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Microscopic nuclear structurecalculations with unitarily
transformed effective interactionsShinichiro Fujii (Kyushu Univ.)
1. Unitary transformation2. Unitary-model-operator approach (UMOA)3. “No-core” shell model 4. Results g.s. energies, s.p. levels in p- and sd-shell nuclei neutron-rich C isotopes , O5. Summary
18
INT Seminar, Seattle, Sep. 27, 2007
0 1 2−100
0
100
200V
(r)
[MeV
]
r [fm]
singlet central(T=1, S=0, Paris pot.)
Derivation of effective interaction (Hamiltonian)by means of unitary transformation
HamiltonianH = H + V0
Unitary transformation of HH = U HU-1~
U = e , (S : anti-Hermitian, S = -S)S †
Decoupling equationQ(e He )P = 0-S S
Solution
(with the restrictive condition PSP = QSQ = 0)K. Suzuki, Prog. Theor. Phys. 68 (1982), 246
S = arctanh(ω - ω ), ω = QωP†
Effective HamiltonianH = PHPeff
~Effective interaction
V = PHP - PH P0~
eff
Unitary transformation operator U in terms of ωU = (1 + ω - ω )(1 + ω ω + ωω )† † † -1/2
P(1 + ω ω) P − Pω (1 + ωω ) QQω(1 + ω ω) P Q(1 + ωω ) Q
-1/2 -1/2
-1/2 -1/2
† † †
† †( (
=
S. Okubo, Prog. Theor. Phys. 12 (1954), 603-
Derivation of effective interaction
ρ1 = 2n +l +2n +l (n , l and n , l : setsof h.o. quantum numbers of the two-body states)
a a b b a a b b
Eff. int. in a huge model space
・S. F., T. Mizusaki, T. Otsuka, T. Sebe, and A. Arima, Phys. Lett. B650, 9 (2007).・S. F., R. Okamoto, and K. Suzuki, Phys. Rev. C 69, 034328 (2004).( (For details,
p
n
pf
pf
Q2
ρ1
P2
ρ1
decouplingQ v P = 0~(2)
22
p
n
Q1
ρ1
ρ1
1P
decouplingQ v P = 0~(1)
11
the ab initio NCSM not be applicable tothe present work( (
p
nρp
ρ
v~
v~= = 0
(2)
(2)
XpP
XnP2Q
ρ1
P2
decouplingQ v P = 0~(2)
2 2
n
Eff. int. for the UMOA
Eff. int. for the 0s-1p0f shell model
ρ1
12 14 16−100
−90
−80
(MeV
)
(MeV)
E
hΩ
O16Nijm 93
g.s.
12 14 16
−100
−90
−80
(MeV
)
(MeV)
E
hΩ
O16Nijm I
g.s.
12 14 16−110
−100
−90
(MeV
)
(MeV)
E
hΩ
O16N3LO
g.s.
12 14 16
−110
−100
(MeV
)
(MeV)
E
hΩ
O16CD Bonn
g.s.
= 6ρ1= 8ρ
1= 10ρ
1
= 12ρ1= 14ρ
1= 16ρ
1= 18ρ
1
Ground−state energies of O16
Nijm 93 Nijm I N3LO CD Bonn Expt.Eg.s. −99.69 −104.25 −110.00 −115.62 −127.62
BE/A 6.23 6.52 6.88 7.23 7.98 (in MeV)
0
10
20
Expt.
Expt.
(MeV
)
O15
Nijm 93
O16 O17
(g.s.)
3/2−
5/2+
1/2−
1/2+
3/2+
Nijm IN3LOCD Bonn
Nijm 93Nijm IN3LOCD Bonn
Comparison of Expt. and UMOA resultsfrom modern NN interactions
5.085.145.576.036.78
6.184.51
4.845.42
5.93
Ex
0
2
4
6
UMOA Expt. UMOA Expt.
Nijm I17F 23F
X (
MeV
)E
5/2+1/2+
3/2+
5/2+
1/2+
3/2+
Single−particle levels in 17F and 23F
12 14 16−340
−320
−300
−280
−260
(MeV
)
(MeV)
E
hΩ
Ca40CD Bonn
g.s.
= 8ρ1= 10ρ
1ρ = 121ρ = 141
= 161
ρ
The 1 and dependences of calculatedground−state energies of 40Ca
ρ hΩ
−20
−10
(MeV
)
Nijm 93
Nijm 93
Expt.
Esp
39CaExpt.
39K
5/2 (0 )+ d5/2
3/2 (0 )+ d3/2
1/2 (1 )+ s1/2
5/2 (0 )+ d5/2
1/2 (1 )+ s1/2
3/2 (0 )+ d3/2
Single−particle energies for hole states in 40Ca
New approach to neutron-rich C isotopes
Large-scale shell model・Code: newly developed version of MSHELL・Model space: the 0s-1p0f shells・Nucleon excitation: up to 2 nucleons from the occupied shells for C up to 2 nucleons to the 1p0f shells・Bare transition operator
Microscopic effective interactionDerived from a high-precision NN interaction (CD Bonn, …)and the Coulomb force in the neutron-proton formalism forthe given model space through a unitary-transformation theory
14
Low-lying energy levels in 16C
B E( 2; 2 → 0+) in 2fm4
1.30 0.63e
-+ 0.11(stat)
-+ 0.16(syst)
1 1
0
2
4
6
original Expt.
(MeV
)
0+
C16
(0+)
2+
2+
0+
CD Bonn
EX
4+
3(+)
2+
0+
2+4+
3+= 15 MeVhΩ
+
0
5
10
15
5/2 (0 )+ d5/2
1/2 (1 )+ s1/2
(MeV
)
unpt.
