Transcript Folie 1 - Middle East Technical University

Modern description of nuclei
Covariant density functional theory
far from stability
of the dynamics
of nuclei far from stability
Barcelona, Dec. 10, 2007
Istanbul, July 2/3, 2008
Peter Ring
Universidad Autónoma de Madrid
Peter RingMünchen
Technische Universität
Technische Universität München
02.03.2008
Summer School IV on Nuclear Collective Dynamics
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Content
Content
II ------------------Motivation
Density Functional Theory
The Nuclear Density Functional
Covariant Density Functional
Ground state properties
Nuclear dynamics and excitations
Outlook
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proton number Z
U
Pb
Au
Fe
H
neutron number N
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neutron number N
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magic numbers
2 8 20 28 50
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126
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Forces acting in the nucleus:
the Coulomb force repels the protons
the strong interaction ("nuclear force") causes binding
is stronger for pn-systems than nn-systems
neutrons alone form no bound states
exception: neutronen stars (gravitation!)
e
the weak interaction causes β-decay:
n
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p
ν5
the nucleon-nucleon interaction:
distance > 1 fm
attractive
π-meson
1 fm
distance < 0.5 fm
repulsive
?
three-body forces ?
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binding energy per particle
sun energy
H
reactor energy
fusion
B
fission
8
He
U
(MeV)
Fe
particle number
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7
β+
β-
N-Z
β+ decay
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β- decay
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β+
β-
N-Z
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the nuclear density: ρ(r)
simplified representation:
ρ
r
ρ=1.6 nucleons/fm3
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proton and neutron densities
or?
ρ
ρ
n
p
p
n
r
ρ
ρ
r
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r
r
small neutronSummer
excess
large neutron excess
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Nuclei far from stability: what can we learn?
- the origin of more than half of the elements with Z>30
- constraints on effective nuclear interactions
- evolution of shell structure
- reduction of the spin-orbit interaction
- properties of weakly-bound and open quantum systems
- exotic modes of collective excitations
(pygmy, toroidal resonances)
- possible new forms of nuclei (molecular states,
bubble nuclei, neutron droplets...)
- asymmetric nuclear matter equation of state and
the link to neutron stars
- applications in astrophysics
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Abundancies of elements in the solar system
Fe
Au
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synthesis of heavy elements beyond Fe
neutron capture and successive β-decay:
en
(N,Z)
(N+1,Z)
(N,Z+1)
Z+1
Z
N N+1
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Study of Nucleosynthesis
r process
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What do the astrophysicists need ?
•
•
•
•
•
•
•
•
•
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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
…..
…..
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nuclei and QCD?
QCD
Scales:
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1 GeV
NN- forces
in the vacuum
effective forces
in the nucleus
100 keV
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Content
Content
II ------------------Motivation
Density Functional Theory
The Nuclear Density Functional
Covariant Density Functional
Ground state properties
Nuclear dynamics and excitations
Outlook
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Summer School IV on Nuclear Collective Dynamics
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density functional theory:
theorem of Hohenberg und Kohn:
Hohenberg
Kohn
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Many-body system in an external field U(r):
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 system
depends on U(r). It is a functional of U(r):
in the same way we obtain the density:
Inverting this relation we can introduce a Legendre transformation
replacing the independent function U(r) by the density ρ(r):
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Decomposition of HK-functional
In practical applications the functional EHK[ρ] is decomposed
into three parts:
The Hartree term is simple:
The non interacting part:
The exchange-correlation
part is the rest:
Exc is less important and often approximated,
but for modern calculations it plays a essential rule.
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Thomas Fermi approximation:
Thomas Fermi
Thomas 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
where γ is the spin/isospin degeneracy. Using this expression at
the local density they find:
This is not very good (molecules are never bound) and therefore one
added later on gradient terms containing ∇ρ and Δρ. This method
is 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.
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Example for Thomas-Fermi approximation:
exact
Thomas-Fermi appr.
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Kohn-Sham theory
Kohn-Sham theory:
In 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
 2

  Veff r  k r    k  k r 

 2m

the density obtained as r 

A
  r 
i 1
i
2
is the exact density
Obviously to each density ρ(r) there exist such a potential Veff(r).
The non interacting part of the energy functional is given by:
Eni   

