Global fitting of pairing density functional; the isoscalar-density dependence revisited

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