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Jacek Dobaczewski
Density functional theory and energydensity functionals in nuclear physics
Jacek DobaczewskiUniversity of Warsaw & University of Jyväskylä
The 9th CNS-EFES International Summer School18-24 August 2010
Center for Nuclear StudyUniversity of Tokyo in RIKEN
Jacek Dobaczewski
Reading material:
Jacek Dobaczewski:2004 RIA Summer Schoolhttp://www.fuw.edu.pl/~dobaczew/RIA.Summer.Lectures/slajd01.htmlJacek Dobaczewski:2005 Ecole Doctorale de Physique, Strasbourghttp://www.fuw.edu.pl/~dobaczew/Strasbourg/slajd01.htmlWitek Nazarewicz:2007 Lectures at the University of Knoxvillehttp://www.phys.utk.edu/witek/NP622/NuclPhys622.htmlJacek Dobaczewski: 2008 the 18th Jyväskylä Summer Schoolhttp://www.fuw.edu.pl/~dobaczew/JSS18/JSS18.htmlJacek Dobaczewski: 2008 Euroschool on Exotic Beamshttp://www.fuw.edu.pl/~dobaczew/Euroschool/Euroschool.htmlJacek Dobaczewski: 2008 Lectures at University of Jyväskylähttp://www.fuw.edu.pl/~dobaczew/FYSN305/FYSN305.htmlJacek Dobaczewski: 2009 Lectures at University of Stellenboschhttp://www.fuw.edu.pl/~dobaczew/Stellenbosch/dobaczewski_lecture.pdf
Home page: http://www.fuw.edu.pl/~dobaczew/
Jacek Dobaczewski
Nuclear Structure
Energy scales in nuclear physics
G.B
erts
chet
al.,
Sci
DA
CR
evie
w 6
, 42
(200
7)
Jacek Dobaczewski
Jacek Dobaczewski
UN
EDF
Col
labo
ratio
n, h
ttp://
uned
f.org
/
Universal Nuclear Energy Density FunctionalUniversal Nuclear Energy Density Functional
Jacek Dobaczewski
Mean-Field Theory ⇒ Density Functional Theory
• mean-field ⇒ one-body densities
• zero-range ⇒ local densities
• finite-range ⇒ gradient terms
• particle-hole and pairing channels
• Has been extremely successful. A broken-symmetry generalized product state does surprisingly good job for nuclei.
Nuclear DFT
• two fermi liquids• self-bound• superfluid
Jacek Dobaczewski
Price of land in Poland per voivodship
Energy density
functional
245
647 Price voivodshipfunctional
654 763
295 580 446
842
631 356
549 548
490
287
623 362
Jacek Dobaczewski
Price of land in Poland per
district
Energy density
functional
Price district functional
Jacek Dobaczewski
Price of land in Eurpe per
country
Energy density
functional
Price country functional
Jacek Dobaczewski
What is DFT?
Density Functional Theory:
A variational method that uses observables as variational
parameters.
Jacek Dobaczewski
Which DFT?
Jacek Dobaczewski
What is the DFT good for?
1) Exact: Minimization of E(Q) gives the exact E and exact Q
2) Impractical: Derivation of E(Q) requires the full variation (bigger effort than to find the exact ground state)
3) Inspirational: Can we build useful models E’(Q) of the exact E(Q)?
4) Experiment-driven: E’(Q) works better or worse depending on the physical input used to build it.
Energy E is afunction(al) of Q
Jacek Dobaczewski
Nuclear Energy Density
Functionals
Jacek Dobaczewski
How the nuclear EDF is built?Local energy density is afunction of
local densityLDA
Non-local energy density is afunction of
non-local densityGogny, M3Y,…
Jacek Dobaczewski
How the nuclear EDF is built?
Quasi-local energy density is a function of
local densities and gradientsSkyrme, BCP,
point-coupling,…
Non-local energy density is a function of
local densitiesRMF (Hartree)
Jacek Dobaczewski
Neutrons in external Woods-Saxon well
R.B
. Wir
inga
& U
NED
F C
olla
bora
tion
Jacek Dobaczewski
Hohenberg-Kohn theorem
Jacek Dobaczewski
Hohenberg-Kohn theorem (trivial version)
Jacek Dobaczewski
Nuclear Energy Density Functional (physical insight)
Jacek Dobaczewski
Effective Theories
Jacek Dobaczewski
Hydrogen atom perturbed near the center
Relative errors in the S-wave binding energies are plotted versus:(i) the binding energy for the Coulomb theory(ii) the Coulomb theory augmented with a delta function in first-order perturbation theory(iii) the non-perturbativeeffective theory through a2, and(iv) the effective theory through a4.
