1. Angular Momentum Projection in TDHF Dynamics : application to Coulomb excitation and fusion C. Simenel 1,2 In collaboration with M. Bender 2, T. Duguet 2, F. Nunes 2 1) CEA-SPhN, Saclay 2) MSU/NSCL, US

2. Motivations Nuclear reactions elastic, inelastic, deep-inelastic, transfer, break-up, fusion, fission… Coulomb + nuclear interactions, couplings, multistep process, tunnel effect… Whole nuclear chart from light to superheavy elements from proton to neutron drip lines Fully microscopic theory effective interaction (Skyrme, Gogny…) Beyond mean-field long range dynamical correlations mixing of trajectories

3. Present status Time dependent mean field Time-Dependent Hatree-Fock theory P.A.M. Dirac, Proc. Camb. Phil. Soc. 26, 376 (1930) First application to nuclear physics Y.M. Engel et al., NPA 249, 215 (1975) P. Bonche, S. Koonin and J.W. Negele, PRC 13, 1226 (1976) 3D calculations of nuclear reactions H. Flocard, S.E. Koonin and M.S. Weiss, PRC 17, 1682 (1978) K.-H. Kim, T. Otsuka and P. Bonche, JPG 23, 1267 (1997) C. S., P. Chomaz and G. de France, PRL 86, 2971 (2001) C. S., P. Chomaz and G. de France, PRL 93, 102701 (2004) no pairing (except QRPA)

4. Present status Beyond time dependent mean field Extended TDHF D. Lacroix, P. Chomaz and S. Ayik, PRC 58, 2154 (1998) Time Dependent Density Matrix S.J. Wang and W. Cassing, Ann. Phys. 159, 328 (1985) M. Tohyama, PRC 36, 187 (1987) Stochastic TDHF O. Juillet and P. Chomaz, PRL 88, 142503 (1998) Time Dependent Generator Coordinate Method P.-G. Reinhard, R.Y. Cusson and K. Goeke, NPA 398, 141 (1983) J.F. Berger, M. Girod and D. Gogny, NPA 428, 23c (1984) H. Goutte, J.F. Berger, P. Casoli and D. Gogny, PRC 71, 024316 (2005) Projected TDHF (present work)

5. Today's objectives: reactions with a deformed projectile effect of the initial orientation on the reaction  many TDHF trajectories Coulomb excitation (rotation) angular momentum projection to calculate the J-population Interferences between initial orientations Fusion incoherent mixing of TDHF trajectories  realistic fusion probability (between 0 and 1) Effect of Coulomb excitation in the approach

6. I) Projected TDHF : formalism A) Projection on angular momentum static case angular momentum "projector": rotated Slater determinant: evolution no feed back of the correlations - on the Slater evolution  TDHF trajectories - on the superposition functions  f(  ) is constant  initial state correlations in the observation only

7. I) Projected TDHF : formalism B) Projection on angular momentum exact J-population we use  high computationnal cost approximated J-population  accurate if the vibration and the rotational speed are small

8. I) Projected TDHF : formalism C) The code: symmetries and numerical tests initial condition: isotropic distribution of (J=0) axial symmetry of  small impact parameter or small dynamical deformation of  (t) or sudden approximation the HF g.s. has an axial symmetry, a time-reversal symmetry and a good parity the evolved Slater determinants have a plane of symmetry  only one collision partner can be deformed no charge mixing in the s.p. wave functions

9. I) Projected TDHF : formalism C) The code: symmetries and numerical tests explicit expression of the JM-population - exact with and - approximated

10. I) Projected TDHF : formalism C) The code: symmetries and numerical tests orthonormalization: convergence with the number of rotational angles 1- | ‹0|0› |, | ‹J|0› |

11. II) Coulomb excitation : rotational band A) Classical calculation rotation due to Coulomb repulsion effects on induced fission 130 Xe + 238 U (E<B)    f(D) D Theoretical calculation Holm et al., PLB 29, 473 (1969)

12. II) Coulomb excitation : rotational band A) Classical calculation K.Alder and A. Winther, Electromagnetic Excitation (1978) C. S., P. Chomaz and G. de France, PRL 93, 102701 (2004) - point like target - small - small - differential equation : with and - solution : - reorientation (  =1) :  Z 1, A 1 Z 2, A 2 D

