The low energy spectra of heavy transitional even nuclei from an effective nucleon-nucleon interaction

Volume 233, number 1,2

PHYSICS LETTERS B

21 December 1989

T H E L O W E N E R G Y S P E C T R A O F HEAVY T R A N S I T I O N A L E V E N N U C L E I FROM AN E F F E C T I V E N U C L E O N - N U C L E O N I N T E R A C T I O N I D E L O N C L E a, j LIBERT a, L. B E N N O U R a,b.l a n d P Q U E N T I N b a C S N S M 2, Bdttment 104, F-91405 Orsay Campus, France b Laboratotre Phystque Thbonque 3, rue du Solarium, F-33170 Gradtgnan, France

Received 24 July 1989, revised manuscript received 3 October 1989

The seven scalar functions offl and y defining a Bohr quadrupole collective hamlltonian have been calculated microscopically from an effective nucleon-nucleon Interaction In a tractable numerical approach corresponding to clean cut approximations of the adiabatic time dependent Hartree-Fock-Bogohubov theory Without any free parameter adjustment and using the SIII Skyrme force fitted once for all mostly on saturation properties, one gets a quantitatively good reproduction of the low lying collective spectra offour transitional nuclei 74Ge,76Se, tl°Cdand t86pt

dealing with simpler cases as tn the purely vibrational or rotational hmlts

1. Introduction

Years ago, Bohr and Mottelson [ 1 ] estabhshed a general framework for the study of collective and ln& v l d u a i degrees of freedom in non-spherical nuclei This so-called "unified model" description has proven to be a mighty source of inspiration for a wealth of theoretical and experimental studies O n the other hand, systematical microscopic assessments of the structure of spherical and deformed nucle~ over the whole nuchdlc chart (based in a more or less loose way u p o n the B r u e c k n e r - H a r t r e e - F o c k approximat i o n ) have emerged as rather predictive tools for a variety of static properties [2] The aim of this work is to bridge the gap between these two, seemingly disconnected, physical approaches for the description of low excitation energy spectra As a starting point, we wdl concentrate here on the spectroscopy of nonspherical even nuclei In order to explore the full richness of the collective dynamics, we will consider here so-called "transltxonal nuclei", even though it is quite clear that our approach is a fortlort capable of Present address ServicePhysique Nucl6aire Th6onque, ULB, CP 229, B-1050 Bruxelles, Belgium 2 Unit6 propre du CNRS 3 Unit6 associee au CNRS 16

2. The theoretical frame The unified model assumptions can be s u m m a rized as follows. First, the collective m o t i o n is supposed to be consistent with the existence of a m e a n field at each time Then, the time dependence of the wavefunctlon is assumed to originate in the time dependence of a limited set of collective variables Furthermore, these collective variables are chosen as the five c o m p o n e n t s of the quadrupole surface deformation tensor Last but not least, the resulting collective m o t i o n is treated as being adiabatic with respect to single particle excitations Dealing with heavy nuclei on the one hand, and with possibly large amplitude collective motion on the other hand, one should of necessity incorporate palrmg correlations in some appropriate fashion. The time dependent H a r t r e e - F o c k - B o g o h u b o v ( T D H F B ) approach in the a&abatlc limit appears therefore to be a well suited starting point for a microscopic description of such collective motions One then considers the equations of motion:

l~ ~ = [ ~:, ~ ]

(1)

0370-2693/89/$ 03 50 © ElsevierSciencePublishers B V (North-Holland)

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where ~ and ~ are the usual [ 3 ] generalized density and hamlltonian matrices of the T D H F B approach Due to the hermltian projector character of ~ , one can use the Baranger-V6n6ronl theorem [4] to yield two time-even hermitlan matrices ~ and Z such that ,,¢=e'X ~

(2)

e -IX

Truncating up to second order in X, the above density # and the resulting hamdtonlan ~¢, one gets two equations of motion (odd and even under time reversal) Now we make a constrained H a r t r e e - F o c k Bogohubov ( C H F B ) ansatz for the adiabatic path 4 , where the constraining fields Q, define the collective variables q, (as their expectation value for ~o) This choice is consistent through well known sum rule theorems (see e g ref [5] ) with the emphasis put here on the low energy part of the strength functions assocmted to Q, operators With such a choice for ~o, one is only left with the solution of the time-odd part of the adiabatic equations of motion In the non-correlated case, it has been shown [5] that this could be achieved by solving a C H F set of equations upon adding to the time-even Q, constraining field, a time-odd constraint - q , P, (where q, is the collective velocity and P, is directly determined from the density defining the admbat~c path) This has been recently generalized for the pairing correlated case [6], in the form of a CHFB approach where two time-odd fields p}uF) and p}acs) are to be considered (for the HF-hke and the BCS-hke CHFB equations respectively).

