Pairing schemes for HFB calculations: Results and discussions K. - - PowerPoint PPT Presentation

Pairing schemes for HFB calculations: Results and discussions

• K. Bennaceur, IPNL/UCB Lyon-1 – CEA/ESNT
• T. Duguet, NSCL – MSU
• P. Bonche, CEA/SPhT
• Zero range pairing:

– density dependence – regularization

• Regularization scheme and pairing at low density
• Microscopic zero range pairing force along the Cr isotopic chaine
• Conclusion
• ✁✂

• ☎✆

• ✁✂

Pairing – Density dependence

SLy4ρ (volume)

• r

SLy4δρ (surface)

20 40 60 80 100 120 140 160 180 200 220 240 260

N

10 20 30 40 50 60 70 80 90 100 110

Z

Particle drip lines

26 30 34 38 42 46 50 8 10 12 14 16 18 20

µN → 0 large density of states around µN

• 60
• 50
• 40
• 30
• 20
• 10

10 20 30 eq [MeV] 0.0 0.2 0.5 0.8 1.0

v2( eq)

168Sn with SLy4ρ

µN = −0.608 MeV ∆N = 1.031 MeV

• ✁✂

• ☎✆

• ✁✂

Microscopic pairing Finite range (FR)and zero range (ZFR)

k|D(kF , P, 0)|k′ = λv(k)h(kF , P, 0)v(k′) → Density dependence: h(kF , P, 0)

Finite range Zero range approximation

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

kF (fm

• 1)

1 2 3 4 5 6 7

C (kF)

C (kF) = h (kF,0,0) Fit in (kF)

n/2 => (5 terms)

Fit in (ln kF)

n => (3 terms)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

kF (fm

• 1)

C

zr (kF)

Fit in (kF)

n/2 => (2 terms)

Fit in (lnkF)

n => (3 terms)

∼ cste ∼ Surf. + Vol.

h(low density) = DDDI

• ✁✂

• ☎✆

• ✁✂

Zero range effective interaction

V pp

eff (r) = t′

• 1 − η

ρ(r) ρ0 γ δ(r)

• η = 0

→ “volume” pairing

• η = 1

→ “surface” pairing

• η = 1/2

→ “mixed” pairing

• Divergence of E → cut-off Ec

DFT (V, S or M) pairing: mixed with Ec = 60 MeV

(Dobaczewski, Flocard, Treiner, NPA ’84)

ULB pairing: surface with Ec = ±5 MeV

• ✁✂

• ☎✆

• ✁✂

Regularization

• Cf. A. Bulgac : nucl-th/0109083, nucl-th/0302007
• Vpp ∝ δ(r) =

⇒ Ep = +∞

⇐ = ˜ ρ(r1, r2) ∝

r1→r2 1 |r1−r2|

• Infinite matter

˜ ρ(r1, r2) − →

r1→ r2

+∞ = ˜ ρreg(r1, r2) +

m∆eikF |r1−r2| 4π2|r1−r2|

< +∞ +∞

• Nuclei

∆(r) = t′

0 ˜

ρreg(r) ≡ t′

0,eff[ρ] ˜

ρ(r) = ⇒ more complex density dependence

Regularized DFT pairing : “RDFT” (V, S or M)

• ✁✂

• ☎✆

• ✁✂

Link to an effective pairing interaction

• V ≡ V

1S0

sep → ... Cf. Thomas(D. & L.)’s presentations ...

∆i = −

• m

i¯ ı|T (0)|m ¯ m2(1 − ρm)ρm ∆m 2Em = −

• m

i¯ ı|D(0)|m ¯ m2ρm ∆m 2Em V eff(ρq) cut-off → 2v2

m

“ZFR” pairing = lim

α→0 FR(range α) K + Ep < +∞ , but K and Ep diverge

• ✁✂

• ☎✆

• ✁✂

Convergence

10 30 50 70 90 110 130

Emax [MeV]

• 40
• 20

20 40 E = -1018.968 MeV FR

• 40
• 20

20 40

Etot(Emax) - Etot(130 MeV) [keV]

E = -1019.470 MeV ZFR

• 40
• 20

20 40 E = -1018.356 MeV DFT

• 40
• 20

20 40 E = -1017.862 MeV RDFT 10 30 50 70 90 110 130

Emax [MeV]

