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Kilka słów o strukturze jądra atomowego
dla matematyków
Wojciech Satuła
From ab initio toward „rigorous” effective theory for light nuclei
Principles of low-energy nuclear physics  effective theories for
medium-mass and heavy systems
Nuclear DFT  coupling constants & fitting strategies
 single-particle fingerprints of tensor interaction
Extensions of the nuclear DF up to N3LO
Multi-reference DFT  beyond mean-field theories
 isospin (and angular momentum) projection
Summary
ab initio
+ NNN + ....
tens of MeV
Argonne V18 NN potential
short range is phenomenological
long range – pion exchange
Paramters (~40) are fitted to two-nucleon data
Triton binding: th: 7.62MeV exp: 8.48MeV
A=4-10 GFMC calculations using Argonne V18 NN potential
and Illinois-2 NNN interaction
Pion three-body sector of Urbana 3-body potential
plus phenomenological short-range 3-body
From „infinite basis” ab initio towards finite basis
„rigorous” effective theory
Strategy: adopt a Hamiltonian and a basis, compute matrix elements and diagonalize
In ab initio many-body
theory H acts in „infinite”
Hilbert space
finite subspace
(P-space)
Heff
N=5
N=4
N=3
N=2
N=1
N=0
Select HO Slater determinat
basis and retain all A-body
determinants below an oscillator
Ecutoff (Nmax)
„P – space”
Realistic local NN interaction
VNN
Expansion in (Q/L)n
Heff
0
Repulsive core in VNN
cannot be accommodated
in this truncated HO basis
needs regularization
Vlow-k, PT
and further renormalization
to the finite basis
(Lee-Suzuki-Okamoto)
0
QXHX-1Q
H : E1 , E2 , E3 , KEPd , KE
Heff : E1 , E2 , E3 , KEd P
model space
dimension
• Properties of Heff for A-nucleon system
•A-body operator even if H is 2- or 3-body
•For P  1 Heff  H
Effective theories for low-energy nuclear physics in
heavy(ier) nuclei:
Hohenberg-Kohn-Sham
Modern Mean-Field Theory  Energy Density Functional
r,
t,





J,
j,
T,
s,
F,
The nuclear effective theory
is based on a simple and very intuitive assumption that low-energy
nuclear theory is independent on high-energy dynamics
hierarchy of scales:
Long-range part of the NN interaction
(must be treated exactly!!!)
where
local
correcting
potential
2roA1/3
~ 2A1/3
ro
~ 10
denotes an arbitrary Dirac-delta model
przykład
Gogny interaction
Fourier
Coulomb
regularization
ultraviolet
cut-off
There exist an „infinite” number
of equivalent realizations
of effective theories
Skyrme interaction - specific (local) realization of the
lim da
nuclear effective interaction:
a
0
LO
NLO
10(11)
parameters
density dependence
spin-orbit
relative momenta
spin exchange
Spin-force inspired local energy density functional
Y | v(1,2) | Y
Slater determinant
(s.p. HF states are equivalent to the Kohn-Sham states)
local energy density functional
Skyrme-inspired functional
is a second order expansion
in densities and
currents:
density rg
dependent CC
tensor
spin-orbit
20 parameters are fitted to:
Symmetric NM:
- saturation density ( ~0.16fm-3)
- energy per nucleon (-16+ 0.2MeV)
- incompresibility modulus (210 +20MeV)
- isoscalar effective mass (0.8)
Asymmetric NM:
- symmetry energy ( 30+ 2MeV)
- isovector effective mass
(GDR sum-rule enhancement)
- neutron-matter EOS
(Wiringa, Friedmann-Pandharipande)
Finite, double-magic nuclei
[masses, radii, rarely sp levels]:
-surface properties
-ZOO–
How many parameters are really needed?
How many parameters can be constrained by fitting global properties?
linear (re)fit to masses
av
Bertsch, Sabbey,
and Uusnakki
Phys. Rev. C71,
054311 (2005)
Can we learn more from the sp
properties?
