1. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop1 Pairing vibrations study in the Time-Dependent Hartree-Fock Bogoliubov approach Benoît AVEZ – CEA/Irfu/SPhN Philippe CHOMAZ – CEA/Irfu Cédric SIMENEL – CEA/Irfu/SPhN

2. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop2 Introduction : Pairing vibrations Pairing correlations → « pairing vibrations » L=0 + –Low energy modes –« Giant Pairing Vibrations (GPV) » (analogous to giant resonances) → No experimental confirmation –Probes : Pair transfer reactions Tool : TDHFB formalism Results –Oxygen isotopes (comparison with QRPA) Khan et al., PRC 69, 014314 (2004) –Calcium isotopes Bès and Broglia, Nucl Phys. 80, 289 (1966) n ħω Fermi (n-1)ħω (n+1)ħω n ħω

3. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop3 TDHFB formalism : general considerations Energy density functional formalism : Approximate solution of the Time-dependent Schroedinger equation Microscopic approach Can account for nuclear superfluidity Structure and dynamics treated on an equal footing Few parameters (Skyrme, Gogny), large pannel of applications (structure, collective excitations, heavy ions collisions…)

4. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop4 Evolution of the generalized density matrix R(t) TDHFB formalism  quasiparticule vacuum at each time t 

5. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop5 Evolution of the generalized density matrix R(t) Energy functional E[ R(t) ] Variational method on the action → TDHFB equations TDHFB formalism H Bogoliubov Hamiltonien

6. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop6 TDHFB formalism : Implementation Implementation of TDHFB assuming spherical symmetry Energy density functional E[ R ] –ph channel : Sly4 parameterization Chabanat et al., NPA 635, 231-256 (1998) –pp (hh) channel : Local, density-dependent pairing bulk+surface Cut-off E qp at 80 MeV,  c =0.32 fm -3, V 0 =-275.25 MeV

7. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop7 TDHFB formalism : Implementation Spherical symmetry + energy defined only from local densities Iterative solution :

8. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop8 TDHFB formalism : Particle number conservation HFB states are NOT eigenstate of N –Fixed in average through a Lagrange constraint « - N » TDHFB : (t) = constant of motion What about ? If stationnary evolution, dropping :

9. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop9 TDHFB formalism : stationnary evolution → Evolution of normal and anomalous densities E (MeV) Example : 18 O ( |2  =14.77 MeV )

10. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop10 TDHFB formalism : Nambu-Goldstone mode Nambu-Goldstone mode → Evolution the initial Lagrange constraint Pot. landscape

11. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop11 TDHFB formalism : Linear response Perturbation at t=0 Fourier transform of « Strength » function Excitations in the next A±2 nuclei (with ground-state energies ≈ E A ±2 ) Linear response : TDHFB ↔ QRPA

12. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop12 Example : the 18 O case Small amplitude excitation t=0 Higher order effects (> QRPA) Width of the 1st oscillation

13. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop13 « Strength » function 18 O±2n TDHFB Example : the 18 O case Small amplitude excitation Fourier

14. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop14 s 1/2 f 7/2 Continuum Level scheme, neutrons d 3/2 d 5/2 s 1/2 εFεF p 3/2 p 1/2 « Strength » function 18 O±2n TDHFB Example : the 18 O case Small amplitude excitation Fourier s 1/2 d 3/2, (p 1/2 ) f 7/2, (p 3/2 ) s 1/2 d 3/2, (p 1/2 ) f 7/2, (p 3/2 )

15. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop15 18 O+2n E. Khan et al., PRC69, 014314 (2004) (dotted : box-QRPA – solid : c-QRPA) QRPA s 1/2 d 3/2 f 7/2 s 1/2 d 3/2, (p 1/2 ) f 7/2, (p 3/2 ) s 1/2 f 7/2 Continuum Level scheme, neutrons d 3/2 d 5/2 s 1/2 εFεF p 3/2 p 1/2 « Strength » function 18 O±2n TDHFB Example : the 18 O case Good agreement with QRPA -Possible differences : ≠ Pairing functionnal ≠ implementation s 1/2 d 3/2, (p 1/2 ) f 7/2, (p 3/2 )

16. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop16 Results : Oxygen isotopes TDHFB response HFB response (unperturbed spectrum) 18 O±2n 20 O±2n 22 O±2n Unperturbed spectrum : 2qp excitations (from the static ground state solution)

17. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop17 Results : Oxygen isotopes « residual interaction » effect –S F (ħω) increases dynamical pairing → collectivity –Shift towards low energy (attractive character of the pairing) 18 O±2n 20 O±2n 22 O±2n TDHFB response HFB response (unperturbed spectrum)

18. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop18 Results : Oxygen isotopes GPV 18 O±2n 20 O±2n 22 O±2n « residual interaction » effect –S F (ħω) increases dynamical pairing → collectivity –Shift towards low energy (attractive character of the pairing) GPV +2n mainly f 7/2 TDHFB response HFB response (unperturbed spectrum)

19. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop19 Results : Oxygen and Calcium isotopes GPV 18 O±2n 20 O±2n 22 O±2n 42 Ca±2n 46 Ca±2n 44 Ca±2n

20. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop20 Conclusions Effect of the dynamical pairing correlations on pairing vibrations Applications on Oxygen and Calcium isotopes Good comparison with QRPA (Oxygen isotopes) Presence of the GPV confirmed in Oxygen isotopes Not seen in Calcium isotopes Outlook Heavier nuclei (Sn, Pb) Treatment of the continuum Application of TDHFB to nuclear reactions

21. CEA DSM Irfu 14 Oct. 2008- Benoît Avez - [Pairing vibrations with TDHFB] - ESNT Workshop21 Acknowledgements and references Michael BENDER, CENBG, Bordeaux Karim BENNACEUR, IPN, Lyon Thomas DUGUET,CEA/Irfu/SPhN, Saclay Hubert FLOCARD, CSNSM, Orsay Elias KHAN, IPN, Orsay Denis LACROIX, GANIL, Caen Initial conditions for the TDHFB code : K. Bennaceur, J. Dobaczewski, Comp. Phys. Com. 168, 96 (2005) Pairing vibrations B. Avez, C. Simenel and Ph. Chomaz, arXiv:nucl-th/0808.3507 (2008) (accepted for publication in Phys. Rev. C) D.R. Bès and R.A. Broglia, Nucl. Phys. 80, 289 (1966) E. Khan, N. Sandulescu, N. Van Giai, and M. Grasso, PRC 69, 014314 (2004)