147 

© Published by College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad. This 

 is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License 

Nuclear Energy Levels Scheme of Sc46 by Using FPD6 and KB3G Interaction 

 

Hasan Ali Kadhim1*  and  Firas Zuhair Majeed2   
1,2

Department of Physics College of Sciences, University of Baghdad, Baghdad, Iraq. 

*Corresponding author.  

 

Received:30 May 2023                               Accepted:3 July 2023                             Published:20 July 2024 

doi.org/10.30526/37.3.3514 

 

Abstract 

      Nuclear energy levels in Sc46  with modest occupancy of the fp-LS shell have been studied 

within shell model calculations. The interactions used to determine the nuclear energy levels are 

FPD6, KB3G, and FPY using the fp shell and d3f7cospn for 1d3/21f7/2 model space. Results are 

compared with each other, and with the existing experimental data, there is clear agreement with 

some results. Aside from the excellent agreements in the replicated values of energy levels scheme, 

using model space interactions is the best-fitted two-body matrix element in the fp shell model 

space. Particularly below 3MeV, the overall estimation of the replicated data is excellent. The wave 

vectors and analysis are represented in the so-called diagrammatic notation, and all inscriptions are 

provided in this style. A single particle vector is generated by utilizing the oscillator's potential and 

considering 𝐶𝑎20
40

  as the core for the fp shell model space and 𝑆32  as the inactive core for the d3f7 

model space. Results are obtained for all examined nuclei using the OXFORD BEUNES AIRES 

SHELL MODEL CODE.  

Keywords: Energy levels - Sc46 - Fp shell - OXBASH Code -   Fpd6. 

1. Introduction 

     The features and internal structure of the nucleus have been the subject of much research. There 

is no all-encompassing theory that can adequately describe the operations, characteristics, and 

structures of nuclei[1]. The shell theory provides several benefits and features, including model 

independence, the applicability of the practical N-N potential, and a broad variety of nuclei, in 

addition to the conventional Hamiltonian related to various kinds of eigenvectors. Obtaining all 

observable nuclei follows naturally from the shell theory; hence, it is still applicable today[1]. Using 

the LS shell (1f7/2 1f5/2 2p3/2 2p1/2) as a reference, they determined the excitation energies, 

binding energies, and spectral factors necessary to generate the effective N-N matrix elements [2]. 

Using the nuclear shell model, the energy levels of different Cu and Ga nuclei were analyzed. An 

inert 𝑁𝑖56  nucleus was used as a model in the theoretical application, with pf single particle orbitals 

https://creativecommons.org/licenses/by/4.0/
https://orcid.org/0003-0007-3164-0333
mailto:hasan.ali2204m@sc.uobaghdad.edu.iq
https://orcid.org/0000-0001-6527-3913
mailto:Firas.majeed@ sc.uobaghdad.edu.iq


IHJPAS. 2024, 37 ( 3 ) 

148 
 

serving as the model space. Using the f5pvh nucleon-nucleon interaction, the effective interactions 

between two nucleons were computed. Space-time dynamics have been modeled using orbits with 

radii of f5/2, p3/2, and p1/2 [3]. The goal of this study is to provide a method for efficiently and 

correctly disseminating computational results for shell models, with an emphasis on fp-shell. These 

estimations were based on the evaluation of Hamiltonian eigenvalues, which are compatible with 

positive parity of energy levels up to (10MeV) for most Ca isotopes, and the Hamiltonian 

eigenvectors transition strength probability, and inelastic electron-nucleus scattering. The 

Hamiltonian has proven to be helpful in the experimental settings we've used so far. The 

experimental data were used to propose a new nuclear level. To do the calculations the OXBASH, 

code [4] is utilized. 

Calculations using the interaction of GXFP1 in the fp model space by adopting the single-particle 

wave functions of the harmonic oscillator. All of the isotopes involved have valence nucleons 

moving in 1f7/2, 2p3/2, 1f5/2, and 2p1/2 orbitals, while the 𝐶𝑎40  nucleus is thought to have a stable 

nucleophile in the fp-form. The consequences of core polarization may be derived using 

microscopic theory in the case of first-order core polarization. To account for core polarization, the 

quadrupole and magnetic moments were calculated using the effective charge and effective g 

factors, respectively [5]. 

