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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

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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), ( 3 He,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 the B(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.

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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.

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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.

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

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

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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.

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Comparison with pure single-particle configurations T=0 T=3 (1g9/2) 2 (1h11/2) 2 pair transition density P (r,r)= (r )= Lotti, Vitturi etal

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r (fm) (r,r) (1d5/2) (1d5/2) (2s1/2) 2 (2s1/2) 2 18 O

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

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

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

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

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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.

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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)

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Simple example to test different discretization procedures Two valence particles, moving in a one-dimensional Woods-Saxon potential V 0, 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(x 1,x 2 ) = V 0 (x 1 -x 2 ) ((x 1 +x 2 )/2)/ 0 can be chosen in such a way that the final correlated wave function is however bound. Such a system is normally called “Borromean”

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

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positive energy states bound states

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

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

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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?

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Energy already practically correct with R box =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? Radial dependence (x,x)

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Logaritmic scale

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

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particle 1

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

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

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

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

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

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linear logaritmic

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Correlated two-particle wave-function expanded over discretized two-particle positive energy states (amplitudes **2)

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

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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?

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Two-particle transfer reactions

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

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

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A way to define a pairing “enhancement” factor P1P1 P2P2 (P 1 ) 2 distance of closest approach

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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).

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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

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

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

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

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58 Mn(p,t) 56 Mn

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

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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 120 Sn(p,t) 118 Sn* reaction

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GPV

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

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Q-value effect Keeping fixed any other parameter, the probability for populating a definite final channel depends on the Q-value of the reaction. The dependence is very strong in the case of heavy- ion induced reactions, weaker in the case of light ions. In the specific case of L=0 two-neutron transfer, the optimal Q-value is zero.

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RPA TDA sp

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RPA TDA sp

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Example (t,p) Quantal DWBA: one step di-neutron transfer Microscopic construction of the di-neutron transfer form factor (Glendenning or Bayman-Kallio methods) Options: zero range : only relative cross-sections or finite range : absolute cross sections (but needs the use of proper triton wf)

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Models for two-particle transfer reactions

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Example Semi-microscopic approach Reaction mechanism: one-step di-neutron (cluster) transfer Microscopy: Formfactor obtained by double-folding the microscopic pair densities of initial and final states with some nucleon-nucleon interaction or Simple folding of microscopic pair density in the target with the one-body mean field of the projectile

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Macroscopic approach Complete parallelism with inelastic excitation of collective surface modes Reaction mechanism: one step transfer produced by a new generalized pair field F ( r ) = P dU/dA = P (R/3A) dU/dr Where the ‘’deformation’’ parameter P is the pair- transfer matrix element and contains all the microscopy of the approach Very simple, appropriate for situations with many other coupled open channels Problem: recoil? Relative cross sections?

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Fully microscopic approach Reaction mechanism: 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 Building blocks: single-particle formfactors and wf’s Problems: quantal calculations rather complex (taking into account full recoil), semiclassical more feasible (but approximate treatment of recoil)

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A A+1 A+2 j ….. j3 j2 j1 E=0 Normal well-bound systems (intermediate bound states) Example |A=2> = i X i [a i + a i + ] 0 |A> one-particle transfer

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Example 208 Pb( 16 O, 18 O) 206 Pb 3p1/2 2s1/2 1d5/2 208 Pb 16 O 18 O 17 O (1d5/2) (2s1/2) 2

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1-particle transfer (d5/2) 2-particle transfer (d5/2) 2 2-particle transfer (correlated) 208 Pb( 16 O, 17,18 O) 207,206 Pb

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208 Pb( 16 O, 18 O) 206 Pb gs two-step one-step Maglione, Pollarolo, Vitturi, Broglia, Winther

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Basic problem: how is changed the picture as we move closer or even beyond the drip lines?

