Otuka_s_2s

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Transcript Otuka_s_2s 

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
Structure of exotic nuclei
Takaharu Otsuka
University of Tokyo / RIKEN / MSU
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
Proton
Neutron
2-body
interaction
3-body
intearction
Aim:
To construct many-body systems
from basic ingredients such as
nucleons and nuclear forces
(nucleon-nucleon interactions)
Introduction to the shell model
What is the shell model ?
Why can it be useful ?
How can we make it run ?
Potential
hard core
0.5 fm
-100 MeV
Schematic picture of nucleonnucleon (NN) potential
1 fm
distance between
nucleons
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
1S
From a book by R. Tamagaki (in Japanese)
0
Spin singlet (S=0) 2S+1=1
L = 0 (S)
J=0
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
neutron
range of nuclear force
from
Due to constant density, potential
energy felt by
is also constant
Mean potential
(effects from other
nucleons)
r
-50 MeV
Distance from the center
of the nucleus
proton
neutron
range of nuclear force
from
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
Proton 陽子
Neutron 中性子
Mean
potential
Harmonic Oscillator (HO)
potential
HO is simpler,
and can be treated
analytically
Eigenvalues of
HO potential
5hw
4hw
3hw
2hw
1hw
Spin-Orbit splitting by the (L S) potential
An orbit with the
orbital angular
momentum l
j = l - 1/2
j = l + 1/2
Orbitals are grouped into shells
20
magic
number
shell gap
8
2
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 )
Eigenvalues of
HO potential
Magic numbers
Mayer and Jensen (1949)
126
5hw
82
4hw
50
3hw
28
20
2hw
8
1hw
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
v
ij,kl
: two-body interaction matrix element
( i j k l : orbits)
A nucleon does not stay in an orbit for ever.
The interaction between 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 determinants f1, 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.
Step 1:
Calculate matrix elements
< f1 | H | f1 >,
< f1 | H | f2 >,
< f1 | H | f3 >, ....
where f1 , f2 , f3 are Slater determinants
In the second quantization,
f1 = aa+ ab+ ag+ ….. | 0 >
n valence particles
+ a + a +
a
….. | 0 >
f2 = a’
g’
b’
f3 = ….
closed shell
Step 2 : Construct matrix of Hamiltonian,
and diagonalize it
H
=
H,
< f1 |H| f1 > < f1 |H| f2 >
< f1 |H| f3 > ....
< f2 |H| f1 > < f2 |H| f2 >
< f2 |H| f3 > ....
< f3 |H| f1 >
< f3 |H| f2 >
< f3 |H| f3 > ....
< f4 |H| f1 >
.
.
.
.
.
.
.
Diagonalization of Hamiltonian matrix
diagonalization
Conventional Shell Model calculation
c Slater determinants
All
diagonalization
Quantum Monte Carlo Diagonalization method
Important bases are selected
(about 30 dimension)
Thus, we have solved the eigenvalue problem :
HY=EY
With Slater determinants f1, f2, f3 ,…,
the eigenfunction is expanded as
Y = c1 f1 + c2 f2 + c3 f3 + …..
ci probability amplitudes
M-scheme calculation
f1 = aa+ ab+ ag+ ….. | 0 >
Usually single-particle state with good j, m (=jz )
Each of fi ’s has a good M (=Jz ),
because M = m1 + m2 + m3 + .....
Hamiltonian conserves M.
fi ’s having the same value of M are mixed.
But, fi ’s having different values of M are not mixed.
The Hamiltonian matrix is decomposed into sub matrices
belonging to each value of M.
M=0
H
=
*
*
*
*
*
*
*
*
0
0
0
*
*
*
*
*
*
*
*
M=1
M=-1
M=2
0
0
0
* * *
* * *
* * *
0
0
* * *
* * *
* * *
0
0
0
0
.
.
.
How does J come in ?
An exercise : two neutrons in f7/2 orbit
J+ : angular momentum raising operator
J+ |j, m >
m1
7/2
5/2
3/2
1/2
m2
-7/2
-5/2
-3/2
-1/2
M=0
m1
J+
|j, m+1 >
m2
7/2 -5/2
5/2 -3/2
3/2 -1/2
M=1
J=0 2-body state is lost
m1
J+
m2
7/2 -3/2
5/2 -1/2
3/2 1/2
M=2
J=1 can be elliminated,
but is not contained
Dimension
Components of J values
M=0
4
J = 0, 2, 4, 6
M=1
3
J = 2, 4, 6
M=2
3
J = 2, 4, 6
M=3
2
J = 4, 6
M=4
2
J = 4, 6
M=5
1
J=6
M=6
1
J=6
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,
M = 0
H
=
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
eJ=0 0
0
0
0
eJ=2 0
0
0
0
eJ=4 0
0
0
0
eJ=6
eJ means the eigenvalue with the angular momentum, J.
