1. Modern description of nuclei far from stability
Covariant density functional theory of the dynamics of nuclei far from stability
Barcelona, Dec. 10, 2007
Peter Ring
Universidad Autónoma de Madrid
Technische Universität München
Istanbul, July 2/3, 2008
Isotopen-Tafel mit Feldern einbauen
Peter Ring
Technische Universität München
Summer School IV on Nuclear Collective Dynamics

2. Content II --------------------
Motivation
Density Functional Theory
The Nuclear Density Functional
Covariant Density Functional
Ground state properties
Nuclear dynamics and excitations
Outlook
Summer School IV on Nuclear Collective Dynamics

3. Summer School IV on Nuclear Collective DynamicsPb
proton number Z Au Fe
neutron number N H
neutron number N
Summer School IV on Nuclear Collective Dynamics

4. Summer School IV on Nuclear Collective Dynamics
magic numbers ?
Summer School IV on Nuclear Collective Dynamics

5. 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 p ν -
Summer School IV on Nuclear Collective Dynamics

6. the nucleon-nucleon interaction:
distance > 1 fm
attractive
π-meson
1 fm
distance < 0.5 fm ?
repulsive
three-body forces ?
Summer School IV on Nuclear Collective Dynamics

7. binding energy per particle
sun energy He H U Fe A
fusion
fission
reactor energy B 8
(MeV)
particle number
Summer School IV on Nuclear Collective Dynamics

8. Summer School IV on Nuclear Collective Dynamicsβ+ β-
N-Z
β+ decay
β- decay
Summer School IV on Nuclear Collective Dynamics

9. Summer School IV on Nuclear Collective Dynamicsβ+ β-
N-Z
Summer School IV on Nuclear Collective Dynamics

10. the nuclear density: ρ(r)
simplified representation: ρ r
ρ=1.6 nucleons/fm3
Summer School IV on Nuclear Collective Dynamics

11. proton and neutron densities
or? n p ρ ρ p n r r ρ r
small neutron excess
large neutron excess
Summer School IV on Nuclear Collective Dynamics

12. 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
Summer School IV on Nuclear Collective Dynamics

13. Abundancies of elements in the solar systemAu
Summer School IV on Nuclear Collective Dynamics

14. synthesis of heavy elements beyond Fe
neutron capture and successive β-decay:
(N,Z+1) e- n
(N,Z)
(N+1,Z)
Z+1 Z
N N+1
Summer School IV on Nuclear Collective Dynamics

15. Summer School IV on Nuclear Collective Dynamics
Study of Nucleosynthesis
r process
Summer School IV on Nuclear Collective Dynamics

16. What do the astrophysicists need ?
nuclear masses (bindung energies – Q-values)
equation of state (EOS) of nuclear matter: E(ρ)
isospin dependence E(ρp, ρn)
nuclear matrix elements (life times of β-decay ..)
cross section for neutron or electron capture ….
fission probabilities
cross sections for neutrino reactions
…..
Summer School IV on Nuclear Collective Dynamics

17. Summer School IV on Nuclear Collective Dynamics
nuclei and QCD?
effective forces
in the nucleus
QCD
NN- forces
in the vacuum
Scales: GeV keV
Summer School IV on Nuclear Collective Dynamics

18. Content II --------------------
Motivation
Density Functional Theory
The Nuclear Density Functional
Covariant Density Functional
Ground state properties
Nuclear dynamics and excitations
Outlook
Summer School IV on Nuclear Collective Dynamics

19. density functional theory:
theorem of Hohenberg und Kohn:
Hohenberg
Kohn
Summer School IV on Nuclear Collective Dynamics

20. 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):
Summer School IV on Nuclear Collective Dynamics

21. 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.
Summer School IV on Nuclear Collective Dynamics

22. Thomas Fermi Thomas Fermi approximation:
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.
Summer School IV on Nuclear Collective Dynamics

23. Example for Thomas-Fermi approximation:
exact
Thomas-Fermi appr.
Summer School IV on Nuclear Collective Dynamics

24. Summer School IV on Nuclear Collective Dynamics
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
the density obtained as 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:
and obviously we have:
Summer School IV on Nuclear Collective Dynamics

