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Transcript of Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from...
Summer School IV on Nuclear Collective Dynamics 02.03.2008 1
Modern description of nucleifar from stability
Modern description of nucleifar from stability
Peter Ring
Universidad Autónoma de Madrid
Technische Universität München
Barcelona, Dec. 10, 2007
Covariant density functional theoryof the dynamics
of nuclei far from stability
Peter Ring
Technische Universität München
Istanbul, July 2/3, 2008
Summer School IV on Nuclear Collective Dynamics 02.03.2008 2
The Nuclear Density Functional
Nuclear dynamics and excitations
Content II --------------------
ContentContent
Outlook
Motivation
Density Functional Theory
Ground state properties
Covariant Density Functional
Summer School IV on Nuclear Collective Dynamics 02.03.2008 3
prot
on n
umbe
r Z
prot
on n
umbe
r Z
H
Fe
Au
Pb
U
neutron number N neutron number N neutron number Nneutron number N
Summer School IV on Nuclear Collective Dynamics 02.03.2008 4
magic numbersmagic numbers
2 8 20 28 50 82 126 168 ?
Summer School IV on Nuclear Collective Dynamics 02.03.2008 5
the weak interaction causes β-decay:
the Coulomb force repels the protons
neutrons alone form no bound statesexception: neutronen stars (gravitation!)
the strong interaction ("nuclear force") causes bindingis stronger for pn-systems than nn-systems
p
e
n ν-
Forces acting in the nucleus:Forces acting in the nucleus:
Summer School IV on Nuclear Collective Dynamics 02.03.2008 6
the nucleon-nucleon interaction:the nucleon-nucleon interaction:
π-meson
distance > 1 fm
attractive
distance < 0.5 fm
?
1 fm
three-body forces ?three-body forces ?
repulsive
Summer School IV on Nuclear Collective Dynamics 02.03.2008 7
8
sun energy
He
H
U
Fe
A
fusion
fission
reactor energy
binding energy per particlebinding energy per particle
particle number
(MeV)
B
Summer School IV on Nuclear Collective Dynamics 02.03.2008 8
β+ decay β- decay
β+
β-
N-Z
Summer School IV on Nuclear Collective Dynamics 02.03.2008 9
β+
β-
N-Z
Summer School IV on Nuclear Collective Dynamics 02.03.2008 10
the nuclear density: ρ(r)the nuclear density: ρ(r)
ρ=1.6 nucleons/fm3
simplified representation:
ρ
r
Summer School IV on Nuclear Collective Dynamics 02.03.2008 11
proton and neutron densitiesproton and neutron densities
or?
r r
ρ ρ
p
n
pn
ρ ρ
r r
small neutron excess large neutron excess
Summer School IV on Nuclear Collective Dynamics 02.03.2008 12
- the origin of more than half of the elements with Z>30the origin of more than half of the elements with Z>30
- constraints on effective nuclear interactionsconstraints on effective nuclear interactions
- evolution of shell structureevolution of shell structure
- reduction of the spin-orbit interactionreduction of the spin-orbit interaction
- properties of weakly-bound and open quantum systems properties of weakly-bound and open quantum systems
- exotic modes of collective excitations exotic modes of collective excitations (pygmy, toroidal resonances)(pygmy, toroidal resonances)
- possible possible new forms of nucleinew forms of nuclei ( (molecularmolecular states states, , bubble bubble nucleinuclei, neutron droplets...), neutron droplets...)
- asymmetric nuclear matter equation of state and asymmetric nuclear matter equation of state and the link to neutron starsthe link to neutron stars
- applications in astrophysics- applications in astrophysics
Nuclei far from stability: what can we learn?Nuclei far from stability: what can we learn?
Summer School IV on Nuclear Collective Dynamics 02.03.2008 13
Abundancies of elements in the solar systemAbundancies of elements in the solar system
Fe
Au
Summer School IV on Nuclear Collective Dynamics 02.03.2008 14
neutron capture and successive β-decay:
(N+1,Z) (N,Z+1)
e-
N N+1
Z+1Z
n
(N,Z)
synthesis of heavy elements beyond Fe synthesis of heavy elements beyond Fe
Summer School IV on Nuclear Collective Dynamics 02.03.2008 15
r process
Study of Nucleosynthesis
Summer School IV on Nuclear Collective Dynamics 02.03.2008 16
What do the astrophysicists need ?