Jacek Dobaczewski
Dimensional analysis - regularization
Jacek Dobaczewski
Dimensional analysis – the hydrogen-like atom
Jacek Dobaczewski
Emission of long electromagnetic waves
Jacek Dobaczewski
Emission of long electromagnetic waves
Jacek Dobaczewski
Blue-sky problem - Compton scattering
Jacek Dobaczewski
N3LO in the chiral perturbation effective field theory
W.C
. Hax
ton,
Phy
s.R
ev. C
77, 0
3400
5 (2
008)
Jacek Dobaczewski
EFT phase-shift analysis
np phase parameters below 300 MeV lab. energy for partial waves with J=0,1,2. The solid line is the result at N3LO. The dotted and dashed lines are the phase shifts at NLO and NNLO, respectively, as obtained by Epelbaum et al. The solid dots show the Nijmegen multi-energy np phase shift analysis and the open circles are the VPI single-energy np analysis SM99. D
.R. E
ntem
and
R. M
achl
eidt
Phys
.Rev
. C68
(200
3) 0
4100
1
Jacek Dobaczewski
Many-fermion Hilbert space
Jacek Dobaczewski
Indistinguishability principle
Jacek Dobaczewski
Fock space
Jacek Dobaczewski
Creation and annihilation operators
Jacek Dobaczewski
Operators in the Fock space
Jacek Dobaczewski
d d
-10
0
10
20
30
40
50
-100
0
100
200
300
400
500
0 1 2 3
0 0.4 0.8 1.2
n-n distance (fm)
n-n
pote
ntia
l (M
eV)
O -O
pot
entia
l (m
eV)
22
O -O distance (nm)22
O -O system22
n-n system (1S0)
O2 O2
n n
n-n versusO2 -O2 interaction
Jacek Dobaczewski
http
://w
ww
.phy
.anl
.gov
/theo
ry/r
esea
rch/
forc
es.h
tml
Jacek Dobaczewski
Lessons learned
1) Energy density functional exists due to the two-step variational method and gives exact ground state energy and its exact particle density.
2) Whenever the energy scales (or range scales) between the interactions and observations are different, the observations can be described by a series of pseudopotentials with coupling constants adjusted to data (an effective theory).
Jacek Dobaczewski
Density-matrix expansion
Jacek Dobaczewski
Density matrices and Wick theorem
Jacek Dobaczewski
Coulomb force – the direct self-consistent potential
Jacek Dobaczewski
Coulomb force – the exchange self-consistent potential
Jacek Dobaczewski
B. G
ebre
mar
iam
et a
l.,Ph
ys. R
ev. C
82, 0
1430
5 (2
010)
(R,r)
s(R,r)
Particle and spin densities in 208Pb
Jacek Dobaczewski
Density-matrix expansion (Negele-Vautherin)(or do we need the non-local density)
Nonlocal energy density
J. D
., B.
G C
arls
son,
M. K
orte
lain
en, J
. Phy
s. G
37,
075
106
(201
0)
Jacek Dobaczewski
100Sn
0.00
0.05
0.10
0 2 4 6 8
Part
icle
den
sity
(fm
-3)
(p)
(n)100Zn
2 4 6 8 10R (fm)
(p)
(n)
mean field one-body densitieszero range local densities
finite range non-local densities
Hohenberg-KohnKohn-ShamNegele-VautherinLandau-MigdalNilsson-Strutinsky
Modern Mean-Field Theory Energy Density Functionalj,, , J,
T,
s,
F,
Nuclear densities as composite fields
Jacek Dobaczewski
Density-matrix expansion (2)
Localenergy density
Jacek Dobaczewski
Density-matrix expansion (3)
Jacek Dobaczewski
Density-matrix expansion (4)
Localenergy density
Jacek Dobaczewski
Density-matrix expansion (6)
0.0
0.5
1.0
0 5 10 15 20
0
0 O(4)
0 Gauss
2
2 O(2)
2 Gauss
Infin
ite-m
atte
r 0(a
) &
2(a)
a = kFr J.
D.,
B.G
Car
lsso
n, M
. Kor
tela
inen
, J. P
hys.
G 3
7, 0
7510
6 (2
010)
Jacek Dobaczewski
Exchange interaction energy in infinite matter
Jacek Dobaczewski
Lessons learned
1) Energy density functional exists due to the two-step variational method and gives exact ground state energy and its exact particle density.
