. 11.0 MeV just above the9

Be 1a-threshold. We find in fact in the compilations of Ajzenberg-Selove a state at 11.08 MeV which is very

904

Fig. 7. – Energy diagrams of molecular chain states in carbon isotopes. The energy scales are aligned to the same level for the decay threshold into 3 a-particles and x-neutrons. The binding energy for the isomeric chain states can be read from this common reference line. Isomeric states are indicated in boxes with black corners, if properties (excitation energies, populating reactions) etc. are known. Predictions for rotational bandheads are given in boxes without black corners.

narrow and which decays by g-emission (!) as well as n- and a-emission. This state in

13_{C* at 11.08 MeV must be the corresponding state to the “chain state” in} 12_C.

Population of this state and possible other states relevant for the chain configuration should be observed in various reactions starting with 9_{Be as target. The (}6_{Li, d) and}

(7Li, t) reactions at ELi4 23.8 MeV on 9Be studied by Ogloblin et al. [13] populate

(among others) two states very strongly at 10.8 MeV and 12.0 MeV.

14_{C. The threshold in} 14_{C for the} 12

C* 12n channel is quite high—at 20.7 MeV (see fig. 7). Similarly the threshold for the “unbound chain” (10_{Be* 1a) is high—at} 18.0 MeV. We may thus expect resonances in the10

Be* 1a system just above 18.0 MeV; these would correspond to a chain state where the two valence neutrons are kept in one covalent bond (configuration “Y” in fig. 7). We expect a series of states connected to the group of states at 6.0 MeV in10_{Be* ( 0}1_{, 2}1_{, 1}2_{, 2}2_{) ranging up to 19.0 MeV. There is a}

clear possibility to observe g-decay branches from and between those states just like for the 01

2 in 12C, because the (9Be 1a1n ) channel is closed, other channels like

(12

C 12n ) or (13

C 11n ) imply a very strong structural change, because of the very different shapes. The configuration where the valence bond is distributed equally

(configuration X) is expected to come below the configurations (Y), because of the stronger binding effect for each valence neutron ( 2 31.66 MeV). I predict this state at ca 17.4 MeV.

15_{C. For}15_{C with three valence neutrons, the isomeric structure can be based on the} 10

Be* dimer and the 9Be valence bond. Starting from the threshold for (12_{C* 13n ) at} 22.0 MeV we can predict that states with this structure should be observed from approximately 17.5 MeV excitation energy. The group of 4 states of the 10_Be*-dimer

with spins 01_{, 2}1_{, 1}2_{, 2}2 _{will couple with the spin of 3 O2}2 _{of the valence neutron in}

the 9_{Be bond and will form a dense cluster of states in this excitation energy region.}

Possible reactions, which can populate these states are the 9Be(7Li , p) reaction, and inelastic scattering. Final-state interactions of neutron-rich fragments, e.g. 9

Be 16_He

can be studied and many more reactions can be conceived with radioactive (neutron-rich) beams.

16_{C. The}16_{C chain isomer will consist of two bonds defined by the}10_{Be* dimer. The}

total binding energy of the two bonds is expected to be approximately 5 MeV! The corresponding states are thus well below the thresholds for the decays into channels with the same structural properties (9_{Be ,}10_{Be* ). These states should have a small}

width for the decay into a’s and neutrons and again the possible existence of g-decays is a very interesting possibility. These chain states can possibly be populated directly using radioactive beams, for example in a9Be(8Li , p) reaction. Higher excitations of the chain will eventually decay into10

Be 16_{He .}

Heavier isotopes of carbon and oxygen should also have states with the properties

of the chain states, (polymers) in particular18_{C and}20_{C. The former will be based on the}

structure of 11_{Be* and will have two times 3 valence neutrons at the covalent bond}

positions; the total binding will be 10 MeV! A very stable chain configuration is expected at an excitation energy of ca. 20 MeV. Similarly in 20_{C such isomeric chain}

states must be expected, which are related to the dimers (excited states) of12Be*. The extrapolation to polymers in neutron-rich oxygen isotopes like 22_{O up to} 26_{O is quite}

obvious. Rather exotic neutron-rich beams will be needed to populate such chain states.

