THE ROLE OF THE CORE POLARIZATION ON C2 AND C4 FORM FACTORS OF fp-SHELL NUCLEI

FOUAD A. MAJEED1,a, FATIMA M. HUSSAIN2


1,2Department of Physics, College of Education for Pure Sciences, University of Babylon, P.O. Box 4, Hilla-Babylon, Iraq
E-mail: fouadalajeeli@yahoo.com
Received May 2, 2013

Inelastic electron scattering form factors for fp-shell nuclei 42Ca, 44Ca, 46Ti, 48Ti, 50Cr and 54Fe are investigated taking into account higher energy configurations outside the fp-shell. Higher energy configurations, which are called core polarization (CP) effects, are included through a microscopic theory that includes excitations from the core 1s-1p, 2s-1d orbits and also from 2p-1f shell to the higher allowed orbits with
6~! excitations. The calculations are performed in the FP model space by employing GXPF1 and FPD6 effective interactions while the core polarization are calculated with Skyrme-Hartree Fock (Skx) and harmonic oscillator (HO) as residual interactions. The predicated total form factors are compared with the available experimental data and
it is shown that the inclusion of the higher excited configuration are very essential in both the transition strength and momentum transfer dependence to obtain reasonable description of the data with no adjustable parameters.
Key words: fp-shell nuclei (e,´e), form factors, shell model, calculated first-order
core polarization effects.
PACS: 25.30.Dh; 21.60.Cs; 27.40.+z.


1. INTRODUCTION

Nuclear structure models can be successfully tested by comparing the calculated and measured electron scattering form factors [1]. The success of such a model reveals a valuable information about the charge and current distributions of nuclei. We recently used this microscopic calculation [2] in order to study the CP effects on
the C2 longitudinal form factors of 10B. Inelastic electron scattering from factors for the excitation of the 2+ states in the 1f7=2 nuclei are successfully described in terms of the shell model within the fn
7=2+fn??1 7=2 p3=2 configurations and with the effective interactions [3]. Radhi et al. [4–8] have successfully proved that the inclusion of CP effects in the p-shell and sd-shell are very essential to improve the calculations
of the form factors. The form factors for the inelastic electron scattering to 2+, 4+ and 6+ states in 46;48;50Ti, 50;52;54Cr and 54;56Fe are studied in the framework of the projected Hartree-Fock model which gave a reasonably good agreement with the experimental form factors using constant effective charges with no adjustable parameters
for the calculations of the form factors [9]. Lightbody, Jr. et al. [10] measured the elastic and inelastic scattering cross sections for 50;52;54Cr at momentum transfers between 0.5 and 2.6 fm??1 along with the comparison between the experimental charge distributions and density dependent Hartree-Fock-Bogolyubov calculations.
The aim of the present work is to study the effect of core polarization on the calculations of C2 and C4 form factors for some selected states for 42Ca, 44Ca, 46Ti, 48Ti, 50Cr and 54Fe. The core is taken at 40Ca for 42Ca, 44Ca, 46Ti, 48Ti and 50Cr, while 48Ca are taken for 54Fe in the framework of the the fp-shell model. The
one body density matrix (OBDM) element used in the present work are calculated by adopting the effective interactions GXPF1 [11] for 42Ca, 44Ca, 46Ti, FPD6 [12] effective interaction is for 48Ti,48Ca, 50Cr and HO [13] for 54Fe, by generating the wave functions of a given transition in the known nuclei using the modified version of
shell model code Oxbash [14]. Higher-energy configurations are included as a firstorder core polarization through a microscopic theory which combines shell model wave functions and highly excited states. Transitions from the core 1s-1p, 2s-1d orbits and also from 2p-1f orbits to all the higher allowed orbits with excitations up
to 6~! are taking into account. The form factors are calculated without introducing any state dependent parameters such as effective charges, which were introduced in the previous work of authors in this mass region. The CP calculation are performed with Skyrme-Hartree-Fock [15] and harmonic oscillator (HO) potentials as residual
interactions. The single-particle wave functions are those of the harmonic-oscillator (HO) potential with size parameter b chosen to reproduce the measured root mean square (rms) charge radii of these nuclei.

3. RESULTS AND DISCUSSION
The CP effects are calculated with Skx [15] and HO as effective residual interactions. The parameters of the Skx and HO are obtained from the fit to the binding energies, rms charge radii, and single-particle energies. In all of the following diagrams (see figure 1), the dotted curve gives the results obtained using the fp-shell model calculations without CP effects. The results with the inclusion of the CP effects are shown by the dashed curve with Skx as residual interaction and the solid curve are those calculated with HO as residual interaction.

4. CONCLUSIONS
A conclusion were drawn that the inclusion of the CP effects are found to be
very essential in the calculations of the C2 and C4 form factors and gives remarkably
good agreement over the fp-shell model calculations for the form factors and
the absolute strengths. The fp-shell models, are able to predicts the static properties
and energy levels of nuclei lies in the fp-shell region but it had shortfall in describing
the dynamic properties such as C2 transition rates and electron scattering form factors.
The choice of Skx as residual effective interaction are more adequate than HO
for core polarization calculations. The inclusion of higher-excited configurations by
means of CP enhances the form factors and brings the theoretical results closer to the
experimental data. These calculations can be extended to cover the entire fp-shell
region depending on the availability of the experimental data, and also can be used
even for higher shells.

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RJP 59(Nos. 1-2), 95–105 (2014)



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