The Restricted Dynamics Approach for A <FONT FACE=Symbol>£</font> 60 Nuclei and the Estimation of Microscopic Values of Phenomenological Collective Model Parameters

Abstract

The results of the evaluation of the Generalized Interacting Boson Model (GIBM) Hamiltonian in the framework of Restricted Dynamics Approach have been used in order to estimate the microscopical values for the parameters of two phenomenological collective models - the Bohr-Mottelson Model (BMM) and the standard Interacting Boson Model (IBM). The microscopically derived parameter values are calculated, using the effective NN-interaction potential parameter values obtained from the Strictly Restricted Dynamics Model (SRDM) energy calculations for light <FONT FACE="Symbol">a</font>-cluster type nuclei with 8 <FONT FACE="Symbol">£ </font>A <FONT FACE="Symbol">£ </font>60, and compared with the phenomenological model parameter values obtained from the direct fit to the experimental level energies of the same nuclei.

The Restricted Dynamics Approach for A £ 60 Nuclei and the Estimation of Microscopic Values of Phenomenological Collective Model Parameters

J.A. Castilho Alcaras

Instituto de Fisica Teórica, UNESP, 01405-900, São Paulo, Brazil

Nuclear Research Center, LV-2169, Salaspils, Latvia

Institute of Theoretical Physics and Astronomy, 2600 Vilnius, Lithuania

Received May 8, 1997

The results of the evaluation of the Generalized Interacting Boson Model (GIBM) Hamiltonian in the framework of Restricted Dynamics Approach have been used in order to estimate the microscopical values for the parameters of two phenomenological collective models - the Bohr-Mottelson Model (BMM) and the standard Interacting Boson Model (IBM). The microscopically derived parameter values are calculated, using the effective NN-interaction potential parameter values obtained from the Strictly Restricted Dynamics Model (SRDM) energy calculations for light a-cluster type nuclei with 8 £ A £ 60, and compared with the phenomenological model parameter values obtained from the direct fit to the experimental level energies of the same nuclei.

I. Introduction

From the point of view of the non-relativistic quantum mechanics one can regard the free nucleus in three-dimensional space as a closed system of well localized A particles - nucleons. The dynamics of this system is defined by its general microscopic Hamiltonian H. At the present stage of nuclear theory development, however, one cannot solve the Schrödinger equation for it. Therefore, H is substituted by the model Hamiltonian H₀, for which it should be possible to solve the corresponding Schrödinger equation.

The traditional way of nuclear model construction is to substitute the original microscopic Hamiltonian H by another - phenomenological Hamiltonian H_ph. In most cases H_ph is taken from the theory of other physical systems (e.g. as in shell model, or Bohr-Mottelson model), or based on some simplified assumptions about the nuclear structure (IBM, IBFM, etc.). Usually, it is impossible to relate these Hamiltonians with microscopic H. Besides, some of these H_ph do not conserve all nuclear integrals of motion.

The Restricted Dynamics Approach (RDA) [1, 2] provides one with the powerful tool for the description of collective phenomena in nucleus via the approximate solution of the microscopic many-body Schrödinger equation. Employing RDA, one can project from the general translationally invariant nuclear Hamiltonian for the system of A nucleons, its collective O_A-1-invariant part, depending only on six microscopic collective variables.

Although the principles how to obtain microscopic collective models from the general microscopic nuclear Hamiltonian are known, the technique of such evaluations is very complicated. At present the collective part of the microscopic central NN-interaction potential has been evaluated analytically only for the multipole-Gauss type Wigner pontentials. This collective part can be rewritten in the form analogous to the potential energy term of Bohr-Mottelson type models. Again, rewriting it in terms of creation-annihilation operators, one can obtain from it the Generalized Interacting Model Hamiltonian.

The relationships between macroscopic phenomenological collective models and microscopic collective models in RDA for the first time have been studied in Refs.[3, 4] and in more details in Refs.[5, 6].

Let us consider the correspondence chain for nuclear collective Hamiltonians, starting with the General Microscopic Nuclear Hamiltonian (GMNH):

where

GRDCM - the general, O_A-1-invariant Restricted Dynamics Collective Model; SRDCM - the Strictly Restricted Dynamics Collective Model - the collective (U_A-1-invariant) part of the full Strictly Restricted Dynamics Model (SRDM); GBMM - the Generalized Bohr-Mottelson Model in 6-dimensional model space; BMM - the standard phenomenological Bohr-Mottelson Model for nuclear collective motion in 5-dimensional model space (obtained by freezing collective variable r²); GIBM - the Generalized Interacting Boson Model in 6 dimensions, which conserves U₆ symmetry; IBM - the standard phenomenological Interacting Boson Model of Arima and Iachello for s,d-bosons in 6 dimensions (denoted in literature as IBM-1).

