# Shell Model for Study Quadrupole Transition Rates in B2 in Some Neon Isotopes in sd-shell with Using Different Interactions

^{*}

**Corresponding Author:**Ahmed H Ali, PhD in Theoretical Nuclear Physics and Medical Physics, College of Medicine, University of Fallujah, Iraq, Tel: + 964- 7815262642, Email: [email protected]

*
Received Date: Dec 10, 2017 /
Accepted Date: Apr 23, 2018 /
Published Date: Apr 27, 2018 *

### Abstract

Quadrupole transition rates B(E2) and Bohr-Mottelson effective charges (B-M) were calculated for even-even 18,20,22,24,26,28Ne isotopes based on sd shell model space. One body transition matrix (OBTM) was calculated using the code [email protected] with different interactions. Our calculation for the reduced transition probabilities B(E2) are compared with available experimental data.

**Keywords:**
Quadrupole transition rates; Effective interaction; Shell model; Effective charges

#### Introduction

Solving the nuclear many-body problem is a fundamental task in nuclear structure studies. The spherical shell model has continually been a reliable tool when comparing with experimental observables. In practical shell model calculations, the valence space is limited within one or several adjacent major shells. The bulk of its wave function is presumably contained in this restricted configuration space [1]. Nuclear shell model is one of the most powerful tools for giving a quantitative interpretation to the experimental data. The two main ingredients of any shell model calculations are the N-N interaction and the configuration space for valence particles. In principle one can either perform shell model calculations with realistic N-N interaction in unlimited configuration space or with renormalized effective interaction limited configuration space [2]. Shell model calculations are carried out within a model space in which the nucleons are restricted to occupy a few orbits. If appropriate effective operators are used taking into account the effect of the larger model space, the shell model provides a reasonable description of these observables [3]. The calculations of shell model, carried out within a model space in which the nucleons are restricted to occupy a few orbits are unable to reproduce the measured static moments or transition strengths without scaling factors. Calculations of transition strengths using the model space wave function alone are inadequate for reproducing the data. Therefore, effects out of the model space, which are called core polarization effects, are necessary to be included in the calculations [4]. A study of nuclei in the *sd* shell can thus lead to a better understanding between a microscopic description of the nucleus (shell model) and a macroscopic (collective) description [5]. The sd-shell nuclei are considered as an inert ^{16}O core and the valence nucleons are distributed in 1d_{5/2}, 2s_{1/2} and 1d_{3/2} shell. Higher configurations can be included through perturbation theory, where particle-hole excitations are allowed from the core and the valence nucleons to all allowed orbits with nћω excitations. The number n depends on the convergence of the calculations. The deformation can be investigated experimentally and theoretically, through their electromagnetic transitions. The general trend of the 2^{+} excitation energy* e *( 2_{1}^{+} ) and the reduced electric quadrupole transition strength between the first excited 2^{+} state and the 0^{+} ground state, B (E 2, 0_{1}^{+} →2_{1}^{+} for even-even nuclei are expected to be inversely proportional to one another [6]. States of mixed configurations the situation differs in the valence shell *sd* shell model for N (neutron) > 8 and p (proton) > 8). **Figure 1** indicates how nucleons move via the nucleon–nucleon interaction. The occupancy pattern of nucleons over different orbits is called configuration [7].

#### Theory

The theoretical calculations of the reduced quadrupole transition probability B performed from calculated the reduced electric matrix element between the initial and final nuclear states is [8]:

(1)

where* e *(k) is the electric charge for the k-th nucleon. Since* e *(k)=0 for neutron, there should appear no direct contribution from neutrons; however, this point requires further attention: The addition of a valence neutron will induce polarization of the core into configurations outside the adopted model space. Such core polarization effect is included through perturbation theory which gives effective charges for the proton and neutron. The reduced electric matrix element can be written in terms of the proton and neutron contributions:

(2)

where is the electric matrix element which is expressed as the sum of the products of the one-body transition matrix (OBTM) times the single-particle matrix elements,

(3)

With j and j’ label single-particle states for the shell model space.

