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Structure of exotic nuclei. Takaharu Otsuka University of Tokyo / RIKEN / MSU. A presentation supported by the JSPS Core-to-Core Program  “ International Research Network for Exotic Femto Systems (EFES)”. 7 th CNS-EFES summer school Wako, Japan August 26 – September 1, 2008. Outline.

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structure of exotic nuclei

Structure of exotic nuclei

Takaharu OtsukaUniversity of Tokyo / RIKEN / MSU

A presentation supported by the JSPS Core-to-Core

Program “International Research Network for Exotic Femto Systems (EFES)”

7th CNS-EFES summer school

Wako, Japan

August 26 – September 1, 2008

slide2
Outline

Section 1: Basics of shell model

Section 2: Construction of effective interaction and

an example in the pf shell

Section 3: Does the gap change ? - N=20 problem -

Section 4: Force behind

Section 5: Is two-body force enough ?

Section 6: More perspectives on exotic nuclei

slide3
2-body

interaction

Proton

Neutron

Aim:

To construct many-body systems

from basic ingredients such as

nucleons and nuclear forces

(nucleon-nucleon interactions)

3-body

intearction

slide4
Introduction to the shell model

What is the shell model ?

Why can it be useful ?

How can we make it run ?

slide5
Schematic picture of nucleon-

nucleon (NN) potential

Potential

hard core

1 fm

distance between

nucleons

0.5 fm

-100 MeV

slide6
Actual potential

Depends on quantum numbers

of the 2-nucleon system

(Spin S,

total angular momentum J,

Isospin T)

Very different from

Coulomb, for instance

1S0

Spin singlet (S=0) 2S+1=1

L = 0 (S)

J = 0

From a book by R. Tamagaki (in Japanese)

slide7
Basic properties of atomic nuclei

Nuclear force = short range

Among various components, the nucleus should

be formed so as to make attractive ones

(~ 1 fm )work.

Strong repulsion for distance less than 0.5 fm

Keeping a rather constant distance (~1 fm) between

nucleons, the nucleus (at low energy)is formed.

 constant density : saturation (of density)

 clear surface despite a fully quantal system

Deformation of surface Collective motion

slide8
proton

range of nuclear force from

neutron

Due to constant density, potential

energy felt by is also constant

Mean potential

(effects from other nucleons)

r

Distance from the center

of the nucleus

-50 MeV

slide9
proton

range of nuclear force from

neutron

At the surface, potential

energy felt by is weaker

Mean potential

(effects from other nucleons)

r

-50 MeV

slide10
Eigenvalue problem of single-particle motion

in a mean potential

 Orbital motion

Quantum number : orbital angular momentum l

total angular momentum j

number of nodes of radial wave function n

E

r

Energy eigenvalues

of orbital motion

slide11
Neutron 中性子

Proton 陽子

slide12
Mean potential

Harmonic Oscillator (HO)

potential

HO is simpler,

and can be treated

analytically

slide13
5hw

4hw

3hw

2hw

1hw

Eigenvalues of

HO potential

slide14
Spin-Orbit splitting by the (L S) potential

An orbit with the

orbital angular

momentum l

j = l - 1/2

j = l + 1/2

slide15
magic

number

20

2

8

Orbitals are grouped into shells

shell gap

closed shell

fully occupied orbits

The number of particles below a shell gap :

magic number (魔法数)

This structure of single-particle orbits

shell structure (殻構造)

slide16
5hw

4hw

3hw

2hw

1hw

Eigenvalues of

HO potential

Magic numbers

Mayer and Jensen (1949)

126

82

50

28

20

8

2

Spin-orbit splitting

slide18
From very basic nuclear physics,

density saturation

+ short-range NN interaction

+ spin-orbit splitting

Mayer-Jensen’s magic number

with rather constant gaps

Robust mechanism

- no way out -

slide19
Back to standard shell model

How to carry out the calculation ?

slide20
Hamiltonian

ei :single particle energy

vij,kl : two-body interaction matrix element

( i j k l : orbits)

slide21
A nucleon does not stay in an orbit for ever.

The interactionbetween nucleons changes

their occupations as a result of scattering.

Pattern of occupation : configuration

mixing

valence

shell

closed shell

(core)

slide22
How to get eigenvalues and eigenfunctions ?

Prepare Slater determinantsf1, f2, f3 ,…

which correspond to all possible configurations

The closed shell (core) is treated as the vacuum.

Its effects are assumed to be included in

the single-particle energies and

the effective interaction.

Only valence particles are considered explicitly.

slide23
< f1 | H | f2 >,

< f1 | H | f1 >,

< f1 | H | f3 >, ....

aa+

ag+

ab+

f1 =

….. | 0 >

aa’+

ag’+

ab’+

….. | 0 >

f2 =

Step 1:

Calculate matrix elements

where f1 , f2 , f3 are Slater determinants

In the second quantization,

closed shell

n valence particles

f3 = ….

slide24
< f1 |H| f3 > ....