e~ "~> = ~ p}acs)=lh

([t ]ro ,po

[

+ L-OqTq,'~

])

'

l

0-o po ) + ~ q , j 2 (0P° \Oq, ~ o - -O-~q

(3)

where Po and ~o are the normal and abnormal densities defining ~o The second order expansion in Z o f the HFB energy around J¢o, yields two terms a zero order term which is the collective potential energy, a second order term which ~s the collective kinetic energy. The latter, quadratic in the collective velocities (1 e the Lagrange multipliers for the time-odd constraints) q,, provides thus inertial parameters in so far as Z is given by the solution of the above double CHFB equations

21 December 1989

3. Details of the present calculations In order to allow for a forthcoming systematical study of spectroscopic properties, various approximations have been made out of the general scheme sketched in the above section 2 First of all, we have limited ourselves to Hartree-Fock ( H F ) plus BCS instead of H a r t r e e - F o c k - B o g o h u b o v ( H F B ) calculations. We have also used a different force for the normal and the abnormal parts of the HFB energy, which could be in principle justified microscopically. For the former part, we have used the SIII Skyrme force [ 7] whose spectroscopic relevance up to moderate deformations is rather well known (at least for heavy nuclei) For the latter part now, we have made a constant pairing matrix element approximation for any single particle states below a given cut-offenergy This matrix element is fixed - as usual - in a given nuclear region by the reproduction of o d d - e v e n binding energy differences [8] - see, however, the discussion in section 4 below Self-consistency corrections ~ la Thouless-Valatln [9 ] for the pairing field have been neglected due to the crudeness of our approximation to this field already at zero order. They have also not been considered for the normal (HF-hke) field This may be - at least partially -justified by some previous results In ref. [5], for an effective mass of ~ 0 75m (m being the nucleon mass), such self-consistent contributions to the mass of the axial mode for axially symmetrical solutions were found not to exceed ~ 10% in the uncorrelated case These results have been confirmed in the presence of pairing correlations in a variety of heavy nuclei along full axial deformation energy curves [ 6 ] As a result our mass parameters are nothing but Inghs cranking masses [ 10 ] including pairing correlations [ 11 ] However, it IS important to stress that, by construction, we have generated cranking operators that are consistent with the choice of our adiabatic path This is a marked difference with the HFB approach o f r e f [ 12 ] To define these operators one has to numerically evaluate the derivatives Opo/ 0q, One should also need in principle O~o/Oq, However, one may deduce its matrix elements from the projector identity ~ = To evaluate the adiabatic path ~o, one replaces the H F variational solution by a self-consistent solution for the semi classical part #o o f the normal density Po 17

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within the so-called extended T h o m a s - F e r m i ( E T F ) a p p r o a c h corresponding to a W l g n e r - K l r k w o o d expansion up to o r d e r 4 in h [ 13 ]. Convoluting properly the denslty~o with the two b o d y interaction, one generates a one body averaged H F hamiltonian whose diagonallzation leads to an a p p r o x i m a t e BCS solution I~g) It is o f course quite clear that I~') contains most o f the shell effects o f the exact H F plus BCS solution The expectation value o f E - a n d similarly o f any other o p e r a t o r - is then a p p r o x i m a t e d as

E " (q/[HI~) ,

(4)

