Quantum Many-body Dynamics in low-energy heavy-ion reactions

1. Quantum Many-body Dynamics in low-energy heavy-ion reactions Kouichi Hagino Tohoku University, Sendai, Japan hagino@nucl.phys.tohoku.ac.jp www.nucl.phys.tohoku.ac.jp/~hagino

2. Heavy-Ion Fusion Reactions around the Coulomb Barrier Key Points: • Fusion and quantum tunneling • Basics of the Coupled-channels method • Concept of Fusion barrier distribution • Quasi-elastic scattering and quantum reflection

3. Basic of nuclear reactions Shape, interaction, and excitation structures of nuclei scattering expt. cf. Experiment by Rutherford (a scatt.) Notation b Target nucleus detector X Y a measures a particle intensity as a function of scattering angles Projectile （beam） X(a,b)Y 208Pb(16O,16O)208Pb : 16O+208Pb elastic scattering 208Pb(16O,16O)208Pb* : 16O+208Pb inelastic scattering 208Pb(17O,16O)209Pb : 1 neutron transfer reaction

4. Scattering Amplitude Motion of Free particle: In the presence of a potential: Asymptotic form of wave function (scattering amplitude)

5. ＝(incident wave) + (scattering wave) （flux conservation） If only elastic scattering: ：phase shift

6. Differential cross section dW The number of scattered particle through the solid angle of dW per unit time: (flux for the scatt. wave)

7. Optical potential and Absorption cross section Reaction processes • Elastic scatt. • Inelastic scatt. • Transfer reaction • Compound nucleus formation (fusion) Loss of incident flux (absorption) Optical potential (note) Gauss’s law

8. Total incoming flux Total outgoing flux r r Net flux loss: Absorption cross section:

9. Overview of heavy-ion reactions Heavy-ion:Nuclei heavier than4He Two forces: 1. Coulomb force Long range, repulsive 2. Nuclear force Short range, attractive Inter-nucleus potential Potential barrier due to the compensation between the two (Coulomb barrier)

10. Three important features of heavy-ion reactions 1. Coulomb interaction: important 2. Reduced mass: large(semi-) classical picture concept of trajectory 3. Strong absorption inside the Coul. barrier rtouch 154Sm 16O rtouch Automatic Compound nucleus formation once touched (assumption of strong absorption) Strong absorption

11. Strong absorption Access to the region of large overlap • High level density (CN) • Huge number of d.o.f. Relative energy is quickly lost and converted to internal energy ：can access to the strong absorption ：cannot access cassically Formation of hot CN (fusion reaction) (note) In the case of Coul. Pocket: disappears at l = lg Reaction intermediate between Direct reaction and fusion: Deep Inelastic Collisions (DIC) Scattering at relatively high energy a/o for heavy systems

12. Fusion reaction and Quantum Tunneling 154Sm 16O rtouch rtouch Automatic CN formation once touched (assumption of strong absorption) Probability of fusion = prob. to access to rtouch Strong absorption Penetrability of barrier Fusion takes place by quantum tunneling at low energies!

13. V(x) x Quantum Tunneling Phenomena V(x) V0 x -a a Tunnel probability:

14. Vb x For a parabolic barrier……

15. Energy derivative of penetrability (note) Classical limit

16. Potential Model: its success and failure Asymptotic boundary condition: Fusion cross section: Mean angular mom. of CN: rabs Strong absorption

17. Wong’s formula C.Y. Wong, Phys. Rev. Lett. 31 (’73)766 i) Approximate the Coul. barrier by a parabola: ii) Approximate Pl by P0: (assume l-independent Rb and curvature) iii) Replace the sum of l with an integral

18. (note) For (note)

20. Comparison between prediction of pot. model with expt. data Fusion cross sections calculated with a static energy independent potential 16O+27Al 40Ar+144Sm 14N+12C L.C. Vaz, J.M. Alexander, and G.R. Satchler, Phys. Rep. 69(’81)373 • Works well for relatively light systems • Underpredicts sfus for heavy systems at low energies

21. Potential model: Reproduces the data reasonably well for E > Vb Underpredicts sfus for E < Vb What is the origin? Inter-nuclear Potential is poorly parametrized? Other origins?

22. Potential Inversion (note) Vb E r1 r2 r

23. A.B. Balantekin, S.E. Koonin, and J.W. Negele, PRC28(’83)1565

24. Fusion cross sections calculated with a static energy independent potential Potential model: Reproduces the data reasonably well for E > Vb Underpredicts sfus for E < Vb What is the origin? Inter-nuclear Potential is poorly parametrized? Other origins?