< f1 |H| f2 >

< f1 |H| f1 >

< f2 |H| f3 > ....

< f2 |H| f2 >

< f2 |H| f1 >

< f3 |H| f1 >

< f3 |H| f2 >

< f3 |H| f3 > ....

.

.

.

< f4 |H| f1 >

.

.

.

.

Step 2 : Construct matrix of Hamiltonian, H,

and diagonalize it

H=

slide25
diagonalization

Conventional Shell Model calculation

All Slater determinants

c

diagonalization

Quantum Monte Carlo Diagonalization method

Important bases are selected

(about 30 dimension)

Diagonalization of Hamiltonian matrix

slide26
Thus, we have solved the eigenvalue problem :

H Y = E Y

With Slater determinantsf1, f2, f3 ,…,

the eigenfunction is expanded as

Y = c1f1 + c2f2 + c3f3 + …..

ci probability amplitudes

slide27
aa+

ag+

ab+

f1 =

….. | 0 >

Usually single-particle state with good j, m (=jz)

fi ’s

has a good M (=Jz),

Each of

because M = m1 + m2 + m3 + .....

fi ’s

having the same value of M are mixed.

M-scheme calculation

Hamiltonian conserves M.

fi ’s

But,

having different values of M are not mixed.

slide28
The Hamiltonian matrix is decomposed into sub matrices

belonging to each value of M.

M=0

M=1

M=-1

M=2

* * * *

* * * *

* * * *

* * * *

0

0

0

H=

* * *

* * *

* * *

0

0

0

* * *

* * *

* * *

0

0

0

. . .

0

0

0

slide29
m1 m2

m1 m2

m1 m2

7/2 -3/2

5/2 -1/2

3/2 1/2

7/2 -7/2

5/2 -5/2

3/2 -3/2

1/2 -1/2

7/2 -5/2

5/2 -3/2

3/2 -1/2

J+

J+

How does J come in ?

An exercise :

two neutrons in f7/2 orbit

J+ : angular momentum raising operator

J+|j, m >

|j, m+1 >

M=2

M=0

M=1

J=1 can be elliminated,

but is not contained

J=0 2-body state is lost

slide30
J = 0, 2, 4, 6

J = 2, 4, 6

J = 2, 4, 6

J = 4, 6

J = 4, 6

J = 6

J = 6

Dimension

Components of J values

M=0

4

M=1

3

M=2

3

2

M=3

M=4

2

M=5

1

1

M=6

slide31
M = 0

eJ=0 0 0 0

0 eJ=2 0 0

0 0 eJ=4 0

0 0 0 eJ=6

* * * *

* * * *

* * * *

* * * *

H=

By diagonalizing the matrix H, you get wave functions

of good J values by superposing Slater determinants.

In the case shown in the previous page,

eJ means the eigenvalue with the angular momentum, J.

slide32
M

eJ 0 0 0

0 eJ’ 0 0

0 0 eJ’’ 0

0 0 0 eJ’’’

* * * *

* * * *

* * * *

* * * *

H=

This property is a general one : valid for cases with

more than 2 particles.

By diagonalizing the matrix H, you get eigenvalues and

wave functions. Good J values are obtained

by superposing properly Slater determinants.

slide33
Some remarks

on the two-body matrix elements

slide34
A two-body state is rewritten as

| j1, j2, J, M >

= Sm1, m2(j1, m1, j2, m2 |J, M) |j1, m1> |j2,m2>

Clebsch-Gordon coef.

Two-body matrix elements

= Sm1, m2( j1, m1, j2, m2 |J, M)

xSm3, m4( j3, m3, j4, m4 |J’, M’)

x

Because the interaction V is a scalar with respect to the

rotation, it cannot change J or M.

Only J=J’ and M=M’ matrix elements can be non-zero.

slide35
Two-body matrix elements

X

X

are independent of M value, also because V is a scalar.

Two-body matrix elements are assigned by

j1, j2, j3, j4 and J.