25. limitations of exact density functionals:
in practice
formally exact
Hohenberg-Kohn: Kohn-Sham: Skyrme: Gogny:
no shell effects no l•s, no pairing no config.mixing
generalized mean field: no configuration mixing, no two-body correlations
local density:
kinetic energy density:
pairing density:
twobody density:
Summer School IV on Nuclear Collective Dynamics

26. Content II --------------------
Motivation
Density Functional Theory
The Nuclear Density Functional
Covariant Density Functional
Ground state properties
Nuclear dynamics and excitations
Outlook
Summer School IV on Nuclear Collective Dynamics

27. Density functional theory
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.
Summer School IV on Nuclear Collective Dynamics

28. Density functional theory in nuclei
D.Brink
D.Vauterin
Skyrme
Slater determinant
density matrix
Mean field:
Eigenfunctions:
Interaction:
Extensions: Pairing correlations, Covariance
Relativistic Hartree Bogoliubov (RHB)
Summer School IV on Nuclear Collective Dynamics

29. 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
Summer School IV on Nuclear Collective Dynamics

30. Content II --------------------
Motivation
Density Functional Theory
The Nuclear Density Functional
Covariant Density Functional
Ground state properties
Nuclear dynamics and excitations
Outlook
Summer School IV on Nuclear Collective Dynamics

31. Dirac equation in atoms:
Coulomb potential: (r)
with magnetic field:
magnetic potential: (r)
Summer School IV on Nuclear Collective Dynamics

32. Dirac equation in nuclei:
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
Summer School IV on Nuclear Collective Dynamics

33. Relativistic potentials
continuum
V-S ≈ 50 MeV
Fermi sea
2m* ≈ 1200 MeV
2m ≈ 1800 MeV
Dirac sea
V+S ≈ 700 MeV
Summer School IV on Nuclear Collective Dynamics

34. Elimination of small components:
for
Summer School IV on Nuclear Collective Dynamics

35. Summer School IV on Nuclear Collective Dynamics
Why covariant ?
no relativistic kinematic necessary:
non-relativistic DFT works well
3) technical problems:
no harmonic oscillator
no exact soluble models
double dimension
huge cancellations V-S
no variational method
conceptual problems:
treatment of Dirac sea
no well defined many-body theory
Summer School IV on Nuclear Collective Dynamics

36. Summer School IV on Nuclear Collective Dynamics
Why covariant
Why covariant?
Large spin-orbit splitting in nuclei
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
Summer School IV on Nuclear Collective Dynamics

37. Summer School IV on Nuclear Collective Dynamics
Walecka model
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
Summer School IV on Nuclear Collective Dynamics

38. Summer School IV on Nuclear Collective Dynamics
Lagrangian density
Lagrangian
free Dirac particle
free meson fields
free photon field
interaction terms
Parameter:
meson masses: mσ, mω, mρ
meson couplings: gσ, gω, gρ
interaction terms
Summer School IV on Nuclear Collective Dynamics

39. Equations of motion equations of motion
for the nucleons we find the Dirac equation
No-sea approxim. !
for the mesons we find the Klein-Gordon equation
Summer School IV on Nuclear Collective Dynamics

40. Static limit (with time reversal invariance)
for the nucleons we find the static Dirac equation
No-sea approxim. !
for the mesons we find the Helmholtz equations
Summer School IV on Nuclear Collective Dynamics

41. 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.
In the non-relativistic case, Hartree with Yukawa forces
would lead to collapse
Summer School IV on Nuclear Collective Dynamics

42. Summer School IV on Nuclear Collective Dynamics
Equation of state (EOS):
EOS-Walecka
σω-model
J.D. Walecka, Ann.Phys. (NY) 83, (1974) 491
Summer School IV on Nuclear Collective Dynamics

43. Effective density dependence:
non-linear potential:
NL1,NL3..
Boguta and Bodmer, NPA 431, 3408 (1977)
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)
g  g(r(r))
DD-ME1,DD-ME2
Summer School IV on Nuclear Collective Dynamics

44. Point-Coupling Modelsσ ω δ ρ
J=0, T=0
J=1, T=0
J=0, T=1
J=1, T=1
Point-coupling model
Manakos and Mannel, Z.Phys. 330, 223 (1988)
Bürvenich, Madland, Maruhn, Reinhard, PRC 65, (2002)
Summer School IV on Nuclear Collective Dynamics

