Structure of exotic nuclei

Takaharu Otsuka 　　 University of Tokyo / RIKEN / MSU

7th CNS-EFES summer schoolWako, Japan

August 26 – September 1, 2008

A presentation supported by the JSPS Core-to-Core Program 　“ International Research Network for

Exotic Femto Systems (EFES)”

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

Outline

Proton

Neutron

2-body interaction

Aim:To construct many-body systems from basic ingredients such as nucleons and nuclear forces (nucleon-nucleon interactions)

3-body intearction

What is the shell model ?

Why can it be useful ?

Introduction to the shell model

How can we make it run ?

0.5 fm

1 fm

distance betweennucleons

PotentialSchematic picture of nucleon-nucleon (NN) potential

-100 MeV

hard core

Actual potential 　

Depends on quantum numbers of the 2-nucleon system

(Spin S, total angular momentum J,

Isospin T)

Very different fromCoulomb, for instance

1S0 Spin singlet (S=0) 2S+1=1L = 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) betweennucleons, 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

neutron range of nuclear force from

Due to constant density, potential energy felt by is also constant

Mean potential(effects from other nucleons)

Distance from the center of the nucleus-50 MeV

r

proton

neutron range of nuclear force from

At the surface, potential energy felt by is weaker

Mean potential(effects from other nucleons)

-50 MeV

r

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

Energy eigenvaluesof orbital motion

E

r

Proton 陽子

Neutron 中性子

Harmonic Oscillator (HO)potential

Mean potential

HO is simpler, and can be treatedanalytically

Eigenvalues of HO potential

5h

4h

3h

2h

1h

Spin-Orbit splitting by the (L S) potential

An orbit with the orbital angular momentum l j = l - 1/2

j = l + 1/2

The number of particles below a shell gap :magic number ( 魔法数 )

This structure of single-particle orbits shell structure ( 殻構造 )

magic number

shell gap

2

20

8

Orbitals are grouped into shells

closed shellfully occupied orbits

Spin-orbit splitting

Eigenvalues of HO potential

Magic numbersMayer and Jensen (1949)

126

8

20

28

50

82

2

5h

4h

3h

2h

1h

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 ?

i : single particle energy

v ij,kl : two-body interaction matrix element

( i j k l : orbits)

Hamiltonian

A nucleon does not stay in an orbit for ever.The interaction between nucleons changes their occupations as a result of scattering.

mixing

Pattern of occupation : configuration

valence shell

closed shell(core)

配位

Prepare Slater determinants 1, 2, 3 ,…

which correspond to all possible

configurations

How to get eigenvalues and eigenfunctions ?

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.

配位

Calculate matrix elements

where 1 , 2 , 3 are Slater determinants

< 1 | H | 1 >, < 1 | H | 3 >, ....< 1 | H | 2 >,

Step 1:

In the second quantization,

1 = ….. | 0 >a+ a

+ a+

n valence particles

2 = ….. | 0 >a’+ a’

+ a’+

3 = ….

closed shell

Step 2 : Construct matrix of Hamiltonian, H, and diagonalize it

< 3 |H| 3 > ....

< 1 |H| 1 > < 1 |H| 3 > ....< 1 |H| 2 >

< 2 |H| 1 > < 2 |H| 3 > ....< 2 |H| 2 >

< 3 |H| 1 > < 3 |H| 2 >

.. .

. .< 4 |H| 1 >

.

.

H =

Diagonalization of Hamiltonian matrix

(about 30 dimension)

cConventional Shell Model calculation All Slater determinants

diagonalization

diagonalization

Quantum Monte Carlo Diagonalization method Important bases are selected

With Slater determinants 1, 2, 3 ,…,

the eigenfunction is expanded as

H

Thus, we have solved the eigenvalue problem :

= c1 1 + c2 2 + c3 3 + …..

ci probability amplitudes

M-scheme calculation

1 = ….. | 0 >a+ a

+ a+

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

Each of i ’s has a good M (=Jz ),because M = m1 + m2 + m3 + .....

Hamiltonian conserves M.

i ’s having the same value of M are mixed.

i ’s having different values of M are not mixed.

But,

H = * * ** * ** * *

* * * ** * * ** * * ** * * *

* * ** * * * * *

. . .

0

0 0

0 0

0

00 0

The Hamiltonian matrix is decomposed into sub matricesbelonging to each value of M.

0 0 0

M=0 M=1 M=-1 M=2

How does J come in ?

two neutrons in f7/2 orbitAn exercise :

M=0 M=2M=1

m1 m2

7/2 -5/25/2 -3/23/2 -1/2J+

m1 m2

7/2 -7/25/2 -5/23/2 -3/21/2 -1/2

m1 m2

7/2 -3/25/2 -1/23/2 1/2J+

J+ : angular momentum raising operator

J+ |j, m > |j, m+1 >

J=0 2-body state is lost J=1 can be elliminated,but is not contained

Dimension

M=0

M=1 3

M=2

M=3

M=4

M=6

M=5

3

2

2

1

1

Components of J values

4

J = 2, 4, 6

J = 2, 4, 6

J = 4, 6

J = 4, 6

J = 6

J = 6

J = 0, 2, 4, 6

By diagonalizing the matrix H, you get wave functionsof good J values by superposing Slater determinants.

