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)

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