Ab Initio Calculations of Three and Four Body Dynamics M. Tomaselli a,b Th. Kühl a, D. Ursescu a a Gesellschaft für Schwerionenforschung, D-64291 Darmstadt,Germany

Ab Initio Calculations of Three and Four Body Dynamics

M. Tomasellia,b

Th. Kühla, D. Ursescua

a Gesellschaft für Schwerionenforschung, D-64291 Darmstadt,Germanyb Technical University Darmstadt, D-64289 Darmstadt, Germany

Marco Tomaselli 2006Brasil: Few body 18

MotivationMotivation


The Equation of Motion (EoM) in the zero The Equation of Motion (EoM) in the zero order dynamic linearization (GLA)order dynamic linearization (GLA)


The Equation of Motion (EoM) in The Equation of Motion (EoM) in the second order GLA-IIIthe second order GLA-III


The Equation of Motion (EoM)-IVThe Equation of Motion (EoM)-IV


The Hamilton's OperatorThe Hamilton's Operator


The Non-Linear Eigenvalue EquationThe Non-Linear Eigenvalue Equation


Analogy with cluster theoryAnalogy with cluster theory

0...)1(01

3211

N

ii

N

ii

iS aSSSae

211 aaS 6543213 aaaaaaS

iSiSeff HeeH

iSiSeff OeeO

.....43212 aaaaS Perturbation approximation possible. We prefer to calculate the

effective operators.

Correlations can be introduced via eiS method

The perturbative terms of the correlation operators Si correspond

to the diagram of the dynamic theory. The particle–hole terms

generated by the S3 operator are put to zero in the ladder perturbation.


Configuration mixing wave functions Configuration mixing wave functions (CMWF)(CMWF)


Effective Hamiltonian SEffective Hamiltonian S22


The wavefunction of the deuteronThe wavefunction of the deuteron


Cluster model based on Dynamic Cluster model based on Dynamic Correlation Model (DCM)Correlation Model (DCM)

000],[1

11

11

N

ii

N

ii

N

ii aaaaH

000],[2

1'

2

1'

1

11

1

11

N

ii

ii

N

ii

N

ii aaaaaaH

......

01

N

iiN a

...000...2

1

2

1''

1

11

12,21,1

N

i iii

N

ii

N

iiNNNtot aaaaa


Effect of linearisation on commutator Effect of linearisation on commutator chain for two body clusterschain for two body clusters

Collect the resulting terms

Within the GLA the higher order terms (4p-2h) are calculated with the Wick's theorem by neglecting the normal order.

01313213

13222

13

2

13

2

hp

p

hp

p

hpVhpEEpVhp

hpVppVpEE

Dynamic eigenvalue equations for mixed mode amplitudes 2 particles => 3 particles – 1 hole


Dynamics eigenvalue equation for one dressed Dynamics eigenvalue equation for one dressed dressed nucleon clustersdressed nucleon clusters

which is solvable self-consistentlywhich is solvable self-consistently


Degree of spuriousityDegree of spuriousity


Charge distributions of Charge distributions of 66HeHe


Charge radii of Charge radii of 66HeHe


Charge distributions of Charge distributions of 66LiLi


Charge form factor for Charge form factor for 66LiLi


Elastic proton scattering on Elastic proton scattering on 66LiLi


Medium effects on the two body Medium effects on the two body matrix elements (matrix elements (1818O)O)


Positive and Negative parity states in Positive and Negative parity states in 1818OO


Comparison with VComparison with Vlow-klow-k potential: potential: 1818OO


The EoM of the Three Nucleon ClustersThe EoM of the Three Nucleon Clusters


Three particle Dynamic modelThree particle Dynamic model


Nuclear results for Li isotopesNuclear results for Li isotopes


Elastic proton scattering on Elastic proton scattering on 1111LiLi

Summary of Charge RadiiSummary of Charge Radii

References: Method:[1] I. Tanihata, Phys. Lett B 206,592 (1988) Interaction Cross Sections with Glauber model,

HO distributions[2] P. Navratil, PRC 57,3119 (1998) Large-basis shell-model calculations[3] S. Pieper, Annu.Rev.Nucl.Part.Sci. 51, 53 (2001) Greens Function Monte Carlo AV18/IL2[4] S. Pieper, PRC 66, 044310 (2002) Greens Function Monte Carlo AV18/IL2[5] Suzuki, Progr.Theo.Phys.Suppl. 146, 413 (2002) Stochastic Variational Multicluster Method

on a correlated gaussian basis[6] M. Tomaselli et al., Can. J. Phys. 80, 1347 (2002) Dynamic Correlation model[7] Penionzhkevich, Nucl.Phys. A 616, 247 (1997) coupled channel calculations, double-folding

optical potential, M3Y effective interaction[8] C.W. de Jager, At.Dat.Nucl.Dat.Tab. 14, 479 (1974) Electron Scattering

Exp.[GSI] Exp.[8] Exp.+Th.[1] Exp+Th[1] Th.[2] Th.[3] Th.[4] Th.[5] Th.[6] Exp+Th[7]

Li - rms R c rms R c rms R p rms R c rms R p rms R p rms Rp rms R p rms R c rms R c

6 2,55 2.55 (4) 2.32 (3) 2.46 (2) 2,045 2,39 2,557 2,46 2.37 (3) 2.27 (2) 2.40 (2) 1,941 2,25 2,27 2,418 2,37 2.26 (2) 1,946 2,09 2,18 2,409 2,30 2.18 (2) 1,986 2,04 2,10 2,42

11 2,47 2.88 (2) 2,67 2,235

Rc = charge radius Rp = point radius


Charge distributions for A=3 nucleiCharge distributions for A=3 nuclei


Charge distribution: alpha particle Charge distribution: alpha particle

RMS (Bonn)=1.50 fm

RMS (Yale)=1.51 fm


Energy splitting and BE(E2;2Energy splitting and BE(E2;2+ + 0 0++) ) transition for transition for 1616CC


Cluster Factorization Theory ICluster Factorization Theory I


Cluster Factorization Theory IICluster Factorization Theory II


Cluster Factorization Theory IIICluster Factorization Theory III


Cluster Factorization Theory IVCluster Factorization Theory IV


Factorisation of the model CMWFs in terms of Factorisation of the model CMWFs in terms of cluster coefficientscluster coefficients

The factorisation method is presently applied to reduce complex Feynman diagrams to simple form

Particle line Hole line Interaction between nucleons


Documents

Ab Initio Calculations of Three and Four Body Dynamics M. Tomaselli a,b Th. Kühl a, D. Ursescu a a Gesellschaft für Schwerionenforschung, D-64291 Darmstadt,Germany