In the framework of this scheme it is possible to derive the microscopic expressions of the phenomenological collective nuclear model parameters in terms of the nucleon-nucleon interaction potential parameters. The corresponding expressions, accounting for the nuclear kinetic energy as well as the collective potential energy expansion up to the sixth-order terms, have been obtained in Ref.[6]. The aim of present investigation was to estimate numerically the microscopical values of the standard Bohr-Mottelson Model and the standard Interacting Boson Model parameters, using the NN-interaction potential parameters obtained on the basis of SRDM calculations for a-cluster type nuclei with 8 £ A £ 60. The comparison of these estimated parameter values with their values, obtained via the direct level energy fit using phenomenological nuclear models, can give us information about the plausibility of such estimations.

II. General Restricted Dynamics Collective Model (GRDCM) in the case of effective NN-potential of multipole and multipole-Gauss type

Let us consider the general translationally invariant nuclear Hamiltonian for the system of A nucleons, interacting by two-body central forces

where for the sake of simplicity we have neglected the difference between protons and neutrons in the operator T^micr. The subscripts W, M, B and H in Eq.(1) refer to Wigner, Majorana, Bartlett and Heisenberg interactions, correspondingly. The are the projection operators of spatial, spin and isospin states of nucleon pairs and .

Now we shall project the collective part from the Hamiltonian Eq.(1), which, according to RDA, is its O_A-1-invariant part. The projection will be carried out in three steps:

1) Instead of the interparticle position vectors , we shall use the A-1 translationally invariant Jacobi vectors [7]

2) Next, we shall express these vectors in terms of Dzublik-Zickendraht variables [8, 9]

Here D^(1₃) and D^(1_A-1) are Wigner matrices of groups O⁺₃ and O_A-1, correspondingly. These matrices are associated with the defining representations of these groups. r^(k) are three radial variables; Q,F, Y are three Euler angles; q₁,q₂,¼,q_3A-9 is a set of internal angles, and s denotes two discrete parameters, referring to O_A-1 group [1, 2]. Let us note, that from all 3(A-1) variables, introduced in Eq.(3), only six:

(and only these), are scalars with respect to the symmetric group S_A^(r), acting on the space variables (Ref.[10]). Therefore, they are the collective microscopic variables.

3) Finally, the resulting collective part of the Hamiltonian (Eq.(1)), depending only on six collective variables, one can obtain if one integrates Eq.(1) over (3A-9) internal variables q₁,q₂,¼,q_3A-9, i.e.

The Hamiltonian H_coll (Eq.(5)) is the General Restricted Dynamics Collective Model Hamiltonian. The evaluation of the kinetic energy term is still an open problem. In order to obtain its expression one should integrate the T^micr expression in Dzublik-Zickendraht variables over internal variables, which is a formidable task.

The problem, how to obtain from Eq.(5) explicit expression for the collective part of the general microscopic central potential V^micr, is still open as well. At present it is solved only for central Wigner potentials presented as

Expression Eq.(6), despite its simple form, encompasses quite a large class of interactions. Among others the potential Eq.(6) includes also the superposition of multipole-Gauss type interactions:

where a = 0,1,2,¼ and a are the potential parameters. In the case of potential Eq.(7) the coefficients b_t of Eq.(6) have the form

where t = a,a+1,a+2,¼

In Refs.[11, 7] the following result was obtained for the collective potential energy term with potential Eq.(6):

III Derivation of the Microscopic Generalized Bohr-Mottelson Model (GBMM) and the standard Bohr-Mottelson Model (BMM) Hamiltonians from the General Restricted Dynamics Collective Model

Now we shall consider how to obtain the Bohr-Mottelson type collective Hamiltonians (in the generalized 6-dimensional and standard 5-dimensional versions) from the General Restricted Dynamics Collective Model (GRDCM) Hamiltonian described in previous section. In order to associate the GRDCM collective potential energy term Eq.(9) with that of the Bohr-Mottelson model [12], first of all we replace ( r⁽¹⁾)², ( r⁽²⁾)² and ( r⁽³⁾)² in Eq.(9) by three new variables (see Refs.[13, 3, 4, 7]):

Then one obtains the final expression (see Ref.[6]) for the collective potential energy term via r², b², b³cos3g (which are O⁺₃-scalars, as we shall see below):

where coefficients C_t(i,l,m) one can evaluate as

The collective part of the central Wigner interaction (Eq.(6)), written in the form of Eq.(11), is the microscopic generalization of the potential energy term of Bohr-Mottelson type phenomenological models in the sense of RDA, or the potential energy term of the Generalized Bohr-Mottelson Model.

The expression for the kinetic energy term of the GBMM, as was told above, is yet to be obtained. For this reason we postulate that the GBMM kinetic energy term is the usual kinetic energy operator in 6-dimensional space.

