COMPACT FORMULAS FOR ROTATIONAL BANDS ENERGIES IN TRANSURANIC REGION

Transcription

1 NUCLEAR PHYSICS COMPACT FORMULAS FOR ROTATIONAL BANDS ENERGIES IN TRANSURANIC REGION R. BUDACA 1, A. A. RADUTA 1, 1 Department of Theoretical Physics, Institute of Physics and Nuclear Engineering,POBox MG, Bucharest 7715, Romania Academy of Romanian Scientists, 5 Splaiul Independentei, Bucharest 59, Romania Received November 7, 11 Analytical formulas for the excitation energies as well as for the electric quadrupole reduced transition probabilities in the ground, beta and gamma bands provided by the asymptotic limit of coherent state model were applied for description of strongly deformed nuclei from the transuranic region. Numerical calculations were performed for 1 nuclei which share some common features. Comparison of the calculation results with the corresponding experimental data shows a very good agreement. The parameters involved in the proposed model satisfy evident regularities being interpolated by smooth curves. The formulas provided for E transition probabilities are very simple and reproduce quiet well the experimentally observed behavior of intraband and interband transitions in case of strongly deformed systems. PACS: 1.1.Re, 3..Lv, 1.. Ev. 1. INTRODUCTION Since the liquid drop model was developed [1, the quadrupole shape coordinates were widely used by both phenomenological and microscopic formalisms to describe the basic properties of nuclear systems. Based on these coordinates, one defines quadrupole boson operators in terms of which model Hamiltonians and transition operators are defined. Since the original spherical harmonic liquid drop model was able to describe only a small amount of data for spherical nuclei, several improvements have been added. Thus, the Bohr-Mottelson model was generalized by Faessler and Greiner [ in order to describe the small oscillations around a deformed shape which results in obtaining a flexible model, called the Vibration Rotation Model (VRM), suitable for the description of deformed nuclei. Later on [3, this picture was extended by including anharmonicities as low order invariant polynomials in the quadrupole coordinates. With a suitable choice of the parameters involved in the model Hamiltonian, the equipotential energy surface may exhibit several types of minima [ like spherical, deformed prolate, deformed oblate, deformed triaxial, etc. To each equilibrium shape, specific properties for excitation energies and electromagnetic transition probabilities show up. Due to this reason, one customarily says that static values of intrinsic coordinates determine a phase for the nuclear system. RP Rom. 57(Nos. ourn. Phys., 7-8), Vol , Nos. 7-8, (1) P , (c) 1-1 Bucharest, 1

2 Compact formulas for rotational bands energies in transuranic region 189 A weak point of the boson description with a complex anharmonic Hamiltonian consists of the large number of the structure parameters which are to be fitted. A much smaller number of parameters is used by the coherent state model (CSM) [5 which uses a restricted collective space generated through angular momentum projection by three deformed orthogonal functions of coherent type. The model is able to describe in a realistic fashion transitional and well deformed nuclei of various shapes including states of high and very high angular momentum. Various extensions to include other degrees of freedom like isospin [, single particle [7 or octupole degrees [8 of freedom have been formulated [9. It has been noticed that a given nuclear phase may be associated to a certain symmetry. Hence, its properties may be described with the help of the irreducible representation of the respective symmetry group. Thus, the gamma unstable nuclei can be described by the O() symmetry [1, the gamma triaxial nuclei by the rigid triaxial rotor D symmetry [11, the symmetric rotor by the SU(3) symmetry and the spherical vibrator by the U(5) symmetry. Thus, even in the 5 s, the symmetry properties have been greatly appreciated. However, a big push forward was brought by the interacting boson approximation (IBA) [1, 13, which succeeded to describe the basic properties of a large number of nuclei in terms of the symmetries associated to the system of quadrupole (d) and monopole (s) bosons which generate a U() algebra. The three limiting symmetries U(5), O() and SU(3) mentioned above, are dynamic symmetries for U(). Moreover, for each of these symmetries, a specific group reduction chain provides the quantum numbers characterizing the states, which are suitable for a certain region of nuclei. Besides the virtue of unifying the group theoretical descriptions of nuclei exhibiting different symmetries, the procedure defines very simple reference pictures for the limiting cases. For nuclei lying close to the region characterized by a certain symmetry, the perturbative corrections are to be included. In Refs. [1, 15, it was shown that the critical points of some transitions correspond themselves to certain symmetries which may be described by the solutions of specific differential equations. The present paper is devoted to a systematic study of the transuranic region of nuclei through asymptotic limit of the CSM approach. The exact formulas for the matrix elements of the model Hamiltonian as well as of the E transition operator have been expanded in power series of 1/x(= d, where d is a real parameter which simulates the nuclear deformation). As a result, analytical compact formulas are obtained for both excitation energies and quadrupole electric transition probabilities. These formulas are used to describe the main features of transuranic nuclei. The above mentioned project was achieved according to the following plan. The basic ideas of the CSM approach are shortly reviewed in Section II. The asymptotic expansion for large deformations is given in Section III. Numerical applications are presented in Section IV and the final conclusions are summarized in Section V.