• nuclear masses (bindung energies – Q-values)• equation of state (EOS) of nuclear matter: E(ρ)
• isospin dependence E(ρp, ρn)
• nuclear matrix elements (life times of β-decay ..)• cross section for neutron or electron capture ….• fission probabilities• cross sections for neutrino reactions• …..• …..
Summer School IV on Nuclear Collective Dynamics 02.03.2008 17
nuclei and QCD?nuclei and QCD?
Scales: 1 GeV 100 keV
effective forcesin the nucleusQCD NN- forces
in the vacuum
Summer School IV on Nuclear Collective Dynamics 02.03.2008 18
The Nuclear Density Functional
Nuclear dynamics and excitations
Content II --------------------
ContentContent
Outlook
Motivation
Density Functional Theory
Ground state properties
Covariant Density Functional
Summer School IV on Nuclear Collective Dynamics 02.03.2008 19
theorem of Hohenberg und Kohn:
density functional theory:density functional theory:
Hohenberg
Kohn
Summer School IV on Nuclear Collective Dynamics 02.03.2008 20
in the same way we obtain the density:
We consider now a realistic manybody system in an external field U(r) and a two-body interaction V(ri,rk). The total energy Etot of the systemdepends on U(r). It is a functional of U(r):
Many-body system in an external field U(r):
Inverting this relation we can introduce a Legendre transformationreplacing the independent function U(r) by the density ρ(r):
Summer School IV on Nuclear Collective Dynamics 02.03.2008 21
Decomposition of HK-functional
In practical applications the functional EHK[ρ] is decomposedinto three parts:
The Hartree term is simple:
Exc is less important and often approximated,but for modern calculations it plays a essential rule.
The non interacting part:
The exchange-correlation part is the rest:
Summer School IV on Nuclear Collective Dynamics 02.03.2008 22
This is not very good (molecules are never bound) and therefore one added later on gradient terms containing ∇ρ and Δρ. This methodis called Extended Thomas Fermi (ETF) theory. However, these are all asymptotic expansions and one always ends up
with semi-classical approximations. Shell effects are never included.
where γ is the spin/isospin degeneracy. Using this expression at the local density they find:
Thomas FermiThomas and Fermi used the local density approximation (LDA) in order to get an analytical expression for the non-interacting term.They calculated the kinetic energy density of a homogeneous system with constant density ρ
Thomas Fermi approximation:
Summer School IV on Nuclear Collective Dynamics 02.03.2008 23
Example for Thomas-Fermi approximation:
exactThomas-Fermi appr.
Summer School IV on Nuclear Collective Dynamics 02.03.2008 24
Obviously to each density ρ(r) there exist such a potential Veff(r).
The non interacting part of the energy functional is given by:
the density obtained as is the exact density
rrr2m
2
kkkeffV
rr
A
ii
1
2
rdVrdm
rdm
E eff
A
ii
A
iini
3
1
3
1
22
32
22rr rr
Kohn-Sham theoryIn order to reproduce shell structure Kohn and Sham introduced a single particle potential Veff(r), which is defined by the condition, that after the solution of the single particle eigenvalue problem
Kohn-Sham theory:
xcHHKnieff EEEEV
)( r
and obviously we have:
Summer School IV on Nuclear Collective Dynamics 02.03.2008 25
limitations of exact density functionals:limitations of exact density functionals:
local density:
kinetic energy density:
pairing density:
twobody density:
Hohenberg-Kohn:Kohn-Sham: Skyrme:Gogny:
in practiceformally exactno shell effectsno l•s,no pairingno config.mixinggeneralized mean field: no configuration
mixing, no two-body correlations
Summer School IV on Nuclear Collective Dynamics 02.03.2008 26
The Nuclear Density Functional
Nuclear dynamics and excitations
Content II --------------------
ContentContent
Outlook
Motivation
Density Functional Theory
Ground state properties
Covariant Density Functional
Summer School IV on Nuclear Collective Dynamics 02.03.2008 27
Density functional theoryDensity functional theory in nuclei:Density functional theory in nuclei:
1) The interaction is not well known and very strong
2) More degrees of freedom: spin, isospin, relativistic, pairing
3) Nuclei are selfbound systems. The exact density is a constant. ρ(r) = const Hohenberg-Kohn theorem is true, but useless
4) ρ(r) has to be replaced by the intrinsic density:
5) Density functional theory in nuclei is probably not exact, but a very good approximation.