Jargon : Two-Body Matrix Element = TBME

Because of complexity of nuclear force, one can not

express all TBME’s by a few empirical parameters.

slide36
Actual potential

Depends on quantum numbers

of the 2-nucleon system

(Spin S,

total angular momentum J,

Isospin T)

Very different from

Coulomb, for instance

1S0

Spin singlet (S=0) 2S+1=1

L = 0 (S)

J = 0

From a book by R. Tamagaki (in Japanese)

slide37
Determination of TBME’s Later in this lecture

An example of TBME : USD interaction

by Wildenthal & Brown

sd shell d5/2, d3/2 and s1/2

63 matrix elemeents

3 single particle energies

Note : TMBE’s depend on the isospin T

Two-body matrix elements

slide38
USD

interaction

1 = d3/2

2= d5/2

3= s1/2

slide39
Effective

interaction

Higher shell

Excitations from lower shells

are included effectively by

perturbation(-like) methods

Effects of core

and higher shell

valence shell

Partially occupied

Nucleons are moving around

Closed shell

Excitations to higher shells are

included effectively

slide40
Configuration Mixing Theory

Departure from the independent-particle model

Arima and Horie 1954

magnetic moment

quadrupole moment

This is included

by renormalizing the

interaction and effective charges.

closed shell

+

Core polarization

slide41
Probability that a nucleon is in the valence orbit

~60%

A. Gade et al.

Phys. Rev. Lett. 93, 042501 (2004)

No problem ! Each nucleon carries correlations

which are renormalized into effective interactions.

On the other hand, this is a belief to a certain extent.

slide42
In actual applications,

the dimension of the vector space is

a BIG problem !

It can be really big :

thousands,

millions,

billions,

trillions,

....

pf-shell

slide43
This property is a general one : valid for cases with

more than 2 particles.

By diagonalizing the matrix H, you get eigenvalues and

wave functions. Good J values are obtained

by superposing properly Slater determinants.

M

eJ 0 0 0

0 eJ’ 0 0

0 0 eJ’’ 0

0 0 0 eJ’’’

* * * *

* * * *

* * * *

* * * *

H=

dimension

Billions, trillions, …

4

slide44
Dimension of shell-model calculations

Dimension of Hamiltonian matrix

(publication years of “pioneer”

papers)

Dimension

billion

Birth of shell model

(Mayer and Jensen)

Floating point operations per second

Year

Year

slide45
Shell model code

Name Contact person Remark

OXBASH B.A. Brown Handy (Windows)

ANTOINE E. Caurier Large calc. Parallel

MSHELL T. Mizusaki Large calc. Parallel

These two codes can handle up to 1 billion dimensions.

(MCSM) Y. Utsuno/M. Honma not open Parallel

slide46
Monte Carlo Shell Model

Auxiliary-Field Monte Carlo (AFMC) method

general method for quantum many-body problems

For nuclear physics,Shell Model Monte Carlo

(SMMC)calculation has been introduced by Koonin

et al. Good for finite temperature.

- minus-sign problem

- only ground state, not for excited states in principle.

Quantum Monte Carlo Diagonalization (QMCD) method

No sign problem. Symmetriescan be restored.

Excitedstates can be obtained.

 Monte Carlo Shell Model

slide47
References of MCSM method

"Diagonalization of Hamiltonians for Many-body Systems by Auxiliary Field Quantum Monte Carlo Technique",

M. Honma, T. Mizusaki and T. Otsuka,

Phys. Rev. Lett. 75, 1284-1287 (1995).

"Structure of the N=Z=28 Closed Shell Studied by Monte Carlo Shell Model Calculation",

T. Otsuka, M. Honma and T. Mizusaki,

Phys. Rev. Lett. 81, 1588-1591 (1998).

“Monte Carlo shell model for atomic nuclei”,

T. Otsuka, M. Honma, T. Mizusaki, N. Shimizu and Y. Utsuno,

Prog. Part. Nucl. Phys. 47, 319-400 (2001)

slide48
diagonalization

Conventional Shell Model calculation

All Slater determinants

c

diagonalization

Quantum Monte Carlo Diagonalization method

Important bases are selected

(about 30 dimension)

Diagonalization of Hamiltonian matrix

slide49
Progress in shell-model calculations and computers

Lines : 105 / 30 years

Dimension of Hamiltonian matrix

(publication years of “pioneer”

papers)

Dimension

More cpu time for

heavier or more exotic nuclei

238Uone eigenstate/day

in good accuracy

requires 1PFlops

Conventional

Monte Carlo

Year

Birth of shell model

(Mayer and Jensen)

Floating point operations per second

京速計算機

(Japanese challenge)

GFlops

Blue Gene

Earth Simulator

Our parallel computer

Year

slide50
Outline

Section 1: Basics of shell model

Section 2: Construction of effective interaction and

an example in the pf shell

Section 3: Does the gap change ? - N=20 problem -

Section 4: Force behind

Section 5: Is two-body force enough ?

