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α-cluster structure in 62,64Ge described by the local potential approach

The unstable nuclei 62Ge and 64Ge are analyzed in terms of the α + core structure applying a nuclear potential with (1 + Gaussian)×(W.S. + W.S.3) shape. The ground state bands of 62Ge and 64Ge and first negative parity band of 64Ge are calculated and compared with experimental data. The calculated 64Ge ground state band gives a good account of the experimental energies from 0+ to 8+ state. The rms intercluster separations and B(E2) transition rates are obtained for the 64Ge ground state band and a discussion is presented.


Introduction
The α-cluster model has been applied successfully to nuclei near the doubly closed shells assuming an α + core configuration [1,2]. Previous works [3][4][5] indicate that the α-cluster model with local potential approach is also adequate to describe the spectroscopic properties of nuclei farther from double shell closures. However, there are many nuclei not studied in this viewpoint yet, particularly in the region between 40 Ca and 90 Zr. The recent work of Souza and Miyake [4] discusses the α + core structure in 46 Cr and 54 Cr using the (1 + Gaussian)×(W.S. + W.S. 3 ) nuclear potential shape and a good account of the respective ground state bands and B(E2) transition rates is obtained. The present work aims to apply the same potential shape in Ge isotopes to test its efficiency in different isotopic chains of this mass region.

Preferential Ge isotopes for α-clustering
The preferential nuclei for α-clustering were selected applying the same criterium used in previous work [3] focusing the intermediate mass region. The variation of binding energy per nucleon was revealed to be an appropriate quantity to pick the preferential α + core configuration among different nuclei. This value is given by Q α /A T , where Q α is the Q-value for α-separation and A T is the mass number of the total nucleus. The chain of even-even Ge isotopes is shown  Figure 1. Q α /A T values obtained for the α + core decomposition of even-even Ge isotopes as a function of the mass number A T . The Q α /A T maximum corresponding to 62 Ge is indicated.
in figure 1 pointing that 62 Ge is the preferential nucleus for α-clustering in this set followed by 64 Ge, with the second highest Q α /A T value.

The Model
The α-cluster model regards the total nucleus as an α-particle orbiting an inert core. The 62 Ge and 64 Ge nuclei are assumed as α+ 58 Zn and α+ 60 Zn systems, respectively. The α + core potential is the sum of nuclear and Coulomb terms. The Coulomb potential is described by an α-particle interacting with an uniformly charged spherical core of radius R. The nuclear potential is expressed by: where V 0 , λ, a, and b are fixed parameters, and R and σ are variable parameters. The description of the ground state bands of the two nuclei is obtained with the fixed values V 0 = 220 MeV, a = 0.65 fm, b = 0.3, and λ = 0.14, while R and σ are fitted for each nucleus. The employed R and σ values are: R = 4.621 fm and σ = 0.366 fm for 62 Ge, and R = 4.647 fm and σ = 0.278 fm for 64 Ge. The values of V 0 , λ, a, and b were adjusted previously to describe the ground state bands of the nuclei 20 Ne, 44 Ti, 94 Mo, and 212 Po, in which the α-cluster structure is recognized in preceding studies [1][2][3]. The depth V 0 is increased to 241 MeV only for the calculation of the 64 Ge negative parity band. The nucleons of the α-cluster, according to Pauli principle, occupy shell-model orbitals out of the core. This restriction is defined by the global 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. In the case of the α+ 58 Zn and α+ 60 Zn systems we have G ≥ 12, where G = 12 corresponds to the ground state band. This value is obtained from the Wildermuth condition [6] and yields an appropriate description of the 62 Ge and 64 Ge ground state bands. The two α + core systems are solved numerically to obtain the properties of the bound and quasi-bound states. MeV with the 2 + and 4 + states, respectively, is suggested.

Results and conclusions
The α-cluster model was applied to 62 Ge and 64 Ge; the energy levels of the ground state band (G = 12), in comparison with experimental data [7,8], are presented in figures 2 and 3. The results show a good description of the 64 Ge ground state band from 0 + to 8 + state. One should take into account that the fixed parameters V 0 , λ, a, and b are the same applied for nuclear spectra description of different mass regions [1,3,4]. The results for the negative parity band of 64 Ge are not conclusive, since the respective experimental levels have undefined assignments.
Unfortunately there are few experimental data on 62 Ge for a consistent comparison; therefore we suggest a correspondence of the experimental energy levels 0.964 MeV and 2.285 MeV with the 2 + and 4 + states, respectively. The calculated B(E2) values for the 64 Ge ground state band are presented in table 1. The formulae used to determine the B(E2) values are detailed in Ref. [3]. Ref. [9] presents the experimental value B(E2) = 27(4) W.u. for the 2 + 1 → 0 + 1 transition, which implies an effective charge δe ≈ 0.5 e so that the model reproduces the experimental value. However, more B(E2) experimental data are needed for a complete evaluation of the theoretical values. Excitation energy (MeV) Figure 3. Calculated energy levels for the ground state band (G = 12) and negative parity band (G = 13) of the α+ 60 Zn system in comparison with experimental energies of 64 Ge [8].
The calculated intercluster rms radii for 64 Ge (see table 1) indicate that the α-cluster character is stronger for the first members of the ground state band. Such behavior (antistretching effect) has already been observed in the 46 Cr and 54 Cr ground state bands with the same α + core potential shape [4].
The present work, in addition to our previous article on Cr isotopes [4], indicates that the (1 + Gaussian)×(W.S. + W.S. 3 ) potential shape is suitable for a global description of the α-cluster structure in the nuclei of transition region. New experimental data, mainly from α-transfer reactions, will be welcome for comparison with our results.