Three isotopes, 𝐶𝑎46−45 , 𝑇𝑖46−45    and 𝑆𝑐 46−45 , had their potential energies calculated. They 

conducted computations in the f7-shell region while accounting for the spin-parity of the valence 

nucleons using the shell model code OXBASH for Windows and the effective interactions, f748pn. 

It was shown that the energy level values tend to converge around the usable norm [6]. The nuclear 

energy levels of the even-even 𝐶𝑎 sotopes were calculated using the nuclear shell model and the 

KShell computer code. This software allows us to do nuclear shell-model calculations using the M-

scheme representation. The very potent 𝐶𝑎40  isotope was employed as the backbone of the 

calculations. It is possible for neutrons to simulate orbits at 0f7/2, 1p3/2, 0f5/2, and 1p1/2 in their 

model space. Nuclear spin, parity, and energy have all been determined for excited Ca isotopes [7]. 

Using the interacting boson model-1 (IBM-1), they collected data on the A50 mass region of even-

even Ti isotopes and examined their collective properties. In this study, the energy levels and rates 

of electromagnetic transitions for the isotopes 𝑇𝑖44−48  and 𝑇𝑖52−60  are determined. The IBM-1 

models did not include the isotope 𝑇𝑖 50 since its neutron count is 28 and 28 is considered the "magic 

number." The energy ratios were originally examined by comparing the normal values of the U(5), 

SU(3), and O(6) dynamical symmetries and the E(5), X(5) critical point symmetries. Based on the 

observable properties of the isotopes in the issue, a subsequent model Hamiltonian was constructed. 

Using the parameters of the fitted Hamiltonian, the lowest energy levels of all the Ti isotopes were 

calculated [8]. 

The shell model based on large-scale unconstrained fp-model space has been used to determine low-

lying energy levels and Gamow-Teller B(GT) transition strengths for the transitions ( 𝐶𝑎 42 𝑆𝑐42 , 

𝐶𝑎 42 𝑇𝑖42 , 𝑆𝑐45  𝐶𝑎45 , 𝑆𝑐45  𝑇𝑖45 , 𝑉45  𝑇𝑖45 ). The intensities of the Gamow-Teller B(GT) transition 

were computed and compared to experimental data on concentrations. The low-lying energy levels 

of the studied nuclei were accurately reproduced. For several of the nuclei examined, the spin and 

parity of previously unconfirmed energy levels had been verified [9]. Energy levels and the 



IHJPAS. 2024, 37 ( 3 ) 

149 
 

probability of electric transition B(E2) in the fp shell of the 𝑁𝑖60  and 𝐹𝑒60  nucleus with the effective 

interactions f5pvh, gx1, and kb3 were calculated using OXBASH. OXBASH is a computer 

application for calculating nuclear installations using a shell mode [10]. They computed the energy 

levels and the reduced electric quadrupole transition probability B(E2) for the 50Fe isotope using 

the OXBASH algorithm and the F754 and F7mbz effective interactions. There are six protons and 

four neutrons in the shell around the 𝐶𝑎40  nucleus of the 𝐹𝑒50  isotope. All of these isotopes had 

identical total angular momentum and ground state parity, and although this produced new energy 

levels, none of them was currently useful [11]. Within shell model calculations had been interested 

in nuclear energy levels; total angular momenta and even-even parity for nucleons that are presented 

outside closed and no core for the elements 𝐶𝑎42 ,  𝐶𝑎44 , 𝐶𝑎46 , and 𝐶𝑎48 ) that filled fp-shells 

(1f7/2,1f5/2,2p3/2,2p1/2). The atomic energy distribution over the periodic table 𝐶𝑎42 ,  𝐶𝑎44 , 𝐶𝑎46 , 

and 𝐶𝑎48  were determined via a total of four interactions. We compare the findings from the FPD6, 

GXPF1, and KB3G interactions with the GOGNY-P2 (fp, fpg, and fpgd model space) interactions 

and find some agreement. 

