Covariant density functional theory: applications in exotic nuclei

Covariant density functional theory: applications in exotic nuclei 18.12.2007 1

Peter Ring

Madrid, 18./19.12. 2007

Technical University MunichUniversidad Autónoma de Madrid

ISTANBUL-06

Covariant density functional theory:applications in exotic nuclei

Covariant density functional theory:applications in exotic nuclei


The Nuclear Density Functional

Nuclear Response Theory

Methods beyond mean field

Exotic rotational excitations

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

ContentContent

Outlook


Density functional theory

Φ Slater determinant ⇔ density matrixρ

∑=

=A

iii

1

)()(),(ˆ r'rr'r ϕϕρ))()(( 11 AA rr ϕϕ ⋅⋅⋅=Φ A

ˆ

δδEh =ρ iiih ϕεϕ =ˆ

Mean field: Eigenfunctions:

ˆ

2

δδδ EV =ρ ρ

Interaction:

Density functional theory in nucleiDensity functional theory in nucleiD.BrinkD.Vauterin

Skyrme

Extensions: Pairing correlations, Covariance Relativistic Hartree Bogoliubov (RHB)


Many-body system with Hamiltonian

Hohenberg-KohnHohenberg-Kohn theoremHohenberg-Kohn theorem

We consider a realistic manybody system with the kinetic energyand two-body interaction in an external field .

In this case the expectation value of the exact energy

The ground state is determined by minimizing with respect to

is given by a universal functional , which depends onlyon the local density , and not on the external potential .

P. Hohenberg, W. Kohn, Phys.Rev. 136B (1964) 864


free energy:

partition function:

expectation values:

differential form:

some thermodynamicsSome basic thermodynamics:

Hamiltonian:

T=1/β: We consider a many-body system in a finite Volume V

H


Gibbs potential:

it can be inverted:

derivative:

Now we replace the volume V by an external potential V → -U(r)and the pressure P by the density P → ρ(r)

LegendreTransformation

P is a monotonic function of V:

Legendre Transformation: (P↔V)


Inverting this relation we can introduce a Legendre transformationreplacing the independent function U(r) by the density ρ(r)

Considering that

the functional derivative of F with respect to U(r) is the density:

many-body system

We consider now a realistic manybody system in an external field U(r) and a two-body interaction V(ri,rk). The free energy depends now on U(r) instead of the volume V, i.e. the energy is a functional of U(r):

Many-body system in an external field U(r)


[ ] [ ][ ] [ ] rdUUFG 3ρρρρ ∫−=

where the independent variable is ρ(r). The potential G, which wecall in the following EHK, equation does not depend on U(r). It is a universal functional of ρ(r) alone:

This is the

)()(

rr

δρδ HKEU −=

[ ] [ ] rdUFEHK3)r()r()r()r( ∫−= ρρρ

VTUUVT ˆˆˆˆˆˆ +=−++=

Hohenberg-Kohn theorem.

The derivative of G with respect to ρ(r) is U(r) :

Kohn-Hohenberg

We find the potential G(T,U[ρ(r)]) (neglecting for simplicity T)

Hohenberg-Kohn theorem


Decomposition of KH-functional

In practical applications the functional EHK[ρ(r)] is decomposedinto three parts:

[ ] [ ] [ ] [ ]ρρρρ xcHniHK EEEE ++=

The Hartree EH is simple:

Exc is less important and often approximated,but for modern calculations it plays a essential rule.

[ ] [ ]0=

=VHKni EE ρρ

[ ] ')()()( rrddVEH33

21 r'r'r, r ρρρ ∫=

The non interacting part:

The exchange-correlationpart is the rest: [ ] [ ] [ ] [ ]ρρρρ HniHKxc EEEE −−=


This is not very good (molecules are never bound) and therefore one added later on gradient terms containing ∇ρ and ∆ρ. 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.

( ) 353

2222

3

3 653

2m2m)2( ρ

γπ

πγ ⎟⎟

⎠

⎞⎜⎜⎝

⎛== ∫

<

hhkkdVE

Fkk

where γ is the spin/isospin degeneracy. Using this expression at the local density they find:

( )∫⎟⎟⎠

⎞⎜⎜⎝

⎛= rdETF

3353

226

53 r

2m

2

ργπh

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 ϕεϕ =⎭⎬⎫

⎩⎨⎧

+Δ−h

( ) ( ) rr ∑=

=A

ii

1

2ϕρ

[ ] ( ) ( ) ( ) ( ) rdVrdm

rdm

E eff

A

ii

A

iini

3

1

3

1

22

32

22rr rr ∫∑∫ ∑∫ −=∇==

==

ρεϕτρ hh

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 −−−=−= δρδ

δρδ ρ)( r

and obviously we have:


with the Hartree potential

and the exchange-correlation potential

( ) ( ) ( ) ( )rrrr xcHeff VVUV ++=

( ) ( ) ( )∫= 'rdVVH3r'r'r, r ρ

( ) [ ]ρδρδ

xcxc EV )( r:r =

[ ] ( ) [ ] [ ]ρρττρ xcHKS EErdm

E ++= ∫ 32

2rh,

of course it all depends on the knowledge or on the approximationof the functional for the exchange-correlation energy

