Transcript Prezentacja programu PowerPoint

Global nuclear structure aspects of
tensor interaction
Wojciech Satuła
in collaboration with
J.Dobaczewski, P. Olbratowski, M.Rafalski, T.R. Werner, R.A. Wyss, M.Zalewski
Kazimierz Dolny 2008
Principles of low-energy nuclear physics effective theories
Coupling constants & fitting strategies
Single-particle fingerprints of tensor interaction
- SO splittings & magic gaps
Influence of tensor fields on:
- the total binding energy & S2n
- nuclear deformability
- high-spin (terminating) states
Summary
ab initio
+ NNN + ....
tens of MeV
Effective theories for low-energy nuclear physics:
Modern Mean-Field Theory  Energy Density Functional
r,
t,





J,
j,
T,
s,
F,
Hohenberg-Kohn-Sham
The nuclear effective theory
is based on a simple and very intuitive assumption that low-energy
nuclear theory is independent on high-energy dynamics
hierarchy of scales:
Long-range part of the NN interaction
(must be treated exactly!!!)
where
local
correcting
potential
2roA1/3
~ 2A1/3
ro
~ 10
denotes an arbitrary Dirac-delta model
przykład
Gogny interaction
Fourier
Coulomb
regularization
ultraviolet
cut-off
There exist an „infinite” number
of equivalent realizations
of effective theories
Skyrme interaction - specific (local) realization of the
lim da
nuclear effective interaction:
a
10(11)
parameters
0
density dependence
spin-orbit
relative momenta
spin exchange
Spin-force inspired local energy density functional
Y | v(1,2) | Y
Slater determinant
(s.p. HF states are equivalent to the Kohn-Sham states)
local energy density functional
Skyrme-inspired functional
is a second order expansion
in densities and
currents:
density rg
dependent CC
tensor
spin-orbit
20 parameters are fitted to:
Symmetric NM:
- saturation density ( ~0.16fm-3)
- energy per nucleon (-16+ 0.2MeV)
- incompresibility modulus (210 +20MeV)
- isoscalar effective mass (0.8)
Asymmetric NM:
- symmetry energy ( 30+ 2MeV)
- isovector effective mass
(GDR sum-rule enhancement)
- neutron-matter EOS
(Wiringa, Friedmann-Pandharipande)
Finite, double-magic nuclei
[masses,radii, rarely sp levels]:
-surface properties
-ZOO–
Binding energy-dictated fit:
MSk1
SkM*
SIII
SkO
SkXc
SkP
SkI1
5
120 130 140 150 160 170 W*
0
SkXc
0.8
0.9
m*/m
1
and contradicting scalings
in the single-particle splittings
7.5
6.5
5.5
std. so
90% so
SIII
SkI1
SLy4
SLy5
SkM*
8.5
SkXc
SkP
MSk1
scales with Wo*
*
oW )
(Wo* = m
o
m
SIII
SkO
0.7
SLy4
6
std. so
90% so
SkP
SkO
SLy4
SkM*
7
De(f7/2-f5/2) [MeV]
W0
190
180
170
160
150
SLy5
De(d3/2-f7/2) [MeV]
superficial m* dependence
in the spin-orbit strength:
experiment
experiment
140 150 160 170 180 190 W0
scales with Wo
(two-body SO interaction)
Fitting strategies of the tensorial coupling constants (I)
De(f5/2-f7/2) [MeV]
bare SkO spectra
8
7
6
5
8
7
6
5
a)
neutrons
protons
b)
40Ca
48Ca
56Ni
Fitting strategies of the tensorial coupling constants (II)
- the details 7
1) Fit of the isoscalar SO strength
j>
2) Fit of the isoscalar tensor strength:
j<
56Ni
j>
F
3) Fit of the isovector tensor strength
or, more precisely, C1J/C1 J
48Ca
D
f7/2 f5/2 splittings around
48Ni
or 78Ni are needed
in order to fix SO-tensor sector
Single-particle energies [MeV]
5

j< F
40Ca
f7/2-d3/2
3
1
40Ca
f7/2-f5/2
p3/2-p1/2
0.7
7
f7/2-p3/2
0.8
0.9
1
f7/2-f5/2
5
3
56Ni
p3/2-p1/2
1
-40
-30
-20
-10
8
6
4
0 CJ0
f7/2-f5/2
48Ca
f7/2-d3/2
2
-80
-60
-40
-20
0 CJ1
SkPT T0=-39(*5);T1=-62(*-1.5);SO*0.8
„World” CC overview
- strategy dependence -
OUR VALUES OF COUPLING CONSTANTS:
SLy4
SKO
SKP
SIII
SkM*
0,69
0,90
1,00
C0∇J
-60
-62
-60
C 0J
-45
-33
-38
C 1J
SLy5
-92
-61
0,76
-58
-51
-65
0,67
-56
-42
-68
all CC are in [MeV fm5]
Standard: C0∇J C1∇J = 3
SkO:
= -0,78
50
-60
SkP
MSk1
Skxc
C1J [MeV fm5]
m*
0
-50
-100
SkO’
SLy4
Skxta
SLy4T
BSF
triangle
Colo
et al.
