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Paramters (~40) are fitted to two-nucleon data 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
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Kilka sw o strukturze jdra atomowego
dla matematykw Wojciech Satua 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 Paramters (~40) are fitted to two-nucleon
data
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 N =0 =1 =2 =4 =3 =5
In ab initio many-body theory H acts in infinite Hilbert space
Select HO Slater determinat basis and retain all A-body
determinants below an oscillator Ecutoff (Nmax) finite subspace
(P-space) Heff P space H : E , K Expansion in (Q/L)n VNN Repulsive
core in VNN
Realistic local NN interaction H : E 1 , 2 3 K d eff P model space
dimension Properties of Hefffor A-nucleon system A-body operator
even if H is 2- or 3-body For P 1 Heff H Expansion in (Q/L)n VNN
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) Modern Mean-Field Theory
Energy Density Functional
Effective theories for low-energy nuclear physics in heavy(ier)
nuclei: Hohenberg-Kohn-Sham Modern Mean-Field Theory Energy Density
Functional j, r,t, 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 ultraviolet
cut-off regularization Fourier Coulomb Long-range part of the NN
interaction (must be treated exactly!!!) hierarchy of scales:
2roA1/3 ~ 2A1/3 ro correcting potential local ~ 10 There exist an
infinite number of equivalent realizations of effective theories
where denotes an arbitrary Dirac-delta model przykad Gogny
interaction Spin-force inspired local energy density
functional
Skyrme interaction - specific (local) realization of the nuclear
effective interaction: lim da a 0 LO NLO 10(11) density dependence
parameters 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 ZOO 20 parameters are fitted to:
Skyrme-inspired functional
is a second order expansion in densities and currents: tensor
spin-orbit density rg dependent CC 20 parameters are fitted to:
Symmetric NM: - saturation density( ~0.16fm-3) - energy per nucleon
( MeV) - incompresibility modulus ( MeV) + - isoscalar effective
mass (0.8) Asymmetric NM: isovector effective mass (GDR sum-rule
enhancement) - symmetry energy( 302MeV) + 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? SLy4 (original) Can we learn more from the sp
properties? linear (re)fit to masses Bertsch, Sabbey, and Uusnakki
Phys. Rev. C71, (2005) De(f5/2-f7/2) [MeV] 5 6 7 8 40Ca 48Ca 56Ni
a) b) neutrons protons SkO av as + asym + s-o Fitting strategies of
the tensorial coupling constants
- the idea - C1 J C0 1 3 5 7 0.7 0.8 0.9 40Ca -40 -30 -20 -10 56Ni
f7/2-f5/2 p3/2-p1/2 f7/2-d3/2 2 4 6 8 -80 -60 from binding energies
48Ca f7/2-p3/2 Single-particle energies [MeV] 1) Fit of the
isoscalar SO strength j< j> F 40Ca 2) Fit of the isoscalar
tensor strength: j> F j< 56Ni 3) Fit of the isovector tensor
strength or, more precisely, C1J/C1 D J 48Ca 48Ni or 78Ni are
needed in order to fix SO-tensor sector f7/2 f5/2 splittings around
SkPTT0=-39(*5);T1=-62(*-1.5);SO*0.8 SLy4T Single-particle
fingerprints of tensor interaction
spin-orbit splittings Spin-orbit splittings [MeV]
SLy4TT0=-45;T1=-60; SO*0.65 n 1h 1i f7/2-f5/2 g9/2-g7/2 1 3 5 7 16O
40Ca 48Ca 56Ni 90Zr 132Sn 208Pb p SLy4T M.Zalewski, J.Dobaczewski,
WS, T.Werner, PRC77, (2008) f7/2 f5/2 p3/2 4p-4h Nilsson DE [MeV]
SkO SkOTX SkOT DEtensor [MeV]
neutrons protons 4p-4h [303]7/2 [321]1/2 Nilsson Rudolph et al.