Section 6: More perspectives on exotic nuclei

slide51
How can we determine

ei :Single Particle Energy

: Two-Body Matrix Element

Effetcive interaction

in shell model calculations

slide52
Determination of TBME’s

Experimental levels of

2 valence particles + closed shell

Early time

TBME

Example : 0+, 2+, 4+, 6+ in 42Ca : f7/2 well isolated

vJ = < f7/2, f7/2, J, T=1 | V | f7/2, f7/2, J, T >

are determined directly

Experimental energy of state J

E(J) = 2 e( f7/2) + vJ

Experimental single-particle energy of f7/2

slide53
5hw

4hw

3hw

2hw

1hw

Eigenvalues of

HO potential

Magic numbers

Mayer and Jensen (1949)

126

82

50

28

20

8

2

Spin-orbit splitting

slide54
The isolation of f7/2 is special. In other cases,

several orbits must be taken into account.

  • In general,c2fit is made
  • TBME’s are assumed,
  • energy eigenvalues are calculated,
  • c2 is calculated between theoretical and
  • experimental energy levels,
  • (iv) TBME’s are modified. Go to (i), and iterate
  • the process until c2 becomes minimum.

Example : 0+, 2+, 4+ in 18O (oxygen) : d5/2 & s1/2

< d5/2, d5/2, J, T=1 | V | d5/2, d5/2, J, T >,

< d5/2, s1/2, J, T=1 | V | d5/2, d5/2, J, T >, etc.

Arima,Cohen, Lawson and McFarlane (Argonne group)), 1968

slide55
At the beginning, it was a perfect c2 fit.
  • As heavier nuclei are studied,
  • the number of TBME’s increases,
  • shell model calculations become huge.

Complete fit becomes more difficult and finally

impossible.

Hybrid version

slide56
Hybrid version

Microscopically calculated TBME’s

for instance, by G-matrix (Kuo-Brown, H.-Jensen,…)

G-matrix-based TBME’s are not perfect,

direct use to shell model calculation is only

disaster

Use G-matrix-based TBME’s as starting point,

and do fit to experiments.

Consider some linear combinations of TBME’s, and

fit them.

slide57
Hybrid version - continued

The c2 fit method produces, as a result of minimization,

a set of linear equations of TBME’s

Some linear combinations of TBME’s are sensitive

to available experimental data (ground and low-lying).

The others are insensitive. Those are assumed to be

given by G-matrix-based calculation (i.e. no fit).

First done for sd shell: Wildenthal and Brown’s USD

47 linear combinations (1970)

Recent revision of USD : G-matrix-based TBME’s have

been improved  30 linear combinations fitted

slide58
Summary of Day 1
  • Basis of shell model and magic numbers
  • density saturation + short-range interaction
  • + spin-orbit splitting
  •  Mayer-Jensen’s magic number
  • How to perform shell model calculations
  • How to obtain effective interactions
structure of exotic nuclei1

Structure of exotic nuclei

Takaharu OtsukaUniversity of Tokyo / RIKEN / MSU

A presentation supported by the JSPS Core-to-Core

Program “International Research Network for Exotic Femto Systems (EFES)”

7th CNS-EFES summer school

Wako, Japan

August 26 – September 1, 2008

Day 2

slide60
Outline

Section 1: Basics of shell model

Section 2: Construction of effective interaction and

an example in the pf shell

Section 3: Does the gap change ? - N=20 problem -

Section 4: Force behind

Section 5: Is two-body force enough ?

Section 6: More perspectives on exotic nuclei

slide61
Day-1 lecture :

Introduction to the shell model

What is the shell model ?

Why can it be useful ?

How can we make it run ?

Basis of shell model and magic numbers

density saturation + short-range interaction

+ spin-orbit splitting

 Mayer-Jensen’s magic number

 Valence space (model space)

For shell model calculations, we need also

TBME (Two-Body Matrix Element) and

SPE (Single Particle Energy)

slide62
An example from pf shell (f7/2, f5/2, p3/2, p1/2)

Phenomenological

Microscopic

G-matrix + polarization correction + empirical refinement

  • Start from a realistic microscopic interaction

M. Hjorth-Jensen, et al., Phys. Repts. 261 (1995) 125

        • Bonn-C potential
        • 3rd order Q-box + folded diagram
  • 195 two-body matrix elements (TBME) and 4 single-particle energies (SPE) are calculated

 Not completely good(theory imperfect)

  • Vary 70 Linear Combinations of 195 TBME and 4 SPE
  • Fit to699 experimental energy data of 87 nuclei

GXPF1 interaction

M. Honma et al., PRC65 (2002) 061301(R)

g matrix vs gxpf1
G-matrix vs. GXPF1

two-body matrix element

output

7= f7/2, 3= p3/2, 5= f5/2, 1= p1/2

  • T=0 … attractive
  • T=1 … repulsive
  • Relatively large modifications in
    • V(abab ; J0) with large J
    • V(aabb ; J1) pairing

input

slide64
Systematics of 2+1
  • Shell gap
    • N=28
    • N=32 for Ca, Ti, Cr
    • N=34 for Ca ??
  • Deviations in Ex
    • Cr at N≧36
    • Fe at N≧38
  • Deviations in B(E2)
    • Ca, Ti for N≦26
    • Cr for N≦24
    • 40Ca core excitations
    • Zn, Ge
    • g9/2 is needed
slide65
GXPF1 vs. experiment

th. exp.

th. exp.