Kohn-Sham functional:

Kohn-Sham functionalDetermination of Veff :

In principle we can find Veff(r) by calculating the functional derivative of

( ) [ ] [ ] [ ]ρρρ δρδ

δρδ

δρδ

xcHHKeff EEEV ++−=r


Practical Applications:

Summarizing the Kohn-Sham scheme has the following steps

a) determine a good approximation for the functional EXC[ρ]b) start with some initial guess for ρ₀c) calculate from this ρ₀ the potentials VH(r) and Vxc(r) and Veff(r)d) solve the single particle Schrödinger equation for Veff(r) and

obtain the wave functions ϕi(r)e) use these single particle wave functions to calculate the density

ρ₁(r) in the next step of the iteration f) repeat this circle until convergence is achieved.


Remarks to Kohn-Sham method:

We have the following remarks to the Kohn-Sham method1) The method is exact under the condition that Vxc[ρ] is known. 2) The single particle wave functions ϕi(r) and the single particle

energies εi are only auxiliary quantities. They have nothing to do with experiment. We only obtain the exact total energy and for the density, i.e. quantities accessible by the density ρ(r).

3) The method works rather well even for shell structures

Methods to get a good approximation for the functional EXC[ρ]1) phenomenological formulas2) in the local density approximation (LDA) the Exc is calculated

exactly by Monte-Carlo techniques for a homogeneous electron gas with density ρ. In the inhomogeneous system the LDA is used. An example: The binding energy of the Ar-atom is reproduced by the Thomas Fermi method with an accuracy of 20 %, by Kohn-Sham method with LDA approximation of 0.5 %.

3) there exist many more sophisticated techniques nowadays


DFT: density of Ar-atom

units: radius: Bohr radiidensities x r2 in inverse Bohr radii

N.Argaman, G. Makov,Am. J. Phys. 68, 69 (2000)


limitations of exact density functionals:limitations of exact density functionals:

local density:

kinetic energy density:

pairing density:

twobody density:

Kohn-Hohenberg:Kohn-Sham:Skyrme:Gogny:

in practiceformally exact

no shell effectsno l•s,no pairingno config.mixing

generalized mean field: no configuration mixing, no two-body correlations


Non-relativistic density functional theory in nuclei:

nuclear densitiesThe building blocks of the nuclear energy density functional are various densities and currents:

( )τσ,,rx rr=For we have the density matrix:

( ) ( ) ( )∑=i

ii xxxx '', rrrrϕϕρ

( ) ( ) ( ) ( )[ ]{ } τσρρσ ρρ 00 qiqiqii rrrrrrrr ',',',', rrrrrrrr+++= 004

1

isoscalar isovector


( ) ( ) ( )∑==στ

30 τστστρρρ rrrrr rrrrr ,,1 isovector density:

pn ρρρ +=0

pn ρρρ −=1

( ) ( )∑=τσσ'

σσ'τσ'στρ σrrrrr rrrs ,0

isoscalar spin density

( ) ( )∑=τσσ'

σσ'ττσ'στρ σrrrrr rrrs ,1 isovector spin density

( ) ( ) ( ) '',' rrTT rrirj rrrrrr

=−∇∇= ρ

2current density T=0,1

( ) ( ) ( ) '',' rrTT rrsirJ rrrrrrr

=×−∇∇=

2spin current density

( ) ( ) ( ) '',' rrTT rrr rrrrr

=∇⋅∇= ρ τ kinetic energy density

( ) ( ) ( ) '',' rrTT rrsrT rrrrrrr

=∇⋅∇= kinetic spin density

local quantities( ) ( ) ( )∑==

στ000 στστρρρ rrrrr rrrrr ,, isoscalar density:

Local quantities:


Energy functional:

Skyrme forceThe Skyrme functional can be derived from a densitydependent two-body force

y


aaa

( )ρτ 161ρ

161ρ

83τ 2

212

30 532

ttttm

++++= +α2h

Skyrme functionalx

( )( ) ( ) 221021 59

641 JttJWtt

rrrr−+∇−∇−+

161ρ

43ρ

2

AE

kkK

ff 2

22

∂∂

=∞ 3

32

fk2πρ =

equation of state (EOS)

incompressibility

effective mass ( )ρ, 21

22

53161

22tt

mm++=

hh*

130

20

253 +++== α

f ttkmρ

HAE ρ

161ρ

83

*

2h

4 parameter

Energy functional for N=Z:


We start from the Skyrme functionaland obtain the equations of motion:

Variation of the Skyrme functional:

and find for spin-saturated nuclei:

( ) ( ) ( )[ ] r,rτ,rρ JEr

[ ] 0, τ,ρ, 23* =⎥

⎦

⎤⎢⎣

⎡− ∫∑ n

nn

k

rdJE ϕεδϕδ r

[ ]( )

( ) 0, rτr

τ,ρ, =⎥⎦

⎤⎢⎣

⎡++= ∫ JWU

mrdJE

rrhrδδρδδ *2

23

with:effective mass:

normal potential

spin-orbit potential

( )( ) ( ),rρ

r 21

22

53161

22tt

mm++=

hh*

( ) ( )

( )

( ) ρWrW

,W-ρ-

τρρr

0

0

2

∇=

∇Δ−

+++=

rr

rr

43

4359

321

53161

163

43

21

2130

Jtt

ttttU


with

and we find the Schroedinger equation:

( )

( )

( ) ( )∑

∑

∑

=

=

=

×∇=

∇∇=

=

A

iii

A

iii

A

iii

J1

1

1

ϕσϕ

ϕϕ

ϕϕ

rrr

rr

*

*

*

r

rτ

rρ

( )( ) r

r kϕρ

δϕδ ∑ ∫=

⎥⎦

⎤⎢⎣

⎡∂∂

++∇∇−=A

ik sl

rrWU

mrdE

10

23 1

23

2rrrhr

**

this yields

( )( ) r

r kkk ϕεϕρ

=⎥⎦

⎤⎢⎣

⎡∂∂

++∇∇− slrr

WUm

rrrhr 123

2 0

2

*


1) no relativistic kinematic necessary:

2) non-relativistic DFT works well

3) technical problems:no harmonic oscillatorno exact soluble modelsdouble dimensionhuge cancellations V-Sno variational method

4) conceptual problems:treatment of Dirac seano well defined many-body theory

0750122 .+=+ NNF mmp

Why not Covariant

Why covariant ?

Covariant density functional theory:Covariant density functional theory:


1.2 1.4 1.6 1.8 2.0kF [fm

−1]

−25

−20

−15

−10

−5

E/A

[M

eV]

Tuebingen (Bonn)BM (Bonn)Bonn A, psReidCD−BonnBonnAV18

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?


(σ) (ω) (ρ)(σ) (ω) (ρ)ψψ ψγψ μ ψτγψ μ r

relativistic densitiesRelativistic densities:Relativistic densities:

( ) ( )rr ψΓψ


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:Dirac equation:


V-S ≈ 50 MeV

V+S ≈ 750MeV

2m* ≈ 1200 MeV

2m ≈ 1880 MeV

relativistic potsRelativistic potentialsRelativistic potentials


(ε → m+ε)

( ) ( )rp1r ii

i gWmε

frr

σ+−+

=2

( )( ) ( )rrp

r1p iii

i

gεgWmε

=⎭⎬⎫

⎩⎨⎧

++ −

rrrrσσ ~2

( ) +−= Wmm21r~

( ) Smm −=r*

( ) ( )rr p1p iii gεgWslr

Wrmm

≈⎭⎬⎫

⎩⎨⎧ +

∂∂

+ −+ rrrr 1

41

2 2~~

SVW ±=±

for mi~2<<ε

elimination of small componentsElimination of small components:Elimination of small components:


Walecka model

Nucleons are coupled by exchange of mesons through an effective Lagrangian (EFT)

(Jπ,T)=(0+,0) (Jπ,T)=(1-,0) (Jπ,T)=(1-,1)

Sigma-meson:attractive scalar field

Omega-meson: short-range repulsive

Rho-meson:isovector field

)()( rr σσgS = )()()()( rrrr eAggV ++= ρτω ρωrr

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 ψγ μμμ

for the mesons we find the Klein-Gordon equation

( )( )( )

)(em

s

ejA

jgm

jgm

gm

μμνν

μρμρμμ

μωμωνν

σσνν

ρ

ω

ρσ

=∂∂

=+∂∂

=+∂∂

−=+∂∂

rr

2

2

2

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )∑

∑

∑

∑

=

=

=

=

−=

=

=

=

A

iii

em

A

iii

A

iii

A

iiis

xxxj

xxxj

xxxj

xxx

13

1

1

1

121

ψγτψ

ψγτψ

ψγψ

ψψρ

μμ

μμ

μμ

)(

rr

No-sea approxim. !

equations of motion( ) .0=

∂∂

∂∂∂

∂kk q

Lq

L -μ

μEquations of motionEquations of motion


for the nucleons we find the static Dirac equation

( )( ) .ε p iiiSmV ψψβα =−++rr

for the mesons we find the Helmholtz equations

( )( )( )

)(em

B

s

eA

gm

gm

gm

ρ

ρρ

ρω

ρσ

ρρ

ωω

σσ

=Δ−

=+Δ−

=+Δ−

−=+Δ−

0

330

20

2

2

( )∑

∑

∑

∑

=

+

=

+

=

+

=

−=

=

=

=

A

iii

em

A

iii

A

iiiB

A

iiis

13

13

3

1

1

121

ψτψρ

ψτψρ

ψψρ

ψψρ

)(

No-sea approxim. !

