Nicolas Michel CEA / IRFU / SPhN Shell Model approach for two-proton radioactivity Nicolas Michel (CEA / IRFU / SPhN) Marek Ploszajczak (GANIL) Jimmy Rotureau

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


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)


(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


• 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


• 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


• 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


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


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))


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))


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)


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


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


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::


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:


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


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..





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

Documents

Nicolas Michel CEA / IRFU / SPhN Shell Model approach for two-proton radioactivity Nicolas Michel (CEA / IRFU / SPhN) Marek Ploszajczak (GANIL) Jimmy Rotureau