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Jerry P. Draayer , Tomas Dytrych, Kristina D. Sviratcheva, Chairul Bahri (LSU) James P. Vary (ISU). Status: Abinito Symplectic Nocore Shell Model. From quarks/gluons to UNIVERSE. Quantum Chromodynamics. quarks gluons. Universe. equation of state, dark/dense matter, nucleosynthesis.
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James P. Vary (ISU)
Status: Abinito
Symplectic Nocore
Shell Model
From quarks/gluons to UNIVERSE
Quantum Chromodynamics
quarks
gluons
Universe
equation of state,
dark/dense matter,
nucleosynthesis
Manynucleon systems
shell structure,
cluster modes,
collective rotations
lowenergy
regime: non
perturbative
Challenge: QCDtied Nuclear Models
Nucleosyntesis
Nucleosynthesis Stellar lifecycles
chandra.harvard.edu
Detector ingredients
Manynucleon systems
12C & 16O:
neutrino experiments
29Si (37Ge):
dark matter search
“femtoLab”
Quantum Chromodynamics
quarks
gluons
parity violation in weak hadronic physics
Symplectic Sp(3,R)
realistic interaction(s)
either possess Sp(3,R) symmetry …
or / and
… the nuclear manybody system
filters out Sp(3,R) symmetrybreaking effects
NoCore
ShellModel
(multi

4h
( ) limit

2h
( ) limit

0h
( ) limit
Solve
Schrödinger equation
in infinite space

h
horizontal
slices
…
Valence shell
Filled shells
12C
ħ=15MeV
Interaction: JISP16 (other results available  ahead)
A. M. Shirokov et al., Phys. Letts. B 621, 96(2005)
Highly deformed modes,
cluster structures,
B(E2) with NO effective charge
‘Larger’ model Spaces
Heavier Nuclei
can reach…
SymplecticNCSM
Multishell
symplectic
slice
NoCore
ShellModel
(multi

4h
( ) limit

2h
( ) limit

0h
( ) limit
SU(3) limit ( )