000 eAggVgS s ++=−= ρωσ ρω ,

static limitStatic limit (with time reversal invariance)Static limit (with time reversal invariance)


( )∑∑=

++

=

−−=−=A

iiiii

A

iii ffggggm

11

2σσσ ψψσ

relativistic saturationRelativistic saturation mechanism:Relativistic saturation mechanism:

We consider only the σ-field, the origin of attractionits 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. ( ) ( )rk1r i

ii g

mεf

rrσ~2+

=

∑∑=

+

=

+ ∇∇+−=−−≈A

iiiB

A

iiiB gg

mgffgm

11

2 12 ~ρρσ σσσ

In the non-relativistic case, Hartree with Yukawa forces would lead to collapse


2nd, 3th ,5th and 8th

order in kf/m*

Fouldy-Wouthousen

Fouldy-Wouthousen:Fouldy-Wouthousen:


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

41

31

21)(

21 σσσσσ σσ ggmUm ++=⇒

)(),(),(,, ρρρ ρωορωο gggggg ⇒

g → g(ρ(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


Density dependence

Three relativistic models:Three relativistic models:

Meson exchange with non-linear meson couplings:

Meson exchange with density dependent coupling constants:

NL1,NL3,TM1,..

DD-ME1,DD-ME2

Boguta and Bodmer, NPA. 431, 3408 (1977)

R.Brockmann and H.Toki, PRL 68, 3408 (1992)

Point-coupling models with density dependent coupling constants:Manakos and Mannel, Z.Phys. 330, 223 (1988)

PC-F1,….


number of param.

How many parameters ?How many parameters ?

symmetric nuclear matter: E/A, ρ0

finite nuclei (N=Z): E/A, radiispinorbit for free

σ

σ

mg

ω

ω

mg

σm

Coulomb (N≠Z): a4 ρ

ρ

mg

density dependence: T=0 K∞

7 parameters

rn - rpT=1

g2 g3

aρ


Fit: DD-ME2Nuclei used in the fit for DD-ME2Nuclei used in the fit for DD-ME2

(%) (%)

Nuclear matter: E/A=-16 MeV (5%), ρo=1,53 fm-1 (10%)

K = 250 MeV (10%), a4 = 33 MeV (10%)


parameterization

Parameterization of denstiy dependenceParameterization of denstiy dependence

PHENOMENOLOGICAL:

saturation densityMICROSCOPIC: Dirac-Brueckner calculations

gσ(ρ)

gω(ρ)

gρ(ρ)

Typel and Wolter, NPA 656, 331 (1999)Niksic, Vretenar, Finelli, Ring, PRC 66, 024306 (2002)Lalazissis, Niksic, Vretenar, Ring, PRC 71, 024312 (05)

4 parameters for density dependence


NM: DD-ME2

Nuclear matter properties:Nuclear matter properties:


EOS for DD-ME2

Neutron Matter

Nuclear matter equation of stateNuclear matter equation of state


Symmetry energy

Symmetry energySymmetry energy

empirical values:empirical values: 30 MeV ≤ a4 ≤ 34 MeV2 MeV/fm3 < p0 < 4 MeV/fm3

-200 MeV < ΔK0 < -50 MeV

saturation density

LombardoLombardo


General remarks to nuclear pairing:1) There is plenty of experimental evidence

2) In principle pairing is a small effect (∆<<M)

3) Most important close to the Fermi surface

4) Smearing of the Fermi surface (v2)

5) Gap in the spectrum:

6) Influence on response functions (e.g. moments of inertia)

7) Phase transition normal fluid → superfluid (with λ,ω,T)

8) Few exp. data on details of pairing (one parameter ∆)

9) Crucial quantity: pair-transfer matrix elements

General Remarks aboout Pairing:General Remarks aboout Pairing:

( )∑∑< +

−≈

−=

' '

'')( '

kk kk

kkkkxx

EEuvvukJk

EEJ

J2

0

22 0

ν ν

ν

( ) 22 Δ+−= λεkkE


Seniority schemeSeniority scheme:Seniority scheme: valence shell (j)

S+ creates a Cooper-paircore

exact ground state:

with the generalized product state: and the quasi-particles:


BCS-theoryBCS-theory


∑=ΦΦ= +

kknnknnnn VVcc '

*''ρ

This simple model can be generalized to the full space and toarbitrary interactions. The HFB-wavefunction |HFB> =|Φ> is defined as the quasi-particlevaccum to the quasiparticles:

