Transcript Nicolas Michel - CEA-Irfu

Isospin mixing and the
continuum coupling in
weakly bound nuclei
Nicolas Michel (University of Jyväskylä)
Marek Ploszajczak (GANIL)
Witek Nazarewicz (ORNL – University of Tennessee)
April 26-29, 2011
CEA / IRFU / SPhN / ESNT
Nicolas Michel
Plan
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Experimental motivation
Berggren completeness relation and Gamow Shell Model
Cluster orbital shell model and Hamiltonian definition
Spectroscopic factor definition
Treatment of Coulomb interaction and recoil term
Isospin symmetry breaking in 6He, 6Be and 6Li
Spectroscopic factors, energies, T+/- and T2 expectation values
Conclusion
April 26-29, 2011
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Halos, resonant states
5
He +n
1867.5
2  1797
4


10
Li + n
9
Li +2n
11


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
 
3/2


9 
Be +2n
Li
10
He +2n
325

300
Be +n
11



505
1/2 
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
He
7316

Be
6 
0
964
1/2 320
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April 26-29, 2011
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Gamow states
• Georg Gamow : simple model for a decay
G.A. Gamow, Zs f. Phys. 51 (1928) 204; 52 (1928) 510
• Definition :
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Complex scaling
• Calculation of radial integrals: exterior complex scaling
• Analytic continuation : integral independent of R and θ
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Complex energy states
Im(k)
Berggren completeness relation
bound states
narrow resonances
Re(k)
antibound states
L+ : arbitrary contour
capturing states
April 26-29, 2011
broad resonances
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Completeness relation
with Gamow states
• Berggren completeness relation (l,j) :
T. Berggren, Nucl. Phys. A 109, (1967) 205 (neutrons only)
Extended to proton case (N. Michel, J. Math. Phys., 49, 022109 (2008))
• Continuum discretization:
• N-body completeness relation:
April 26-29, 2011
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Cluster orbital shell model
• Shell model : 3A degrees of freedom (particles coordinates)
3(A-1) physically (translational invariance) → spurious states
• Lawson method (standard shell model) :
Nħω spaces only : unavailable for Berggren bases
• Solution : cluster orbital shell model, core coordinates.
Relative coordinates: no center of mass excitation
April 26-29, 2011
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Hamiltonian definition
•
6He, 6Be, 6Li:
valence particles, 4He core :
H = T1b + WS(5Li/5He) + MSGI + Vc + Trec
0p3/2 (resonant), contours of s1/2, p3/2, p1/2, d5/2, d3/2 scattering states, recoil included
MSGI : Modified Surface Gaussian Interaction:
•
6Be:
Coulomb interaction necessary
Problem: long-range, lengthy 2D complex scaling, divergences
Solution: one-body long-range / two-body short-range separation
H1b one-body basis:
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Spectroscopic factors in GSM
• One particle emission channel: (l,j,p/n)
• Basis-independent definition:
• Experimental : all energies taken into account
• Standard : representation dependence (n,l,j,p/n)
• 5He / 6He, 5Li / 6Be, 5He / 6Li, 5Li / 6Li
non resonant components necessary.
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Coulomb interaction
and recoil term
Harmonic oscillator expansion
Physical precision
of the order of 1 keV
Sufficient for practical applications
N. Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315
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6Be/5Li
– 6He/5He
Cusps (π)
6Li/5He
– 6Li/5Li
Cusps (ν)
π asymptotic ≠ ν asymptotic
N. Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315
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Spectroscopic factors distribution
Re[S2] > 1, Im[S2] ≠ 0
Large occupation of non-resonant continuum
N. Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315
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Observables of 0+, 2+ (T=1) states