The frozen orbitals and limited occupations method [12] were developed for use in full-space 

computations where the GOGNY-P2 interaction was used as the effective full-space two-body 

interaction. A nuclear shell model with ca as an inert core and FPD6, HW, and FPY as model space 

effective interactions was used to examine 𝑆𝑐44  [13] and 𝐶𝑎44  [14] nuclear energy levels, and the 

replicated results were compared to experimental data. In this investigation, they employed the 

OXBASH method to produce model space wave vectors and to receive the equivalent model space 

effective interaction. By considering the single-particle potential (harmonic oscillator) and the 

elastic magnetic electron scattering form factors, they established a model space factor for 𝐶𝑎41  

using F7MBZ, with 𝐶𝑎40  as the inert core. Then, the Ca quadrupole moments (41, 43, 45, and 47) 

are calculated using the shell model [16], and the nuclear shell theory is applied to the entire system 

[15]. The theoretical level schemes of the selected nuclear states provide the basis for our FP-space 

model computations of different effective interactions. Spins and association energies that are 

common in GX1A, KB3G, FPD6, and GX1A are present in several experimentally observed levels 

in 𝐶𝑎52 . Spin spectra 𝐶𝑎52−54  and 𝐶𝑎56−58  were produced for efficient interactions [17].  Some 

physical properties, like the electromagnetic properties' effects on the nuclear structure of certain, 

cobalt (Co) isotopes with mass numbers A = 56-60, have been analyzed. These properties include 

elastic longitudinal form factors, electric quadrupole moments, and magnetic dipole moments. All 

GXFP1 interaction calculations are performed in the fp model space, using the harmonic oscillator's 

single-particle wave functions [18]. The modified-surface one-boson exchange potential 

(MSOBEP) model developed for the sd-shell [19,20] was successfully applied to generate a six-

parameter density-dependent interaction for the bottom section of the fp shell FPD6 [21-25]. Even 

when used in the contemporary GXPF1 method, the FPD6 interaction remains extremely effective. 

While FPD6 already has monopole interactions, including them may improve the effective SPE 

attained for [26]. Recently, the GXPF1 has made it possible to apply the entire Effective two-body 

matrix elements + group interaction (ETBME+G) technique to the fp shell interaction, which has 



IHJPAS. 2024, 37 ( 3 ) 

150 
 

been done for all fp shell nuclei in the mass range (47-65)[27,28]. Future theoretical and 

experimental studies will focus on neutron-rich fp shell nuclei.  

The aim of this research is to use the OXBASH code[29] to calculate the nuclear energy levels in 

𝑆𝑐46  isotopes, with 𝐶𝑎40  serving as the inert core, FPD6, KB3G, and FBY representing the fp shell, 

FZMBZ representing the 1f7/2 subshell orbit, and D3F7COSPN representing the 1d3/2 1f7/2 model 

space. 

2. Materails and Methods 

     A minimum of the total energy describes the ground state of the core and two additional nucleon 

systems. In general, the two extra nucleons are, therefore, in the lowest available single-particle 

orbit and are related to the total spin and isospin value for which 𝐸𝛤
(1)(𝜌2) assumes the smallest 

value. All other states indicate excited states, in which the two particles are connected to distinct 

values or one or both of the additional nucleons are excited into a separate single-particle orbit. As 

shown in eq. (1), the Hamiltonian of the core-plus-two-nucleon system may be divided into two 

parts [30]:  

𝐻 =  𝐻𝑐𝑜𝑟𝑒 +  𝐻12                                                                                                                           (1) 

With [30]  

𝐻𝑐𝑜𝑟𝑒 =  ∑ [𝑇(𝑘) + 𝑈(𝑘)]𝐴
𝑘=3 + [∑ 𝑊(𝑘, 𝑙)𝐴

3=𝑘<𝑙 −  ∑ 𝑈(𝑘)𝐴
𝑘=3 ],                                                 (2) 

𝐻12 =  ∑ [𝑇(𝑘) + 𝑈(𝑘)]𝐴
𝑘=1 + [ ∑ ∑ 𝑊(𝑘, 𝑙) + 𝑊(1, 2)𝐴

𝑙=3 − ∑ 𝑈(𝑘)2
𝑘=1

2
𝑘=1 ]                             (3) 

where 𝐻𝑐𝑜𝑟𝑒 denotes the interaction of the core particles (designated by k=3, ..., A). Assuming the 

closed-shell core is inert, the contribution of H to total energy is constant. H denotes the contribution 

of the two extra particles. It is more precisely expressed as [30].  

