Transcript Folie 1 - Department

ISTANBUL-06
Covariant density functional theory
for masses and fission barriers
Kazimierz Dolny, Sept. 25, 2008
Peter Ring
Technical University Munich
Sept.
25, 2008
www.ph.tum.de
XV Nuclear Physics Workshop Kazimierz 2008: "75 years of nuclear fission"
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Content
Relativistic density functional theory
Calculation of masses
Fission barriers
Conclusions
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25, 2008
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D.Brink
D.Vautherin
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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Covariant density functional theory
The Walecka model is used as a vehicle
to implement a covariant density functional theory
through an effective Lagrangian.
(J,T)=(0+,0)
S (r)  g  (r)
Sigma-meson:
attractive scalar field
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(J,T)=(1-,0)
(J,T)=(1-,1)

V (r)  g (r)  g   (r)  eA(r)
Omega-meson:
short-range repulsive
Rho-meson:
isovector field
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Density models:
Three relativistic
dependence
Meson exchange with non-linear meson couplings:
Boguta and Bodmer, NPA. 431, 3408 (1977)
Lalazissis, Koenig, Ring, PRC 55. 540 (1997)
1 2 2
m 
2

NL1,NL3,TM1,..
1
1
1
U ( )  m2 2  g 2 3  g 3 4
2
3
4
Meson exchange with density dependent coupling constants:
R.Brockmann and H.Toki, PRL 68, 3408 (1992)
Lalazissis, Niksic, Vretenar, Ring, PRC 71, 024312 (2005)
DD-ME1,DD-ME2
g(ρ)
8 parameters
Point-coupling models with density dependent coupling constants:
Manakos and Mannel, Z.Phys. 330, 223 (1988)
Buervenich, Madland, Maruhn, Reinhard, PRC 65, 44308 (2002)
G(ρ)
PC-F1,….
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Stragegies to determine the parameters:
Nuclear matter data (from masses)
- binding energy E/A
- saturation density ρ0
- surface energy as
- symmetry energy a4
(- compressibility ???)
Conclusions:
Conclusions
1 ------Specific finite nuclei: spherical
- binding energies,
- radii - spherical closed shell or single closed shell
- no single particle energies!
How to avoid shell effects?
- fit to deformed nuclei (at the beginning)
- fit to fission barriers (future)
Ab initio ?
(far future)
- fit to exact calculations for nuclear matter
- fit to exact calculations for half infinite nuclear matter
- problem of accuracy
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DD-ME: parameterization of denstiy dependence
parameterizatio
saturation density
n
MICROSCOPIC: Dirac-Brueckner calculations
g()
PHENOMENOLOGICAL:
g()
4 parameters for density dependence
Typel and Wolter, NPA 656, 331 (1999)
Niksic, Vretenar, Finelli, Ring, PRC 66, 024306 (2002)
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25, 2008
Nuclear
Physics
Workshop
Kazimierz 2008: "75 years of nuclear fission"
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Lalazissis,
Niksic, Vretenar,XVRing,
PRC
71, 024312
(05)
g()
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How many parameters ?
number of
param.
symmetric nuclear matter:
finite nuclei (N=Z):
g
m
E/A, ρ0
E/A,
radii
spinorbit for free
g
m
m
g
Coulomb (N≠Z):
m
a4
density dependence: T=0
T=1
Sept.
25, 2008
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7-8 parameters
K∞
g2
rn - rp
aρ
g3
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rms-deviations:
masses:
Dm = 900 keV
Masses:
900
Dr = 0.015 fm
keVradii:
Lalazissis, Niksic, Vretenar, Ring, PRC 71, 024312 (2005)
256No:
1899.21 (1889.65) MeV
ΔE = 560 keV → 0.3 ‰
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Superheavy Elements: Qa-values
SH-Elements
Exp: Yu.Ts.Oganessian et al, PRC 69, 021601(R) (2004)
Th: G.A Lalazissis et al, PRC 71, 024312 (2005)
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U. Garg: Monopole-resonance and compressibility
-200
FSUGold
-300
K(MeV)
-400
SG2
SkM*
SkI4
SG1
SLy4
SkM
SkI3
NLC
Sa
-500
MSkA
TM1
DDME2
DDME1
S3
SkX
S1
S6
S4
-600
200
TM2
NL3
-700
220
240
260
280
300
320
340
360
380
KOO (MeV)
c ≈ -1
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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 < DK0 < -50 MeV
Lombardo
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Pb:
GMR/GDR
G.A. Lalazissis et al, PRC 71, 024312 (2005)
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DD-ME2
13
Sn
IsoscalarGMR
GiantinMonopole
in Sn-isotopes
Isoscalar GMR in spherical nuclei → nuclear matter compression modulus Knm.
Sn isotopes:
DD-ME2 / Gogny pairing
Theory: Lalazissis et al
Exp:
U. Garg, unpublished
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Giant resonances in deformed nuclei:
IVGDR in 100Mo
pygmy:
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Exp: Rossendorf
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isovector-dipole response in 100Mo
IV-GDR
ρ0 + δρ(t)
K=0Sept.
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K=1XV Nuclear Physics Workshop Kazimierz 2008: "75 years of nuclear fission"
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pygmy modes in 100Mo
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fission barriers
+ binding energies
DD-ME2/D1S
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Fission barrier
depends on pairing:
Stepien, Szymanski, PLB 26, 182 (1968)
Rutz, Bender, Reinhard et al, PLB 468, 1 (1999)
Samyn, Goriely, Pearson, PRC 72, 044316 (2005)
Staszczak, Dobaczewski, Nazarewicz, Int. J.Mod. Phys.E
Gogny D1S:
constant G:
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20% -> 1.5 MeV
20% -> 2.5 MeV
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Simplified pairing energy functional:
Y. Tian, Z.Y. Ma, P.R.
Gogny:
separable:
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Determination of the function P(r) in nuclear matter
perfect agreement for nuclear matter
P(r) Gaussian form
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P(r) obtained by mapping
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In finite nuclei:
comparison with Gogny:
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dynamic results for the new force:
comparison with Gogny:
Y. Tian, Z.Y. Ma, P.R. (2007)
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comparison with experiment:
A. Ansari PLB 623, 37 (2005)
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investigation of the bare force:
Sn
Pb
there is room left for phonon-exchange force
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Comparison of δ-force and separable force with Gogny D1S:
matrix elements:
<nn'|V|mm'>
pairing field Δnn'
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Conclusions:
Relativistic density functionals
- provide a high quality description of nuclear structure at low costs
Density dependence is essential
Conclusions:
- in the isoscalar (NL3) and in the isovector channel (DD-ME2)
Conclusions
1 -------
Masses are comparable with modern mass models
Fission barriers depend on the pairing
- on the pairing strength, but also on the form of pairing
Ab initio determination of the nuclear density functionals
- in principle possible by fitting the density dependence to nuclear matter calculations
- however problems with accuracy.
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Colaborators:
S. Karatzikos
G. A. Lalazissis
Z.Y. Ma
T. Niksic
D. Pena
Y. Tian
D. Vretenar
Sept.
25, 2008
www.ph.tum.de
(Thessaloniki)
(Thessaloniki)
(Beijing)
(Zagreb)
(Orsay)
(Beijing)
(Zagreb)
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