∑ += ++

nnnknnkk cVcUα 0=Φkα

with the normal density:

the pairing tensor: ∑=ΦΦ=k

knnknnnn UVcc '*

''κ

The density functional depends on two densities:

Hartree-Fock Bogoliubov TheoryHartree-Fock Bogoliubov Theory

E[ρ,κ] = ERMF[ρ] + EGogny[κ]


The variation of E’[ρ,κ]=E[ρ,κ]-λTr(ρ) with respect to ρ and κ yealdstwo coupled equations for the HFB wave functions Uk(r) and Vk(r)

kk

k

k

k EVU

VU

))

))

-- * ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

ΔΔ

rr

rr

((

((

ˆˆˆ

* hh

with two potentials: the normal mean field the pairing field

ρδδ

ˆ'ˆ Eh =

κδδ

ˆˆ E=Δ

we have no longer a sharp Fermi surface (ρ2 ≠ ρ)but there is still a constraint: *ˆˆˆˆ κκρρ =−2

we have independent quasiparticles with theoccupation numbers 0 or 1

Hartree-Fock Bogoliubov EquationsHartree-Fock Bogoliubov Equations


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 fields

4) in realistic nuclei one needs a density dependence- non-linear coupling of mesons- density dependent coupling-parameters

5) modern parameter sets (7 parameter) provideexcellent description of ground state properties- binding energies (1 ‰)- radii (1 %)- deformation parameters

6) 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 fields

4) in realistic nuclei one needs a density dependence- non-linear coupling of mesons- density dependent coupling-parameters

5) modern parameter sets (7 parameter) provideexcellent description of ground state properties- binding energies (1 ‰)- radii (1 %)- deformation parameters

6) pairing effects are non-relativisitic

Conclusions IConclusions part I:Conclusions part I:


Density functional theory in nuclei

Ground state properties


Nuclear dynamics and excitations

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

ContentContent

Conclusions


proton radioactivityproton radioactivity

shell quenching N=28shell quenching N=28

halo phenomenahalo phenomena

nuclearchart

deformed nucleideformed nuclei

superheavysuperheavy elementselements


The spin-orbit potential originates fromthe addition of two large fields: the

field of the vector mesons (short range repulsion), and the scalar field of the sigma meson (intermediate attraction).

weakening of the effective single-neutron spin-orbit potential in neutron-rich isotopes

Energy splittingsbetween spin-orbit partner states

reduced energy spacings betweenspin-orbit partners

G. A. Lalazissis, D. Vretenar, P. Ring, NPA 57 (1998) 2294

Reduction of lsRadial dependenceof the spin-orbit term of the single neutron potential

reduction of the spin-orbit potentialreduction of the spin-orbit potential

+


Neutron and Proton SkinsNeutron and Proton Skins


nuclearchart

halo phenomenahalo phenomena


Neutron halo‘s

11Li

Mean field theory of halo‘s:(RHB in the continuum)

advantages:

* residual interaction by pairing

* self-consistent description

* universal parameters

* polarization of the core

* treatment of the continuum

problems:

*center of mass motion

*boudary conditions at infinity

Neutron halo‘sNeutron halo‘s


Density in Li-nuclei

J. Meng and P. Ring , PRL 77, 3963 (1996)J. Meng and P. Ring , PRL 80, 460 (1998)

rel. Hartree-Bogoliubov

in the continuum

density dependent δ−pairing

Densities in Li-isotopesDensities in Li-isotopes


canonical basis in Li

* eigenstates of the density matrix

* wavefunction has BCS-type

J. Meng and P. Ring , PRL 77, 3963 (1996)

nnnhn

aavu

nn

nnnnn

Δ=Δ=

−+=Φ ++Π ,

)(

ε

canonical basis in Li-isotopescanonical basis in Li-isotopes


Giant halo in Zr-nuclei

J. Meng and P. Ring , PRL 80, 460 (1998)

Giant halo in the Zr region:Giant halo in the Zr region:


proton radioactivityproton radioactivity

nuclearchart


Ground-state proton emitters

Self-consistent RHB calculations -> separation energies, quadrupole deformations, odd-proton orbitals, spectroscopic factors Lalazissis, Vretenar, Ring

Phys.Rev. C60, 051302 (1999)

Proton emitters I

characterized by exotic ground-state decay modes such as the direct emissionof charged particles and β -decays with large Q-values.

Vretenar, Lalazissis, Ring, Phys.Rev.Lett. 82, 4595 (1999)

Nuclei at the proton drip line:Nuclei at the proton drip line:


How far is the proton-drip line from the experimentallyknown superheavy nuclei?

G. A. Lalazissis et al., PRC 59 (2004) 017301

Proton drip-line in the sub-Uranium region Proton drip-line in the sub-Uranium region

Possible ground-state protonemitters in this mass region?