𝐻12 =  𝐻12
(0)

+  𝐻12
(1)

.                                                                                                                        (4) 

where 𝐻12
(0)

 Specifies the single-particle Hamiltonian given by [30]  

𝐻12
(0)

= [𝑇(1) + 𝑈(1)] + [𝑇(2) + 𝑈(2)] =  𝐻𝑠.𝑝.(1) + 𝐻𝑠.𝑝.(2)                                                   (5) 

A and 𝐻12
(1)

 denotes the residual interaction and it is given by [30] 

𝐻12
(1)

= [∑ 𝑊(1, 𝑙)𝐴
𝑙=3 − 𝑈(1)] + [∑ 𝑊(2,   𝑙)𝐴

𝑙=3 − 𝑈(2)] + 𝑊(1, 2)                                                        (6) 

If we now take   

𝑈(𝑘) =  ∑ 𝑊(𝑘, 𝑙)𝐴
𝑙=3          For k= 1, 2,                                                                                            (7) 

Only the two-particle term W (1,2) persists after the single-particle terms in the residual interaction 

𝐻12
(1)

 exactly vanish. In other words, it gives the residual interaction 𝐻12
(1)

= 𝑊(1, 2), where W (1,2) 

lacks any words with a single particle. 



IHJPAS. 2024, 37 ( 3 ) 

151 
 

According to the definition in equation (7), which sums up all of the core's particles, the single-

particle states are defined by the core nucleus. The Hartree-Fock theory provides the general 

derivation of a self-consistent single-particle potential U. However, because the later approximation 

is so complicated, it won't be covered here. In the majority of shell-model computations, the 

assumption is made that the Saxon-Woods potential or the simple harmonic oscillator can be used 

to describe the single-particle potential. 𝐻12
(1)

  is no longer equivalent to W(1, 2) in that scenario. It 

is now assumed that the residual interaction, as provided in eq. (6) may still be represented by a 

two-body interaction for such a simplified single-particle potential U. The remaining two-body 

interaction will now be defined by ∑ 𝑉(𝑖, 𝑗)𝑖<𝑗 . Consequently, one has [30] in the scenario of two 

active particles outside of a core: 

𝐻12
(1)

=

𝑉(1, 2                                                                                                                                                                  (8) 

For the entire Hamiltonian, the following can be written from equations (3), (4), (5), and (6): 

𝐻 =  𝐻𝑐𝑜𝑟𝑒 +  𝐻𝑠.𝑝.(1) +  𝐻𝑠.𝑝.(2) + 𝑉(1, 2).                                                                                (9) 

The expected value [30] provides the binding energy of the nucleus when two particles outside the 

core are in orbit and related to spin and isospin :  

𝐸 Γ 
𝑏 (𝐴) =  ⟨𝛷𝛤

(0)
 (1, … , 𝐴)|𝐻|𝛷𝛤

(0)
 (1, … , 𝐴)⟩                                                                               (10) 

As the anti-symmetrized product of the core wave function, Φ00 (core), and the wave function 

𝛷𝛤
(0)

 (1, … , 𝐴) characterizing the additional two nucleons, the total Hamiltonian in the state 

𝛷𝛤
(0)

 (1, … , 𝐴)  of the whole nucleus can be stated as follows [30]: 

𝛷𝛤
(0)

 (1, … , 𝐴) = 𝒜{𝛷00 (core)𝛷𝛤
(0)

 (1, 2)}.                                                                                (11) 

        In the particles 3,..., A and 1, 2, respectively, it is thought that the functions Φ00 (core) and 

𝛷𝛤
(0)

 (1, 2) are antisymmetric. By permuting particle coordinates and selecting suitable linear 

combinations, the antisymmetrizer 𝒜  must finish antisymmetrizing all particles. The right 

outcomes are likewise achieved using the more straightforward product function 

𝛷00 (core)𝛷𝛤
(0)

 (1, 2) for the evaluation of the matrix element in equation (11). Since the whole 

Hamiltonian in equation (11) has been divided into terms that operate on particles 1, 2, or 3, A, one 

can derive the following from the orthonormality of the wave function: 𝛷00 (core) and   𝛷𝛤
(0)

 (1, 2)  

[30]: 

