tens of mev + nnn +.... ab initio intro: define fundaments my model is „standing on” sp...
TRANSCRIPT
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
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)
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
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
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
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
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
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
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
~
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
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
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%
~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
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
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
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
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=0.97447(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
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 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
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
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
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