Let us introduce six new spherical coordinates q^[l]m(l = 0,2), replacing six collective variables r²,b,g,Q, F, Y by O⁺₃-irreducible bilinear forms of the relative Jacobi vectors, summed on the particle index, namely:

which turn out to be the only functions of collective variables one needs. Replacing in Eq.(3) Carthesian index s by the spherical normalized index m = 0,±1 and using the explicit expressions for D^(1₃) matrices, one obtains from Eq.(13) following results:

where are usual O⁺₃ Wigner matrices and r,b and g are given by Eq.(10).

Using Eqs.(14) and (15) one can relate r²,b²,b³cos3g to O⁺₃-scalars as follows:

where

Therefore we see, that the set of six new collective variables q^[l]m is equivalent to the set of previous collective variables r²,b,g,Q,F,Y.

Now one can define the kinetic energy operator of the GBMM in the canonical way:

where C_k is the kinetic energy constant. Using Eqs.(14) and (15) one can present this operator as

where are the components of the orbital momentum operator in the internal reference frame of the nucleus. J_p¢(b,g) is the momentum of inertia

and T^BM(¼) denotes the usual (5-dimensional) Bohr-Mottelson model kinetic energy operator [13, 3, 12].

Now, the full GBMM Hamiltonian [3] can be presented as

where the potential energy operator is the same one as in the General Restricted Dynamics Model Hamiltonian (see Eq.(11)):

The standard Bohr-Mottelson Model (BMM) Hamiltonian in 5-dimensional model space one can evaluate from the GBMM Hamiltonian if one "freezes" variable r². Then one obtains

where the kinetic energy term T^BM(¼) is given by the corresponding term in Eq.(19), but V^BM(¼) has form

which can be evaluated as a particular case from Eqs.(22),(11).

IV. Derivation of the Microscopic Generalized Interacting Boson Model (GIBM) Hamiltonian

In order to derive from the GBMM Hamiltonian the microscopic GIBM Hamiltonian let us introduce six creation (h^lm) and six annihilation (h^+lm) operators as usual:

By taking second order expressions in h^lm and h^+lm one can realize the algebra of the group Sp(6,R), whose compact unitary subgroup U₆ generators are

The contravariant and covariant indices in Eq.(26) are related by

Taking linear combinations of U₆ generators (Eq.(26)) one constructs all the generators of three U₆ subgroup chains, ending with O₃⁺ (Ref.[14]):

If one inverts Eq.(25), one gets

which allows one to express, using Eq.(16), the invariants r², b², b³cos3g in terms of creation and annihilation operators as

where e.g.

and similarly for other terms of the b³cos3g expression in Eq.(30).

Now, let us rewrite the GBMM Hamiltonian Eq.(21) in terms of creation-annihilation operators Eq.(25)

where

1) the kinetic energy operator is obtained from the canonic expression (Eq.(18)), in which the replacements, given by Eq.(29), are made:

2) for the potential energy operator from Eq.(11) one obtains

where

and r²(¼), b²(¼), b³cos3g(¼) denote the corresponding expressions, given by Eq.(30).

Now, we split the microscopic GBMM Hamiltonian (Eq.32)) in two parts, the first of which contains only U₆ generators and constant terms, i.e.,

In Eq.(36)

H^GIBM( h^lm,h^+l¢m¢ ) = H^GIBM is the Generalized Interacting Boson Model (GIBM) Hamiltonian - the U₆-restricted part of the GBMM Hamiltonian H^GBMM.

Again, one can write H^GIBM as

The kinetic part easily can be written as U₆-restriction of the kinetic operator (Eq.(33)):

Here is the operator of the total number of U₆-quanta. Concerning the potential energy operator Eq.(34), when one takes into account Eq.(30), one can easily see, that from the terms r²(¼), b²(¼), b³cos3g(¼) only those with i+2l+3m = t = even have U₆-restricted terms.

Finally, from Eq.(37), taking into account Eq.(38) and the t = 0,2,4,6 terms of the potential energy expansion Eq.(34), one can write the GIBM Hamiltonian in the following form (for details see Ref.[6]):

where we have used the notation

for the U₆-restriction of the corresponding monomials of O⁺₃-scalars. Coefficients R_t in Eq.(39) are given by Eq.(35). Parameters b_t describe the features of the chosen NN-potential, taken in the form of Eq.(6). In the case of multipole-Gauss type potential Eq.(7) b_t are given by Eq.(8). For the superposition of two multipole-Gauss type potentials (Eq.(A.6)) the coefficients b_tR_t are given by Eq.(A.20).

Besides the kinetic energy constant C_k, there is one more constant in Eq.(39) - the potential energy constant C_p. For the Wigner type central NN-interaction it assumes simple form

One can easily obtain the C_p expression also in more general case, when the central NN-interaction contains Wigner, Majorana, Bartlett and Heisenberg forces of equal potential depth and radial form [4], given by expressions Eq.(6), or Eq.(7):

where

A₀ = A(A-1)/2 and L (f), L (S), L (T) are the eigenvalues of the symmetric group class operators, consisting of orbital, spin and isospin exchange operators [15, 16]. c_W, c_M, c_B, c_H are the exchange constants of the central interaction.