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
H
=
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
eJ
0
0
0
0
eJ’
0
0
0
0
eJ’’
0
0
0
0
eJ’’’
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>
Two-body matrix elements
Clebsch-Gordon coef.
<j1, j2, J, M | V | j3, j4, J’, M’ >
= Sm1, m2 ( j1, m1, j2, m2 | J, M )
x Sm3, 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
X | V | j3, j4, J, M
X>
<j1, j2, 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.
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
1S
From a book by R. Tamagaki (in Japanese)
0
Spin singlet (S=0) 2S+1=1
L = 0 (S)
J=0
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
Effects of core
and higher shell
Higher shell
Excitations from lower shells
are included effectively by
perturbation(-like) methods
Effective
interaction
～
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
closed shell
This is included
by renormalizing the
interaction and
effective charges.
+
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,
pf-shell
billions,
trillions,
....
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
H
=
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
dimension
4
eJ
0
0
0
0
eJ’
0
0
0
0
eJ’’
0
0
0
0
eJ’’’
Billions, trillions, …
Dimension of shell-model calculations
Dimension
Dimension of Hamiltonian matrix
(publication years of “pioneer”
papers)
billion
Floating
operations per second
Birth of
shellpoint
model
(Mayer and Jensen)
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. Symmetries can be restored.
Excited states 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 of Hamiltonian matrix
diagonalization
Conventional Shell Model calculation
c Slater determinants
All
diagonalization
Quantum Monte Carlo Diagonalization method
Important bases are selected
(about 30 dimension)
Dimension
Progress in shell-model calculations and computers
Lines : 105 / 30 years
Dimension of Hamiltonian matrix
(publication years of “pioneer”
papers)
More cpu time for
heavier or more exotic nuclei
Conventional
Monte Carlo
Birth of shell model
(Mayer and Jensen)
Year
GFlops
Floating point operations per second
238U one eigenstate/day
in good accuracy
requires 1PFlops
京速計算機
(Japanese challenge)
Blue Gene
Earth Simulator
Year
Our parallel computer
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
Effetcive interaction
in shell model calculations
How can we determine
ei : 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
Experimental energy of state J
E(J) = 2 e( f7/2) + vJ
Experimental single-particle energy of f7/2
Eigenvalues of
HO potential
Magic numbers
Mayer and Jensen (1949)
126
5hw
82
4hw
50
3hw
28
20
2hw
8
1hw
2
Spin-orbit splitting
The isolation of f7/2 is special. In other cases,
several orbits must be taken into account.
In general, c 2 fit is made
(i) TBME’s are assumed,
(ii) energy eigenvalues are calculated,
(iii) 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,
(i) the number of TBME’s increases,
(ii) 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
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
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
Structure of exotic nuclei
Day 2
Takaharu Otsuka
University of Tokyo / RIKEN / MSU
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)
Microscopic
Phenomenological
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 to 699 experimental energy data of 87 nuclei
GXPF1 interaction
M. Honma et al., PRC65 (2002) 061301(R)
two-body matrix
element
<ab; JT | V | cd ; JT >
output
G-matrix vs. GXPF1
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.
56Ni
exp.
th.
exp.
57Ni
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
⇒ Measure of breaking of
this conventional idea
doubly magic
Ni
neutron
Ni
proton
48Cr
total
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
States of different nature
can be reproduced within a
single framework
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
In the shell model, single-particle properties are
considered by the following quantities …….
Effective single particle energy
• Monopole part of the NN interaction
VabT
(2 J  1)V

=
 (2 J  1)
JT
abab
J
J
Angular averaged interaction
Isotropic component is extracted
from a general interaction.
• Effective single-particle energy (ESPE)
ESPE is changed by N
vm
Monopole interaction, vm
N
particles
ESPE :
Total effect on singleparticle energies due to
interaction with other
valence nucleons
Effective single-particle energies
Z=20
Z=22
f5/2
n-n
new
magic
numbers ?
p3/2
Z=24
34
p-n
p1/2
32
Lowering of f5/2 from Ca to Cr
- weakening of N=34 48Ca
Rising of f5/2 from
to
- emerging of N=34 -
54Ca
Why ?