25. Target dependence of fusion cross section Strong target dependence at E < Vb

26. Low-lying collective excitations in atomic nuclei Low-lying excited states in even-even nuclei are collective excitations, and strongly reflect the pairing correlation and shell strucuture Taken from R.F. Casten, “Nuclear Structure from a Simple Perspective”

27. Effect of collective excitation on sfus: rotational case Comparison of energy scales Tunneling motion: 3.5 MeV (barrier curvature) Rotational motion: The orientation angle of 154Sm does not change much during fusion (note) Ground state (0+ state) when reaction starts 16O 154Sm Mixing of all orientations with an equal weight

28. 16O 154Sm The barrier is lowered for q=0 because an attraction works from large distances. Def. Effect: enhances sfus by a factor of 10 ~ 100 The opposite for q=p/2. The barrier is highered as the rel. distance has to get small for the attraction to work Fusion: interesting probe for nuclear structure

29. } excited states ground state More quantal treatment: Coupled-Channels method Coupling between rel. and intrinsic motions 0+ Entrance channel 0+ 0+ 0+ Excited channel 2+ 0+

30. Schroedinger equation: or Coupled-channels equations

31. 0+ Entrance channel 0+ 0+ 0+ Excited channel 2+ 0+ Boundary condition

32. Coupling Potential: Collective Model (note) coordinate transformation to the rotating frame ( ) • Vibrational case • Rotational case Coordinate transformation to the body-fixed rame (for axial symmetry) In both cases

33. Deformed Woods-Saxon model:

34. Deformed Woods-Saxon model (collective model) CCFULL K.H., N. Rowley, and A.T. Kruppa, Comp. Phys. Comm. 123(’99)143 Nuclear coupling: Coulomb coupling: Rotational coupling: Vibrational coupling:

35. 0+,2+,4+ 2+ 0+ Vibrational coupling Rotational coupling 4+ 2+ 0+

36. Coupled-channels equations and barrier distribution Calculate sfus by numerically solving the coupled-channels equations Let us consider a limiting case in order to understand (interpret) the numerical results • enI: very large • enI = 0 Adiabatic limit Sudden limit

37. 16O 154Sm Slow intrinsic motion Barrier Distribution C.C. in the sudden limit Coupled-channels: diagonalize

38. w B1 B2 B3 P0 E B E B1 B2 B3 E B Barrier distribution

39. Barrier distribution: understand the concept using a spin Hamiltonian Hamiltonian (example 1)： For Spin-up For Spin-down x x

40. Wave function (general form) Asymptotic form at (the C1 and C2 are fixed according to the spin state of the system) (flux at ) Tunnel probability ＝ (incoming flux at )

41. Tunneling prob. is a weighted sum of tunnel prob. for two barriers

42. Tunnel prob. is enhanced at E < Vb and hindered E > Vb • dP/dEsplits to two peaks　　　　　“barrier distribution” • The peak positions of dP/dE correspond to each barrier height • The height of each peak is proportional to the weight factor

43. Hamiltonian (example 2): in case with off-diagonal components If spin-up at the beginning of the reaction

44. Hamiltonian (example 3): more general cases x dependent E dependent K.H., N. Takigawa, A.B. Balantekin, PRC56(’97)2104 (note) Adiabatic limit:

45. Sub-barrier Fusion and Barrier distribution method • Fusion takes place by quantum tunneling at low energies • C.C. effect can be understood in terms of distribution of many barriers • sfus is given as an average over the many distributed barriers Tunneling of a spin system The way how the barrier is distributed can be clearly seen by taking the energy derivative of penetrability Can one not do a similar thing with fusion cross sections?

46. One important fact: experimental observable is not penetrability, but fusion cross section (Fusion barrier distribution) N. Rowley, G.R. Satchler, P.H. Stelson, PLB254(’91)25 (note)Classical fusion cross section

47. Fusion Test Function Classical fusion cross section: Tunneling effect smears the delta function • Fusion test function: • Symmetric around E=Vb • Centered on E=Vb • Its integral over E is • Has a relatively narrow width

48. Barrier distribution measurements Fusion barrier distribution Needs high precision data in order for the 2nd derivative to be meaningful (early 90’s)

49. Experimental Barrier Distribution 16O 154Sm Requires high precision data M. Dasgupta et al., Annu. Rev. Nucl. Part. Sci. 48(’98)401

50. Investigate nuclear shape through barrier distribution Nuclear shapes