8
7
6
5
De(f5/2-f7/2) [MeV]
SLy4
(original)
8
7
6
5
as + asym + s-o
SkO
a)
neutrons
protons
b)
40Ca
48Ca
56Ni
Fitting strategies of the tensorial coupling constants
- the idea 7
1) Fit of the isoscalar SO strength
j>
2) Fit of the isoscalar tensor strength:
j<
56Ni
j>
F
3) Fit of the isovector tensor strength
or, more precisely, C1J/C1 J
48Ca
D
f7/2 f5/2 splittings around
48Ni
or 78Ni are needed
in order to fix SO-tensor sector
Single-particle energies [MeV]
5

j< F
40Ca
f7/2-d3/2
3
1
40Ca
f7/2-f5/2
p3/2-p1/2
0.7
7
f7/2-p3/2
0.8
0.9
1
f7/2-f5/2
5
3
56Ni
p3/2-p1/2
1
-40
-30
-20
-10
8
6
4
0 CJ0
f7/2-f5/2
48Ca
f7/2-d3/2
2
-80
-60
-40
-20
0 CJ1
SkPT T0=-39(*5);T1=-62(*-1.5);SO*0.8
Single-particle fingerprints of tensor interaction
spin-orbit splittings
Spin-orbit splittings [MeV]
SLy4T
SLy4T T0=-45;T1=-60; SO*0.65
n
1h
7
1i
p
7
5
5
3
3
1
g9/2-g7/2
f7/2-f5/2
1
1h
g9/2-g7/2
f7/2-f5/2
M.Zalewski, J.Dobaczewski, WS, T.Werner, PRC77, 024316 (2008)
f5/2
p3/2
f7/2
Nilsson
[321]1/2
[303]7/2
neutrons protons
SkOTX
SkO
SkOT’
-3
-4
8
tensor
spin-orbit
DE [MeV]
10
DEtensor [MeV]
12
Rudolph et al. PRL82, 3763 (1999)
4p-4h
6
4
-5
-6
2
0
0
0.1 0.2 0.3 0.4
0
0.1 0.2 0.3 0.4
deformacja b2
0
0.1 0.2 0.3 0.4
0
0.1 0.2 0.3 0.4 b
2
1.8
Fits of s.p. energies
1.6
1.4
1.2
SLy5
SkP
SkO'
SIII
1.0
Singular value
1
(b)
0.1
0.01
0.001
0
2
4
6
8
10
Number of singular value
12
M. Kortelainen et al., Phys. Rev. C77, 064307 (2008)
RMS deviation of
s.p. energies (MeV)
(a)
EXP: M.N. Schwierz, I. Wiedenhover,
and A. Volya, arXiv:0709.3525
Singular value
decomposition
=
Possible extensions:
explicit reconstruction of the NDF
N2LO, N3LO – higher order derivatives
Mass dependent coupling constants
New higher-order physics-motivated terms: ~t( r)2
D
Beyond mean-field
multi-reference density functional theory
Spontaneous Symmetry Breaking (SSB)
Total energy
(a.u.)
Symmetry-conserving
configuration
Symmetry-breaking
configurations
Elongation (q)
Restoration of broken symmetry
Euler angles
rotated Slater
determinants
are equivalent
solutions
gauge angle
where
Motivation:
Determination of Vud matrix element of the CKM matrix
d5/2
from superallowed beta decay
 test of unitarity  test of three generation quark
Standard Model of electroweak interactions
J=0+,T=1
+
N-Z=-2 T+ J=0 ,T=1
vector (Fermi) cc
Tz=-1
N-Z=0
Tz=0
nucleus-independent
p1/2
p3/2
s1/2
p
8
8
2
2
n
p
n
Isospin symmetry breaking and restoration –
general principles
There are two sources of the isospin symmetry breaking:
Engelbrecht & Lemmer,
- unphysical, related to the HF approximation itself
PRL24, (1970) 607
- physical, caused mostly by Coulomb interaction
(also, but to much lesser extent, by the strong force isospin noninvariance)
Solve SHF (including Coulomb) to get isospin symmetry broken
(deformed) solution |HF>:
See: Caurier, Poves & Zucker,
PL 96B, (1980) 11; 15
Apply the isospin projection operator:
in order to create good-isospin „basis”:
Compute projected (PAV) energy
and Coulomb mixing before
rediagonalization:
BR
BR
aC = 1 - |bT=|Tz||2
Rediagonalize the Hamiltonian in
the good-isospin „basis” |a,T,Tz>
in order to remove spurious
isospin-mixing:
AR
n=1 2
aC = 1 - |aT=T
|
z
Isospin breaking: isoscalar, isovector & isotensor
Isospin invariant
Few numerical results:
(I) Isospin mixing in ground states of e-e nuclei
aC [%]
Here the HF is solved without
Coulomb |HF;eMF=0>.
Here the HF is solved with
Coulomb |HF;eMF=e>.