500 1000 1500 2000 Ep = -13.912 MeV FR 500 1000 1500 2000

Ep(Emax) - Ep(130 MeV) [keV]

Ep = -21.992 MeV ZFR 500 1000 1500 2000 Ep = -12.845 MeV DFT 500 1000 1500 2000 Ep = -11.798 MeV RDFT

SLy5

120Sn

• ✁✂

• ☎✆

• ✁✂

Summary and recipes

• FR

→ Emax ∼ 30 MeV Forces with regularization:

• ULB

→ Emax ∼ EF + 5 MeV

• DFTx

→ Emax ∼ Ec + 30 MeV ∼ 90 MeV (= Ec if direct integration)

• RDFTx →

Emax ∼ 60 MeV

(staggering)

• ZFR

→ Emax ∼ 90 MeV Strengths adjusted to minimize

• ∆Nκ − ∆N(5)
• for 120Sn, 198Pb and 212Pb.
• ✁✂

• ☎✆

• ✁✂

Computation time

10 30 50 70 90 110 130

Emax [fm]

10 20 30 40 50 60 70 80

Time [s]

FR ZFR ULB DFT RDFT 20 30 40

Rbox [fm]

50 100 150 200 250 300

Time [s]

FR ZFR ULB DFT RDFT

• ✁✂

• ☎✆

• ✁✂

So...

Microscopic regularizations: – require modifications of the codes (not too hard...) – Converge at rather high energy (60 to 90 MeV) Are they really useful ? Do they change the physics ?

• ✁✂

• ☎✆

• ✁✂

Effect of the different regularization schemes

ULB, DFT(V,S,M), ZFR → different density dependences... ZFR very strong at low density

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

kF (fm

• 1)

C

zr (kF)

Fit in (kF)

n/2 => (2 terms)

Fit in (lnkF)

n => (3 terms)

• use of the same density dependence V pp

eff ≡ t′

1 − η ρ(r) ρ0

γ

δ(r) for each regularization scheme:

ULB = cut off (narrow window) DFT = cut off (wide window) “R” = Bulgac & Yu. “2v2” = same as ZFR

• γ < 1

⇒ pairing enhancement at low density

• ✁✂

• ☎✆

• ✁✂

Density dependences – η = 1/2

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

[fm-3]

200 400 600 800 1000 1200

f( ) [MeV]

= 1 = 1/2 = 1/4 = 1/6 fZFR

5 10

r [fm]

200 400 600 800

f[ (r)] [MeV]

In a nucleus

η = 1/2 (mixed pairing): → the main part of the gap in nuclei comes from the inside → the strength can not be very strong at low density (surface)

• ✁✂

• ☎✆

• ✁✂

Density dependences – η = 1

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

[fm-3]

400 800 1200 1600 2000 2400

f( ) [MeV]

= 1 = 1/2 = 1/4 = 1/6 fZFR

5 10

r [fm]

1500 3000 4500

f[ (r)] [MeV]

In a nucleus

γ < 1 − → stronger pairing at    low density nucleus surface

• ✁✂

• ☎✆

• ✁✂

Density dependence – Isospin

2 4 6 8 10 12

r [fm]

0.00 0.04 0.08 0.12 0.16

(r) [fm-3]

0.16 0.16 x 70/120 0.16 x 50/120

• 1 −

ρ ρ0

• γ
• r
• 1 −

ρq ρq0

• γ
• ✁✂

• ☎✆

• ✁✂

Density dependence – Isospin

2 4 6 8 10 12

r [fm]

0.00 0.04 0.08 0.12 0.16

(r) [fm-3]

0.16 0.16 x 70/120 0.16 x 50/120

• 1 −

ρ ρ0

• γ
• r
• 1 −

ρq ρq0

• γ
• ✁✂

• ☎✆

• ✁✂

Surface pairing – γ = 1

Effect of the regularization scheme: Gaps in tin isotopes

50 60 70 80 90 100 110 120

N

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

< N> [MeV]

ULB DFTS RDFTS

(Bulgac & Yu)

DFTS+v2

• ULB ≃ DFTS ≃ RDFTS
• DFTS+2v2 → pp Vs. hh asymmetry
• ✁✂

• ☎✆

• ✁✂

Effect of the regularization scheme: Pairing fields

2 4 6 8 10 12 14 16

r [fm]