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Data from GANIL, Navin etal, 2011

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Extremely difficult to extract the fundamental 2/ ratio

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11 Li+p -> 9 Li+t Data from ISAC-2, TRIUMF Isao Tanihata etal 9 Li+t 10 Li+d (absent)

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Sensitivity to the pairing function in 11 Li P0: 3% of (s 1/2 ) 2 P2: 31% of (s 1/2 ) 2 P3: 45% of (s 1/2 ) 2

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A A+1 A+2 j2 j1 E=0 Systems closer to the drip lines (intermediate bound and unbound states) Example |A=2> = { i X i [a i + a i + ] 0 + ∫ dE X(E) [a + (E)a + (E)] 0 } |A> one-particle transfer to continuum

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208 Pb( 16 O, 17 O*) 207 Pb gs 1d5/2 2s1/2 d3/2 continuum bound

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continuum one-particle transfer process A A+1 A+2 Two-particle trasfer will proceed mainly by constructive interference of successive transfers through the (unbound) continuum intermediate states Systems at the drip lines (intermediate unbound states) |A=2> = ∫ dE X(E) [a + (E)a + (E)] 0 |A>

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Discretized continuum one-particle transfer process A A+1 A+2 The integration over the continuum intermediate states can becomes feasible by continuum discretization: but how many paths should we include? Thousands or few. for example only the resonant states?

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Basic problem Different discretization procedures lead to single-particle states with different behaviours on the tail. This produces one-particle transfer formfactors which are rather different. So even if the properties of the pair wave functions seem to be equally described by all methods with relatively small basis, the predicted probabilities for two-particle transfer turn out to converge only if an extremely large basis (often impossible to treat) is used. Alternative approaches?

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Break-up of a two-particle halo system is a rather complex 4-body process. Let us first start from break-up of a one-particle halo system, and and to make it simpler let us consider an one-dimensional case

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Simple modelling of one-particle halo break-up Single particle, initially moving in a one-dimensional Woods-Saxon potential V 0, perturbed by a time-dependent interaction V(x,t), assumed to be of gaussiam shape V(x,t)=V exp(-t 2 / t ) exp( -(x-x 0 ) 2 / x ) x0x0 x Obs: simulation of the nuclear field generated in a collision with a heavy partner

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The particle is assumed to be initially in one of the bound states N (x) of V N

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Exact full evolution of the system obtained by solving the time-dependent Schroedinger equation ih ∂ (x,t )/∂ t = [H 0 + V(x,t)] (x,t) with H 0 = -(h 2 /2 ) d 2 /dx 2 + V 0 (x)

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First example: initial bound state N=1 The time evolution of the wave function in this case is such that the final wave function is only distributed over different bound states (no role of continuum states) N

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Initial state Obs: parity non conservation (x,t)| P N t ∞

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Stronger coupling with excitation to the continuum (still with some final probability of being bound): partial break-up N

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Obs: change of scale bound states continuum states initial bound state N=3 Final wave function NOT confined in potential well

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Excitation from the last (weakly-bound) orbital: complete breakup N

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(x,t)| breakup initial state: bound state N=5 final state: moving fragments

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momentum distribution of the two fragments k (fm -1 )

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The solution of the problem has so far not involved any expansion into any basis. All continuum effects are automatically completely taken into account. However, to give the final Q-value distribution, one needs to look at the final wave function by expanding it into a set of basis states, including the continuum.

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E (MeV) Left fragment Right fragment dP/dE bound Asymmetric partial breakup Q-value final distribution E (MeV)

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Let us move now to the break-up of a two-particle halo system

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The perturbing interaction (that produces the break-up) is a one-body field (i.e. acting individually on each of the two particles). The enhanced two- particle break-up originates from the correlation in the two-particle wave function, and not from the reaction mechanism

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Time evolution

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Some conclusions: Correlated sequential one-particle transfer seems to be the proper reaction mechanism present in two-particle transfer processes and to offer a formalism able to directly test the microscopic features of the pair wave functions. This implies a knowledge of the pairing states expressed in a single particle basis, since the same single-particle states enter as intermediate states in the sequential reaction process For systems at the drip lines this means that it is mandatory a proper description of the continuum states Different continuum discretization procedures have to be consistently used in the structure and reaction sectors, and the behaviour outside of the nuclear radii has to carefully checked, being fundamental for the reaction process

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