𝛷00 (core) and   𝛷𝛤
(0)

 (1, 2) [14] : 



IHJPAS. 2024, 37 ( 3 ) 

152 
 

⟨𝛷00 (𝑐𝑜𝑟𝑒)𝛷𝛤
(0)

 (1, 2)|𝐻|𝛷00 (core)𝛷𝛤
(0)

 (1, 2)⟩ =  ⟨𝛷00 (𝑐𝑜𝑟𝑒)|𝐻𝑐𝑜𝑟𝑒|𝛷00 (𝑐𝑜𝑟𝑒)⟩ +

 ⟨𝛷𝛤
(0)

 (1, 2)|𝐻𝑠.𝑝.(1) +  𝐻𝑠.𝑝.(2)|𝛷𝛤
(0)

 (1, 2)⟩ +  ⟨𝛷𝛤
(0)

 (1, 2)|𝑉(1, 2)|𝛷𝛤
(0)

 (1, 2)⟩                     (12) 

The following Eqs. [20] provide a definition for the linked two-particle wave function 𝛷𝛤(1. 2) 

[14]: 

Φ𝐽𝑀(𝑗𝑎(1)𝑗𝑏(2)) =  ∑ ⟨𝑗𝑎𝑚𝑎𝑗𝑏𝑚𝑏|𝐽𝑀⟩ ∅𝑗𝑎𝑚𝑎(1)  ∅𝑗𝑏𝑚𝑏(2)𝑚𝑎𝑚𝑏
                                           (13) 

Φ𝐽𝑀
± =  √

1

2
(1 ± 𝑃12)Φ𝐽𝑀(𝑗𝑎(1)𝑗𝑏(2)) = √

1

2
{Φ𝐽𝑀(𝑗𝑎(1)𝑗𝑏(2)) ±

(−1)𝐽−𝑗𝑎−𝑗𝑏 Φ𝐽𝑀(𝑗𝑏(1)𝑗𝑎(2))}                                                                                                     (14) 

One may integrate out the coordinates of particle 2 for 𝐻𝑠.𝑝.(1)   matrix element evaluation in 

equation (12), and one can do the same for 𝐻𝑠.𝑝.(2)   matrix element evaluation in equation (12). 

The matrix elements for single-particle eigen states ρ, each reduce to expectation values of 𝐻𝑠.𝑝.  due 

to the orthonormality of the Clebsch-Gordan coefficients. It is discovered that Eq. (12) and Eq. (12) 

with the single-particle energies provided by [30] are identical. 

2𝑒𝜌 =  ⟨𝛷𝛤
(0)

 (1, 2)|𝐻𝑠.𝑝.(1) +  𝐻𝑠.𝑝.(2) |𝛷𝛤
(0)

 (1, 2)⟩ =  ⟨𝜌2|𝐻12
(0)

|𝜌2⟩
𝛤
                                         (15) 

In [14], the residual interaction is proveded : 

𝐸𝛤
(1)

(𝜌2) =  ⟨𝛷𝛤
(0)

 (1, 2)|𝑉(1, 2)|𝛷𝛤
(0)

 (1, 2)⟩ =  ⟨𝜌2|𝑉(1, 2)|𝜌2⟩𝛤                                                 (16)       

And the core's binding energy, as indicated by [14] 

3. Results  

        For 46 sc, as explained in Figure 1, the system is pure, the experimental data has 30 values, 

the repeated data for the interaction (FPD6, FPY, KB36, F7MBZ, and d3F7COSPN) are good, and 

others have some problems. The interaction (FPD6 and KB3G) has good reproductions and minor 

differences from that of experimental data, and the ground state for this interaction is (4+), the first 

excited 6+, and the second is 3+, and the order of series is accorded with the KB3G interaction and 

F7MBZ up to 1Mev, but the differences between interactions are apparent. The best results are fit 

to KB3G, FPY, and F7MBZ, but the result of FPY is shifted upward, which reveals that the 

interaction in the same model space has changed actions in the value of the study below. The energy 

levels below E = 3 meV are more interesting, while values above these ranges are difficult to 

duplicate because not all of the processes inside the nucleus are represented by the model used in 

our study. The d3F7cosPN interaction failed to 2 reproduce experimental data due to an apparent 

overlap in energy levels, which is less than 200 keV as shown in Figure 2, and it does not resemble 

previous interactions where it has a different model space from other interactions.  