3 19 R. Budaca, A. A. Raduta 3. THE COHERENT STATE MODEL FOR THREE INTERACTING BANDS The model proposed in Ref. [5 is known under the name of CSM (coherent state model) and aims at describing in a realistic fashion, the lowest three rotational bands, ground, beta and gamma. Here we describe briefly the basic ingredients. First, one builds up a collective boson space being guided by the experimental picture. Thus, each band is generated by projecting the angular momentum from three orthogonal deformed states which modeled the ground, beta and gamma bands respectively, in the intrinsic frame of reference. The deformed ground band state is an axially deformed coherent state ψ g, while the other two intrinsic states are orthogonal polynomial excitations of the ground band model function. These excitations were chosen such that they are mutually orthogonal both before and after angular momentum projection. All three states are dependent on a real parameter d which simulates the nuclear deformation. In the limit d, the projected states must go to the first three highest seniority states of the boson multiplets while in the asymptotic region, i.e. large value for d, the states written in the intrinsic frame of reference have expressions similar to those associated to the liquid drop model, in the strong coupling regime. By this requirement, we assure that the model states have a behavior which is consistent with the so called Sheline-Sakai scheme [1, 17 which makes a continuous link between the vibrational and rotational spectra. These properties are satisfied by the following three sets of projected states: [ φ g M (d) = N g P Mψ g, ψ g = exp d(b b ), (1) ( φ β M (d) = N β P MΩ β ψ g, Ω β = b b b ) + 3d ( b b ) d3, () 1 7 ( φ γ M (d) = N γ P MΩ γ, ψ g, Ω γ,m = b b ) + d,m 7 b m. (3) The normalization factors of the above states are given in Appendix A. Within the restricted space just defined, one constructs an effective Hamiltonian by requiring a maximal decoupling, i.e. the off diagonal matrix elements are equal or close to zero. Ideal would be to have a diagonal Hamiltonian but this is not possible due to the gamma band. However, one solution is given by the six order quadrupole boson Hamiltonian: H = A 1 ( ˆN + 5Ω β Ω β ) + A Ĵ + A 3 Ω β Ω β, Ω β = (b b ) + d. () 5 where ˆN denotes the quadrupole boson number operator: ˆN = b µ b µ. (5) µ=

