Description of Nucleiin Real & Model Spaces

Yoshinori AKAISHI

RIKEN LectureJanuary 6, 2006

Nuclear Physics

Finite number of nucleonsinteracting with strong interaction

Rich structures on the basis of the nuclear Saturation

Nuclear saturation

0

-5

-10

-20

E/A(MeV)

kF (fm-1)1.0 2.0

1.36

-15.7Exp

E/A = -avA + asA2/3+ …

Infinite nuclear matter

-7.8

23.9

-15.9

-31.7

E/AKE/A

PE/A

1S3S1P3P1D3D

-15.83.20.3

-2.2-1.3

H-J(MeV)

TT veQv

is included.

~0

Y. Akaishi, K. Takada & S. Takagi,Prog. Theor. Phys. 36 (1966) 113533

3

34

244 LAkLA F

kk F

/, =⎟⎠⎞

⎜⎝⎛=∑=

≤ρπ

π

323

2Fk

πρ =Fermi gas

V

1 2 3

∞

r fm

ϕ

Theory of nuclear matter

ψHealing

K.A. Brueckner & C.A. LevinsonPhys. Rev. 97 (1955) 1344

J. GoldstoneProc. Roy. Soc. A239 (1957) 267

H.A. BethePhys. Rev. 103 (1958) 241

Cor

rela

tion

te

vvt0

1+=

Independent-pair scattering mode

L.C. Gomes, J.D. Walecka & V.F. WeisskopfAnn. Phys. 3 (1958) 241

Foundation of Shell Model

QTQE/A = aρ + bρ2+…

2121 tte −−+= εε

Y. Akaishi, H. Bando, A. Kuriyama & S. NagataProg. Theor. Phys. 40 (1968) 288

Hole-line expansion method

h≥Δ⋅Δ rp

kF

geQvvg +=⇒

3S/1S ratio 1.0 1.7

Matter 4He

Tensor enhancement

ΦΦ+ΦΦ TTT VeQVVX

Infinite matter

Few-body systemsY. Akaishi & S. Nagata,

Prog. Theor. Phys. 48 (1972) 133

Potential matrix elements

ν=ωM

2hh

4He

ΦΦ+ΦΦ TTT VeQVVX

Nuclear saturation(1) Tensor force(2) Exchange force(3) Repulsive core

(1-m)+mPr ,Saturation condition, m > 0.8,

is not satisfied.

Majorana exchange

Energy of nuclear matterH.A. Bethe, Ann. Rev. Nucl. Sci. 21 (1971) 93

kF fm-10.5 1.0 1.5

3S1

1S0

-10

-20

0

E/A MeV

M. Serra, T. Otsuka et al., Prog. Theor. Phys. 113 (2005) 1009:g-matrix Relativistic Mean Field

Lower densities

Central ~ -100 MeV at 1 fm

Tensor force effects on clusterization

Y. Akaishi, H. Bando & S. Nagata,Prog. Theor. Phys. Suppl. 52 (1972) 339

“Essential 4-body”ω =ε1+ε2

Enhanced

Suppressed

12

22

1212 28 SVVSVeQSV

ΔΔTT

TT +−≈

( ) MeVav 20021 −≈+−= ttωΔ

“Model space”

S. Nag

ata

H. T

anak

a

“Real space”

Alpha particle (0s)4

Model

KE

E

PE

1E central

3E central

ATMS

Real

Real space vs. model space

Multiple scattering process

Two-body scattering in medium

Amalgamation of Two-body correlations into Multiple Scattering process

ATMS

Two-body scattering in medium

ijijijij geQvvg +=

)1ˆ('')1ˆ()()(

−∑+∑=− klklkl

klkl

ij FgeQg

eQF

)1( −klu

klklkl

ij FgeQF ˆ'ˆ

)(∑=

)'1ˆ('''

)'1ˆ(

)()()()(

)(

mnmn

klklkl

mnmn

klkl

klkl

ij

geQFg

eQg

eQg

eQ

geQF

∑−−∑+∑∑=

∑−−

)1( −klu

)1( −klu

)1( −klu )1( −mnu )1~( −klu

0th ATMS

1st ATMS

2nd ATMS

ATMS can improve the wave function in a systematic way.

klkl

uF ∏=)(

Jastrow

Day’sapprox.

