Investigating nuclear pairing correlations via microscopic
two‐particle transfer reactions
Andrea Vitturi
Dipartimento di Fisica e Astronomia, Padova
1. Pairing correlations and space correlations
2. Reaction mechanism for two-particle transfer reactions
3. Two-particle transfer as a spectroscopic tool. One
example: the case of 68Ni
4. Q-value effects and the search for high-lying collective
pairing states
1. Pairing correlations and space correlations
Coulomb excitation, inelastic to collective states and singleparticle transfer reactions are to large extent well described by
first-order one-step mechanism. The extraction of structure
information is most of the times unique. But two-nucleon transfer
reactions are instead more complicated processes
(and multi-pair transfer are even more complicate …..)
It is however widely accepted that pairing correlations strongly
effect (and enhance) two-particle transfer reactions. But the
quantitative connection is not obvious. Will cross section scale with
the square of the two-particle transfer matrix elements? Or the
radial dependence of the two-particle transition densities contain
more information? And how this information enter into the reaction
mechanism? Available structure information is usually provided in terms
of T0, the square of the matrix element of the pair creation
(or removal) operator
P+ =∑j [a+ja+j]00
and it is often assumed that the cross section for twoparticle transfer just scale with T0. For this reason the
traditional way to define and measure the collectivity of
pairing modes is to compare with single-particle pair
transition densities and matrix elements to define some
“pairing” single-particle units and therefore “pairing”
enhancement factors.
Obs: We discuss here monopole pairing modes, i.e. 0+states
Typical “pairing” response
excited
states
gs
A
A+2
enhancement
g.s. in
210Pb Excited
0+ states Giant Pairing
Vibration in
210Pb 10
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 processes involving pairs of particles (pair
transfer, pair beak-up, pair knock-out, … ).
OBS: Pairs are correlated in space, but the correlation
depends on the specific involved orbitals
δρP(R,r) r Catara, Insolia, Maglione, Vitturi R 206Pb R R larger R, smaller r
R r (3p1/2)-2 (2f5/2)-2 Correlated g.s. (RPA)
Cooper pair wave-function (pair density) δρP(R,r) R
r
Pillet, Sandulescu, Schuck Other example: the case of 11Li
Hagi Hagino, Sagawa, Schuck
The physics underlying the space correlations can be clarified in a simple
one-dimensional model
Two valence particles, moving in a one-dimensional Woods-Saxon potential
V0, interacting via a residual density-dependent short-range attractive
interaction.
Modeling 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 δ(x1-x2) ρ((x1+x2)/2)/ρ0
can be chosen in such a way that the final correlated wave function is
however bound ( “Borromean” system)
Curro Perez-Bernal, Laura Moschini and A.V.
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 machinery used with bound states, the most popular
approach is the discretization of the continuum.
All discretization procedures are equivalent as long as a full complete
basis is used.
In practice all
procedures 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
• diagonalization in a basis given by HO wave functions
• impose boundary conditions in a BOX
• the case of discretized wave functions with scattering boundary
conditions (CDCC)
• Gamow states (complex energies)
Diagonalization in a box
WS single-particle states obtained imposing boundary
conditions at a box (R=20 fm)
Single-particle energies
positive energy
states
bound
states
Correlated energy of the two-particle system
(as a function of the box radius)
unphysical twoparticle states
(basis dependent)
physical two-particle
bound state
The value of the binding energy is converging
(with some oscillations) to the final value
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? Radial dependence
δρ(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? Logaritmic scale
Correlated
two-particle
wave-function
expanded over
discretized
two-particle
positive energy
states
R=15fm
OBS Enormous
number of
components
R=40fm
|Φ (x1,x2)|2
X2 (fm)
Two-particle correlated wave function: correlation clearly favors the situation
with the two particles on the same side
X1 (fm)
For comparison the situation with uncorrelated wave function
|Φ (x1,x2)|2
Effect of pairing correlation on the pairing density
ρP (x) =< A + 2|a† (x)a† (x)|A >= Ψ(x, x)
pairing
enhancement
factor
Space correlation strongly effected by mixing of states with different parity
Example: using only even
single particle states to
construct the two-particle
correlated wave function
A natural extension: alpha-clustering
Catara, Insolia, Maglione, Vitturi
Alpha-cluster probabilities in 232 Th, displayed in the intrinsic frame
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)
2. Reaction mechanism for two-particle transfer reactions
But if the qualitative behavior may be clear, the
quantitative aspects of two-particle 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
OBS: Mechanism well established
Cf talk by Enrico Vigezzi for
up-to-date implementations
Pairing will effect in a similar way different reactions, all involving
nucleon pairs (not only two-particle transfer, but also break-up of
two-particle halo nuclei or two-particle knock-out reactions). In all
cases the process is induced by the external one-body field, but
pairing correlations in the initial nucleus enhances it. Break-up of a two-particle halo system is a rather complex 4-body process.
