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MICROSCOPIC CALCULATIONS OF ISOSPIN IMPURITIES AND ISOSPIN-SYMMETRY-BREAKING CORRECTIONS USING ISOSPIN AND ANGULAR-MOMENTUM PROJECTD DFT
Wojciech Satuła
ab initio
Intro: effective low-energy theory for medium mass and heavy nuclei mean-field (or nuclear DFT) beyond mean-field (projection)
Summary
Symmetry (isospin) violation and restoration: unphysical symmetry violation isospin projection Coulomb rediagonalization (explicit symmetry violation)
in collaboration with J. Dobaczewski, W. Nazarewicz & M. Rafalski
structural effects SD bands in 56Ni superallowed beta decay
isospin impurities in ground-states of e-e nuclei
symmetry energy – new opportunities of study
Effective theories for low-energy (low-resolution) nuclear physics (I):
Low-resolution separation of scales which isa cornerstone of all effective theories
Fou
rier
local
corre
ctin
gp
ote
ntia
l
hierarchy of scales:
2roA1/3
ro~ 2A1/3
is based on a simple and very intuitive assumption that low-energy
nuclear theory is independent on high-energy dynamics
~ 10
The nuclear effective theory
Long-range part of the NN interaction
(must be treated exactly!!!)
where
reg
ula
rizatio
nCoulomb
ultravioletcut-off
denotes an arbitrary Dirac-delta model
Gogny interaction
przykład
There exist an „infinite” number
of equivalent realizationsof effective theories
lim daa 0
Skyrme interaction - specific (local) realization of the nuclear effective interaction:
spin-orbit
density dependence10(11)
parameters
Y | v(1,2) | Y
Slater determinant(s.p. HF states are equivalent to the Kohn-Sham states)
Skyrme-force-inspired local energy density functional
local energy density functional
relative momenta spin exchange
LO
NLO
SV
Elongation (q)
Tota
l en
erg
y
(a.u
.)
Symmetry-conserving
configuration
Symmetry-breaking
configurations
Skyrme (nuclear) interaction conserves such symmetries like: rotational (spherical) symmetry isospin symmetry: Vnn = Vpp = Vnp (in reality
approximate) parity…
LS LS LS
Mean-field solutions (Slater determinants) break (spontaneously) these symmetries
Euler angles gauge angle
Restoration of broken symmetry
rotated Slater determinantsare equivalent
solutions
where
Beyond mean-field multi-reference density functional theory
There are two sources of the isospin symmetry breaking:- unphysical, caused solely by the HF approximation- physical, caused mostly by Coulomb interaction (also, but to much lesser extent, by the strong force isospin non-invariance)
Find self-consistent HF solution (including Coulomb) deformed Slater determinant |HF>:
Calculate the projected energy andthe Coulomb mixingBefore Rediagonalization:
BR
aC = 1 - |bT=|Tz||2
BR
in order to create good isospin„basis”:
Apply the isospin projector:
Isospin symmetry restoration
Engelbrecht & Lemmer, PRL24, (1970) 607
See: Caurier, Poves & Zucker, PL 96B, (1980) 11; 15
Diagonalize total Hamiltonian in„good isospin basis” |a,T,Tz> takes physical isospin mixing
Isospin invariant
Isospin breaking: isoscalar, isovector & isotensor
aC = 1 - |aT=Tz
|2AR n=1
0
0.2
0.4
0.6
0.8
1.0
aC
[%
]
40 44 48 52 56 60Mass number A
0.01
0.1
1
44 48 52 5640 60
0
0.2
0.4 BRARSLy4
Ca isotopes:
eMF = 0
eMF = e
Numerical results:(I) Isospin impurities in ground states of e-e nuclei
Here the HF is solved without Coulomb |HF;eMF=0>.
Here the HF is solved with Coulomb |HF;eMF=e>.
