Global fitting of pairing density functional; the isoscalar-density dependence revisited
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Global fitting of pairing density functional;the isoscalar-density dependence revisited
Masayuki YAMAGAMI (University of Aizu)
MotivationConstruction of energy density functional for description of static and dynamical properties across the nuclear chart
⇒ Focusing on the pairing part (pairing density functional)
a. Determination of –dependence (Not new problem, but one of bottlenecks in DF calc.)
b. Connection to drip-line regions
Our discussionDensity dependence of pairing in nuclei
• NN scattering of 1S0 (strong @low-• Many-body effects (e.g. phonon coupling)
Standard density functional for pairing
2104
,0
-30
0
0
1
depende
0.16 fm
nce
pairn p
n p
H r Vr
r
fo
(Isoscalar densi
parameter
ty),
r
Our question: How to determine 0 ??
phonon coupling
Difficulty for 0 (-dependence)Mass number A dependence of pairing
J. Dobaczewski, W. Nazarewicz, Prog. Theor. Phys. Supp. 146, 70 (2002)
A
0=1 0=0
A
0 dep.
But...A
Neutron excess =(N-Z)/A dependence
Mass data: G. Audi et al., NPA729, 3 (2003)n,exp: 3-point mass difference formula
,exp 2 ( ) 1/3( )
1 7.74 , 75 / 6.AA
n
nn A
(same dependence for proton pairing)
1 & dependence
simultaneously for ,
n p
A
Our model
-33 01 , 1 (n), 1 (p), , 0.16 fmp pn n
2
, ,e,
0 2 x1 0 p, , ,tot HFBn p
V
Pairing density functional with isoscalar & isovector density dep.
Pairing density functional with isoscalar & isovector density dep.
Parameter optimization Parameter optimization 0 1 2 0, , ,V
Theoretical framework Theoretical framework
• Hartree-Fock-Bogoliubov theory (Code developed by M.V. Stoitsov et al.)• Axially symmetric quadrupole deformation• Skyrme forces (SLy4, SkM*, SkP, LNS) • Energy cutoff = 60 MeV for pairing
2
21 34
, 0
10 1
10
0 021pair
n p
rVH r
Procedures for parameter optimization
Data: G. Audi et al., NPA729, 3 (2003)exp: 3-point mass difference formula
0Determinat
Our go
ion
al
of
0
1 0 02 0
0 1 2
0
0
0
For each
, ,
optimizing
in regions of open-shell nuclei
,
, , ,
tot
totAt las
V
V
t
0 1 2 04 parameters; , , , V
2
21 34
, 0
10 1
10
0 021pair
n p
rVH r
2
21 34
, 0
10 1
10
0 021pair
n p
rVH r
21 1, ,
21 1, ,
21 1, ,
Extrapolation: Zone1 → Zone2, 3
0 00.8 minimum of tot
- Skyrme SLy4 case -
Specific examples in Zone3 (outside fitting)
Sn
Pb
0
1 0 2 0 0 0
0.75 (SLy4 force)
, , Zon i en 1V
Verifying for typical Skyrme forces0 0.8
Connection to drip-line region (low- limit)
2 2
2
Pairing strength in vaccum
22,
2nn cut
vac cutnn cut
a mEV k
m a k
n-n scattering length
18.5 MeVnna
(à la Bertsch & Esbensen)
Validity of assumption V0=Vvac
ComparisonProcedure 1; V0=Vvac + optimized (0, 1, 2)Procedure 2; Optimized (0, 1, 2, V0)
Results m*/m=0.7~0.8 ⇒ Good coincidence Procedure 1 ~ Procedure 2
m*/m=1.0 ⇒ tot of 1 & 2 are comparable, although the minimum positions are different.
☺☹
Conclusion
a. Strong –dep. (0 ～ 0.8) for typical Skyrme forcesb. 1–tems should be included.c. Connection to drip-line regions, if m*/m=0.7~0.8.
0 global fitting 18.5 fmvac nnV V a
1. -dependence of the pairing part of local energy density functional is studied.
2. All even-even nuclei with experimental data are analyzed by Skyrme-HFB.
☹☺
Definition of pairing gap
3 3
3 1 1
2
(same for proton)
N NN
Pairing gap: A-dependence only
, parametersA
Survey of 1(opt.) : pairing and effective mass
-dependence of effective masses
*
*
**
*
*
* *
s s
s s
v
v
n
p
m
m
m
m
m
m
m m
m m
m m m
mm m
12 Skyrme parametersSKT6 (=0.00), SKO’ (0.14), SKO (0.17), SLy4 (0.25), SLy5 (0.25), SKI1 (0.25), SKI4 (0.25), BSK17 (0.28), SKP (0.36), LNS (0.37), SGII (0.49), SkM* (0.53)
.1 0 2 @ , 0.5,2.5opt
*
1*
/
:
/
)
1
(v
sm m
m m
Isoscalar effective mass :
Isovector effective mass :
Enhancment factor of TRK sum rule
n p
1*
( :
1
)
/vm m
Isovector effective mass :
Enhancment factor of TRK sum rule
† †''
, ' 0
*1
*
*
,exp
ˆ
exp 1/
( : constant single-particle level density)
sensitive to
Schematic model
[our case] ; ;
PPpair k kk k
k k
L
L
L
V
H G c c c c
G
mG G
m
mmG
m
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