2) Whenever the energy scales (or range scales) between the interactions and observations are different, the observations can be described by a series of pseudopotentials with coupling constants adjusted to data (an effective theory).
3) In nuclei, the non-local energy density functionals can be replaced by the local ones. This is because the range of the interaction is shorter than the range of variations in the local and non-local density matrix.
Jacek Dobaczewski
Density-matrix expansion (5)
Jacek Dobaczewski
J. D
obac
zew
skie
t al.,
J. P
hys.
G: 3
7, 0
7510
6 (2
010)Negele-Vautherin density-matrix expansion
Jacek Dobaczewski
J. D
obac
zew
skie
t al.,
J. P
hys.
G: 3
7, 0
7510
6 (2
010)Negele-Vautherin density-matrix expansion
Jacek Dobaczewski
J. D
obac
zew
skie
t al.,
J. P
hys.
G: 3
7, 0
7510
6 (2
010)Negele-Vautherin density-matrix expansion
Jacek Dobaczewski
Density-matrix expansion and the Skyrme force
Jacek Dobaczewski
Negele-Vautherin density-matrix expansion
Based on the Negele-Vautherin density-matrix expansion, we have derived the NLO Skyrme-functional parameters corresponding to the finite-range Gogny interaction. The method has been extended to derive the coupling constants of local N3LO functionals
J. D
obac
zew
skie
t al.,
J. P
hys.
G: 3
7, 0
7510
6 (2
010)
Jacek Dobaczewski
Negele-Vautherin density-matrix expansion
J. D
obac
zew
skie
t al.,
J. P
hys.
G: 3
7, 0
7510
6 (2
010)
Jacek Dobaczewski
0
400
800
1200
1600
0 50 100 150 200
Bin
ding
Ene
rgy
(MeV
)
Number of nucleons A
0
1
2
3
4
5
0 50 100 150 200
Prot
on R
MS
radi
us R
p (fm
)
Number of nucleons A
Quasi-local vs. non-local functionals
40Ca48Ca
56Ni78Ni
100Sn132Sn
208Pb
8.0
8.5
0 100 200
B /
A (M
eV)
40Ca
48Ca56Ni
78Ni
100Sn
132Sn
208Pbquasi-local (Gogny+DME NLO)non-local (Gogny)
0.90
0.95
1.00
0 100 200
Rp
/ A1/
3 (fm
)
The Negele-Vautherin density-matrix expansion (DME) up to NLO (2nd order) applied to the Gogny non-local functional gives a
Skyrme-like quasi-local functional. The results for self-consistent observables, obtained for both functionals are very similar.
J. D
obac
zew
skie
t al.,
J. P
hys.
G: 3
7, 0
7510
6 (2
010)
Jacek Dobaczewski
Negele-Vautherin density-matrix expansion
Open symbols show the results obtained directly by using the Negele-Vautherin DME, whereas the full symbols show the results inferred from the time-even sector by using the Gogny-equivalent Skyrme force. Solid and dashed lines (left and right panels) show the values of the isoscalar and isovector coupling constants, respectively.
J. D
obac
zew
skie
t al.,
J. P
hys.
G: 3
7, 0
7510
6 (2
010)
The quasi-local EDF derived from the finite-range force does not correspond to a zero-range force
Jacek Dobaczewski
Nuclear densities as composite fields
Jacek Dobaczewski
Local energydensity:(no isospin,no pairing)
Jacek Dobaczewski
Nuclear Energy Density Functional
Jacek Dobaczewski
Complete local energy density
Mean field Pairing
E. Perlińska, et al.,Phys. Rev. C69 (2004) 014316
Jacek Dobaczewski
Mean-field equations
Jacek Dobaczewski
Lessons learned1) Energy density functional exists due to the two-step
variational method and gives exact ground state energy and its exact particle density.
2) Whenever the energy scales (or range scales) between the interactions and observations are different, the observations can be described by a series of pseudopotentials with coupling constants adjusted to data (an effective theory).
3) In nuclei, the non-local energy density functionals can be replaced by the local ones. This is because the range of the interaction is shorter than the range of variations in the local and non-local density matrix.
4) Systematic energy density functionals with derivative corrections can be constructed and the resulting self-consistent equations solved.
Jacek Dobaczewski
Applications
Jacek Dobaczewski
Phenomenological effective interactions
Jacek Dobaczewski
0
20
40
60
80
100
0 20 40 60 80 100 120 140 160 180Neutron Number
N=Z
N=2Z
deformation Prot
on N
umbe
r
Theory
< -0.2-0.2 ÷ -0.1
-0.1 ÷ +0.1+0.1 ÷ +0.2
+0.2 ÷ +0.3+0.3 ÷ +0.4
> +0.4
SLy4volume pairing
M.V
. Sto
itsov
, et a
l., P
hys.