6He
0+ state
6Be
(V1)
6Be
(V2)
6Li
6Li
(V1)
(V2)
Ecalc (MeV)
-0.974
1.653
1.371
0.0866
-0.0706
Eexp (MeV)
-0.973
1.371
1.371
0.136
0.136
Γcalc (keV)
0
41
14
8.85 ·10-3
9.13 ·10-3
Γexp (keV)
0
92
92
8.2 ·10-3
8.2 ·10-3
S2 (π)
————
1.015-i0.147
1.015-i0.177
1.061-i0.280
1.028-i0.300
S2 (ν)
0.87-i0.383
—————
—————
0.911-i0.361
0.898-i0.369
6He
2+ state
6Be
(V1)
6Be
(V2)
6Li
(V1)
6Li
(V2)
Ecalc (MeV)
0.823
2.887
2.679
1.667
1.569
Eexp (MeV)
0.824
3.041
3.041
1.667
1.667
Γcalc (keV)
89
986
804
404
329
Γexp (keV)
113
1160
1160
541
541
S2 (π)
—————
0.973-i0.014
0.978-i0.016
0.987-i0.003
0.993-i0.003
S2 (ν)
1.061+i0.001
—————
—————
1.034-i0.024
1.043-i0.022
V1 : WSnucl(π) = WSnucl(ν)
V2 : WS(π) fitted to 6Be binding energy
N. Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315
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Configuration mixing
of 0+ (T=1) states
(Ck)2
6He
6He (rig. core)
6Be
(0p3/2)2
0.750-i 0.692
0.798-i 0.732
1.090-i 0.243
1.107-i 0.288
0.994-i0.587
0.949-i0.614
S1(πp3/2)
—————
—————
-0.115+i0.218
-0.143+i0.255
-0.084+i0.226
-0.050+i0.244
S1(νp3/2)
0.243+i0.619
0.244+i0.668
—————
—————
0.066+i0.308
0.0797+i0.314
S2(s1/2)
0.009+i0.0
0.0+i0.0
0.022+i0.0
0.023+i0.004
0.011+i0.0
0.010+i0.0
S2(p1/2)
0.012+i0.0
0.013+i0.0
0.008+i0.001
0.009+i0.0
0.011+i0.0
0.012+i0.0
S2(p 3/2)
-0.049-i0.074
-0.063+i0.065
-0.030+i0.029
-0.028+i0.034
-0.033+i0.054
-0.033+i0.055
S2(d3/2)
0.002+i0.0
0.001+i0.0
0.002+i0.0
0.002-i0.0
0.002+i0.0
0.002+i0.0
S2(d5/2)
0.032+i0.0
0.006+i0.0
0.025-i0.0
0.031-i0.04
0.031+i0.0
0.031+i0.0
(V1)
6Be
(V2)
6Li
(V1)
6Li
(V2)
V1 and V2 fits, recoil : slight change of basis states occupation
Redistribution of basis states occupation from Coulomb Hamiltonian
N. Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315
April 26-29, 2011
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Nicolas Michel
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Isospin operators
expectation values
• Isospin operators:
Same basis demanded for protons and neutrons
Coulomb infinite-range part in 1/r to diagonalize
N. Michel, Phys. Rev. C, 83 (2011) 034325
• 1/r matrix representation with Berggren basis
Infinities appear on the diagonal with scattering states :
•
Possible treatments:
Cut after r >R : no infinities but very crude
Analytical subtraction of integrable singularities :
Off-diagonal method : replacement of diverging
April 26-29, 2011
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by
Nicolas Michel
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1/r treatment precision
Cut method
Subtraction
method
Off-diagonal
method
Numerical precision obtained
with off-diagonal method
April 26-29, 2011
N. Michel, Phys. Rev. C, 83 (2011) 034325
CEA / IRFU / SPhN / ESNT
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Application to 0+ (T=1) states
IAS:
Isobaric
analog
state
6Li(V1)
6Be(V1)
‹0+ | 0+ IAS›2
0.995
0.951-i0.050
Tav
0.9994
1
0+ of 6Li almost isospin invariant
0+ of 6Be shows large isospin asymmetry
6Be : two valence protons → T=1 exactly
Partial dynamical symmetry
N.Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315
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Conclusion
• GSM: Exact calculations with valence protons and neutrons
Recoil exactly taken into account with COSM formalism
Coulomb interaction: exact asymptotic via Z = Zval potential introduction
Theoretical and numerical errors of the model controlled
• Isospin asymmetry: Proton and neutron spectroscopic factors
0+ and 2+ T=1 triplets of 6He, 6Li and 6Be
Same separation energies for all A=6 systems
Differences from Coulomb Hamiltonian only: continuum coupling
Spectroscopic factors : neutron with cusps, proton without cusps
Different configuration mixings for isobaric analog states
T2 and T- expectation values : partial dynamical symmetry
Origin : Coulomb+continuum , no charge-dependent effective forces
April 26-29, 2011
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