+2p1hExpt.
Esp
C
CD Bonn
Single-particle energies in 15C
Calculated results bythe unitary-model-oparator approach
(UMOA)
The calculated results are denoted by "dressed"
In the present shell model without any adjustable parameters → wrong ordering for the 1/2 and 5/2 states in C due to the small model- space size
15
+ +
To remedy the wrong ordering and re-produce the binding energies for the 1/2 and 5/2 states of the UMOA results → introduce a minimal refinement of the one-body energies for the 0d and 1s orbits of the neutron
+
+
5/2
1/2
correctordering
Low-lying energy levels in 16C
B E( 2; 2 → 0+) in 2fm4
1.30 0.84 0.63e
-+ 0.11(stat)
-+ 0.16(syst)
1 1
0
2
4
6
original dressed Expt.
(MeV
)
0+
C16
(0+)
2+
2+
0+
CD Bonn
EX
4+
3(+)
2+
0+
2+4+
3+
2+
2+3+4+
0+0+
0+0+
= 15 MeVhΩ
+
the 2+ and 0+ states in 18CEnergy differences between
B E( 2; 2 →+ 0+) in 2fm4
1.19 2.10 ?1 1 e
0
2
original dressed Expt.
(MeV
)
0+
C18
2+
0+
CD Bonn
EX
hΩ = 15 MeV
2+
0+
2+
1 1
correctordering
Low-lying energy levels in 16C
B E( 2; 2 → 0+) in 2fm4
1.38 0.91 0.63e
-+ 0.11(stat)
-+ 0.16(syst)
1 1+
0
2
4
6
original dressed Expt.
(MeV
)
0+
C16
(0+)
2+
2+
0+
N3LO
EX
4+
3(+)
2+
0+
2+4+
3+
2+
2+3+4+
0+0+
0+0+
= 14 MeVhΩ
the 2+ and 0+ states in 18CEnergy differences between
B E( 2; 2 →+ 0+) in 2fm4
1.38 2.47 ?1 1 e
1 1
0
2
original dressed Expt.
(MeV
)
0+
C18
2+
0+
N3LO
EX
hΩ = 14 MeV
2+
0+
2+
14
16
18
+-
CD Bonn N LO Expt.C 3.42 4.11 3.74 0.50C 0.84 0.91 0.63C 2.10 2.47 ?
3
++-
0.11(stat)0.16(syst)
for "dressed"
B(E2; 2 → 0 ) in C14,16,181 1+ +
in e fm2 4
0
1
2
3
4
5
14C 16C(2+→0+) (2+
→0+)
CD Bonn
expt.calc.
1 1 1 1
( 2
) in
2 fm
4B
E
e
( 2) of 14-16 CB E
(bare charge)
0
1
2
3
4
5
14C 15C 16C(2+→0+) (5/2+
→1/2+) (2+→0+)
CD Bonn
expt.calc.
1 1 1 1 1 1
( 2
) in
2 fm
4B
E
e
( 2) of 14-16 CB E
(bare charge)
0
1
2
3
4
5
14C 15C 16C(2+→0+) (5/2+
→1/2+) (2+→0+)
CD Bonn
expt.calc.
1 1 1 1 1 1
( 2
) in
2 fm
4B
E
e
( 2) of 14-16 CB E
( = 0.164 )en e
0 0.2 0.4
5
10
15
( 2
;2+→
0+)
in
fm
41
1B
Ee
2
neutron effective charge ( )e
Dependence of B(E2) onthe neutron effective charge
CD Bonn
14C
16C
18C
14
16
18
+-
e = 0 e = 0.164e Expt.C 3.42 3.66 3.74 0.50C 0.84 2.62 0.63C 2.10 4.98 ?
n
++-
0.11(stat)0.16(syst)
for "dressed"
B(E2; 2 → 0 ) in C14,16,181 1+ +
in e fm2 4
CD Bonn
n
14
16
18
+-
e = 0 e = 0.164e Expt.C 3.42 3.66 3.74 0.50C 0.84 2.62 0.63C 2.10 4.98 4.7
n
++-
0.11(stat)0.16(syst)
for "dressed"
B(E2; 2 → 0 ) in C14,16,181 1+ +
in e fm2 4
CD Bonn
n
+0.8 -0.6(stat)+1.4
-0.9(syst)
0
2
4
6
8
0+
2+
4+0+2+
2+0+3+
0+
2+
4+
2+
3+
0+
Expt.Original
Energy levels in 18O
CD BonnhΩ = 15 MeV
X (
MeV
)E
0
2
4
6
8
0+
2+
4+0+2+
2+0+3+
0+
2+
4+
2+
3+
0+
?
Expt.DressedOriginal
Energy levels in 18O
CD BonnhΩ = 15 MeV
X (
MeV
)E
SummaryDerived the effective interaction for a given model space from the high-precision modern NN force through a uni-tary transformation theory toward the microscopic de-scription of structure of exotic nuclei
・Unitary-model-operator approach (UMOA) g.s. energy for closed-shell nuclei, s.p. energy, … ・Large-scale shell model (with information about s.p. states obtained by the UMOA) complicated structure, transition, …
Including the genuine three-body force and diminishingthe approximations in the calculation
まとめCollaboratorsUMOA Ryoji Okamoto (Kyushu Inst. of Tech.) Kenji Suzuki (Kyushu Inst. of Tech.)“No-core” shell model Takahiro Mizusaki (Senshu Univ.) Takaharu Otsuka (Univ. of Tokyo) Takashi Sebe (Hosei Univ.) Akito Arima (Japan Science Foundation)Application of the “No-core” shell model to O Bruce R. Barrett (Univ. of Arizona)
18