Neutron-Rich Nuclei within a realistic shell-model approach Angela Gargano Napoli A. Gargano...
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Transcript of Neutron-Rich Nuclei within a realistic shell-model approach Angela Gargano Napoli A. Gargano...
Neutron-Rich Nuclei within
a realistic shell-model approach
Angela Gargano
Napoli
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
L. Coraggio (Napoli)
A. Covello (Napoli)
N. Itaco (Napoli)
T.T.S. Kuo (Stony Brook)
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
Plan of the talk
Theoretical framework: Renormalization of the bare NN potential by means of
the Vlow-k approach
Derivation of the model space effective interaction by means of the plus folded diagram method
Outline of calculations
Results: neutron-rich nuclei northeast of 132Sn
and comparison with conterpart nuclei in the stable 208Pb regionneutron-rich Ca isotopes
neutron-rich C isotopes
Summary
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
Shell-model calculations
i jj ijV(i)HH 0iiisoici ls)(rU)(rUT(i)H
0
3. Two-body matrix elements
1. Model space
2. Single-particle energies (taken from the experimental spectra of nuclei with one-valence nucleon or treated as free parameters)
4. Construction and diagonalization of the energy matrices
Traditional approach:
Two-body matrix elements treated as free parametersTwo-body matrix elements treated as free parameters Empirical effective interactions containing adjustable Empirical effective interactions containing adjustable
parametersparameters [e.g., s-d shell nuclei, B. A. Brown and B. H. Wildenthal, Ann. Rev. Nucl. Part. Sci.38, 28(1988)]
defined within a reduced model space and acting only between valence nucleons
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
... VVTH NNNNN
ααα ψEHψ Ab-initio calculations:
Green’s function Montecarlo method, no-core shell model, coupled cluster method, UCOM
(three-nucleon interactions have been also taken into account)
nuclear properties, such as binding and excitation energies, are calculated directly from first principles of quantum mechanics, using an appropriate computational scheme
Challenge for nuclear physics: understand the properties of nuclei starting from the forces among nucleons
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
Realistic shell-model calculations:We start from
PΨE)P ΨVP(HPΨPH eff0eff
10NNNN HHU)(VU)(TVTH
ααα ψEHψ where U is a one-body auxiliary potential introduced to define a convenient single-paticle basis
and define the effective shell-model Hamiltonian
eff0 VHH
through the model-space Schrödinger equation
where the E and the corresponding are required to be a subset of the eigenvalues and eigenvectors of the original Hamiltonian
The P operator projects into the chosen model space
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
all modern NN potentials fit equally well the deuteron properties and the NN scattering data up the inelastic
threshold 2/Ndata ~ 1
● Choice of the nucleon-nucleon potential
Derivation of Veff
NoteNote these potentials cannot be used directly in the derivation of Vthese potentials cannot be used directly in the derivation of Veff eff
due to their strong short-range repulsion, but adue to their strong short-range repulsion, but a
• • Renormalization procedure is neededRenormalization procedure is needed
CD-Bonn, Argonne V18 , Chiral potentials,…
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
Renormalization of the NN interaction
Traditional approach: Brueckner G-matrix method
Vlow-k approach: construction of a low-momentum NN potential Vlow-k confined within a momentum-space cutoff
S. Bogner,T.T.S. Kuo,L. Coraggio,A. Covello,N. Itaco, Phys. Rev C 65, 051301(R) (2002)S. Bogner, T.T.S. Kuo, A. Schwenk, Phys. Rep. 386, 1 (2003).L. Coraggio et al, Prog. Part. Nucl. Phys. 62 (2009) 135
Λk
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
Vlow-k approach
Derived from the original VNN by integrating out the high-momentum components of the original VNN potential down to the
cutoff momentum
Vlow-k decouples high- and low-energy degrees of freedom
preserves the physics of the original NN interaction
the deuteron binding energy
scattering phase-shifts up to the cutoff momentum ΛHow to contruct Vlow-k?
Effective interaction technique based on the the Lee-Suzuki similariry transformation (Prog. Theor Phys 64 (1980) 2091)
low-momentum spaceQ complementary space
X similarity transformation
Decoupling equation solved by the iterative procedure proposed by Andreozzi (Phys Rev. C 54 (1996) 684)
T- V
0Q
HXX
klow
1
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
Features of Vlow-k
real effective potential in the momentum space (indipendent from the starting energy or from the model space, as instead the case of the G matrix defined in the nuclear medium)
eliminates sources of non-perturbative behavior can be used directly in nuclear structure calculations
gives an approximately unique representation of the NN potential for 2 fm-1 ELab 350 MeV - Vlow-k’s extracted from various phase-shift equivalent potentials are very similar to each other
NoteNote VVlow-k low-k is developed for the two-body systemis developed for the two-body system for A>2 the low-energy observables are not the samefor A>2 the low-energy observables are not the same
of the original NN potential of the original NN potential and depend (to a certain extent) on the value of and depend (to a certain extent) on the value of
This may be removed complementing the two-body Vlow-k with three- and higher-body components
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
+ folded diagram method
1i
ieff FQV ˆ
collection of irreducible valence-linked diagrams with Vlow-k
replacing VNN in the interaction vertices
Fi
i-folded diagrams (expressed in terms of derivatives)
T.T.S. Kuo and E. Osnes, Lecture Notes in Physics, vol 364 (1990)L. Coraggio et al, Prog. Part. Nucl. Phys. 62 (2009) 135
Veff ,constructed for two valence particles, is defined
-in the nuclear medium
-in a subspace of the Hilbert space
accounts perturbatively for
• configurations excluded from the chosen model space
• excitations of the core nucleons
developed within the framework of the time-dependent perturbative approach by Kuo and co-workers
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
2-body diagrams up to 2nd order:V V1p1h V2p V2p2h
1-body diagrams up to 2nd order S-box
Calculation of : • inclusion of diagrams up to a finite order in the interaction • truncation of the intermediate-state summation
Sum of the folded series by the Lee-Suzuki method [Prog. Theor. Phys. 64, 2091 (1980)]
+ + …
Construction of Veff
NoteNote
NoteNote
(2)eff
(1)effeff VVV
Diagramatic expression of the
(1)eff0 VH single-particle energies
TB component of the shell-model Hamiltoniam
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
Results
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
neutrons
pro
tons
nuclei beyond the N=82 shell
n-rich nuclei
n-rich Ca isotopes
n-rich C isotopes
magic nature of 132Sn?