Rev
. C68
, 054
312
(200
3)
Jacek Dobaczewski
Gogny D1S
J.-P.
Del
aroc
heet
al.,
Phy
s. R
ev. C
81, 0
1430
3 (2
010)
Jacek Dobaczewski
S. G
orie
ly e
t al.,
Phys
. Rev
. Let
t. 10
2, 2
4250
1 (2
009)
S. G
orie
ly e
t al.,
Phys
. Rev
. Let
t. 10
2, 1
5250
3 (2
009)
Nuclear binding energies (masses)
The first Gogny HFB mass model. An explicit and self-consistent account of all the quadrupole correlation energies are included within the 5D collective Hamiltonian approach.
2149 masses
RMS = 798 keV
The new Skyrme HFB nuclear-mass model, in which the contact-pairing force is constructed from microscopic pairing gaps of symmetric nuclear matter and neutron matter.
2149 masses
RMS = 581 keV
Jacek Dobaczewski
B. S
abbe
y et
al.,
Phys
. Rev
. C75
, 044
305
(200
7)
J.-P.
Del
aroc
he e
t al.,
Phys
. Rev
. C81
, 014
303
(201
0)537 2+ energies 359 2+ energies
First 2+ excitations of even-even nuclei
Skyrme HF+BCS calculations plus the particle-number and angular-momentum projection and shape mixing.
Gogny HFB calculations plus the 5D collective Hamiltonian approach.
Jacek Dobaczewski
UNEDF collaboration: SciDAC Review 6, 42 (2007)
Spontaneous fission
A. S
tasz
czak
et a
l.,Ph
ys. R
ev. C
80, 0
1430
9 (2
009)
Symmetry-unrestricted Skyrme HF+BCS calculations.
Jacek Dobaczewski
M. B
ende
ret a
l.,Ph
ys. R
ev. C
78, 0
2430
9 (2
008)
Collective states in even-even nuclei
J.M. Y
ao e
t al.,
Phys
. Rev
. C81
, 044
311
(201
0)
Jacek Dobaczewski
S. P
éru
et a
l.,Ph
ys.R
ev. C
77, 0
4431
3 (2
008)
J. Te
rasa
kiet
al.,
arXi
v:10
06.0
010
Giant resonances in deformed nuclei
K. Y
oshi
da e
t al.,
Phys
.R
ev. C
78, 0
6431
6 (2
008)
24Mg
24Mg
174Yb
Jacek Dobaczewski
Fast RPA and QRPA + Arnoldi method
J. Toivanen et al., Phys. Rev. C 81, 034312 (2010)
Jacek Dobaczewski
Jacek Dobaczewski
Jacek Dobaczewski
Jacek Dobaczewski
Jacek Dobaczewski
0+
2+
132Sn
J. To
ivan
enet
al.,
Phys
.Rev
. C 8
1, 0
3431
2(2
010)
Fast RPA and QRPA + Arnoldi method
Jacek Dobaczewski
132Sn
Isoscalar 1- strength functions
0
500
1000
1500
0 10 20 30 40 50
standard RPAArnoldi (spurious)Arnoldi
E [MeV]
e2 fm
6 / M
eV
Removal of spurious modes
J. To
ivan
enet
al.,
Phys
.Rev
. C 8
1, 0
3431
2(2
010)
Jacek Dobaczewski
QRPA timing
Spherical QRPA+Arnoldiscales linearly with the size of the single-particle space
Deformed QRPA+Arnoldiexpected to scale quadratically, that is, as
Standard QRPA scales quartically, that is, as
Scaling properties
Future plans: Full implementation and testing of the spherical QRPA + Arnoldi methodin the code HOSPHE with new-generation separable pairing interactions. Systematic calculations of multipole giant-resonance modes to be used in the EDF adjustments.Deformed QRPA + Arnoldi method implemented in the code HFODD. Systematic calculations of -decay strengths functions and -delayed neutron emission probabilities to be used in the EDF adjustments.
Jacek Dobaczewski
T. Inakura et al., Phys. Rev. C80, 044301 (2009)
Iterative methods to solve (Q)RPA
Photoabsorption cross sections
Theory
Exp
Finite-amplitude method to solve the QRPA equations.
