Study of Nuclear Structure of 24,26 Na isotopes by using USDB interaction

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Study of Nuclear Structure of

24,26

Na isotopes by using

USDB interaction

Ali .K. Hasan1 and Azhar.N.Rahim2

Department of Physics, College of Education for Girls, University of Kufa, Iraq [email protected] and [email protected] Abstract:

In this article, we assess the accuracy of theoretical shell model in calculating the excited states of Sodium isotopes24,26Na on the

basis of recently reported experimental results. The assessments rely on the calculations of the energy levels, reduced electric quadrupole transition probabilities B(E2) and reduced magnetic dipole transition probabilities B(M1) are based on OXBASH shell model code by applying USDB interaction. Applying the program for above isotopes using the defined codes introduces several files which each file contains a set of data. Mean while the ground state of excitation energy evaluated by OXBASH code together with energy levels and reduced electric quadruple transition probability B(E2)and Magnetic dipole transition probability B(M1) and also probable places for nucleons’ placements in each energy level. A compilation of SD-shell energy levels calculated with the USD Hamiltonian and has been published around 1988.A comparison had been made between our results and the available experimental data to test theoretical shell model description of nuclear structure in Sodium isotopes. The calculated energy spectrum is in good agreement with the available experimental data.

1.Introduction:

Obtaining the nuclear structure and energy levels of nuclei is one of the criteria to improve investigations of nuclei properties. Nuclear models have the property to help us to better understanding of nuclear structure which contains main physical properties of nuclei, and shell-model is one of the most prominent and successful nuclear models. This model can be compared with the electron shell model for atoms. As atomic behavior and properties can be described with valance electrons which exist out of a closed shell, similarly, valance nucleons (protons or neutrons) in a nucleus which are placed out of close shells (with magic numbers 2,8,20,28,50,82 and 126) play important roles in determining nuclear properties. Nuclei with magic numbers are very stable and have completely different properties comparing with their neighbors[1].The nuclear shell model has been very successful in our understanding of nuclear structure: once a suitable effective interaction is found, the shell model can predict various observables accurately and systematically. For light nuclei, there are several “standard” effective interactions such as the Cohen- Kurath and the USD interactions for the p and SD shells, respectively. Analysis of neutron-rich SD nuclei has been of intense curiosity in recent years as they present new aspects of nuclear structure [2].Traditional shell-model studies have recently received a renewed interest through large scale shell-model computing in no-core calculations for light nuclei .Because of the quite importance of the( 0d5\2,1s1\2,0d3\2) space for variety of problems in nuclear structure, this space is a region where the shell model can play an indispensable role and is at the frontier of our computational abilities[3].The shell model calculations of the neutron-rich Sodium isotopes have been developed using the OXBASH code[4].

2.Theory:

The nuclear shell model, introduced almost 50 years ago by Mayer Haxel, Jensen, and Suess, has been very successful in describing the properties of nuclei with few valence nucleons[5]. These properties include the energy levels, magnetic and quadruple moments, electromagnetic transition probabilities, beta decay, and cross section for various reactions. The basic assumption of the nuclear shell model is that, to a first approximation each nucleon moves independently in a potential that represents the average interaction with the other nucleons in a nucleus. The complete Schrodinger equation for A nucleons reads as [6].

In the realistic shell model, we have to take into account( H ) this part of the nuclear Hamiltonian that was omitted in the

mean-field description. Nucleon configurations are mixed by this residual interaction. Interactions between nucleons make them

jump from one orbital to another with conserve (𝑇,𝐽𝜋), so that the wave function contains several configurations. So, we should

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𝐻|𝛹𝑖⟩=𝐸𝑖|𝛹𝑖⟩

Configuration mixing leads to the wave functions to consist of more than just one Slater determinant. So, we are looking for the wave function of the system in the form.

|𝛹𝑖⟩=� 𝑎𝑘𝑖|Φk⟩ ,𝑖= 1,2, … . .𝑛 n

𝑘=1

where g the number of pure configurations considered and it is related to the valance space used, and ak is

amplitude(weight) the wave function|Φ_k⟩Usually, the valance space incorporates all possible configurations of valence protons

and valence neutrons in the partially filled orbitals, while the rest is considered as an inert core(usually, we take a double magic numbers). So we treat only the valance nucleons. This theory efficient for few numbers of valance nucleons(smaller than five valance nucleons). It is clear that the valance space becomes quickly huge for numerical treatment as the number of valance nucleons increases [8].