13. II) Coulomb excitation : rotational band B) TDHF approach self consistent mean field theory independant s.p. wave functions mean values of one body observables (ex : orientation) quantal treatment of inertia P. Bonche code K.-H. Kim, T. Otsuka and P. Bonche, JPG 23, 1267 (1997)  deformed projectile + Coulomb potential of the target Skyrme forces (SLy4d) T. Skyrme, Phil. Mag. 1 (1956)

14. II) Coulomb excitation : rotational band B) TDHF approach 24 Mg (+ 208 Pb) E CM = 112 MeV (≈B) D init. = 220fm head on collision` Rutherford trajectory approach phase only

15. II) Coulomb excitation : rotational band B) TDHF approach 24 Mg (+ 208 Pb)  ∞ = 45 deg. population of J ? C. S., P. Chomaz and G. de France, PRL 93, 102701 (2004) D0D0  ∞ = 45

16. II) Coulomb excitation : rotational band C) PTDHF: excitation probability set of projected states - A. Valor, P.-H. Heenen and P. Bonche, NPA 671, 145 (2003) - M. Bender, H. Flocard and P.-H.Heenen, PRC 68, 044321 (2003) et al., 24 Mg

17. II) Coulomb excitation : rotational band C) PTDHF: excitation probability time evolution of the J-population 24 Mg (+ 208 Pb) E CM = 112 MeV (≈B) head on collision approaching phase only Time (fm/c) P J (t)

18. II) Coulomb excitation : rotational band C) PTDHF: excitation probability 24 Mg (+ 208 Pb) @ E CM =690 MeV ~ 6B angular distribution interferences between orientations still need nuclear potential (target) and interferences between scattering angles P J (t  ∞) c.m. scattering angle (deg.) J=2 J=0 Semi-classical PTDHF "improved" TDHF

19. III) Fusion of deformed nuclei at the barrier 24 Mg+ 208 Pb @ 94 MeV head-on collision initial distance: 20 fm the fusion probability depends on the initial orientation

20. III) Fusion of deformed nuclei at the barrier 24 Mg+ 208 Pb initial distance: 20 fm  no long range Coulomb excitation Isotropic model : red line  B 0 ~ 97 MeV  ~ 0.06 ~  SLy4d /2 barrier ~ 10% lower than expected (collision term ?) CM Energy (MeV) Orientation  at 20fm FUSION SCATTERING 0  /4  /2

21. III) Fusion of deformed nuclei at the barrier Fusion probability - Isotropic distribution at D=20fm : blue line - Isotropic distribution at D=220fm : red points reduction of the fusion due to Coulomb excitation no concluding effect of nuclear excitation on the fusion probability P fus P fus /P 0 E CM (MeV)

22. Conclusions and perspectives PTDHF approximated angular momentum projection on TDHF trajectories beyond mean field for the observation, ex: P J (t) Coulomb excitation  strong effect of the interferences fusion  reduction of the fusion due to Coulomb excitation  exact projection  interferences between scattering angles  effect of interferences (orientations) on fusion ?  feed back of the correlations on the evolution  TDGCM  pairing (TDHF)

23. annexe

24. Plan I) Projected TDHF: Formalism Time Dependent Generator Coordinate Method Projection on angular momentum The code: symmetries and numerical tests II) Coulomb excitation: 24Mg rotationnal band Classical calculation TDHF approach PTDHF: excitation probability III) Fusion of deformed nuclei at the barrier Rotationnal couplings in the entrance channel Beyond mean field results IV) Conclusions and perspectives

25. I) Projected TDHF : formalism A) TDGCM wave function q: collective variable f : superposition function  : Slater determinant P.-G. Reinhard et al.  (t)   TDHF (t) J.F. Berger et al. + H. Goutte et al.  (t)   HFB time dependent Griffin-Hill-Wheeler equation

26. I) Projected TDHF : formalism C) The code: symmetries and numerical tests explicit expression of the JM-population - exact with and - approximated - without interferences

27. I) Projected TDHF : formalism C) The code: symmetries and numerical tests test of the overlap between Slater determinants

28. 24 Mg (+ 208 Pb) @ E CM =690 MeV ~ 6B