The electric matrix element can be represented in terms of only the model space matrix elements by assigning effective charges (e^{eff} (t_{z})) to the neutrons and protons

(4)

They formulated an expression for the effective charges to explicitly include neutron excess via [9]

(5)

The reduced electric transition probability from j_{i} to j_{f} be defined as [8]:

(6)

#### Results and Discussion

The calculation of the reduced electric transition probability B(*E2*) from the ground 0^{+} state to the first excited 2^{+} state for some neon even-even ^{18,20,22,24,26,28}Ne isotopes and which were performed by using equation (6). The one body transition matrix element *(OBTM*) values were obtained by the shell model calculations that performed via the computer code NuShellX [10] MSU and using different interactions such as *USDB* (Universal sd-shell interaction B) [11], USDA interaction (Universal sd-shell interaction A) [11] and Bonn-A interaction [12]. The reduced quadrupole transition probability is calculated using different effective charges such as conventional effective charges (*CEF*) [13], Bohr-Mottelson effective charges (B-M) [9,14] and standard effective charges (ST)* e *_{p}=1.36 and* e *_{n}=0.45 [14,15]. The radial wave functions for the single-particle matrix elements were calculated with the harmonic oscillator (HO) potential with size parameters for each isotope are calculated as

with as shown in **Table 1** [16].

A_{10}Ne |
b(fm) |
(E)_{x}_{theo}.(MeV) |
(E)_{x}_{exp}.(MeV) |
B(E2)_{CEF} |
B(E2)_{ST} |
e_{p}, e_{n (B-M)} |
B(E2)_{B-M} |
B(E2)_{exp} |
---|---|---|---|---|---|---|---|---|

18 | 1.750 | 1.99 | 1.887 | 136 | 149 | 1.24, 0.94 | 123 | 243 ± 16 |

20 | 1.773 | 1.746 | 1.663 | 243.5 | 246.2 | 1.18, 0.82 | 300.6 | 333 ± 16 |

22 | 1.794 | 1.363 | 1.274 | 246.6 | 244.8 | 1.13, 0.72 | 279.1 | 229.8 ± 42 |

24 | 1.814 | 2.111 | 1.981 | 202 | 202.6 | 1.09, 0.63 | 191.4 | 143+57 -24 |

26 | 1.833 | 2.063 | 2.018 | 195.8 | 204 | 1.06, 0.56 | 147 | 155 ± 32 |

28 | 1.850 | 1.623 | 1.30 | 175 | 122.9 | 1.03, 0.50 | 180.2 | 136 ± 32 |

**Table 1:** The reduced electric transition probability B(*E2*) in units of e^{2} fm^{4} and excitation energies for Ne isotopes (Z=10). Experimental Ex and B(*E2*) are taken from Reference [17]. Calculations B(*E2*) using USDB interaction [11] and set effective charges, conventional effective charges (*CEF*) *ep*=1.3 and *en*=0.5 [13], Bohr-Mottelson effective charges *(B-M)* [9], and standard effective charges (ST)*ep*=1.36 and *en*=0.45 [15].

The presented results for B(*E2*) values in this work were compared with the available experimental values give in reference [17].

**USDB Interaction**

Reduced transition probabilities in units of *e ^{2}fm^{4}* are calculated for Neon Ne isotopes (Z=10) with mass number A=18, 20, 22, 24, 26, 28 and with neutron number N=8, 10, 12, 14, 16, 18, respectively. Shell model calculations in

*sd*model space and USDB interaction [11] was used to generate the OBTM elements for the ground state with J=0 and excited state with J=2. The harmonic oscillator size parameter

*b*[16] was calculated for each isotope and tabulated in

**Table 1**. All isotopes in the present work composed of the core

^{16}O nucleus plus two protons surrounding the core. These outer two protons are considered to move in the

*sd*shell model space. The calculated reduced electric transition probability B(E 2; ) using USDB interaction and different effective charges these results of the B(

*E2*)

_{CEF}, B(

*E2*)

_{ST}and B(

*E2*)

_{B-M}are displayed in

**Table 1**and plotted in

**Figure 2a**as a function of neutron number

*N*and mass number A in comparison with the experimental values [17]. The Bohr-Mottelson effective charges (B-M) [9] were calculated for

^{18,20,22,24,26,28}Ne isotopes as shown in

**Table 1**, Conventional effective charges (CEF) [13] which are for proton 1.3

*and for the neutron 0.5*

*e**and standard effective charges (ST) [15] which are 1.36*

*e**e*for the proton and 0.45

*e*for the neutron. The B(

*E2*) were calculated for 18Ne where B(

*E2*)

_{CEF}=136, B(

*E2*)

_{ST}=149, and B(

*E2*)

_{B-M}=123, these values underestimates the measured data (experimental value) 243 ± 16

*e*[17]. The B(

^{2}fm^{4}*E2*) were calculated of 20Ne isotope where B(

*E2*)