h
0h
Symplectic Sp(3,R)
ShellModel
(multiħ
vertical
slices
monopole & quadrupole collective excitations
horizontal
slices
…
Valence shell
Filled shells
20Ne
G. Rosensteel and D.J. Rowe (1980)
J.P. Draayer, K.J. Weeks, G. Rosensteel (1984)
20year “IBM” Pause
.…
.…
n
2
0
2
0
n
n
0
2
2
n
0
:
:
:
:
.…
.…
P
Q
:
:
U
U†=
P
Q
Q
:
:
Spherical harmonic oscillatorbasis
Symplecticbasis
Revisit below  SRG …
k
l
m
m
Familiar onedimensional harmonic oscillator problem
Coupled equations of motion (ignore symmetry):
Solve
eigenvalue problem
Uncoupled equations of motion (with symmetry):
Easy as !
physicsanimations.com
Normal modes ... associated with the symmetry of the pendulum motion
Use symmetry to reduce a twovariable problem to a onevariable (S) problem
…
…
Canonical coordinates…
Hamiltonian:
+ potential energy
Nucleus with A nucleons
( l, m
( b, g)
m
b
g
l
nucleon system
HO Hamiltonian
(p2+x2)/2
BohrMottelson Model
SU(3)
Model
vorticity
(from irrotational
to rigid rotor flows)
mass quadrupole
moment
angular momentum
SO(3)
multishell monopole
and quadrupole
collective vibrations
manyparticle
kinetic energy
collective
microscopic
G. Rosensteel and D.J. Rowe, Phys. Rev. Lett. 38 (1977) 10
Symplectic {Sp(3,R) SU(3) SO(3)} Model
( l, m
( b, g)
m
b
g
l
Elliott Model (single shell)
collective
microscopic
(11)
…….
Angular Momentum
Quadrupole Moment
Number Operator
Multishell Coupling
SO(3)
L1,M
Q2,M
N
AL,M
BL,M
(11)
………
SU(3)
xi, pi
……
(00)
essentially HO Hamiltonian
[Hamiltonian H0 = (p2+x2)/2]
(20)
…
(02)
[Monopole, L = 0 & Quadrupole, L = 2]
3body interaction!
G. Rosensteel and D.J. Rowe, Phys. Rev. Lett. 38 (1977) 10
Symplectic {Sp(3,R) SU(3) SO(3)} Model
(11)
L1,M
Q2,M
N
AL,M
BL,M
A’s raise 2ħ
1+5 = 6
(11)
N & L & Q
valence space
xi, pi
(00)
1+3+5 = 9
(20)
1+5 = 6
B’s lower 2ħ
(02)
9+6+6 = 21
SO(3)
SU(3)
U(3)
Sp(3,R)
collective
microscopic
12C
Vertical slices: 2shell L=0 and L=2 excitations
.
.
.
6ħ
N=6 sdgi
N=5 pfh
4ħ
N=4 sdg
N=3 pf
2ħ
N=2 sd
N=1 p
Valence shell
Filled shell
N=0 s
Examples for protons and protonneutron excitations
In addition to the 2ħ 1p1h excitations: small (~1/A) 2ħ 2p2h correction for removing spurious centerofmass motion
12C
Examples for proton excitations
e.g., 2ħ2p2h, 4ħ4p4h, …
.
.
.
N=6 sdgi
N=5 pfh
4ħ
N=4 sdg
N=3 pf
2ħ
N=2 sd
N=1 p
Valence shell
Filled shell
N=0 s
12C
.
.
.
6ħ
N=6 sdgi
Model space of all possible Sp(3,R) vertical slices = NCSM space
N=5 pfh
4ħ
N=4 sdg
N=3 pf
2ħ2p2h vertical slice
N=2 sd
N=1 p
Valence shell
Filled shell
N=0 s
Revisit below CM
… or, CCX theory …
12C
ħ=15MeV
Interaction: JISP16 (other results available  ahead)
A. M. Shirokov et al., Phys. Letts. B 621, 96(2005)
12C
Only 3 vertical slices: ~80%
NCSM wave function projected
onto Symplectic basis
6ħ
4ħ
2ħ
NoCore ShellModel wave function probability distribution
(0+gs)
ħ=15MeV
0ħ
12C
16O
12C
Only a few symplectic slices:
6ħ
4ħ
0+gs, 2+1, 4+1
2ħ
0+gs
0ħ
NCSM wave function projected
onto Symplectic basis
6ħ
4ħ
2ħ
NoCore ShellModel wave function probability distribution
(0+gs)
1 vertical slice
(most deformed, S=0) ~65%
ħ=15MeV
0ħ
Probability Distribution: Ground State – 8590%
100%
of 0ħ
Only 3 0p0h symplectic irreps: ~80%
(04) “slice”
Sp(3,R)
NCSM
(00) “slice”
0gs Probability distribution (%)
Validity of Elliott’s SU(3)
N ()
2ħ 2p2h Sp(3,R) irreps:
~4% (12C)
~10% (16O)
24.5(04) : most deformed
24.5(12)2: spin one states
Dimension of Model Space
1012
3 0p0h
all 0p0h
12C
16O
1010
NCSM
dominant
NCSM
0p0h + 2p2h
108
NCSM
106
Sp(3,R)
Sp(3,R)
104
102
1
Compared to NCSM
0ħ
4ħ
8ħ
12ħ
0ħ
4ħ
8ħ
12ħ
Model Space
Model Space
Dimension of model space
0.009% for 12C
0.0004% for 16O
T. Dytrych. KDS,
C. Bahri, J.P. Draayer, J.P. Vary,
Phys. Rev. Lett. 98 (2007) 162503
12C
g.st.
( l, m
( b, g)
m
b
g
l
Area = Probability of dominant Sp(3,R) slices
ħ
=15 MeV
Dominant modes:
0p0h: (0 4) [oblate]
2p2h: (2 4)
12C
16O
Ground state
(0p0h + 2p2h symplectic slices)
0+2
(0p0h + 2p2h symplectic slices)
Larger the probability for a given shape, longer the time it is dispalyed
ħ
( =15 MeV)
Spin Distribution in NCSM Eigenstates
2+
0+
12C
4+
90
90
80
80
70
70
60
60
50
50
40
40
Bare
Bare
Bare
11
11
11
12
12
12
13
13
13
14
14
14
15
15
15
16
16
16
17
17
17
18
18
18
30
30
20
20
10
10
0
0
Spin=0
Spin=1
Spin=2
Probability amplitude (%)
Probability amplitude (%)
Independence of Oscillator Strength
0+
2+
4+
0+
Spin components of converged states
Only 6 Sp(3,R) irreps (3 0p0h and 3 2p2h )
Overlaps (%) of NCSM wavefunctions with dominant Sp(3,R) states
Spin=0
Spin=1
Symplectic structure is not altered by LeeSuzuki transformation
Spin=0
Spatial wavefunctions: independent of whether bare or effective interaction is used
Sp(3,R) + Complementary (spinisospin) symmetry
collective
microscopic
(11)
L1,M
Q2,M
N
AL,M
BL,M
A’s raise 2ħ
1+5 = 6
(11)
N & L & Q
valence space
xi, pi
(00)
1+3+5 = 9
(20)
1+5 = 6
B’s lower 2ħ
(02)
9+6+6 = 21
Sp(3,R)
Spin
X
S
HSp(2)=a{XX}+bL.S+cS2
Symplectic Sp(3,R) symmetry preserving Hamiltonian
(the most general)
Compare with
JISP16 realistic interaction (and others)
1f7/2
1
Perfect!
0.9
0.8
0.7
‘good’
0.6
0.5
0.4
0.3
0.2
‘poor’
0.1
0
nearly perfect
very large
Correlation coefficient
large
medium
small
trivial
‘very large’
N=5 pfh
N=4 sdg
N=3 pf
N=2 sd
N=1 p
N=0 s
Effective JISP16, 6 shells (ħ=15 MeV)
Bare JISP16
Effective JISP16, 4 shells (ħ=15 MeV)
JISP16 NN interaction
Symmetry breaking
Symplectic interaction
Correlation coefficient
JISP16 NN interaction
Effective JISP16, 6 shells (ħ=15 MeV)
Bare JISP16
Effective JISP16, 4 shells (ħ=15 MeV)
Symmetry breaking
Symplectic interaction
‘very large’
Correlation coefficient
N=5 pfh
N=4 sdg
N=3 pf
N=2 sd
N=1 p
N=0 s
Pairing interaction
Bare JISP16
Effective JISP16, 4 shells (ħ=15 MeV)
Pairing interaction
Symmetry breaking
Symplectic interaction
‘very large’
Correlation coefficient
N=5 pfh
‘poor’
N=4 sdg
N=3 pf
N=2 sd
N=1 p
N=0 s
Where do we stand? … We’ve learned a lot; we’ve go lots to learn!
Similarity Renormalization Group and VNN
SRG
dHs
ds
Hs
d
ds
O
[O, Hs]
Renormalized interaction Hs
Bare interaction (VNN, VNNN, …) Hs=0
Unitary
transformation
where+= (antihermitian)
s is called the 'flow parameter'
typically choose= [O, Hs]where O is a physically relevant operator, e.g. K.E.
'flow equation'
SRG
Renormalized interaction Hs
Bare interaction (VNN, VNNN, …) Hs=0
Unitary
transformation
flow equation
flow parameter
Secondorder invariant operator of SU(3) (diagonal in SU(3) basis)
SU(3) basis states:
Flow equation solved for interaction matrix elements in SU(3) basis
Trel+NN interaction (bare CDBonn), ħ=15 MeV
SRG
6
MeV
4
2
iHifor the i SU(3) state:
( )=(0 0) L=0 S=0
0