Proton drip-line for super-heavy elements: Proton drip-line for super-heavy elements:


superheavysuperheavy elementselementsnuclearchart


Synthesis of super-heavy elements

neutron capture and successive β—decay:

n

(N,Z) (N+1,Z) (N,Z+1)

e-

fusion of two nuclei:

compound nucleus

α-decay

fission

n


Deformation Deformation

Clalssical nuclear droplet

Z>100Z<100


deformation

Quantum mechanical shell effects

Shell effects lead to enhanced stabilityat specific proton and neutron numbers(magic numbers)


Deformation

GS Saddle Scission

Pote

ntia

l Ene

rgy

Quantenmechanical shell effects

Elements with Z>100 arestabilized by shell effects

Bfdecay byfission !!!

classical dropletclassical droplet

shell effectsshell effects


208Pb

Region of spherically sabilized nuclei („Island of Stability“)

Region of deformed stabilized nuclei with Z=108 and N=162

GSI: Elemente 107-112first synthesisedand uniquelyidentified:

107 – Bh108 – Hs109 – Mt

Shell correction Eshell in the region of superheavy elements

P. Möller et al.

Name accepted for element 110 (16.8.2003):Darmstadtium– Ds

for element 111 (18.5.2005):Röntgenium- Rg


SH-Elements

Exp: Yu.Ts.Oganessian et al, PRC 69, 021601(R) (2004)

Superheavy Elements: Qα-valuesSuperheavy Elements: Qα-values


Covariant density functional theory




Content III --------------------

ContentContent

Conclusions


TDRMF: Eq.

and similar equations for the ρ- and A-field

( ) ( ) ( )tSmi

ti iit ψβαψ ⎟⎠

⎞⎜⎝

⎛ −++⎟⎠⎞

⎜⎝⎛ −∇=∂ VV

rrr 1

[ ] ( ) ( )[ ] ( ) ( )[ ] ( ) ( )tgtm

tgtm

tgtm

B

B

s

j

ρ

ρ

rvωω

ωω

σσ

ω

ω

σ

=+Δ−

=+Δ−

−=+Δ−

2

02

2

∑∑∑

=

=+

=

=

=

=

A

i iiB

A

i iiB

A

i iis

1

1

1

ψαψ

ψψ

ψψ

rrj

ρ

ρ

Time dependent mean field theory:Time dependent mean field theory:

No-sea approxim. !


K∞=271

K∞=355

Monopole motion

K∞=211

)()( trt ΦΦ 2Breathing mode: 208PbBreathing mode: 208Pb

δδδ EV

2

=ˆρ ρ

Interaction:


GMR: T=0

GMR: T=1

D. Vretenar et al., PRE 56(1997) 6418

Order and Chaos Order and ChaosOrder and Chaos


RRPARelativistic RPA for excited statesRelativistic RPA for excited states

RRPA matrices:

the same effective interaction determines the Dirac-Hartree single-particle spectrum and the residual interaction

δρph, δραh

δρhp, δρhα

Small amplitude limit:

ground-state density

δδδ EV

2

=ˆρ ρ

Interaction:


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

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


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 informationson 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


GMR in SnIsoscalar Giant Monopole in Sn-isotopesIsoscalar Giant Monopole in Sn-isotopes

Isoscalar GMR in spherical nuclei → nuclear matter compression modulus Knm.

Theory: Lalazissis et alExp: U. Garg, unpublished

Sn isotopes: DD-ME2 / Gogny pairing


Pygmy resonances


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 dipolestrength 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 configurationsfor 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-


Spurious modes in Ne-20

Eph=hω

Goldstone modes in 20NeGoldstone modes in 20Ne

NL3

1_ 1+

1+


evolution of the GDR in deformed Ne isotopesevolution of the GDR in deformed Ne isotopes

δρ(r,z)


IV E1 in 26Ne

0

0.1

0.2

0.3

0.4

0.5R

(E1)

[e2fm

2/

MeV

]R

(E1)

[e2fm

2/

MeV

]

0 10 20 30 40

E [MeV]E [MeV]

Kπ = 1−

Kπ = 0−

IV-dipole in 26NeIV-dipole in 26Ne

β=0.29, Δn=1.3 MeV, sumrule=109 % NL3( <12 MeV:= 5 % )


pygmy-resonance in Ne-26Pygmy-Resonance in deformed 26NePygmy-Resonance in deformed 26Ne

GANILGANIL

THEORYTHEORY


Kπ = 0− 8.51 MeV

−0.04

−0.02

0

0.02

0.04

r2δρ

[fm

−1]

r2δρ

[fm

−1]

0 2.5 5 7.5 10 12.5 15

r [fm]r [fm]

neutronproton Kπ = 1− 9.99 MeV

−0.04

−0.02

0

0.02

0.04

r2δρ

[fm

−1]

r2δρ

[fm

−1]

0 2.5 5 7.5 10 12.5 15

r [fm]r [fm]

neutronproton

Pygmy in 26Ne ?Pygmy in 26Ne ?


