1. Unblocking of the Gamow-Teller Strength for Stellar Electron CaptureBy
Mavra Ishfaq
Supervised by
Prof. Dr. Jameel-Un Nabi
Faculty Of Engineering Sciences
GIK Institute Of Engineering Sciences and Technology
Topi, Swabi, KPK, Pakistan

2. Outline Introduction Formalism Results and Discussion Conclusions
References

3. Introduction Weak Decay Processes: β- decay: β+ decay:
The most general processes in nuclear decay in which weak interaction occur are given below
β- decay:
β+ decay:
Electron capture:
Positron capture:

4. Gamow-Teller (GT) and Fermi Transitions
Let us consider the example of β- decay in which one neutron is converted into a proton and emits an electron and antineutrino.
Gamow-Teller transitions take place when the emitted electron and antineutrino spins are parallel. i.e. S = 1.
The β-decay transitions in which the ejected electron and antineutrino are anti-parallel for which S = 0 are known as Fermi transitions.
Transitions with total orbital angular momentum of electron-antineutrino pair, L = 0 are termed as Allowed transitions.

5. Selection rules for Fermi and GT transitions
Selection rules for allowed GT and Fermi transition are:
ΔJ=0
for Fermi transitions
ΔJ=0, ±1
Gamow-Teller transitions
( but not )
If L > 0 then these types of decays are Forbidden.

6. Applications of GT transitions Strength
GT strength distribution connects the parent nucleus states to the daughter nucleus states in the GT- (GT+) direction, in which a neutron (proton) is converted into proton (neutron), they are being considered because the simplicity of the excitation make them an ideal probe for testing the nuclear structure models.
The GT transitions strength is an important physical quantity for understanding of nuclear structures as well as for calculation of Astrophysical process(e.g., evolution of massive stars, nucleosynthesis process).
GT transitions strength are the key ingredients for calculating the weak interaction rates.
Gt applications…???

7. Model Description (pn-QRPA)
Different nuclear models are used to calculate the GT strength function. The pn-QRPA and Shell Model are widely used for the large scale calculations of these GT Strength.
The pn-QRPA model was first developed by Halbleib and Sorensen[1]. An extension of the model to deformed nuclei was given by Mӧller and Krumlinde [2]. In the present work in addition to the well known particle-hole (ph) force, the particle-particle (pp) interaction, first consider by Cha [3], is taken into account by adding GT interaction to QRPA Hamiltonian.
The Hamiltonian of the pn-QRPA model is given by,
[1] J. A. Halbleib, R. A. Sorensen, Nucl. Phys. A. 98, 542, (1967).
[2] P. Mӧller and J. Krumlinde, Phys. Rev. Letters, 55, , (1985).
[3] D. Cha, B. Schwesinger, J. Wambach and J. Speth, Nucl. Phys. A, 430, , (1984).
[4] S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk 29, 16,(1955).
………(10)
Where is the single-particle Hamiltonian, is the pairing force, is the ph GT force, and is the pp GT force.[4]

8. Model Parameters The pn-QRPA model requires a numbers of parameters,
The Nilsson Potential parameter taken from [5].
Pairing interaction strengths.
The Gamow-Teller interaction strengths χ (for ph interaction) and κ(for pp interaction) [6].
The nuclear deformation parameter [7].
The Q-value of the reaction.
For the Q-values, the mass formulas from [8] were adopted.
[5] S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk 29, 16, (1955). [6] M. Hirsch, A. Staudt, K. Muto, H. V. Klapdor-Kleingrothaus, ADNT 53, 165, (1993).
[7] S. Raman, C. H. Malarkey, W. T. Milner, C. W. Nestor, Jr, And P. H. Stelson, At.Data. Nucl. Data. Tables, 36, 1-96, (1987). [8] A. Staudt, E. Bender, K. Muto, and H. V. Klapdor AT. Data & Nuc. Data Tables, 44, 79, (1990).

