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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


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


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


Introduction to the shell model

What is the shell model ?

Why can it be useful ?

How can we make it run ?


Schematic picture of nucleon-

nucleon (NN) potential

Potential

hard core

1 fm

distance between

nucleons

0.5 fm

-100 MeV


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)


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


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


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


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


Neutron 中性子

Proton 陽子


Mean potential

Harmonic Oscillator (HO)

potential

HO is simpler,

and can be treated

analytically


5hw

4hw

3hw

2hw

1hw

Eigenvalues of

HO potential


Spin-Orbit splitting by the (L S) potential

An orbit with the

orbital angular

momentum l

j = l - 1/2

j = l + 1/2


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)


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


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 -


Back to standard shell model

How to carry out the calculation ?


Hamiltonian

ei :single particle energy

vij,kl : two-body interaction matrix element

( i j k l : orbits)


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)


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.


< 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 = ….


< 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=


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


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


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.


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


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


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


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.


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.


Some remarks

on the two-body matrix elements


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

<j1, j2, J, M | V | j3, j4, J’, M’ >

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

xSm3, m4( j3, m3, j4, m4 |J’, M’)

x<j1, m1, j2, m2 | V | j3, m3, j4, m4 >

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.


Two-body matrix elements

<j1, j2, J, M | V | j3, j4, J, M >

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.


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)


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

<j1, j2, J, T | V | j3, j4, J, T >


USD

interaction

1 = d3/2

2= d5/2

3= s1/2


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


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


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.


In actual applications,

the dimension of the vector space is

a BIG problem !

It can be really big :

thousands,

millions,

billions,

trillions,

....

pf-shell


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


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


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


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


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)


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


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


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


How can we determine

ei :Single Particle Energy

<j1, j2, J, T | V | j3, j4, J, T >

: Two-Body Matrix Element

Effetcive interaction

in shell model calculations


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


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


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


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


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.


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


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


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


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)


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

    <ab;JT | V | cd ; JT >

    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


    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


    GXPF1 vs. experiment

    th. exp.

    th. exp.

    57Ni

    56Ni


    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


    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


    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


    N=32, 34 magic numbers ?

    Issues to be clarified

    by the next generation RIB machines


    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 …….


    ESPE is changed by Nvm

    Monopole interaction, vm

    N particles

    ESPE :

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


    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 -


    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


    Exotic Ti Isotopes

    53Ti

    54Ti

    2+

    56Ti

    55Ti

    2+


    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


    Changing magic numbers ?

    We shall come back to this problem

    after learning under-lying mechanism.


    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


    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


    About same

    radius

    11Li

    208Pb

    Strong tunneling of loosely bound

    excess neutrons

    Neutron halo

    Nakamura’s lecture


    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


    Also in the 1980’s,

    32Mg

    low-lying 2+


    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


    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)


    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


    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)


    Level scheme of na isotopes by sdpf m interaction compared to experiment

    N=17

    N=16

    N=18

    N=19

    Level scheme of Na isotopesby SDPF-M interaction compared to experiment


    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.


    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


    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


    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


    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



    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


    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 -


    Key to understand it :

    Tensor Force

    One pion exchange ~ Tensor force


    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


    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


    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


    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 …….


    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)


    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)


    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’<


    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


    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


    Monopole Interaction

    of the tensor force

    is considered

    to see the connection

    between the tensor force

    and the shell structure


    tensor

    no s-wave to

    s-wave

    coupling

    differences in

    short distance :

    irrelevant

    Tensor potential


    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.


    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, …


    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


    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)


    64

    Weakening of Z=64 submagic structure for N~90

    Single-particle levels of 132Sn core


    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


    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


    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.


    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)



    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

    <ab;JT | V | cd ; JT >

    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


    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


    T=0 monopole interactions in the pf shell

    Tensor force

    (p+r exchange)

    GXPF1A

    G-matrix

    (H.-Jensen)

    Tensor

    component

    is

    subtracted


    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 ?


    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 -



    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’


    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’


    (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)


    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


    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 -


    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’


    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


    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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