1. Collective Response of Atom Clusters and Nuclei: Role of Chaos Trento April 2010 Mahir S. Hussein University of Sao Paulo

2. Contents  Metal Clusters  Nuclei  Giant dipole resonances  Exit doorway model  Damping width and Chaos  Random matrix theory

3. Metal Clusters  Aggregate of N atoms  Excite plasmon-type Mie resonances  Electrons oscillate out of phase with respect to ions.  Probes: Laser, electrons, other clusters.  Multi-plasmon resonances

4. PRL, 68, 3916 (1992). PRL,70, 2036 (1993).

5. Energy and width of cluster plasmon resonances PRL,70, 2036 (1993).

6. Multiphonon excitation in Na clusters PRL, 80, 1194 (1998). A perfect harmonic oscillator : n=4

7. Plasmon excitation energy vs. size of Xe and Ar clusters Phys. Rev. Lett. 67, 3290 (1991).

8. Plasmon excitation energy and width vs. size of Hg clusters Phys. Rev. Lett. 69, 3212 (1992).

9. Nuclei  Aggregate of N neutrons and Z protons  Collective excitation of giant dipole and quadrupole resonances  Probes: photons, electrons, other nuclei  Neutrons oscillate against protons (dipole)  Collective state is damped to chaotic configuration.

10. J. G. Woodworth et al., Phys. Rev. C, 19, 1667 (1979); G. J. O’keefe et al., Nucl. Phys. A 649, 239 (1987).

11. R. Schmidt et al. Phys. Rev. Lett. 70, 1767 (1993) Single- and double-phonon excitation in Xe nucleus.

12. Rev.Mod.Phys., 47, 713 (1975). Shape is Lorentzian (spherical).

13. Rev. Mod. Phys., 55, 287 (1983) Widths of giant dipole and quadrupole resonances in nuclei

14. Rev.Mod.Phys., 47, 713 (1975). Excitation energy of giant dipole resonances in nuclei

15. Pygmy giant resonances!

16. Phys. Rev. Lett., 95, 132501 (2005).

17. Split dipole state: Schrődinger cat? Z. Phys. A 355. 165 (1996)

18. Exit doorway model Consider the many-body Hamiltonian with time-dependent interaction Intrinsic Hamiltonian of the system External perturbation System responds to action of and is excited to some collective state (not an eigenstate of )

19. Calculation of excitation probabilities Write Use Get (taking ) Initial conditions Excitation probabilities

20. Consider : dipole operator : electric field of external probe.

21. Excitation of collective state with finite lifetime can be treated using the exit-doorway (ED) model Expanding

22. Since, We get Model Uniform spectrum get

23. Spreading or damping width, measures the degree of mixing of the collective state with the background (chaotic) states.

24. Take does not mix excited states With ED hypothesis, get Write

25. Get three coupled differential equations With Excitation probabilities

26. And cross-section

27. Ann. Phys. (New York), 284, 178 (2000). Calculation of excitation cross-section of a damped plasmon in sodium clusters

28. Role of chaos Response function, spreading width Dynamical enhancement of multiphonon excitations.

29. Dynamical enhancement of multiphonon excitation Owing to, to go from ground state to 2 phonon state (via 1 phonon) one may, if collision time, excite a single phonon on top of background which gives rise to

30. Excitation probability Harmonic (Poissonian) Dynamical (Brink-Axel) Ann. Phys.(New York), 276, 111 (1999). Nucl. Phys. A 731, 163 (2004).

31. Ann. Phys. (NY) 276,111 (1999); Nucl. Phys. A 690, 382 (2001). Coupling between collective and chaotic states.

32. Evolution of density matrix with damping and dynamical chaos enhancement

33. Phys. Rev. C, 60, 014604 (1999). Dynamical chaos enhancement of cross-section

34. Ann. Phys.(New York), 276, 111 (1999). Nucl. Phys. A 731, 163 (2004). Dynamical enhancement could be as large as 80%.

35. Conclusions Collective response of finite many-body systems is affected by the degree of chaoticity of the internal degrees of freedom. The system acquires a damping width: Damped harmonic oscillator Chaos leads to an enhanced excitation owing to the damping.

36. Thanks to my collaborators: Brazil: C. A. Bertulani, L. F. Canto, B. V. Carlson, R. Donangelo, J. X. de Carvalho, M. P. Pato, A. F. R. de Toledo Piza Germany: T. Aumann, H. Emling USA: H. Feshbach, A. K. Kerman, V. Kharchenko, E. Timermmans

37. Thank you!