Transcript Unified description of pf

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
解剖（学）
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
Summary of Day2-1
Single-particle properties (shell structure) are one of
the most dominant elements of nuclear structure.
Example : The deformation (of low-lying states)
is a Jahn-Teller effect to a good extent.
We discussed on how single-particle levels change
(shell evolution) as functions of Z and N in exotic nuclei
due to various components of the nuclear force.
1. Tensor force
2. Central force
Summary of Day2 - 2
Some components of NN interactions
show characteristic patterns of the shell evolution
Tensor force : - variation of spin-orbit splitting
- strong
 unexpected oblate deformation of a doubly-magic 42Si
Central force : - differentiates different radial nodal
structure of single-particle orbits
- stronger (< tensor)
Tensor + Central combined (typically in the ratio 2:1)
 lowering of neutron f5/2 in Ca-Cr-Ni
inversion of proton f5/2 and p3/2 in Cu for N~46
54Ca, 78Ni, 100Sn
Day-One experiments of RIBF