There are many programs that implement the calculations of the shell model, which differ in their methods of calculation as well as the language in which they were built, and differ in the systems that work on them and their speed in the completion of calculations, including OXBASH. And to run the calculations of the shell model using OXBASH. It was necessary to know how to install the program and how to operate it to ensure the accuracy of calculations and results. In order to calculate the nuclear structure properties of both ground and excited states based on the nuclear shell model one needs to have wave functions of those states. These wave functions are obtained by using the shell-model code OXBASH [9]. OXBASH (Oxford- Buenos Aires Shell Model Code) is a powerful computer code to calculate the energy levels, reduced electric quadrupole transition probabilities B(E2)and Magnetic quadrilateral transition probability B(M1)of light and medium nuclei. By using it, we can measure the energy levels of the nucleus and compare it with experimental data as well [10].

3.Shell model calculations:

The calculations have been conducted using the code OXBASH for Windows. The code uses an m-scheme Slater determinant

basis. Using a projection technique, wave functions with good angular momentum J and isospin T are constructed. The effective

interactions of the USDB Hamiltonians with SD model space. The SD model spaces consists of 0d5/2, 1s1/2, and 0d3/2 above the Z

= 8 and N = 8 closed shells for protons and neutrons, respectively .single-particle energies (SPEs) for every Hamiltonian used in

this work (MeV).Use in this work USDB interaction and single-particle energies (SPEs) are {0d5/2=-3.926,1s1/2=-3.208and

0d3/2=2.112} respectively [11].

4.Results and Discussion:

Shell model calculations for low lying energy states of 24,26 Na isotopes have been performed for the space model( 0d5/2, 1s1/2, and

0d3/2) with neutrons(N=13 and 15) above the 16O close core for above isotopes. The calculations are based on the

Universal(sd-shell) Hamiltonian (USDB) .we have used the OXBASH code in both m-scheme and jj-coupling .The object of this present study is to calculate energy levels and reduced electric quadrupole transition probabilities B(E2)and Magnetic quadrilateral transition probability B(M1) by employing harmonic oscillator potential (HO, b), b<0 all isotopes. The effects of core polarization have been taken into account in the calculations by effective charges of both protons and neutrons.

4.1 Energy Levels:

4.1.1.24Na nucleus:

According to the shell model, the ground state of 24Na nucleus is a closed 16O core with eight nucleons distributed as three

protons and five neutron in sd space, which is similar for other Na isotopes (13 ≤ N ≤ 11) for the closed core and proton distribution .Excited states are formed by the configuration of these nucleons in the sd-shell model space .Table (1) show the

comparison between theoretical and available experimental data of 24Na nucleus[12] by using the USDB interaction.

From the down table both USDB Hamiltonians agree reasonable well with empirical data compared with the 24Na energy levels

with the experimental values. The ground state was confirmed 41+.The agreement is good for the states of with empirical values.

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17+,18+and 05+levels confirmed as positive parity. This study also confirmed the Jπ =(24+),(43+), (25+)and 26+ levels with the calculated energies (2.787, 3.001, 3.604 and 4.045) MeV, respectively. The levels of for which the angular momentum and parity are yet unknown Jπ=33+,15+,54+,27+,29+,71+,62+,02+,48+,19+,57+,410+,66+,58+,72+,59+,510+,67+,73+,68+,74+,81+,06+,

82+,76+,77+,83+,84+,91+,86+,92+and 87+, respectively. New energy levels were predicted at the states

(Jπ=88+،94+،89+،102+،95+،810+،96+،97+،98+،103+،104+،111+،99+،910+،،105+،106+،112+،107+،108+،109+،113+،1010+،114+،115+،121+،116+،117+، 118+،119+،122+،1110+and 123+ ).