2
A
A
2

2
3


 r d 3r 


r
d
r   i 

i
2m
2m i 1
i 1
3





r
V
r
d
r
eff

and obviously we have:
Veff(r)    Eni      EHK  EH  Exc 
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limitations of exact density functionals:
in practice
formally exact
Hohenberg-Kohn:
Kohn-Sham:
Skyrme:
Gogny:
no
no
no
no
shell effects
l•s,
pairing
config.mixing
generalized mean field: no configuration mixing,
no two-body correlations
local density:
kinetic energy density:
pairing density:
twobody density:
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Content
Content
II ------------------Motivation
Density Functional Theory
The Nuclear Density Functional
Covariant Density Functional
Ground state properties
Nuclear dynamics and excitations
Outlook
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Density functional
theory
in nuclei:
Density
functional
theory
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.
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D.Brink
D.Vauterin
Density functional theory in nuclei
Skyrme

Slater determinant
  A (1 (r1 )     A (rA ))
Mean field:
E
ˆ
h
ˆ
 ˆ
density matrix
A
ˆ (r, r' )   i (r ) i (r' )
i 1
Eigenfunctions:
hˆ  i  i  i
Interaction:
2

E
ˆ
V 
ˆˆ
Extensions: Pairing correlations, Covariance
Relativistic Hartree Bogoliubov (RHB)
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General properties of self-consistent mean field theories:
•
•
•
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
Virtues:
(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
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Content
Content
II ------------------Motivation
Density Functional Theory
The Nuclear Density Functional
Covariant Density Functional
Ground state properties
Nuclear dynamics and excitations
Outlook
02.03.2008
Summer School IV on Nuclear Collective Dynamics
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Dirac equation in atoms: Dirac equation
Coulomb potential:
(r)
with magnetic field:
magnetic potential:
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(r)
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Dirac equation in nuclei:Dirac equation
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
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Relativistic potentials
continuum
V-S ≈ 50 MeV
Fermi sea
2m* ≈ 1200 MeV
2m ≈ 1800 MeV
Dirac sea
V+S ≈ 700 MeV
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Elimination of small components:
(ε → m+ε)
W  V  S

1
f i r  
pgi r 
εi  2m  W
 


1
p  W  gi r   εi gi r 
p
~
 εi  2mr 

~
for i  2m
1
~
mr   m  W
2
m*r   m  S
1 1 W 
 1 

l s  W  gi r   εi gi r 
p ~ p  ~2
4m r r
 2m

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Why covariant ?
1) no relativistic kinematic necessary:
p F2  mN2  mN 1  0.075
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
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Why covariant?
1)
2)
3)
4)
5)
6)
7)
8)
9)
Large spin-orbit
splitting in nuclei
Why covariant
Large fields V≈350 MeV , S≈-400 MeV
Success of Relativistic Brueckner
Success of intermediate energy proton scatt.
relativistic saturation mechanism
consistent treatment of time-odd fields
Pseudo-spin Symmetry
Connection to underlying theories ?
As many symmetries as possible
Coester-line
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Walecka
model
Walecka
model
Nucleons are coupled by exchange of mesons
through an effective Lagrangian (EFT)
(J,T)=(1-,0)
(J,T)=(0+,0)
S (r)  g  (r)
Sigma-meson:
attractive scalar field
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(J,T)=(1-,1)

V (r)  g (r)  g   (r)  eA(r)
Omega-meson:
short-range repulsive
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Rho-meson:
isovector field
37
Lagrangian
Lagrangian
density
free Dirac particle
free meson fields
free photon field
interaction terms
Parameter:
meson masses:
mσ, mω, mρ
meson couplings:
gσ, gω, gρ
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interaction terms
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Lmotion
L
equations
of

 0.
Equations of motion
  qk  qk
for the nucleons we find the Dirac equation
  i  V   m  S 
i
 0.
No-sea approxim. !
for the mesons we find the Klein-Gordon equation
 
 
 







 m2     g   s
 m2    g  j

2 
 m     g  j
(em)

   A  ej
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s x 
A
   x   x 
i
i
j  x  
   x 


j  x  




x

 i  i x
A
i
i 1
i x
A
i 1
A
1
 x    i  x  1   3    i  x 
2
i 1
(em)

j
i 1
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statictime
limitreversal invariance)
Static limit (with
for the nucleons we find the static Dirac equation

p  V  m  S  i  εi i .
S   g s , V  g0  g0  eA0
No-sea approxim. !
for the mesons we find the Helmholtz equations
   m     g 
   m    g 
   m    g 
2


2

0
2

3
0
(em)
 A0  e
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

3
s
B
s 
B 
 
3
A

i
i 1
A
 

i
i 1
A

i 1
(em)

i

A

i 3
 i

i 1
i

i
1
1   3  i
2
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Relativistic saturation mechanism:
We consider only the σ-field, the origin of attraction its source is the
scalar density
m2    g   i  i   g   gi gi  f i  f i 
A
A
i 1
i 1
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.