V. Microscopic derivation of the standard Interacting Boson Model (IBM) Hamiltonian and its microscopic extension

The standard IBM Hamiltonian can be written in several equivalent forms. For our purposes the most convenient form is the one written directly via Casimir invariants (Ref.[4])

where e₀,e₁,e₂,e,a,e₁ ,b,g,d,h are model parameters.

In Eq.(43) all two-boson interaction operators are presented using first and second order Casimir operators (expressed via U₆ generators Eq.(26)) of three U₆ subgroup chains (Eq.(28)). Then,

Four remaining Casimir operators in Eq.(43) one can express via combinations of boson operators in the similar way

(see expressions (49)-(54) in Ref.[6]).

The next, most difficult, step in the study of relationships between GIBM and IBM Hamiltonians (Eq.(39) and Eq.(43)) is to obtain, in terms of Casimir invariants of the U₆ subgroup chains Eq.(28) (Eqs.(44)-(48)), the explicit expressions for U₆-restricted monomials F_t(i,l,m) (see Eq.(40)) of O⁺₃-scalars, entering in the potential energy terms of Eq.(39):

The calculation methods, examples and results of F_t(i,l,m) evaluation for t = 2,4,6 values are given in the appendix of our paper [6].

If one inserts the F_t(i,l,m) expressions for t = 0,2,4,6 values into the GIBM Hamiltonian (Eq.(39)) and compares the coefficients at corresponding Casimir invariants in Eq.(43) and H^GIBM (Eq.(39)) one can derive (see Ref.[6]) following microscopic expressions for the parameters of the standard IBM Hamiltonian (Eq.(43)), via b_t(t = 0,2,4,6) values associated with NN-interaction potential:

These results generalize the microscopic IBM Hamiltonian parameter calculations presented in Ref.[4], because we have accounted for the contribution of the kinetic energy term (characterized by the kinetic energy constant C_k), included the t = 6 order terms in the potential energy expansion, as well as corrected some errors in t = 4 order terms.

VI. The Strictly Restricted Dynamics Model (SRDM)

It is a rather difficult task to solve the Schrödinger equation for the full Restricted Dynamics Model Hamiltonian H_RDM with realistic NN-interaction potentials. Because of this, in Refs.[1, 7] the simplified version of H_RDM - the so called Strictly Restricted Dynamics Model (SRDM) Hamiltonian, was proposed. In this case O_A-1 group is enlarged to the unitary group U_A-1 and SO₃ group - to U₃. The detailed description of SRDM and its application for a-cluster nuclei with A £ 40, employing effective NN-interaction potential of multipole-Gauss-exponent type, one can find in our review paper [18].

The general structure of SRDM Hamitonian is

where H^{0 coll} is the U_A-1-collective (scalar) part of the general microscopic nuclear Hamiltonian H (Eq.(1)) and H^{0 a} - the U₃-scalar part of H, which is responsible for the anticollective effects in the nucleus. The basis function of SRDM can be written as Y(NL₀| ₁, ₂,¼, _A-1;Q), where: _i are Jacobi vectors (Eq.(2));

Q - the set of A-nucleon spin-isospin variables;

L₀ - the set of nuclear integrals of motion, which must be conserved according to the symmetry requirements of the many-nucleon system: L₀ = (pJM_JM_TAa);

N - the additional quantum number, denoting both U₃ and U_A-1 IRs.

In order to separate the orbital and spin-isospin parts in the basis function of SRDM we shall use the Wigner supermultiplet scheme approximation [19, 20]. For the calculations in the framework of SRDM it is more convenient to use the unitary scheme basis functions [21], characterized by the following group reduction chain:

In this case quantum numbers of the orbital part of Wigner supermultiplet scheme basis functions have the following meaning:

N₀ - quantum number of the symmetric representation [N₀,0,¼,0] of group U_3(A-1), determining the parity of SRDM state;

N º (N₁N₂N₃) - quantum numbers corresponding to IRs of groups U₃ and U_A-1 in the chain U_3(A-1)É U₃×U_A-1, where N₁+N₂+N₃ = N₀;

L - quantum number corresponding to IR of group SO₃, with K denoting it's multiplicity in the chain U₃É SO₃;

w = (w₁w₂w₃) - quantum numbers denoting the IR of O_A-1 group, and d - the multiplicity index for w in the chain U_A-1É O_A-1. If N₀ = N_min, then w₁ = N₁,w₂ = N₂,w₃ = N₃, and, therefore, d is redundant;

f - denotes the IRs of group S_A in the chain O_A-1É S_A with multiplicity a.