SkPT
SkOT
Skxtb
Brink &
Stancu
-40
0
40
80
C0J [MeV fm5]
Brown et al. PRC74, 061303 (2006)
Colo et al. PLB646, 227 (2007)
Brink & Stancu,
PRC75, 064311 (2007)
Selected single-particle fingerprints of tensor interaction:
(I) spin-orbit splittings
Spin-orbit splittings [MeV]
SLy4T
SLy4T T0=-45;T1=-60; SO*0.65
n
1h
7
1i
p
7
5
5
3
3
1
g9/2-g7/2
f7/2-f5/2
1
1h
g9/2-g7/2
f7/2-f5/2
M.Zalewski, J.Dobaczewski, WS, T.Werner, PRC77, 024316 (2008)
8
20
28
50
(III) „Otsuka mechanism”:
Neutrons filling j>’ subshell
influence proton s.p.
energies:
Otsuka et al., PRL87, 082502 (2001); PRL95, 232502 (2005)
8
20
h9/2 s1/2
g9/2 g7/2
g9/2 p1/2
p1/2 d5/2
82 126
p
f7/2 p3/2
n
48Ca
d3/2 f7/2
2
g9/2 p1/2
4
56Ni
g9/2 d5/2
p1/2 d5/2
6
d3/2 f7/2
8
SkP
SkPT
exp
f7/2 d3/2
f7/2 p3/2
10
28
40
50
82
M.Zalewski et al., PRC77, 024316 (2008)
Magic gaps [MeV]
Selected single-particle fingerprints of tensor interaction:
(II) magic-gap energies
Z
The tensorial „magic structure”
Z
isoscalar
isoscvector
N=Z
Z
N
N
total
14 – d5/2
32 – f7/2 p3/2
56 – g9/2 d5/2
90 – h11/2 f7/2
90
56
56
32
32
14
N
Tensor forces in neutron rich nuclei
Z~56
N~90
Z~32
N~56
Z~14, N~32
Baumann et al. Nature Vol 449, 1022 (2007)
40Mg, 42Al
0
SLy4
-5
-10
-15
-20
SLy4T
40Ca
48Ca 56Ni
90Zr 132Sn 208Pb
SkOT’:
SO reduced by 15%
C0J=-44.1MeVfm5
C1J=-91.6MeVfm5
SkOT’’:
1.00015C0r & 0.99C1r
ETH – EEXP [MeV]
ETH – EEXP [MeV]
M.Zalewski et al., PRC77, 024316 (2008)
SLy4Tmin
5
3 SkOT’’
2
1
0
-1
-2
E>0
40Ca
16O
48Ca
20 shells
SLy4
SkOT’
90Zr
56Ni
80Zr
132Sn
100Sn
208Pb
bare
8
7
6
5
8
7
6
5
time-even
8
7
6
5
8
7
6
5
TE&TO
8
7
6
5
40Ca
48Ca
56Ni
40Ca
8
7
6
5
48Ca
56Ni
De(pf5/2-pf7/2) [MeV]
De(nf5/2-nf7/2) [MeV]
Polarisation effects in a presence of strong tensor fields
SkO versus SkOT’
Influence of tensor on two-neutron separation energy
in oxygen isotopes
S2n [MeV]
18
20
22
26
28
30
oxygen
10
5
0
d5/2
dS2n [MeV]
24
s1/2
( )
d3/2 ( )
1
0
-1
-2
AME03
SkO
SkOT’
18
20
22
24
A
26
28
30
Deformation properties
in a presence
of strong tensor fields
f5/2
p3/2
f7/2
Nilsson
[321]1/2
[303]7/2
neutrons protons
SkOTX
SkO