• 1.8
• 1.3
• 0.8
• 0.3

0.2

N(r) [MeV]

102Sn

ULB DFTS RDFTS DFTS+v2

2 4 6 8 10 12 14 16

r [fm]

• 3.0
• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

0.0 0.5

N(r) [MeV]

120Sn

ULB DFTS RDFTS DFTS+v2

2 4 6 8 10 12 14 16

r [fm]

• 3.5
• 3.0
• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

0.0 0.5

N(r) [MeV]

150Sn

ULB DFTS RDFTS DFTS+v2

2 4 6 8 10 12 14 16

r [fm]

• 3.5
• 3.0
• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

0.0 0.5

N(r) [MeV]

170Sn

ULB DFTS RDFTS DFTS+v2

Longer tail of the pairing field with DFTS in neutron rich nuclei

• ✁✂

• ☎✆

• ✁✂

Effect of the regularization scheme: 170Sn

2 4 6 8 10 12 14 16

r [fm]

• 3.5
• 3.0
• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

0.0 0.5

N(r) [MeV]

170Sn

ULB DFTS RDFTS DFTS+v2

2 4 6 8 10 12 14 16

r [fm]

0.000 0.004 0.008 0.012 0.016 0.020

[fm-3]

n

~

ULB DFTS RDFTS DFTS+v2

2 4 6 8 10 12 14 16

r [fm]

0.00 0.02 0.04 0.06 0.08 0.10

[fm-3]

n

ULB DFTS RDFTS DFTS+v2

2 4 6 8 10 12 14 16

r [fm]

10-6 10-5 10-4 10-3 10-2 10-1

[fm-3]

n

ULB DFTS RDFTS DFTS+v2

almost no effect on the normal density

• ✁✂

• ☎✆

• ✁✂

... So, the question is: DFTS leads to a more extended pairing field. (cut off at 60 MeV) Is it a bug or a feature ?

What happens if we eventually go to a stronger interaction at low density ? → γ from 1 to 1/6 (stronger pairing at low density) → the cut off Ec from a ULB type (5 MeV above EF ) to 60 MeV

• ✁✂

• ☎✆

• ✁✂

170Sn γ = 1/2

The strength of the pairing is readjusted for larger Ec

2 4 6 8 10 12 14 16 18 20

r [fm]

• 3.5
• 3.0
• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

0.0

N(r) [MeV] EC = 5 MeV EC = 10 MeV EC = 20 MeV

2 4 6 8 10 12 14 16 18 20

r [fm]

0.000 0.005 0.010 0.015

[fm-3]

n

~

EC = 5 MeV EC = 10 MeV EC = 20 MeV

2 4 6 8 10 12 14 16 18 20

r [fm]

0.00 0.02 0.04 0.06 0.08 0.10

[fm-3]

n

EC = 5 MeV EC = 10 MeV EC = 20 MeV

2 4 6 8 10 12 14 16 18 20

r [fm]

10-6 10-5 10-4 10-3 10-2 10-1

[fm-3]

n

EC = 5 MeV EC = 10 MeV EC = 20 MeV

• ✁✂

• ☎✆

• ✁✂

170Sn γ = 1/6

The strength of the pairing is reduced for larger Ec

2 4 6 8 10 12 14 16 18 20

r [fm]

• 4.0
• 3.5
• 3.0
• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

0.0

N(r) [MeV] EC = 5 MeV EC = 10 MeV EC = 20 MeV

2 4 6 8 10 12 14 16 18 20

r [fm]

0.000 0.005 0.010 0.015 0.020

[fm-3]

n

~

EC = 5 MeV EC = 10 MeV EC = 20 MeV

2 4 6 8 10 12 14 16 18 20

r [fm]

0.00 0.02 0.04 0.06 0.08 0.10

[fm-3]

n

EC = 5 MeV EC = 10 MeV EC = 20 MeV

2 4 6 8 10 12 14 16 18 20

r [fm]

10-5 10-4 10-3 10-2 10-1

[fm-3]

n

EC = 5 MeV EC = 10 MeV EC = 20 MeV

• ✁✂

• ☎✆

• ✁✂

170Sn γ = 1/6

The strength of the pairing is reduced for larger Ec

2 4 6 8 10 12 14 16 18 20

r [fm]