57Ni

56Ni

slide66
56Ni (Z=N=28) has been considered to be a doubly magic nucleus where proton and neutron f7/2 are fully occupied.

Probability of closed-shell in the ground state

doubly magic

  ⇒ Measure of breaking of

this conventional idea

Ni

neutron

Ni

proton

48Cr

total

slide67
States of different nature

can be reproduced within a

single framework

54Fe yrast states

  • 0p-2h configuration
    • 0+, 2+, 4+, 6+ …p(f7/2)-2
    • more than 40% prob.
  • 1p-3h … 1st gap
    • One-proton excitation
    • 3+, 5+
    • 7+~11+
  • 2p-4h … 2nd gap
    • Two-protons excitation
    • 12+~

p-h : excitation from

f7/2

slide68
58Ni yrast states
  • 2p-0h configuration
    • 0+, 2+…n(p3/2)2
    • 1+, 3+, 4+…n(p3/2)1(f5/2)1
    • more than 40% prob.
  • 3p-1h … 1st gap
    • One-proton excitation
    • 5+~8+
  • 4p-2h … 2nd gap
    • One-proton &
    • one-neutron excitation
    • 10+~12+

p-h : excitation from

f7/2

slide69
N=32, 34 magic numbers ?

Issues to be clarified

by the next generation RIB machines

slide70
Effective single particle energy
  • Monopole part of the NN interaction
  • Angular averaged interaction

Isotropic component is extracted

froma general interaction.

In the shell model, single-particle properties are

considered by the following quantities …….

slide71
Effective single-particle energy (ESPE)

ESPE is changed by Nvm

Monopole interaction, vm

N particles

ESPE :

Total effect on single-particle energies due to interaction with other valence nucleons

slide72
Effective single-particle energies

Z=22

Z=20

Z=24

f5/2

n-n

p-n

new

magic

numbers ?

34

32

p1/2

p3/2

Lowering of f5/2 from Ca to Cr

- weakening of N=34 -

Why ?

Rising of f5/2 from 48Ca to 54Ca

- emerging of N=34 -

slide73
Exotic Ca Isotopes : N = 32 and 34 magic numbers ?

51Ca

53Ca

52Ca

54Ca

2+

2+

?

exp. levels :Perrot et al. Phys. Rev. C (2006), and earlier papers

slide74
Exotic Ti Isotopes

53Ti

54Ti

2+

56Ti

55Ti

2+

slide75
ESPE

(Effectice Single-

Particle Energy)

of neutrons

in pf shell

G

f 5/2

GXPF1

f 5/2

Why is neutron f 5/2

lowered by filling

protons intof 7/2

Ca

Ni

slide76
Changing magic numbers ?

We shall come back to this problem

after learning under-lying mechanism.

slide77
Outline

Section 1: Basics of shell model

Section 2: Construction of effective interaction and

an example in the pf shell

Section 3: Does the gap change ? - N=20 problem -

Section 4: Force behind

Section 5: Is two-body force enough ?

Section 6: More perspectives on exotic nuclei

slide78
nuclei

(mass number)

stable

exotic

-- with halo

A

Proton number 

Neutron number 

Studies on exotic nuclei in the 80~90’s

Left-lower part of

the NuclearChart

Stability line and drip lines are not so far from each other

 Physics of loosely bound

neutrons, e.g., halo

while other issues like 32Mg

proton halo

neutron halo

リチウム11

11Li

neutron skin

slide79
About same

radius

11Li

208Pb

Strong tunneling of loosely bound

excess neutrons

Neutron halo

Nakamura’s lecture

slide80
In the 21st century, a wide

frontier emerges between the

stability and drip lines.

Stability line

Drip line

nuclei

(mass number)

stable

exotic

Riken’s work

A

Neutron number 

Proton number 

中性子数 

(同位元素の種類)

huge area

slide81
Also in the 1980’s,

32Mg

low-lying 2+

slide82
Basic picture was

deformed

2p2h state

energy

intruder ground state

stable

exotic

pf shell

gap ~

constant

N=20

sd shell

Island of Inversion

9 nuclei:

Ne, Na, Mg with N=20-22

Phys. Rev. C 41, 1147 (1990),

Warburton, Becker andBrown

slide83
One of the major issues over the millennium was

to determine the territory of

the Island of Inversion

  • Are there clear boundaries in all directions ?
  • Is the Island really like the square ?

Which type of boundaries ?