In both cases rediagonalizationis performed for the total Hamiltonian including Coulomb
W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRL103 (2009) 012502
0123456
00.20.40.60.81.0
20 28 36 44 52 60 68 76 84 92A
ARBR
SLy4
aC [
%]
E-E
HF [
MeV
]
N=Z nuclei
100
This is not a single Slater determinatThere are no constraints on mixing coefficients
AR
AR
BR
BR
(II) Isospin mixing & energy in the ground states of e-e N=Z nuclei:
~30%DaC
HF tries to reduce the isospin mixing by:
in order to minimize the total energy
Projection increases the ground state energy(the Coulomb and symmetryenergies are repulsive)
Rediagonalization (GCM)
lowers the ground state energy but only slightlybelow the HF
Position of the T=1 doorway state in N=Z nuclei
20
25
30
35
20 40 60 80 100A
SIII SLy4
SkP
E(T
=1)
-EH
F [
MeV
]
meanvalues
Sliv & Khartionov PL16 (1965) 176
based on perturbation theoryDE ~ 2hw ~ 82/A1/3 MeV
Bohr, Damgard & Mottelsonhydrodynamical estimate
DE ~ 169/A1/3 MeV
31.5 32.0 32.5 33.0 33.5 34.0 34.5
y = 24.193 – 0.54926x R= 0.91273
doorway state energy [MeV]
4
5
6
7a
C [
%]
100Sn
SkO
SIIIMSk1
SkP SLy5
SLy4
SkO’
SLySkPSkM*
SkXc
Dl=0, Dnr=1 DN=2
D. Rudolph et al. PRL82, 3763 (1999)
f7/2
f5/2p3/2
neutrons protons
4p-4h
[303]7/2
[321]1/2
Nilsson
1
space-spin symmetric
2
f7/2
f5/2p3/2
neutrons protons
g9/2 pp-h
two isospin asymmetricdegenerate solutions
Isospin symmetry violation insuperdeformed bands in 56Ni
4
8
12
16
20
5 10 15 5 10 15
Exp. band 1Exp. band 2Th. band 1Th. band 2
Angular momentum Angular momentum
Exc
itat
ion
en
ergy
[M
eV] Hartree-Fock Isospin-projection
aC [
%]
band 12468 band 2
56Ni
Mean-field
pph
nph
T=0
T=1
centroiddET
dET
Isospin projection
W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRC81 (2010) 054310
Primary motivation of the project isospin corrections
for superallowed beta decay
s1/2
p3/2
p1/2
p
2
8
n p
2
8
n
d5/2
14O 14NHartree-Fock
Experiment:Fermi beta decay:
f statistical rate function f (Z,Qb)
t partial half-life f (t1/2,BR)
GV vector (Fermi) coupling constant
<t+/-> Fermi (vector) matrix element
|<t+/->|2=2(1-dC)
Tz=-/+1 J=0+,T=1
J=0+,T=1
t+/-
BR
(N-Z=-/+2)
(N-Z=0)Tz=0
Qb
t1/2
Experiment world data survey’08
10 cases measured with accuracy ft ~0.1%
3 cases measured with accuracy ft ~0.3%
nucleus-independent
~2.4%Marciano & Sirlin, PRL96 032002 (2006)
~1.5% 0.3% - 1.5%
T&H, PRC77, 025501 (2008)
What can we learn out of it?From a single transiton we can determine experimentally:
GV2(1+DR) GV=const.
From many transitions we can:
test of the CVC hypothesis (Conserved Vector Current)
exotic decays Test for presence of a Scalar Currentse
e J.H
ard
y,
EN
AM
’08
pre
sen
tati
on
one can determine
mass eigenstates
CKMCabibbo-Kobayashi-Maskawaweak eigenstates
With the CVC being verified and knowing Gm (muon decay)
test unitarity of the CKM matrix
0.9490(4) 0.0507(4) <0.0001
|Vud|2+|Vus|2+|Vub|2=0.9997(6)
|Vud| = 0.97418 + 0.00026
test of three generation quark Standard Model of electroweak interactions
Tow
ner &
Hard
yPhys. R
ev. C
77, 0
2550
1 (2
008)
Hardy &TownerPhys. Rev. C77, 025501 (2008)
Model dependence
Liang & Giai & MengPhys. Rev. C79, 064316 (2009)
spherical RPACoulomb exchange treated in the
Slater approxiamtion
dC=dC1+dC2
shellmodel
meanfield
Miller & SchwenkPhys. Rev. C78 (2008) 035501;C80 (2009) 064319
radial mismatch of the wave functions
configuration mixing
Isobaric symmetry violation in o-o N=Z nuclei
ground stateis beyond mean-field!
