Vlasov eq. for finite Fermi systems with pairing

1. Vlasov eq. for finite Fermi systems with pairing V.I.Abrosimov, D.M.Brink, A.Dellafiore, F.Matera Kazimierz Dolny, September‘07

2. Outline: • Introduction • Semiclassical TDHFB eqs. • Extended Vlasov eq. • One-dimensional systems • Conclusions .

3. Normal system (nopairing) • TDHF eq. • Normal Vlasov eq. .

4. Linear approx. Linearized Vlasov eq.

5. Correlated sys.(with pairing) • TDHFB eqs. where .

6. from Wigner-transf. TDHFB eqs. where we • have kept only terms of first order in • Semiclassical TDHFB eqs.: . • here

7. Semiclassical TDHFB eqs.-Static limit • (Bengtsson and Schuck (1980)) where • quasiparticle energy, • the chemical potential is determined by the • number of particles , • is related to by the energy gap eq.

8. Constant- approx. • Approximate then semiclass.TDHFB eqs.become

9. In static limit one finds • theequilibrium solution: • with the particle energy and • the quasiparticle energy • Note that in the limit of , • we have and .

10. Equilibrium distributions

11. EXTENDED linearizedVlasov eq. • From semiclass. TDHFB eqs. follows (with constant- approx.) with

12. One-dimensional systems: zero-order solution • Zero-order approx.: • Eigenfrequencies: • where • frequency with • Note that eigenfrequencies of normal • system:

13. To ensure particle-number conservation and • to eliminate the spurious strength, we express • density fluctuations in terms current density • through continuity eq.: Correlated zero-order propagator A propagator is defined by • Then gives gives and

14. Uncorrelated Eigenfrequencies No gap No spreading EWSR Correlated Eigenfrequences Gap Spreading EWSR Spurious strength is subtracted! Uncorr. vs. corr.

15. Response function • External field

16. Small size:

17. Medium size:

18. Large size:

19. Collective solution Coll. propagator satisfies integral eq. Separable interaction gives collective response function by where

20. Collective response(medium size: )

21. Conclusions • Semiclassical TDHFB eqs. have been studied in a simplified model, in which the pairing field is treated as a constant phenomenological parameter. • A simple prescription for restoring both global and local particle-number conservation is proposed. • We have shown in one-dimensional system that our model represents the main effects of pairing correlations. • It is of interest to extend the present method to three-dimensional systems. Work on this problem is in progress. .