4 Compact formulas for rotational bands energies in transuranic region 191 Indeed H has the property that its matrix elements between a beta band state and a ground band or a gamma band state are vanishing. The matrix elements of H between the states presented above are expressed in terms of the basic overlap integral and its k-th derivatives (see Appendix A), defined by (d ) = 1 P (y)e d P (y) dy, I (k) (x) = dk dx k, x = d, () where P denotes the Legendre polynomial of rank. Taking into account the composition rule as well as the recurrence relations for the Legendre polynomial one can prove that the basic integrals satisfy the differential equation: d dx x 3 x d dx x + ( + 1) x =, (x = d ). (7) The solution for this equation is: (d ) = ( (!) ( ) (d ) e d 1 1F 1!( + 1)! ( + 1), + 3 ; 3 ), d (8) where 1 F 1 (a,b;z) is the hypergeometric function of the first kind. The excitation energies of the ground, beta and gamma bands are functions of the the ratio d I(1) and its first three derivatives with respect to d. 3. LARGE DEFORMATION REGIME One salient feature of CSM is the behavior of the projected states as function of the deformation parameter especially for the extreme limits of d and large d. While in the vibrational limit these are just multiphonon states in the rotational regime, i.e. for asymptotic values for deformation parameter d, the wave functions of the ground, beta and gamma band states predicted by the liquid drop model [1 in the large deformation regime are nicely simulated. Indeed as proved in Ref. [5, writing the projected states in the intrinsic reference frame and then considering a large deformation d, one obtains: ( ) ϕ i M = C β 1 e d kβ [δ i,g DM(Ω d ) + δ i,β 9 11 D M(Ω ) +δ i,γ βf kγ(d M(Ω ) + ( ) D M, (Ω )), (9) where k is a constant defining the canonical transformation relating the quadrupole bosons and the quadrupole collective conjugate coordinates: α µ = 1 k (b µ + ( ) µ b µ ), π µ = ik (( ) µ b µ b µ), (1)

5 19 R. Budaca, A. A. Raduta 5 while the constants C and f are C = 3 π 1 k /3 ( + 1) 1/, f = (8 + ( ) +1 ) 1/. (11) It is worth noticing that the model Hamiltonian yields for the ground band similar excitation energies as the effective Hamiltonian H eff = 11A 1 ˆN + A Ĵ. (1) Averaging this Hamiltonian on a vibrational ground band state, one obtains a quadratic expression in N, the number of bosons in the considered state: H eff = N(11A 1 + A + A N). (13) In the asymptotic region for d the average matrix element of H eff is [18 proportional to ( + 1): ( ) 11A1 H eff = ( + 1) d + A. (1) For the intermediate situation for the deformation parameter d, we may use for energies either rational functions of d with the coefficients being functions of the angular momentum as given in the previous section, or asymptotic expansion for the matrix elements in power of 1/x. The later version was developed in Ref. [19. Here we sketch the ideas and give the final results. The asymptotic expressions for the matrix elements are obtained by considering the behavior of the overlap integral for large d. This is obtained by using the asymptotic expression for the hypergeometric function: F (a,c;z) = Γ(c) Γ(a) ez z a c [1 + O( z 1 ), (15) One finds that the dominant term of the asymptotic form of ex 3x. (1) This suggests as trial function for the quantity satisfying the differential equation (7) the following series: = e x n x n=1a n. (17) This series expansion together with the differential equation offer a recurrence relation for the series coefficients: A n+1 = A n (n + )(n 1). (18) n Using the asymptotic form (1) as the limit condition, which infers A 1 = 1 3, solution (17) is completely determined. is

6 Compact formulas for rotational bands energies in transuranic region 193 For big values of the deformation parameter, the series can be approximated by a truncation, such that one arrives at the following expression = x 1 1 3x 5 9x 37 ( 1 7x 3 + x x + 13 ) 18x 3 ( + 1) x I(1) 1 5x 3 ( + 1) + O(x ). (19) This approximation can be substantially improved. Indeed, let us write the differential equation (7) in the form x ( x I(1) ) + ( x I(1) ) ( x 1 x I(1) ) x + ( + 1) = () and replace the first term by the derivative of expression (19). Obviously, one obtains a quadratic equation for the quantity xi (1) /I() whose positive solution is x I(1) = 1 [ x + G, (1) where G = 9 ( x(x ) ) ( x x + 37 ) x + ( x 3x + 13 )( x + 1) 9x 3 ( + 1). () Note that the mixing m.e. between ground and γ states are negligible within the approximation of large deformation. Using approximation (19), the energies of the β and γ bands can be written as follows: E β = 1 P β E γ = A 1 S γ P γ [ A 1 S β + A 3F β + A ( + 1), (3) + A ( + 1). () The polynomials P,S,F, in ( + 1), are given in Ref.[19. To these equation we add the equations determining the excitation energies in the ground band: [ E g x = 11A 1 + G + A ( + 1). (5) This expression for the ground band energies was obtained in Ref. [19. A similar expression was obtained, based on the variational moment of inertia method, by