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+
55Ti
56Ti
2+
ESPE
(Effectice SingleParticle Energy)
G
f 5/2
of neutrons
in pf shell
f 5/2
GXPF1
Why is neutron f 5/2
lowered by filling
protons into f 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
Studies on exotic nuclei in the 80~90’s
Left-lower part of
the Nuclear Chart
proton halo
Stability line and drip lines
Proton number 
are not so far from each
other
 Physics of loosely bound
neutrons, e.g., halo
while other issues like
32Mg
neutron halo
11Li
リチウム１１
Neutron number 
neutron skin
A nuclei
(mass number)
stable
exotic
-- with halo
Strong tunneling of loosely bound
excess neutrons
Neutron halo
About same
radius
11Li
208Pb
Proton number 
In the 21st century, a wide
frontier emerges between the
stability and drip lines.
Stability line
Drip line
huge area
A nuclei
中性子数
Neutron
number 
（同位元素の種類）
(mass number)
stable
exotic
Riken’s work
Also in the 1980’s,
32Mg
low-lying 2+
Basic picture was
Island of Inversion
energy
deformed
2p2h state
intruder ground state
stable
9 nuclei:
pf shell
Ne, Na, Mg with N=20-22
Phys. Rev. C 41, 1147 (1990),
Warburton, Becker and
Brown
exotic
N=20
sd shell
gap ~
constant
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)
Steep (sharp)
Straight lines
Small gap vs. Normal gap
v ~ < f (Qp Qn) >
dv=large
v=0
For larger gap,
f must be larger
 sharp boundary
normal
Max pn force
For smaller gap,
f is smaller
 diffuse boundary
open-shell
N
intruder
semi-magic
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
― normal dominant : N=16, 17
Q
― strongly mixed : N=18
― intruder dominant : N=19, 20
Onset of intruder dominance
before arriving at N=20
m
Monte Carlo Shell Model calculation
with full configuration mixing :
Config.
Phys. Rev. C 70, 044307 (2004),
Utsuno et al.
Exp.: Keim et al. Euro. Phys. J.
A 8, 31 (2001)
Level scheme of Na isotopes
by SDPF-M interaction compared to experiment
N=16
N=18
N=17
N=19
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.
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
WBB (1990)
Expansion
of the
territory
SDPF-M (1999)
~5MeV
~2MeV
O Ne Mg
Ca
Neyens et al. 2005 Mg
Tripathi et al. 2005 Na
Dombradi et al. 2006 Ne
Terry et al. 2007 Ne
Phys. Rev. Lett. 94, 022501 (2005), G. Neyens, et al.
Tokyo
MCSM
Strasbourg
unmixed
USD (only sd shell)
2.5 MeV
0.5 MeV
31Mg
19
New picture
energy
deformed
2p2h state
intruder ground state
stable
exotic
pf shell
gap ~
constant
sd shell
energy
Conventional picture
spherical
normal state
?
intruder ground state
stable
exotic
pf shell
N=20
deformed
2p2h state
gap
changing
sd shell
N=20
Effective N=20 gap
between sd and pf shells
constant gap
~2MeV
O Ne Mg
Island of Inversion
SDPF-M
(1999)
Expansion
of the
territory
?
~6MeV
Ca
Shallow
(diffuse & extended)
Island of Inversion
is like a paradise
?
Steep (sharp)
Straight lines
Why ?
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 effective central forces in NN interaction
(as represented by s meson, etc.)
 nuclear binding
This talk : First-order tensor-force effect
(at medium and long ranges)
One pion exchange  Tensor force
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 q with respect to the total spin S.
V ~ Y2,0 ~ 1 – 3 cos2q
q
S
q=0
attraction
q=p/2
repulsion
relative
coordinate
Deuteron : ground state J = 1
Total spin S=1
Relative motion : S wave (L=0) + D wave (L=2)
proton
neutron
Tensor force does mix
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
In the shell model, single-particle properties are
considered by the following quantities …….
Effective single particle energy
• Monopole part of the NN interaction
VabT
(2 J  1)V

=
 (2 J  1)
JT
abab
J
J
Angular averaged interaction
Isotropic component is extracted
from a general interaction.
