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Cluster Interpretation of Properties of Alternating Parity Bands in Heavy Nuclei

The properties of the states of the alternating parity bands in actinides, Ba, Ce and Nd isotopes are analyzed within a cluster model. The model is based on the assumption that cluster type shapes are produced by the collective motion of the nuclear system in the mass asymmetry coordinate. The calculated spin dependences of the parity splitting and of the electric multipole transition moments are in agreement with the experimental data.


I. INTRODUCTION
The low-lying negative parity states observed in actinides and in heaviest known Ba, Ce, Nd and Sm isotopes are definitely related to reflection-asymmetric shapes [1,2]. There are several approaches to treat collective motion leading to reflection-asymmetric deformations. One of them is based on the concept of a nuclear mean field which has a static octupole deformation or is characterized by large amplitudes of reflection-asymmetric vibrations around the equilibrium shape [2,3,4,5,6]. In this approach the parity splitting is explained by octupole deformation. Another approach [7,8,9,10] is based on the assumption that the reflection-asymmetric shape is a consequence of alpha-clustering in nuclei [11,12,13]. In the algebraic model [7,8,9,10] the corresponding wave functions of the ground and excited states consist of components without and with dipole bosons (in addition to the quadrupole bosons), which are related to mononucleus and alpha-cluster components, respectively. The variant of algebraic model including the octupole bosons in addition to the dipole bosons has been applied in [14,15] to the description of the low-lying negative parity states in actinides. In [16,17,18,19] a cluster configuration with a lighter cluster heavier than 4 He was used in order to describe the properties of the low-lying positive and negative parity states. In both models [7,8,9,10] and [16,17,18,19] the relative distance between the centers of mass of clusters at fixed mass asymmetry is the main collective coordinate for the description of the alternating parity bands.
Nuclear cluster effects are mostly pronounced in the light even-even N = Z nuclei with alpha-particle as the natural building block. There is a nice relationship between alphacluster description and deformed shell model [11]. It is known from Nilsson-Strutinsky type calculations for light nuclei that nuclear configurations corresponding to the minima of the potential energy contain particular symmetries which are related to certain cluster structures [20,21,22]. By using antisymmetrized molecular dynamics approach [23,24], the formation and dissolution of clusters in light nuclei, like 20 Ne and 24 Mg, are described. The idea of clusterization applied to heavy nuclei does not contradict the mean field approach. 2 The coexistence of the clustering and of the mean field aspects is a unique feature of nuclear many body system. The problem of existence of a cluster structure in a ground state of heavy nuclei has attracted much attention, especially, because of the experimentally observed cluster decay [25]. The available experimental and theoretical results provide the evidence for existence of fission modes created by the clustering of the fissioning nuclei [26]. Indications of clusterization of highly deformed nuclei are demonstrated in [27,28].
The aim of the present paper is a development of the cluster-type model which provides not only a qualitative but also a quantitative explanation of the properties of alternating parity bands. The description of the excitation spectra, Eλ-transition probabilities (λ=1,2,3) and the angular momentum dependence of the parity splitting [29,30] are the main subjects of this paper. Our model is based on the assumption that the reflection-asymmetric shapes are produced by the collective motion of the nuclear system in the mass asymmetry coordinate [31]. The values of the odd multipolarity transitional moments (dipole and octupole) are strongly correlated with the mass asymmetry deformation of nucleus. In general, the value of the quadrupole moment is related to the degree of the quadrupole correlations (deformation) in nucleus. However, the collective motion in the mass asymmetry degree of freedom simultaneously creates a deformation with even and odd multipolarities. Therefore, the calculations of Eλ-transition moments are of interest in the proposed model. The single particle degrees of freedom are not taken explicitly into consideration since our aim is to show that the suggested cluster model gives a good quantitative explanation of the observed properties of the low-lying negative parity states. If it is so, this model can serve as a good ground for development of an extended model with additional degrees of freedom.
It should be noted that the first results of calculations of the alternating parity spectra for a few actinides within the cluster model are already presented in the Letter [31]. Besides Ra, Th and U isotopes, in the present paper we present the results of calculations of the energies of alternating parity bands in 240,242 Pu, 144,146,148 Ba, 146,148 Ce, and 146,148 Nd. The electromagnetic transitions are described in this paper with the cluster model for many nuclei and the spin dependence of the intrinsic quadrupole transition moment is predicted for 238 U.

3
Simple analytical expressions obtained for the parity splitting and the spectra of alternating parity bands are useful for the estimations. The dependence of alpha-clusterization in actinides on the angular momentum is shown for the first time.