7 19 R. Budaca, A. A. Raduta 7 Holmberg and Lipas in Ref. [. A generalization of the Holmberg-Lipas formula was obtained in [1 by a semiclassical description of a second order quadrupole boson Hamiltonian. Taking the asymptotic limit of the exact m.e. of the quadrupole operator given in Ref.[5, one obtains very simple formulas for transition m.e. for large deformation case [19. The asymptotic expressions for the reduced m.e. of the harmonic quadrupole transition operator are [19 φ i Q h φ i = dq hc K i K i, i = g,β,γ, K i = δ iγ, φ γ Qh φ g = q h C φ β Qh φ γ = 3 19 q hc, while the β and ground band states are connected by anharmonic part of Q µ : φ β Qanh φ g =, () 7 19 q anhc. (7) Note that in the asymptotic limit of the deformation parameter d, the projected functions are similar to that of the liquid drop model in the strong coupling regime. The Clebsch-Gordan factorization of the transition probabilities is known in the literature as Alaga s rule [. Thus, we may say that our description of the deformed nuclei is consistent with the Alaga s rule.. NUMERICAL RESULTS The analytical expressions for energies and transition probabilities presented in the previous sections were applied to 1 transuranic nuclei which are known as well deformed. The results are compared with the available data for both energies and reduced transition probabilities. Excitation energies of the three rotational bands have been calculated with Eqs.(5), (3) and (). The parameters involved were calculated by a least squares procedure. The results are listed in Table I. Therein we also give the root mean square for the deviations of the calculated excitation energies from the corresponding experimental data, denoted by χ, the total number of states in the three bands and the ratio E + 1 /E + 1. The mentioned ratio indicates how far we are from the rotational limit 3.33 associated with the SU(3) dynamical symmetry. Excitation energies in ground, beta and gamma bands are presented in Fig.1 as function of angular momentum. Theoretical fits are compared with the corresponding experimental data which are abundantly available in some cases. Except for 8,3 Th, which exhibit a triaxial shape [3, the listed nuclei have a ratio E + /E 1 + close to the rotational limit. The 1 results concerning the fitted parameters are given in Table I, from which one remarks

8 8 Compact formulas for rotational bands energies in transuranic region 195 the high accuracy for the theoretical description. Except for 8 Th, 3 U and Pu where the two excited bands are only slightly split apart, for other nuclei the excited states relative position reclaim on ideal SU(3) symmetry. Table 1 The fitted parameters, d,a 1,A and A 3, determining the ground, γ and β energies in the limit of large d, i.e. asymptotic regime, for 1 nuclei from the transuranic region. Also we give the r.m.s. for the deviations of the calculated and experimental energies, denoted by χ, the total number of states in the considered three bands and the ratio E + /E 1 +. In the first column are listed the nuclei and the 1 reference from where the experimental data were taken. Nucleus E +/E + d A 1 [kev A [kev A 3 [kev χ [kev Number of states 8 Th [, Th [ Th [ U [ U [ U [ U [ Pu [ Pu [ Pu [ Cm [ Cm [ Now we would like to comment on the change of parameters when we pass from one nucleus to another. This is done by plotting the deformation parameter as function of the nuclear deformation β, Fig., and the structure coefficients A 1,A and A 3 as functions of A + (N Z)/, in Figs.3-5. The atomic mass number was corrected by the third component of the isospin in order to infer also the result dependence on Z. In Fig. it is shown that there exists a linear correspondence between the deformation parameter d and the quadrupole deformation β. The linear dependence of d on the quadrupole nuclear deformation β has been studied analytically, in a different context, in Ref. [3. The structure coefficient A 1 depends on A + (N Z)/ according to what is shown in Fig.3. The fitted A 1 values are interpolates by a fourth order polynomial. Two nuclei, 3 Th and 3 Th, fall apart from the interpolating curve. As a matter of fact 3 Th is known as being the best candidate for triaxial rigid shape phase. The parameters A yielded by the fitting procedure are interpolated by a line,

