Ab Initio Calculations of Three
and Four Body Dynamics
M. Tomasellia,b
Th. Kühla, D. Ursescua
a Gesellschaft
für Schwerionenforschung, D-64291 Darmstadt,Germany
b Technical University Darmstadt, D-64289 Darmstadt, Germany
Motivation
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The Equation of Motion (EoM) in the zero
order dynamic linearization (GLA)
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The Equation of Motion (EoM) in
the second order GLA-III
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The Equation of Motion (EoM)-IV
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The Hamilton's Operator
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The Non-Linear Eigenvalue Equation
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Analogy with cluster theory
Correlations can be introduced via eiS method
N
N
i 1

1 2
i 1
eiS  ai 0  (1  S1  S 2  S3 ...) ai 0
S2  a1 a2a3 a4
S1  a a
S3  a1 a2a3 a4a5 a6 .....
Perturbation approximation possible. We prefer to calculate the
effective operators.
H eff  e
 iS 
He
iS
Oeff  e
 iS 
Oe
iS
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.
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Configuration mixing wave functions
(CMWF)
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Effective Hamiltonian S2
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The wavefunction of the deuteron
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Cluster model based on Dynamic
Correlation Model (DCM)
N
N   ai 0
i 1
N
N
N 1
[ H ,  a ] 0   a 0   ai a1 0

i

i
i 1
N 1
i 1
N 1
i 1
N 2 2
i 1
i 1
i 1 i '1
[ H ,  ai a1 ] 0   ai a1 0   ai ai ' 0
......
N
N 1
N 2 2
i 1
i 1
i 1 i '1
tot  N  N 1,1  N 2,2  ...   ai 0   ai a1 0   ai ai ' 0 ...
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Effect of linearisation on commutator
chain for two body clusters
Within the GLA the higher order terms (4p-2h) are calculated with
the Wick's theorem by neglecting the normal order.
Collect the resulting terms
E  E2 p  2 p V 2 p
3 p1h V 2 p
2 p
0
E  E3 p1h  3 p1h V 3 p1h  3 p1h
2 p V 3 p1h
Dynamic eigenvalue equations for mixed mode amplitudes
2 particles => 3 particles – 1 hole
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Dynamics eigenvalue equation for one dressed
dressed nucleon clusters
which is solvable self-consistently
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Degree of spuriousity
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Charge distributions of 6He
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Charge radii of 6He
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Charge distributions of 6Li
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Charge form factor for 6Li
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Elastic proton scattering on 6Li
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Medium effects on the two body
matrix elements (18O)
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Positive and Negative parity states in 18O
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Comparison with Vlow-k potential: 18O
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The EoM of the Three Nucleon Clusters
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Three particle Dynamic model
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Nuclear results for Li isotopes
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Elastic proton scattering on 11Li
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Summary of Charge Radii
Li -
Exp.[GSI]
rms R c
Exp.[8]
rms R c
6
7
8
9
11
2,55
2,46
2,37
2,30
2,47
2.55 (4)
2.37 (3)
Rc = charge radius
Exp.+Th.[1] Exp+Th[1]
rms R p
rms R c
2.32 (3)
2.27 (2)
2.26 (2)
2.18 (2)
2.88 (2)
2.46 (2)
2.40 (2)
Th.[2]
rms R p
2,045
1,941
1,946
1,986
Th.[3]
rms R p
Th.[4]
rms Rp
2,39
2,25
2,09
2,04
Th.[5]
rms R p
2,27
2,18
2,10
Th.[6]
rms R c
2,55
2,41
2,40
2,42
2,67
Exp+Th[7]
rms R c
2,235
Rp = point radius
References:
[1] I. Tanihata, Phys. Lett B 206,592 (1988)
Method:
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
Charge distributions for A=3 nuclei
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Charge distribution: alpha particle
RMS (Bonn)=1.50 fm
RMS (Yale)=1.51 fm
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Energy splitting and BE(E2;2+
transition for 16C
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Cluster Factorization Theory I
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Cluster Factorization Theory II
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Cluster Factorization Theory III
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Cluster Factorization Theory IV
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Factorisation of the model CMWFs in terms of
cluster coefficients
The factorisation method is presently applied to reduce complex Feynman diagrams to simple form
Particle line
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Interaction between nucleons
Hole line
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