Jacek Dobaczewski
J. Toivanen et al., Phys. Rev. C81, 034312 (2010)
Iterative methods to solve (Q)RPA
Monopole resonance
Arnoldi method to solve the RPA equations.
132Sn
Jacek Dobaczewski
Dipole oscillationsin 14Be
A.S
. Um
ar e
t al.,
Phys
. Rev
. Let
t. 10
4, 2
1250
3 (2
010)
T. N
akat
suka
sa e
t al.,
Nuc
l. Ph
ys. A
788,
349
(200
7)
UNEDF collaboration: Quantum Dynamics with TDSLDA,
A. Bulgac, et al.,http://www.phys.washington.edu/gr
oups/qmbnt/index.html
Time-dependent solutions
– 8Be collision(tip configuration)
neutrons
protons
Jacek Dobaczewski
Nuclear Energy Density Functional
Jacek Dobaczewski
Fits of s.p. energies
Singular value decomposition
=
EXP: M.N. Schwierz, I. Wiedenhover,and A. Volya, arXiv:0709.3525
(a)
(b)1.0
1.2
1.4
1.6
1.8
RM
S de
viat
ion
ofs.
p. e
nerg
ies
(MeV
)
0.001
0.01
0.1
1
0 2 4 6 8 10 12
SLy5SkPSkO'SIII
Number of singular value
Sing
ular
val
ue
M.K
orte
lain
enet
al.,
Phys
.Rev
. C77
, 064
307
(200
8)
Jacek Dobaczewski
Bef
ore
Afte
r
Fits of single-particle energies
05
101520
-4 -2 0 2 4
SLy5 (fit)
i - iex
p
(MeV)
(e)
-4 -2 0 2 4
SIII(fit)
(MeV)
(h)
-4 -2 0 2 4
SkP(fit)
(MeV)
(f)
-4 -2 0 2 4
SkO'(fit)
(MeV)
(g)05
101520
SLy5
i - iex
p
(a) SkP (b) SkO' (c) SIII (d)
M. Kortelainen et al., Phys. Rev. C77, 064307 (2008)
Jacek Dobaczewski
Fit residuals for centroids of SO partners (SkP)
n n
L L
Before
Before
After
After
M.K
orte
lain
enet
al.,
to b
e pu
blis
hed
Jacek Dobaczewski
Fit residuals for splittings of SO partners (SkP)
n n
L L
Before
Before
After
After
M.K
orte
lain
enet
al.,
to b
e pu
blis
hed
Jacek Dobaczewski
Global (masses)
Bertsch, Sabbey, and Uusnakki
Phys. Rev. C71, 054311 (2005)
How many parameters are really needed?
Jacek Dobaczewski
SciDAC 2 UNEDF Project (USA)
Building a Universal Nuclear Energy Density Functional
• Understand nuclear properties “for element formation, for properties of stars, and for present and future energy and defense applications”
• Scope is all nuclei, with particular interest in reliable calculations of unstable nuclei and in reactions
• Order of magnitude improvement over present capabilities Precision calculations
• Connected to the best microscopic physics• Maximum predictive power with well-quantified uncertainties
FIDIPRO Project (Finland)
http://www.unedf.org/
http://www.jyu.fi/accelerator/fidipro/
Jacek Dobaczewski
UNEDF Skyrme Functionals
M. K
orte
lain
en, e
t al.,
arXi
v:10
05.5
145
Jacek Dobaczewski
UNEDF Skyrme Functionals
M. K
orte
lain
en, e
t al.,
arXi
v:10
05.5
145
Jacek Dobaczewski
New functionals
Jacek Dobaczewski
Nuclear Energy Density Functional
Jacek Dobaczewski
Derivatives of higher order: Negele & Vautherindensity matrix expansion
B.G
.Car
lsso
net
al.,
Phys
. Rev
.C 7
8, 0
4432
6 (2
008)
Jacek Dobaczewski
Energy density functional up to N3LO
B.G
.Car
lsso
net
al.,
Phys
. Rev
.C 7
8, 0
4432
6 (2
008)
Jacek Dobaczewski
Numbers of terms in the density functional up to N3LO
Eq. (30) ≡ density independent CCEq. (28) ≡ density dependent CC
B.G
.Car
lsso
net
al.,
Phys
. Rev
.C 7
8, 0
4432
6 (2
008)
Jacek Dobaczewski
Energy density functional for spherical nuclei (I)
B.G
.Car
lsso
net
al.,
Phys
.Rev
. C 7
8, 0
4432
6 (2
008)
Phys
.Rev
. C 8
1, 0
2990
4(E)
(201
0)
Jacek Dobaczewski
Energy density functional for spherical nuclei (II)
B.G. Carlsson et al., Phys. Rev. C 78, 044326 (2008)Phys. Rev. C 81, 029904(E) (2010)
Jacek Dobaczewski
Convergence of density-matrix expansions for nuclear interactions (the direct term)
B.G
. Cal
sson
, J. D
obac
zew
ski,
arXi
v:10
03.2
543
Jacek Dobaczewski
Convergence of density-matrix expansions for nuclear interactions (the exchange term)
B.G
. Cal
sson
, J. D
obac
zew
ski,
arXi
v:10
03.2
543
Jacek Dobaczewski
0.6
0.8
1.0
1.2
1.4
1.6
1.8 R
MS
devi
atio
n of
s.p.
