Abstract
The 94Mo nucleus is treated as an alpha-cluster interacting with an inert core through a phenomenological local potential. The properties of the ground state band of the alpha+90Zr system, such as the energy levels, intercluster rms radii and B(E2) transition strengths were calculated. These results are compared with previous references and available experimental data. Some predictions were made concerning the negative parity band and the excited positive parity band.
NUCLEAR STRUCTURE
The alpha-cluster bands in 94Mo
Marco Antonio de Souza; H. Miyake
Instituto de Física, Universidade de São Paulo, CP 66318, 05315-970, São Paulo, SP, Brazil
ABSTRACT
The 94Mo nucleus is treated as an a-cluster interacting with an inert core through a phenomenological local potential. The properties of the ground state band of the a+90Zr system, such as the energy levels, intercluster rms radii and B(E2) transition strengths were calculated. These results are compared with previous references and available experimental data. Some predictions were made concerning the negative parity band and the excited positive parity band.
The a-cluster structure is an important feature for light nuclei. The a-cluster model has been successful in reproducing the behaviour of the energy spacings, electromagnetic properties, a-widths and a-nucleus elastic scattering data around the double shell closures such as 16O and 40Ca [1-3]. The search for a-cluster states in heavier nuclei is the natural sequence of investigations. Recent studies [4-6] indicate the persistence of a clustering in the A = 90 mass region.
According to the a-cluster model, the total nucleus is viewed as an a-particle orbiting an inert core. In the case where both constituents have spin zero, no complications arise from the spin-orbit coupling. The 90Zr medium-heavy nucleus presents a closed shell for neutrons and filled subshells for protons up to 2p1/2 and may be considered approximately as an inert core. In this way the proposed model is employed to the 94Mo nucleus, which is regarded as an a+90Zr system. The interaction was described through a local phenomenological potential V(r) = VC(r) + VN(r) containing the nuclear and Coulomb terms. The Coulomb potential VC(r) is taken to be that of a point a-particle interacting with an uniformly charged spherical core of radius R = 5.793 fm. The intercluster nuclear potential
proposed by Buck, Merchant and Perez [4] was used, where V0 = 220 MeV, a = 0.65 fm, b = 0.3 and R is the radius used in the Coulomb potential. The parameters V0, a and b have been fitted to reproduce satisfactorily the experimental excitation energies of the ground state bands of 20Ne, 44Ti, 94Mo and 212Po, as well as the experimental a-decay half-lives for several even-even heavy nuclei, while R is fitted for 94Mo. The form of VN(r) has been adopted because of its similarity to the real parts of the optical potentials determined from a-16O and a-40Ca elastic scattering [4].
The a-90Zr system can be solved numerically to obtain the properties of the bound, quasibound and resonant states. The energy eigenvalues yield the levels of the spectrum and the associated wave functions are used to calculate other nuclear properties. The nucleons of the a-cluster must lie in shell-model orbitals above those already occupied by the core nucleons, according to the Pauli principle. This restriction is defined by the quantum number
G = 2N + L,
where N is the number of internal nodes in the radial wave function and L is the orbital angular momentum. G is a global quantum number which identifies a band of levels. In the case of the a-90 Zr system we have G > 16, where G = 16 corresponds to the ground state band. This value is obtained from the Wildermuth condition [7] and yields an appropriate description of the a-90Zr ground state band [4, 5].
The energy levels of the ground state band (Fig. 1) and the respective radial wave functions were obtained solving directly the Schrödinger equation for the system. These functions were used to determine the associated B(E2) transition rates and the intercluster rms radii for this band (Table I). The calculated ground state band shows a quasirotational spectrum and it is in total agreement with the theoretical values of Ref. [4], where the spectrum is obtained by using the Bohr-Sommerfeld quantization rule. The calculated energies give a satisfactory description of the experimental spectrum, if one takes into account that the parameters of this local potential are not L dependent or energy dependent.
The rms intercluster distance is seen to decrease from 5.13 fm for the 0+ state to 4.86 fm for the 10+ state. This antistretching effect also appears in calculations for the a-16O [1] and a-40Ca [2, 3] cluster systems. A comparison of the calculated rms radii with the sum of the experimental radii [10] for a (1.674 fm) and 90Zr (4.244 fm) suggests that the ground state band has a compact a-cluster structure, in agreement with calculations of Ohkubo [5]. The calculated B(E2) transition strengths have the same magnitude of the experimental values. These results may be considered satisfactory, once the B(E2) values are calculated without any effective charge.
The degree of shell-model-like character in the ground state band was estimated through the expansion of the corresponding radial wave functions in the harmonic oscillator basis with the parameter n = µw/2
, where µ is the reduced mass of the system and w is given by w = 41A-1/3 MeV. This expansion shows a large overlap between these radial wave functions and the respective harmonic oscillator wave functions with similar quantum numbers G and L (the overlap increases from 89.1% for the 0+ state to 95.0% for the 10+ state). Such result indicates that the ground band states present a shell-model trend.
The intercluster potential V(r) has also been used to make some predictions about higher lying bands which are above the a+90Zr threshold. The functions dL(E) for L = 0, 1, 3 and 5 were calculated, where dL is the scattering phase shift for a given L (see Figs. 2 and 3). The total a-width Ga can be extracted from the functions dL(E) by [3, 8]
where ER is the resonance energy for which the phase shift crosses p/2 up to a constant multiple of p. The vicinity of each resonance was located previously by using the bound state approximation and the resonance energies were determined through the functions dL(E). In the numerical calculation of these functions, an energy step DE = 0.4 MeV was applied for energies far from the resonances; however, for energies in the vicinity of each resonance, DE must be reduced in orders of magnitude to provide an accurate value for ddL/dE in ER.
The calculated resonance energies and total a-widths are given in Table II. The values refer to the first members of the negative parity band (1-, 3- and 5-) and the first member of the excited positive parity band (0+). The very small widths of the states of the negative parity band indicate that the properties of such states may be investigated within the bound state approximation. Therefore, it is suggested that the adequate treatment for the G = 17 band is analogous to the treatment given for the negative parity band of the a-40Ca system [3]. The level 0+ of the G = 18 band lies at about 1.5 MeV below the Coulomb barrier, resulting in a larger width (87 keV). This indicates that the G = 18 band presents a typical resonance feature. Higher spin members of the G = 17 and G = 18 bands are expected above the calculated levels, as in lighter a-core systems.
The present calculations confirm that the a-cluster model is still an important tool for the spectroscopic analysis of 94Mo. The procedures described in this work may provide other predictions concerning the higher spin states of the G = 16, 17 and 18 bands. More experimental data are necessary to verify our calculated energy levels for the negative parity band and the excited positive parity band.
Acknowledgments
The authors thank the members of the Nuclear Spectroscopy with Light Ions Group of IFUSP for the useful discussions. This work was partially supported by CAPES.
Received on 6 May, 2005
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Publication Dates
-
Publication in this collection
07 Nov 2005 -
Date of issue
Sept 2005
History
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Received
06 May 2005