Table 1: Comparison of the experimental excitation energies[13] and excitation energies predictions for 24Na nucleus by using USDB interactions

J π(exp.) Energy (exp.)

(MeV) Energy (OXBASH)

USDA (MeV) Jπ (OXBASH)

4+ 0.000

0.000 41+

1+ 0.472

0.540 11+

2+ 0.563

0.629 21+

2+ 1.341

1.107 22+

(3)+ 1.344

1.338 31+

1+ 1.346

1.324 12+

5+ 1.512

1.546 51+

2+ 1.846

1.807 23+

3+ 1.885

1.803 32+

---1.960

2.348 33+

3+ 2.513

2.627 34+

4+ 2.562

2.649 42+

(2+ ,3+) 2.977

2.787 24+

(4+,2+) 3.216

3.001 43+

1+ 3.413

3.346 13+

1+ 3.589

3.621 14+

3+ 3.628

3.459 35+

(2+ , 1+) 3.655

3.604 25+

---3.527 01+

---3.707 44+

---3.860 52+

---4.003 61+

---3.936

3.844 36+

---4.112 53+

(1-,2+) 3.977

4.045 26+

---4.220

4.366 +

15

3-4.526

4.679 37+

---4.690

4.721 46+

---4.772

4.782 54+

---4.897 55+

---4.781 38+

---4.891

4.908 27+

2+,3,4+ 5.031

5.020 28+

(1,2,3)-5.045

4.816 +

16

---5.180

5.323 29+

---5.308

5.313 71+

(1,3)-5.397

5.184 +

17

---5.408

5.399 +

62

---5.340 47+

1-5.479

5.659 +

62

---5.571

5.567 02+

---5.585

5.556 63+

---5.720

5.820 03+

---5.896

5.979 48+

1-,3+ 5.918

5.904 310+

---6.530 64+

---5.953

5.995 56+

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---6.183

6.192 19+

---6.222

6.139 57+

---6.724 110+

---6.794 65+

---6.578

6.586 410+

---6.715

7.442 66+

---6.787

6.638 58+

---6.770 04+

---6.846

6.695 72+

---7.090

7.236 59+

0-7.085

7.103 05+

---7.324

7.483 510+

---7.433

7.442 67+

---7.511

7.497 73+

---7.708

7.826 68+

---7.832

7.843 74+

---8.565 75+

---8.390

8.482 81+

---8.610

8.554 06+

---8.860

8.722 69+

---8.772 82+

---9.008 76+

---9.022 77+

---9.065 610+

---9.280

9.094 07+

---9.630

9.870 83+

---10.031 78+

---10.360 08+

---10.342 79+

---10.519 09+

---10.724 84+

---10.583 710+

---10.966 85+

---10.845 010+

---11.610

11.762 86+

---12.190

12.185 91+

---11.900

12.953 92+

---13.396 93+

---12.540

12.592 87+

---12.611 88+

---14.097 94+

---12.997 89+

---16.507 102+

---14.606 95+

---13.207 810+

---14.097 96+

---14.870 97+

---15.161 98+

---15.014 103+

---16.092 104+

---17.019 111+

---15.161 99+

---15.748 910+

---17.375 105+

---17.577 106+

---18.461 112+

---18.136 107+

---18.485 108+

---19.287 109+

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---19.482 1010+

---20.540 114+

---21.396 115+

---22.031 121+

---22.675 116+

---23.884 117+

---24.330 118+

---24.577 119+

---25.057 122+

---25.211 1110+

---29.988 123+

4.1.2. 26Na nucleus:

The ground state of 26Na is nucleus is a close 16O core plus six nucleons distributed as three protons and three neutron in sd space

at0d5/2, 1s1/2, and 0d3/2configurations. At table (2) show the comparison between theoretical and available experimental data of

Na16 nucleus[12] by using the USDB interaction.