Shallow

(diffuse & extended)

Straight lines

Steep (sharp)

slide84
Small gap vs. Normal gap

v ~ < f(Qp Qn) >

dv=large

For larger gap,

fmust be larger

 sharp boundary

v=0

normal

For smaller gap,

f is smaller

 diffuse boundary

intruder

Max pn force

N

semi-magic

open-shell

dv=smaller

The difference dv is modest

as compared to “semi-magic”.

Inversion occurs for

semi-magic nuclei most easily

slide85
Na isotopes :

What happens

in lighter ones

with N < 20

Original Island of Inversion

electro magnetic moments and wave functions of na isotopes
Electro-magnetic moments and wave functions of Na isotopes

Q

― normal dominant : N=16, 17― strongly mixed : N=18― intruder dominant : N=19, 20Onset of intruder dominance

before arriving at N=20

m

Monte Carlo Shell Model calculation

with full configuration mixing :

Phys. Rev. C 70, 044307 (2004),

Utsuno et al.

Config.

Exp.: Keim et al. Euro. Phys. J.

A 8, 31 (2001)

slide88
Major references on MCSM calculations for N~20 nuclei

"Varying shell gap and deformation in N~20 unstable nuclei studied by

the Monte Carlo shell model",

Yutaka Utsuno, Takaharu Otsuka, Takahiro Mizusaki and

Michio Honma,

Phys. Rev. C60, 054315-1 - 054315-8 (1999)

“Onset of intruder ground state in exotic Na isotopes and evolution of

the N=20 shell gap”,

Y. Utsuno, T. Otsuka, T. Glasmacher, T. Mizusaki and M. Honma,

Phys. Rev. C70, (2004), 044307.

Many experimental papers include MCSM results.

slide89
WBB (1990)

SDPF-M (1999)

~5MeV

~2MeV

Ne

O

Mg

Ca

Monte Carlo Shell Model (MCSM) results have been obtained

by the SDPF-M interaction for the full-sd + f7/2 + p3/2 space.

Effective N=20 gap

between sd and pf shells

Expansion

of the

territory

Neyens et al. 2005Mg

Tripathi et al. 2005Na

Dombradi et al. 2006Ne

Terryet al.2007Ne

slide90
Phys. Rev. Lett. 94, 022501 (2005), G. Neyens, et al.

Strasbourg

unmixed

Tokyo

MCSM

USD (only sd shell)

2.5 MeV

0.5 MeV

31Mg19

slide91
New picture

Conventional picture

deformed

2p2h state

deformed

2p2h state

spherical

normal state

energy

energy

?

intruder ground state

intruder ground state

stable

stable

exotic

exotic

pf shell

pf shell

gap ~

constant

gap

changing

N=20

N=20

sd shell

sd shell

slide92
Effective N=20 gap

between sd and pf shells

Island of Inversion

Expansion

of the

territory

constant gap

SDPF-M

(1999)

?

~6MeV

~2MeV

Ne

O

Mg

Ca

?

Shallow

(diffuse & extended)

Straight lines

Steep (sharp)

Island of Inversion

is like a paradise

slide94
Outline

Section 1: Basics of shell model

Section 2: Construction of effective interaction and

an example in the pf shell

Section 3: Does the gap change ? - N=20 problem -

Section 4: Force behind

Section 5: Is two-body force enough ?

Section 6: More perspectives on exotic nuclei

slide95
From undergraduate nuclear physics,

density saturation

+ short-range NN interaction

+ spin-orbit splitting

Mayer-Jensen’s magic number

with rather constant gaps

(except for gradual A dependence)

Robust mechanism

- no way out -

slide96
Key to understand it :

Tensor Force

One pion exchange ~ Tensor force

slide97
Key to understand it : Tensor Force

p meson : primary source

r meson (~ p+p) : minor (~1/4) cancellation

Ref:Osterfeld, Rev. Mod. Phys. 64, 491 (92)

p, r

Multiple pion exchanges

 strong effectivecentral forces in NN interaction

(as represented bysmeson, etc.)

 nuclear binding

This talk : First-order tensor-force effect

(at medium and long ranges)

One pion exchange  Tensor force

slide98
V ~ Y2,0~ 1 – 3 cos2q

q=p/2

q=0

repulsion

attraction

How does the tensor force work ?

Spin of each nucleon is parallel, because the total spin must be S=1

The potential has the following dependence on the angle qwith respect to the total spin S.

q

S

relative

coordinate

slide99
Deuteron : ground state J = 1

Total spin S=1

Relative motion : S wave (L=0) + D wave (L=2)

proton

Tensor force does mix

neutron

The tensor force is crucial to bind the deuteron.

Without tensor force, deuteron is unbound.

No S wave to S wave coupling by tensor force

because of Y2 spherical harmonics

slide100
Effective single particle energy
  • Monopole part of the NN interaction
  • Angular averaged interaction

Isotropic component is extracted

froma general interaction.

