1. Allowed Gamow-Teller (GT) strengths and β+-decay rates for Odd-A nuclei
Jameel-Un Nabi
Ghulam Ishaq Khan Institute Of Engineering Sciences and Technology
Pakistan
New Aspects of the Hadron and Astro/Nuclear Physics
National University of Uzbekistan
(5-10 November, 2018)

2. Collaboration This work was done in collaboration with;
Dr. Muhammad Majid
And
Mr. Muhammad Riaz
Ghulam Ishaq Khan Institute Of Engineering Sciences and Technology
Pakistan.

3. Outlines Motivation of current work Stellar evolution
Weak-decay processes
Formalism used in the calculations of weak rates
pn-QRPA model description
Ikeda Sum Rule
Comparison of calculated rates with measured data and previous calculations.
Conclusions.

4. Motivation of Current Work
In a new study by Cole et al. [1], it was concluded that quasi-particle random phase approximation (QRPA) calculations show large deviations and overestimate the total experimental Gamow–Teller (GT) strength for odd-A nuclei. It was also concluded that QRPA calculated electron capture rates exhibit larger deviation than those derived from the measured GT strength distributions. The main purpose of this study is to probe the findings of the Cole et al. work. This study gives useful information on the performance of QRPA-based nuclear models.
Our findings show that this is not the case for all kind of QRPA calculations. [1] A. L. Cole et al., Phys. Rev. C 86 (2012)

5. The Life Cycle of a High Mass Star
Stars are born, live and die. They have a beginning, evolutionary phases and end.
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6. Development of a Heavy Mass Star of (10-25) Solar Masses
Main sequence
star
Red super giant
Advanced burning phases
Hydrostatic life of the star
There are two characteristic phases of a star development
At the beginning we imagine a collection of gas particles. They feel a self-gravity which tends to make the gas collapse. This is balanced by the thermal pressure due to motions of particles in the gas.
Gravitational collapse
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pogge/Ast162/Unit2/structure.html
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7. Pre-Supernova Stage 4/9/2019
At pre-supernova stage the star shows approximately a shell structure composed of a nickel-iron core surrounded by layers of burning silicon, neon, oxygen, carbon, and helium, and a vast mantle of hydrogen.
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8. Supernova Catastrophic Collapse ………..Supernova
Stellar Core becomes so heavy
Electron degeneracy pressure support
Gravitational force increases
If the Core mass is ≥ 1.4 Mo
The core is so heavy that it cannot withstand its own gravitational force.
As the core contracts and heats up electron degeneracy pressure becomes important but is only sufficient to halt the collapse if the core mass is less than 1.4 Solar Masses (called the Chandrasekhar Mass).
The electron degeneracy pressure support is lost due to electron capture and the catastrophic collapse begins, known as Supernova explosion.
Catastrophic Collapse
………..Supernova
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9. Evolutionary Stages of 25 MΘ Star
Temperature Density
Stages Time scale (T9) (g-cm-3)
Hydrogen burning x 106 y
Helium burning x 105 y x 102
Carbon burning y x 105
Neon burning y x 106
Oxygen burning months x 107
Silicon burning d x 107
Core collapse Seconds x 109
Core bounce milliseconds ≈ 3 x 1014
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John W. Negele and E. Vogt, eds.ÂăAdvances in nuclear physics. Vol. 27. Springer Science & Business Media, (2003).
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10. Weak Interaction Processes
The most general processes in nuclear decay in which weak interaction occur are given below:
β--decay:
β+-decay:
Electron capture:
Positron capture:
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11. Fermi and Gamow-Teller Transitions
Fermi Transitions:
S=0 -  -
n p e
Gamow-Teller Transitions:
S=1
In the Fermi decay mode the conservation of angular momentum requires that the spin of the baryons to point in the same direction before and after the decay. p - - 
n e
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12. Formalism used in our calculation
The comparative half life of an ordinary decay from the state ‘i’ of the parent nucleus to the state ‘j‘ of the daughter nucleus is related to the nuclear reduced transition probability as:
………(1)
where
……..(2)
The value of D=6143 s is adopted in the calculations [2].
Nuclear reduced transition probability is given by:
………(3)
gA/gV = [3].
[2] J. C. Hardy and I.S. Towner, Phys. Rev. C 79(5), (2009),
[3] K. Nakamura, J. Phys. G, Nucl. Part. Phys., 37, (2010)

13. Continue….. Ji represent the total spin of the level |i> ,
where
………(4)
………(5)
Ji represent the total spin of the level |i> ,
shows Pauli spin matrix and the term
is known as the iso-spin raising and lowering operator .
The capture (decay) rates of a transition from the ith state of a parent nucleus (Z, N) to the jth state of the daughter nucleus is given by
………(6)

14. Nuclear Models
Electron captures rates and beta decays, are important in the modeling of astrophysical processes like hydrostatic burning phases, late presupernova stages of the stellar evolution, and synthesis of heavy elements above iron group nuclei in stellar furnaces.
Different nuclear models are used to calculate the weak decay rates. The QRPA and Shell model are widely used for the large scale calculations of these weak rates in stellar environment.