ene
rgie
s (M
eV)
0.001
0.01
0.1
1
0 2 4 6 8 10 12
EXP-1EXO-2EXP-1 (NM)EXP-2 (NM)
Number of singular value
Sing
ular
val
ue
SLy4
2nd orderEXP-1:M.N. Schwierz,I. Wiedenhover, andA. Volya, arXiv:0709.3525
EXP-2:M.G. Porquet et al.,to be published
NM:Nuclear-matterconstraints on: saturation density energy per particle incompressibility effective mass
B.G
.Car
lsso
net
al.,
to b
e pu
blis
hed
NLO
Fits of s.p. energies – regression analysis
Jacek Dobaczewski
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8 R
MS
devi
atio
n of
s.p.
ene
rgie
s (M
eV)
0.0001
0.001
0.01
0.1
1
0 4 8 12 16 20 24 28
EXP-1EXP-2EXP-1 (NM)EXP-2 (NM)
Number of singular value
Sing
ular
val
ue
SLy4
4th orderEXP-1:M.N. Schwierz,I. Wiedenhover, andA. Volya, arXiv:0709.3525
EXP-2:M.G. Porquet et al.,to be published
NM:Nuclear-matterconstraints on: saturation density energy per particle incompressibility effective mass
B.G
.Car
lsso
net
al.,
to b
e pu
blis
hed
(Galilean invariance)
N2LO
Fits of s.p. energies – regression analysis
Jacek Dobaczewski
EXP-1:M.N. Schwierz,I. Wiedenhover, andA. Volya, arXiv:0709.3525
EXP-2:M.G. Porquet et al.,to be published
NM:Nuclear-matterconstraints on: saturation density energy per particle incompressibility effective mass
B.G
.Car
lsso
net
al.,
to b
e pu
blis
hed
0.0
0.4
0.8
1.2
1.6 R
MS
devi
atio
n of
s.p.
ene
rgie
s (M
eV)
0.0001
0.001
0.01
0.1
1
0 10 20 30 40 50 60
EXP-1EXP-2EXP-1 (NM)EXP-2 (NM)
Number of singular value
Sing
ular
val
ue
SLy4
6th orderN3LO
(Galilean invariance)
Fits of s.p. energies – regression analysis
Problem of insta
bilities u
nsolved yet
M. Korte
lainen and T. Lesin
ski
J. Phys. G
: Nucl.
Part. Phys. 3
7 064039
Jacek Dobaczewski
Program HOSPHE
Solution of self-consistent
equations for the N3LO nuclear
energy density functional in
spherical symmetry
B.G
.Car
lsso
net
al.,
arX
iv:0
912.
3230
Jacek Dobaczewski
Program HOSPHE
Solution of self-consistent
equations for the N3LO nuclear
energy density functional in
spherical symmetry
B.G
.Car
lsso
net
al.,
arX
iv:0
912.
3230
HFODD
HOSPHE
Jacek Dobaczewski
Spontaneous symmetry breaking
Jacek Dobaczewski
Nitrogen atom
Hydrogen atom
Ammonia molecule NH3
left state right state
Jacek Dobaczewski
Ammonia molecule NH3
Distance of N from the H3 plane (a.u.)
Tota
l ene
rgy
(a.u
.)Symmetry-conserving
configuration
Symmetry-breakingconfigurations
Jacek Dobaczewski
Jacek Dobaczewski
225Ra
ExperimentR.G. Helmer et al., Nucl. Phys. A474 (1987) 77
Skyrme-Hartree-FockJ. Dobaczewski, J. Engel,
Phys. Rev. Lett. 94, 232502 (2005)
1/2+
1/2- 55
0
Jz=1/2
Jacek Dobaczewski
7.6×10-63.2 ×10125.5 ×1081.8 ×10-9
ratio
~0.1 e× fm0.76 e×nmD~5 ns16 ksT1/2 (E.M.)