In the shell model, single-particle properties are

considered by the following quantities …….

slide101
Intuitive picture of monopole effect of tensor force

wave function of relative motion

spin of nucleon

large relative momentum

small relative momentum

repulsive

attractive

j> = l + ½, j< = l – ½

TO et al., Phys. Rev. Lett. 95, 232502 (2005)

slide102
Monopole Interaction

of the Tensor Force

j<

neutron

j>

j’<

proton

j’>

Identity for tensor monopole interaction

( j’j>)

( j’j<)

(2j> +1) vm,T+ (2j<+1)vm,T= 0

vm,T: monopole strength for isospin T

T. Otsuka et al., Phys. Rev. Lett. 95, 232502 (2005)

slide103
Major features

spin-orbitsplitting varied

Opposite signs

T=0 : T=1 = 3 : 1 (same sign)

Only exchange terms (generally for spin-spin forces)

neutron, j’<

proton, j>

tensor

proton, j>

neutron, j’<

slide104
Tensor Monopole Interaction :

total effects vanished for

spin-saturated case

j<

neutron

no change

j>

j’<

proton

j’>

Same Identity with different interpretation

( j’j>)

( j’j<)

(2j> +1) vm,T+ (2j<+1)vm,T= 0

vm,T: monopole strength for isospin T

slide105
j<

neutron

Tensor Monopole Interaction

vanished for s orbit

j>

s1/2

proton

For s orbit, j> and j< are the same :

( j’j>)

( j’j<)

(2j> +1) vm,T+ (2j<+1)vm,T= 0

vm,T: monopole strength for isospin T

slide106
Monopole Interaction

of the tensor force

is considered

to see the connection

between the tensor force

and the shell structure

slide107
tensor

no s-wave to

s-wave

coupling

differences in

short distance :

irrelevant

Tensor potential

slide108
Proton effective single-particle levels

(relative to d3/2)

Tensor monopole

f7/2

d3/2

d5/2

neutron

proton

p + rmeson tensor

exp.

Cottle and Kemper,

Phys. Rev. C58, 3761 (98)

neutrons in f7/2

spectroscopic factor for 1p from 48 ca probing proton shell gaps
Spectroscopic factor for -1p from 48Ca:probing proton shell gaps

w/ tensor

w/o tensor

d5/2-s1/2 gap

d3/2-s1/2 gap

Kramer et al. (2001) Nucl PHys A679 NIKHEF exp.

slide110
N=16 gap : Ozawa, et al., PRL 84 (2000) 5493;

Brown, Rev. Mex. Fis. 39 21 (1983)

only

exchange

term

d3/2

Tensor

force

d5/2

Example : Dripline of F isotopes is 6 units away from O isotopes

Sakurai et al., PLB 448 (1999) 180, …

slide111
WBB (1990)

SDPF-M (1999)

~5MeV

~2MeV

Ne

O

Mg

Ca

Monte Carlo Shell Model (MCSM) results have been obtained

by the SDPF-M interaction for the full-sd + f7/2 + p3/2 space.

Effective N=20 gap

between sd and pf shells

Expansion

of the

territory

Neyens et al. 2005Mg

Tripathi et al. 2005Na

Dombradi et al. 2006Ne

Terryet al.2007Ne

slide112
1h11/2 protons

1g7/2 protons

51Sb case

Opposite monopole

effect from

tensor force

with neutrons

in h11/2.

Z=51 isotopes

Tensor by

  • + r meson

exchange

h11/2

+ common effect

(Woods-Saxon)

g7/2

No mean field theory,

(Skyrme, Gogny, RMF)

explained this before.

1h11/2 neutrons

Exp. data from J.P. Schiffer et al., Phys. Rev. Lett. 92, 162501 (2004)

slide113
64

Weakening of Z=64 submagic structure for N~90

Single-particle levels of 132Sn core

slide114
Proton collectivity

enhanced at Z~64

Weakening of Z=64 submagic structure for N~90

1h9/2

2d3/2

64

2d5/2

8 neutrons in 2f7/2

reduces the Z=64 gap

to the half value

8 protons in 1g7/2

pushes up 1h9/2

by ~1 MeV

slide115
Neutron single-particle energies

Mean-field models

(Skyrme or Gogny)

do not reproduce this

reduction.

Tensor force effect

due to vacancies of

proton d3/2 in 4718Ar29 :

650 (keV) by p+r meson

exchange.

f 5/2

f 7/2

slide116
RIKEN RESEARCH, Feb. 2007

Magic numbers do change, vanish and emerge.