Summer School IV on Nuclear Collective Dynamics 02.03.2008 28
Slater determinant density matrix̂
A
iii
1
)()(),(ˆ r'rr'r ))()(( 11 AA rr A
ˆ
E
h ̂ iiih ˆ
Mean field: Eigenfunctions:
ˆ
2
E
V ̂ ̂
Interaction:
Density functional theory in nucleiDensity functional theory in nucleiD.BrinkD.Vauterin
Skyrme
Extensions: Pairing correlations, Covariance Relativistic Hartree Bogoliubov (RHB)
Summer School IV on Nuclear Collective Dynamics 02.03.2008 29
• the nuclear energy functional is so far phenomenological
and not connected to any NN-interaction.
• it is expressed in terms of powers and gradients of the nuclear ground state density using the principles of symmetry and simplicity
• The remaining parameters are adjusted to characteristic properties of nuclear matter and finite nuclei
General properties of self-consistent mean field theories:
(i) the intuitive interpretation of mean fields results in terms of intrinsic shapes and of shells with single particle states
(ii) the full model space is used: no distinction between core and valence nucleons, no need for effective charges
(iii) the functional is universal: it can be applied to all nuclei throughout the periodic chart, light and heavy, spherical and deformed
Virtues:
Summer School IV on Nuclear Collective Dynamics 02.03.2008 30
The Nuclear Density Functional
Nuclear dynamics and excitations
Content II --------------------
ContentContent
Outlook
Motivation
Density Functional Theory
Ground state properties
Covariant Density Functional
Summer School IV on Nuclear Collective Dynamics 02.03.2008 31
Dirac equationDirac equation in atoms:Dirac equation in atoms:
with magnetic field:
Coulomb potential: (r)
magnetic potential: (r)
Summer School IV on Nuclear Collective Dynamics 02.03.2008 32
scalar potential
vector potential (time-like)
vector potential (space-like)
vector space-like corresponds to magnetic potential (nuclear magnetism)is time-odd and vanishes in the ground state of even-even systems
Dirac equationDirac equation in nuclei:Dirac equation in nuclei:
Summer School IV on Nuclear Collective Dynamics 02.03.2008 33
Fermi sea
Dirac sea
2m* ≈ 1200 MeV
V+S ≈ 700 MeV
V-S ≈ 50 MeV
2m ≈ 1800 MeV
continuum
Relativistic potentialsRelativistic potentials
Summer School IV on Nuclear Collective Dynamics 02.03.2008 34
(ε → m+ε)
rp1
r ii
i gWmε
f
2
rrp
r
1p iii
i
gεgWmε
~2
Wmm2
1r~
Smm r*
rr p1
p iii gεgWslr
W
rmm
1
4
1
2 2~~
SVW
for mi~2
Elimination of small components:Elimination of small components:
Summer School IV on Nuclear Collective Dynamics 02.03.2008 35
1) no relativistic kinematic necessary:
2) non-relativistic DFT works well
3) technical problems: no harmonic oscillator no exact soluble models double dimension huge cancellations V-S no variational method
4) conceptual problems: treatment of Dirac sea no well defined many-body theory
0750122 . NNF mmp
Why covariant ?Why covariant ?
Summer School IV on Nuclear Collective Dynamics 02.03.2008 36
Coester-line
Why covariant1) Large spin-orbit splitting in nuclei2) Large fields V≈350 MeV , S≈-400 MeV3) Success of Relativistic Brueckner4) Success of intermediate energy proton scatt.5) relativistic saturation mechanism6) consistent treatment of time-odd fields7) Pseudo-spin Symmetry8) Connection to underlying theories ? 9) As many symmetries as possible
Why covariant?Why covariant?