PRL82, 3763 (1999) 2 4 6 8 10 12 0.1 0.2 0.3 0.4 DE [MeV] tensor
spin-orbit deformacja b2 SkO SkOTX SkOT -6 -5 -4 -3 DEtensor [MeV]
0.1 0.2 0.3 0.4 b2 Singular value decomposition
Fits of s.p. energies EXP: M.N. Schwierz, I. Wiedenhover, and A.
Volya, arXiv: Singular value decomposition M. Kortelainen et al.,
Phys. Rev. C77, (2008) = Possible extensions: N2LO, N3LO higher
order derivatives
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 Total energy
(a.u.) Elongation (q)
multi-reference density functional theory Spontaneous Symmetry
Breaking (SSB) Elongation (q) Total energy (a.u.)
Symmetry-conserving configuration Symmetry-breaking configurations
Restoration of broken symmetry
Euler angles gauge angle rotated Slater determinants are equivalent
solutions where Determination of Vud matrix element of the CKM
matrix
Motivation: Determination of Vud matrix element of the CKM matrix
from superallowed beta decay test of unitarity test of three
generation quark Standard Model of electroweak interactions
J=0+,T=1 N-Z=-2 N-Z=0 T+ Tz=-1 vector (Fermi) cc Tz=0 d5/2
nucleus-independent 8 8 p1/2 p3/2 2 2 s1/2 p n p n Isospin symmetry
breaking and restoration general principles
There are two sources of the isospin symmetry breaking: unphysical,
related to theHF approximation itself physical, caused mostly by
Coulomb interaction (also, but to much lesser extent, by the strong
force isospin non-invariance) Engelbrecht & Lemmer, PRL24,
(1970) 607 Solve SHF (including Coulomb) to get isospin symmetry
broken (deformed) solution |HF>: See: Caurier, Poves &
Zucker, PL 96B, (1980) 11; 15 in order to create good-isospin
basis: Apply the isospin projection operator: BR Compute projected
(PAV) energy and Coulomb mixing before rediagonalization: aC= 1 -
|bT=|Tz||2 BR aC = 1 - |aT=Tz|2 Rediagonalize the Hamiltonian
in
the good-isospin basis |a,T,Tz> in order to remove spurious
isospin-mixing: aC= 1 - |aT=Tz|2 AR n=1 Isospin breaking:isoscalar,
isovector & isotensor Isospin invariant Few numerical
results:
Isospin mixing in ground states of e-e nuclei 0.2 0.4 0.6 0.8 1.0
aC[%] 40 44 48 52 56 60 Mass number A 0.01 0.1 1 BR AR SLy4 Ca
isotopes: eMF = 0 eMF = e Here the HF is solved without Coulomb
|HF;eMF=0>. Here the HF is solved with Coulomb |HF;eMF=e>. In
both cases rediagonalization is performed for the total Hamiltonian
including Coulomb aC [%] (II) Isospin mixing & energy in the
ground states of
e-e N=Z nuclei: HF tries to reduce the isospin mixing by: 1 2 3 4 5
6 0.2 0.4 0.6 0.8 1.0 20 28 36 44 52 60 68 76 84 92 A AR BR SLy4 aC
[%] E-EHF [MeV] N=Z nuclei 100 ~30% DaC in order to minimize the
total energy Projection increases the ground state energy (the
Coulomb and the symmetry energies are repulsive) Rediagonalization
(GCM) 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 Position of the T=1 doorway state in N=Z
nuclei
Bohr, Damgard & Mottelson hydrodynamical estimate DE ~ 169/A1/3
MeV 20 25 30 35 mean values E(T=1)-EHF [MeV] Sliv & Khartionov
