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
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
配位
mixing
valence
shell
closed shell
(core)
How to get eigenvalues and eigenfunctions ?
Prepare Slater determinants f1, f2, f3 ,…
which correspond to all possible configurations
配位
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 All Slater determinants
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
<j1, j2, J, M
M>
X | V | j3, j4, J, 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
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
配位混合理論
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 of Hamiltonian matrix
(publication years of “pioneer”
papers)
Dimension
billion
operations per second
Birth ofFloating
shellpointmodel
(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
補助場（量子）モンテカルロ法
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
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 All Slater determinants
diagonalization
Quantum Monte Carlo Diagonalization method
Important bases are selected
(about 30 dimension)
Dimension
Progress in shell-model calculations and computers
Dimension of Hamiltonian matrix
(publication years of “pioneer”
papers)
Monte Carlo
Conventional
Lines : 105 / 30 years
More cpu time for
heavier or more exotic nuclei
238U
Birth of shell model
(Mayer and Jensen)
Year
Floating point operations per second
one eigenstate/day
in good accuracy
requires 1PFlops
京速計算機
(Japanese challenge)
GFlops
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

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