_{CEF}=243.5 and B(

*E2*)

_{ST}=246.2, these values underestimate the experimental value 333 ± 16

*e*except the calculated value of B(

^{2}fm^{4}*E2*)

_{B-M}=300.6 is very close to experimental value. The B(

*E2*) were calculated of

^{22}Ne isotope where B(

*E2*)

_{CEF}=246.6 and B(

*E2*)

_{ST}=244.8, these values agree well with the experimental value 229 ± 42

*e*while the calculated value of B(

^{2}fm^{4}*E2*)

_{B-M}=279.1 is very close to experimental value. The B(

*E2*) were

**Figure 2:** Calculated of B (*E2*; 0^{+} → 2^{+}) and excitation energy of eve- even Ne isotopes. The experimental values are taken from Reference [17].

calculated of 24Ne isotope where B(*E2*)_{CEF}=202, B(*E2*)_{ST}=202.6, these values close to experimental value *e ^{2}fm^{4}* while the calculated value of B(

*E2*)

_{B-M}=191.4 agree with the experimental value. Also, B(

*E2*) were calculated for

^{26}Ne isotope where B(

*E2*)

_{B-M}which agree with the experimental value 155 ± 32, while the calculated value of B(

*E2*)

_{CEF}is close to the experimental value and the calculated value of B(

*E2*)

_{ST}overestimate the experimental value. The B(

*E2*) were calculated

^{28}Ne isotope B(

*E2*)

_{CEF}and B(

*E2*)

_{B-M}values close to the experimental value 136 ± 32 while the calculated value of B(

*E2*)

_{ST}agree with the experimental value. The excitation energies were calculated for

^{18,20,22,24,26,28}Ne isotopes and are compare with the experimental values [17] and tabulated in

**Table 1**and plotted in

**Figure 2b**which shows an inverse relation between the excitation energy and transition rate B(

*E2*) [5]. Theoretical values overestimate the experimental values where the excitation energy for some isotopes were high when fill orbit such as N=14 and 16 or isotope has the magic property such as N=8 as shown in

**Figure 2b**while it is decreasing when N=10, 12 and 18.

**USDA Interaction**

Reduced transition probabilities in units of *e ^{2}fm^{4}* are calculated for Neon Ne isotopes (Z=10) with mass number A=18, 20, 22, 24, 26, 28 and with neutron number N=8, 10, 12, 14, 16, 18, respectively. Shell model calculations in

*sd*model space and USDA interaction [11] was used to generate the OBTM elements for the ground state with J=0 and excited state with J=2. The harmonic oscillator size parameter [16] was calculated for each isotope and tabulated in

**Table 1**. All isotopes in the present work composed of the core

^{16}O nucleus plus two protons surrounding the core. These outer two protons are considered to move in the

*sd*shell model space. The calculated reduced electric transition probability B(E 2; ) using USDA interaction and different effective charges these results of the B(

*E2*)

_{CEF}, B(

*E2*)

_{ST}and B(

*E2*)

_{B-M}are displayed in

**Table 2**and plotted in

**Figure 3a**as a function of neutron number N and mass number A in comparison with the experimental values [17]. The Bohr-Mottelson effective charges (B-M) [9] were calculated for

^{18,20,22,24,26,28}Ne isotopes as shown in

**Table 2**, Conventional effective charges (CEF) [13] which are for proton 1.3

*e*and for the neutron 0.5

*e*and standard effective charges (ST) [15] which are 1.36

*e*for the proton and 0.45

*e*for the neutron. The B(

*E2*) were calculated of 18Ne isotope where B(

*E2*)

_{CEF}=136, B(

*E2*)

_{ST}=149, and B(

*E2*)

_{B-M}=123, these values underestimate the experimental value 243 ± 16

*e*[17]. The B(

^{2}fm^{4}*E2*) were calculated of 20Ne isotope where B(

*E2*)

_{CEF}=242 and B(

*E2*)

_{ST}=244.6, these values underestimate the experimental value 333 ± 16

*e*except the calculated value of B(

^{2}fm^{4}*E2*)

_{B-M}=298.7 is close to the experimental value. The B(

*E2*) were calculated of

^{22}Ne isotope where B(

*E2*)

_{CEF}=249.2 and B(

*E2*)

_{ST}=247.7, these values agree with the experimental value 229 ± 42

*e*while the calculated value of B(

^{2}fm^{4}*E2*) B-M=280.4 is close to the experimental value. The B(

*E2*) were calculated of 24Ne isotope where B(

*E2*)