Today’s perspectives

Conventional picture (since 1949)

A city works its magic. … N.Y.

slide117
Effect of tensor force on (spherical)

superheavy magic numbers

1k17/2

2h11/2

N=184

Neutron

Tensor force

added

Woods-Saxon

potential

Occupation of neutron

1k17/2 and

2h11/2

Proton single particle levels

Otsuka, Suzuki and Utsuno,

Nucl. Phys. A805, 127c (2008)

slide119
Shell evolution by realistic effective interaction : pf shell

Phenomenological

Microscopic

G-matrix + polarization correction + empirical refinement

  • Start from a realistic microscopic interaction

M. Hjorth-Jensen, et al., Phys. Repts. 261 (1995) 125

        • Bonn-C potential
        • 3rd order Q-box + folded diagram
  • 195 two-body matrix elements (TBME) and 4 single-particle energies (SPE) are calculated

 Not completely good(theory imperfect)

  • Vary 70 Linear Combinations of 195 TBME and 4 SPE
  • Fit to699 experimental energy data of 87 nuclei

GXPF1 interaction

M. Honma et al., PRC65 (2002) 061301(R)

g matrix vs gxpf11
G-matrix vs. GXPF1

two-body matrix element

output

7= f7/2, 3= p3/2, 5= f5/2, 1= p1/2

  • T=0 … attractive
  • T=1 … repulsive
  • Relatively large modifications in
    • V(abab ; J0) with large J
    • V(aabb ; J1) pairing

input

slide121
f-p

f-f

p-p

T=0 monopole interactions in the pf shell

Tensor force

(p+r exchange)

GXPF1A

G-matrix

(H.-Jensen)

“Local pattern”  tensor force

slide122
T=0 monopole interactions in the pf shell

Tensor force

(p+r exchange)

GXPF1A

G-matrix

(H.-Jensen)

Tensor

component

is

subtracted

slide123
The central force is modeled by a Gaussian function

V = V0exp( -(r/m) 2) (S,T dependences)

with V0 = -166 MeV, m=1.0 fm,

(S,T) factor (0,0) (1,0) (0,1) (1,1)

--------------------------------------------------

relative strength 1 1 0.6 -0.8

Can we explain the difference between f-f/p-p and f-p ?

slide124
f-p

f-f

p-p

T=0 monopole interactions in the pf shell

Tensor force

(p+r exchange)

GXPF1

G-matrix

(H.-Jensen)

Central (Gaussian)

- Reflecting

radial overlap -

slide126
j = j’

T=1monopole

interactions

in the pf shell

GXPF1A

G-matrix

(H.-Jensen)

Tensor force

(p+r exchange)

Basic scale

~ 1/10 of T=0

Repulsive

corrections

to G-matrix

j = j’

slide127
j = j’

T=1monopole

interactions

in the pf shell

GXPF1A

G-matrix

(H.-Jensen)

Tensor force

(p+r exchange)

Central (Gaussian)

- Reflecting

radial overlap -

j = j’

slide128
(Effective) single-particle energies

n-n

p-n

KB3G

Lowering of f5/2 from Ca to Cr :

~ 1.6 MeV = 1.1 MeV (tensor) + 0.5 MeV (central)

Rising of f5/2 from 48Ca to 54Ca :

p3/2-p3/2 attraction p3/2-f5/2 repulsion

KB interactions : Poves, Sanchez-Solano, Caurier and Nowacki, Nucl. Phys. A694, 157 (01)

slide129
Major monopole components of GXPF1A interaction

T=0 - simple central (range ~ 1fm) + tensor

- strong (~ 2 MeV)

-attractive modification from G-matrix

T=1 - More complex central (range ~ 1fm) + tensor

- weak ~ -0.3 MeV (pairing), +0.2 MeV (others)

-repulsive modification from G-matrix

even changing the signs

Also in sd shell….

Central force : strongly renormalized

Tensor force : bare p + r meson exchange

slide130
T=0 monopole interactions in the sd shell

Tensor force

(p+r exchange)

G-matrix

(H.-Jensen)

SDPF-M

(~USD)

Central (Gaussian)

- Reflecting

radial overlap -

slide131
T=1monopole

interactions

in the sd shell

SDPF-M(~USD)

G-matrix

(H.-Jensen)

Tensor force

(p+r exchange)

Basic scale

~ 1/10 of T=0

Repulsive

corrections

to G-matrix

j = j’

j = j’

slide132
This is not a very lonely idea Chiral Perturbation of QCD

Short range central forces

have complicated origins and

should be adjusted.

S. Weinberg,

PLB 251, 288 (1990)

Tensor force is explicit

slide133
Outline

Section 1: Basics of shell model

Section 2: Construction of effective interaction and

an example in the pf shell

Section 3: Does the gap change ? - N=20 problem -

Section 4: Force behind

Section 5: Is two-body force enough ?

Section 6: More perspectives on exotic nuclei

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