The Nuclear Structure for Exotic Neutron-Rich of 42 , 43 , 45 , 47 K Nuclei

In this paper the proton, neutron and matter density distributions and the corresponding root mean square (rms) radii of the ground states and the elastic magnetic electron scattering form factors and the magnetic dipole moments have been calculated for exotic nucleus of potassium isotopes K (A= 42, 43, 45, 47) based on the shell model using effective W0 interaction. The single-particle wave functions of harmonic-oscillator (HO) potential are used with the oscillator parameters b. According to this interaction, the valence nucleons are asummed to move in the d3f7 model space. The elastic magnetic electron scattering of the exotic nuclei 42 K (J π T= 2 2), 43 K(J π T=3/2 + 5/2), 45 K (J π T= 3/2 + 7/2) and 47 K (J π T= 1/2 + 9/2) investigated through Plane Wave Born Approximation (PWBA). The inclusion of core polarization effect through the effective g-factors is adequate to obtain a good agreement between the predicted and the measured magnetic dipole moments.


Introduction:
One of the key issues in current nuclear physics research is to investigate the properties of so-called `exotic nuclei' and of `exotic nuclear structures'.Exotic nuclei are nuclei with a proton-to-neutron ratio that is very different from the proton-toneutron ratio in stable nuclei (a technical term related to this ratio is the `isospin').The exotic nuclear structures can be defined as excitation modes of nuclei that have a very different structure than the structure (or shape) of the nuclear ground state [1].Because of the rapid decay of exotic nuclei, it is rather difficult to make targets with them, therefore, experiments have been done in inverse kinematics with a beam of exotic nuclei incident on a target.
The electron scattering from exotic nuclei is not presently available; the technical proposal for an electron-ion collider has been incorporated in the GSI/Germany physics program at FAIR [2].A similar program exists for RIKEN/Japan facility [3].In both cases the main purpose is to study the structure of nuclei far from the stability line.Such facilities in future will explore the charge density distributions for nuclei far from the valley of Open Access stability line, having skins or halos.Therefore, the halo nuclei are an extreme case of exotic nuclei with almost zero binding energy.
The electric and magnetic properties of a nuclear state, namely on what the static magnetic dipole and electric quadrupole moments can teach us about the nucleus as a system of independently moving particles in a central potential or as a system of collectively moving nucleons.The magnetic moment is very sensitive to the single particle orbits occupied by the unpaired nucleons, while the quadrupole moment is a unique instrument to study the deformation and collective behavior of nuclei both at low and high excitation energy.Both quantities can be directly compared with the predicted values in different nuclear models.Consequently, the magnetic moment may also give a means to distinguish between spherical and deformed states [4,5].
Karataglidis and Amos [6] presented elastic and inelastic electron scattering form factors for several neutron-rich exotic nuclei.The results have been obtained using large space no-core shell models.While the elastic scattering form factors are insensitive to the details of the neutron density, it is found that inelastic scattering may be influenced by extensive neutron distributions Sieja and Nowacki [7] investigated the neutron rich nuclei which can be described by shell model calculations in the p-sd and sd-pf model spaces.They quantified the effects of the core polarization on the multipole part (pairing and quadrupole) of the effective Hamiltonians.They showed that proton core polarization contributions are responsible for the reduction of the neutron-neutron nuclear matrix elements which, in the recent shell model studies, appeared necessary between p-sd carbon and oxygen and sd-pf silicon and calcium nuclei.
Neyens [8] measured the ground state magnetic moments and spins of the exotic isotopes 49,51 K at the ISOLDE facility at CERN using bunched-beam high-resolution collinear laser spectroscopy.The reinversion of the ground state spin from I = 1/2 in 47,49 K back to the normal I = 3/2 in 51 K.At GANIL (Caen, France) the quadrupole moment of the 33 Al ground state has been measured using the continuous-beam -nuclear magnetic resonance method applied to a spinpolarized beam produced at the LISE fragment separator.The large value establishes a very mixed wave function with about equal amounts of normal and neutron particle-hole excited configurations contributing to its ground state wave function.
Kreim et al., [9] reported and deduced on the measurement of optical isotope shifts for 38,39,42,44,46- 51 K relative to 47 K from which changes in the nuclear mean square charge radii across the N=28 shell closure.The investigation was carried out by bunched-beam collinear laser spectroscopy at the CERN-ISOLDE radioactive ion-beam facility.Mean square charge radii are now known from 37 K to 51 K, covering all νf 7/2 -shell as well as all νp 3/2 -shell nuclei.These measurements, in conjunction with those of Ca, Cr, Mn and Fe, provide a first insight into the Z dependence of the evolution of nuclear size above the shell closure at N=28.
The aim of the present work is to study the magnetic elastic electron scattering form factors and to calculate the magnetic dipole moments of exotic nucleus 42,43,44,45 K (neutron-rich) using W0 interaction [10] in d3f7 -model space.The elastic magnetic electron scattering of the exotic 42,43,45,47 K nuclei are investigated through Plane Wave Born Approximation (PWBA).Also the proton, neutron and matter density distributions and the corresponding root mean square (rms) radii of the ground states are calculated.Calculations are presented with model space and with corepolarization (CP) effects by using effective g-factors.