9 19 R. Budaca, A. A. Raduta 9 E() [MeV E() [MeV E() [MeV 8 Th 3 Th 3 Th E() [MeV E() [MeV E() [MeV 8 1 [h - 3 E() [MeV 8 1 [h - 3 E() [MeV 8 1 [h - 3 E() [MeV 8 1 [h - 3 E() [MeV 8 1 [h - 3 E() [MeV 8 1 [h [h U 3 U 3 U 8 1 [h U 38 Pu Pu E() [MeV 8 1 [h - 3 Pu Cm 8 Cm 8 1 [h [h [h - 3 Fig. 1 Energy spectra of ground, γ and β bands described by means of asymptotic formulas with parameters for SU(3) nuclei from transuranic region. Open symbols denote uncertain or with possible band assignment experimental points, which were not taken into account in the fitting procedure. Experimental data are taken form [ 3.

10 1 Compact formulas for rotational bands energies in transuranic region 197 Fig. The deformation parameter d as function of the nuclear deformation β taken from Ref. [35. Fig. 3 The parameter A 1 as yielded by the fitting procedure as a function of A + (N Z)/. The numerical values are interpolated with a forth order polynomial.

11 198 R. Budaca, A. A. Raduta 11 Fig. The linear fit of the A values in respect to A + (N Z)/. Fig. 5 The parameter A 3 as yielded by the fitting procedure as function of A + (N Z)/.

12 1 Compact formulas for rotational bands energies in transuranic region 199 as is shown in Fig.. The slope of this line is very small, such that one can say that a specific value for structure parameter A is associated to transuranic nuclei treated here. Incidently the highest value of this parameter is obtained for 3 Th. On the contrary, the obtained values for the structure parameter A 3 are very scattered, being mostly concentrated around the minimum of a parabola which interpolates the best all values. Table B(E) transition probabilities in the asymptotic limit for few deformed transuranic nuclei which have absolute experimental values also for inter-band transitions. Values in square braces were not taken into account for the fitting procedure. Only for 3 Th and 38 Pu were used the uncertain β-ground transition probabilities in order to fix the q anh parameter. Experimental data are taken from [, 7, 3. B(E) tr. prob. 3 Th 3 Th 38 U 38 Pu π i π f Exp. Th. Exp. Th. Exp. Th. Exp. Th. + g + g g + g g + g g + g [ g 8 + g [ g 1 + g [ g 1 + g [ g 1 + g g 1 + g [ g 18 + g [ g + g [ g + g g + g g + g β + g [ [.38.7 [ β + g β + g [ [ 3..7 [ [ γ + g γ + g γ + g [ d q h [(W.u.) q anh [(W.u.) Concluding, except for a few nuclei which exhibit some features of triaxial shapes, all structure coefficients show a smooth dependence on A + (N Z)/. The deformation parameter d is related linearly with the nuclear deformation β. For each considered nucleus the two parameters defining the quadrupole tran-

