tens of mev + nnn +.... ab initio intro: define fundaments my model is „standing on” sp...
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tens of MeV+ NNN + ....
3D angular momentum and isospin restored calculationswith the Skyrme EDF
Wojciech Satuła
ab initio
Intro: define fundaments my model is „standing on” sp mean-field (or nuclear DFT) beyond mean-field (projection after variation)
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 ISB corrections to superallowed beta decay
isospin impurities in ground-states of e-e nuclei
Results
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Skyrme-force-inspired local energy density functional(without pairing)
Skyrme (nuclear) interaction conserves: rotational (spherical) symmetry isospin symmetry: Vnn = Vpp = Vnp (in reality
approximate)
LS LS LS
Mean-field solutions (Slater determinants) break (spontaneously) these symmetries
Y | v(1,2) | Y
average Skyrme interaction (in fact a functional!) over
the Slater determinant
local energy density functional
Deformation (q)
Tota
l en
erg
y
(a.u
.)
Symmetry-conserving
configurtion
Symmetry-breaking
configurations
SV is the only Skyrme interactionBeiner et al. NPA238, 29 (1975)
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Euler anglesin space or/and isospace
gauge angle
Restoration of broken symmetry
rotated Slater determinantsare equivalent
solutions
where
Beyond mean-field multi-reference density functional theory
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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>:
in order to create good isospin„basis”:
Apply the isospin projector:
Isospin symmetry restoration
Engelbrecht & Lemmer, PRL24, (1970) 607
Diagonalize total Hamiltonian in„good isospin basis” |a,T,Tz> takes physical isospin mixing
aC = 1 - |aT=Tz
|2AR n=1
aC = 1 - |bT=|Tz||2
BR
See: Caurier, Poves & Zucker, PL 96B, (1980) 11; 15
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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
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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
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Excitation energy 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
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aligned configuration
p pnn
anti-aligned configurationn p or n porn p n p
T=0
Isospin projection
T=1
T=0n p n p
Mean-field
nn p p
four-fold degeneracy of the sp levels
Spontaneous isospin mixing in N=Z nuclei in other but isoscalar configs
yet another strong motivation for isospin projection
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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
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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
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SVD
SVD eigenvalues(diagonal matrix)
Isospin-projection is non-singular:
singularity (if any) at =b p
is inherited by r
1 + |N-Z| - h + |N-Z|+2kk is a multiplicity of zero singular values
h > 3in the worst case
W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRC81 (2010) 054310
ij
-1 r =S yi* Oij jj
~
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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
Isospin and angular-momentum projected DFT is ill-defined except for the hamiltonian-driven functionals
there is no alternative but Skyrme SV
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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
inverse of theoverlap matrix
space & isospin rotatedsp state
HF sp state
Why we have to to use Skyrme-V?
T
ij
-1 r =S yi* Oij jj
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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
10 cases measured with accuracy ft ~0.1% 3 cases measured with accuracy ft ~0.3%
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~2.4%Marciano & Sirlin,
PRL96, 032002, (2006)
nucleus-independent
~1.5% 0.3% - 2.0%
eg
n
NS-independent
The 13 precisely known transitions, after including theoretical corrections, are used to
NS-dependent
test the CVC hypothesis
Towner & HardyPhys. Rev. C77, 025501 (2008)
Towner, NPA540, 478 (1992)PLB333, 13 (1994)
eg
n
courtesy of J.Hardy
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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
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Towner & HardyPhys. Rev. C77, 025501 (2008)
„Hidden” model dependence
Liang & Giai & MengPhys. Rev. C79, 064316 (2009)
spherical RPACoulomb exchange treated in the
Slater approxiamtion
dC=dC2+dC1
shellmodel
meanfield
Miller & SchwenkPhys. Rev. C78 (2008) 035501;C80 (2009) 064319
radial mismatch of the wave functions
configuration mixing
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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
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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)
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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=0.97447(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)
W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski,
PRL106 (2011) 132502
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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
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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 hypothesisTowner & Hardy, 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
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0
0.5
1.0
1.5
2.0
0 10 20 30 40 50 60 70 80
2.5Tz Tz+1
Tz = -1Tz = 0
dC [
%]
A
Ncutoff=10 Ncutoff=12
SUMMARY OF THE CALCULATIONS
A=18
A=38
A=58
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Summary
[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 0+0+beta decay: encomaps 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?
Pairing & other (shape vibrations) correlations can be„realtively easily” incorporated into the scheme by combining projection(s) with GCM
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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