• 4.0
• 3.5
• 3.0
• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

0.0

N(r) [MeV] EC = 5 MeV EC = 10 MeV EC = 20 MeV Surf.+2v2

2 4 6 8 10 12 14 16 18 20

r [fm]

0.000 0.005 0.010 0.015 0.020

[fm-3]

n

~

EC = 5 MeV EC = 10 MeV EC = 20 MeV Surf.+2v2

2 4 6 8 10 12 14 16 18 20

r [fm]

0.00 0.02 0.04 0.06 0.08 0.10

[fm-3]

n

EC = 5 MeV EC = 10 MeV EC = 20 MeV Surf.+2v2

2 4 6 8 10 12 14 16 18 20

r [fm]

10-5 10-4 10-3 10-2 10-1

[fm-3]

n

EC = 5 MeV EC = 10 MeV EC = 20 MeV Surf.+2v2

• ✁✂

• ☎✆

• ✁✂

170Sn – Pairing fields Vs. Rbox

Surface pairing with ρ1/6, cut off EF ± 5 MeV

5 10 15 20 25 30 35 40

r [fm]

• 2.0
• 1.5
• 1.0
• 0.5

0.0

N(r) [MeV]

Rbox =

= -0.189 MeV, Ep = -3.742 MeV

20 fm

= -0.206 MeV, Ep = -4.882 MeV

25 fm

= -0.240 MeV, Ep = -6.921 MeV

30 fm

= -0.278 MeV, Ep = -8.734 MeV

35 fm

= -0.311 MeV, Ep = -10.246 MeV

40 fm

• same behaviour with ρ1/2
• worse if the pairing active space is enlarged...
• ✁✂

• ☎✆

• ✁✂

Occupation of the canonical states

170Sn, Ec = 5 MeV

• 50
• 30
• 10

10 30 50 70

can [MeV]

10-6 10-5 10-4 10-3 10-2 10-1 100

v2

• Surf. + Ec = 5 MeV
• Surf. + 2v2
• ✁✂

• ☎✆

• ✁✂

Occupation of the canonical states

170Sn, Ec = 5 MeV

• 20
• 15
• 10
• 5

5 10 15 20

can [MeV]

0.0 0.2 0.4 0.6 0.8 1.0

v2

• Surf. + Ec = 5 MeV
• Surf. + 2v2
• ✁✂

• ☎✆

• ✁✂

Occupation of the canonical states

170Sn, Ec = 10 MeV

• 50
• 30
• 10

10 30 50 70

can [MeV]

10-6 10-5 10-4 10-3 10-2 10-1 100

v2

• Surf. + Ec = 10 MeV
• Surf. + 2v2
• ✁✂

• ☎✆

• ✁✂

Occupation of the canonical states

170Sn, Ec = 20 MeV

• 50
• 30
• 10

10 30 50 70

can [MeV]

10-6 10-5 10-4 10-3 10-2 10-1 100

v2

• Surf. + Ec = 20 MeV
• Surf. + 2v2
• ✁✂

• ☎✆

• ✁✂

Occupation of the canonical states

170Sn, Ec = 40 MeV

• 50
• 30
• 10

10 30 50 70

can [MeV]

10-6 10-5 10-4 10-3 10-2 10-1 100

v2

• Surf. + Ec = 40 MeV
• Surf. + 2v2
• ✁✂

• ☎✆

• ✁✂

ZFR density dependence and cut off Ec = ±5 MeV

4 8 12 16 20

r [fm]

• 2.0
• 1.5
• 1.0
• 0.5

0.0

N(r) [MeV]

102Sn

4 8 12 16 20

r [fm]

• 2.0
• 1.5
• 1.0
• 0.5

0.0

N(r) [MeV]

120Sn

4 8 12 16 20

r [fm]

• 2.0
• 1.5
• 1.0
• 0.5

0.0

N(r) [MeV]

140Sn

4 8 12 16 20

r [fm]

• 2.0
• 1.5
• 1.0
• 0.5

0.0

N(r) [MeV]

142Sn

4 8 12 16 20

r [fm]

• 2.0
• 1.5
• 1.0
• 0.5

0.0

N(r) [MeV]

150Sn

4 8 12 16 20

r [fm]

• 2.0
• 1.5
• 1.0
• 0.5

0.0

N(r) [MeV]

170Sn

• ✁✂

• ☎✆

• ✁✂

Surface pairing with γ = 1

2 4 6 8 10 12 14 16

r [fm]

• 3.5
• 3.0
• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

0.0 0.5

N(r) [MeV]

150Sn

ULB DFTS RDFTS DFTS+v2

2 4 6 8 10 12 14 16

r [fm]

• 3.5
• 3.0
• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

0.0 0.5

N(r) [MeV]

170Sn

ULB DFTS RDFTS DFTS+v2

not a feature...