II. MODEL
A. Hamiltonian in mass asymmetry coordinate Dinuclear systems consisting of a heavy cluster A 1 and a light cluster A 2 were first introduced to explain data on deep inelastic and fusion reactions with heavy ions [32,33,34].
The mass asymmetry coordinate η, defined as η = ( which describes a partition of nucleons between the nuclei forming the dinuclear system and the distance R between the centers of clusters are used as relevant collective variables [35]. The wave function in η can be thought as a superposition of different cluster-type configurations including the mononucleus configuration with |η|=1, which are realized with certain probabilities. The relative contribution of each cluster component in the total wave function is determined by the collective Hamiltonian described below. Our calculations have shown that in the considered cases the dinuclear configuration with an alpha cluster (η = η α ) has a potential energy which is close or even smaller than the energy of the mononucleus at |η| = 1 [28,31]. Since the energies of configurations with a light cluster heavier than an αparticle increase rapidly with decreasing |η|, we restrict our investigations to configurations with light clusters not heavier than Li (η = η Li ), i.e. to cluster configurations near |η| = 1.
The Hamiltonian describing the dynamics in η has the following form where B η is the effective mass and U(η, I) is the potential. In order to calculate the dependence of parity splitting on the angular momentum and the electric dipole, quadrupole and octupole transition moments we search for solutions of the stationary Schrödinger equation describing the dynamics in η: 4 HΨ n (η, I) = E n (I)Ψ n (η, I). (2) The eigenfunctions Ψ n of this Hamiltonian have a well defined parity with respect to the reflection η → −η. Before we come to the results of Eq. (2), we discuss the calculation of the potential U(η, I), the mass parameter B η and the moments of inertia ℑ(η) appearing in H.

B. Potential energy
The potential U(η, I) in Eq. (1) is taken as a dinuclear potential energy for |η| < 1 Here, the internuclear distance R = R m is the touching distance between the clusters and is set to be equal to the value corresponding to the minimum of the potential in R for a given η. The quantities B 1 and B 2 (which are negative) are the experimental binding energies of the clusters forming the dinuclear system at a given mass asymmetry η, and B is the binding energy of the mononucleus. The quantity V (R, η, I) in (3) is the nucleus-nucleus interaction potential. It is given as with the Coulomb potential V coul , the centrifugal potential V rot =h 2 I(I + 1)/(2ℑ(η, R)) and the nuclear interaction V N . In the realization of the cluster model developed in this paper, where overlap of clusters is much smaller than in the model of [16], the choice of the relevant cluster configuration follows the minimum of the total potential energy of the system with a cluster-cluster interaction taken additionally into consideration. As the result we describe the same nuclear properties as in [16] with configurations of clusters having larger mass asymmetry and a smaller overlap.
The potential V (R, η, I) and the moment of inertia ℑ(η, R)) are calculated for special cluster configurations only, namely for the mononucleus (|η|=1) and for the two cluster configurations with the α-and Li -clusters as light clusters, respectively. These calculated points are used later to interpolate the potential smoothly by a polynomial. The energies of the Li-cluster configurations are about 15 MeV larger than the binding energies of the mononuclei considered. Therefore, for small excitations only oscillations in η are of interest which lie in the vicinity of |η|=1, i.e. only cluster configurations up to Li -clusters need to be considered. The potential V N is obtained with a double folding procedure with the ground state nuclear densities of the clusters. Antisymmetrization between the nucleons belonging to different clusters is regarded by a density dependence of the nucleon-nucleon force which gives a repulsive core in the cluster-cluster interaction potential. Details of the calculation of V N are given in [36]. The parameters of the nucleon-nucleon interaction are fixed in nuclear structure calculations [37]. Other details are presented in [31].
Our calculations show that the potential energy has a minimum at |η|=η α in 218,220,222,224,226 Ra and 222,224,226 Th isotopes. In order to demonstrate the dependence of the potential on the neutron number, we present in Fig. 1 calculated values of U(η α , I = 0) ≡ U(η α ) of configurations with an α -cluster taking the long chain of Ba isotopes as an example. In the neutron deficient isotopes U(η α ) is smaller than zero and an α-clusterization is more likely. When the neutron number approaches the magic value of 82, the nucleus becomes stiffer with respect to vibrations in η and U(η α ) is larger than zero. The appearance of two neutrons above shell closure is in favor for an α -clusterization. In this case U(η α ) drops much and again becomes smaller than zero. Further addition of neutrons increases the nuclear stiffness with respect to η vibrations.

