shell model approach for two-proton radioactivity
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Nicolas MichelNicolas MichelCEA / IRFU / SPhN CEA / IRFU / SPhN
Shell Model approach Shell Model approach for two-proton radioactivityfor two-proton radioactivity
Nicolas MichelNicolas Michel (CEA / IRFU / SPhN)(CEA / IRFU / SPhN)Marek Ploszajczak Marek Ploszajczak (GANIL)(GANIL)
Jimmy Rotureau Jimmy Rotureau (ORNL – University of Tennessee)(ORNL – University of Tennessee)Witek Nazarewicz Witek Nazarewicz (ORNL – University of Tennessee)(ORNL – University of Tennessee)
October 11-13, 2008October 11-13, 2008
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN 22
Plan Plan
• Experimental dataExperimental data
• R-matrix for diproton emissionR-matrix for diproton emission
• Shell Model Embedded in the Continuum (SMEC)Shell Model Embedded in the Continuum (SMEC)
• SMEC with one and two particles in the continuumSMEC with one and two particles in the continuum
• Used approximations for diproton emission and resultsUsed approximations for diproton emission and results
• Gamow Shell Model with valence protonsGamow Shell Model with valence protons
• Berggren completeness relation and Coulomb interactionBerggren completeness relation and Coulomb interaction
• Mirror effects in Mirror effects in 66He and He and 66BeBe : spectroscopic factors: spectroscopic factors
• Conclusion et perspectivesConclusion et perspectives
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN 33
Experimental dataExperimental data
Three Three diproton emitters diproton emitters discovered:discovered: 4545Fe,Fe,5454Zn, (Zn, (4848Ni)Ni)Theoretical description: Theoretical description: new models new models to be to be developpeddevelopped
B. Blank et al., Phys. Rev. Lett., B. Blank et al., Phys. Rev. Lett., 9494, 232501 (2005), 232501 (2005) C. Dossat et al., Phys. Rev. C, C. Dossat et al., Phys. Rev. C, 7272, 054315 (2005), 054315 (2005)
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN 44
(A. Brown, F.C. Barker, Phys. Rev. C (A. Brown, F.C. Barker, Phys. Rev. C 6767, 041304(R) (2003)), 041304(R) (2003))
• Standard Shell Model: Standard Shell Model: spectroscopic factors spectroscopic factors onlyonly
• R-matrix reaction formulas:R-matrix reaction formulas: single particle fit, s-wave phase shifts (p+p)single particle fit, s-wave phase shifts (p+p)
• No mixing between channels:No mixing between channels: no continuum couplingno continuum coupling
• Extension of R-matrix standard formulas:Extension of R-matrix standard formulas:
Q,P,S: Q,P,S: available energy, penetration and shift factorsavailable energy, penetration and shift factors
½½(U): (U): density of p+p states from s-wave phase shiftsdensity of p+p states from s-wave phase shifts
M,a,M,a,µµspsp22 : reduced mass, channel radius, single particle reduced width: reduced mass, channel radius, single particle reduced width
R-matrix formulationR-matrix formulation
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN
• Feshbach space separation:Feshbach space separation:
Q:Q: A A bound and quasi-bound (narrow resonant) bound and quasi-bound (narrow resonant) one-body statesone-body states
P:P: A-1A-1 bound and quasi-bound bound and quasi-bound one-body states, 1 one-body states, 1 scattering statescattering state
• Hamiltonian, wave functions:Hamiltonian, wave functions:
55
(K.Bennaceur, N.Michel, F. Nowacki, J. Okolowicz and M. Ploszajczak, Phys. Lett. B, (K.Bennaceur, N.Michel, F. Nowacki, J. Okolowicz and M. Ploszajczak, Phys. Lett. B, 488,488, 75 (2000)) 75 (2000))(J. Okolowicz, M. Ploszajczak, and I. Rotter, Phys. Rep., (J. Okolowicz, M. Ploszajczak, and I. Rotter, Phys. Rep., 374374, 271 (2003)), 271 (2003))
SMEC: SMEC: one particle in the continuumone particle in the continuum
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN
• Two-body clusters added:Two-body clusters added:
Q:Q: A A bound and resonant bound and resonant one-body statesone-body states
P:P: A-1A-1 bound and resonant bound and resonant one-body states, 1 one-body states, 1 scattering statescattering state