15. Model Description (pn-QRPA)
The pn-QRPA model was first developed by Halbleib and Sorensen [4]. An extension of the model to deformed nuclei was given by Moller and Krumlinde [5]. In the present work in addition to the well known particle-hole (ph) force, the particle-particle (pp) interaction, first consider by Cha [6], is also taken into account by adding GT interaction to QRPA Hamiltonian.
The Hamiltonian of the pn-QRPA model is given by,
Single-particle energies and wave functions were calculated in the Nilsson model, which takes into account nuclear deformation [7].
[4] J. A. Halbleib and R. A. Sorensen, Nucl. Phys. A, 98 (1967) 542.
[5] J. Krumlinde and P. Moller, Nucl. Phys. A , 417 (1984) 419.
[6] D. Cha, Phys. Rev. C 27 (1983) 2269.
[7] S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk, 29 (1955) No. 16.
………(A)
Where is the single-particle Hamiltonian, is the pairing force, is the ph GT force, and is the pp GT force.

16. Model Parameters The pn-QRPA model requires a numbers of parameters
The Nilsson Potential parameter taken from [8].
Pairing interaction strengths.
The Gamow-Teller interaction strengths χ (for ph interaction) and κ (for pp interaction).
The Nuclear Deformation parameter.
Q2 is electric quadruple moment taken from [9].
The Q-value of the reaction were taken from [10].
[8] S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk 29 (1955) No. 16.
[9] P. M¨oller et al., At. Data Nucl. Data Tables, 1 (2016) 109.
[10] G. Audi, et al., Chinese Physics C 36, (2012)1157.

17. Results and Discussion
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18. Ikeda Sum Rule ISRre = ΣB(GT-) - ΣB(GT+) = 3fq 2 (N-Z).
Ikeda Sum Rule (Ikeda 1964) is a mathematical tool which connects the nucleon numbers (neutrons and protons) with the microscopic structure of a given nucleus through the total Gamow Teller strengths in either direction. It is formulated as:
This rule is model independent. fq is the quenching factor.
It provides a guideline for all theoretical calculations of GT strength function.
Theoretical calculations of GT strength functions are supposed to fulfill the Ikeda sum rule.
In our model the Hamiltonian (Eq. (A)) was tuned to reproduce the Ikeda sum rule.
ISRre = ΣB(GT-) - ΣB(GT+) = 3fq 2 (N-Z).
Nuclei ΣB(GT-) ΣB(GT+) ISRre (cal) ISRre (th)
45Sc
55Mn

19. GT transitions from 45Sc to 45Ca
(a) (n,p) data [W. P. Alford, et al., Nucl. Phys. A (1991)] (b) pn-QRPA calculations (c) QRPA calculations using the Skyrme interactions [P. Sarriguren, Phys. Rev. C (2013)] (d) FRDM [P. Moller and J. Randrup, Nucl. Phys. A (1990)] (e) Shell model [A. Poves et al., Nucl. Phys. A (2001)] (f) Shell model [M. Honma et al., Phys. Rev. C (R) (2004)].

20. GT transitions from 55Mn to 55Cr

21. Comparison of electron capture (EC) rates for 45Sc

22. Comparison of EC rates for 55Mn

23. Comparison of EC rates with FFN and LSSM for 45Sc
FFN [G. M. Fuller, W. A. Fowler and M. J. Newman, Ap. J. Suppl Ser (1980); Ap. J. Suppl Ser (1982); Ap. J. Suppl . Ser. 252 (1982) 715].
LSSM[K. Langanke and G. Martinez-Pinedo, At. Data Nucl. Data Tables 79 (2001) 1].

24. Comparison of EC rates with FFN and LSSM for 55Mn

25. Calculated electron capture (EC) and positron emission (PE) rates for 45Sc

26. Calculated EC and PE rates for 55Mn

27. Conclusion The pn-QRPA model fulfilled the Ikeda Sum Rule.
For stellar applications the EC and PE rates over wide range of astrophysical density (10 – 1011 g/cm3) and temperature (0.01 GK to 30 GK) were calculated.
We compared our calculated GT strength and EC rates both with the measured and previous calculations.
Our results were in decent comparison with the measured data.
We also compared our calculated EC rates with the FFN and LSSM results.
The overall mutual comparisons show that at low temperatures the pn-QRPA computed EC rates are generally in agreement with LSSM.
However at high temperature, the pn-QRPA calculated rates are enhanced than the FFN and LSSM rates.
It was also concluded that at low temperature and high stellar density regions the PE rates may be neglected in simulation codes.
From astral viewpoint these enhanced EC rates may have substantial impact on the late stage evolution of massive stars and the shock waves energetics. Results of simulations illustrate that EC rates have a solid effect on the core collapse trajectory and on the properties of the core at bounce.

28. Thank you!
Thank you !