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


The Nuclear Density Functional

Nuclear dynamics and excitations

Content II --------------------

ContentContent

Outlook

Motivation

Density Functional Theory

Ground state properties

Covariant Density Functional


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


magic numbersmagic numbers

2 8 20 28 50 82 126 168 ?


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:


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


8

sun energy

He

H

U

Fe

A

fusion

fission

reactor energy

binding energy per particlebinding energy per particle

particle number

(MeV)

B


β+ decay β- decay

β+

β-

N-Z


β+

β-

N-Z


the nuclear density: ρ(r)the nuclear density: ρ(r)

ρ=1.6 nucleons/fm3

simplified representation:

ρ

r


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


- 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?


Abundancies of elements in the solar systemAbundancies of elements in the solar system

Fe

Au


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


r process

Study of Nucleosynthesis


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• …..• …..


nuclei and QCD?nuclei and QCD?

Scales: 1 GeV 100 keV

effective forcesin the nucleusQCD NN- forces

in the vacuum




Content II --------------------

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Outlook

Motivation





theorem of Hohenberg und Kohn:

density functional theory:density functional theory:

Hohenberg

Kohn


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):


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:


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:


Example for Thomas-Fermi approximation:

exactThomas-Fermi appr.


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:


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




Content II --------------------

ContentContent

Outlook

Motivation





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.


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)


• 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:




Content II --------------------

ContentContent

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Motivation





Dirac equationDirac equation in atoms:Dirac equation in atoms:

with magnetic field:

Coulomb potential: (r)

magnetic potential: (r)


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:


Fermi sea

Dirac sea

2m* ≈ 1200 MeV

V+S ≈ 700 MeV

V-S ≈ 50 MeV

2m ≈ 1800 MeV

continuum

Relativistic potentialsRelativistic potentials


(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:


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 ?


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?


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


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


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


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)


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


EOS-Walecka

J.D. Walecka, Ann.Phys. (NY) 83, (1974) 491

σω-model

Equation of state (EOS):Equation of state (EOS):


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


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)


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




Content II --------------------

ContentContent

Outlook

Motivation





EOS for DD-ME2

Neutron Matter

Nuclear matter equation of stateNuclear matter equation of state


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



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:




Content II --------------------

ContentContent

Outlook

Motivation





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. !


K∞=271

K∞=355

Monopole motion

K∞=211

)()( trt 2

Breathing mode: 208PbBreathing mode: 208Pb

E

V2

ˆ̂ ̂

Interaction:


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:


A. Ansari, Phys. Lett. B (2005)

Ansari-Sn2+-excitation in Sn-isotopes: 2+-excitation in Sn-isotopes:


Relativistic (Q)RPA calculations of giant resonances

Isovector dipole response

Sn isotopes: DD-ME2 effectiveinteraction + Gogny pairing

protons neutrons

Isoscalar monopole response


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)


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


Soft dipole modes and neutron skinSoft dipole modes and neutron skin


Exp: pygmy O


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


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


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-


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


pygmy modes in 100Mopygmy modes in 100Mo


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


neutrino reactions neutrino reactions e-

ν

(N,Z)→(N-1,Z+1)

pe-

n

+

e-

nspin-isospin-wave

p

νν

W-


beta-decay beta-decay e-

(N,Z)→(N-1,Z+1)

pe-

n

+

e-

nspin-isospin-wave

p

W±

ν-

ν-ν-


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


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)


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


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


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]


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


RHB+RQRPA

RHB+RQRPA NEUTRINO-NUCLEUS (56Fe) CROSS SECTION RHB-RQRPA neutrino-nucleus 56Fe cross sectionRHB-RQRPA neutrino-nucleus 56Fe cross section


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




Content II --------------------

ContentContent

Outlook

Motivation





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


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


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


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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