H =

M = 0

* * * ** * * ** * * ** * * *

eJ=0 0 0 0

0 eJ=2 0 0

0 0 eJ=4 0

0 0 0 eJ=6

In the case shown in the previous page,

eJ means the eigenvalue with the angular momentum, J.

This property is a general one : valid for cases with more than 2 particles.

H =

M

* * * ** * * ** * * ** * * *

eJ 0 0 0

0 eJ’ 0 0

0 0 eJ’’ 0

0 0 0 eJ’’’

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

Because the interaction V is a scalar with respect to therotation, it cannot change J or M.

A two-body state is rewritten as

| j1, j2, J, M >

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

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

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

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

x m3, m4 ( j3, m3, j4, m4 | J’,

M’ )

Two-body matrix elements

<j1, j2, J, M | V | j3, j4, J’, M’ >

Clebsch-Gordon coef.

Two-body matrix elements

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

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

Two-body matrix elements are assigned by

j1, j2, j3, j4 and J.

Because of complexity of nuclear force, one can notexpress all TBME’s by a few empirical parameters.

Jargon : Two-Body Matrix Element = TBME

X X

Actual potential 　

Depends on quantum numbers of the 2-nucleon system

(Spin S, total angular momentum J,

Isospin T)

Very different fromCoulomb, for instance

1S0 Spin singlet (S=0) 2S+1=1L = 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 >

USDinteraction

1 = d3/2

2= d5/2

3= s1/2

Closed shellExcitations to higher shells areincluded effectively

valence shellPartially occupiedNucleons are moving around

Higher shell Excitations from lower shellsare included effectively by perturbation(-like) methods

～

Effective interaction

Effects of coreand higher shell

Arima and Horie 1954 magnetic moment quadrupole moment

Configuration Mixing Theory

Departure from the independent-particle model

+

closed shell

This is includedby renormalizing theinteraction and effective charges.

Core polarization

配位混合理論

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 correlationswhich 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.

H =

M

* * * ** * * ** * * ** * * *

eJ 0 0 0

0 eJ’ 0 0

0 0 eJ’’ 0

0 0 0 eJ’’’dimensio

n 4 Billions, trillions, …

Dim

en

si

on

Dimension of Hamiltonian matrix(publication years of “pioneer” papers)

Year

Floating point operations per secondBirth of shell model(Mayer and Jensen)

Year

Dimension of shell-model calculations

billion

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

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

These two codes can handle up to 1 billion dimensions.

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 負符号問題 - only ground state, not for excited states in principle.

Quantum Monte Carlo Diagonalization (QMCD) method No sign problem. Symmetries can be restored. Excited states can be obtained. Monte Carlo Shell Model

補助場（量子）モンテカルロ法

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 of Hamiltonian matrix

(about 30 dimension)

cConventional Shell Model calculation All Slater determinants

diagonalization

diagonalization

Quantum Monte Carlo Diagonalization method Important bases are selected

Our parallel computer

More cpu time forheavier or more exotic

nuclei

238U one eigenstate/day

in good accuracyrequires 1PFlops

京速計算機 (Japanese challenge)

Blue Gene

Earth Simulator

Dim

en

sio

n

Birth of shell model(Mayer and Jensen)

Year

Dimension of Hamiltonian matrix(publication years of “pioneer” papers)

Lines : 105 / 30 years

Year

Floating point operations per second

Progress in shell-model calculations and computers

GFlo

ps

Monte CarloConventional

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

Outline

Effetcive interaction in shell model calculations

How can we determine

i : Single Particle Energy

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

: Two-Body Matrix Element

Determination of TBME’s

Early time Experimental levels of 2 valence particles + closed shell

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

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

Experimental energy of state J

Experimental single-particle energy of f7/2

Spin-orbit splitting

Eigenvalues of HO potential

Magic numbersMayer and Jensen (1949)

126

8

20

28

50

82

2

5h

4h

3h

2h

1h

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

The isolation of f7/2 is special. In other cases, several orbits must be taken into account.

In general, 2 fit is made(i) TBME’s are assumed,(ii) energy eigenvalues are calculated,(iii) 2 is calculated between theoretical and experimental energy levels,(iv) TBME’s are modified. Go to (i), and iterate the process until 2 becomes minimum.

At the beginning, it was a perfect 2 fit.

As heavier nuclei are studied, (i) the number of TBME’s increases,(ii) shell model calculations become huge.

Complete fit becomes more difficult and finallyimpossible.

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, andfit them.

Hybrid version - continued

Some linear combinations of TBME’s are sensitiveto available experimental data (ground and low-lying).

The others are insensitive. Those are assumed to begiven by G-matrix-based calculation (i.e. no fit).

The 2 fit method produces, as a result of minimization, a set of linear equations of TBME’s

First done for sd shell: Wildenthal and Brown’s USD 47 linear combinations (1970)

Recent revision of USD : G-matrix-based TBME’s havebeen improved 30 linear combinations fitted

Summary of Day 1

1. Basis of shell model and magic numbers density saturation + short-range interaction + spin-orbit splitting Mayer-Jensen’s magic number

2. How to perform shell model calculations

3. How to obtain effective interactions