Nicolas Michel (University of Jyväskylä)
Marek Ploszajczak (GANIL)
Witek Nazarewicz (ORNL – University of Tennessee)
April 26-29, 2011
CEA / IRFU / SPhN / ESNT
Nicolas Michel

•
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 6 He, 6 Be and 6 Li
•
Spectroscopic factors, energies, T +/and T 2 expectation values
•
Conclusion
CEA / IRFU / SPhN / ESNT
April 26-29, 2011 Nicolas Michel 2




April 26-29, 2011

10 Li +n

11
9 Li +2n
Li

9


Be +2n
5 He +n
4 He +2n
325
300 

6 

He

7316
1867.5
2 
1797
0 
964
 10 Be +n
11 Be
CEA / IRFU / SPhN / ESNT 


1/2 

505
1/2  320
Nicolas Michel 3


April 26-29, 2011
CEA / IRFU / SPhN / ESNT
Nicolas Michel 4

•
Georg Gamow : simple model for a decay
G.A. Gamow, Zs f. Phys. 51 (1928) 204; 52 (1928) 510
•
Definition :
April 26-29, 2011
CEA / IRFU / SPhN / ESNT
Nicolas Michel 5

•
Calculation of radial integrals: exterior complex scaling
•
Analytic continuation : integral independent of R and θ
April 26-29, 2011
CEA / IRFU / SPhN / ESNT
Nicolas Michel 6

Berggren completeness relation
Im(k) bound states narrow resonances
Re(k) antibound states
L + : arbitrary contour capturing states
April 26-29, 2011 broad resonances
CEA / IRFU / SPhN / ESNT
Nicolas Michel 7

•
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
CEA / IRFU / SPhN / ESNT
Nicolas Michel 8

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
CEA / IRFU / SPhN / ESNT
Nicolas Michel 9

Hamiltonian definition
•
6 He, 6 Be, 6 Li: valence particles, 4 He core :
H = T
1b
+ WS( 5 Li/ 5 He) + MSGI + V c
+ T rec
0p
3/2
(resonant), contours of s
1/2
, p
3/2
, p
1/2
, d
5/2
, d
3/2 scattering states, recoil included
MSGI : Modified Surface Gaussian Interaction:
•
6 Be: Coulomb interaction necessary
Problem: long-range, lengthy 2D complex scaling, divergences
Solution: one-body long-range / two-body short-range separation
H
1b one-body basis:
April 26-29, 2011
CEA / IRFU / SPhN / ESNT
Nicolas Michel 10

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
)
•
5 He / 6 He, 5 Li / 6 Be, 5 He / 6 Li, 5 Li / 6 Li non resonant components necessary .
April 26-29, 2011
CEA / IRFU / SPhN / ESNT
Nicolas Michel 11

Coulomb interaction and recoil term
Harmonic oscillator expansion
April 26-29, 2011
Physical precision of the order of 1 keV
Sufficient for practical applications
N. Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315
CEA / IRFU / SPhN / ESNT
Nicolas Michel 12

6 Be/ 5 Li – 6 He/ 5 He
Cusps (π)
6 Li/ 5 He – 6 Li/ 5 Li
Cusps (ν)
N. Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315
April 26-29, 2011
CEA / IRFU / SPhN / ESNT
Nicolas Michel 13

Spectroscopic factors distribution
Re[S 2 ] > 1, Im[S 2 ] ≠ 0
Large occupation of non-resonant continuum
N. Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315
April 26-29, 2011
CEA / IRFU / SPhN / ESNT
Nicolas Michel 14

