Two-particle transfer and pairing correlations (for both T-0 and T=1): interplay of reaction mechanism and structure properties Andrea Vitturi and Jose Antonio Lay (Padova, Catania, Sevilla) ECT*, September 2016
Basic point to discuss: how reactions involving pair of nucleons (e.g. two-particle transfer, two-particle breakup, two-particle knock-out) can provide quantitative signature of collectivity in T=0 and T=1 pairing modes This dynamical source of information should be complementary to the one associated to static properties (as correlation energies, for example)
How to use dynamics to study pairing correlations? The main road is clearly provided by the study of those processes where a pair of particles in involved, e.g. transferred from/to another nucleus (two-particle transfer) or ejected onto the continuum (two-particle break-up or two-particle knock-out). Clearly the probabilities for such processes must be influenced by the particle-particle correlations, but the quantitative connection is not obvious.
Unfortunately, the situation is different, for example, from low-energy one-step Coulomb excitation, where the excitation probability is directly proportional to the B(Eλ) 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.
It is often assumed that the cross section for two-particle transfer just scale with T 0, the square of the matrix element of the pair creation (or removal) operator P + = j [a + ja + j] 00 For this reason the easiest way to define and measure the collectivity of pairing modes is to compare with singleparticle pair transition densities and matrix elements to define some pairing single-particle units and therefore pairing enhancement factors. Obs: We discuss for the moment monopole pairing modes, i.e. 0+states
Typical pairing response excited states A gs A+2 enhancement g.s. in 210 Pb Excited 0 + states Giant Pairing Vibration in 210 Pb 10
Pair strength function 22 O (d3/2) 2 enhancement (f7/2) 2 Khan, Sandulescu,Van Giai, Grasso
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.
Comparison with pure single-particle configurations pair transition density ρ ν P (r,r)=κν (rσ)=<0 c(rσ)c(rσ) ν> T=0 T=3 (1g9/2) 2 (1h11/2) 2 Lotti, Vitturi etal
18 O (2s1/2) 2 δρ (r,r) 0.8 (1d5/2) 2 + 0.6(2s1/2) 2 (1d5/2) 2 r (fm)
206 Pb (3p1/2) 2 (2f5/2) 2 Ψ(r 1,r 2 ) 2 as a function of r 2, for fixed r 1 Correlated ground state OBS: mixing of configurations with opposite parity position of particle 1
Lotti etal particle-particle spatial correlations Ψ(r 1,r 2 ) 2 as a function of r 2, for fixed r 1 Neutron addition mode: ground state of 210 Pb position of particle 1 (1g9/2) 2
r R δρ P (R,r) Catara etal, 1984 206 Pb R R R r (3p1/2) 2 (2f5/2) 2 Correlated g.s.
δρ P (R,r) R r Pillet, Sandulescu, Schuck
Interesting problem: how is changed the picture as we move closer or even beyond the drip lines? Example: the case of 6 He R r Oganessian, Zagrebaev, Vaagen, 1999
Other example: the case of 11 Li
A natural extension: alpha-clustering Catara, Insolia, Maglione, Vitturi (1984) Alpha-cluster probabilities in 232 Th, displayed in the intrinsic frame by projecting over an alpha particle wave function 4 particles in pure (time-reversed) Nilsson orbits Nilsson+BCS for both protons and neutrons OBS: 1. Alpha-probability distributed over the entire surface 2. Total alpha spectroscopic factor Sα increases orders of magnitude (although still a factor 10 smaller than experiment)
A possible phenomenological approach to define enhancement factors from transfer reactions The classical example: Sn+Sn (superfluid on superfluid) +1n -1n Von Oertzen, Bohlen etal +2n +3n +4n
A way to define a pairing enhancement factor, by plotting transfer probabilities not as function of the scattering angle, but as function of the distance of closest approach of the corresponding classical trajectory P 1 P 2 (P 1 ) 2 distance of closest approach
General problem: how separate the contribution of 0+ states? How to separate the contribution from the collective ground state?