For orbital basis functions of Wigner supermultiplet scheme the reduction rules in the chain U_3(A-1)É U₃×U_A-1 are well-known [15, 7]. In the U₃É SO₃ chain the reduction rules are equivalent to those of SU₃É SO₃ chain [7]. If IRs of U₃ group are labelled by [N₁N₂N₃], then one can introduce for SU₃ group IRs the Elliott's notation

In this case, according to Ref.[22], the multiplicity index K assumes following values:

and

Note, that SU₃ basis functions |(lm)KLM > in the chain SU₃É SO₃É SO₂ are not orthogonal with respect to K, therefore, one must perform the orthogonalization procedure before the diagonalization of SRDM Hamiltonian matrix.

In the chain U_A-1É O_A-1É S_A the reduction rules are more complicated [15]. Taking into account that in the unitary scheme (Eq.(52)) groups U₃ and U_A-1 are complementary [7] (i.e. IRs of U₃ and U_A-1 are determined by the same set of quantum numbers [N₁N₂N₃]), one can use for unitary scheme basis the well-known SU₃ classification scheme for nuclei, introduced by Elliott [22].

The problem of the classification of spin-isospin functions of Wigner supermultiplet scheme is reduced to the classification of (S,T) multiplets, entering in IRs of group S_A^(st). Here, (S,T) denotes the pair of IRs l_s and l_t of groups S_A^(s) and S_A^(t), which for fixed A value are related by expressions [l_s] = [(A/2)+S,(A/2)-S],[l_t] = [(A/2)+T,(A/2)-T]. Therefore, the classification of (S,T) multiplets is reduced to the reduction

where is the multiplicity index of equal pairs (l_s,l_t) in [f\tilde], which can be calculated using formula given in Ref.[23].

According to Eq.(51), the SRDM Hamiltonian consists of two parts, accounting for collective and anticollective effects, correspondingly. The general SRDM Hamiltonian matrix structure in the employed Wigner supermultiplet basis functions is:

Indices, placed in brackets (...) at Hamiltonian terms, denote quantum numbers which are conserved. Therefore, the general SRDM Hamiltonian matrix is numerated by nine quantum numbers: K, L, S, d, w, a, f, , T.

The evaluation of SRDM Hamiltonian matrix Eq.(57) is described in Ref.[18]. The general structure of its matrix elements is

where separate terms denote collective (coll) and anticollective (a) kinetic (kin) and Coulomb (e) energies as well as the central (c), vectorial (v) and tensorial (t) NN-interactions.

VII. The results of a-cluster type nuclei energy calculations in the framework of SRDM

The a-cluster type nuclei with A = 4,8,¼ are characterized by the Young pattern [f] = [4^A/4] and the spin-isospin pair (S,T) = (0,0). The SRDM quantum numbers for a-cluster type nuclei with A = 4,8,¼,80 are given in Table 1.

Table 1:
The characteristics of a-cluster type nuclei with 4 £ A £ 80

The SRDM Hamiltonian for a-cluster type nuclei, following from Eq.(58) if one notices that for the states given in Table 1 the vectorial and tensorial terms of NN-interaction vanish, has the form:

Joining together all expressions for particular terms of SRDM Hamiltonian Eq.(59) (for details see Ref.[18]), the following final expression for the SRDM matrix elements in the case of a-cluster type nuclei can be obtained:

The quantities entering in Eq.(60) have the following meaning:

The oscillator frequency parameter n_A (or the wave function scaling factor r_y) is adopted according to Ref.[16], and it is equal to

where r₀₀, entering in the nuclear radius expression R = r₀₀A^1/3, we shall regard as a model parameter.

The model parameters C₁ and C₂ are defined by expressions:

These parameters are two independent combinations of central force exchange constants (subjected to the normalization condition [24] c_W+c_M+c_B-c_H = -1) together with the common NN-interaction potential depth V_c0, assuming that the effective central NN-interaction potential has the same shape for Wigner (W), Majorana (M), Bartlett (B) and Heisenberg (H) forces.

E_kin denotes the kinetic energy term

The collective submatrix elements of orbital operators for Wigner interaction are given by the expression:

where (KL,K¢L) denotes the collective (U_A-1-invariant) density matrices in compact form and are Talmi integrals of central (), or Coulomb () interaction. Some details about collective density matrices are given in Ref.[18], and their calculation technique is described in Refs.[25, 26, 27, 28, 29, 30]. The evaluation of Talmi integrals for the effective NN-potential employed in our calculations is described in Ref.[31] and ^{Appendix A}.

in Eq.(60) denotes the anticollective (SU₃-invariant) orbital matrix elements of central (c) and Coulomb (e) interactions (see Eq.(B.12)). The details of their evaluation one can find in ^{Appendix B}.

The SRDM Hamiltonian matrices are given in the Elliott's basis |(lm)KL > , and the quantities , entering in Eq.(60), can be expressed [18] via the overlap integrals < (lm)KL|(lm)K¢L > of SU₃ basis functions (see Ref.[32] and Appendix D of Ref.[18]). Applying to Eq.(60) the orthogonalization and diagonalization procedures, we obtain eigenvalues (energies E) and mixing amplitudes of the SRDM Hamiltonian (Eq.(59)).