IHJPAS. 2024, 37 ( 3 ) 

153 
 

 

Figure 1. The energy levels system of 𝑆𝑐46  by using FPD6 , FPY , KB3G , F7MBZ and D3F7COSPN  interactions with close 

core 𝐶𝑎40
  and 𝑆32

, positive parity , ten order 

 

 

Figure 2. The density of states scheme of 𝑆𝑐46  per 1MeV 



IHJPAS. 2024, 37 ( 3 ) 

154 
 

4. Discussion 

The terms of the two body matrix elements and the fitting parameter are anticipated by shell 

theory and residual interaction. Depending on the term constituting the interaction and the 

parameters under consideration, the intrusions either succeeded or failed to reproduce the 

experimental data. Some interactions are best fitted to the arrangement of (A), and their fitting 

parameters are normalized on (A-42) terms drawn from real (N-N) interactions and taking into 

consideration the channel of interaction (ST), where the meson particles represent the link between 

interacting nuclei. The density of energy levels, as shown in Figure 2, shows that the states are 

condensed in the range (2–3) MeV and reduced fast for all the interactions, and the value of the 

density of energy levels is different between the interactions. However, all of them indicate that the 

analyzers must be modified in order to obtain the energy level experimentally with a hyperfine 

structure. 

 

5. Conclusion  

     The quality and amount of recreated energy are both improved by Modern's effective interaction. 

In some model spaces and for some isotopes, the best choice of interaction is determined by the 

value of binding energies and separation energies for both neutrons and protons that can be 

reproduced by the chosen interaction, and then the interaction will succeed in model space in 

accordance with shell theory, which is still valid and confident in the reproduced theoretical results. 

 

Acknowledgment 

     All thanks and appreciation to the College of Sciences, University of Baghdad for their valuable 

comments on this article, and thanks go to the journal’s management and members. 

 

Conflict of Interest 

     The authors declare that they have no conflicts of interest. 

 

Funding 

      None. 

 

References 

  

1. Honma, M.; Otsuka, T.; Brown, B.A.; Minzusaki, T. Effective interaction for pf-shell nuclei. 

Physical Review Journals. 2002, 65, 12-23. https://doi.org/10.1103/physrevc .65.061301                   

2. Crannell, H.; Helm, R.; Kendall, H.; Oeser, J.; Yearian, M. Electron-Scattering Study of Nuclear 

Levels in Cobalt, Nickel, Lead, and Bismuth. Physical Review Journals. 1961, 123(3), 923. 

https://doi.org/10.1103/physrev.123.923 .  

3. Akkoyun, S.; Bayram, T.; Böyükata, M. Shell Model Calculations for some pf Shell Nuclei. 

ALKU Journal of Science. 2019, 18(2), 34-45. 

4. Ali, A.M.; Khamees, A.A. Study of the Structure of Exotic 52,54,56,58Ca Isotopes Using 

OXBASH Code. Iraqi Journal of Science. 2019, 60(57), 55-67.  

https://doi.org/10.1103/physrevc%20.65.061301
https://doi.org/10.1103/physrev.123.923


IHJPAS. 2024, 37 ( 3 ) 

155 
 

5. Hameed, B.S.; Rejah, B.K. Study the Nuclear Structure of Some Cobalt Isotopes. Baghdad 

Science Journal. 2022, 19(6), 1566. https://doi.org/10.21123/bsj.2022.753 .  

6. Hasan, A.K.; Kareem, R.M. Calculation of Energy Levels and B(E2) for 𝐶𝑎46−45 , 𝑆𝑐 46−45 and 

𝑇𝑖46−45  Using Shell Model Code OXBASH. International Journal of Scientific and Research 

Publications. 2019, 9, 269. https://doi.org/10.29322/ijsrp.9.08. 2019.p9243.  

7. Akkoyun, S.; Ayhan, Y.; Bayram, T. Nuclear Shell Model Calculations for Ca Isotopes. BEU 

Journal of Science. 2020, 9(3), 1102. https://doi.org/10.17798/bitlisfen.665872.    

8. ŞAHİN, Y.; BÖYÜKATA, M.; Description of even-even Ti isotopes within IBM-1 model. 