To make it simpler let us consider again an one-dimensional case
(Hagino, Vitturi, Sagawa, Perez Bernal)
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 twoparticle break-up originates from the correlation in
the two-particle wave function, and not from the
reaction mechanism
Initial time
Final time
3. Two-particle transfer as a spectroscopic tool:
the case of 68Ni
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 E2+ energies and B(E2) values)
One example associated with possible breaking of shell closure:
68Ni
(N=40)
The case of 68Ni ( possible sub-shell at N=40 )
68Zn 69Zn 70Zn 71Zn 72Zn 73Zn 65Cu 66Cu 67Cu 68Cu 69Cu 70Cu 71Cu 72Cu 40 Z = 28 64Ni 65Ni 66Ni 67Ni 68Ni 69Ni 70Ni 71Ni N = 40 63Co 64Co 65Co 66Co 67Co 68Co 69Co 70Co g9/2 28 If our universe was more neutron-rich
HO magic
numbers would have not been present
279>($4&(:%')7*;>(
N292%*(
N=14,
16 new
shells only for smaller Z ……..
68Ni closed
The N=40 shell seems
to survive
forlocal
and disappear
e.g. Stanoiu PRC 69 (2004), Hoffmann PLB 672(2009)
Which mechanisms -> dramatic changes ?
Ni isotopes
e.g. Sorlin, Porquet PPNP 61 (2008), Phys. Scr T 152 (2013)
Otsuka Phys. Scr T152 (2013)
64Cr
6000
E(2+) (keV)
12Be
32Mg
32Mg
4000
2000
2000
6C
1000
500
0
0
8O
20Ca
14Si
1500
4Be
4 6 8 Ni
10 1268Ni
Neutron Number
56
0
68Ni
2000
3000
2500
=C<=->
B(E2;2+
%15(";@<AB
>( 0+)
(
3500
8O
E(2+) (keV)
40
E(2+)
10Ne
16S
12Mg
12 16 20 24
Neutron Number
1500
1000
500
0
28Ni
24Cr
26Fe
32 36 40 44
Neutron Number
Hannawald PRLS
82
2n (1999), Sorlin, EPJA16 (2003), Aoi, PRL 102 (2009)
Gade PRC 81 (2010), Ljungvall PRC81 (2010), Lenzi PRC82 (2010), W. Rother PRL106 (2011)
40
Neutron number
6*7"#$7(8%)"9(.:*#9%';*9*%(<'%=>?':/(@'%=(ABCADEAFEGAHG(
Weakening of the N=40 shell. 67Co between spherical (?)
OBS Quantum phase transitions in odd systems
67Co
68Ni
and deformed 66Fe
Spherical ?
Ni
Co
Fe
N=35
36
37
38
39
40
Deformed ?
……but what about two-particle transfer intensities ?
Jytte Elseviers (Leuven, ISOLDE)
815 2743 709 2512 E0 478 t (66Ni,68Ni) p
5-‐ 2+ 2849 0.86ms 2743 0+ 2512 2+ 21+ 02+ σ (0+exc) = 0.05 x σ (gs)
02+ 0+ 01+ 0+ 2033 1770 E0 01+ 2033 1770 276ns 0 68Ni Assuming N-40 closure and
neutron nature of excited 0+ state
weak
g9/2
0+exc
40 40 p1/2
p1/2
strong
gs
40 g9/2 p1/2 g9/2 p1/2 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
Excited state =
40 - β
40 g9/2 p1/2 g9/2 p1/2 g9/2 + β
40 p1/2 g9/2 + α
40 66
p1/2 68
Ni(t,p) Ni
0.2
2
gs: !(p1/2) +#(g9/2)
2
+
exc
0
2
: -#(p1/2) + !(g9/2)
Mixing coefficient will
be determined from
fitting experimental
value. But need for
different experiment for
discrimination (e.g. with
heavy ions)
"exc/"g.s.
0.15
pure
configuration
0.1
0.05
0
0.5
2
possible values
experiment
0.6
0.7
α!2
2
0.8
0.9
1
The relative cross sections transferring two particles in the p1/2 and g9/2 is
reversed in the case of (14C,12C) reaction, the latter being favored in this case.