Summer School IV on Nuclear Collective Dynamics 02.03.2008 37
Walecka model
Nucleons are coupled by exchange of mesons through an effective Lagrangian (EFT)
(J,T)=(0+,0) (J,T)=(1-,0) (J,T)=(1-,1)
Sigma-meson: attractive scalar field
Omega-meson: short-range repulsive
Rho-meson:isovector field
)()( rr gS )()()()( rrrr eAggV
Walecka modelWalecka model
Summer School IV on Nuclear Collective Dynamics 02.03.2008 38
interaction terms
Parameter:
meson masses: mσ, mω, mρ
meson couplings: gσ, gω, gρ
Lagrangian
free photon fieldfree Dirac particle free meson fields
Lagrangian densityLagrangian density
interaction terms
Summer School IV on Nuclear Collective Dynamics 02.03.2008 39
for the nucleons we find the Dirac equation
.0 iSmVi
for the mesons we find the Klein-Gordon equation
)(em
s
ejA
jgm
jgm
gm
2
2
2
A
iii
em
A
iii
A
iii
A
iiis
xxxj
xxxj
xxxj
xxx
13
1
1
1
12
1
)(
No-sea approxim. !
equations of motion .0
kk q
L
q
L-
Equations of motionEquations of motion
Summer School IV on Nuclear Collective Dynamics 02.03.2008 40
for the nucleons we find the static Dirac equation
.ε p iiiSmV
for the mesons we find the Helmholtz equations
)(em
B
s
eA
gm
gm
gm
0
330
2
02
2
A
iii
em
A
iii
A
iiiB
A
iiis
13
13
3
1
1
12
1
)(
No-sea approxim. !
000 eAggVgS s ,
static limitStatic limit (with time reversal invariance)Static limit (with time reversal invariance)
Summer School IV on Nuclear Collective Dynamics 02.03.2008 41
A
iiiii
A
iii ffggggm
11
2
Relativistic saturation mechanism:Relativistic saturation mechanism:
We consider only the σ-field, the origin of attraction its source is the scalar density
for high densities, when the collapse is close, the Dirac gap ≈2m* decreases, the small components fi of the wave functions increase and reduce the scalar density, i.e. the source of the σ-field, and therefore also scalar attraction. rk
1r i
ii g
mεf
~2
A
iiiB
A
iiiB gg
mgffgm
11
2 12 ~
In the non-relativistic case, Hartree with Yukawa forces would lead to collapse
Summer School IV on Nuclear Collective Dynamics 02.03.2008 42
EOS-Walecka
J.D. Walecka, Ann.Phys. (NY) 83, (1974) 491
σω-model
Equation of state (EOS):Equation of state (EOS):
Summer School IV on Nuclear Collective Dynamics 02.03.2008 43
Density dependence
Effective density dependence:Effective density dependence:
non-linear potential:
density dependent coupling constants:
43
32
2222
4
1
3
1
2
1)(
2
1 ggmUm
)(),(),(,, gggggg
g g(r))
NL1,NL3..
DD-ME1,DD-ME2
Boguta and Bodmer, NPA 431, 3408 (1977)
R.Brockmann and H.Toki, PRL 68, 3408 (1992)S.Typel and H.H.Wolter, NPA 656, 331 (1999)T. Niksic, D. Vretenar, P. Finelli, and P. Ring, PRC 56 (2002) 024306
Summer School IV on Nuclear Collective Dynamics 02.03.2008 44
Point-coupling model
Point-Coupling ModelsPoint-Coupling Models
σ ω δ ρ
J=0, T=0 J=1, T=0 J=0, T=1 J=1, T=1
Manakos and Mannel, Z.Phys. 330, 223 (1988)Bürvenich, Madland, Maruhn, Reinhard, PRC 65, 044308 (2002)
Summer School IV on Nuclear Collective Dynamics 02.03.2008 45
interaction terms
Parameter:
point couplings: Gσ, Gω, Gδ , Gρ,
derivative terms: Dσ
Lagrangian
photon field
free Dirac particle
Lagrangian density for point couplingLagrangian density for point coupling
interaction terms
Summer School IV on Nuclear Collective Dynamics 02.03.2008 46
The Nuclear Density Functional
Nuclear dynamics and excitations
Content II --------------------
ContentContent
Outlook
Motivation
Density Functional Theory
Ground state properties
Covariant Density Functional
Summer School IV on Nuclear Collective Dynamics 02.03.2008 47
EOS for DD-ME2
Neutron Matter
Nuclear matter equation of stateNuclear matter equation of state
Summer School IV on Nuclear Collective Dynamics 02.03.2008 48
Symmetry energy
Symmetry energySymmetry energy
empirical empirical values:values:
30 MeV a4 34 MeV2 MeV/fm3 < p0 < 4 MeV/fm3
-200 MeV < K0 < -50 MeV
saturation density
LombardoLombardo
Summer School IV on Nuclear Collective Dynamics 02.03.2008 49