The choice of NN-potential is limited by the consideration that its expression should not be too complicated, so that one can evaluate Talmi integrals. But, on the other hand, it must reproduce the experimentally known features of NN-interaction (e.g. it must be able to describe a hard core at small distances, and its tail must not extend outside the size of the nucleus). One can see, that the single multipole-Gauss potential Eq.(7) does not satisfy these requirements. The next in complexity potential, which allows one to reproduce partly the characteristics of the realistic NN-interaction potential, is the superposition of two multipole-Gauss type potentials [31], which we have employed for the calculations of a-cluster type nuclei with A £ 60. The analytical expressions of this potential, its parametrization, as well as the details of Talmi integral calculations one can find in ^{Appendix A}.

The results for a-cluster type nuclei with 8 £ A £ 60, obtained with the NN-interaction potential taken in the form of Eq.(A.6), are given in Table 2. In this table

with m - the number of experimentally known levels (including binding energy). The values of following parameters have been obtained by the fit to the available experimental binding and excited level energies:

r₀₀ - the nuclear radii parameter (see Eq.(61)); r₀,r₁,k - the parameters defining the shape of the potential formed by the superposition of two multipole-Gauss type potentials (see ^{Appendix A}); C₁,C₂ - parameters combining potential depth V_c0 and exchange constants (see Eq.(62)).

Table 2:
The SRDM parameter values for a-cluster type nuclei with 8 £ A £ 60

These results seem to be quite satisfactory with regard to the fact, that the ground state binding energies (whose values vary in the interval ~ 30-350 MeV) and the low-lying excited level energies (which are in the energy interval ~ 1-15 MeV) were calculated together at the same time. Contrary to the results obtained for light a-cluster type nuclei with A £ 40 (Ref.[33, 18]), the effective NN-interaction potential parameter values for 40 < A £ 60 region a-cluster nuclei reveal significant dependence from the nuclear mass number A. The obtained values of nuclear radii parameters r₀₀ are in reasonable agreement with the known calculation results for other neighbouring nuclei.

VIII. Numerical estimation of microscopic values for the parameters of phenomenological collective models

The relationship chain between nuclear collective Hamiltonians is based on the General Restricted Dynamics Collective Model (GRDCM) Hamiltonian Eq.(5) evaluated from the General Microscopic Nuclear (GMNH) Hamiltonian Eq.(1). Since GRDCM Hamiltonian remains unsolved as yet in the case of many-nucleon system, we shall employ for our numerical estimation of microscopical values for phenomenological collective model parameters the results of a-cluster type nuclei energy calculations presented in Sect. VII.

We shall consider the microscopic estimations for parameters entering in both phenomenological collective models - the standard BMM and the standard IBM Hamiltonians. The microscopical values of two standard Bohr-Mottelson Model (BMM) Hamiltonian parameters (the mass parameter - B and the nuclear stiffness parameter - C), obtained using the results of SRDM energy calculations for a-cluster type nuclei, can be compared with their phenomenological values, which follow from the BMM rotational band energy formula. The Standard Interacting Boson Model (IBM) Hamiltonian is characterized by 10 parameters e₀,e₁,e₂,¼,h. The microscopical values of these parameters, estimated using the NN-interaction potential parameter values obtained from the SRDM energy calculations for a-cluster type nuclei, can be compared with their phenomenological values, obtained by means of standard IBM energy formulas for limiting cases. The preliminary results of such comparison have been reported in Ref.[34].

First of all, we need to derive the microscopic values of the kinetic energy constant C_k (see Eq.(38)) and the potential energy constant C_p (see Eq.(39)). The rough estimation for the kinetic energy constant microscopic value one can obtain assuming that the kinetic energy terms of GIBM and SRDCM Hamiltonians are approximately equal. This assumption gives us the following relationship between corresponding matrix elements:

Then

and the microscopical value of kinetic energy constant is

where we have used the microscopic value of the total number of U₆-quanta (bosons)

According to the prescription of V.Vanagas [3, 2] such N^micr value can be deduced from the following branching rule for the reduction chain U₆É SU₃É O⁺₃: if N is IR of U₆, then one can enumerate all SU₃ IRs contained in N by counting all the partitions of number 2N into no more than 3 parts, i.e. 2N = 2N^micr = N₁+N₂+N₃ = N₀, where N₁³ N₂³ N₃ and all N₁,N₂,N₃ are even. The representations [N₁N₂N₃] obtained in such a way denote all SU₃ IRs contained in N.

The microscopic value of the potential energy constant also is adopted from the collective central NN-interaction energy term of SRDM Hamiltonian (Eq.(59)). Then, for a-cluster type nuclei we have

where C₁ and C₂ are parameters, whose values are determined from the SRDM energy fit calculations.