T:T: A-2A-2 bound and resonant bound and resonant one-body states, 2 one-body states, 2 scattering statesscattering states
• Approximations necessary:Approximations necessary:
Full problem Full problem currently impossible currently impossible to treat (to treat (zero-range interaction divergencezero-range interaction divergence))
• Cluster emission:Cluster emission: diproton considered as a diproton considered as a closed systemclosed system
• Sequential emission:Sequential emission: Two Two independentindependent protons emitted, two protons emitted, two two-body decaystwo-body decays
system system resonant: resonant: standardstandard sequential emissionsequential emission
system system scattering: scattering: virtual sequential emissionvirtual sequential emission
66
(J. Rotureau, J. Okolowicz and M. Ploszajczak, Phys. Rev. Lett., (J. Rotureau, J. Okolowicz and M. Ploszajczak, Phys. Rev. Lett., 9595, 042503 (2005) ; , 042503 (2005) ; Nucl. Phys. A, Nucl. Phys. A, 767,767, 13 (2006)) 13 (2006))
SMEC: SMEC: two particles in the continuumtwo particles in the continuum
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN
• Effective Hamiltonian:Effective Hamiltonian: HHPTPT and H and HTPTP couplings couplings neglectedneglected
• Two-body cluster treatment:Two-body cluster treatment:
Internal diproton degrees of freedom: Internal diproton degrees of freedom: phenomenologicalphenomenological
Integration over energy of cluster U, Integration over energy of cluster U, weightedweighted by p+p by p+p s-wave s-wave density of states density of states ½½(U)(U)
Effective two-body reaction Effective two-body reaction
77
SMEC: SMEC: cluster approximationcluster approximation
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN
• Effective Hamiltonian:Effective Hamiltonian: HHQTQT and H and HTQTQ couplings couplings neglectedneglected
• Two independent decays:Two independent decays:
h: h: mean field mean field of the of the first emitted proton first emitted proton on the A-1 daughter nucleuson the A-1 daughter nucleus
Q’,P’ : subspaces associated to systemQ’,P’ : subspaces associated to system
All interactions All interactions between the emitted protons between the emitted protons averaged or suppressedaveraged or suppressed
88
SMEC: sequential decaySMEC: sequential decay
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN 99
SMEC: diproton decaySMEC: diproton decay
• Q space:Q space: 1s 0d 0f 1p1s 0d 0f 1p
• Interaction in Q space:Interaction in Q space: USD, KB3, G-matrixUSD, KB3, G-matrix
• Interaction in P space:Interaction in P space:
B. Blank, M. Ploszajczak, to be publishedB. Blank, M. Ploszajczak, to be published
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN 1010
SMEC: SMEC: 4545FeFe
(J. Rotureau, J. Okolowicz and M. Ploszajczak, Nucl. Phys. A ,(J. Rotureau, J. Okolowicz and M. Ploszajczak, Nucl. Phys. A ,767,767, 13 (2006)) 13 (2006))
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN 1111
SMEC: SMEC: 4848NiNi
(J. Rotureau, J. Okolowicz and M. Ploszajczak, Nucl. Phys. A, (J. Rotureau, J. Okolowicz and M. Ploszajczak, Nucl. Phys. A, 767,767, 13 (2006)) 13 (2006))
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN 1212
Gamow statesGamow states
• Georg GamowGeorg Gamow : simple model for : simple model for decay decay G.A. Gamow, Zs f. Phys. G.A. Gamow, Zs f. Phys. 5151 (1928) 204; (1928) 204; 5252 (1928) 510 (1928) 510
• DefinitionDefinition : :
• Straightforward generalization to non-local potentials (HF)Straightforward generalization to non-local potentials (HF)
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN 1313
Complex scalingComplex scaling
• Calculation of radial integrals:Calculation of radial integrals: exteriorexterior complex scalingcomplex scaling
• Analytic continuation Analytic continuation : integral : integral independentindependent of R and of R and θθ
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN 1414
Complex energy statesComplex energy states
bound statesbound states
broad resonancesbroad resonances
narrow resonancesnarrow resonances
