T=0 Pairing in Coordinate space

1. Workshop ESNT, Paris T=0 Pairing in Coordinate space Shufang Ban Royal Institute of Technology (KTH) Stockholm, Sweden

2. Outline • 1. Introduction: delta force in HFB • 2. Symmetry of the s.p. wave function • 3.Delta matrix can be real • 4.If real kappa is possible? • 5.Summary and further work

3. HFB Equation: Delta force Anti-symmetric 1. Introduction 1.1 Algorithm for using delta force in HFB calculations:

4. Local in coordinate space, we can calculate the value at every point r. T=1 pairing (nn, pp) Local Delta potential

5. T=1 paring: T=0 paring: All the possible pairing correlations: Alan L. Goodman, Phys. Rev. C 58(1998)R3051

6. Wave function: Delta force 1.2 using delta force in generalized HFB calculations including np-pairing: Local in Coordinate space

7. Parity: z-signature: Time-reversal: Global Phase convention: 2. Symmetries of the s.p. wave function Four real components:

8. 1/8 space 2. Symmetries of the s.p. wave function [1] P. Bonche, H. Flocard, and P. H. Heenen, Nucl. Phys. A 443 (1985) 39

9. P. Bonche, et. al., Nucl. Phys. A 467 (1987) 475 Y. Engel, et. al., Nucl. Phys. A 249 (1975) 215 Time-reversal symmetry is broken by cranking A. L. Goodman, Nucl. Phys. A 186 (1972) 475 Axial symmetry is broken by np pairing Signature symmetry is broken by np pairing 2. Symmetries of the s.p. wave function

10. 2. Symmetries of the s.p. wave function [1] P. Bonche, H. Flocard, and P. H. Heenen, Nucl. Phys. A 443 (1985) 39 [2] P. Bonche, H. Flocard, and P. H. Heenen, Nucl. Phys. A 467 (1987) 475

11. Phase convention: (1) 3.Pairing matrix can be real

12. Assume real and using the wave function symmetry (1) (2) The integrand {…} is anti-symmetric under inversion y— -y, there for we have Paring matrix can be real

13. 4.If real is possible?

14. 4.If real is possible? is real.

15. 4.If real is possible? Re Complex Im

16. 4.If real is possible? Chose complex wave function and assume real Remained question: 1. If complex wave function, real kappa are equivalent to real wave function, complex kappa? Is there any transformation between them? 2. How we can construct the input wave functions of general case from the wave function of T=1 case? the np pairing can be described in general.

17. 5. Summary and further work • Make sure if the kappa can be real? • Construct the pairing matrix • Construct the calculation space by the symmetries • …… • Aim: develop the code cr8 with np pairing included. Further work: • Using delta force, we can get the local pairing matrix, • for both with or without np pairing cases. 2. The np pairing breaks axial and signature symmetries, we must calculate it in ¼ space when parity and phase convention are required. • Chose complex wave function, assume real kappa, • the pairing matrix can be real. 4. Using complex wave function and real kappa, the np pairing can be described without lose generality. There are still remained questions.

18. Thank you !

22. Static: Cranking: 2. Symmetries of the s.p. wave function

23. 1.2 using delta force in generalized HFB calculations including np-pairing: Wave function: Delta force

24. 5. Summary and further work Cranking triaxial-deformed wave function as input: Construct density matrix and T=1 pairing matrix in ¼ space Assure starting values for the delta potential, T=0 paring Get the H for HFB equation, solve it and get first U, V Mixing neutron proton in quasi-particle basis Calculate new density matrix and pairing matrix Calculate new density matrix and diagonalizes in canonical basis