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Pairing & low-lying continuum states in 6He
Lorenzo Fortunato
Dip. Fisica e Astronomia «G.Galilei»,
University of Padova
&
I.N.F.N. – Sez. di Padova
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Collaboration & Acknowledgements
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Work in collaboration with :
• Jagjit Singh – Padova Univ. (Italy)
Special thanks
• Rajdeep Chatterjee – I.I.T. Roorkee (India)
• Andrea Vitturi – Padova Univ. (Italy)
L. Fortunato
Motivation
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Both experimentally and theoretically there are still large
uncertainties on the structure of light systems close to the dripline. The case I want to discuss is 6He and its connections with
5He.
•
4He
is very tightly bound (core exc. are at very high energy)
•
5He
is unbound (two low-lying resonances)
•
6He
is weakly bound in its g.s. and it has a number of
resonances that have been recently re-investigated at
GANIL. 6He is borromean it has halo features and one would
like to understand the role of the pairing interaction in making
it bound.
L. Fortunato
Outline
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1. Comparison of 5He and 6He spectra
2. Calculation of unbound resonant p-states in 5He
3. Construction of a basis for two-particle states made up on
unbound single-particle states
4. Calculation of pairing matrix elements
5. Diagonalization of the hamiltonian with this simple pairing
Results
1. Bound 6He J=0+ ground state, continuum J=0 +,1 +,2 +
2. Electromagnetic λ=2 response and identification of
resonances in the continuum
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Comparison of spectra
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New data!
p(8He,t)
X.Mougeot
et al., PLB
718 (2012)
441-446
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Recent experiment
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p(8He,t)
Picture from
X.Mougeot et al., PLB 718 (2012) 441-446
Comparison of experiments and theories
Picture from
X.Mougeot et al., PLB 718 (2012) 441-446
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Another way of representing these data
Data in black from TUNL and NNDC
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L. Fortunato
5He resonances
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The p3/2 and p1/2
resonances of 5He
are reproduced with a
Wood-Saxon
potential plus spinorbit that gives
correct energy
centroids and widths.
They range from
0<r<100 fm and from
0<EC<10 MeV
L. Fortunato
Discretizing the continuum
Piyadasa et al. PRC 60, 044611 (1999)
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Alternative ...
With a different program we checked that these wavefunctions
are OK, by calculating the phase-shifts for similar potentials
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Poles of the S-matrix
As a test, we peform countour integration (residues) on the Smatrix in the complex plane to pinpoint the position of the poles.
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Two-particle system
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Each single-particle unbound orbital reads :
The two-particle states can be constructed as :
p3/2
p1/2
p3/2
p1/2
0+, 2+
1+, 2+
0+
Total of 5 states built from p2 configurations
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Contact delta-interaction
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Generalization of
Slater integral
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Procedure
Construct
5He p
3/2 and
p1/2 states 010 MeV
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Construct
the twoparticle J=0
basis states
Calculate the
matrix elem.
with Pairing
interaction
( ~ 9 Gb !! )
(2.4 Gb each !)
( 0.5 Mb )
Get eigenvalues
and eigenvectors
( ~ 9.7 Gb !! )
Diagonalize the
total hamiltonian:
g
H= ε1+ε2+<|V|>
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The basis is built like this ... for each J
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Results of diagonalisation for J=0, various basis sizes
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J=0 ground state wavefunction
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J=0 ground state probability density
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Composition in terms of basis states
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J=2 states
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J=2 two-particle continuum state (oscillating both in r1 and r2)
with EC = 8.0 MeV - picture of w.f. yet to be antisymmetrized L. Fortunato
Preliminary calculation of E2 Response - 1
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Centroid ~0.8 MeV
1.0E+02
Width ~0.11 MeV
8.0E+01
G=147
G=300
G=475
G=500
6.0E+01
4.0E+01
2.0E+01
0.0E+00
0
0.5
1
1.5
2
2.5
This is a calculation limited to a reduced model space containing
only (p3/2)2 configurations (that is 0+ and first 2+), used to find the
appropriate value for the pairing strength that reproduces the
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narrow 2+ resonance.
Preliminary calculation of E2 response - 2
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500
450
The narrow 2+ resonance is
obtained at the right energy and
with a consistent width.
400
350
300
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There is another bump
200
150
100
50
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
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Preliminary calculation of E2 response - 3
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50
Unfinished
calculations
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40
35
30
25
20
15
10
5
0
0
0.5
1
1.5
2
2.5
3
3.5
Second resonance at ~ 2.7 MeV with larger
width (maybe ~ 1.1 MeV)
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4.5
5
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Conclusions and perspectives
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1. We have shown how the bound borromean ground state of
6He emerges from the coupling of two unbound p-waves in
the 5He continuum, due to the presence of the pairing
interaction. Other similar studies have used artificially bound
p-states or have used a box to discretize the continuum.
2. We obtain a well-behaved 6He ground state and we are
studying the electromagnetic response to continuum states
(E2 and M1 are feasible within our model space).
3. The 2+ resonances look good, though the second does not
match with the recent experiment.
Future plans: J.Singh will perform more tests and calculations to see
whether the predictions are modified by different choice of pairing interaction
(density dependent?), energy cuts, model space (inclusion of s-states?), etc.
L. Fortunato