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Summer School IV on Nuclear Collective Dynamics 02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische Universität München Barcelona, Dec. 10, 2007 Covariant density functional theory of the dynamics of nuclei far from stability Peter Ring Technische Universität München Istanbul, July 2/3, 2008
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Page 1: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 2: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 3: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 4: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

Summer School IV on Nuclear Collective Dynamics 02.03.2008 4

magic numbersmagic numbers

2 8 20 28 50 82 126 168 ?

Page 5: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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:

Page 6: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 7: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 8: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

Summer School IV on Nuclear Collective Dynamics 02.03.2008 8

β+ decay β- decay

β+

β-

N-Z

Page 9: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

Summer School IV on Nuclear Collective Dynamics 02.03.2008 9

β+

β-

N-Z

Page 10: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 11: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 12: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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?

Page 13: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 14: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 15: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

Summer School IV on Nuclear Collective Dynamics 02.03.2008 15

r process

Study of Nucleosynthesis

Page 16: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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• …..• …..

Page 17: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 18: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 19: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

Summer School IV on Nuclear Collective Dynamics 02.03.2008 19

theorem of Hohenberg und Kohn:

density functional theory:density functional theory:

Hohenberg

Kohn

Page 20: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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):

Page 21: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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:

Page 22: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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:

Page 23: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

Summer School IV on Nuclear Collective Dynamics 02.03.2008 23

Example for Thomas-Fermi approximation:

exactThomas-Fermi appr.

Page 24: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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:

Page 25: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 26: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 27: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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.

Page 28: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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)

Page 29: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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:

Page 30: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 31: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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)

Page 32: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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:

Page 33: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 34: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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:

Page 35: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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 ?

Page 36: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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?

Page 37: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 38: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 39: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 40: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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)

Page 41: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 42: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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):

Page 43: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 44: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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)

Page 45: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 46: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 47: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 48: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 49: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

Summer School IV on Nuclear Collective Dynamics 02.03.2008 49

Page 50: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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:

Page 51: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 52: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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. !

Page 53: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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:

Page 54: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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:

Page 55: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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:

Page 56: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 57: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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)

Page 58: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 59: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

Summer School IV on Nuclear Collective Dynamics 02.03.2008 59

Soft dipole modes and neutron skinSoft dipole modes and neutron skin

Page 60: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

Summer School IV on Nuclear Collective Dynamics 02.03.2008 60

Exp: pygmy O

Page 61: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 62: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 63: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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-

Page 64: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 65: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

Summer School IV on Nuclear Collective Dynamics 02.03.2008 65

pygmy modes in 100Mopygmy modes in 100Mo

Page 66: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 67: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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-

Page 68: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

ν-

ν-ν-

Page 69: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 70: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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)

Page 71: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 72: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 73: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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]

Page 74: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 75: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 76: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 77: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 78: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 79: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 80: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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

Page 81: Summer School IV on Nuclear Collective Dynamics02.03.2008 1 Modern description of nuclei far from stability Peter Ring Universidad Autónoma de Madrid Technische.

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 .”


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