Ca isotopes:
BR
SLy4
AR
0.4
0.2
eMF = 0
0
1.0
1
0.1
0.8
0.01
0.6
40
0.4
44
48
52 56
60
eMF = e
0.2
In both cases rediagonalization
is performed for the total
Hamiltonian including
Coulomb
0
40
44
56
48
52
Mass number A
60
E-EHF [MeV]
aC [%]
(II) Isospin mixing & energy in the ground states of
e-e N=Z nuclei:
6
5
4
3
2
1
0
1.0
0.8
0.6
0.4
0.2
0
HF tries to reduce the
isospin mixing by:
DaC ~30%
in order to minimize
the total energy
N=Z nuclei
SLy4
BR
AR
Projection increases the
ground state energy
(the Coulomb and the symmetry
energies are repulsive)
Rediagonalization (GCM)
20 28 36 44 52 60 68 76 84 92 100
A
lowers the ground state
energy but only slightly
below the HF
This is not a single Slater determinat
There are no constraints on mixing coefficients
Bohr, Damgard & Mottelson
hydrodynamical estimate
DE ~ 169/A1/3 MeV
35
mean
values
30
Sliv & Khartionov PL16 (1965) 176
Dl=0, Dnr=1  DN=2
DE ~ 2hw ~ 82/A1/3 MeV
25
20
SIII
SLy4
SkP
20
based on perturbation theory
40
60
A
80
100
aC [%]
E(T=1)-EHF [MeV]
Position of the T=1 doorway state in N=Z nuclei
7
6
5
4
SkP SLy5
MSk1
100Sn
SLy SkP
SkM*
SLy4
SkO’ SkXc
SIII
SkO
y = 24.193 – 0.54926x R= 0.91273
31.5 32.0 32.5 33.0 33.5 34.0 34.5
doorway state energy [MeV]
Isobaric symmetry breaking in odd-odd N=Z nuclei
Let’s consider N=Z o-o nucleus disregarding, for a sake of simplicity,
time-odd polarization and Coulomb (isospin breaking) effects
4-fold
degeneracy
n
n
p
p
n
n
CORE
CORE
aligned configurations
n p or n p
p
p
anti-aligned configurations
n p or n p
After applying „naive” isospin projection we get:
T=1
n p
T=0
Mean-field can differentiate between
n p and n p
only through time-odd polarizations!
n p
T=0
ground state
is beyond mean-field!
no time-odd polarizations included
10C
10C
-60
42Sc
42Sc
E [MeV]
-61
T=0
-354
-62
10B
T=1
E [MeV]
-63
-356
10B
4275
-64
907
T=1
-65
-358
-66
T=0
10B
Isospin projection &
Hartree-FockCoulomb rediagonalization
-360
42Ca 42
Ca
Hartree-Fock
T=1
Isospin projection &
Coulomb rediagonalization
Qb values in super-allowed transitions
time-odd
time-even
th
exp
Qb – Qb [MeV]
4
3
0,2%
0,9%
1
2,5%
1,5%
4,1%
10,1%
-1
20 30 40 50
Atomic number
29,9%
26,3%
21,7%
7,9%
Hartree-Fock
Hartree-Fock
10
15,1%
0,9%
3,7%
0,8%
0
0
isospin projected
isospin projected
2
60 0
10
20 30 40 50
Atomic number
60
Isospin symmetry violation due to time-odd fields in the intrinsic system
time-odd
Isobaric analogue
states:
T=1,Tz=1
T=1,Tz=-1
T=1,Tz=0
e-e
o-o
e-e
AMP+IP projection from the „anti-aligned” Slater determinant
(very preliminary tests – no Coulomb rediagonalization)
10C
Energy [MeV]
-59
J=0+
J=0+,T=1
-60
J=0+
-61
J=0+,T=1
-62
-63
-64
-65
10B
J=1+,T=0
J=1+
J=3+,T=0
Hartree
Fock
AMP
AMP
+IP
|OVERLAP|
1
0.1
only AMP
0.01
0.001
IP+AMP
0.0001
0.0 0.5
1.0
r=S
ij
original sp state
1.5
yi*
2.0
Oij jj
2.5
3.0
bT [rad]
p
-1
inverse of the
overlap matrix
space-isospin rotated
sp state
1
Isospin symmetry violation in
superdeformed bands in 56Ni
f5/2
p3/2
f7/2
4p-4h
Nilsson
[321]1/2
neutrons protons
[303]7/2
space-spin symmetric
2
g9/2
f5/2
p3/2
f7/2
D. Rudolph et al. PRL82, 3763 (1999)
pp-h
neutrons protons
two isospin asymmetric
degenerate solutions
Mean-field versus isospin-projected mean-field
T=1
interpretation
dET
Excitation energy [MeV]
pph
20
centroid
aC [%]
nph
dET
T=0
8
6
4
2
band 2
band 1
Isospin-projection
Hartree-Fock
16
56Ni
12
Exp. band 1
Exp. band 2
Th. band 1
Th. band 2
8
4
5
10
15
Angular momentum
5
10
15
Angular momentum
SUMMARY
(verbal)
Local Density Functional Theory for Superfluid Fermionic Systems:
The Unitary Gas
Aurel Bulgac, Phys. Rev. A 76, 040502 (2007)
running coupling constant
in order to renormalize....
ultraviolet
divergence in
pairing tensor
ab initio
calculations by:
Chang & Bertsch
Phys. Rev. A76, 021603
von Stecher, Greene & Blume,
E-print:0705.0671v1
From two-body, zero-range tensor interaction
towards the EDF:
mean-field
averaging