For the microscopic estimation of the standard BMM parameters we shall employ now the phenomenological version of the standard BMM Hamiltonian (see also Eq.(23))

which is characterized by the nuclear mass parameter B and the stiffness parameter C. The solution of in diagonal approximation for axially symmetric quadrupole deformation, characterized by deformation parameter b₀, yields the well known rotational band energy formula [35]:

from which one can obtain the phenomenological values of parameters B and C:

The microscopic value B^micr of the mass parameter B one can obtain from the comparison of microscopically evaluated (see Eqs.(23),(19)) and phenomenological (see Eq.(71)) kinetic energy terms of the standard BMM Hamiltonian, which gives

where is determined according to the formula Eq.(68). Analogously, the microscopic value of the nuclear stiffness parameter C^micr can be obtained by comparing the phenomenological potential energy term of Eq.(71) with corresponding microscopic potential energy term in the expansion of H^GBMM (Eqs.(22),(11)):

which gives

where values of , b₂ and R₂ are determined according to formulae Eqs.(70), (8), (35), correspondingly, or Eqs.(70) and (A.20) in the case of superposition of two multipole-Gauss type potentials (Eq.(A.6)).

The comparison of the microscopical values of the standard BMM Hamiltonian parameters B^micr,C^micr with their phenomenological counterparts for a-cluster type nuclei is presented in Table 3.

Table 3:
The microscopical and phenomenological values of the standard BMM Hamiltonian parameters for a-cluster type nuclei

The microscopic values of the standard IBM Hamiltonian parameters e₀,e₁,¼,h can be calculated using formulae Eqs.(50), inserting microscopic values of kinetic and potential energy constants , (see Eqs.(68),(70)). The corresponding phenomenological values of the standard IBM Hamiltonian parameters one can obtain using the diagonal energy formulae of standard IBM in SU₅,SU₃ and O₆ limiting cases (see Ref.[36]):

1) in SU₅ case

2) in SU₃ case

3) in O₆ case

where A(N) = e₀+e₁N+e₂N(N+5).

The results of comparison between microscopical and phenomenological values of the standard IBM Hamiltonian parameters for a-cluster type nuclei one can find in Table 4. In both cases the microscopic value (Eq.(69)) of the total boson number N = N^micr was used.

Table 4: The microscopical and phenomenological values of the standard IBM Hamiltonian parameters for a-cluster type nuclei

IX. Conclusions

For the first time the numerical estimation and comparison of microscopical and phenomenological values of the standard BMM and the standard IBM parameters have been performed, using the Restricted Dynamics Approach based relationship chain between collective models. The Strictly Restricted Dynamics Model (SRDM) parameter values, used as input data for the evaluation of microscopical values for BMM and IBM parameters, have been obtained in SRDM energy calculations for a-cluster type nuclei with A £ 60.

The comparison of microscopically derived values of phenomenological model parameters with their values obtained from the direct fit to the experimental level energies shows considerable discrepancies in the case of standard BMM (2-3 orders of magnitude). This fact can be explained remembering that standard BMM usually is employed for the description of heavy nuclei (A > 100), characterized by the stable deformation of the nuclear core. In the case of standard IBM the microscopically estimated parameter values agree with their phenomenological counterparts much better.

These results indicate that our approach can give only a first very rough estimation for the parameters of phenomenological collective models. One needs more elaborated efforts for the real success in this direction.

Appendix A. The NN-interaction potentials and Talmi integrals

In order to calculate the collective and anticollective matrix elements of NN-interaction one needs to calculate interaction integrals for the chosen NN-interaction potential. In the collective matrix elements Eq.(64) interaction integrals I_el,e¢l¢ are already reduced to Talmi integrals I_ss (since I_sl,s¢l¢ = I_ss,ssº I_ss).

For the evaluation of the anticollective matrix elements we shall use the SU₃-averaged interaction integrals I_[00]e (see Ref.[15]), depending on the interaction integrals I_el (I_elº I_el,el):

where d_e = (e+1)(e+2)/2. Interaction integrals I_el, in turn, for values e = 0,1,2,3,4,5,6, which we need in our calculations, can be reduced to Talmi integrals I_ee, using formulae of Table 15.1 in Ref.[15]. Then we have (see Ref.[24]):

For the convenience of practical calculations, we introduce into chosen NN-interaction potential expression the wave function scaling factor r_y (see Eq.(61)), which leads to the replacement

for the particular exchange components of the central NN-interaction. Talmi integrals of SRDM, taking into account Eq.(82), have the form

= . Integrals Eq.(83) are defined with respect to the unknotted three-dimensional harmonic oscillator radial wave functions

The NN-interaction potential, written as the superposition of two multipole-Gauss type potentials [31], which we have employed for the calculations of a-cluster type nuclei with A £ 60, can be presented as follows:

where

and

In Gauss potential case (when q₁ = q₂ = 0 and b₁ = b₂ = 2) the potential Eq.(85) is characterized by 4 parameters: r_1p,r_2p,V₀₁,V₀₂. Taking into account the replacement Eq.(82), the NN-interaction potential expression Eq.(85) becomes

where

Such superposition of two multipole-Gauss type potentials allows to obtain wide variety of potential shapes, which are more convenient to investigate if one introduces new parameters:

where

New parameters (r₀, r₁, V₁, k) have the following meaning:

r₀ - the distance , where V(r_a) = 0. (In our calculations this parameter was fixed, since the potential shape is defined by the r₀ and r₁ ratio.) r₁ - the distance, where [(¶)/( ¶r_a)]V(r_a) = 0. V₁ - the total potential depth defined as V₁ = V(r₁). k - the ratio of the first and second Gauss potential scales .