Theory:
The interaction of the electron with the spin and currents distributions of nuclei can be considered as an exchange of a virtual photon with angular momentum 1  along momentum transfer q direction.This is called transverse scattering [11].From the parity and angular momentum selection rules, only electric multipoles can have longitudinal components, while both electric and magnetic multipoles can have transverse components [11,12].
The squared magnetic form factors for electron scattering between nuclear states J i and J f involving angular momentum transfer J are given by [13]: is the magnetic electron scattering multipole operator.For a magnetic operator T J T the reduced matrix elements are written as the sum of the product of the one-body density matrix elements (OBDM) times the single-particle transition matrix elements [14]: where is the multipolarity and the states are initial and final states of the nucleus.While α and β denote the final and initial single-particle states, respectively (isospin is included).The nuclear shell model calculations were performed using the OXBASH Shell model code [15], where the one body density matrix (OBDM) elements given in eq.(2) were obtained.
The single-nucleon form factor [16] and the center-of-mass form factor [17] are given by: (1), we obtain: The single particle matrix element in spin-isospin state is given by [14]: The reduce matrix element of the magnetic transition operator expressed as the sum of the product of the elements of the one-body density matrix (OBDM) times the singleparticle matrix elements, and is gives by: The magnetic dipole moment  of a state of total angular momentum J is given by [14]: The neutron skin occurs as a consequence of the neutron excess in heavy nuclei.The neutron skin in general is defined as the radial difference (root-mean square-radius difference) of the neutron and proton distributions with surface thickness ( ) [18]: Where

Results and Discussion:
The radial wave functions for the single-particle matrix elements were calculated with harmonic oscillator (HO) potential with size parameters b are adjusted for each isotope of potassium to reproduce the measured root mean square matter radius (R m ).The calculations of the proton, neutron, and matter rms radii and magnetic form factors are carried out using d3f7shell model space with effective W0 interaction [10] in OXBASH code [15].The core polarization (CP) effect is included by using effective g-factors.
The values of calculated matter (R m ), proton (R p ) and neutron (R n ) rms radii, magnetic dipole moments ( ), and oscillator size parameter (b) of potassium isotopes nuclei are displayed in table 1.
The calculated total magnetic form factors for 42 K ground state (J π T= 2 -2) using W0 interactions in d3f7-model space and with free g s -factors are shown in Fig. ( [20].
The calculated total magnetic form factors for 43 K ground state (J π T= 3/2 + 5/2) using W0 interactions in d3f7model space and with free g s -factors are shown in Fig. (2).The individual multipoles contributions M1 and M3 are denoted by dashed and dashed-dot curves respectively, while the E2 is disappeared because it has negligible contribution.The diffraction minimum for M1 component located at momentum transfer q=1.9fm -1 , but for M3 component the diffraction minimum located at q=0.6 fm -1 .The total form factors in d3f7-shell model space are included by solid curve.
The calculated total magnetic form factors for 47 K ground state (J π T= 1/2 + 9/2) using W0 interactions in d3f7model space and with free g s -factors

Conclusions:
Shell-model calculations are performed for K isotopes including core-polarization effects through firstorder perturbation theory.The magnetic dipole moments µ of the 42,43,45,47 K nuclei depend clearly on assigned configurations and their experimental data will be useful to determine the deformations of the ground states of nuclei near the neutron drip line.The inclusion of core polarization (the effective g-factors) is adequate to obtain a good agreement between the predicted and measured magnetic dipole moments.The elastic magnetic form factors are influenced by details of nuclear wave function and the center of mass correction which depends on the mass number and the size parameter b.

References:
[1] Neyens, G. 2003 where A is the nuclear mass number and b is the harmonic-oscillator size parameter, for halo nuclei b equal to the average of b core and b halo .Introducing these corrections into Eq.

Fig. ( 1
Fig. (1): The magnetic form factors for ground state of 42 K calculated in d3f7-model space.The individual multipoles contributions of M1 and M3 are shown.
g s -factors are shown in Fig.(3).The individual multipoles contributions M1 and M3 are denoted by dashed and dashed-dot curves, respectively, while the E2 multipole is disappeared because it has a negligible contribution.The diffraction minimum for M1 component located at momentum transfer q= 2.0fm -1 .The different values of g s -factors, which give different values for magnetic dipole moment.The choice for free g-factors gives effective g-factors with g(eff)=0.9g(free) decreased the magnetic moment value by

Fig. ( 3
Fig. (3): The magnetic form factors for ground state of 45 K calculated in d3f7-model space.The individual multipoles contributions of M1 and M3 are shown.

Fig. ( 2
Fig. (2): The magnetic form factors for ground state of 43 K calculated in d3f7-model space.The individual multipoles contributions of M1 and M3 are shown.

Fig. ( 4 )
Fig. (4): The magnetic form factors for ground state of 47 K calculated in d3f7-model space.The individual multipole contribution of M1 is shown.