Summer School IV on Nuclear Collective Dynamics 02.03.2008 50
1) density functional theory is in principle exact2) microscopic derivation of E(ρ) very difficult3) Lorentz symmetry gives essential constraints - large spin orbit splitting - relativistic saturation - unified theory of time-odd fields4) in realistic nuclei one needs a density dependence - non-linear coupling of mesons - density dependent coupling-parameters5) modern parameter sets (7 parameter) provide excellent description of ground state properties - binding energies (1 ‰) - radii (1 %) - deformation parameters6) pairing effects are non-relativisitic
1) density functional theory is in principle exact2) microscopic derivation of E(ρ) very difficult3) Lorentz symmetry gives essential constraints - large spin orbit splitting - relativistic saturation - unified theory of time-odd fields4) in realistic nuclei one needs a density dependence - non-linear coupling of mesons - density dependent coupling-parameters5) modern parameter sets (7 parameter) provide excellent description of ground state properties - binding energies (1 ‰) - radii (1 %) - deformation parameters6) pairing effects are non-relativisitic
Conclusions IConclusions part I:Conclusions part I:
Summer School IV on Nuclear Collective Dynamics 02.03.2008 51
The Nuclear Density Functional
Nuclear dynamics and excitations
Content II --------------------
ContentContent
Outlook
Motivation
Density Functional Theory
Ground state properties
Covariant Density Functional
Summer School IV on Nuclear Collective Dynamics 02.03.2008 52
TDRMF: Eq.
and similar equations for the ρ- and A-field
tSmi
ti iit
VV
1
tgtm
tgtm
tgtm
B
B
s
j
ρ
ρ
2
02
2
A
i iiB
A
i iiB
A
i iis
1
1
1
j
ρ
ρ
Time dependent mean field theory:Time dependent mean field theory:
No-sea approxim. !
Summer School IV on Nuclear Collective Dynamics 02.03.2008 53
K∞=271
K∞=355
Monopole motion
K∞=211
)()( trt 2
Breathing mode: 208PbBreathing mode: 208Pb
E
V2
ˆ̂ ̂
Interaction:
Summer School IV on Nuclear Collective Dynamics 02.03.2008 54
RRPARelativistic RPA for excited states Relativistic RPA for excited states
RRPA matrices:
the same effective interaction determines the Dirac-Hartree single-particle spectrum and the residual interaction
ph, h
hp, h
Small amplitude limit:
ground-state density
E
V2
ˆ̂ ̂
Interaction:
Summer School IV on Nuclear Collective Dynamics 02.03.2008 55
A. Ansari, Phys. Lett. B (2005)
Ansari-Sn2+-excitation in Sn-isotopes: 2+-excitation in Sn-isotopes:
Summer School IV on Nuclear Collective Dynamics 02.03.2008 56
Relativistic (Q)RPA calculations of giant resonances
Isovector dipole response
Sn isotopes: DD-ME2 effectiveinteraction + Gogny pairing
protons neutrons
Isoscalar monopole response
Summer School IV on Nuclear Collective Dynamics 02.03.2008 57
IS-GMRIsoscalar Giant Monopole: IS-GMRIsoscalar Giant Monopole: IS-GMR
The ISGMR represents the essential source of
experimental information on the nuclear
incompressibility
constraining the nuclear matter compressibility
RMF models reproduce the experimental data only if
250 MeV K0 270 MeV
Blaizot-concept:
T. Niksic et al., PRC 66 (2002) 024306
ρ(t) = ρ0 + δρ(t)
Summer School IV on Nuclear Collective Dynamics 02.03.2008 58
IV-GDRIsovector Giant Dipole: IV-GDRIsovector Giant Dipole: IV-GDR
the IV-GDR represents one of the sources of experimental informations on the nuclear matter symmetry energy
constraining the nuclear matter symmetry energy
32 MeV a4 36 MeV
the position of IV-GDR isreproduced if
T. Niksic et al., PRC 66 (2002) 024306
Summer School IV on Nuclear Collective Dynamics 02.03.2008 59
Soft dipole modes and neutron skinSoft dipole modes and neutron skin
Summer School IV on Nuclear Collective Dynamics 02.03.2008 60
Exp: pygmy O
Summer School IV on Nuclear Collective Dynamics 02.03.2008 61
Pygmy: O-chain
Effect of pairing correlations on the dipole strength distribution
What is the structure of low-lying strength below 15 MeV ?