When one introduces parameters r₀, r₁, V₁ and k, then the old parameters, entering in the potential expression Eq.(85), can be evaluated as follows:

and

where

For NN-potentials, written as a superposition of two multipole-Gauss type potentials (Eq.(85)), corresponding Talmi integrals are obtained as the weighted sum of two Talmi integrals for each multipole-Gauss potential V¢ and V¢¢ (for details see Ref.[31]):

where q, r_p, z₁, b for the first part of the potential assume values q₁, r_1p, Z₀₁, b₁, and for the second part - q₂, r_2p, Z₀₂, b₂, correspondingly. Then, the summed Talmi integrals can be presented as follows:

where the weight factors:

one can express using Eqs.(91) and (92). Since the common part of the potential depth V₁º V_c0 we include in parameters C₁ and C₂ (see Eq.(62)), then from Eqs.(96), (91) and (92) we finally obtain following expressions for the weight factors between two parts of the Talmi integral Eq.(95):

which should be used also when evaluating the potential part of H^GIBM (Eq.(39)). Then, in the case of potential Eq.(85), we have

where a₁ and a₂ are defined by Eqs.(89) and (90).

Appendix B. The expressions of SRDM anticollective density matrices

The resulting formulae for the anticollective orbital matrix elements of Wigner and Majorana interactions are following [24]:

where e₀ = 0,1,2,¼ denotes SU₃-shells (s,p,sd,¼) and i_ex - ground (i_ex = 0) and excited (i_ex = 1,2,¼) SU₃-configurations. In Eqs.(100) and (101) Q_[00]e(¼) denotes the anticollective (U₃-scalar) two-particle density matrix. Its general expressions are known only in the case of p-shell (e₀ = 1) and sd-shell (e₀ = 2) nuclei. One can find them in Refs.[15, 1, 24].

For other shells (e₀³ 3), which we need in the case of a-cluster nuclei with A > 40, we must use the approximate expressions for the two-particle anticollective density matrix components, which have been evaluated employing the approach described in Refs.[15, 37]. According to this method, the two-particle anticollective density matrix Q_[00]eº Q_e can be approximately expressed using "four universal relationships" (see Eqs.(25.14), (25.15), (25.17) and (25.32) in Ref.[15]):

In Eq.(103) L (f) is the eigenvalue of the orbital coordinate exchange operator, its expression one can find, e.g. in Ref.[18]. Quantity G in Eq.(105) is the eigenvalue of the second (quadratic) Casimir operator of the SU₃ group:

Expressions for functions S₁^(Z,N), S₂^(Z,N), one can find in Refs.[15, 37].

From the sums of both sides of these four relationships with coefficients b₁^W, b₂^W, b₃^W, b^W₄ and b₁^M, b₂^M, b₃^M, b₄^M, correspondingly, one obtains:

1) the approximate equation in the case of Wigner forces:

2) and, analogously, in the case of Majorana forces:

Coefficients b_i^W,b_i^M (i = 1-4) one can find solving the overdetermined linear equation systems [37]:

and

where for A £ 80 e = 0,1,¼,6. The expressions for SU₃-averaged integrals I_[00]e one can find in ^{Appendix A}. Then, taking into account relationships Eqs.(102)-(106) and using evaluated coefficients b_i^W,b_i^M, we can express the total anticollective energies entering in Eq.(60) as follows:

using Coulomb and central NN-interaction potentials, correspondingly.

For the first time the approximate expressions for the SU₃-invariant two-particle density matrix have been obtained in Ref.[37]. Though, empolying these expressions in our SRDM calculations of 44 £ A £ 80 nuclei, we have found that these expressions contain errors. Therefore, we have solved systems Eqs.(109) and (110) again and obtained the corrected formulae for the coefficients b_i^W,b_i^M, which we have used in our calculations (in the following expressions e = e_max = 2e₀+i_ex for 40 < A £ 80, since e₀ = 3 and i_ex = 0, therefore, e_max = 6):

Coefficients b_i^M one can obtain replacing in all expressions I_i with (-1)ⁱI_i.

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Appendix A

Appendix B

Publication Dates

• Publication in this collection
  14 June 1999
• Date of issue
  Dec 1997

History

• Received
  08 May 1997