N=34 shell closure?
location of the neutron drip line?
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
132132Sn regionSn region132132Sn regionSn region
CD-Bonn + Vlow-k
: second-order calculationSingle-particle energies from the experimental spectra of 133Sb and 133Sn
• 132Sn coreValence neutron levels: 1f7/2, 2p3/2, 0h9/2, 2p1/2, 1f5/2, 0i13/2Valence proton levels: 0g7/2, 1d5/2, 1d3/2, 0h11/2, 2s1/2
n-rich Ca isotopesn-rich Ca isotopesn-rich Ca isotopesn-rich Ca isotopes
CD-Bonn + Vlow-k
: third-order calculationSingle-neutron energies from a fit to exp energies of 47Ca and 49Ca
• 40Ca coreValence neutron levels: 0f7/2, 0f5/2, 1p3/2, 1p1/2
n-rich C isotopesn-rich C isotopesn-rich C isotopesn-rich C isotopes
N3LOW [chiral potential with a sharp momentum cutoff at 2.1 fm-1] : third-order calculationTheoretical single-neutron energies
• 14C coreValence neutron levels: 0d5/2, 0d3/2, 1s1/2
Input of our calculations
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
Theory
B(E2;0+ 2+) = 0.033 e2b2
134Sn Coulex (Oak Ridge)
B(E2;0+ 2+) = 0.029(4) e2b2
Theory
0.726
134Sn
Expt.
from the f7/2p1/2 configurationtheir location below the 8+
due to the new position of
the p1/2 level measured @
ORNLp1/2=1.36 MeV
(old value: 1.66 MeV)[Nature 465 (2010)]
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
0
0,5
1
1,5
2
0+ 2+ 4+ 6+
E(M
eV
)
134Sn Expt 134Sn Calc
134Sn 132Sn + 2
0
0,5
1
1,5
2
0+ 2+ 4+ 6+ 8+
E(M
eV
)
210Pb Expt 210Pb Calc
(f7/2)2 multiplet
210Pb 208Pb + 2 (g9/2)2
multiplet
J
J
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
0
0,5
1
1,5
2
0+ 2+ 4+ 6+ 8+
E(M
eV
)136Sn Calc 212Pb Calc 212Pb Expt
J
136Sn 132Sn + 4
212Pb 208Pb + 4
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
A 134Sn 135Sn 136Sn 137SnBE Calcrelative to 132Sn
5.91 8.30 11.86 14.18
BE Exptrelative to 132Sn
5.92*
BE/A Calc 8.27 8.23 8.20 8.15
N/Z 1.70 1.72 1.74 1.68
124Sn(stable) with N/Z=1.48 BE/A=8.46
* M. Dworschak et al. Phys. Rev. Lett. 100, (2008) 072501Old value (Fogelberg et al., 1999): 6.365 MeV
neutron shell gap at N= 82 restored
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
B(E2;42 ) = 1.64 W.u.
B(E2;64) = 0.81 W.u.
B(E2;222) = 0.34 W.u.
B(E2;224) = 0.22 W.u.
Q(2) = -1.3 efm2
µ(2) = -0.56 nm
134Sn (Theoretical predictions)
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
136Sb is at present the most exotic open-shell nucleus beyond 132Sn for whichinformation exists on excited states
136Sb
Expt Theory
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
L. Coraggio et al. Phys. Rev. C 80, (2009) 044311
J.J. Valiente Dobón et al. PRL 102, 242502 (2009)
Expt Theory Expt Theory
50Ca 52Ca
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
Ca isotopes - Ground-state energy per valence neutron
A
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
A * M. Honma et al. EPJ A 25, 499 (2005)
*
no shell gap at N=34
Ca isotopesExcitation energies of the J =21
+ states
Effective single particle energies of the f5/2 and p1/2 levels (relative to the p3/2 level )
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
J 1)J(2JjjVjj1)J(2
jjV ;
jjV
j jn
jεESPE(j)
f0 5/2
34Np1 1/2
f1 7/2
p1 3/2
-427 keV*
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
C isotopes from 16C to 24C – ground-state energy (relative to 14C )
*to reproduce the exp g.s. energy of 15C relative to 14C Egs(calc)=-0.79 ; Egs(exp)= -1.22 MeV
22C is the last bound isotopeK. Tanaka et al PRL 104, 062701 (2010)
S2n(evaluation)=420 keV S2n(calc)=601 keV
L. Coraggio et al. PR C 81, 064303 (2010)
•
•
•
• •
•
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
C isotopesExcitation energies of the J =21
+ states
ES
PE (
MeV
)
N
no subshell closure at N=14g.s. in 20C:14% of (d5/2)6 configuration
A. Gargano JAPEN-ITALY EFES Workshop Torino- 2010Napoli
Reliability of “realistic shell-model calculations” for light heavy nuclei
This outcome gives confidence in its predictive power, and may stimulate and be helpful to future experiments.
Three-body forces seem to contribute mainly to the absolute energy of the single-particle. Role of three-body forces needs futher investigations
It is of key importance to gain more experimental information
Summary