LL++ : arbitrary contour : arbitrary contour
antibound statesantibound states
capturing statescapturing states
Im(k)Im(k)
Re(k)Re(k)
Berggren completeness relationBerggren completeness relation
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN 1515
Completeness relation Completeness relation with Gamow stateswith Gamow states
• Berggren completeness relation Berggren completeness relation (l,j) : (l,j) :
T. Berggren, Nucl. Phys. A T. Berggren, Nucl. Phys. A 109109, (1967) 205 (, (1967) 205 (neutrons onlyneutrons only))
Extended to Extended to proton case proton case (N. Michel, J. Math. Phys., (N. Michel, J. Math. Phys., 4949, 022109 (2008)), 022109 (2008))
• Continuum discretizationContinuum discretization::
• N-body completeness relationN-body completeness relation::
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN 1616
Model for Model for 66He and He and 66BeBe
• 66He, He, 66Be:Be: valence particles, valence particles, 44He coreHe core : : HHnn = T + WS( = T + WS(55He) + SGIHe) + SGI
HHpp = T + WS( = T + WS(55Li) + SGI + VLi) + SGI + Vcc
WS(WS(55Li) = WSLi) = WSnuclnucl + U + Ucc(Z=2)(Z=2)
0p0p3/23/2, 0p, 0p1/21/2 (resonant), contours of p (resonant), contours of p3/23/2 and p and p1/21/2 scattering states scattering states
SGI : Surface Gaussian Interaction:SGI : Surface Gaussian Interaction:
• 66Be:Be: Coulomb interaction necessaryCoulomb interaction necessary
Problem:Problem: long-range, lengthy 2D complex scaling, divergences long-range, lengthy 2D complex scaling, divergences
Solution:Solution: one-bodyone-body long-range / long-range / two-bodytwo-body short-range short-range separationseparation
HH00 one-body basis: one-body basis:
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN 1717
Nuclear energiesNuclear energies
• WS potentials:WS potentials: VV00 = 47 MeV ( = 47 MeV (55He/He/66He), VHe), V00 = 47.5 MeV ( = 47.5 MeV (55Li/Li/66Be)Be)
• SGI interaction:SGI interaction: V(J=0) = -403 MeV fmV(J=0) = -403 MeV fm33, V(J=2) = -610 MeV fm, V(J=2) = -610 MeV fm33
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN 1818
Spectroscopic factors in GSMSpectroscopic factors in GSM
• One particle emission channelOne particle emission channel: (l,j,: (l,j,))
• Basis-independent definitionBasis-independent definition::
• ExperimentalExperimental : all energies taken into account : all energies taken into account
• StandardStandard : representation : representation dependencedependence ( (nn,l,j,,l,j,))
• 55He / He / 66He, He, 55Li / Li / 66BeBe : non resonant components : non resonant components necessarynecessary..
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN 1919
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN 2020
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN 2121
October 11-13, 2008October 11-13, 2008 Nicolas MichelNicolas MichelCEA / IRFU / SPhNCEA / IRFU / SPhN 2222
Conclusion et perspectivesConclusion et perspectives
ConclusionConclusion• SMEC and GSM:SMEC and GSM: Two Two complementarycomplementary models models
• SMEC:SMEC: convenient for proton emitters, convenient for proton emitters, very small widths very small widths can be calculatedcan be calculated ApproximationsApproximations needed for the moment, needed for the moment, very complicated very complicated formulasformulas Emission channels Emission channels interfereinterfere: : model-dependentmodel-dependent description ( description (cluster, sequentialcluster, sequential))
• GSM:GSM: First calculations with several valence protons First calculations with several valence protons Simple and powerful Simple and powerful modelmodel Spectroscopic factors: Spectroscopic factors: mirror effects mirror effects due to due to continuum continuum
PerspectivesPerspectives• More realistic interactions to be used with SMEC and GSMMore realistic interactions to be used with SMEC and GSM• SMEC:SMEC: full problem with full problem with three-body asymptotics three-body asymptotics possiblepossible• GSM:GSM: study of a study of a largerlarger set of light nuclei set of light nuclei
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