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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 1
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
L. L. FortunatFortunatoo
Work in collaboration with :
•Jagjit Singh – Padova Univ. (Italy)
•Rajdeep Chatterjee – I.I.T. Roorkee (India)
•Andrea Vitturi – Padova Univ. (Italy)
Special thanks
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Motivation
L. L. FortunatFortunatoo
Both experimentally and theoretically there are still large uncertainties on the structure of light systems close to the drip-line. 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.
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Outline
L. L. FortunatFortunatoo
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
1. Bound 6He J=0+ ground state, continuum J=0 +,1 +,2 +
2. Electromagnetic λ=2 response and identification of resonances in the continuum
Results
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Comparison of spectra
L. L. FortunatFortunatoo
New data!
p(8He,t)
X.Mougeot et al., PLB 718 (2012)
441-446
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Recent experiment
Picture from X.Mougeot et al., PLB 718 (2012) 441-446
p(8He,t)
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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
L. L. FortunatFortunatoo
Data in black from TUNL and NNDC
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5He resonances
L. L. FortunatFortunatoo
The p3/2 and p1/2 resonances of 5He are reproduced with a Wood-Saxon potential plus spin-orbit that gives correct energy centroids and widths.
They range from 0<r<100 fm and from 0<EC<10 MeV
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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 S-matrix in the complex plane to pinpoint the position of the poles.
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Two-particle system
L. L. FortunatFortunatoo
Each single-particle unbound orbital reads :
The two-particle states can be constructed as :
p3/2 p1/2
p3/2 0+, 2+ 1+, 2+
p1/2 0+
Total of 5 states built from p2 configurations
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Contact delta-interaction
L. L. FortunatFortunatoo
Generalization of Slater integral
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Procedure
L. L. FortunatFortunatoo
Construct 5He p3/2 and p1/2 states 0-
10 MeV
Construct the two-
particle J=0 basis states
Calculate the matrix elem. with Pairing interaction
(2.4 Gb each !) ( ~ 9 Gb !! )
Diagonalize the total hamiltonian:
H= ε1+ε2+<|V|>
Get eigenvalues and eigenvectors
( 0.5 Mb )
( ~ 9.7 Gb !! )g
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The basis is built like this ... for each J
L. L. FortunatFortunatoo
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Results of diagonalisation for J=0, various basis sizes
L. L. FortunatFortunatoo
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J=0 ground state wavefunction
L. L. FortunatFortunatoo
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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
L. L. FortunatFortunatoo
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 -
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Preliminary calculation of E2 Response - 1
L. L. FortunatFortunatoo
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 narrow 2+ resonance.
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Centroid ~0.8 MeV
Width ~0.11 MeV
Preliminary calculation of E2 response - 2
L. L. FortunatFortunatoo
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The narrow 2+ resonance is obtained at the right energy and
with a consistent width.
There is another bump
Preliminary calculation of E2 response - 3
L. L. FortunatFortunatoo
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Second resonance at ~ 2.7 MeV with larger width (maybe ~ 1.1 MeV)
Unfinished calculations
Conclusions and perspectives
L. L. FortunatFortunatoo
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.
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