Observables of 0 + , 2 + (T=1) states
0+ state
E calc
(MeV)
E exp
(MeV)
Γ calc
Γ exp
(keV)
(keV)
S 2 (π)
S 2 (ν)
6 He 6 Be (V1) 6 Be (V2) 6 Li (V1) 6 Li (V2)
-0.974
1.653
1.371
0.0866
-0.0706
-0.973
0
1.371
41
1.371
14
0.136
8.85 ·10 -3
0.136
9.13 ·10 -3
0 92 92 8.2 ·10 -3 8.2 ·10 -3
———— 1.015-i0.147
1.015-i0.177
1.061-i0.280
1.028-i0.300
0.87-i0.383
————— ————— 0.911-i0.361
0.898-i0.369
2+ state
E calc
(MeV)
E exp
(MeV)
Γ calc
Γ exp
(keV)
(keV)
S 2 (π)
6 He 6 Be (V1) 6 Be (V2) 6 Li (V1) 6 Li (V2)
0.823
0.824
2.887
3.041
2.679
3.041
1.667
1.667
1.569
1.667
89 986 804 404 329
113 1160 1160 541 541
————— 0.973-i0.014
0.978-i0.016
0.987-i0.003
0.993-i0.003
S 2 (ν) 1.061+i0.001
————— ————— 1.034-i0.024
1.043-i0.022
V1 : WS nucl
(π) = WS nucl
(ν) V2 : WS(π) fitted to 6 Be binding energy
N. Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315
April 26-29, 2011
CEA / IRFU / SPhN / ESNT
Nicolas Michel 15

Configuration mixing of 0 + (T=1) states
(C k
) 2 6 He 6 He (rig. core)
6 Be (V1) 6 Be (V2) 6 Li (V1) 6 Li (V2)
(0p
3/2
) 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(πp
3/2)
————— —————
-0.115+i0.218
-0.143+i0.255
-0.084+i0.226
-0.050+i0.244
S1(νp
3/2)
0.243+i0.619
0.244+i0.668
————— —————
0.066+i0.308
0.0797+i0.314
S2(s
1/2
) 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(p
1/2
) 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(d
3/2
) 0.002+i0.0
S2(d
5/2
) 0.032+i0.0
0.001+i0.0
0.006+i0.0
0.002+i0.0
0.025-i0.0
0.002-i0.0
0.031-i0.04
0.002+i0.0
0.031+i0.0
0.002+i0.0
0.031+i0.0
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
CEA / IRFU / SPhN / ESNT
Nicolas Michel 16

•
Isospin operators:
Isospin operators expectation values
Same basis demanded for protons and neutrons
Coulomb infinite-range part in 1/r to diagonalize
•
1/r matrix representation with Berggren basis
Infinities appear on the diagonal with scattering states :
N. Michel, Phys. Rev. C, 83 (2011) 034325
•
Possible treatments:
Cut after r >R : no infinities but very crude
Analytical subtraction of integrable singularities :
Off-diagonal method : replacement of diverging by
April 26-29, 2011
CEA / IRFU / SPhN / ESNT
Nicolas Michel 17

Cut method
1/r treatment precision
Subtraction method
Off-diagonal method
Numerical precision obtained with off-diagonal method
April 26-29, 2011
CEA / IRFU / SPhN / ESNT
N. Michel, Phys. Rev. C, 83 (2011) 034325
Nicolas Michel 18

Application to 0 + (T=1) states
‹
0+ | 0+ IAS
›
2
T av
6 Li(V1)
0.995
0.9994
6 Be(V1)
0.951-i0.050
1
0 + of 6 Li almost isospin invariant
0 + of 6 Be shows large isospin asymmetry
6 Be : two valence protons → T=1 exactly
Partial dynamical symmetry
N.Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315
April 26-29, 2011
CEA / IRFU / SPhN / ESNT
Nicolas Michel
IAS:
Isobaric analog state
19

•
GSM: Exact calculations with valence protons and neutrons
Recoil exactly taken into account with COSM formalism
Coulomb interaction : exact asymptotic via Z = Z val potential
Theoretical and numerical errors of the model controlled introduction
•
Isospin asymmetry: Proton and neutron spectroscopic factors
0 + and 2 + T=1 triplets of 6 He, 6 Li and 6 Be
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
T 2 and T expectation values : partial dynamical symmetry
Origin : Coulomb+continuum , no charge-dependent effective forces
CEA / IRFU / SPhN / ESNT
April 26-29, 2011 Nicolas Michel 20