Proton transfer
Another example : multinucleon transfers at Legnaro 40 Ca+ 208 Pb el+inel 1n 2n Neutron transfer channels (odd-even transfer effect?) 3n
Corradi etal, LNL -2n channel ( 42 Ca) enhancement factor
Q-value effect Aside from the precise description of the reaction mechanism (and therefore from the absolute values of the cross sections) one has to take into account Q-value effects. In fact, keeping fixed any other parameter, the probability for populating a definite final channel depends on the Q-value of the reaction. The dependence (in first approximation a gaussian distribution centered in the optimum Q-value) is very strong in the case of heavy-ion induced reactions, weaker in the case of light ions. The optimum Q-value depends on the angular momentum transfer and on the charge of the transferred particles. In the specific case of L=0 two-neutron transfer, the optimal Q-value is approximately zero. But the actual Q-value for two-particle transfer to the ground states may be different from zero..
Experimental evidence: one example 96 Zr+ 40 Ca Selecting final 42 Ca mass partition In this case the population of the collective ground state is hindered by the Q-value window. What information on pairing correlations? counts gs excited states Total kinetic energy loss (MeV)
As a result, the correlated states may be populated in a much weaker way than uncorrelated states Example: 96 Zr+ 40 Ca, leading to 42 Ca In this case is favored the excitation of an uncorrelated 0+ state at about 6 MeV Corradi, Pollarolo etal, LNL gs
Playing with different combinations of projectile/target (having different Q gg -value) one can favour different energy windows Example: Target 208 Pb Final 210 Pb (at bombarding energy E cm = 1.2 E barrier ) gs excited states
The pairing strength (as a function of the excitation energy) is therefore modulated by the Q-value cut-off to yield the final two-particle cross section RPA TDA sp Sofia, Vitturi
But the Q-value window can be used also with some profit. For example the occurrence of large positive Q-value for the ground state transition leads to optimal kinematics conditions for high-lying states. This is the region of the still hunted Giant Pairing Vibration (GPV) RPA TDA sp Obs: What about GPV for T=0 modes?
But let us move to two-particle transfer reactions and the associated reaction mechanism. In fact, if the qualitative behavior may be clear, the quantitative aspects of twoparticle transfer require a proper treatment of the reaction mechanism. 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), to more tractable frameworks.
To provide deep structure information, the approach must be microscopic. The fully microscopic approach is based on correlated sequential two-step process (each step transfers one particle) Microscopy: Pairing enhancement comes from the coherent interference of the different paths through the different intermediate states in (a-1) and (A+1) nuclei, due to the correlations in initial and final wave functions Basic idea: dominance of mean field, which provides the framework for defining the single-particle content of the correlated wave functions
Example 208 Pb( 16 O, 18 O) 206 Pb 2s1/2 3p1/2 0.6 0.8 1d5/2 16 O 17 O 18 O 208 Pb 0.8 (1d5/2) 2 + 0.6(2s1/2) 2
208 Pb( 16 O, 17,18 O) 207,206 Pb 1-particle transfer (d5/2) 2-particle transfer (correlated) σ T =3.90 mb 2-particle transfer (d5/2) 2 σ T =1.98 mb Obs: to get cross sections one needs optical potentials