where H is the microscopic hamlltonlan including the effecuve force. This approach constitutes therefore an i m p r o v e d version o f a well known a p p r o x i m a t i o n scheme - often referred to as the expectation value m e t h o d [ 14 ] - which leads in these refs [ 14 ] as in the present work to relative ( d e f o r m a t i o n ) energies c o m p a t i b l e within up to 0 5 MeV with exact H F plus BCS results. Details o f the whole purely static part o f our a p p r o a c h will be given elsewhere [ 15 ] Here, we will limit ourselves by m e n t i o n i n g that to find/~o, we m i n i m i z e the semi-classical energy by a variation within a limited space o f d e f o r m e d F e r m i type denslty functions In practice, axial and non-axial purely elllpsoldal shapes connected through the semi-axis values to the usual Bohr parameters fl and 7 [ 16 ] have been considered F o r the collective variables fl and 7 we have thus p e r f o r m e d calculations for 56 points o f a sextant corresponding to the spherical point and to f l = 0 05, 0 10, , 0.55 a n d 7 = 0 °, 15 °, , 60 ° At each o f these points, four other static calculations have be~n perf o r m e d to define by a three-points m e t h o d the two d e r l v a t w e s ofpo Stable derivatives have been found u p o n choosing the steps A p = 0 001 a n d d ~ = 0 25 ° One is therefore able to compute, for the 56 points, along with the potential energy V(fl, 7), 1 e ( q/I HI ~u), the three mass p a r a m e t e r s Map(fl, ~,), M~(fl, 7) a n d Mre(fl, 7) with obvious n o t a t i o n The c o m p u t a t i o n o f the three m o m e n t s o f inertia ~3,(fl, ~,) is far more easy due to the scalar character o f H and is p e r f o r m e d by using s t a n d a r d formulae [ 17 ] involving matrix elements o f the single particle angular m o m e n t u m componentsjk The Bohr collective hamlltonlan, constructed microscopically as sketched above, has been diagonalized by projecting ItS solutions in a suitable basis, ac18

21 December 1989

cording to the m e t h o d p r o p o s e d in ref. [18] It corresponds to the P a u h requantizatlon ansatz N o zero-point m o t i o n corrections have been taken into account at this stage. O u r numerical d l a g o n a h z a t l o n has been shown to be numerically equivalent to the finite difference a p p r o a c h initiated long ago in ref [ 19 ] A given basis state corresponds to a sum over each even K-value (with usual n o t a t i o n ) o f fl'~ sln(n7) or tim c o s ( n y ) according to the parity o f ½K, m u l t i p l i e d by Wlgner rotation matrices a n d suitable geometrical factors O u r basis states satisfy the required s y m m e t r y conditions discussed in ref [ 19 ] An exponential overall factor ensures the vanishing o f the wavefunctlons at largefl-values It corresponds to the exponential o f a weighted sum o f the scalar invariants f12, ]~4, f13 COS 37 whose p a r a m e t e r s are adj u s t e d to m i n i m i z e the ground-state 0 + energy Due to the integration weight, one constructs ~ la S c h m i d t an o r t h o n o r m a l set out o f the previous basis before d l a g o n a h z a t i o n The versatile character o f our basis states allows us to reach a numerical convergence for a tractable matrix size (typically one needs about 19 basis state for 0 + elgensolutlons which corresponds to tim terms up to m = 12)

4. Discussion of results Potential energy surfaces V(fl, 7) are displayed in fig 1. The 74Ge nucleus appears very soft with a shallow m i n i m u m at y-~ 30 ° and fl_~ 0 25. This pattern is q u a h t a t l v e l y consistent with similar results using the Gogny force [20 ] or the SKM* Skyrme force. A d d lng two protons - 1 e for 7 6 S e - the soft character o f V(fl, 7) is preserved but its m i n i m u m has been shifted to the oblate edge. The l~°Cd potential energy surface IS typical o f a transitional nucleus with two local (prolate and oblate ) m i n i m a - the former being lower in energy than the latter. These results are in agreem e n t with those o b t a i n e d in H F plus BCS calculations using the same force [21] F o r the 186pt nucleus, one gets a typical W i l e t s - J e a n [22] potential energy surface (1 e almost degenerate in the y-direct i o n ) This feature was also o b t a i n e d in exact selfconsistent calculations [23] with the S i l l force, even though a small dip was found there In the prolate side o f the constant-7 valley Calculated mass p a r a m e t e r surfaces generally ex-

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PHYSICS LETTERSB

0

0

1

2

05

0

05

05

0

OS

1l oCd

/ 0

Fig 1 Potential energysurfacesfor four transitional nuclei Contour lines are labelled by their energy (in MeV) relative to the absolute minimumrepresented by a black dot The energydifference between two adjacent such lines is l MeV The radial coordinate varying from 0 to 0 5, is the quadrupole deformation parameter fl hlbit a rather rich structure The existence of peaks in them reflects the presence of single particle levels in the vicinity of the chemical potential (i.e. situations where there is a large c o n t r i b u t i o n of diagonal matrix element in the Inghs cranking s u m ) as exphcltly demonstrated for Map (/7, y = 0 ° and 60 ° ) in ref. [ 6 ] M o m e n t s of inertia, quite on the contrary, are found to be globally very close Indeed to what could be deduced from a hydrodynamic-hke formula [ 16 ]. The spectra resulting from the dlagonahzatlon of the Bohr h a m d t o m a n are shown m fig. 2 for the 74Ge, 765e and 1lOCd nuclei in comparison wtth experimental data [24-26 ]. As a general rule, one reproduces the experimental energies within less than 2 0 0 - 3 0 0 keV, for the first nine or ten levels Assigning each calculated level to the ground-band, the quasi-beta b a n d or the quasi-gamma band, is generally quite obvious from mere energy considerations In doubt, one could decide u p o n checking the m a i n K-component In principle, specific assignments of levels in bands should of course result from calculations of transition probabilities which are yet to be performed within our approach. In this context, it is interesting to discuss the probabilities In the (fl, 7) sextant, as-