RHB + RQRPA calculations with the NL3 relativistic mean-field plus D1S Gogny pairing interaction.
Transition densities
Evolution of IV dipole strength in Oxygen isotopes
Summer School IV on Nuclear Collective Dynamics 02.03.2008 62
Pygmy: 132-SnMass dependence of GDR and Pygmy dipole states in Sn isotopes. Evolution of the low-lying strength.
Isovector dipole strength in 132Sn.
GDR
Nucl. Phys. A692, 496 (2001)
Distribution of the neutron particle-hole configurations for the peak at7.6 MeV (1.4% of the EWSR)
Pygmy state
exp
Summer School IV on Nuclear Collective Dynamics 02.03.2008 63
Vibrations in deformed nucleiVibrations in deformed nuclei
K
J
Goldstone modesTranslations:
Rotations:
Gauge rotations:
Giant dipole modes:
Scissor modes:
T=0 T=1
K=1-
K=1+
K=0-
K=1+
K=0+
K=0- K=1-
Summer School IV on Nuclear Collective Dynamics 02.03.2008 64
IV-GDR in 100MoIV-GDR in 100Mo
IV-GDR
K=0- K=1-
ρ0 + δρ(t)
isovector-dipole response in 100Moisovector-dipole response in 100Mo
Summer School IV on Nuclear Collective Dynamics 02.03.2008 65
pygmy modes in 100Mopygmy modes in 100Mo
Summer School IV on Nuclear Collective Dynamics 02.03.2008 66
scattering at a single nucleon excitation of the entire nucleuswe need the nuclear spectrum
response of the nucleus to an incoming particleresponse of the nucleus to an incoming particle
Summer School IV on Nuclear Collective Dynamics 02.03.2008 67
neutrino reactions neutrino reactions e-
ν
(N,Z)→(N-1,Z+1)
pe-
n
+
e-
nspin-isospin-wave
p
νν
W-
Summer School IV on Nuclear Collective Dynamics 02.03.2008 68
beta-decay beta-decay e-
(N,Z)→(N-1,Z+1)
pe-
n
+
e-
nspin-isospin-wave
p
W±
ν-
ν-ν-
Summer School IV on Nuclear Collective Dynamics 02.03.2008 69
IAR-GTRSpin-Isospin Resonances: IAR - GTRSpin-Isospin Resonances: IAR - GTR
Z,N Z+1,N-1
isospin flip
Z,N Z,NT IAR
spin flip
Z,NTS GTR -
r-rskin neutrondrdV
slEE pnIARGTR ~ ~ ~ - p n
Summer School IV on Nuclear Collective Dynamics 02.03.2008 70
S=0 T=1 J = 0+
S=1 T=1 J = 1+
ISOBARIC ANALOG AND GAMOW-TELLER RESONANCES (RQRPA)
SPIN-FLIP & ISOSPIN-FLIP EXCITATIONS
SPIN-FLIP & ISOSPIN-FLIP EXCITATIONS
ISOSPIN-FLIP EXCITATIONS
Spin-Isospin Resonances: IAR - GTRSpin-Isospin Resonances: IAR - GTR
PR C69, 054303 (2004)
Summer School IV on Nuclear Collective Dynamics 02.03.2008 71
β-decay: Sn,Te
h9/2->h11/2G. Martinez-Pinedo and K. Langanke,
PRL 83, 4502 (1999)
T. Niksic et al, PRC 71, 014308 (2005)
β-decayβ-decay
Summer School IV on Nuclear Collective Dynamics 02.03.2008 72
important:
1. we learn about the reaction mechanism
2. we calculate the detector response for neutrino reactions
3. neutrinos play also a role in nuclear synthesis
so far there exist ony few data: → deuteron, 12C, 56Fe
ZXN
0+ 1+
1-
2+3-
2-
0+
Z+1XN-1
Eneutrino-nucleus reactionsneutrino-nucleus reactions
Summer School IV on Nuclear Collective Dynamics 02.03.2008 73
Supernova neutrino flux isgiven by Fermi-Dirac spectrum
Cross section averaged over Supernova neutrino flux
Cross section averaged over supernova neutrino fluxCross section averaged over supernova neutrino flux
4
3
2
1
0
5
2 4 6 8 10 T [MeV]
Summer School IV on Nuclear Collective Dynamics 02.03.2008 74
distribution of cross sections overmultipolarities is strongly model dependent