C. Moments of inertia
The calculation of the moment of inertia ℑ(η) = ℑ(η, R m ) needed to determine the potential energy at I = 0 has been described in [31]. For completeness, we repeat in this subsection the most important information. As was shown in [28], the highly deformed states are well described as cluster systems and their moments of inertia are about 85% of 6 the rigid-body limit. Following this, we assume that the moment of inertia of the cluster configurations with α and Li as light clusters can be expressed as Here, ℑ r i , (i = 1, 2) are the rigid body moments of inertia for the clusters of the dinuclear system, c 1 =0.85 [28,31] for all considered nuclei and m 0 is the nucleon mass.
It should be noted that the angular momentum is treated in this paper as the sum of the angular momentum of the collective rotation of the heavy cluster and of the orbital momentum of the relative motion of the two clusters. Single particle effects, like alignment of the single particle angular momentum in the heavy cluster, are presently disregarded.
For |η| = 1, the value of the moment of inertia is not known from the data because the experimental moment of inertia is a mean value between the moment of inertia of the mononucleus (|η|=1) and the ones of the cluster configurations arising due to the oscillations in η. We assume that where ℑ r is the rigid body moment of inertia of the mononucleus with A nucleons calculated with deformation parameters from [38] and c 2 is a scaling parameter which is fixed by the energy of the first 2 + or other positive parity state, for example 6 + . The chosen values of c 2 vary in the interval 0.1 < c 2 < 0.3. So, in our calculations there is a free parameter c 2 .
However, this parameter is used to describe the rotational energies averaged over the parity and not the parity splitting studied in this paper.

D. Mass parameter
The method of the calculation of the inertia coefficient B η used in this paper is given in [39]. Our calculations show that B η is a smooth function of the mass number A. As a consequence, we take nearly the same value of B η =20 ×10 4 m 0 fm 2 for almost all considered actinide nuclei with a variation of 10%. However, for 222 Th and 220,222 Ra we varied B η in the range B η =(10-20)×10 4 m 0 fm 2 to obtain the correct value of E 0 (I = 0). These variations of B η lead to better results for light Ra isotopes than those in [31], where the obtained values of the parity splitting at the beginning of the alternating parity band are smaller than the experimental ones. Using a smooth mass dependence of B η [39] we get B η =4.5×10 4 m 0 fm 2 in the Ba, Ce and Nd region. However, better results we obtain for B η =3×10 4 m 0 fm 2 .
For very asymmetric dinuclear systems, we can use simple analytical expressions to establish a connection between the relative distance and mass asymmetry coordinates on one side and the multipole expansion coefficients β 2 and β 3 on the other side [28] Here, R 0 is the spherical equivalent radius of the corresponding compound nucleus. One In the actinide region for an α -particle configuration, η ≈0.96 and (dβ 3 /dη) 2 ≈11.25. Then the mass parameters for β 3 and η-variables are related as If we take the value of B β 3 = 200h 2 MeV −1 known from the literature [4], then B η ≈9.3 ×10 4 m 0 fm 2 . This value is compatible with the one used in our calculations.

III. INTRINSIC ELECTRIC MULTIPOLE MOMENTS
Solving the eigenvalue equation (2), we obtain the wave functions of the positive and negative parity states for different values of the quantum number I of angular momentum.
These wave functions are used then to calculate transition matrix elements of the electric multipole operators by integration over η. The electric multipole operators for a system of a dinuclear shape have been calculated [28] by using the following expression For slightly overlapping clusters when the intercluster distance R m is about or larger than the sum of the radii of clusters (R 1 + R 2 ), the nuclear charge density ρ Z can be taken as a sum of the cluster charge densities Using (11) and assuming axial symmetry of the nuclear shape, we obtain [28] the following expressions for the intrinsic electric multipole moments (2)), (14) where Of course, other effects related to degrees of freedom, which are not included in the model, like the alignment of the single particle momenta or interaction with other negative parity bands with different K quantum number can contribute as well. However, a general agreement between the experimental data and the results of calculations shows that the simple cluster model used in this paper gives a firm ground for the consideration of the alternating parity bands.
In the considered nuclei the ground state energy level lies near the top of the barrier in η, if exists, and the weight of the α−cluster configuration (Fig. 2) estimated as that contribution to the norm of the wave function which is located at |η| ≤ η α is about 5 × 10 −2 for 226 Ra, which is close to the calculated spectroscopic factor [25]. This means that our model is in qualitative agreement with the known α−decay widths of the nuclei considered.
The spectra of those considered nuclei whose potential energy has a minimum at the alpha cluster configuration can be well approximated by the following analytical expression Here, the parity splitting δE(I) is given as with . were obtained by fitting the experimental spectra for the nuclei considered (see Fig. 3).
These formulae clearly demonstrate that there are two important quantities which prede-  However, we found numerically that Eq. (17) works quite well also at low I. 16 To combine the two limits at I=0 and for I ≫ 1, we use the following expression Substituting this result into (A3), we obtain The last expression can be rewritten as