SkOT’
-3
-4
8
tensor
spin-orbit
DE [MeV]
10
DEtensor [MeV]
12
Rudolph et al. PRL82, 3763 (1999)
4p-4h
6
4
-5
-6
2
0
0
0.1 0.2 0.3 0.4
0
0.1 0.2 0.3 0.4
deformacja b2
0
0.1 0.2 0.3 0.4
0
0.1 0.2 0.3 0.4 b
2
constrained HFB calculations
in spin-saturated 80Zr
6
80Zr
SkO
SkOT’
4
3
2
1
spin-orbit
tensor
DE [MeV]
5
SkOTX
0.4
0.5
0
0.1
0.2
0.3
b2
Further tests in simple-situations:
terminating states around A~50:
-7/2
(n=6)
f7/2 28
f7/2
+1/2
+3/2
+5/2
20
-1
+1/2
+3/2
+5/2
+3h
0h
+1/2
+3/2
protons
neutrons
-5/2
-3/2
-1/2
p-h
across
the gap
-3/2
-1/2
d3/2 filled
24
14h
+7/2
fully
46Ti
f7/2 d3/2
-5/2
-3/2
-1/2
partially
filled
-7/2
(n=7) -1
DE = E( d3/2 f7/2
-3/2
-1/2
+1/2
+3/2
cranking: -wjz
n+1
+7/2
n
) - E( f7/2 I )
Imax
max
PRC71, 024305 (2005) H.Zduńczuk, W.Satuła, R.Wyss
Standard parameterizations:
-1.0
-1.5
0
mo*
0.70
1.5
1.0
0.5
0
-0.5
DEth-DEexp [MeV]
SLy4
SLy5
SkXc
-0.5
0.76
0.91
1.00
SIIILT
SkPLT
SLy4LT
SkOLT
-0.5
-2.0
-1.0
42Ca 44Ca 44Sc 45Sc 45Ti 46Ti 47V
f7/2
p-h
DEth-DEexp [MeV]
SIII
SkM*
SkP
SkO
Spin-orbit and tensor
modified parameterizations
20
~5MeV
d3/2
rescaled by m*
42Ca44Ca44Sc 45Sc 45Ti 46Ti 47V
„spectroscopic-quality” functionals
must have large (>0.9)
effective mass!!!
~
SUMMARY & OUTLOOK
Simple three-step procedure is proposed in
order to fit the SO & tensor CC
The method leads to strong attractive tensor
fields and week SO potentials:
 improvement of the s.p. properties
The tensor interaction influences:
 binding energies („magic structure”)
 S2n energies
 nuclear deformability (novel mechanisms)
......
 high-spin properties
in an extremely neat and robust manner...
Amenable to further generalizations...
From two-body, zero-range tensor interaction
towards the EDF:
mean-field
averaging
Local Density Functional Theory for Superfluid Fermionic Systems:
The Unitary Gas
Aurel Bulgac, Phys. Rev. A 76, 040502 (2007)
running coupling constant
in order to renormalize....
ultraviolet
divergence in
pairing tensor
ab initio
calculations by:
Chang & Bertsch
Phys. Rev. A76, 021603
von Stecher, Greene & Blume,
E-print:0705.0671v1