T=0n pT=0
T=1n p
Mean-field can differentiate between
n p and n ponly through time-odd polarizations!
aligned configurationsn p
nn p p
n panti-aligned configurations
or n por n p
nn p pCORE CORE
Tz=-/+1 J=0+,T=1
J=0+,T=1
t+/-
BR
(N-Z=-/+2)
(N-Z=0)Tz=0
Qb
t1/2
ISOSPIN PROJECTION
MEAN FIELD
0
10
20
30
40
1 3 5 7
aC
[%
]
2K
isospin
isospin & angular momentum
0.586(2)%
42Sc – isospin projection from [K,-K] configurations with K=1/2,…,7/2
-7/2
7/2
-5/2-3/2-1/2 1/2 3/2 5/2
f7/2
0.0001
0.001
0.01
0.1
1
0.0 0.5 1.0 1.5 2.0 2.5 3.0
|OV
ER
LA
P|
bT [rad]
only IP
IP+AMP
p r =S yi
* Oij jjij
-1
inverse of theoverlap matrix
space & isospin rotatedsp state
HF sp state
T
Singularities force us to use interaction-driven functional SV
Hartree-Fock
ground statein N-Z=+/-2 (e-e) nucleus
antialigned statein N=Z (o-o) nucleus
Project on good isospin (T=1) and angular momentum (I=0)
(and perform Coulomb rediagonalization)
<T~1,Tz=+/-1,I=0| |I=0,T~1,Tz=0>T+/-
CPU~ few h
~ few years
14O 14NH&T dC=0.330%
L&G&M dC=0.181%
our: dC=0.303% (Skyrme-V; N=12)
~ ~
Project on good isospin (T=1) and angular momentum (I=0)
(and perform Coulomb rediagonalization)
10 14 18 22 30 34 42
dC [
%]
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
A
Tz= 1 Tz=0
26 42 50 66 74 A0
0.5
1.0
1.5
2.0
dC [
%]
Tz=0 Tz=1
26 38
34 58
Vud=0.97418(26)
Vud=0.97444(23)
Ft=3071.4(8)+0.85(85)
Ft=3070.4(9)our (no A=38):
H&T:
|Vud|2+|Vus|2+|Vub|2==1.00031(61)
0.970
0.971
0.972
0.973
0.974
0.975
0.976
|Vu
d|
superallowed b-decay
p+-decay
n-decay
T=1/2 mirrorb-transitions
H&T’08
Liang et al.
ourmodel
0 5 10 15 20 25 30 35 40
dC(SV)
dC(EXP)
dC [%
]
Z of daughter
0
0.5
1.0
1.5
2.0
2.5
Confidence level test based on the CVC hypothesisT&H PRC82, 065501 (2010)
dC = 1+dNS - Ft
ft(1+dR)‚
(EXP)
Minimize RMS deviationbetween the caluclated and experimental dC withrespect to Ft
c2/nd=5.2for Ft = 3070.0s
75% contribution to thec2 comes from A=62
0
2
4
6
10 20 30 40 50
a’ s
ym [
MeV
]
SV
SLy4L
SkML*
SLy4
A (N=Z)
„NEW OPPORTUNITIES” IN STUDIES OF THE SYMMETRY ENERGY:
T=0
T=1n p
E’sym = a’symT(T+1)
12a’sym
asym=32.0MeV
asym=32.8MeV
SLy4:
In infinite nuclear matter we have:
SV:
asym=30.0MeVSkM*:
asym= eF + aintmm*
SLy4: 14.4MeV SV: 1.4MeV SkM*: 14.4MeV
Summary and outlook
[Isospin projection, unlike the angular-momentum and particle-number projections, is practically non-singular !!!]
Elementary excitations in binary systems may differfrom simple particle-hole (quasi-particle) exciatationsespecially when interaction among particles posseses additional symmetry (like the isospin symmetry in nuclei)
Superallowed beta decay: encomapsses extremely rich physics: CVC, Vud, unitarity of the CKM matrix, scalar currents… connecting nuclear and particle physics … there is still something to do in dc business …
Projection techniques seem to be necessary to account for those excitations - how to construct non-singular EDFs?
How to include pairing into the scheme?
-1.5
-1.0
-0.5
0.0
0.5
20 25 30 35 40 45A
Fig43:110427D
Eexp-D
Eth [
MeV
]
Nolen-Schiffer anomaly in mirror symmetric nucleiVpp-Vnn
SLy4 (HF)SV (HF)SV (PROJ)
Mirror-symmetric nuclei (preliminary)