1
f i r  
A
m2    g  B  2
i 1
kg i r 
~
εi  2m
A
1
f i  f i   g  B  ~  gigi
m i 1
In the non-relativistic case, Hartree with Yukawa forces
would lead to collapse
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Equation of state (EOS):
EOS-Walecka
σω-model
J.D. Walecka, Ann.Phys. (NY) 83, (1974) 491
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Density
Effective
density dependence:
dependence
non-linear potential:
NL1,NL3..
Boguta and Bodmer, NPA 431, 3408 (1977)
1 2 2
m 
2

1 2 2 1
1
3
U ( )  m   g 2  g 3 4
2
3
4
density dependent coupling constants:
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
g , g , g   g (  ), g (  ), g  (  )
g  g(r))
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DD-ME1,DD-ME2
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Point-Coupling Models
σ
J=0, T=0
ω
J=1, T=0
δ
J=0, T=1
ρ
J=1, T=1
Point-coupling model
Manakos and Mannel, Z.Phys. 330, 223 (1988)
Bürvenich, Madland, Maruhn, Reinhard, PRC 65, 044308 (2002)
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Lagrangian density forLagrangian
point coupling
free Dirac particle
interaction terms
interaction terms
Parameter:
photon field
point couplings:
Gσ, Gω, Gδ , Gρ,
derivative terms:
Dσ
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Content
Content
II ------------------Motivation
Density Functional Theory
The Nuclear Density Functional
Covariant Density Functional
Ground state properties
Nuclear dynamics and excitations
Outlook
02.03.2008
Summer School IV on Nuclear Collective Dynamics
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NuclearEOS
matter
for equation
DD-ME2 of state
Neutron Matter
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Symmetry
Symmetry
energy
energy
saturation density
empirical values:
30 MeV  a4  34 MeV
2 MeV/fm3 < p0 < 4 MeV/fm3
-200 MeV < K0 < -50 MeV
Lombardo
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Conclusions part I:
Conclusions I
1) density functional theory is in principle exact
2) microscopic derivation of E(ρ) very difficult
3) 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) provide
excellent description of ground state properties
- binding energies (1 ‰)
- radii (1 %)
- deformation parameters
6) pairing effects are non-relativisitic
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Content
Content
II ------------------Motivation
Density Functional Theory
The Nuclear Density Functional
Covariant Density Functional
Ground state properties
Nuclear dynamics and excitations
Outlook
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TDRMF: Eq.
Time dependent mean field theory:

 1 

i t i t       V   V  m  S i t 

 i

   m  t    g ρ t 
   m  t   g ρ t 


   m  t   g j t 
2

2

2


0
s
 B
 B

 
 
A
ρs 
i 1
ρB

jB


i i
i 1

A


i i
i 1
No-sea approxim. !
i  i
A
and similar equations for the ρ- and A-field
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(t)r 2 (t)Monopole
motion
Breathing
mode: 208Pb
K∞=211
K∞=271
K∞=355
Interaction:
2