PL16 (1965) 176 Dl=0, Dnr=1 DN=2 DE ~ 2hw ~ 82/A1/3 MeV SIII SLy4
SkP based on perturbation theory 31.5 32.0 32.5 33.0 33.5 34.0 34.5
y = x R= doorway state energy [MeV] 4 5 6 7 aC [%] 100Sn SkO SIII
MSk1 SkP SLy5 SLy4 SkO SLy SkM* SkXc 20 40 60 80 100 A Isobaric
symmetry breaking in odd-odd N=Z nuclei
Lets consider N=Z o-o nucleus disregarding, for a sake of
simplicity, time-odd polarization and Coulomb (isospin breaking)
effects 4-fold degeneracy n p n p CORE CORE aligned configurations
anti-aligned configurations n p n p n p or or n p T=0 After
applying naive isospin projection we get: T=1 ground state is
beyond mean-field! n p Mean-field can differentiate between and
only through time-odd polarizations! no time-odd polarizations
included
-66 -65 -64 -63 -62 -61 -60 E [MeV] Hartree-Fock 10C 10B Isospin
projection & Coulomb rediagonalization 4275 2098 T=0 T=1 exp:
1908 Isospin projection & Coulomb rediagonalization T=1 exp:
6424 T=0 42Sc 42Ca 7784 907 Hartree-Fock 42Ca 42Sc -360 -358 -356
-354 E [MeV] Qb values in super-allowed transitions
time-even time-odd 4 Qb Qb[MeV] th exp 3 isospin projected isospin
projected 2 0,2% 4,1% 0,9% 2,5% 29,9% 10,1% 1 15,1% 21,7% 1,5% 3,7%
0,9% 26,3% 0,8% 7,9% Hartree-Fock -1 Hartree-Fock 10 20 30 40 50 60
10 20 30 40 50 60 Atomic number Atomic number time-odd T=1,Tz=-1
T=1,Tz=0 T=1,Tz=1 e-e o-o Isospin symmetry violation due to
time-odd fields in the intrinsic system Isobaric analogue states:
AMP+IP projection from the anti-aligned Slater determinant
(very preliminary tests no Coulomb rediagonalization) 10C very
preliminary (qualitative) -65 -64 -63 -62 -61 -60 -59 J=0+ J=0+,T=1
J=0+,T=1 J=0+ Energy [MeV] very preliminary (qualitative) 10B
J=1+,T=0 J=1+ J=3+,T=0 Hartree Fock AMP +IP AMP |OVERLAP| p r =S
yi* Oij jj only AMP IP+AMP bT [rad] 0.0001 0.001 0.01
0.1 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 |OVERLAP| bT [rad] only AMP
IP+AMP p r =S yi* Oij jj ij -1 original sp state space-isospin
rotated sp state inverse of the overlap matrix Isospin symmetry
violation in superdeformed bands in 56Ni
1 Isospin symmetry violation in superdeformed bands in 56Ni 4p-4h
f7/2 f5/2 p3/2 neutrons protons [303]7/2 [321]1/2 Nilsson
space-spin symmetric 2 f7/2 f5/2 p3/2 neutrons protons g9/2 pp-h
two isospin asymmetric degenerate solutions D. Rudolph et al.
PRL82, 3763 (1999) Mean-field versus isospin-projected
mean-field
interpretation pph nph T=0 T=1 centroid dET 2 4 6 8 band 2 aC [%]
band 1 Hartree-Fock Isospin-projection 4 8 12 16 20 56Ni Excitation
energy [MeV] Exp. band 1 Exp. band 2 Th. band 1 Th. band 2 5 10 15
5 10 15 Angular momentum Angular momentum SUMMARY (verbal) Local
Density Functional Theory for Superfluid Fermionic Systems:
The Unitary Gas Aurel Bulgac, Phys. Rev. A 76, (2007) running
coupling constant in order to renormalize.... ultraviolet
divergence in pairing tensor ab initio calculations by: Chang &
Bertsch Phys. Rev. A76, von Stecher, Greene & Blume, E-print:
v1 From two-body, zero-range tensor interaction towards the
EDF:
mean-field averaging