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### Structure of exotic nuclei

### Structure of exotic nuclei

195 two-body matrix elements (TBME) and 4 single-particle energies (SPE) are calculated Vary 70 Linear Combinations of 195 TBME and 4 SPE Fit to699 experimental energy data of 87 nuclei

195 two-body matrix elements (TBME) and 4 single-particle energies (SPE) are calculated Vary 70 Linear Combinations of 195 TBME and 4 SPE Fit to699 experimental energy data of 87 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

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

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 ?

nucleon (NN) potential

Potential

hard core

1 fm

distance between

nucleons

0.5 fm

-100 MeV

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

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

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

Spin-Orbit splitting by the (L S) potential

An orbit with the

orbital angular

momentum l

j = l - 1/2

j = l + 1/2

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 -

How to carry out the calculation ?

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=

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 = 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.

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.

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.

<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.

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 >

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

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.

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

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

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

"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)

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

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

ei :Single Particle Energy

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

: Two-Body Matrix Element

Effetcive interaction

in shell model calculations

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

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

- 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

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

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

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

Not completely good（theory imperfect)

GXPF1 interaction

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

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

- 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

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

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

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

(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

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

(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

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

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)

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

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)

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.

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

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

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

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

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

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)

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)

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

of the tensor force

is considered

to see the connection

between the tensor force

and the shell structure

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 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, …

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)

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

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

Not completely good（theory imperfect)

GXPF1 interaction

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

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-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-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 -

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’

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

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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