E
ˆ
V 
 ˆ  ˆ
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RRPA
Relativistic RPA for excited states
ph, h
Small amplitude limit:
ground-state density
hp, h
RRPA matrices:
the same effective interaction determines
the Dirac-Hartree single-particle spectrum
and the residual interaction
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Interaction:
Vˆ 
E
 ˆ  ˆ
2
54
2+-excitation
in Sn-isotopes:
Ansari-Sn
A. Ansari, Phys. Lett. B (2005)
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Relativistic (Q)RPA calculations of giant resonances
Sn isotopes: DD-ME2 effective
interaction + Gogny pairing
Isovector dipole response
protons
neutrons
Isoscalar
monopole
response
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IS-GMR
Isoscalar Giant
Monopole: IS-GMR
The ISGMR represents the
essential source of
experimental information on
the nuclear incompressibility
Blaizot-concept:
constraining the nuclear
matter compressibility
ρ(t) = ρ0 + δρ(t)
RMF models reproduce the
experimental data only if
250 MeV  K0  270 MeV
T. Niksic et al., PRC 66 (2002) 024306
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IV-GDR
Isovector
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
the position of IV-GDR is
reproduced if
32 MeV  a4  36 MeV
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Niksic
et al.,
PRC 66
Summer School IV on T.
Nuclear
Collective
Dynamics
(2002) 024306
58
Soft dipole modes and neutron skin
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Exp: pygmy O
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Pygmy: Ochain
Evolution of IV dipole
strength in Oxygen isotopes
RHB + RQRPA calculations with
the NL3 relativistic mean-field
plus D1S Gogny pairing interaction.
Transition densities
What is the structure of lowlying strength below 15 MeV ?
Effect of pairing
correlations on the
dipole strength
distribution
02.03.2008
Summer School IV on Nuclear Collective Dynamics
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Pygmy: 132-Sn
Mass dependence of GDR and Pygmy
dipole states in Sn isotopes. Evolution
of the low-lying strength.
Isovector dipole strength in
132Sn.
Nucl. Phys. A692, 496 (2001)
GDR
Distribution of
the neutron
particle-hole
configurations
for the peak at
7.6 MeV (1.4%
of the EWSR)
Pygmy state
exp
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Vibrations in deformed nuclei
J
T=0
K
T=1
Goldstone modes
Translations:
K=0-
Giant dipole modes:
K=1-
Rotations:
K=0-
K=1-
Scissor modes:
K=1+
Gauge rotations: K=0+
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Summer School IV on Nuclear Collective Dynamics
K=1+
63
100Mo 100
IV-GDR
in
isovector-dipole response in Mo
IV-GDR
ρ0 + δρ(t)
K=002.03.2008
K=1Summer School IV on Nuclear Collective Dynamics
64
pygmy modes in 100Mo
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response of the nucleus
to an incoming particle
scattering at a single nucleon
02.03.2008
excitation of the entire nucleus
we need the nuclear spectrum
Summer School IV on Nuclear Collective Dynamics
66
e-
neutrino reactions
(N,Z)→(N-1,Z+1)
Wν
e-
p
p
e-
+
ν
02.03.2008
n
n
ν
spin-isospin-wave
Summer School IV on Nuclear Collective Dynamics
67
e-
beta-decay
ν-
(N,Z)→(N-1,Z+1)
W±
e-
p
ν-
p
e-
+
n
02.03.2008
n
spin-isospin-wave
Summer School IV on Nuclear Collective Dynamics
68
IAR-GTR
Spin-Isospin Resonances:
IAR - GTR
Z+1,N-1
Z,N
GTR  S- T Z,N
spin flip 
IAR  T Z,N
Z,N
isospin flip 
EGTR - EIAR
02.03.2008
dV
~ l  s  ~
~ neutron skin  rn - rp
dr
Summer School IV on Nuclear Collective Dynamics
p
n
69
ISOBARIC ANALOG AND GAMOW-TELLER RESONANCES
Spin-Isospin
Resonances: IAR - GTR
(RQRPA)
ISOSPIN-FLIP
EXCITATIONS
A
T F   i1

S=0 T=1 J = 0+
S=1 T=1 J = 1+
SPIN-FLIP &
ISOSPIN-FLIP
EXCITATIONS
A
T GT
  i1  

PR C69, 054303 (2004)
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β-decay
β-decay: Sn,Te
h9/2->h11/2
G. Martinez-Pinedo and K. Langanke,
PRL 83, 4502 (1999)
T.
Niksic et al, PRC 71, 014308 (2005)
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neutrino-nucleus reactions
E
0+
ZXN
32+
210+
1+
Z+1XN-1
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
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Cross section averaged over supernova neutrino flux
Supernova neutrino flux is
given by Fermi-Dirac spectrum
5
4
Cross section averaged over
Supernova neutrino flux
3

2
1
0
2
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Summer School IV on Nuclear Collective Dynamics
4
6
T [MeV]
8
10
73
DISTRIBUTION OF CROSS SECTIONS OVER
Distribution of cross section over multipolarities
MULTIPOLARITIES
distribution of
cross sections over
multipolarities is
strongly model
dependent
02.03.2008
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74
RHB+RQRPA NEUTRINO-NUCLEUS
(56Fe) CROSS SECTION
56
RHB-RQRPA neutrino-nucleus Fe cross section
RHB+RQRPA
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CROSS SECTIONS AVERAGED OVER NEUTRINO FLUX
Cross section (νe,e-) averaged over supernova neutrino flux
muon
decay
at rest
e flux
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Content
Content
II ------------------Motivation
Density Functional Theory
The Nuclear Density Functional
Covariant Density Functional
Ground state properties
Nuclear dynamics and excitations
Outlook
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construction areas
Is density functional theory exact
in self-bound systems as nuclei?
beyond mean field
tensor-forces and single particle stucture?
improvement of the functional
derivation of the functional from the NN-force ?
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Colaborators:
A. Ansari (Bubaneshwar)
G. A. Lalazissis (Thessaloniki)
D. Vretenar (Zagreb)
T. Niksic
N. Paar
(Zagreb)
(Zagreb)
D. Pena Arteaga
A. Wandelt
02.03.2008
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References
Literature
Books on Nuclear Structure Theory




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”
Review Articles on Covariant Density Functional Theory







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
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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), “Application 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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