Live fish “Sashimi”

Structure-dependent3S1 interaction

Alpha particle

MeV621.=ωh

ATMS M. Sakai, I. Shimodaya,Y. Akaishi, J. Hiura & H. Tanaka,

Prog. Theor. Phys. Suppl. 56 (1974) 32

Short-range correlation

Tensor renormalization

0s0p

0d,1s

Since then, 20 years have passed.Y. En’yo:

What is “Tensor Force”?

Realistic NN interaction

21212 LVWVSLrVSrVrVV

rrrLLWLSTC )()()( ++++=

( )( )212

2112 3 σσσσ rr

rrrr−=

rrrS

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−++

+−−

+=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+

−

+

−

JJ

JJ

JJ

JJ

JJ

JJ

YY

Y

JJJJ

JJJ

JYY

YS

,,

,,

,,

,,

,,

,,

)()()(

)()(

11

1

11

11

1

11

12

2201601220

16012

121

r =1.4 fm

OPEP

VC

VT

( ) ( ) ( )1211122

2 4 rrfirrrrr

−⎭⎬⎫

⎩⎨⎧ ∇−=−∇ −+ δτσ

κπφκ )()(

( ) ( ) ( )12

1211112 rr

rrfirr rr

rrrrrr

−−−

⎭⎬⎫

⎩⎨⎧ ∇=− −+ κ

τσκ

φexp)()(

( ) ( ) { } ( ) ( )12

121112

01

02122212 rr

rrfifirrV rr

rrrrrrrr

−−−

⎭⎬⎫

⎩⎨⎧ ∇++

⎭⎬⎫

⎩⎨⎧ ∇=− +−−+ κ

σκ

ττττττσκ

exp)()()()()()(OPEP

( )12122 4 rre

rrr−−=∇ δπφ

121

1rr

e rr−

=φ

( )12

12121

rreerrV rr

rr

−=−Coul

np)( 2=−τ

( ) ( ) ( ) ( )r

rrr

Scmc

frVκ

κκκ

σσττ −⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛+++⎟

⎟⎠

⎞⎜⎜⎝

⎛=

exp)(πOPEP 2122121

22 331

31 rrrr

h

h

rrr cmrrr π, =−= κ12

( )( )3212

21 13rr

rr⎭⎬⎫

⎩⎨⎧ −− μμμμ rr

rrrr

OPEPπ+

n

pn

p1 2

An evidence for OPEP

Alpha particle

Phys. Rev. C64 (2001) 044001Benchmark test calculation of 4N

FYH. Kamada, W. Gloeckle et al.

CRCGVM. Kamimura, E. Hiyama et al.

SVMY. Suzuki, K. Varga et al.

HHM. Viviani, A. Kievsky et al.

GFMCJ. Carlson, R.B. Wiringa et al.

NCSMP. Navratil, B.R. Barrett et al.

Ordinarysingle particle model

AV8’ (Sc&D tr.)-25.354.1

-32.0-47.4

-79.4

K. IkedaTensor pion

0=l

1=l

2002

“Is the effective model unique?”

Ikeda’s idea

Parity- & charge-mixed s.p. stateTensor pion

Charge

0=l

1=l＾

＾

＾

A challenge @ the citadel of standard S. M.

Parity

π+, π0, π- coherence

r2

r fm

Tensor BHF calculation(0s+0p)4 AV8’

Pp=6.0%

fs(r)

r fp(r)

0 1 2 3 4

0.4

0.8

1.2

∞=)D(Qk

sp ωω hh >Y. Kanada-En’yo

Mixed up!

Alpha particle

Phys. Rev. C64 (2001) 044001Benchmark test calculation of 4N

FYH. Kamada, W. Gloeckle et al.

CRCGVM. Kamimura, E. Hiyama et al.

SVMY. Suzuki, K. Varga et al.

HHM. Viviani, A. Kievsky et al.

GFMCJ. Carlson, R.B. Wiringa et al.

NCSMP. Navratil, B.R. Barrett et al.