0.012 as6.6 psT1/2 (Q.M.)55 keV0.1 meV
225RaNH3
Jacek Dobaczewski
Lessons learned1) Energy density functional exists due to the two-step variational
method and gives exact ground state energy and its exactparticle density.
2) Whenever the energy scales (or range scales) between the interactions and observations are different, the observations can be described by a series of pseudopotentials with coupling constants adjusted to data (an effective theory).
3) In nuclei, the non-local energy density functionals can be replaced by the local ones. This is because the range of the interaction is shorter than the range of variations in the localand non-local density matrix.
4) Systematic energy density functionals with derivative corrections can be constructed and the resulting self-consistent equations solved.
5) In finite systems, the phenomenon of spontaneous symmetry breaking is best captured by the mean-field or energy-density-functional methods.
Jacek Dobaczewski
Nuclear deformation
Elongation (a.u.)
Tota
l ene
rgy
(a.u
.)Symmetry-conserving
configuration
Symmetry-breakingconfigurations
Jacek Dobaczewski
Origins of nuclear deformation
Elongation (a.u.)
Sing
le-p
artic
le e
nerg
y (a
.u.) Open-shell system:
8 particles on 8 doublydegenrate levels
Jacek Dobaczewski
0
20
40
60
80
100
0 20 40 60 80 100 120 140 160 180Neutron Number
N=Z
N=2Z
deformation Prot
on N
umbe
r
Theory
< -0.2-0.2 ÷ -0.1
-0.1 ÷ +0.1+0.1 ÷ +0.2
+0.2 ÷ +0.3+0.3 ÷ +0.4
> +0.4
SLy4volume pairing
M.V
. Sto
itsov
, et a
l., P
hys.
Rev
. C68
, 054
312
(200
3)
Jacek Dobaczewski
Tensor effects
Jacek Dobaczewski
M.Honma et al., Eur. Phys. J. 25,s01 (2005) 499B. Fornal, XXIX Mazurian Lakes Conference on Physics (2005)
B. F
orna
let a
l., P
hys.
Rev
. C 7
0, 0
6430
4 (2
004)
D.-C
. Din
caet
al.,
Phy
s. R
ev. C
71,
041
302
(200
5)
Jacek Dobaczewski
0.00
0.05
0.10
0.15
24 26 28 30 32 34 36
Neutron Number N
BE2
(0+
2+ ) (
e2 b2 )
0.0
0.5
1.0
1.5
26 28 30 32 34
E(2+ ) (
MeV
)
Neutron Number N
22TiA
Jacek Dobaczewski
Evolution of the single particle orbitalswith Z going from 28 to 20
V couples j> and j< orbitals and favors charge exchange processes
f7/2 f5/2T. Otsuka et al. Phys. Rev. Lett 87, 082502 (2001)
B. F
orna
l, XX
IX M
azur
ian
Lake
sC
onfe
renc
eon
Phy
sics
(200
5)
Tensor InteractionVT = (2) Y(2 Z(r)
Jacek Dobaczewski
Tensor-even, tensor-odd, and spin-orbit interactions
where
Jacek Dobaczewski
Tensor energy densities
For conserved spherical and time-reversal symmetries,averaged tensor and SO interactions give the followingenergy densities:
where the particle and SO densities read
Jacek Dobaczewski
Single-particle spin-orbit potentials
Variation of the energy densities with respect tothe single-particle wave functions givesform factors of the single-particle spin-orbit potentials:
Jacek Dobaczewski
J (R
) [fm
-4]
n
-0.005
0.000
0.005
0.010
0.015
0 2 4 6 8R (fm)
N=28
N=38
te=200
Z=28
Neutron S-O Density
J. D
obac
zew
ski,
nucl
-th/0
6040
43
Jacek Dobaczewski
J (R
) [fm
-4]
n
-0.005
0.000
0.005
0.010
0.015
0 2 4 6 8R (fm)
N=40
N=50
te=200
Z=28
Neutron S-O Density
J. D
obac
zew
ski,
nucl
-th/0
6040
43
Jacek Dobaczewski
J (R
) [fm
-4]
n
-0.005
0.000
0.005
0.010
0.015
0 2 4 6 8R (fm)
N=28
N=38 N=40
N=50
te=200
Z=28
Neutron S-O Density
J. D
obac
zew
ski,
nucl