DISTRIBUTION OF CROSS SECTIONS OVER MULTIPOLARITIES Distribution of cross section over multipolaritiesDistribution of cross section over multipolarities
Summer School IV on Nuclear Collective Dynamics 02.03.2008 75
RHB+RQRPA
RHB+RQRPA NEUTRINO-NUCLEUS (56Fe) CROSS SECTION RHB-RQRPA neutrino-nucleus 56Fe cross sectionRHB-RQRPA neutrino-nucleus 56Fe cross section
Summer School IV on Nuclear Collective Dynamics 02.03.2008 76
CROSS SECTIONS AVERAGED OVER NEUTRINO FLUX Cross section (νe,e-) averaged over supernova neutrino fluxCross section (νe,e-) averaged over supernova neutrino flux
muon decayat rest
e flux
Summer School IV on Nuclear Collective Dynamics 02.03.2008 77
The Nuclear Density Functional
Nuclear dynamics and excitations
Content II --------------------
ContentContent
Outlook
Motivation
Density Functional Theory
Ground state properties
Covariant Density Functional
Summer School IV on Nuclear Collective Dynamics 02.03.2008 78
Is density functional theory exact
in self-bound systems as nuclei?
derivation of the functional from the NN-force ?
construction areas
tensor-forces and single particle stucture?
beyond mean field
improvement of the functional
Summer School IV on Nuclear Collective Dynamics 02.03.2008 79
ColaboratorsColaborators::
A. Ansari (Bubaneshwar)A. Ansari (Bubaneshwar)G. A. Lalazissis (Thessaloniki)G. A. Lalazissis (Thessaloniki)D. Vretenar (Zagreb)D. Vretenar (Zagreb)
T. Niksic (Zagreb) T. Niksic (Zagreb) N. Paar (Zagreb)N. Paar (Zagreb)
D. Pena Arteaga D. Pena Arteaga A. Wandelt A. Wandelt
Summer School IV on Nuclear Collective Dynamics 02.03.2008 80
A. Bohr and B. Mottelson, “Nuclear Structure, Vol. I and II” P. Ring and P. Schuck, “The Nuclear Many-Body Problem” J.-P. Blaizot and G. Ripka, “Quantum Theory of Finite Systems” V.G. Soloviev, “Theory of Atomic Nuclei”
B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16, 1 (1986) P.-G. Reinhard, Rep. Prog. Phys. 52, 439 (1989) B. D. Serot, Rep. Prog. Phys. 55, 1855 (1992) P. Ring, Progr. Part. Nucl. Phys. 37, 193 (1996) B. D. Serot and J. D. Walecka, Int. J. Mod. Phys. E6, 515 (1997) Lecture Notes in Physics 641 (2004), “Extended Density Functionals in Nuclear Structure” D.Vretenar, Afanasjev, Lalazissis, P.Ring, Phys.Rep. 409 ('05) 101
Books on Nuclear Structure Theory
Review Articles on Covariant Density Functional Theory
LiteratureReferences
Summer School IV on Nuclear Collective Dynamics 02.03.2008 81
Computer Programs
Computer Programs
H. Berghammer et al, Comp. Phys. Comm. 88, 293 (1995), “Computer Program for the Time-Evolution of Nuclear Systems in Relativistic Mean Field Theory.” W. Pöschl et al, Comp. Phys. Comm. 99, 128 (1996), “Application of the Finite Element Method in self-consistent RMF calculations.” W. Pöschl et al, Comp. Phys. Comm. 101, 295 (1997), “Applica- tion of the Finite Element Method in RMF theory: the spherical Nucleus.” W. Pöschl et al, Comp. Phys. Comm. 103, 217 (1997), “Relativistic Hartree-Bogoliubov Theory in Coordinate Space: Finite Element Solution in a Nuclear System with Spherical Symmetry.” P. Ring, Y.K. Gambhir and G.A. Lalazissis, 105, 77 (1997), “Computer Program for the RMF Description of Ground State Properties of Even-Even Axially Deformed Nuclei .”