4. – Conclusions

The present search on information on nuclear dimers and isomeric chain states (polymers) in neutron-rich isotopes of light nuclei is based on the concept of covalent molecular orbitals (ref. [14]) of neutrons. The level scheme of the Be-isotopes can almost exclusively be explained in this approach. This conclusion is also the main result of the work of Seya et al. [7]. We may expect as a general rule that clusterisation and formation of chains will occur for very neutron-rich nuclei close to the threshold to

a-particle substructures (see refs. [1, 15]). The formation of such structures via

condensation from a nuclear medium consisting of a-particles and neutrons may be expected to occur; the formation of chain states of neutron-rich nuclei can possibly be observed in central nuclear collisions using very neutron-rich beams (and targets), following the speculative suggestion of Wilkinson [7] on the formation of nuclear rings. Wilkinson started with the idea to form rings of a-particles bound by the (p_{3 O2}2g )2₀1 bonds in order to have a possible explanation of the “Anomalon” phenomenon. However, as shown in the present work, the ground state of 10_{Be does not show}

906

the necessary properties, it is too compact. The neutron-rich chain states have corresponding isospin analogue states (T_Dand T_Estates) in the neighbouring isotopes, where for example one neutron is replaced by a proton.

These states offer fascinating possibilities for studies using radioactive beam facilities, in particular for g-spectroscopy if the g-branches can be measured.

R E F E R E N C E S

[1] VONOERTZENW., Z. Phys. A, 354 (1996) 37; 357 (1997) 355.

[2] Proceedings of International Conference Atomic and Nuclear Clusters, Santorini (1993), edited by G. S. ANAGNOSTATOSand W. VON OERTZEN(Springer, Berlin, Heidelberg) 1995 (Z. Phys. A, 349 (1994)).

[3] BRINK D. M., Proceedings of International School of Physics “Enrico Fermi”, Varenna, Course XXXVI, edited by C. BLOCH(Academic Press, New York) 1966, p. 247; MERCHANTA. C. and RAEW. D. M., Nucl. Phys. A, 549 (1992) 431; also Z. Phys. A, 349 (1994) 243. [4] CHEVALIERP., SCHEIBLINGF., GOLDRINGG., PLESSARI. and SACHSM. W., Phys. Rev., 160

(1967) 827.

[5] WUOSMAAA. H., Z. Phys. A, 349 (1994) 249, and references therein. [6] LIUQ. K. K., Nucl. Phys. A, 550 (1992) 263.

[7] SEYAM., KOHNOM. and NAGATAS., Prog. Theor. Phys., 65 (1981) 204. [8] WILKINSOND. H., Nucl. Phys. A, 452 (1986) 296.

[9] SCHARNWEBERD., GREINERW. and MOSELU., Nucl. Phys. A, 164 (1971) 257.

[10] AJZENBERG-SELOVE F. and BUSH C. L., Nucl. Phys. A, 336 (1980) 1; 449 (1986) 1; AJZENBERG-SELOVEA., Nucl. Phys. A, 490 (1988) 1; 433 (1985) 1.

[11] EISENBERG J. M. and GREINER W., Nuclear Models, Vol. 1 (North-Holland Amsterdam), 1970, p. 255.

[12] KANADA-ENYO Y. K., HORIUCHI H. and ONO A., Phys. Rev. C, 52 (1995) 628; and private communications. Also contribution to RNB4-conference at Omiya (Japan), June 1996. [13] OGLOBLINA. A., in Proceedings of the Conference on Nuclear Reactions Induced by Heavy

Ions, Heidelberg, 1969, edited by R. BOCKand W. HERING(North-Holland) 1970, p. 231. [14] VONOERTZENW., Nucl. Phys. A, 148 (1970) 529;VONOERTZENW. and BOHLENH. G., Phys.

Rep. C, 19 (1975) 1; IMANISHIB. andVONOERTZENW., Phys. Rep. C, 155 (1987) 29. [15] HORIUCHIH., IKEDAK. and SUZUKIY., Prog. Theor. Phys. Suppl., 52 (1972) 89.