Cumhuriyet Science Journal. 2021, 42(1), 177. https://doi.org/10.17776/csj.856118  

9. Obaid, S.M.; Majeed, F.A.; Mohammed, F.E.  Nuclear structure and Gamow-teller b(gt) 

transition strengths for some selected fp-shell nuclei, Karbala International Journal of Modern 

Science. 2022, 8, 522. https://doi.org/10.17776/csj.856118 .   

10. Hasan, A. K.; Salih, A.A. Calculation of Energy Levels and B(E2) for 60Ni and 60Fe Isotopes 

by Using Nuclear Shell Model. National Qualification in Journalism. 2022, 20, 1050. 

https://doi.org/10.14704/nq.2022.20.8.NQ44115 .   

11. Hasan, A.K.; Salih, A.A. Energy Levels and B(E2) Calculation for 50Fe Isotope Using F754 and 

F7mbz Interactions. National Qualification in Journalism. 2022, 20, 115. 

https://doi.org/10.14704/nq.2022.20.5.nq22154 .   

12. Majeed F.Z.; Mashaan, S.S. Energy Levels Calculation for Ca Isotopes Using Different 

Interactions. Indian Journal of Natural Sciences. 2018, 9(50), 12-34.  

13. Hassan, M.K.; Majeed, F.Z.  FP shell effective interactions and nuclear shell structure of  𝑆𝑐44 . 

East European journal of physics.  2023, 1, 89. doi.org/10.26565/2312-4334-2023-1-07 . 

14. Hassan, M.K.; Majeed, F.Z.  Nuclear energy levels in 𝐶𝑎 44  using fpd6pn interaction. East 

European journal of physics. 2023, 1, 69. doi.org/10.26565/2312-4334-2023-1-10. 

15. Hussien, R.M.; Majeed, F.Z. The Effective M3Y Residual Interaction In 𝐶𝑎41  As a Nuclear 

Diffraction Grating of Electrons. Baghdad Science Journal. 2022, 19(6), 1395. 

https://doi.org/10.21123/bsj.2022.6776.  

16. Ali, A.H. Investigation of the Quadrupole Moment and Form Factors of Some Ca Isotopes. 

Baghdad Science Journal. 2020, 17(2), 507.  https://doi.org/10.21123/bsj. 2020. 17.2.0502  

17. Ali, A.M.; Khamees, A.A. Study of the Structure of Exotic 52,54,56,58Ca Isotopes Using 

OXBASH Code. Iraqi Journal of Science. 2019, 60, 60. https://doi.org/10.24996/ijs. 

2019.60.1.8m  

18. Hameed, B.S.; Rejah, B.K. Study the Nuclear Structure of Some Cobalt Isotopes. Baghdad 

Science Journal. 2022, 19, 1566. https://doi.org/10.21123/bsj.2022.7537.  

19. Schielke, S.; Speidel, K.H.; Kenn, O.; Leske, J.; Gemein, N.; Offer, M.; Sharon, Y.Y.; Zamick, 

L.; Gerber, J.; Maier-Komor, P. First measurement and shell model interpretation of the g factor 

of the 21+ state in self-conjugate radioactive 44Ti. Physics Letters B. 2003, 567, 153–158. 

https://doi.org/10.1016/j.physletb.2003.06.027.  

20. Schielke, S.; Hohn, D.; Speidel, K.H.; Kenn, O.; Leske, J.; Gemein, N.; Offer, M.; Gerber, J.; 

Maier-Komor, P.; Zell, O.; Sharon, Y.Y.; Zamick, L. Evidence for 40Ca core excitation from g 

https://doi.org/10.21123/bsj.2022.753
https://doi.org/10.29322/ijsrp.9.08.%202019.p9243
https://doi.org/10.17798/bitlisfen.665872
https://doi.org/10.17776/csj.856118
https://doi.org/10.17776/csj.856118
https://doi.org/10.14704/nq.2022.20.8.NQ44115
https://doi.org/10.14704/nq.2022.20.5.nq22154
https://doi.org/10.26565/2312-4334-2023-1-07
https://doi.org/10.26565/2312-4334-2023-1-10
https://doi.org/10.21123/bsj.2022.6776
https://doi.org/10.21123/bsj.%202020.%2017.2.0502
https://doi.org/10.24996/ijs.%202019.60.1.8m
https://doi.org/10.24996/ijs.%202019.60.1.8m
https://doi.org/10.21123/bsj.2022.7537
https://doi.org/10.1016/j.physletb.2003.06.027


IHJPAS. 2024, 37 ( 3 ) 

156 
 

factor and B(E2) measurements on the 21+ states of 42,44Ca. Physics Letters B. 2003, 571, 29–

35. https://doi.org/10.1016/j.physletb.2003.08.015.  