13 11 R. Budaca, A. A. Raduta 13 sition operator were determined by a least squares fit of the experimental available data. The B(E) values for some of the well deformed nuclei considered by the present work, have been calculated using the asymptotic expressions for the matrix elements given by Eqs.() and (7). The results are listed in Table. As seen from this table, a very good agreement between the results of our calculations and the corresponding experimental data is obtained. A special mention is deserved by 3 Th where B(E) values are available, and an excellent agreement is obtained. The good agreement between the calculated and experimental values of B(E) transition probabilities is due to the dependence on deformation of the reduced matrix elements of the quadrupole transition operator between states of the same band, which assures the the intraband transitions are more probable than the interband ones. As a final conclusion one can say that the properties of all these transuranic nuclei can be described fairly well by the asymptotic regime of CSM. 5. CONCLUSIONS The present paper considers the CSM approach in the extreme regime of large deformations. Thus, the matrix elements of the model Hamiltonian as well as of the E transition operator between the angular momentum projected states modeling the members of the ground, beta and gamma bands, are expanded in power series of 1/x(= d ). As a result the excitation energies in the three bands are expressed analytically as ratios of polynomials in 1/x with the coefficients depending on angular momentum. Concerning the matrix elements of the E transition operator, for large deformation the whole angular dependence is contained by a Clebsch-Gordan coefficient which is accompanied by a factor depending on d for intraband transitions and independent of deformation for interband transitions. This simple description is used to describe the available data for 1 transuranic nuclei exhibiting with some exceptions the SU(3) symmetry. The results are in a good agreement with the corresponding experimental data for both excitation energies for the three bands and the transition probabilities. The distinct features of the SU(3) are associated with a specific deformation parameter and structure coefficients. Indeed for A structure parameter was obtained a really narrow domain of values, while the values of A 3 are gathered around the minimum of a parabola in the variable A + (N Z)/. Changing the nucleus under consideration the coefficients are changing anyway but obey a certain rule expressed by their dependence on A + (N Z)/. This is the case of the A parameter values, which are interpolated by a fourth order polynomial in the same variable. The nucleus which falls apart from this interpolating polynomial corresponds to 3 Th, which indeed deviates from the traditional rotational behavior inclining more to the triaxial nuclei. In fact this is a

14 1 Compact formulas for rotational bands energies in transuranic region 111 measure of the predictability power of the CSM approach. In this way CSM is able to describe not only axially deformed nuclei, but also nuclei with hints of triaxiallity. Comparing CSM with the Liquid Drop Model (LDM), one may say that CSM is a highly anharmonic model while LDM has a harmonic structure. However, in the large deformation situation, the CSM wave functions are similar to those characterizing LDM in the strong coupling limit. Another successful anharmonic model was proposed by Gneus and Greiner but that uses a large number of parameters and moreover, the quadrupole conjugate momenta contribute to the Hamiltonian only through the quadratic terms. Moreover, energies are obtained by the diagonalization procedure in a spherical basis which may encounter convergence difficulties for large deformations. By contrast CSM projects, over angular momentum, states from a coherent state and two orthogonal polynomial excitations and consequently is especially realistic for the well deformed nuclei. This feature is actually confirmed by the application from this paper where the transuranic nuclei spectra are obtained with a high accuracy. CSM accounts for features which are complementary to those described by IBA. Indeed CSM s model Hamiltonian is not a boson number conserving Hamiltonian. Moreover, while IBA uses a space of states with limited number of bosons, CSM states cover the whole boson space since they are projected from infinite series of bosons. Due to this feature, the IBA approach is concerned with the description of low lying states with angular momentum not exceeding 1 + and with a moderate deformation. By contrast, CSM works quite well for high spin states (in Fig.1 energies for states with 3 are shown). CSM was applied for the description of the triaxial nuclei [37 and the results were compared with those obtained within the Vibration Rotation Model [. Recently, a more extensive study of triaxial nuclei with CSM has been performed [38 and the results were compared with those produced by a solvable model. The results of the present paper show that the asymptotic limit of the CSM is not only analytically consistent with the phenomenological behavior of strongly deformed nuclear systems but also provide an extremely precise description of the available data in this region of the nuclear shape phase space. Acknowledgments. This work was supported by the Romanian Ministry for Education Research Youth and Sport through the CNCSIS project ID-138/8. REFERENCES 1. A. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. (1) (195); A.Bohr and B.Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 7(1) (1953).. A. Faessler and W. Greiner, Z. Phys. 18, 5 (19); 17, 15 (19); 177, 19 (19) ; A. Faessler, W. Greiner and R. Sheline, Nucl. Phys. 7, 33 (195).