Intuitive picture of monopole effect of tensor force
wave function of relative motion
spin of nucleon
large relative momentum
attractive
small relative momentum
repulsive
j> = l + ½, j< = l – ½
TO et al., Phys. Rev. Lett. 95, 232502 (2005)
T. Otsuka 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>)
(2j> +1) vm,T
( j’ j<)
+ (2j< +1) vm,T
vm,T : monopole strength for isospin T
= 0
Major features
Opposite signs
spin-orbit splitting varied
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>)
(2j> +1) vm,T
( j’ j<)
+ (2j< +1) vm,T
vm,T : monopole strength for isospin T
= 0
j<
Tensor Monopole Interaction
vanished for s orbit
j>
proton
s1/2
For s orbit, j> and j< are the same :
( j’ j>)
(2j> +1) vm,T
( j’ j<)
+ (2j< +1) vm,T
vm,T : monopole strength for isospin T
= 0
neutron
Monopole Interaction
of the tensor force
is considered
to see the connection
between the tensor force
and the shell structure
Tensor potential
tensor
no s-wave to
s-wave
coupling
differences in
short distance :
irrelevant
Proton effective single-particle levels
(relative to d3/2)
Tensor monopole
f7/2
d3/2
d5/2
proton
neutron
p  r meson 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
d3/2-s1/2 gap
Kramer et al. (2001) Nucl PHys A679
d5/2-s1/2 gap
NIKHEF exp.
N=16 gap : Ozawa, et al., PRL 84 (2000) 5493;
Brown, Rev. Mex. Fis. 39 21 (1983)
d3/2
d5/2
Tensor
force
only
exchange
term
Example : Dripline of F isotopes is 6 units away from O isotopes
Sakurai et al., PLB 448 (1999) 180, …
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
WBB (1990)
Expansion
of the
territory
SDPF-M (1999)
~5MeV
~2MeV
O Ne Mg
Ca
Neyens et al. 2005 Mg
Tripathi et al. 2005 Na
Dombradi et al. 2006 Ne
Terry et al. 2007 Ne
51Sb
case
Opposite monopole
effect from
tensor force
with neutrons
in h11/2.
1h11/2 protons
1g7/2 protons
Z=51 isotopes
Tensor by
h11/2
g7/2
No mean field theory,
(Skyrme, Gogny, RMF)
explained this before.
p r meson
exchange
+ common effect
(Woods-Saxon)
1h11/2 neutrons
Exp. data from J.P. Schiffer et al., Phys. Rev. Lett. 92, 162501 (2004)
Weakening of Z=64 submagic structure for N~90
Single-particle levels of 132Sn core
64
Weakening of Z=64 submagic structure for N~90
Proton collectivity
enhanced at Z~64
2d3/2
1h9/2
64
2d5/2
8 protons in 1g7/2
pushes up 1h9/2
by ~1 MeV
8 neutrons in 2f7/2
reduces the Z=64 gap
to the half value
Neutron single-particle energies
f
f
5/2
Mean-field models
(Skyrme or Gogny)
do not reproduce this
reduction.
7/2
Tensor force effect
due to vacancies of
proton d3/2 in 4718Ar29 :
650 (keV) by p+r meson
exchange.
RIKEN RESEARCH, Feb. 2007
Magic numbers do change, vanish and emerge.
Conventional picture (since 1949)
Today’s perspectives
A city works its magic. … N.Y.
Effect of tensor force on (spherical)
superheavy magic numbers
Proton single particle levels
Occupation of
neutron
1k17/2 and
2h11/2
1k17/2
2h11/2
N=184
Neutron
Woods-Saxon
potential
Tensor force
added
Otsuka, Suzuki and Utsuno,
Nucl. Phys. A805, 127c (2008)
Anatomy of shell-model interaction
Shell evolution by realistic effective interaction : pf shell
Microscopic
Phenomenological
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 to 699 experimental energy data of 87 nuclei
GXPF1 interaction
M. Honma et al., PRC65 (2002) 061301(R)
two-body matrix
element
<ab; JT | V | cd ; JT >
output
G-matrix vs. GXPF1
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
T=0 monopole interactions in the pf shell
Tensor force
(p+r exchange)
GXPF1A
G-matrix
(H.-Jensen)
f-f
p-p
f-p
“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 = V0 exp( -(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 ?
T=0 monopole interactions in the pf shell
Tensor force
(p+r exchange)
GXPF1
G-matrix
(H.-Jensen)
Central (Gaussian)
- Reflecting
radial overlap f-f
p-p
f-p
T=1 monopole interaction
T=1 monopole
interactions
in the pf shell
GXPF1A
G-matrix
(H.-Jensen)
Tensor force
(p+r exchange)
Basic scale
~ 1/10 of T=0
j = j’
j = j’
Repulsive
corrections
to G-matrix
T=1 monopole
interactions
in the pf shell
GXPF1A
G-matrix
(H.-Jensen)
Tensor force
(p+r exchange)
Central (Gaussian)
- Reflecting
radial overlap j = j’
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
p3/2-p3/2 attraction
54Ca
:
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=1 monopole
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
S. Weinberg,
PLB 251, 288 (1990)
Short range central forces
have complicated origins and
should be adjusted.
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
To be continued