21. Poves, A.; Sánchez-Solano, J.; Caurier, E.; Nowacki, F. Shell model study of the isobaric chains 

A=50, A=51 and A=52. Nuclear Physics A. 2001, 694, 157–198. https://doi.org/ 10.1016/s0375-

9474(01)00967-8.  

22. Honma, M.; Otsuka, T.; Brown, B.A.; Mizusaki, T. Shell-model description of neutron-rich pf-

shell nuclei with a new effective interaction GXPF 1. The European Physical Journal A. 2005, 

25, 499–502. https://doi.org/10.1140/epjad/i2005-06-032-2. 

23. Dinca, D.C.; Janssens, R.V.F.; Gade, A.; Bazin, D.; Broda, R.; Brown, B.A.; Campbell, C.M.; 

Carpenter, M.P.; Chowdhury, P.; Cook, J.M.; Deacon, A.N.; Fornal, B.; Freeman, S.J.; 

Glasmacher, T.; Honma, M.; Kondev, F.G.; Lecouey, J.-L.; Liddick, S.N.; Mantica, P.F.; 

Mueller, W.F.; Olliver, H.; Otsuka, T.; Terry, J.R.; Tomlin, B.A.; Yoneda, K. Reduced transition 

probabilities to the first2+state inTi52,54,56and development of shell closures at N=32,34. 

Physical Review C. 2005, 71, 12. https://doi.org/10.1103/physrevc .71.041302.  

24. Bertsch, G.F.; Brown, B.A.; Sagawa, H. High-energy reaction cross sections of light nuclei. 

Physical Review C. 1989, 39(2),1154–1157. https://doi.org/10.1103/physrevc .39.1154.  

25. Dean, D.J.; Hjorth-Jensen, M. Pairing in nuclear systems: from neutron stars to finite nuclei. 

Reviews of Modern Physics. 2003, 75, 607–656. https://doi.org/10.1103 /revmodphys.75.607. 

26. Schmid, K.W.; Grümmer, F.; Faessler, A. Complex mean fields and unnatural parity pairing in 

the Hartree-Fock-Bogoliubov problem with symmetry-projection before the variation. Annals of 

Physics. 1987, 180, 1–73. https://doi.org/10.1016/0003-4916(87) 90129-1.  

27. Petrovici, A.; Schmid, K.W.; Faessler, A. Microscopic aspects of shape coexistence in 72Kr and 

74Kr. Nuclear Physics A. 2000, 665, 333–350.https://doi.org/10.1016/s0375-9474(99)00811-8.  

28. Brown, B.A. Neutron Radii in Nuclei and the Neutron Equation of State. Physical Review 

Letters. 2000, 85, 5296–5299. https://doi.org/10.1103/physrevlett.85.5296. 

29. Brown, B.A. OXBASH code, Msunscl Report. 1988, 524(23), 34-56. 

30. Brussaard, P.J.; Glademans, P.W.M. Shell-model Applications in Nuclear Spectroscopy. North-

Holland Publishing Company, Amsterdam. 1977, 12(3), 66-76.  

https://doi.org/10.1016/j.physletb.2003.08.015
https://doi.org/%2010.1016/s0375-9474(01)00967-8
https://doi.org/%2010.1016/s0375-9474(01)00967-8
https://doi.org/10.1140/epjad/i2005-06-032-2
https://doi.org/10.1103/physrevc%20.71.041302
https://doi.org/10.1103/physrevc%20.39.1154
https://doi.org/10.1103%20/revmodphys.75.607
https://doi.org/10.1016/0003-4916(87)%2090129-1
https://doi.org/10.1016/s0375-9474(99)00811-8
https://doi.org/10.1103/physrevlett.85.5296