-th/0
6040
43
Jacek Dobaczewski
-30
-20
-10
02s_1/2
1d_3/2
1d_5/2
2p_1/2
2p_3/2
1f_5/2
1f_7/2
102030
N (M
eV)
Proton Number Z
HFB+SLy4
N=28
N=20
te=0N=32
J. D
obac
zew
ski,
nucl
-th/0
6040
43
Jacek Dobaczewski
-30
-20
-10
0 2s
1d
1d
2p
2p
1f
1f
102030
cent
(MeV
)
Proton Number Z
26
10
SO (M
eV)
1d2p1f
HFB+SLy4
te=0 N=32
J. D
obac
zew
ski,
nucl
-th/0
6040
43
Jacek Dobaczewski
26
10
SO (M
eV)
1d2p1f
-30
-20
-10
0 2s
1d
1d
2p
2p
1f
1f
102030
cent
(MeV
)
Proton Number Z
HFB+SLy4
te=-100 N=32
J. D
obac
zew
ski,
nucl
-th/0
6040
43
Jacek Dobaczewski
26
10
SO (M
eV)
1d2p1f
-30
-20
-10
0 2s
1d
1d
2p
2p
1f
1f
102030
cent
(MeV
)
Proton Number Z
HFB+SLy4
te=200 N=32
J. D
obac
zew
ski,
nucl
-th/0
6040
43
Jacek Dobaczewski
Polarizationeffects
for neutronspin-orbitsplitting
M. Zalewski et al.,Phys. Rev. C77,024316 (2008)
exp
Jacek Dobaczewski
Fits of C0J,
C0J , and C1
J
SkPL
M. Zalewski et al.,Phys. Rev. C77,024316 (2008)
Jacek Dobaczewski
Fits of spin-orbit and tensor coupling constants
M. Z
alew
ski e
t al.,
Phy
s. R
ev. C
77, 0
2431
6 (2
008)
Jacek Dobaczewski
Spin
-orb
it sp
littin
gs [M
eV]
1234567
2
4
6
8
16 O 40 Ca48 Ca
56 Ni90 Zr
132 Sn208 Pb
16 O 40 Ca48 Ca
56 Ni90 Zr
132 Sn208 Pb
pp
n n
1g1h
f7/2-f5/2
f7/2-f5/2
1g 1h1i
2g
d5/2-d3/2p3/2-p1/2
d5/2-d3/2p3/2-p1/2
SkPSkP T
SkP originalSkP T
ensor + SO*0.8
T
M. Z
alew
ski e
t al.,
Phy
s. R
ev. C
77, 0
2431
6 (2
008)
Jacek Dobaczewski
Spin
-orb
it sp
littin
gs [M
eV]
1234567
2
4
6
8
pp
n n
f7/2-f5/2
1g1h
1i
2g
SkOSkO T
SkOoriginal
SkO T
ensor + SO*0.81g 1h
d5/2-d3/2p3/2-p1/2
d5/2-d3/2p3/2-p1/2
T
16 O 40 Ca48 Ca
56 Ni90 Zr
132 Sn208 Pb
16 O 40 Ca48 Ca
56 Ni90 Zr
132 Sn208 Pb
f7/2-f5/2
M. Z
alew
ski e
t al.,
Phy
s. R
ev. C
77, 0
2431
6 (2
008)
Jacek Dobaczewski
Challenges
Jacek Dobaczewski
Collectivity
beyond mean field, ground-state correlations, shape coexistence, symmetry restoration, projection on good quantum numbers, configuration interaction, generator coordinate method, multi-reference DFT, etc….
True formean field
Jacek Dobaczewski
I. Range separation and exact long-range effects
II. Derivatives of higher order:
III. Products of more than two densities:
Extensions
Jacek Dobaczewski
Lessons learned1) Energy density functional exists due to the two-step variational
method and gives exact ground state energy and its exact particle density.
2) Whenever the energy scales (or range scales) between the interactions and observations are different, the observations can be described by a series of pseudopotentials with coupling constants adjusted to data (an effective theory).
3) In nuclei, the non-local energy density functionals can be replaced by the local ones. This is because the range of the interaction is shorter than the range of variations in the local and non-local density matrix.
4) Systematic energy density functionals with derivative corrections can be constructed and the resulting self-consistent equations solved.
5) In finite systems, the phenomenon of spontaneous symmetry breaking is best captured by the mean-field or energy-density-functional methods.
6) Energy density functionals up to the second order in derivatives(Skyrme functionals) provide for a fair but not very precise description of global properties of nuclear ground states.
Jacek Dobaczewski
Thank you