But the value of the cross section ratio is still highly selective
66
68
66
14 12 68
Ni(t,p)
Ni( Ni
C, C) Ni
2
0.2
2
gs: !(p1/2) +#(g9/2)
2
+
exc
0
2
: -#(p1/2) + !(g9/2)
2
1.5
"exc/"gs
"exc/"g.s.
0.15
0.1
0.05
1
pure
Exp ?
pure
configuration
configuration
possible values
0.5
experiment 00.5
0.6
0.7
0.8
!
2
0.9
1
4. Q-value effects and the search for highlying collective pairing states
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 optimal Q-value) is very strong in the case of
heavy-ion induced reactions, weaker in the case of light ions.
The optimal 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 close to
zero.
But the actual Q-value for two-particle transfer to the (pairing
collective) ground states may be different from zero …..
Experimental evidence
counts
Negligible transitions to GS
due to Q-value effects.
What information on pairing
correlations?
Example
96Zr+40Ca
Selecting final
42Ca mass partition
(two-neutron transfer)
gs
Corradi etal, LNL
excited
states
Total kinetic energy loss (MeV)
(or excitation energy in 42Ca)
Playing with different combinations of projectile/target
(having different Qgg-value) one can favor different
energy windows
σ(Q) = σ(Qopt ) exp(−(Q − Qopt )2 /a(Ecm )2 )
Example: Target 208Pb Final 210Pb (at bombarding energy
Ecm = 1.2 Ebarrier)
208Pb (AX,A-2X) 210Pb
excited states
gs
As a result, the correlated gs state may be populated in a much
weaker way than (excited) uncorrelated states !!
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)
High-lying pairing resonances (giant pairing resonances) 0+ states
In addition to the lowest collective state
one expects pair transfer strength at higher
energies. This distribution will be strongly
affected by the existence of major shells,
such that the inter-shell distance is
appreciably larger than the distance between
the levels within a shell. In this case a
concentration of pairing strength in a single
state is expected for each major shell
collective non collective gpv 2hω gs A+2 Example
g.s. in
210Pb Collective
Giant Pairing
Vibration in
210Pb Excited
non-collective
0+ states 10
If our goal is to favor the excitation of a high-lying
state, as in the case of the giant pairing
vibration, it may be useful the use of a weakly-bound
projectile (eg 6He) to populate the GPV in a stable
nucleus, to profit from favorable Q-value
matching (cf Fortunato, Vitturi, Von Oertzen) Q00-Qopt ≈ 10 MeV
Projectiles that leads to
large positive Q-values for
two-particle transfer
to the ground state
In more details …….
15
0
5
10
Q00-Qopt
208Pb(AX,A-2X)210Pb
15
15
Q00-Qopt ≈ 10 MeV
15.00
10.00
5.000
Proton number
0
10
10
-5.000
-10.00
-15.00
-20.00
-25.00
-30.00
5
5
6He
11Li
0
0
5
10
Neutron number
0
15
Two-neutron transfer leading to 210Pb
“weakly-bound”
projectiles
6He
100
0
208Pb(11Li,9Li)210Pb
5
10
15
100
10
0
gs
1
0.01
1E-3
Energy [MeV]
10
gs
0
5
1
1
0.1
0.1
0.01
0.01
1E-3
0
10
Energy [MeV]
Obs: in all cases Ecm = 1.2 EB
15
10
T/Tgs
gs
1E-3
10
gs
gs
1
gpv
0.1
0.1
0.01
T/Tgs
0.01
10
15
10
208Pb(18O,16O)210Pb
gpv
0.1
0
10
1
0.1
1E-3
5
T/Tgs
gs
1
gpv
10
gs
gs
208Pb(6He,4He)210Pb
18O
T/Tgs
T/Tgs
10
“normal” projectile
gs
11Li
T/Tgs
Pairing strength ( Pair transfer cross section)
From pairing strength to pair-transfer cross section: modulation
due to Q-value effect
0.01
1E-3
1E-3
0
5
10
15
Energy [MeV]
Dasso, Sofia, ViFuri 20
10
15
20
20
208Pb(AX,A-2X)210Pb
15
Proton number
Using different projectiles,
Q-value effects may favor or not the
excitation of the GPV.
I select in the figures the part of the
nuclear chart where the population
of the GPV is:
5
gs > gpv
10
10
gpv >gs
5
1. larger that that of the
ground state
15
5
10
5
15
20
15
20
20
Neutron number
20
5
10
208Pb(AX,A-2X)210Pb
Proton number
2. larger than 10 percent of the
ground state
15
gs > 10*gpv
15
10
10
5
5
gpv > 0.1*gs
5
10
Neutron number
15
20
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