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

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

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

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

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

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

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

2. Symmetries of the s.p. wave function xyz 1++p 2__p 3_+-p 4+_ [1] P. Bonche, H. Flocard, and P. H. Heenen, Nucl. Phys. A 443 (1985) 39 1/8 space

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

2. Symmetries of the s.p. wave function Static Triaxial-de Crankingnp paring Cranking+ np paring ParityYes Time- reversal YesNoYesNo SignatureYes No Phase convention Yes Calculated Coor-space 1/8 [1]1/8 [2]1/4 [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

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

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

4. If real is possible?

is real.

4. If real is possible? Complex Im Re

Chose complex wave function and assume real 4. If real is possible? the np pairing can be described in general. 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?

5. Summary and further work 1.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. 3.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. Further work: 1.Make sure if the kappa can be real? 2.Construct the pairing matrix 3.Construct the calculation space by the symmetries …… Aim: develop the code cr8 with np pairing included.

Thank you !

2. Symmetries of the s.p. wave function xyz 1++p 2--p 3-+-p 4+- Static:Cranking:

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

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