2

4
Flow parameter s=1/2
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Exact solution for the lowestlying state
0.4
120
0.2
100
0.0
Number of non
diagonal matrix
elements
80

0.2
60
40

0.4
20

0.6
5
10
15
20
25
30
SRG
5
0

5
0.0
0.1
0.2
0.3
0.4
0.5
= 0.71 MeV
Structure dictated by the kinetic energy T& quadrupole operator Q which both belong to Sp(3,R) …
MeV
iHifor the i SU(3) state:
( )=(0 0) L=0 S=0
Exact solution for the lowestlying state
Flow parameter s=1/2
SRG
MeV
iHifor the i SU(3) state:
( )=(0 0) L=0 S=1
500
400
Flow parameter s=1/2
10
300
Exact solution for the lowestlying state
8
0.6
200
6
0.4
100
4
0.2
5
10
15
20
25
30
2
0.0
Number of non
diagonal matrix
elements
0

0.2

2

0.4
0.00
0.01
0.02
0.03
0.04
0.05
SRG
5
0

5
0.00
0.05
0.10
0.15
0.20
= 2.24 MeV
MeV
iHifor the i SU(3) state:
( )=(0 0) L=0 S=1
Exact solution for the lowestlying state
Flow parameter s=1/2
SRG
Comparison with a simple cluster model
CM
Possible SU(3) symmetry of cluster wave functions
Constituent clusters “frozen” to SU(3)symmetric ground states
Relative motion of clusters (carries Q≥4 oscillator quanta)
100% overlap with the leading symplectic bandheads!
Hecht, Phys. Rev C 16 (1977) 2401
Suzuki, Nucl. Phys. A 448 (86) 395
Comparison with cluster wavefunctions
CM
16O
(00) Sp(3,R) slice in NCSM 0+ state
g.st.
Projection onto cluster wave functions
……………31%
..………… ..65%
6 ħ
4 ħ
2 ħ
0 ħ
Probability distribution, %
………… .100%
Project at…
ħ, MeV
Projection of Sp(3,R) slices on cluster states
CM
16O
Probability, %
2p2h
4p4h
Sp(3,R) slices built on most deformed
npnh
bandheads
ħ
ħ
ħ
ħ
ħ
ħ
ħ
(00) Sp(3,R) slice insufficient – (42) & (84) slices must be included also
cluster modes constitute:
30% of [(A(20) A(20))(00)];
67% of [(A(20))(42)];
100% of (84) (the (84) Sp(3,R) bandhead)
CM