Lecture II Pairing correlations tested in heavy-ion induced reactions Outline

1. Lecture II • Pairing correlations tested in heavy-ion • induced reactions • Outline • pairing correlation and correlations in space • systems at the drip lines • role and treatment of continuum states • reaction models for two-particle break-up and • two-particle transfer reactions • connecting reaction data to structure properties

2. How to use dynamics to study pairing correlations?The main road is clearly provided by the two-particle transfer process induced by light ions (reactions as (t,p), (p,t), (3He,n), (,d)) or heavy ions, which are both exploring precisely the pair correlations.Unfortunately, the situation is different, for example, from low-energy one-step Coulomb excitation, where the excitation probability is directly proportional to theB(E) values. Here the reaction mechanism is much more complicated and the possibility of extracting spectroscopic information on the pairing field is not obvious. The situation is actually more complicated even with respect to other processes (as inelastic nuclear excitation) that may need to be treated microscopically, but where the reaction mechanism is somehow well established.

3. It is often assumed that the cross section for two-particle transfer will scale with the square of the matrix element of the pair creation operator P+ =∑j [a+a+]00 (or pair removal). But this is not obviously true.

4. How to define and measure the collectivity of pairing modes? We could compare with single-particle pair transition densities and matrix elements to define some “pairing” single-particle units and therefore “pairing” enhancement factors.

5. Pair strength function 22O (d3/2)2 enhancement (f7/2)2 Khan, Sandulescu,Van Giai, Grasso

6. enhancement Giant Pairing Vibration in 210Pb Excited 0+ states g.s. in 210Pb 10

7. But the two-particle transfer process in not sensitive to just the pair matrix element. We have to look at the radial dependence, which is relevant for the reaction mechanism associated with pair transfer processes.

8. Comparison with pure single-particle configurations pair transition density P(r,r)= (r)=<0|c(r)c(r)|> T=0 T=3 (1h11/2)2 (1g9/2)2 Lotti, Vitturi etal

9. 18O (1d5/2)2 (r,r) 0.8 (1d5/2)2 + 0.6(2s1/2)2 (2s1/2)2 r (fm)

10. Lotti etal particle-particle spatial correlations |r1,r2)|2 as a function of r2, for fixed r1 Neutron addition mode: ground state of 210Pb position of particle 1 (1g9/2)2

11. (3p1/2)2 (2f5/2)2 206Pb Correlated ground state |r1,r2)|2 as a function of r2, for fixed r1 OBS: mixing of configurations with opposite parity position of particle 1

12. P(R,r) r R 206Pb R R R r Correlated g.s. (3p1/2)2 (2f5/2)2

13. Basic problem: how is changed the picture as we move closer or even beyond the drip lines? Example: the case of 6He R r

14. For weakly-bound systems at the drip lines it is mandatory to include in the models the positive energy part of the spectrum. If one wants to still use the same machinary used with bound states, the most popular approach is the discretization of the continuum. But the discretization MUST go in parallel in a consistent way both in the structure and in reaction parts.

15. All discretization procedures are equivalent as long as a full complete basis is used. In practice all procedudes contain a number of parameters and criteria, that make not all procedures equally applicable in practical calculations. Computational constraints may in fact become a severe problem. • As possibilities we can consider • the case of HO wave functions • diagonalization in a BOX • the case of discretized wave functions with scattering boundary conditions (CDCC) • Gamow states (complex energies)

16. Simple example to test different discretization procedures Two valence particles, moving in a one-dimensional Woods-Saxon potential V0, interacting via a residual density-dependent short-range attractive interaction. Modelling a drip-line system, one can choose the Fermi surface in such a way that there are no available bound states, and the two unperturbed particles must be in the continuum. The residual interaction V(x1,x2) = V0(x1-x2) ((x1+x2)/2)/0can be chosen in such a way that the final correlated wave function is however bound. Such a system is normally called “Borromean”

17. Diagonalization in a box WS single-particle states obtained imposing boundary conditions at a box (R=20 fm)

18. positive energy states bound states

19. Correlated energy of the two-particle system (as a function of the box radius)

20. The value of the binding energy is converging (with some oscillations) to the final value

21. Energy already practically correct with a box of 15 fm, but what about the wave function? In particular, how does it behave in the tail?

22. Radial dependence (x,x) Energy already practically correct with Rbox=15fm, but what about the wave function? In particular, how does it behave in the tail, essential for a proper description,e.g., of pair-transfer processes?

23. Logaritmic scale

24. R=15fm Correlated two-particle wave-function expanded over discretized two-particle positive energy states R=40fm

25. particle 1

26. Other option: diagonalization in a harmonic oscillator basis positive energy states bound states

27. WS single-particle states obtained from Harmonic Oscillator basis (N=10)

28. unphysical two-particle states (basis dependent) physical two-particle bound state

29. The value of the binding energy is converging (with some oscillations) to the final value

30. The radial dependence, however …. (x,x)

31. linear logaritmic

32.  Correlated two-particle wave-function expanded over discretized two-particle positive energy states (amplitudes **2) 

33. Another option: slicing the continuum by bunching scattering states (in steps of E)

34. But let us come now to the description of • two classes of reactions where pairing • correlations play a dominant role: • Two-particle transfer reaction • Break-up of a two-particle (Borromean) • halo system • The key point: how the structure properties • and the continuum enter into the reaction • mechanism?

35. Two-particle transfer reactions

36. Example of multinucleon transfers at Legnaro 1n 2n el+inel Neutron transfer channels 3n

37. Another classical example: Sn+Sn (superfluid on superfluid) +1n -1n +2n +3n +4n

38. A way to define a pairing “enhancement” factor P1 P2 (P1)2 distance of closest approach

41. Reaction mechanism and models for two-particle transfer processes Large number of different approaches, ranging from macroscopic to semi-microscopic and to fully microscopic. They all try to reduce the actual complexity of the problem, which is a four-body scattering (the two cores plus the two transferred particles).

42. Aside from the precise description of the reaction • mechanism (and therefore from the absolute values of • the cross sections), some points are more or less • well established • Angular distributions • Role of other multipole states • Q-value

43. Angular distributionWith light ions at forward angles one excites selectively 0+ states The excited states in 114Sn are of proton character at Z=50 closed shell 0+ 0+ 0+ pv

44. 112Sn(p,t)110Sn Lowest 0+,2+,4+ states Guazzoni etal Obs: Cross section to 0+ state order of magnitude larger at 0 degrees

45. l=0 l=0 strong selectivity at forward angle with angular momentum transfer l=2

46. 0+ 58Mn(p,t)56Mn 2+ 4+

47. Angular distribution Situation different for heavy-ions induced pair transfer processes: angular distributions are always peaked around the grazing angle, independently of the multipolarity 208Pb(16O,18O)206Pb gs two-step one-step

48. Higher multipolarities Far from the very forward angles the pairing vibrational states are overwhelmed by states with other multipolarities Example: predicted total cross sections in 120Sn(p,t)118Sn* reaction

49. GPV

50. Bump at 10 MeV does not come from GPV, but from incoherent sum of different multipolarities