21 December 1989

soclated with each of our quantal states This is sketched in fig. 3 for the 74Ge nucleus #1. One observes almost no rotational stretching of the triaxlal ground-band shape. The quasi-beta b a n d exhibits a one-node pattern m fl along a y---30 ° line This is consistent with the expected feature of a o n e - p h o n o n state. The situation is less clear with the quasi-gamma b a n d where no apparent one-node states are found in the y-direction. The third 0 + state corresponds, as expected, to a rather complex structure. Similar conclusions hold for the calculated wavefunctlons of 765e For the l l°Cd nucleus, one finds g r o u n d - b a n d states that are essentially prolate (as opposed to the broad trlaxial distributions in 74Ge). Consequently, the double-humped distribution in the quasi-beta b a n d is found aligned on a y = 0 ° direction. A rather interesting feature appears in this nucleus Contrarily to the previous lighter nuclei, one observes a sizeable centrifugal stretching starting with the 6 + state: the average value offl which was found around 0 19 for the two first m e m b e r of the band, becomes 0 25 and 0.28 for the 6 + and the 8 + states This feature is reflected in the trend of the so-called kinematic mom e n t of inertia whose value is increased by a 2 4 factor when evaluated from the 8+--.6 + t r a n s m o n as opposed to the 2 + ~ 0 + transition In both the 7aGe and 765e, the second 2 + level (quasi-gamma b a n d head) lies below the second 0 ÷ state. This feature, which is reproduced in our calculations, may be attributed to the existence of a rigid triaxlal rotor [ 27 ] It may as well result from a potential energy surface being very soft m the y-direction. In our calculations it seems that the second explanation prevails One notices also the excellent reproductlon of the third 0 + state energy in the 76Se nucleus We may finally add that a correct reproduction of the first 4 +, second 2 + and second 0 + states is strongly contingent upon a correct description of the (/7, 7) dependence of the mass parameters as estabhshed by calculating spectra using a simple constant mass parameter approximation (as given e g. in ref [ 18 ] ) A similar conclusion could be drawn from the

at Due to symmetry properties (reflecting themselves in the integration metric m (,8, 7) variables) the probablhty density for an infimteslmalcartesian surface in the sextant is exactly vanishing on the axial edges (y= 0 °, 60 ° ) 19

Volume 233, number 1,2

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21 December 1989

MeV

/

ZnGe

3 6+

~2+ _ -'--

-

0--~--+-. 0+

Exp

Th

2+ ~"Exp

4+ .-"-(3÷)

Exp

2+

--

4÷

- "l

~

2+

6+-""

{ 2+ , l } / /

Th 2+

Th

Exp

0_..+..- . . . . Th

Exp

Th

2+

0+

0+

Exp Th

Exp

6+

0

5+ (2,3,4+///

4--~-+. . . .

0+__. . . .

I -

8+/'

~

3+

4+

I

Se

41+

..... 2

76

--" . . . . .

4+

3+ 4* ------

4+

0~ - . . . .

2+

2+

Exp

Th

Exp

Th

Ol.+. . . . .

Exp

Th

II°cd

Th

186

Pt

~-6+/"

~

__.. o+

~2+- l _

. . . . . 3+ .

