Investigation of the Quadrupole Moment and Form Factors of Some Ca Isotopes

Nuclear shell model is adopted to calculate the electric quadrupole moments for some Calcium isotopes 20Ca (N = 21, 23, 25, and 27) in the fp shell. The wave function is generated using a two body effective interaction fpd6 and fp space model. The one body density matrix elements (OBDM) are calculated for these isotopes using the NuShellX@MSU code. The effect of the core-polarizations was taken through the theory microscopic by taking the set of the effective charges. The results for the quadrupole moments by using Bohr-Mottelson (B-M) effective charges are the best. The behavior of the form factors of some Calcium isotopes was studied by using Bohr-Mottelson (B-M) effective charges.


Introduction:
The shell structure of single-particle levels in a spherical potential can be affected by the nuclear deformation (1). The first approximation representing closed core nuclei (the atomic number, Z, is a magic number and, the neutron number is a magic number). Additionally, the outer-nucleons can move in a certain configuration space which interact with the core and each other by a residual interaction such particle-particle and particle-core interactions (2). These interactions depend on the selected model space and residual interactions; one can investigate this by comparing several experimental parameters such as spin/parity, excitation energy, the magnetic moment and the quadrupole moment and parametrization of the residual interaction and validity of the model space. The measurement of the quadrupole moment is often a good test to ascertain if the model space and the parameterizations are suitable (2). The nuclear electric quadrupole moment depends on a deflection of nuclear charge distribution from spherical symmetry which gives a useful measure of the core polarization especially if the valence nucleons are neutrons which do not directly participate to the electric quadrupole moment (3). Shell model calculations; adopt fixed measurement of the effective charges. The Bohr-Mottelson (B-M) (4) particle-vibration combines with model for the effective charges. Department  In the present work, we will adopt shell model calculations with a harmonic oscillator (HO) single-particle wave functions to calculate the quadrupole moments of some Calcium isotopes in the fp shell. The calculations adopted different effective charges and were compared with experimental data. Also, the behavior of the form factors for some calcium isotopes by using Bohr-Mottelson (B-M) effective charges is studied.

Theory
The electric quadrupole moment is associated with deformation of nuclei. The nuclear charge distribution in state | ⟩ is given by (7) ( ) = ⟨ | ∑ ( ) ( − ( ))| ⟩ =1 … 1 where ( ) describes the nuclear charge distribution, e(k) denotes the charge of nucleon numbered k and is axial symmetric about the z-axis. Therefore one has also the relation Q xx =Q yy Q zz = -2Q xx = -2Q yy Q xy = Q yz = Q x z= 0 Hence, due to eq. above, only one independent quadrupole component, i.e. Q zz remains, which can be represented by ( ) = ∫ ( )(3 2 − 2 ) …2 when, alternatively, the quadrupole moment ( ) is written in terms of the five spherical harmonics Y 2M () with M=0, ±1, ±2, then, due to the axial symmetry of the charge distribution, the components with M ≠0 vanish. The connection between Y 20 () and Cartesian coordinates are given by: where e(k) denotes the charge of nucleon numbered k, where e(k)=0 for a neutron and e(k)=e for a proton.
Here also, a numerical factor √ 16 /5 occurs, because the quadrupole moment of an axially symmetric body is conventionally defined as 〈3 2 − 2 〉 .
The expectation value in Eq. 4 still depends on the projection quantum number M. For the given J the moments for different values M are not independent, however, since they are defined by Clebsch-Gordan coefficients according to the Wigner-Eckart theorem. In nuclear physics the quadrupole moment of a state of angular momentum J is defined as the expectation value of the state M=J. It is referred to as the spectroscopic, or static, quadrupole moment given by the definition: where the electric transition operator ̂( 2) is defined as … 6 For the calculation of the quadrupole moment initial and final state wave function must be identical. The electric 2 L pole operator () possesses parity (-1) L , and the electric 2 L pole moment should vanish for odd values of L. The intrinsic is defined with respect to the axis of symmetry of the charge distribution. The nuclear matrix element of the electromagnetic operators ⟨ ||̂( 2)|| ⟩ between final nuclear states (J f ) and the initial state (J i ), where multipolarity τ is the sum of the products of single-particle matrix elements times OBDM ⟨ ||̂ ( )|| ⟩ = ∑ ( ′ , , , ′ , ) ⟨ ′ ||̂( )|| ⟩ …7 where the single-particle states represented by ′ and for the shell model space.
The model space matrix elements and sharing the effective charges ( ( )), can be written by the electric matrix element The formulated of the effective charges are written as (8) The quadrupole moment in state | = 2 = 0 > for = can be described by (7): the form factor between nuclear shell model states of final state (J f ) and initial state (J i ) involves the momentum transfer (q) and the angular momentum (τ), between nuclear shell model states of final state (J f ) and initial state (J i ), is given by (5,9) | ( , )| 2 =

Result and Discussion:
Theoretical values of the electric quadrupole moments for fp-shell states, calculated as described, are presented and compared with experimental values as Tables. Results with three different effective charges, are given to illustrate the variations which can arise.
Calculations of the shell model are adopted using NuShellX@MSU code (10) to obtain excitation energy levels and the one body matrix elements (OBME). in the present work, The effective interaction fpd6 (6) and model space fp are used. The single particle-matrix elements can be calculated by size parameters b and the HO. The (Mp = mass of proton) (11).