IV-GDR in 100MoIV-GDR in 100Mo


IV-GDR in 100MoIV-GDR in 100Mo

IV-GDR

K=0- K=1-

ρ0 + δρ(t)



ISD modesIsoscalar dipole compression -- toroidal modesIsoscalar dipole compression -- toroidal modes

Isoscalar GMR in spherical nuclei -> nuclear matter compression modulus Knm.

Giant isoscalar dipole oscillations -> additional information on the nuclear incompressibility.

ISGDR strength distributionsEffective interactions with different Knm.

Compression modeThe low-energy strengthdoes not depend on Knm !

Phys. Lett. B487, 334 (2000)


Toroidalmotion

multipole expansion of a four-current distribution:charge moments magnetic moments electric transverse moments -> toroidal moments

isoscalar toroidal dipole operator:

ISGDR transition densitiesfor 208Pb (NL3 interaction)

Toroidal motionToroidal motion

toroidal dipole moment: poloidal currents on a torus


Currents

Toroidal dipolestrength distributions.

Velocity distributions in 116Sn

Vretenar, Paar, Niksic, Ring,Phys. Rev. C65, 021301 (2002)


IAR-GTRSpin-Isospin Resonances: IAR - GTRSpin-Isospin Resonances: IAR - GTR

Z,N Z+1,N-1

isospin flip τ

Z,N Z,NT IAR +=

spin flip σ

Z,NTS GTR - +=

r-rskin neutrondrdVslEE pnIARGTR =)⋅(Δ ~ ~ ~ -

p n


PN-resonancesSpin-Isospin Resonances: IAS and GTRSpin-Isospin Resonances: IAS and GTR

proton-neutron relativistic QRPA

charge-exchange excitations

π and ρ-meson exchange generate the spin-isospindependent interaction terms

the Landau-Migdal zero-rangeforce in the spin-isospin channel

GAMOW-TELLER RESONANCE:

ISOBARIC ANALOG STATE:

S=1 T=1 Jπ = 1+

S=0 T=1 Jπ = 0+

(g’0=0.55)


IARIsobaric Analog Resonance: IARIsobaric Analog Resonance: IAR

N. Paar, T. Niksic, D. Vretenar, P.Ring, PR C69, 054303 (2004)

experiment


GTRGT-ResonancesGT-Resonances

N. Paar, T. Niksic, D. Vretenar, P.Ring, PR C69, 054303 (2004)


GTR/IAR –neutron skin

Neutron skin and IAR/GRTNeutron skin and IAR/GRT

The isotopic dependence of the energyspacings between the GTR and IAS

direct information on the evolution of the neutron skinalong the Sn isotopic chain


β-decay: Sn,Te

νh9/2->πh11/2G. Martinez-Pinedo and K. Langanke,

PRL 83, 4502 (1999)T. Niksic et al, PRC 71, 014308 (2005)

β-decayβ-decay


Density functional theory in nuclei




Content IV --------------------

ContentContent

Conclusions


Particle-VibrationalCoupling (PVC)

Particle-Vibrational Coupling (PVC):energy dependent self-energyParticle-Vibrational Coupling (PVC):energy dependent self-energy

single particle strength:

+

+

RPA-modes

μ

μ

mean field pole part

=

non-relativistic investigations:Ring, Werner (1973)Hamamoto, Siemens (1976)Perazzo, Reich, Sofia (1980)Bortignon et al (1980)Bernard, Giai (1980)Platonov (1981)Kamerdzhiev, Tselyaev (1986)

Dyson equation


Relativistic diagramsContributions to Σ(ω)

in the relativistic case:Contributions to Σ(ω)in the relativistic case:


-8 -6 -4 -2 0 2 4 60,0

0,2

0,4

0,6

0,8

1,0S

pect

rosc

opic

fact

or

E, MeV

209Bi 1h9/2

-4 -2 0 2 4 6 80,0

0,2

0,4

0,6

0,8

1,0

Spe

ctro

scop

ic fa

ctor

E, MeV

209Bi 1i13/2

10 12 14 16 180,0

0,2

0,4

0,6

0,8

1,0

Spe

ctro

scop

ic fa

ctor

E, MeV

209Bi 2h11/2

-2 0 2 4 6 8 100,0

0,2

0,4

0,6

0,8

1,0

Spe

ctro

scop

ic fa

ctor

E, MeV

209Bi 2f5/2

Fragmentation in 209-BiDistribution of single-particle strength in 209Bi


Single particle spectrumSingle particle spectrum in the Pb regionSingle particle spectrum in the Pb region

meff 0.76 0.92 1.0 0.71 0.85 1.0

E. Litvinova and P. Ring, PRC 73, 44328 (2006)


Width of Giant ResonancesWidth of Giant Resonances

The full response contains energy dependent parts coming from vibrational couplings.

ph-phonon amplitudes(QRPA)