The transfer probabilities with the involved orbital. In addition whether the final wave function only involves a pure orbital, or whether it is correlated 110 Sn(t,p) 112 Sn σ(mb/sr) 10-2 10-3 10-4 correlated BCS wave function (g 7/2 ) 2 σ= 0.028 µb (d 5/2 ) 2 σ= 1.134 µb (d 3/2 ) 2 σ= 0.473 µb (s 1/2 ) 2 σ= 1.078 µb (h 11/2 ) 2 σ= 0.075 µb BCS (1) σ=2.536 µb BCS (2) σ=3.482 µb BCS (3) σ=1.330 µb pure orbitals 10-5 0 10 20 30 40 50 60 θ(deg) OBS: The shape of the angular distribution is the same, being associated with the L=0 transfer
Does the two-particle transfer cross in superfluid systems simply scale with the pairing gap value Δ? The two-particle transfer probability is enhanced by the pairing correlation but does NOT scale simply with the value of the pairing gap Δ. In fact, if we alter the sequence of single-particle levels, and change the value of the pairing strength G to obtain the same Δ, the cross section will be different, due to different interplay of the single-particle states BCS 1 BCS 2 BCS 3 i(mev) B i i (MeV) B i i (MeV) B i 0g 7/2-0.027 0.75-0.027 1.15-2.027 0.64 1d 5/2 0.882 1.13-0.118 0.57 0.882 1.02 2s 1/2 1.330 0.53-0.670 0.33 1.330 0.59 0h 11/2 2.507 0.79 4.507 0.61 5.507 0.46 2d 3/2 2.905 0.39 2.905 0.26 2.905 0.27 σ (mb) 2.5 3.4 1.3
We consider the same case as before, i.e. the transfer of two neutrons from 110 Sn to 112 Sn (0+; gs) using the reactions (14C,12C) or (18O,16O) In addition to the information on the target, we need now to specify on which orbit the particle are transferred in the projectile In the (14C,12C) the two neutrons are assumed to be picked-up from the p1/2 shell. In the (18O,16O) from the pure d5/2 shell, or from a combination of (d5/2) 2 and (s1/2) 2
If we consider the same case as before, i.e. the transfer of two neutrons from 110 Sn to 112 Sn (0+; gs), but using different reactions, e.g. (14C,12C) or (18O,16O) the ranking of the cross sections associated to the different orbitals changes. projectile single-particle orbital in the target (t,p) ( 14 C, 12 C) ( 18 O, 16 O) (0p 1/2 ) 2 (1s 1/2 ) 2 (0p 3/2 ) 2 (0d 5/2 ) 2 (0d 5/2 ) 2 (1s 1/2 ) 2 Conf 1 112 Sn (0g 7/2 ) 2 2.80E-5 1.73E-5 1.19E-4 7.09E-4 9.00E-4 1.19E-3 2.01E-4 1.24E-3 (1d 5/2 ) 2 1.13E-3 3.00E-2 4.71E-3 5.54E-3 1.18E-3 3.55E-3 1.13E-2 1.19E-2 (2s 1/2 ) 2 1.08E-3 1.53E-2 5.38E-3 7.05E-3 1.16E-3 5.02E-3 1.42E-2 1.59E-2 (1d 3/2 ) 2 4.73E-4 1.34E-3 2.79E-3 9.87E-3 4.14E-3 1.26E-2 6.62E-3 1.83E-2 (0h 11/2 ) 2 7.50E-5 7.77E-4 5.29E-5 1.05E-4 7.65E-5 1.10E-4 9.06E-5 1.88E-4 Conf A 2.54E-3 8.77E-2 2.26E-2 3.77E-2 1.21E-2 8.60E-3 1.95E-2 7.53E-2 BCS
Same results shown as histograms Total two-particle transfer cross section σ projectile BCS single-particle orbital in the target
Basic outcome of microscopic description of pair-transfer processes: strong dependence of the two-particle transfer probability on the microscopic structure of the colliding nuclei. Two-particle transfer can therefore to be used as a quantitative spectroscopic tool: in particular the population of the 0 + states will be an indication of the changes in the underlying structure (in addition to the usual E 2+ energies and B(E2) values) One example associated with possible breaking of shell closure: 68 Ni (N=40) J.A. Lay, L. Fortunato and A. Vitturi
The case of 68 Ni ( possible sub-shell at N=40 ) 65 Cu 68 Zn 69 Zn 70 Zn 71 Zn 72 Zn 73 Zn 66 Cu 67 Cu 68 Cu 69 Cu 70 Cu 71 Cu 72 Cu 40 Z = 28 64 Ni 65 Ni 66 Ni 67 Ni 68 Ni 69 Ni 70 Ni 71 Ni 63 Co 64 Co 65 Co 66 Co 67 Co 68 Co 69 Co 70 Co 28 N = 40 g 9/2
The N=40 shell seems to survive for 68 Ni and disappear only for smaller Z.. Ni isotopes E(2+) 2000 68 Ni E(2+) (kev) 1500 28 Ni B(E2;2+ -> 0+) 1000 500 24 Cr 26 Fe 56 Ni 68 Ni 0 32 36 40 44 Neutron Number S 2n ) 40 Neutron number
Weakening of the N=40 shell. 67 Co between spherical (?) 68 Ni and deformed 66 Fe OBS Quantum phase transitions in odd systems 67 Co Spherical? Ni Co Fe N=35 36 37 38 39 40 Deformed?
