Inelastic Form Factor Calculations for 46,48,50 Ti Isotopes Using Tassie Model

Inelastic form factors of electrical transition have been calculated for 46,48,50 Ti isotopes using the Tassie model. The form factors have been calculated for diﬀerent exciting energies. The harmonic oscillator (HO) wave function has been used as a single-particle wave function. The model space has been considered as 1f7/2, 2p3/2, 2p1/2, and 2f5/2. Gx1 has been used as eﬀective interaction in all calculations. In all calculations, the eﬀective charge has been considered as 1.5e for proton and 0.5e for neutron. All obtained results have been compared with data from an experiment. The calculations show the Tassie model gives a good description of longitudinal form factors of 46,48,50 Ti isotopes in E(2 + ) transitions as compared with experimental data, especially in the region below 2 fm − 1 of momentum transfer, but in the E(4 + ), the theoretical results deviated slightly from experimental data especially in the region greater than 1.5fm − 1 of momentum transfer.


Introduction
e gamma transitions and excitation of electrons in the nuclei have been considered in the Tassie model (TM) [1]. e form factors of the inelastic longitudinal scattering are calculated. is model is reduced to the normal liquid drop model for uniform load distribution. TM is a more elastic model that attempts a nonuniform loading and mass density distribution. e transition density depends on the density of the nucleus in the ground state, in these models [1]. e density of ground state for all occupied shells including the center is formulated according to single-particle charge density in TM.
e nucleons restricted within 1f7/2, 2p3/2, 2p1/2, and 2f5/2 only. Comparisons between obtained theoretical results and experimental longitudinal electron scattering form factors have been used as a stringent test of the model for transition density. Microscopic and macroscopic theories can be used to study excitation in nuclei [2]. e shell model within a restricted model space (MS) is one of the models which usefully describe nuclei's static properties, where an effective charge is used. Calculations of form factors using only model space wave function are not always successful in reproducing electron scattering data [3]. e MS has been considered as 1f7/2, 2p3/2, 2p1/2, and 2f5/2. Gx1 has been used as effective interaction in all calculations. e effective charge has been considered as 1.5e for proton and 0.5e for neutron. Different excitation energies are obtained in the calculation of the inelastic electrical transition of 46,48,50 Ti isotopes. e theoretical calculations are compared with experimental data taken from reference [4] and reference [5]. In all figures, the theoretical calculations are shown as continuous curves, and the experimental data are shown as a dotted curve. e single-particle wave function of the harmonic oscillator (HO) potential with size parameter b is used to reproduce the measured ground-state root mean square charges of the radii of nuclei under study. e one body density (OBD) matrix elements χ Λ Γ f Γ i (α f , α i ) have been calculated by using the OXBASH [6] as shell model code.
is research was performed at the Laboratory of the department of Physics, University of Surrey, UK.

Theory
e core-polarization (CP) transition density is given by the Tassie form according to the collective modes of nuclei. e Coulomb form factors for the Tassie model become [1] If the first term gives zero contribution, it is possible to add the second and third term together as −q ∞ 0 dr r J+1 ρ 0 (r) [(d/d(qr)) + ((J + 1)/qr)]J j (qr). From the recursion of spherical Bessel function [7], (2) e form factor, therefore, takes shape [7] e constant proportionality can be measured using the form factor q � k.
. (4) e form factors at q � k is related to the strengths of the transitions B (CJ). It can be shown as [7] N � ∞ 0 dr r 2 j J (kr)ρ ms .  46 Ti is given in Figure 1(a). e theoretical form factors show three peaks. Figure 1(a) shows longitudinal E(2 + ) electron scattering form factors as a function of momentum transfer for 46 Ti. We notice that the first peak occurs at q � 0.6 fm −1 , the second peak at q � 1.7 fm −1 , and the third peak at q � 2.6 fm −1 . It is noticed from Figure 1(a) that the TM gives very good results in the form factor of first and second peaks, and the third maximum gives good results. is indicates that the total form factors are very close to the practical data for the first peak q < 1 fm −1 , while the second and third peaks are overestimated. is explains that the TM is in good agreement with all momentum transfers.

Longitudinal Form Factors of E(4 + ) Transition.
e longitudinal electrical transition form factors of 46 Ti of 2.010 MeV excitation energy have been calculated and compared with practical data. Figure 1(b) shows the form factor calculation of E(4 + ). is result shows two peaks. e 2 Science and Technology of Nuclear Installations calculation of this state is to ensure the theoretical form factors without the experimental data. e E(4 + ) gives two peaks in all momentum transfer extended from 1 to 3.

Longitudinal Form Factors of E(2 + )
Transition. e first maximum of longitudinal electrical transition form factors with excitation energy 0.984 MeV occurs at 0.65 fm −1 momentum transfer, the second maximum occurred at 1.5 fm −1 , and the third maximum occurred at 2.5 fm −1 of momentum transfer as shown in Figure 2(a). e first and second peaks of the theoretical calculations as shown as continuous curve are good agreement with experimental data which shows as dotted curve, but the third peak of theoretical calculations locates under the experimental data. Figure 2(b) shows good agreement with experimental data in the momentum transfer below 0.7 fm −1 of longitudinal electrical transition form factors with excitation energy 2.296 MeV. As shown in Figure 2(b), the theoretical result is slightly underestimated over 0.7 fm −1 momentum transfer.

Longitudinal Form Factors of E(2 + ) Transition.
e theoretical data of this transition with excitation energy 1.554 MeV give three peaks. e first maximum occurred at 0.7 fm −1 , the second maximum occurred at 1.7 fm −1 momentum transfer, and the third maximum occurred at 2.5 fm −1 momentum transfer, as shown in Figure 3(a).

Longitudinal Form Factors of E(4 + )
Transition. e longitudinal electrical transition form factors of 50 Ti with excitation energy 2.657 MeV are given in Figure 3(b). All the theoretical calculations as shown in continuous curve in figure are over the experimental data which can be seen in Figure 3(b). is indicates that the theoretical form factors of E(4 + ) transition do not describe the experimental data because the Tassie model fails to describe E(4 + ) for 50 Ti isotope.

Conclusions
e Tassie model gives a good description of longitudinal form factors of 46,48,50 Ti isotope in E(2 + ) transitions as compared with experimental data especially in the region below 2 fm −1 of momentum transfer, but in the E(4 + ), the theoretical results deviate slightly from experimental data especially in the region greater than 1.5 fm −1 of momentum transfer.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  [4]. (b) E(4 + ) transition, the practical data are taken from reference [5].