O+

Exp Thl

~:::=

2+

Th2

Exp TN

Th2

Exp Thl Th2

Fig 2 Comparison of experimental (Exp) and calculated (Th) spectra for four transition nuclei In the 186pt case, two theoretical results are given They correspond to the pairing strength G used in the static calculations ofref [24] for one of them (Thl) and to reduced G value (namely the prevmus G minus 3%) for the other (Th2) results of the calculation m the same region reported in ref. [28] In the t86pt case (as for the three previously discussed nuclei) the calculations have been performed with pairing matrix elements G already used in previous static calculations (1.e roughly fitted on the lowest quasi-particle energies in nelghbourlng odd nuclei) They yield a first 2 + level energy somewhat too high (see fig 2) In view of the rather poor phenomenologlcal d e t e r m i n a t i o n of the pairing strength parameter, we found it appropriate to decrease the previous G-value by 3%. The corresponding spect r u m is also displayed in fig 2 where a good quantitative agreement with data [ 29 ] is obtained for both the ground b a n d a n d the second 0 + (quasi-beta b a n d head) states The o n e - p h o n o n character of the latter state is ascertained by the one-node pattern of ~ts collective wavefunction Even though it IS Impossible to rule out a priori an explanation in terms oftwo-quaslparhcle modes, it appears that a purely collective na20

ture of such an excitation seems c o m p a u b l e with its observed energy. More definite conclusions on that point should, again, result from a proper description of transition probabilities More generally, one should stress the extreme sensmvity of the low energy collective spectrum (ground b a n d m o m e n t s of inertia a n d p h o n o n energies) with respect to the pairing strength This fact leads to the conclusion that the reproduction of such spectroscopic properties could be considered as a rather rehable way of defining G in a given nuclear region

5. Conclusion This work is a self-consistent and mIcro~oplcally well founded version of previous approaches which made use of phenomenological mean fields (see e g. ref [ 30 ] ) It also corresponds to a clean cut prescrlpn o n for the cranking operators.

Volume 233, number 1,2

PHYSICS LETTERS B

21 December 1989

/

%s

o15

0

/

/

t

o

o5

o5

0

0

/

o,/

9s

o

O5

0

/

0+

/

O5

Fig 3 Calculated probabdlty densities m the collectwe space for some low lyinglevelsof 74Ge Represented from left to right are ground band states, quasi-beta band states, the head of the quasx-gammaband together with the third 0+ state Contour hnes are labelled by the value of the probabdlty density for a cartesmn metric ('gdfldT) whose scale ~sgiven by the ,g-valuewhich vanes in the figure from 0 to 05

Undoubtedly, the remarkable reproductmn of experimental data concerning the low energy spectra of different transitional nuclei xs the mare result of the present paper It confirms the abihty of the Skyrme SIII lnteracUon to yield good spectroscopic properties. Pairing correlations however, should be treated in a more systematic way. The strong dependence of the calculated spectroscopic properues on the global pairing strength qualifies such an approach as a good testing ground for further i m p r o v e m e n t of the pa~ring matrix elements m use Some progress should be made also to u n d e r s t a n d the h m l t a t l o n s of our seml-quantal approach One possible path to do so, would be to compare our results with those obtained m a generator coordinate approach with or without the use of the questionable gausslan overlap approximation [31 ] A direct test

of the collectwe wavefunctmns still remains to be performed, namely the evaluation of electromagnetic transition probabilities Nevertheless the agreement obtained w~th low energy spectra, w~thln an approach ~solatlng consistently the quadrupole degrees of freedom, provides strong i n d l c a u o n s for the collective ( q u a d r u p o l e ) character of the considered excttatlons A systematical study of the collective properUes in the nuclear regions considered here (A ~- 70, A - 110, .4 ~- 190) should be performed to confirm the finding of the present study O f particular interest would be to apply such an approach to the very interesting phen o m e n o n of shape isomerism at no spin [8,32,33 ]. The couphng of our collective states w~th one quasiparticle states to describe odd nuclei has already been attempted upon using approximate soluUons of the 21

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general Bohr hamtltontan dynamics [34] Such a study ts presently extended upon mcludmg present collective solutmns A generallsatlon of these calculattons to two quasi-particle states to describe o d d odd nuclei or non-collective exc~tatlons m even nuclei would complete the global project of revising microscopically the unified model descrtptlon of nonsphertcal nuclei.