Electric quadrupole moment
The present work studies the microscopic structure of Ca nucleus in the fp shell. Calculations of the shell model should be a restricted model space where fp-model space adopted in the present work. Effective charges which are introduced in evaluating quadrupole moments in shell model are related to nucleon excitations in nuclei. Effective charges are important because of the polarization of the core which is not involved in the model space calculations. One set of effective charges, the B-M (8) are calculated according to Eq. 9 for some isotopes in the present work and shown in Table 1. isotopes with Eq.10 and the results were compared with experimental values in Ref. (14,15) as shown in Tab 1and Fig.1.
In Fig.1 the electric quadrupole moments for 20Ca isotopes are calculated with effective charges B-M (8). The calculated value of Q (7/2 -, 41Ca)= -9 e fm 2 is in agreement with the experimental value (14) and shape nucleus of 41Ca is oblate where B-M effective charges 1.17e and 0.79e, for the proton and neutron, respectively. The calculated value of Q (7/2 -, 43Ca) = -3.5 e fm 2 is overestimated with experimental values (14). The B-M effective charges are 1.13e and 0.72e, for the proton and neutron, respectively. The calculated value of Q (7/2 -, 45Ca) = 3.41 e fm 2 is in close agreement with the experimental value (14). This value shows a prolate deformation which number of neutrons are odd (N=25) and have positive parity, where B-M effective charges are 1.12e and 0.7e, for the proton and neutron, respectively. Finally, the calculated of value of Q (7/2 -, 47Ca) = 10.66 e fm 2 is overestimated with experimental values (15) . This value shows a large prolate deformation for N = 27 where 20 neutrons in sd shell (core) and neutrons 7 neutrons in 1f 7/2 orbit and adopted the B-M effective charges are 1.1e and 0.66e, for the proton and neutron, respectively. These calculations are presented in Tab1 as shown in Fig.1.  In Fig. 2 the electric quadrupole moments for 20Ca isotopes are calculated with the standard effective charges ST (e p =1.36, e n =0.45) (12) for all isotopes in the present work and the results are tabulated in Table 2. The calculated value of Q (7/2 -, 41Ca) = -5.11 e fm 2 is overestimated with the experimental value (14). This value shows that the nucleus deformation is oblate and there is one particle (neutron) in 1f 7/2 shell. The calculated value of Q (7/2 -, 43Ca) = -2.1 e fm 2 is overestimated with experimental values (14). This value shows the presence of the deformation nucleus is oblate and three-valance nucleons (neutron) in 1f 7/2 orbit. The calculated value of Q (7/2 -, 45Ca) = 2.21 e fm 2 is in close agreement with the experimental values (14), which shows a nucleus deformation is prolate. Finally, the calculated of value of Q (7/2 -, 47Ca) =7.34 e fm 2 is overestimated with experimental values (15). This value shows a large prolate deformation for neutrons 20 in sd shell where are frozen and seven-valence neutrons in 1f 7/2 -shell these values are presented in Tab 2. proton and neutron 1.36, 0.45,  respectively (12) and by using interaction fpd6 (6).  In Fig.3 the electric quadrupole moments for 20Ca isotopes are calculated with conventional effective charges (Con) are e p = 1.3 e and e n = 0.5 e (13) for all isotopes in the present work. The calculated value of Q (7/2 -, 41Ca) = -5.68 efm 2 is overestimated with the experimental value (14). This value shows deformation nucleus is oblate and one particle (neutron) in 1f 7/2 shell. The calculated value of Q (7/2 -, 43Ca) = -2.37 efm 2 is overestimated with experimental values (14). This value shows deformation nucleus is oblate and three-valance nucleons (neutron) in 1f 7/2 orbit. The calculated value of Q (7/2 -, 45Ca) = 2.45 efm 2 is in close agreement with the experimental value (14). This value shows a nucleus deformation is prolate. Finally the calculated of value of Q (7/2 -, 47Ca) = 8.16 efm 2 (15) is overestimated with experimental values (15). This value shows a large prolate deformation for neutrons 20 in sd shell as frozen and neutrons 7 in 1f 7/2 -shell. These results are presented in Tab 3.

Open Access
Baghdad Science Journal

The Form Factors
The calculations of the form factors for some Ca isotopes where no experimental data are available, the fp model space with the interaction fpd6 (6) was adopted. The model space matrix elements are obtained using Eq.8, where the required one-body density matrix was calculated using code the NuShellX@MSU (10). The form factors C2 is calculated using an expression of the transition charge density operator from Eq.11.
The electric-charge form factors are computed for 41, 43, 45Ca isotopes as the Fourier transform of the charge density. The present work calculated |F(q)| 2 , where the form factors for 41Ca and for 43Ca are shown in Fig.5 and form factor for 45Ca is shown in Fig.6. Calculation of the form factors is done by using set effective charges, the first conventional effective charges (e p =1.3, e n =0. It has been obtained from Eq. 11, and by using the different effective charges. Figure 6. Squared charge form factor for 45Ca as a function of the momentum transfer (q). It has been obtained from Eq. 11, and by using the different effective charges.

Conclusion:
The characteristics of the quadrupole collectivity that develops N ≥20 are discussed by comparing the experimental values of the quadrupole moments with the shell model calculations. Shell model is adopted to calculate the quadrupole moments of Ca (41, 43, 45, and 47). Our calculations include core-polarization effects through three effective charges. The set of effective charges are Bohr-Mottelson (B-M), standard effective charges (ST) and conventional effective charges (Con). The results of the quadrupole moment by using fpd6 interaction are in agreement with the experiment when using B-M effective charges. In the case, the results almost agree with the experimental value. The results show an increase of the quadrupole moments as increasing the neutron number. Also, the results show an increase in the values of the form factors using the effect of core polarization is found essential in both the momentum transfer and transition strengths dependence and gives a good description for the behavior of charge distribution.
Author's declaration: 508 -Ethical Clearance: The project was approved by the local ethical committee in University of Fallujah.