9. Formalism
The weak decay rate from the ith state of the parent to the jth state of the daughter nucleus is given by
where (ft)ij is related to the reduced transition probability Bij of the nuclear transition by
………(11)
………(12)

10. Cont…
The D appearing in Eq. (12) is a compound expression of physical constants,
&
where B(F) and B(GT) are reduced transition probabilities of the Fermi and Gamow-Teller (GT) transitions respectively,
………(13)
………(14)
………(15)
………(16)

11. Cont…
In Eq. (16), σk is the spin operator and stands for the isospin raising and lowering operator. The value of D=6295 s is adopted [9] and the ratio of the axial-vector (gA) to the vector (gV) coupling constant is taken as [10].
[9] Yost., G.P., Barnett., R.M., Hinchliffe., I., et al., Phys. Letter. B 204, 1 (1988).
[10] Rodin, V., Faessler, A., Simkovic, F., Vogel, P. J. Phys. 56, 495 (2006).

12. Re-normalized Ikeda Sum Rule (re-ISR)
Re-normalized Ikeda sum rule in pn-QRPA is given by
ISRre-norm = S(-) − S(+) ∼ = 3fq2 (N − Z)
Our calculated GT strengths are all quenched within the pn-QRPA formalism by the factor of fq2 = (0.55)2 which is used for pf shell nuclei [11].
That’s why the ISRre-norm is said to be model independent. The ISR provides the bench mark values to be followed by all nuclear models as close as possible.
………(9)
[11] S. Cakmak, J.-U. Nabi, T. Babacan and C. Selam, Astrophysics and Space Science 352, (2014)

13. Continued
The appropriateness of the model calculation can be checked through re-ISR.
In case of neutron rich nuclei if calculated ƩBGT- value is less than 3fq2 (N-Z), then we can conclude that the calculated strength is not correct and shows that few strength lies outside the calculated energy domain or some strength present inside the domain is not recognized by the model.

14. Results and Discussion
We compare our calculated Gamow-Teller transitions with the Shell and DQRPA Model calculations.
Plotted graphs depicts the unblocking of GT transition.
We further calculate stellar electron capture rates in supernovae environments (T~ 0.5 – 1.5 MeV) for 76Se and made comparison with those obtained from experimental GT data and previous calculations.
The table 1 shows the comparison of ISR (both theoretical and calculated) and Cut-Off Energy (MeV) for 76Se.
Ratios of calculated electron capture (e-cap) rates to β+-decay for different selected densities and temperatures are shown in table 3.
C Cut-off Energy(MeEergy(

15. Fig1; B(GT+) calculation using pn-QRPA model for the ground state of 76Se compared with experimental data, Shell and DQRPA Model.
[12] Z. Q-Jun, Y. Yan, Z. Qiang. Chin. Phys. C, 35, 1022–1025, (2011). [13 E.-W. Grewe et Al. Phy. Rev. C, 78, , (2008).
[14] E. Ha, M.-K. Cheou, Nucl. Phys. A, 934, , (2015).

16. Fig2; Calculated e-cap rates for 76Se based on the pn-QRPA model compared with the experimental data, Shell and DQRPA Model for different temperatures and ρ = gcm-3

17. Fig3; Calculated e-cap rates for 76Se based on the pn-QRPA model compared with those based on the experimental data, Shell & DQRPA Model for T = 10 GK.

18. Table1; re-ISR rule for 76Se
Cut-off Energy(MeV)
Theoretical
re-ISR
Calculated
15.90
21.78

19. Table 2; Calculated total B(GT+) Strength, Centroid and Cut off Energy for 76Se in e-cap direction.
Model
∑ B(GT+)
Ē+ (MeV)
Cut off Energy(MeV)
EXP
1.00
2.50
10.00
DQRPA
2.04
7.48
25.00
pn-QRPA
1.42
3.15
15.90 SM
1.46
5.3
9.80

20. Table3; Ratio of calculated e-cap rates to β+ decay for different selected densities and temperatures.
ρYe (gcm -3)
R(e-cap/β+)
1GK
5 GK
10 GK
20 GK
8.50
2.60E+23
2.63E+06
3.62E+04
2.98E+04
9.50
6.64E+28
1.60E+09
2.52E+06
4.00E+05
10.50
2.51E+31
5.38E+11
4.65E+08
2.50E+07

21. Conclusion
Recent theoretical predictions show that electron capture on these nuclei is possible due to configuration mixing by the residual interaction.
Another reason for Gamow-Teller transitions to occur is the thermal unblocking of pf- shell single-particle states.
These theoretical predictions using the pn-QRPA model have been verified for the issue of unblocking of Gamow-Teller transitions for 76Se.

22. Thank you !