From the Down table both USDB Hamiltonians agree reasonable well with empirical data compared with the 26Na energy levels

with the experimental values. The ground state was confirmed 31+. The agreement is good for the states of with empirical values.

Jπ=11+,21+and 22+ as compared with the experimental data, respectively. The Jπ = 39+ levels confirmed as positive parity. This study also confirmed the Jπ=(12+),(32+), (41+),(51+),(24+),(25+),(47+)and 210+ levels with the calculated energies (1.281,1.708,1.628,2.977,2.236 ,3.065, 4.683 and 5.303) MeV, respectively. When compared with the process values. The levels of for which the angular momentum and parity are yet unknown at the states Jπ=01+،42+،13+،14+،33+،02+،44+،34+،35+،16+،45+،28+،17+،19+،65+and 08+ ,respectively.New energy levels were predicted at the states Jπ=74+،75+،09+،010+،76+،77+،78+،82+،91+،79+،83+،710+،84+،85+،86+،87+،88+،92+،89+،810+،93+،101+،94+،102+،95+،96+،97+،98+،103+،99+،910+،10

4+،105+،111+،106+،107+،108+،109+،112+،1010+،113+،114+،115+،116+،121+،117+،118+،119+،122+،1110+،123+,respectively.

Table 2: Comparison of the experimental excitation energies[14] and excitation energies predictions for 26Na nucleus by using USDB interactions

J π(exp.) Energy (exp.)

(MeV) Energy (OXBASH)

USDA (MeV) Jπ (OXBASH)