Ordinarysingle particle model

AV8’ (Sc&D tr.)-25.354.1

-32.0-47.4

-79.4

“Ptolemaic”Geocentric

“Copernican”

Parity- & charge-mixedsingle particle model

AV8’ (Sc tr.)-7.658.6

-25.8-9.3

-35.1

-31.1

P(p)=6.0 %

-66.2

Heliocentric

Wave function of 4HeDS Ψ+Ψ=Ψ

{ } { }( )M1,

M,2

M2,

M,1D 2,02,0)0022(

SLSLL

MmMmMmMmM

SL ggMM −∑=Ψ

{ } { } { }( )M1

M2

Mm2

M1

ASS 0,00,00,0 mmm fff −+=Ψ

[ ] S)2(

)(

TE3

10M Φ∑ ⊗=LL M

ijijijij

ijmm,M rrwcg

rr

Principal S Mixed S’

Mixed D

[ ] 2,)]4()3([)]2()1([ 2112 ==Σ Sssss

[ ] 0,)]4()3([)]2()1([ 01101 ==Σ Sssssm

[ ] 0,)]4()3([)]2()1([ 00002 ==Σ Sssssm

Spin functions⎥⎦⎤

⎢⎣⎡ −+→ SNSNS

2,

2partition

Isospin-spin functions [T, S]R

{ } [ ]01

02

02

012

1A0,0 mmmm TT Σ−Σ=

{ } [ ]01

01

02

022

1M10,0 mmmmm TT Σ−Σ=

{ } [ ]01

02

02

012

1M20,0 mmmmm TT Σ+Σ=

{ } 201

M12,0 Σ= mm T

{ } 202

M22,0 Σ= mm T

[4]

[22]

P Q

Momentum distribution in 4He

Model space

k fm-1

log

w(k

)

0 1 2 3 4 5

0

-1

-2

-3

-4

-5

-6

T ~ 50 MeV ΔT ~ 25 + 25 MeV

Tensorcorrel.

Short-rangecorrel.

400 MeV/c

High momentumand short range

at k=0.4 GeV/c

j0(kr)

j2(kr)High momentumbut long range

-1)D(Q fm8.2=k

P Q

Momentum distribution in 4He

Harmonic oscillator

model

Tensor correlationplus

Short-range correlation

k fm-1

log

w(k

)

0 1 2 3 4 5

0

-1

-2

-3

-4

-5

-6

F)S(

Q kk =

P Q

r fm

MeV

vT(r)

gT(r)

-80

40

0

-40

-120

0 1 2 3

-1)D(Q fm8.2=k

∞=)D(Qk

To effective central

kF=1.4, 1.1, 0.7 fm-1

less dependence on kF

r fm

MeVfm2 r2gC

-1)D(Q fm8.2=k

F)D(

Q kk =

0 1 2 3 4

100

50

0

-50

-100

-150

r fm

Tensor BHF calculation(0s+0p)4 AV8’

Pp=6.0%

fs(r)

r fp(r)

0 1 2 3 4

0.4

0.8

1.2

∞=)D(Qk 2.8 fm-1

-23.1 89.1

-29.5-27.4

-56.9

-55.2

P(p)=9.8 %

-112.1

-1)D(Q fm8.2=k

Alpha particle

Phys. Rev. C64 (2001) 044001Benchmark test calculation of 4N

Ordinarysingle particle model

AV8’ (Sc&D tr.)-25.354.1

-32.0-47.4

-79.4

“Ptolemaic”Geocentric

“Copernican”

Parity- & charge-mixedsingle particle model

AV8’ (Sc tr.)-23.189.1

-29.5-27.4

-56.9

-55.2

P(p)=9.8 %

-112.1

Heliocentric

Concluding remarks

Ordinary single particle model

State-dependent effective interactions(Density-, cluster-, halo-, … dependent)

Challenging !

“New generation”

Pions play a leading role in nuclei.

Charge-parity mixedsingle particle model

High-momentum phenomenadue to long-range tensor correlation

(Restoration of chiral symmetry)

Models of new generation

A wide variety of nuclear structures

Effective interaction

Dynamically determined

Antisymmetrized Molecular DynamicsY. Kanada-En’yo, H. Horiuchi et al.

must include physical essencesof strong interaction.

“Pion dominance”

Thank you very much!