• ✁✂

• ☎✆

• ✁✂

ZFR Vs. schematic surface forces

2 4 6 8 10 12 14 16 18 20

r [fm]

• 3.5
• 3.0
• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

0.0

N(r) [MeV]

170Sn

• Surf. + 2v2 ( =1)
• Surf. + 2v2 ( =1/6)
• Surf. + Ec=5 MeV ( =1/6)

ZFR

ZFR: Strong enhancement of the pairing field at the nucleus surface

• ✁✂

• ☎✆

• ✁✂

ZFR Vs. schematic surface forces

2 4 6 8 10 12 14 16 18 20

r [fm]

• 60
• 50
• 40
• 30
• 20
• 10

UN(r) [MeV]

170Sn

• Surf. + 2v2 ( =1)
• Surf. + 2v2 ( =1/6)
• Surf. + Ec=5 MeV ( =1/6)

ZFR

The HF field is almost the same for all pairing forces !

• ✁✂

• ☎✆

• ✁✂

Summary

• The usual regularization methods can not handle a pairing force which is

strong at low density.

• Even the “standard” surface pairing (γ = 1) is strong at low density.
• The two microscopic regulators presented here do the job.
• The “2v2” regulator comes with a density dependence without free

parameter and which is strong at low density.

• ✁✂

• ☎✆

• ✁✂

Results for exotic nuclei

Exotic nucleus with “halo” + strong pairing at low density + microscopic regularization (2v2 or Bulgac & Yu) ⇒ Density more diffuse at the surface ⇒ Increase the pairing fields at the surface ⇒ Drawback: Pairing correlations tend to reduce the halo ⇒ Regularization confine the pairing density

= ⇒ Highly non trivial effect...

• ✁✂

• ☎✆

• ✁✂

Cr isotopes

2 4 6 8 10 12 14 16

r [fm]

• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

0.0

N(r) [MeV]

ZFR

72Cr 76Cr 78Cr 80Cr 2 4 6 8 10 12 14 16

r [fm]

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016

[fm-3]

ZFR

~

72Cr 76Cr 78Cr 80Cr 2 4 6 8 10 12 14 16

r [fm]

0.000 0.020 0.040 0.060 0.080 0.100

[fm-3]

ZFR

72Cr 76Cr 78Cr 80Cr 2 4 6 8 10 12 14 16

r [fm]

10-6 10-5 10-4 10-3 10-2 10-1

[fm-3]

ZFR

72Cr 76Cr 78Cr 80Cr

˜ ρ enhanced at the surface of the nucleus

• ✁✂

• ☎✆

• ✁✂

Cr isotopes

2 4 6 8 10 12 14 16

r [fm]

• 2.5
• 2.0
• 1.5
• 1.0
• 0.5

0.0

N(r) [MeV]

80Cr

• Surf. + 2v2 ( =1)
• Surf. + 2v2 ( =1/6)

ZFR

• ✁✂

• ☎✆

• ✁✂

Neutron radii in Cr

44 46 48 50 52 54 56

N

4.1 4.2 4.3 4.4 4.5 4.6

<rN> [fm]

= 1 = 1/6

Cr

• ✁✂

• ☎✆

• ✁✂

Neutron radii in Cr

44 46 48 50 52 54 56

N

4.1 4.2 4.3 4.4 4.5 4.6

<rN> [fm]

= 1 = 1/6 fZFR[ ]

Cr

• ✁✂

• ☎✆

• ✁✂

Conclusion

• Microscopic finite range pairing (FR) and its zero range limit (ZFR) lead

to a strong pairing interaction at low density.

• Microscopic regulators and cut offs are not equivalent tools.
• Cut offs can not handle the strong intensity of the pairing interaction at

low density.

• A pairing stronger at low density hardly modify the density but has a

dramatic effect on the anomalous density.

• ✁✂

• ☎✆

• ✁✂