3+ 0.000

0.000 31+

1+ 0.082

0.004 11+

2+ 0.232

0.108 21+

2+ 0.406

0.325 22+

(1+) 1.509

1.281 12+

--- 1.660

1.628 41+

(3+) 1.808

1.708 32+

---1.740 01+

(4+) 1.996

1.988 42+

2.045 2.059

23+

(5+) 2.118

2.224 51+

(2+) 2.192

2.236 24+

--- 2.452

2.450 13+

(1+) 2.720

2.677 14+

--- 2.803

2.726 33+

---2.977 52+

---2.998 43+

(2+) 3.222

3.065 25+

---3.304

3.231 02+

---3.417

3.434 44+

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---3.542 26+

---3.603

3.644 34+

---3.814

3.791 35+

---3.966

3.985 45+

---

4.086 16+

---4.188

4.243 36+

---4.269 27+

---4.429 37+

---4.440

4.433 28+

---

4.461 46+

---

4.525 29+

---4.591 61+

---4.683 47+

---4.702

4.703 53+

---4.851 17+

---4.872 38+

---4.873 03+

---4.938 39+

---4.915

4.960 54+

---5.117 62+

(4+) 4.970

5.202 48+

---5.213 18+

---5.243 55+

(2+) 5.080

5.303 210+

--- 5.480

5.349 310+

---5.634 19+

---5.715 63+

---5.724 110+

---5.791 49+

---5.914 71+

---5.940 56+

---6.162 410+

---6.181 04+

---6.257 57+

---6.414 58+

---6.578 59+

---6.583 64+

---6.793 510+

---6.999 72+

---7.009 65+

--- 7.200

7.034 05+

---7.522 66+

---7.802 67+

---7.847 06+

---8.134 68+

---8.150 07+

---8.202 69+

---8.347 73+

---8.513 81+

---8.660 610+

---8.846 74+

--- 9.000

9.060 08+

---9.068 75+

---9.207 09+

---9.357 76+

---9.693 010+

---9.726 77+

---9.859 91+

---9.990 78+

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---10.343 79+

---10.446 83+

---10.581 710+

---11.104 84+

---11.441 85+

---11.549 86+

---11.790 87+

---12.415 92+

---12.428 88+

---12.738 89+

---12.738 93+

---12.839 810+

---13.252 94+

---13.456 101+

---13.605 95+

---14.287 96+

---14.824 97+

---15.222 98+

---15.441 102+

---15.844 99+

---15.886 910+

---16.217 103+

---16.892 104+

---17.297 111+

---17.888 105+

---18.340 106+

---18.565 107+

---19.050 112+

---19.097 108+

---19.293 106+

---19.531 107+

---19.582 112+

---21.628 113+

---22.932 114+

---23.594 115+

---23.669 116+

---23.758 121+

---24.865 117+

---25.336 116+

---25.736 122+

---26.025 118+

---31.664 123+

4.2 : B(E2) and B(M1)

Transition rates are a sensitive indicator for most modern effective interactions developed to describe the sd-shell region. This sensitivity resulting from the adoption of transition rates on the single particle wave function (Hamiltonian eigenvectors).In this

section, the theoretical and experimental reduced electric quadrupole transition probability B(E2) (in units of e2fm4) and reduced

magnetic dipole transition probability B(M1) (in units of μ2, μ Bohr magneto) values for 24,26

Na isotope[15]. The comparison between theoretical and experimental B(E2) shows an advantage for USDB calculations for many states. The reduced magnetic dipole transition probabilities B(M1) results gave a clear advantage to the USDB calculations compared to the other Hamiltonians results. The transition strengths calculated in this work performed using the harmonic oscillator potential HO for each in-band transition by assuming pure B(E2)transition. Core polarization effect were included by choosing the effective charges for proton

and for neutron ep = en = 0.350e. We also calculated magnetic quadrilateral transition probability B(M1),Values of effective

charge are (ep =en = 0.350e) and the free nucleon g factors are gs(p) = 5.586, gs(n) =−3.826, gl(p) =1and gl(n) = 0. New electric

and magnetic{ B(E2),B(M1),} transitions were expected in Our results by using (USDB) interaction are listed in Tables(3and4)

for 24Na nucleus and Tables(5and6) for 26Na nucleus , transition probabilities gives agreement comparing with experimental

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Table 3.Comparison of the B(E2) results in unit e2 fm4 for 24Na nucleus with the experimental data[13].

B (E2; ↓) Exp. Results(e2fm4) B (E2)our Results for USDB

(e2fm4)(ep=0.350, en=0.350) Ji→Jf

---17.71

21+→41+

---9.963

22+→41+

1.685 1.263

31+→41+

---40.84

51+→41+

---6.479

32+→41+

---3.778

42+→41+

---1.618

52+→41+

---7.237

61+→41+

---3.444

62+→41+

---33.81

21+→11+

---10.65

22+→11+

4.934 3.275

23+→11+

---13.38

31+→11+

---5.527

32+→11+

---17.67

22+→21+

1.233 30.73

31+→21+

2.878 2.929

32+→21+

---19.93

42+→21+

---12.59

01+→21+

---0.9825

02+→21+

---0.5084

12+→31+

---2.590

32+→31+

---14.27

51+→31+

---19.24

42+→31+

---0.4727

32+→51+

---2.732

42+→51+

---2.530

52+→51+

---40.85

61+→51+

---10.20

71+→51+

---0.5711

62+→51+

---26.25

71+→61+

---0.9019

62+→61+

---8.791

72+→61+

---16.21

81+→61+

---1.142

82+→61+

---0.6661

62+→71+

---6.243

72+→71+

---15.35

81+→71+

---18.65

82+→71+

Table4. Comparison of the B(M1) results in unit µ2for 24Na nucleus with the experimental data[13].

