ISOSPIN MIXING PHENOMENA IN THE VICINITY OF N=Z LINEWojciech Satuła
+ NNN + .... tens of MeV
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 & M. Borucki
structural effects SD bands in 56Ni superallowed beta decay
isospin impurities in ground-states of e-e nuclei
symmetry energy – new opportunities of studying with the isospin projection
Effective theories for low-energy (low-resolution) nuclear physics (I):
Low-resolution separation of scales which isa cornerstone of all effective theories
Fourierlocal
correctingpotential
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
regularizationCoulomb
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-orbitdensity dependence10(11)
parameters
Y | v(1,2) | YSlater 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 ene
rgy
(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
-EH
F [M
eV]
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
3.0
3.5
4.0
4.5
29.5 30.0 30.5 31.0 31.5
aC
[%]
aCaC
(AR)
(BR)
80Zr
ET=1 [MeV](AR)
MSk
1M
Sk1
SkO
’Sk
O’
SkP
SLy5
SLy4
SLy7
SkM
*Sk
Xc
SkP
SLy5
SLy4 Sk
M*
SLy7 Sk
Xc
SIII
SIII
Isospin mixing in 80Zr from GDR gamma decay studies
communicated byFranco Camera
at the Zakopane’10meeting
doorway state energy
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
itatio
n en
ergy
[MeV
] Hartree-Fock Isospin-projection
aC [%
]
band 12468 band 2
56Ni
Mean-field
pph
nph
T=0
T=1centroiddET
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
p2
8
n p2
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=1t+/-
BR
(N-Z=-/+2)
(N-Z=0)Tz=0Qb
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%
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 Current
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
0.9491(4) 0.0504(6) <0.0001
|Vud|2+|Vus|2+|Vub|2=0.9996(7)
|Vud| = 0.97425 + 0.00023
test of three generation quark Standard Model of electroweak interactions
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+dC2shell
modelmeanfield
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 p
only 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=1t+/-
BR
(N-Z=-/+2)
(N-Z=0)Tz=0Qb
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 mixing in nKpK antialigned configurations for
K=1/2,3/2,5/2, and 7/2
(
( (
(
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~5h
~50000h
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)
0
0.5
1.0
1.5
2.0
30 40 50 60 70
d C [
%]
Tz=0 Tz=1
A
00.20.40.60.81.01.2
10 15 20 25 30 35 40
d C [
%]
Tz=-1 Tz=0
A
our
Vud=0,97418(26)
Vud=0,97466
Ft=3071.4(8)+0.85(85)
Ft=3069.2(8)our (no A=38):
H&T:
0
2
4
6
10 20 30 40 50
a’sy
m [
MeV
]
SV
SLy4LSkML*
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 Elementary excitations in binary systems may differ
from 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?
Isopin projection, unlike angular-momentum and particle-number projections, is practically non-singular !!!
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
pr =S yi
* Oij jjij
-1
inverse of theoverlap matrix
space & isospin rotatedsp state
HF sp state
-1
01
23
4
0 10 20 30 40 50 60
Qb –
Qb
[MeV
]th
exp
Atomic number
0,2%
0,8%
0,9%
1,5%
2,5%
3,7%
4,1%
time-even
Hartree-Fock
isospin projected
0 10 20 30 40 50 60Atomic number
0,9%
7,9%
15,1%10,1%
26,3%
29,9%
21,7%
time-odd
Qb values in super-allowed transitions
time-odd
T=1,Tz=-1 T=1,Tz=0 T=1,Tz=1e-e e-eo-o
Isospin symmetry violation due to time-odd fields in the intrinsic system
Isobaric analogue states:
isospin projected
Hartree-Fock
4567
18 19 20 21 22asym(rNM/2) [MeV]
4567
29 30 31 32asym(rNM) [MeV]
0.70.80.91.01.1
aC [
%]
18 19 20 21 22asym(rNM/2) [MeV]
29 30 31 32asym(rNM) [MeV]
0.70.80.91.01.1
aC [
%]
40Ca
40Ca 100Sn
100Sn
SkO
SkO’SkM*
MSk1SkXc
SIIISLy
SkP
SIII
SkP MSk1
SkXcSkM*
SLy5SLy
SkO’SkO
SkO
SkO’SkM*
SIII
SIII SkO
SkP
SLySLySkXc
MSk1
SkPMSk1
SkXcSkM* SLy
SkO’SLy5
0.20
0.25
0.30
0.35
6 8 10 12Number of shells
d C [
%]
Towner & Hardy 2008
Liang et al. (NL3)
aligned configurationsn p
nn p p
n panti-aligned configurations
Isobaric symmetry breaking in odd-odd N=Z nucleiLet’s consider N=Z o-o nucleus disregarding, for a sake of simplicity,
time-odd polarization and Coulomb (isospin breaking) effects
or n por n p
T=0
After applying „naive” isospin projection we get:
T=0T=1
ground stateis beyond mean-field!
n p n p
Mean-field can differentiate between n p and n p
only through time-odd polarizations!
4-fold degeneracy
nn p pCORE CORE
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 [M
eV]
meanvalues
Sliv & Khartionov PL16 (1965) 176
based on perturbation theoryDE ~ 2hw ~ 82/A1/3 MeV
Bohr, Damgard & Mottelsonhydrodynamical estimateDE ~ 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]
4567
aC [%
] 100Sn
SkO
SIIIMSk1
SkP SLy5
SLy4SkO’
SLySkPSkM*
SkXc
Dl=0, Dnr=1 DN=2