Semiclassical Approach to Pairing of Drip-Line Nuclei P. Schuck IPN - - PowerPoint PPT Presentation

Semiclassical Approach to Pairing of Drip-Line Nuclei

• P. Schuck

IPN Orsay and LPMMC Grenoble

CONTENT Thomas-Fermi approximation for gap, TF-BCS Drip-line situations, drip-line nuclei BCS vs HFB

• corrections to LDA

Applications of TF-BCS and TF-HFB Conclusions

Drip-line nuclei

L R 10 20 30 40 50 60 70 80 90 100

R

20 40 60

U(R) r=ω2/ω1=0.1 r=0.2 r=0.4 r=1

Overflow situations of superfluid fermions in finite mean field potential → nuclei (drip line), nuclei in Wigner Seitz cells in crust of neutron stars, Cold atoms etc.

: TF; black: quantal; slab with pocket, depth = - 40 MeV and cut off + 50 MeV

• 40
• 30
• 20
• 10

10 20 30 40 50

µ (MeV)

0.2 0.4 0.6 0.8 1 1.2 1.4

∆ (µ) MeV

Thomas-Fermi Quantal

X.Vinyas, P.S., PRL 107

Thomas-Fermi approach to pairing. TF-BCS Gap equation ∆n =

• n′

n|v|n′¯ n′ ∆n′ 2

• (en′ − µ)2 + ∆2

n′

(1) Time reversal invariance: n¯ n|rr′ ≡ r|nn|r′ ≡ ρn(r, r′) TF: ρn(r, r′) → δ(En − p2

2m − U(R))

TF-gap equation ∆(E) =

• E ′ g(E ′)V (E, E ′)

∆(E ′) 2

• (E ′ − µ)2 + ∆2(E ′)

κn = unvn → κ(r, p) =

• dEκ(E)δ(E − p2

2m − U(R)) + O(2)

HFB calculation in double HO-potential

10 15 20 25 30 µ

0.4 0.8

Eqp r=0.1 r=0.2 r=0.4 r=1

• X. Vinyas, P.S. et al., PRA 90. Figure prepared by A. Pastore.

Evolution of the lowest HFB-quasi-particle energy as a function of µ. Thomas-Fermi pour HFB later...

HFB for some drip-line nuclei

20 40 60 80 100 N 0.5 1 1.5 ∆

n [MeV]

Ca SLy4

∆LCS <∆ uv> ∆exp

(a)

40 60 80 100 120 N 0.5 1 1.5 ∆

n [MeV]

Mo SLy4 (e)

∆LCS <∆ uv> exp

Vertical broken line corresponds to drip-line. SLy4 force is used. Pastore, Margueron, P.S., Vinyas, PRC 88.

LDA can become very bad ...

5 10 15 20 25 30 35 40 45 R (fm) 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2 ∆(R) (MeV)

LDA TF-BCS

A = 500 Z = 50

Wigner-Seitz cell; SLy4 + pairing

HFB BCS + TF-BCS

10 20 30 40 50 R [fm] 1 2 3 ∆

n LOC(R) [MeV] 250Zr 500Zr 1100Sn 1800Sn

10 20 30 40 50 R [fm] 1 2 3 ∆

n LOC(R) [MeV] 1800Sn 1100Sn 500Zr 250Zr

Wigner-Seitz cells with SLy4. Some differences between HFB (left) and BCS (right) can be seen. TF-BCS: broken lines.

However, BCS can also become quite wrong ...

80 120 160 200

µF [MeV]

2 4 6 8 10 12 14 16 18 20

Eqp [MeV] HFB BCS

Deep Woods-Saxon potential from steep violet to soft (HO-like) red

What is reason for failure?

• Ueff. = U(R) − µ + ∆2(R)

Eqp

4 8 12 16 20

R [fm]

• 150
• 100
• 50

Ueff(R) [MeV] µ=120 MeV µ=140 MeV µ=160 MeV µ=180 MeV µ=200 Mev

4 8 12 16 20

R [fm]

• 100
• 50

Ueff(R) [MeV] µ=120 MeV µ=140 MeV µ=160 MeV µ=180 MeV µ=200 Mev

Pockets of different Woods-Saxon potentials with varying width parameters a1 = 1, 11

5 10 15 20

R

• 0.4
• 0.2

0.2 0.4

vnl(R)

r=0.1 µ=24 µ=26

Anomalous density. Wigner-Kirkwood -expansion: κ = κ0 ≡ κLDA + κ2 (2) κ2 = − h 4E 3 − 3∆2 4E 5 + 5h2∆2 12E 7 2p2 m 2∇2∆ 4m + 3h∆ 4E 5 − 5h∆3 4E 7 + ∆ 12E 5 2p2 m 2(∇∆)2 4m + ∆ 4E 3 − 3∆3 8E 5 2 4m2∇2U − h∆ 2E 5 − 5h∆3 4E 7 2 4m

• (∇U)2 + +1

3 2p2 m ∇2U

• +

1 2E 3 − 5h2∆2 2E 7 2 4m

• ∇U · ∇∆
• ,

(3)

Strinati − 1 g ∆(r) =

• dk

(2π)3 ∆(r) 2E(r, 0, k) + 1 2

• dk

(2π)3 ek − µ(r) 2E 3(r, 0, k)∇2

r

2∆(r) 4m

• .

(4)

0.5 1 5 10 15 20 j(ρ)/jmax (kFaF)-1=-1 T=0 0.5 1 j(ρ)/jmax (kFaF)-1=-1 T=0.5Tc 0.5 1 j(ρ)/jmax (kFaF) =-1 T=0.9Tc 2 4 6 8 kFρ (kFaF)-1=0 T=0 (kFaF)-1=0 T=0.5Tc (kFaF) =0 T=0.9Tc 2 4 6 (kFaF)-1=1 T=0 (kFaF)-1=1 T=0.5Tc (kFaF) =1 T=0.9Tc

0.2 0.4 0.6 0.8 1 1.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Θ/Θcl Ω 1 3 19 43 39 (kFaF)-1= -1 (kFaF)-1= 0 (kFaF)-1= 1 5e-05 0.0001 0.00015 0.0002 0.01 0.02 0.03 0.04 Θ/Θcl Ω x y 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Conclusions: BCS good approximation for containers with steep surface. TF-BCS very good approximation to quantal BCS. BCS not applicable for wide HO. Pairing strongly quenched at the drip.

• expansion for HFB.

Applications of TF-BCS and TF-HFB

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