but what about two-particle transfer intensities? t ( 66 Ni, 68 Ni) p Jytte Elseviers (Leuven, ISOLDE) 0 1 + 815 2743 709 2512 E0 478 5-2 + 0 + 2849 0.86ms 2743 2512 0 2 + 2 + 0 + 2033 1770 E0 2033 1770 276ns 0 2 + 2 1 + 0 1 + 0 + 68 Ni 0 σ (0+ exc ) = 0.05 x σ (gs)
Assuming N-40 closure and neutron nature of excited 0+ state weak g9/2 40 0 + exc 40 g 9/2 p 1/2 p1/2 strong p1/2 gs 40 g 9/2 p 1/2 But microscopic calculation for (t,p) gives a ratio σexc/σgs a factor 3 too large. This may imply mixing of 0p-0h with 2p-2h components
Ground state = α# 40! g 9/2! + β# 40! g 9/2! p 1/2! p 1/2! Excited state = - β# 40! g 9/2! + α# 40! g 9/2! p 1/2! p 1/2! 66 68 Ni(t,p) Ni 0.2 gs:!(p 1/2 ) 2 +#(g 9/2 ) 2 0 + exc : -#(p 1/2 )2 +!(g 9/2 ) 2 0.15 pure configuration Mixing coefficient will be determined from fitting experimental value. But need for different experiment for discrimination (eg with heavy ions) " exc /" g.s. 0.1 possible values experiment 0.05 0 0.5 0.6 0.7 0.8 0.9 1! 2 α 2
Let me try to apply the concepts so far described to the case of a pn transfer, e.g in a ( 3 He,p) reaction + 0 T=0 A X L=0 S=1 L=2 S=1 L=0 S=0 1 + T=0 24 Mg( 3 He,p) 26 Al 3 He@25 MeV at Grand Raiden (GR) high resolution spectrometer A+2 Y 0 + T=1 Calculations within 2nd order DWBA calculations (code FRESCO) Spectroscopic information from Shell Model calculations (USDB) Do we get information on collectivity of T=0 and T=1 modes from just σ(0+)/σ(1+)?
24 Mg( 3 He,p) 26 Al (1 + ) Check of coherence for L=0 transfer dσ/dω (mb/sr) 10 0 10-1 L=0 full (d 5/2 ) 2 +(s 1/2 ) 2 +(d 3/2 ) 2 +( (d 5/2 ) 2 +(s 1/2 ) 2 +(d 3/2 ) 2 (d 5/2 ) 2 +(s 1/2 ) 2 (d 5/2 ) 2 0 10 20 30 θ c.m. (deg)
Effect of contributions from L=0 and L=2 for 1+ states 24 Mg( 3 He,p) 26 Al 10 0 10-1 Preliminary 1+ only jj jj simultaneous All configurations dσ/dω (mb/sr) 10-2 10-3 0 20 40 θ c.o.m. (deg) ) Components with j 6= j increases (1 + ) reducing (0 + )/ (1 + ) ) Test of tensor interactions Small components (2s1/2)(1d3/2) and (1d5/2)(1d3/2) but they further increase the cross section
Comparing enhancement factors for 0+ and 1+ states 24 Mg( 3 He,p) 26 Al (0 + ) 24 Mg( 3 He,p) 26 Al (1 + ) 10 0 10 0 Full wavefunction Pure (d 5/2 ) 2 Full wavefunction Pure (d 5/2 ) 2 dσ/dω (mb/sr) 10-1 dσ/dω (mb/sr) 10-1 10-2 0 10 20 30 θ c.m. (deg) 10-2 0 10 20 30 θ c.m. (deg)