Nuclear Electric Quadrapole Moments ( Q ) in 58 Ni

Nuclear Electric quadrapole moments Q in Ni for some selected levels have been investigated and calculated through Nuclear shell model and considering of 56 Ni as an inert core with two active neutrons in a model space (2p3/2, 1f5/2 and 2p1/2) and the configuration mixing of the original states is also done. F5Pvh interaction has been utilized as a two body interaction to generate model space vectors with harmonic oscillator potential as a single particle wave function. OXBASH code is used to carry this calculations and the program of Core, Valence, Tassie (CVT) written in FORTRAN go language to calculate the Electric quadrapole moments between excited states themselves. All of these calculations have been carried through model space vectors only. One body density matrix elements (OBDM) for ground and Excited states is calculated in order to carry the calculations using single particle Transition matrix elements between excited states theme selves.


Introduction
A large part of the knowledge of nuclei is obtained from the study of electromagnetic transitions, since the electromagnetic interaction is well understood, in contrast with the nuclear forces. It is, for example, the main source of information about the spin assignments of nuclear states. The nuclear multipole moments and the transition rates for the various multipole radiations can be calculated theoretically, once the nuclear wave functions are known.
We shall not give a complete derivation of the required electromagnetic transition operators. Instead only some of the most important steps that lead to explicit expressions for these operators will be summarized. For example, [Blatt and Weisskopf (1952), Jackson (1962), Morse and Feshbach (1953) and Roy and Nigam (1967).
Some of the basic equations related to the electromagnetic interaction are summarized. A derivation of the operators that will be used later to calculate electromagnetic transition and moments, Centre of mass corrections are treated. Measurable quantities such as reduced transition rates, lifetimes, branching and mixing ratios are defined. The much used Weisskopf single particle estimates of transition strengths are derived. It is shown that the isospin formalism allows us to express transition strengths in terms of isoscalar and isovector contributions. This separation can be used for example, to correlate transition rates in isobaric mass multiplets [1].
The nuclear quadrupole moments vary widely in magnitude. In particular the nuclei in the mass regions 150 A 190 and A 225 possess large permanent deformations. Light nuclei usually can be considered as to consist of a spherical core with a small number of extra nucleons. In such a picture the nuclear quadrupole moment derives completely from the extra nucleons. The extra nucleons may be coupled to pairs j 2 with J=0 and then, in this extreme single-particle model, the quadrupole moment is due to the last odd proton [1].
The Nucleus 58 Ni Nickel (Ni) possesses five stable isotopes including 58 Ni, 60 Ni, 61 Ni, 62 Ni and 64 Ni. In addition, 27 radioactive isotopes have been discovered ranging from 48 Ni to 79 Ni, some of them have short-half lives; others have long-half lives. The longest-half lives is 59 Ni with a half-life of 7.6×10+4 years. Most of them are under a minute or a second. The least unstable is 79 Ni with a half-life of 635×10−9 s. [2,3]. The nucleus Ni 58 has 28 protons and 30 neutrons, two neutrons play essential role in the model space shell, outer the closed shell when the inert core 56 Ni is under consideration. Wang and Ren (2005) [4] systematically investigated the elastic electron scattering on both stable and unstable nuclei with the relativistic eikonal approximation, where the charge density distributions of nuclei were from the self-consistent relativistic mean field model. Calculations had shown that the relativistic eikonal approximation can reproduce the experimental data of electron scattering on nuclei ranging from the light region, such as 12 C, to the heavy region, such as 208 Pb. This was the systematic test of the relativistic eikonal approximation for elastic electron scattering for both light and heavy nuclei, including the calculated charge form factors for 48 Ni, 56 Ni, 58 Ni, 64 Ni, 68 Ni, 74 Ni and 78 Ni isotopes. Bespalova et al. (2010) [5] studied the experimental single-particle energies and occupation probabilities for neutron states near the Fermi energy in 58,60,62,64 Ni nuclei which had been obtained from joint evaluation of the data on nucleon stripping and pickup reactions on the same nucleus. The resulting data were analyzed within a mean-field model with dispersive optical-model potential. Good agreement was obtained between the calculated and experimental singleparticle energies of the subshells.  [7] calculated elastic and in elastic form factor and for the transition from the ground state to J+1( L =J = 2,4) state in 58-68Ni and 24Mg, the starting point of method was a set of Hartree-Fock-Bogoliubov wave functions generated with a constraint on the axial quadrupole moment and using a Skyrme energy density functional. Correlations beyond the mean field were introduced by projecting mean-field wave functions on angular-momentum and particle number by mixing the symmetry restored wave functions.