_{CEF}=193, B(

*E2*)

_{ST}=193.2 and B(

*E2*)

_{B-M}=184, These values agree to the experimental value 143

_{−24}

^{+57}

*e*. Also, B(

^{2}fm^{4}*E2*) were calculated for

^{26}Ne isotope where B(

*E2*)

_{CEF}and B(

*E2*)

_{ST}, these values overestimate the experimental value 155 ± 32 while the calculated value of B(

*E2*)

_{B-M}agrees very well with the experimental value. The B(

*E2*) were calculated for

^{28}Ne isotope where B(

*E2*)

_{CEF}and B(

*E2*)

_{ST}values close to the experimental value 136 ± 32 while the calculated value of B(

*E2*)

_{B-M}agree with the experimental value. The excitation energies were calculated for

^{18,20,22,24,26,28}Ne isotopes and are compare with experimental values and tabulated in

**Table 2**and plotted in

**Figure 3b**which shows agreement theoretical values with experimental values except the excitation energies of

^{18,24,28}Ne isotopes. For magic number N=8, the B(

*E2*) value is lower than those of N ≤ 18, which corresponds to a maximum value of the excitation energy. The excitation energy is decreasing when N=12 to become minimum. The excitation energies will increase for

^{24,26}Ne when N=14, 16 to become maximum because the neutrons in

^{24}Ne fill the 0d

_{5/2}orbit and in

^{26}Ne fill the 1s

_{1/2}orbit. The excitation energy is decrease when N=18 to become minimum because neutrons in

^{28}Ne not fill 0d

_{3/2 }orbit. There are Similarities in the behavior of the excitation energies with USDB interaction and of the excitation energies with USDA interaction as shown in

**Figure 2b**.

A_{10}Ne |
b(fm) |
(E)_{x}_{theo}.(MeV) |
(E)_{x}_{exp}.(MeV) |
B(E2)_{CEF} |
B(E2)_{ST} |
ep , e_{n}_{B-M} |
B(E2)_{B-M} |
B(E2)_{exp} |
---|---|---|---|---|---|---|---|---|

18 | 1.750 | 2.023 | 1.887 | 136 | 149 | 1.24, 0.94 | 123 | 243 ± 16 |

20 | 1.773 | 1.696 | 1.663 | 242 | 244.6 | 1.18, 0.82 | 298.7 | 333 ± 16 |

22 | 1.794 | 1.310 | 1.274 | 249.2 | 247.7 | 1.13, 0.72 | 280.4 | 229.8 ± 42 |

24 | 1.814 | 2.181 | 1.981 | 193 | 193.2 | 1.09, 0.63 | 184 | 143+57 -24 |

26 | 1.833 | 2.086 | 2.018 | 195 | 203 | 1.06, 0.56 | 146 | 155 ± 32 |

28 | 1.850 | 1.645 | 1.30 | 175.8 | 181.4 | 1.03, 0.50 | 122.9 | 136 ± 32 |

**Table 2:** The reduced electric transition probability *B*(E2) in units of *e*^{2} *fm*^{4} and excitation energies for Ne isotopes (*Z=*10). Experimental *E*_{x} and B(*E*2) are taken from Reference [17]. Calculations B(E2) using USDAinteraction [11] and set effective charges, conventional effective charges (*CEF*) *ep*=1.3 and *en*=0.5 [13], Bohr-Mottelson effective charges (*B-M*) [9], and standard effective charges (*ST*)*ep*=1.36 and *en*=0.45 [15].

**Figure 3:** Calculated of B (*E2*; 0^{+} → 2^{+}) and excitation energy of eve- even Ne isotopes. The experimental values are taken from Reference [17].

**SDBA interaction**

Reduced transition probabilities in units of *e ^{2}fm^{4}* are calculated for Neon Ne isotopes (Z=10) with mass number A=18, 20, 22, 24, 26, 28 and with neutron number N=8, 10, 12, 14, 16, 18, respectively. Shell model calculations in

*sd*model space and SDBA interaction [12] was used to generate the

*OBTM*elements for the ground state with J=0 and excited state with J=2. The harmonic oscillator size parameter [16] was calculated for each isotope and tabulated in

**Table 1**. All isotopes in the present work composed of the core

^{16}O nucleus plus two protons surrounding the core. These outer two protons are considered to move in the

*sd*shell model space. The calculated reduced electric transition probability B(

*E 2*; 143

_{−24}

^{+57}) using SDBA interaction and different effective charges these results of the B(

