Shell Model calculations
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Transcript Shell Model calculations
Triaxial shapes of the sd and fp shell nuclei
with realistic shell model Hamiltonians
Zao-Chun Gao (高早春)
China Institute of Atomic Energy
Contents
Introduction.
Variation After 3D Angular Momentum Projection.
Examples of 26Mg and 28Si with the USD
interaction.
Triaxial shapes of the sd and pf shell nuclei with
the USD and GXPF1A interactions.
Summary.
Introduction
Nuclear Deformation
1.
2.
3.
A very important concept in the understand
of the nuclear structure.
Determined by various methods:
Macro-micro PES calculations (LDM+SC).
Mean field. ( RMF, HF, HFB).
Beyond mean field. (Variation After Angular
Momentum Projection, Project Energy
Surfaces ).
Nuclear Hamiltonian and Shell Model
As a many-body quantum system, the structure
of the nucleus is determined by the nuclear
Hamiltonian H. The nuclear deformation also
can be obtained from H.
In the shell model calculations, good shell
model Hamiltonians are crucial in the
successful description of various nuclear
properties.
sd shell: USD, USDA,USDB, etc.
fp shell: FPD6, KB3,GXPF1A, etc.
Shell Model calculations:Excitation energies
Shell Model calculations: Binding energies
Even-Z
Odd-Z
GXPF1 interaction
GXPF1 interaction
（Honma etal PRC69 034335(2004)
Shell Model calculations:
Electro-magnetic moments
GXPF1 interaction
GXPF1 interaction
Honma etal PRC69 034335(2004)
Shell Model calculations:
Spetroscopic Factors
M.B. Tsang et al PRL 102 062501(2009)
USDA and USDB
GXPF1A
Where is the nuclear deformation
in the shell model calculations?
In the Shell Model calculations, the basis is
spherical.
There is NO concept of the nuclear deformation
in the Shell Model. Even in the case of describing
rotational bands.
Example: 48Cr [Caurier etal PRL 75 2466(1995)]
Nuclear deformation is not an observable.
Hartree-Fock calculations with SM
Hamiltonians
USD interaction
GXPF1A interaction
Mean Field
Good:
Bad:
Very clear intrinsic structure.
Applied to the whole nuclear region.
No good angular momentum.
Missing correlations beyond mean
field.
Potential Energy Curves from CHFB
and Projected CHFB
32Mg
Gogny Interaction (D1S)
From:
et al.
Motivation of this work:
From SM H，we should also use the
method beyond mean field to obtain the
nuclear shapes , and compare them with
those shapes obtained from the HF
method.
Beyond mean field :Variation After 3Dimensional Angular Momentum Projection.
3-Dimensional Angular Momentum
Projection (3DAMP)
The key tool to transform the mean-field wave
function from the intrinsic to the laboratory
frame of reference.
A intrinsic state
with triaxial
deformation
2I 1
I
I
ˆ
ˆ
PMK
d
D
MK R
2
8
projected states differed by
K I , I 1,...I
Theories relate to the 3DAMP for
24Mg
Skyrme energy
density functional
RMF
M. Bender etc
Phys. Rev. C78,
024309 (2008)
J. M. Yao, etc
Phys. Rev. C81,
044311 (2010)
Gogny force
T. Rodríguez etc
Phys. Rev. C81,
064323 (2010)
Variation After Projection : VAMPIR
The only standard method where variation after angularmomentum projection is exactly considered (together with
restoration of N,Z,and parity).
Very complicated.
Too much time consuming due to the five-fold integration.
(3 for AMP and 2 for N,Z projection)
No explicit discussions about the intrinsic triaxial shapes.
Present VAP is much simpler !
Using HF type Slater determinant
(HFB type in VAMPIR)
Real HF transformation
(Complex HFB in VAMPIR)
Time reversal symmetry is imposed.
Gamma degree of freedom is allowed.
The adopted SM interactions: USD and GXPF1A
Basic idea of the present VAP
HF type Slater determinant
Wik are real here, determined by minimizing EPJ(I)
Where fK satisfy
Algorithm of the present VAP
(L-BFGS quasi-Newton method)
Thouless Theorem:
Problem:The VAP SD may have an
arbitrary orientation in the space.
Treatment: The 3 principle axes of the triaxial shape
have to be in accord with the laboratory axes.
Time reversal symmetry and Q21=0
Quantities of Q and g for the
deformed VAP Slater determinant
There may have several possible solutions in the
VAP calculations.
Needs to try many times to find out all minima.
Ground state
(I=0) of 26Mg
with USD interaction
No EPJ(I=0)
(MeV)
1 -103.954
2 -103.954
3 -103.954
4 -103.954
5 -103.954
6 -103.954
7 -103.287
8 -103.954
9 -103.954
10 -103.287
Q
16.372
16.372
16.372
16.372
16.372
16.372
15.424
16.372
16.372
15.424
g
32.017
-87.982
87.983
-152.018
-32.017
32.017
-52.229
-32.018
32.017
-52.229
All possible VAP (I=0) and HF
solutions for 26Mg
26
90
120
g (deg)
-96
60
20
30
150
Q
10
0 180
0
10
330
210
Energy (MeV)
Mg
-98
300
270
VAP
-97.694
-98.396
-100
-102
-103.287
-104 -103.954
20
240
HF
-105.536
-106
SM
Another example of
28
Si
90
120
g (deg)
-122
60
-124
30
Q
10
0 180
0
10
330
210
20
240
300
270
Energy (MeV)
20
150
28Si
-126
VAP
HF
-126.031
-126.300
-128
-129.612
-130
-132
-132.228
-134.089
-134 -134.133
-135.938
-136
SM
In HF: Most nuclei are axial.
In VAP:No nulceus is axial !
Hartree-Fock
With USD
interaction
VAP
Q and g values in the sd shell nuclei
with 10 N,Z 18
The same situation for
the fp shell nuclei!
Hartree-Fock
With GXPF1A
interaction
VAP
Q and g values in the fp shell nuclei
with 22Z 32, 22 N 38
Summary
Variation After Projection (VAP)
calculations have been performed using
Hartree-Fock type Slater determinant in
the shell model space.
Using USD and GXPF1A interaction, VAP
calculations show that all the sd and fp
shell nuclei are triaxial, while most nuclei
are exactly axial in the HF calculations.
Thanks for your attention!
Including Particle-hole excitations on top
of the VAP SD:
Projected Configuration Interaction (PCI)
HF
VAP
52
Fe
g (deg)
45
30
15
0
0 5 10 15 20 25 30
Q2
Energy (MeV)
60
GXPF1A interaction
-146
GXPF1A
-148
I=6
-150
I=4
I=2
-152
I=0
HF VAP PCI SM
56Ni
60
HF
VAP
56
Ni
With GXPF1A interaction
g (deg) -196
45
30
15
0
0 5 10 15 20 25 30
Q2
Energy (MeV)
PCI for
-198
-200
I=6
-202
I=4
I=2
-204
-206
I=0
HF VAP PCI SM
Comparison with MCSM and VAMPIR
With FPD6 interaction
60
-202.0
-202.2
Energy (MeV)
45
HF
VAP
VAP
g (deg)
30
-202.4
15
-202.6
-202.8
0
5
0
10 15 20 25 30
Q2
-203.0
1 SD
2 SDs
3 SDs
-203.2
PCI
VAMPIR
(2002)
MCSM
(2010)
SM