45. Lagrangian density for point coupling
free Dirac particle
interaction terms
interaction terms
Parameter:
point couplings: Gσ, Gω, Gδ , Gρ,
derivative terms: Dσ
photon field
Summer School IV on Nuclear Collective Dynamics

46. Content II --------------------
Motivation
Density Functional Theory
The Nuclear Density Functional
Covariant Density Functional
Ground state properties
Nuclear dynamics and excitations
Outlook
Summer School IV on Nuclear Collective Dynamics

47. Nuclear matter equation of state
EOS for DD-ME2
Neutron Matter
Summer School IV on Nuclear Collective Dynamics

48. Symmetry energy Symmetry energy
saturation density
empirical values:
30 MeV £ a4 £ 34 MeV
2 MeV/fm3 < p0 < 4 MeV/fm3
-200 MeV < DK0 < -50 MeV
Lombardo
Summer School IV on Nuclear Collective Dynamics

49. Summer School IV on Nuclear Collective Dynamics
Summer School IV on Nuclear Collective Dynamics

50. Summer School IV on Nuclear Collective Dynamics
Conclusions part I:
Conclusions I
density functional theory is in principle exact
microscopic derivation of E(ρ) very difficult
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
Summer School IV on Nuclear Collective Dynamics

51. Content II --------------------
Motivation
Density Functional Theory
The Nuclear Density Functional
Covariant Density Functional
Ground state properties
Nuclear dynamics and excitations
Outlook
Summer School IV on Nuclear Collective Dynamics

52. Time dependent mean field theory:
TDRMF: Eq.
Time dependent mean field theory:
No-sea approxim. !
and similar equations for the ρ- and A-field
Summer School IV on Nuclear Collective Dynamics

53. Summer School IV on Nuclear Collective Dynamics
Monopole motion
Breathing mode: 208Pb
K∞=211
K∞=271
K∞=355
Interaction:
Summer School IV on Nuclear Collective Dynamics

54. Relativistic RPA for excited states Small amplitude limit:
RRPA
Relativistic RPA for excited states
drph, drah
Small amplitude limit:
ground-state density
drhp, drha
RRPA matrices:
Interaction:
the same effective interaction determines
the Dirac-Hartree single-particle spectrum and the residual interaction
Summer School IV on Nuclear Collective Dynamics

55. 2+-excitation in Sn-isotopes:
Ansari-Sn
A. Ansari, Phys. Lett. B (2005)
Summer School IV on Nuclear Collective Dynamics

56. Summer School IV on Nuclear Collective Dynamics
Relativistic (Q)RPA calculations of giant resonances
Sn isotopes: DD-ME2 effective
interaction + Gogny pairing
Isovector dipole response
protons
neutrons
Isoscalar
monopole
response
Summer School IV on Nuclear Collective Dynamics

57. Isoscalar Giant Monopole: IS-GMR
The ISGMR represents the
essential source of experimental information on the nuclear incompressibility
Blaizot-concept:
ρ(t) = ρ0 + δρ(t)
constraining the nuclear
matter compressibility
RMF models reproduce the
experimental data only if
250 MeV £ K0 £ 270 MeV
T. Niksic et al., PRC 66 (2002)
Summer School IV on Nuclear Collective Dynamics

58. Isovector Giant Dipole: IV-GDR 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
T. Niksic et al., PRC 66 (2002)
Summer School IV on Nuclear Collective Dynamics

59. Soft dipole modes and neutron skin
Summer School IV on Nuclear Collective Dynamics

60. Summer School IV on Nuclear Collective Dynamics
Exp: pygmy O
Summer School IV on Nuclear Collective Dynamics

61. Summer School IV on Nuclear Collective Dynamics
Pygmy: O-chain
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 low-lying strength below 15 MeV ?
Effect of pairing
correlations on the
dipole strength
distribution
Summer School IV on Nuclear Collective Dynamics

62. Summer School IV on Nuclear Collective Dynamics
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
Summer School IV on Nuclear Collective Dynamics

63. Vibrations in deformed nucleiK J
T=0
T=1
Goldstone modes
Translations:
Rotations:
Gauge rotations:
Giant dipole modes:
Scissor modes:
K=0-
K=1-
K=0-
K=1-
K=1+
K=1+
K=0+
Summer School IV on Nuclear Collective Dynamics