*E2*)

_{CEF}, B(

*E2*)

_{ST}and B(

*E2*)

_{B-M}are displayed in

**Table 3**and plotted in

**Figure 4a**as a function of neutron number N and mass number A in comparison with the experimental values [17]. The Bohr-Mottelson effective charges (B-M) [9] were calculated for

^{18,20,22,24,26,28}Ne isotopes as shown in

**Table 3**, Conventional effective charges (CEF) [13] which are for proton 1.3

*e*and for the neutron 0.5

*e*and standard effective charges (ST) [15] which are 1.36

*e*for the proton and 0.45

*e*for the neutron. The B(

*E2*) were calculated for 18Ne isotope where B(

*E2*)

_{CEF}=137, B(

*E2*)

_{ST}=147, and B(

*E2*)

_{B-M}=122.4, these values underestimate the measured data (experimental value) 243 ± 16

*e*[17]. The B(

^{2}fm^{4}*E2*) were calculated for 20Ne isotope where B(

*E2*)

_{CEF}=247.5 and B(

*E2*)

_{ST}=250, these values underestimate the experimental value 333 ± 16

*e*except the calculated value of B(

^{2}fm^{4}*E2*)

_{B-M}=305.5 is close to experimental value. The B(

*E2*) were calculated for

^{22}Ne isotope where B(

*E2*)

_{CEF}=242 and B(

*E2*)

_{ST}=244, these values agree with the experimental value 229 ± 42

*e*while the calculated value of B(

^{2}fm^{4}*E2*)

_{B-M}=280 is close to experimental value. The B(

*E2*) were calculated for 24Ne isotope B(

*E2*)

_{CEF}=234, B(

*E2*)

_{ST}=234.3 and B(

*E2*) B-M=242, these values overestimate the experimental value 57 24 143+−

*e*. Also, B(

^{2}fm^{4}*E2*) were calculated for

^{26}Ne isotope where B(

*E2*)

_{CEF}and B(

*E2*)

_{ST}, these values overestimate the experimental value 155 ± 32 while the calculated value of B(

*E2*)

_{B-M}, is close to the experimental value. The B(

*E2*) were calculated for

^{28}Ne isotope where B(

*E2*)

_{CEF}and B(

*E2*)

_{ST}, these values overestimate the experimental value 136 ± 32 while the calculated value of B(

*E2*)

_{B-M}agree with the experimental value. The excitation energies were calculated for

^{18,20,22,24,26,28}Ne isotopes and are compare with experimental values and tabulated in

**Table 3**and plotted in plotted in

**Figure 4b**which shows the theoretical values agree to the experimental values, except the excitation energies of

^{18,24,28}Ne isotopes. For magic number N=8, the B(

*E2*) value is lower, which corresponds to a maximum value of the excitation energy. The excitation energy is decreasing when N=12 to become minimum and the excitation energy will increase when N=14, 16 because the neutrons in 24,26Ne isotopes fill the 0d

_{5/2}orbit and the 1s

_{1/2}orbit, respectively. The excitation energy is decrease when N=18 because the neutrons not filled the d

_{3/2}orbit as shown in

**Figure 4b**[18].

**Figure 4:** Calculated of B (*E2*; 0^{+} → 2^{+}) and excitation energy of eve- even Ne isotopes. The experimental values are taken from reference [17].

#### Conclusion

Shell model was adopted to calculate transition rates B(*E2*) of eveneven Ne (18, 20, 22, 24, 26, and 28) isotopes including core-polarization effects through taken effective charges. Calculations B(*E2*) with USDB interaction are better when using Bohr-Mottelson effective charges (BM). Our results showed a decrease in the transition rates B(*E2*) from the experimental value at the magic number N=8 while increasing the excitation energy. Calculations excitation energy for even-even Ne isotopes which adopted on USDA interaction is best from other interactions in the present work. Excitation energy increasing for isotope which has magic number also when neutrons are fill orbit. Excitation energy is inversely proportional to transition rates B(*E2*) . In SDBA interaction, results of the B(*E2*) are approximatly similar when using standard effective charges or conventional effective charges. Results of the B(*E2*) by using USDA interaction with B-M effective charges are better.

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Citation: Ali AH (2018) Shell Model for Study Quadrupole Transition Rates in B2 in Some Neon Isotopes in sd-shell with Using Different Interactions. J Astrophys Aerospace Technol 6:160. DOI: 10.4172/2329-6542.1000160

Copyright: © 2018 Ali AH. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.