15 11 R. Budaca, A. A. Raduta G. Gneus, U. Mosel and W. Greiner, Phys. Lett. 3 B, 397 (199).. P. Hess,. Maruhn and W. Greiner, Phys. Rev. C3, 335 (1981);. Phys. G: Nucl. Part. Phys. 7, 737 (1981). 5. A. A. Raduta, V. Ceausescu, A. Gheorghe and R. M. Dreizler, Phys. Lett. 99B, (1981); Nucl. Phys. A381, 53 (198).. A. A. Raduta, V. Ceausescu and A. Faessler, Phys. Rev. C3, 111 (1987). 7. A. A. Raduta, C. Lima and A. Faessler, Z. Phys. A 313, 9 (1983). 8. A. A. Raduta, Al. H. Raduta and A. Faessler, Phys. Rev. C55, 177 (1997); A. A. Raduta, D. Ionescu and A. Faessler, Phys. Rev. C5, 3 (). 9. A. A. Raduta, in Recent Res. Devel. Nuclear Phys. 1, 1 7 (), ISBN: X. 1. L. Wilets and M. ean, Phys. Rev. 1, 788 (195). 11. A. S. Davydov and G. F. Filippov, Nucl. Phys. 8, 788 (1958). 1. A. Arima and F. Iachello, Ann. Phys. (N.Y.) 99, 53 (197); 13, 8 (1979). 13. F. Iachello and A. Arima, The Interacting Boson Model (Cambridge University Press, Cambridge, England, 1987). 1. F. Iachello, Phys. Rev. Lett. 85, 358 (). 15. F. Iachello, Phys. Rev. Lett. 87, 55 (1). 1. R. K. Sheline, Rev. Mod. Phys. 3, 1 (19). 17. M. Sakai, Nucl. Phys. A 1 (197) 31; Nucl. Data Tables A 8, 33 (197); A 1, 511 (197). 18. A. A. Raduta, A. Gheorghe and M. Badea, Z. Physik A 83, 79 (1977). 19. A. A. Raduta and C. Sabac, Ann. Phys. (N.Y.) 18, 1 (1983).. P. Holmberg and P. O. Lipas, Nucl. Phys. A 117, 55 (198). 1. A. A. Raduta, R. Budaca, A. Faessler, our. Phys. G: Nucl. Part. Phys. 37, 8518 (1).. G. Alaga, Nucl. Phys., 5 (1957). 3. A. A. Raduta, P. Buganu, Phys. Rev. C 83, 3313 (11).. A. Artna-Cohen, NDS 8, 73 (1997). 5.. Gröger, T. Weber,. de Boer, H. Baltzer,K. Freitag, A. Gollwitzer, G. Graw and C. Günter, Acta Phys. Pol. 9, 35 (1998).. Y, A. Akovali, NDS 9, 155 (1993). 7. M. R. Schmorak, NDS 3, 139 (1991). 8. Y, A. Akovali, NDS 71, 181 (199). 9. M. R. Schmorak, NDS 3, 183 (1991). 3. F. E. Chukreev, V. E. Makarenko, M.. Martin, NDS 97, 19 (). 31. F. E. Chukreev, Balraj Singh, NDS 13, 35 (). 3. Y, A. Akovali, NDS 9, 177 (). 33. Y, A. Akovali, NDS 87, 9 (1999). 3. E. der Mateosian,. K. Tuli, NDS 75, 87 (1995). 35. G. A. Lalazissis, S. Raman and P. Ring, Atomic data and Nuclear Data Tables, 71, 1 (1999). 3. Lo Iudice, A. A. Raduta and D. S. Delion, Phys. Rev. C 5, 17 (199). 37. U. Meyer, A. A. Raduta, A. Faessler, Nucl. Phys. A 37, 31 (1998). 38. A. A. Raduta, P. Buganu, Phys. Rev. C 83, 3313 (11).