References [1 ] • Bohr and B Mottelson, Mat Fys Medd Dan Vld Selsk 27 (1953)no 16 [2 ] See e g P Quentin and H Flocard, Annu Rev Nucl Part Sci 28 (1978) 523 [3] N N Bogohubov, Usp Flz Nauk 67 (1959) 549 [Sov Phys Usp 2 (1959) 236], J G Valatm, Phys Rev 122 (1961) 1012, M Baranger, in Carg6se Lectures in Theoretical Physics, 1962, ed M Levy (BenJamin, New York, 1963) p V-1 [ 4 ] M Baranger and M V6n6ronl, Ann Phys ( N Y ) 1 1 4 (1978) 123 [ 5 ] M J Gmnnom and P Quentin, Phys Rev C 21 (1980) 2060, 2076 [6] L Bennour, J Llbert and P Quentin, to be pubhshed [7 ] M Beiner, H Flocard, Nguyen Van Giai and P Quentin, Nucl Phys A238 (1975) 29 [ 8 ] For a discussion of this point see e g P Bonche, J J Kneger, P Quentin, M S Weiss, J Meyer, M Meyer, N Redon, H Flocard and P -H Heenen, Nucl Phys A 500 (1989) 308 [9 ] D J Thouless and J G Valatin, Nucl Phys 31 (1962) 211 [ 1 0 ] D R Inghs, Phys Rev 96 (1954) 1059, 103 (1956) 1796 [ l l ] S T Belyaev, Mat Fys Medd Dan Vld Selsk l l (1959) 31 [ 12 ] See e g M Glrod, Ph Dessagne, N Langevm, F Pougheon and P Roussel, Phys Rev C 37 (1988) 2600 [ 1 3 ] M Brack, C Guet and H-B Hiikansson, Phys Rep 123 (1985) 276 [ 14 ] C M Ko, H V Pauh, M Brack and G E Brown, Phys Lett B45 (1973) 433, Nucl Phys A236 (1974) 269, M Brack, Phys Lett B 71 (1977) 239 [ 15 ] I Deloncle, K Bencheikh, L Bennour, M Brack, J Llbert, J Meyer and P Quentin, contnbutmn to the 10th Session d'6tudes biennale de Physique Nucl6aire (Aussols, France, March 1989), report LYCEN 89-02, p S18 l

22

21 December 1989

[16]A Bohr, Mat Fys Medd Dan Vid Selsk 26 (1952)no 14 [17] S G Nllsson and O Prior, Mat Fys Medd Dan Vld Selsk 32 (1960) no 16 [ 18 ] J Llbert and P Quentin, Z Phys A 306 (1982) 223 [ 19 ] K Kumar and M Baranger, Nucl Phys A 92 ( 1967 ) 223 [ 20 ] M Girod, contribution to the 8th Session d'6tudes blennale de Physique Nucl6aire (Aussols, France, March 1985), report LYCEN 85-02, p S12 1 [21 ] M Meyer, J Dani6re, J Letessier and P Quentin, Nucl Phys A316 (1979) 93, N Redon, J Meyer, M Meyer, P Quentin, P Bonche, H Flocard and P -H Heenen, Phys Rev C 38 (1988) 550 [22] L Wllets and M Jean, Phys Rev 102 (1956) 788 [23] N Redon, J Meyer, P Quentin, M S Weiss, P Bonche, H Flocard and P -H Heenen, Phys Lett B 181 (1986) 223 [24] B Slngh and D A Vlggars, Nucl Data Sheets 51 (1987) 225 [25] B Singh and D A Viggars, Nucl Data Sheets 42 (1984) 233 [ 26 ] P de Gelder, E Jacobs and D de Frenne, Nucl Data Sheets 38 (1983) 545 [27] A S Davydov and G F Fillpov, Nucl Phys 8 (1958) 237 [28] S E Larsson, G Leander, I Ragnarsson and N G Alenius, Nucl Phys A261 (1976)77 [29] R B Firestone, Nucl Data Sheets 55 (1986) 583 [30] S G Rohozlnskl, J Dobaczewskl, B Nerlo-Pomorska and K Pomorskl, Nucl Phys A 292 (1977) 66 [ 3 1 ] P G Remhard, Nucl Phys A252 (1975)120, 133, A261 (1976) 291, M Girod andB Grammaucos, Nucl Phys A 330 (1979) 40 [32] E F Moore, R V F Janssens, R R Chasman, I Ahmad, T L Khoo, F L H Wolfs, D Ye, K B Beard, U Garg, M W Dngert, Ph Benet, Z W Grabowsla and J A Clzewsla, Phys Rev Lett 63 (1989) 360 [ 33 ] V V Pashkevltch, Dubna preprlnt P4-4383 (1969), quoted in S M Pohkanov, Usp Flz Nauk 107 (1972) 685, R R Chasman, Phys Lett B 219 (1989)227, M Glrod, J P Delaroche, D Gogny and J F Berger, Phys Rev Lett 62 (1989) 2452, J P Delaroche, M Glrod, J Llbert and I Deloncle, Phys Lett B232 (1989) 145 [34] D E Medjadl, P Quentin, M Meyer and J Llbert, Phys Lett B 181 (1986) 185