64. isovector-dipole response in 100Mo
IV-GDR in 100Mo
isovector-dipole response in 100Mo
IV-GDR
ρ0 + δρ(t)
K=0-
K=1-
Summer School IV on Nuclear Collective Dynamics

65. Summer School IV on Nuclear Collective Dynamics
pygmy modes in 100Mo
Summer School IV on Nuclear Collective Dynamics

66. response of the nucleus to an incoming particle
scattering at a single nucleon
excitation of the entire nucleus
we need the nuclear spectrum
Summer School IV on Nuclear Collective Dynamics

67. Summer School IV on Nuclear Collective Dynamics
neutrino reactions e- ν
(N,Z)→(N-1,Z+1) W- p e- n +
spin-isospin-wave ν
Summer School IV on Nuclear Collective Dynamics

68. Summer School IV on Nuclear Collective Dynamics
beta-decay e- ν -
(N,Z)→(N-1,Z+1) W± e- p p ν - e- ν - + n n
spin-isospin-wave
Summer School IV on Nuclear Collective Dynamics

69. Spin-Isospin Resonances: IAR - GTR
Z,N
Z+1,N-1
spin flip s
isospin flip t p n
Summer School IV on Nuclear Collective Dynamics

70. Spin-Isospin Resonances: IAR - GTR
ISOBARIC ANALOG AND GAMOW-TELLER RESONANCES (RQRPA)
ISOSPIN-FLIP EXCITATIONS
S=0 T=1 J = 0+
S=1 T=1 J = 1+
SPIN-FLIP & ISOSPIN-FLIP EXCITATIONS
PR C69, (2004)
Summer School IV on Nuclear Collective Dynamics

71. G. Martinez-Pinedo and K. Langanke,
β-decay
β-decay: Sn,Te
G. Martinez-Pinedo and K. Langanke,
PRL 83, 4502 (1999)
h9/2->h11/2
T. Niksic et al, PRC 71, (2005)
Summer School IV on Nuclear Collective Dynamics

72. neutrino-nucleus reactions3- 2+ 2- 1- 0+ 1+ 0+
ZXN
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
Summer School IV on Nuclear Collective Dynamics

73. Cross section averaged over supernova neutrino flux
Supernova neutrino flux is
given by Fermi-Dirac spectrum 4 3 2 1 5 a
T [MeV]
Cross section averaged over
Supernova neutrino flux
Summer School IV on Nuclear Collective Dynamics

74. Distribution of cross section over multipolarities
DISTRIBUTION OF CROSS SECTIONS OVER MULTIPOLARITIES
distribution of
cross sections over
multipolarities is
strongly model
dependent
Summer School IV on Nuclear Collective Dynamics

75. RHB-RQRPA neutrino-nucleus 56Fe cross section
Summer School IV on Nuclear Collective Dynamics

76. Cross section (νe,e-) averaged over supernova neutrino flux
CROSS SECTIONS AVERAGED OVER NEUTRINO FLUX
Cross section (νe,e-) averaged over supernova neutrino flux
muon
decay
at rest
ne flux
Summer School IV on Nuclear Collective Dynamics

77. Content II --------------------
Motivation
Density Functional Theory
The Nuclear Density Functional
Covariant Density Functional
Ground state properties
Nuclear dynamics and excitations
Outlook
Summer School IV on Nuclear Collective Dynamics

78. Summer School IV on Nuclear Collective Dynamics
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 ?
Summer School IV on Nuclear Collective Dynamics

79. Summer School IV on Nuclear Collective Dynamics
Colaborators:
A. Ansari (Bubaneshwar)
G. A. Lalazissis (Thessaloniki)
D. Vretenar (Zagreb)
T. Niksic (Zagreb)
N. Paar (Zagreb)
D. Pena Arteaga
A. Wandelt
Summer School IV on Nuclear Collective Dynamics

80. Summer School IV on Nuclear Collective Dynamics
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
Summer School IV on Nuclear Collective Dynamics

81. Summer School IV on Nuclear Collective Dynamics
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 .”
Summer School IV on Nuclear Collective Dynamics