Self energy

ph interactionamplitude

δρωδω )()( Σ

=V


E1 photoabsorptioncross section

photoabsorption cross sectionphotoabsorption cross section


4 5 6 7 8 9 100

5

10

15

42

44

cros

s se

ctio

n [m

b]

RQRPA RQTBA

PDR 120Sn

S [ e

2 fm 2 /

MeV

]

E [MeV]5 10 15 20 25 30

0

200

400

600

800

1000

1200

RQRPA RQTBA

GDR 120Sn

E [MeV]4 5 6 7 8 9 10

0

2

4

20

22

24

cros

s se

ctio

n [m

b]

RQRPA RQTBA

PDR 116Sn

E [MeV]5 10 15 20 25 30

0

200

400

600

800

1000

1200 RQRPA RQTBA

GDR 116Sn

E [MeV]

5 6 7 8 9 100

1

2

3

PDR 90Zr

RQRPA RQTBA

E [MeV]5 10 15 20 25

0

200

400

600

800 RQRPA RQTBA

GDR 90Zr

cros

s se

ctio

n [m

b]

E [MeV]5 6 7 8 9 10

0

2

4

6

8 RQRPA RQTBA

PDR 88Sr

S [e

2 fm 2 /

MeV

]

E [MeV]5 10 15 20 25

0

200

400

600

RQRPA RQTBA

cros

s se

ctio

n [m

b]GDR 88Sr

E [MeV]

Electric dipole excitations in stable nuclei


potential surface: Mg-32

Energy surface in 32MgEnergy surface in 32Mg

Beyond Mean Field: Beyond Mean Field:

0ˆˆ =Φ−Φ QqHδ

pure mean field


0ˆˆ =Φ−Φ QqHδConstraint Hartree Fock produces wave functions depending on a generator coordinate q ( )qq Φ=

qqfdq )( ∫=ΨGCM wave function is asuperposition of Slaterdeterminants

[ ] 0)'( '' ' =−∫ qfqqEqHqdqHill-Wheeler equation:

with projection: qPPqfdq IN ˆˆ)( ∫=Ψ

GCM-methodGenerator Coordinate Method (GCM)Generator Coordinate Method (GCM)


Ar-36 surface

36Ar: GCM: N+J projection vs. J-projection36Ar: GCM: N+J projection vs. J-projection


Ar-32 wavefunctions

GCM-wavefunctionsGCM-wavefunctions


Mg-24 spectrum

Spectra in 24MgSpectra in 24Mg


Mg-24 spectrum

Spectra in 24MgSpectra in 24Mg


Quantum phase transitions and critical symmetriesQuantum phase transitions and critical symmetries

X(5) 152SmE(5): F. Iachello, PRL 85, 3580 (2000)X(5): F. Iachello, PRL 87, 52502 (2001)

R.F. Casten, V. Zamfir, PRL 85 3584, (2000)

Interacting Boson Model

Casten Triangle


R. Krücken et al, PRL 88, 232501 (2002)

Transition U(5) → SU(3) in Ne-isotopesTransition U(5) → SU(3) in Ne-isotopes

R = BE2(J→J-2) / BE2(2→0)




GCM: only one scale parameter: E(21)X(5): two scale parameters: E(21), BE2(02→21)

Problem in present GCM: restricted to γ=0

J+N

F. Iachello, PRL 87, 52502 (2001)

R. Krücken et al, PRL 88, 232501 (2002)

Niksic et al PRL 99, 92502 (2007)


Neighboring nuclei:Neighboring nuclei:

BE2(J→J-2)/BE2(2→0)

E(J)/E(2) SU(3)

U(5)

X(5)

SU(3)

U(5)

X(5)



ColaboratorsColaborators::

A. A. AnsariAnsari ((BubaneshwarBubaneshwar))G. A. G. A. LalazissisLalazissis (Thessaloniki)(Thessaloniki)D. D. VretenarVretenar (Zagreb)(Zagreb)

E. E. LitvinovaLitvinova ((ObninskObninsk))T. T. NiksicNiksic (Zagreb) (Zagreb) N. Paar (Darmstadt)N. Paar (Darmstadt)

D. D. PenaPena de de ArteagaArteagaE.E. LopesLopes (BMW)(BMW)A. Wandelt (Telekom)A. Wandelt (Telekom)


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 DensityFunctionals in Nuclear Structure”D. Vretenar et al, Phys. Rep. 409 (2005) 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 inRelativistic Mean Field Theory.”W. Pöschl et al, Comp. Phys. Comm. 99, 128 (1996), “Applicationof 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 sphericalNucleus.”W. Pöschl et al, Comp. Phys. Comm. 103, 217 (1997), “RelativisticHartree-Bogoliubov Theory in Coordinate Space: Finite ElementSolution 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 StateProperties of Even-Even Axially Deformed Nuclei .”

Documents

Covariant density functional theory: applications in exotic nuclei