Exp. results B(M1) Cal. Results

USDB Ji→Jf

0.3401 0.2697

31+→41+

---0.3228

51+→41+

---0.06733

42+→41+

---1.135

21+→11+

---0.6632

22+→11+

0.020406 0.006045

23+→11+

---0.1770

31+→22+

---0.03513

32+→31+

---0.8562

42+→31+

---0.2025

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---0.7856

02+→11+

---0.2719

01+→12+

---0.0002657

02+→12+

---0.3917

42+→51+

---0.2873

61+→51+

---0.02435

62+→61+

---0.3229

71+→61+

---0.02950

72+→61+

---0.0002075

62+→71+

---0.01240

72+→71+

---0.001071

81+→71+

---0.006340

82+→71+

0.4654 0.7838

32+→21+

0.003938

0.002568×10-3

33+→41+

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Table5.Comparison of the B (E2)results in unit e2 fm4 for 26Na nucleus with the experimental data

B (E2)Exp. Results (e2fm4) [16] B (E2)our . Results(e2fm4)

en=0.350 , ep=0.350 Ji→Jf

16.47 19.05

11+→31+

50.325 20.02

21+→31+

13.725 14.05

22+→31+

---18.11

12+→31+

---3.897

32+→31+

---4.745

42+→31+

---2.317

41+→31+

---23.66

51+→31+

---2.499

52+→31+

---5.662

21+→11+

86.925 48.58

32+→11+

---2.121

12+→11+

---10.69

32+→11+

2.287 25.01

22+→11+

---2.342

22+→21+

---0.1018

12+→21+

---25.04

01+→21+

---14.00

32+→21+

---8.920

41+→21+

---14.14

42+→21+

---16.17

12+→22+

---0.6580

01+→22+

---0.7810

32+→22+

---0.1757

41+→32+

---22.25

42+→32+

---1.683

02+→22+

---1.778

51+→32+

---0.8589

52+→32+

---0.3302

42+→41+

---0.9571

51+→41+

---50.15

52+→41+

---13.23

61+→41+

---0.2773

62+→41+

---2.499

52+→51+

---0.3603

61+→51+

---2.326

62+→51+

---23.08

71+→51+

---6.905

72+→51+

---0.02135

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---27.73

71+→61+

---5.222

72+→61+

---30.70

81+→61+

---0.1888

82+→71+

---0.1721

82+→61+

---16.38

81+→71+

---2.497

72+→71+

---25.83

91+→71+

---0.01602

92+→71+

---2.309

82+→81+

---18.54

91+→81+

---1.897

92+→81+

Table6. Comparison of theB(M1) results in unit µ2for 26Na nucleus with the experimental data

B(M1) Exp. Results[16] B(M1) our. Results

Ji→Jf

0.00179 0.1531

21+→31+

0.002685 0.6216

22+→31+

---0.0008276

41+→31+

---0.1328

32+→31+

---0.2971

42+→31+

0.00537 0.08654

21+→11+

0.000716 0.4041

22+→11+

---0.4252

01+→11+

---0.3141

02+→11+

---0.3470

22+→21+

---0.02708

32+→22+

---0.02708

32+→21+

---0.004789

42+→41+

---0.05102

12+→11+

---0.08585

51+→41+

---0.05243

52+→51+

---0.001111

61+→51+

---0.1046

62+→61+

---0.04680

71+→61+

---0.2052

72+→71+

---0.06191

81+→71+

---0.01022

82+→71+

5. Conclusions

Full sd-space shell model calculations were performed using the code OXBASH for Windows. The SD model space are employed with the effective interactions(USDB) to reproduce the level spectra , reduced electric quadrupole transition probability B(E2)

and magnetic quadrilateral transition probability B(M1)for the nuclei 24,26Na. Good agreement were obtained by comparing these

calculations with the recently available experimental data for the level spectra using USDB effective interaction. Calculation of the transition strengths prove that USDB is more consistent in for the(sd-shell) region.

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