Theory
The nuclei are assumed to have a spherical shape. This is a good approximation for nuclei that have magic numbers of neutrons or protons: 2, 8, 20, 28, 50, 82 and 126. These numbers come from the shell structure of the nucleus. Nuclei with magic numbers of neutrons or protons have a "closed shell" that encourages a spherical shape. Nuclei with Z or N far from a magic number are generally deformed. The simplest deformations are so-called quadrupole deformations where the nucleus can take either a prolate shape (rugby ball) or an oblate shape (cushion). The electric quadrupole moment is considered as a criterion of the deviation of the electric charges of the nucleus from the spherical shape or spherical distribution.
The electric quadrupole moment operator is given by [8]: Where and are the position coordinates of th nucleon and is given in Cartesian coordinates as = + + .
After substituting equation 2 with = 2 into equation 3 , one can obtain: The initial and final states of the nucleus can be written as: The matrix element of the electric transition operator in the above equation can be reduced by using Wigner-Eckart theorem as: P C C 9 C 9 Q R 20 R 8 8 9 8 9 S T = −1 ' Q V) Q W C 2 8 − C 0 8 X × P C 9 C 9 Q RB 2 BR 8 9 8 9 S T In nuclear physics the quadrupole moment of a state of angular momentum is defined as the expectation value of the electric quadrupole moment operator in the state = With the use of Wigner-Eckart theorem, the reduced matrix element of equation 7 can be written as: P C 9 C 9 Q RB 2 BR 8 9 8 9 S T = ] −1 ^QV^" Q %/,& _ 9 C 9 9 8 −9 Q 0 9 S` × 〈 C 9 C Bb 2 bB 8 9 8 〉 And the electric quadrupole moment of equation 7 becomes: ^QV^" Q %/,& _ 9 C 9 9 8 −9 Q 0 9 S` × 〈 C 9 C Bb ^b B 8 9 8 〉 With using Wigner-Eckart theorem [9], the reduced electric transition probability can be calculated in terms of the many-particle matrix element of the electric multipole transition operator reduced in spin-isospin as: Where the reduced many-particle matrix element of the electric multipole transition operator is given 9 'î by '^, as: Also, equation 11 can be written in proton-neutron formalism as: Where the reduced many-particle matrix element of the electric transition operator can be calculated in spin-isospin formalism and in proton-neutron formalism with using equations 12 and 13 , respectively. From equations 9 and 10 , one can show that the quadrupole moment is related to the reduced transition probability as: The electric quadrupole moment is to be taken in units of . E• or . €. Where b is the barn, € = 100 E• [10]. The single particle quadrupole moment •.‚. x of a nucleon in an orbit with spin x depends on the radial and angular properties of the orbit as shown in the following equation [8]: Where 〈 | 〉 is the mean square radius for a particle in the orbit >, @, x .

Result and Discussion
The relation between a Clebsch-Gordon coefficient and 3-j symbol as the following [1].  The free neutron has e zero charge but the effective neutron has an effective charge. The measurement proved that the amount (negative) and the value less than one electronic charge.
There is a research and study of the measurements for the nuclear shell model and multipolarities which proved the existence of active charge of neutrons.
The values of the OBDM elements for electric quadrupole moment(Q) for transitions from state to the same state as shown in the tables from (2) to (12).

Conclusion
Very weak values of Electric Quadrapole moment (Q) are generated from 58 Ni due to its active particles (Neutron) which are neutral particles (e n =0) but it is possesses a small values of Electric Quadrapole moment (Q) due to some extent of active charge inherent to the motion and interaction inside the Nucleus.
The calculated one body density matrix element was carried by the use of OXBASH code and then the resulted output of these files had been included in another computer program to finish the calculation and produce the multipole moment.