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 60 U(R) 40 L r= ω 2 / ω 1 =0.1 R r=0.2 r=0.4 20 r=1 0 10 20 30 40 50 60 70 80 90 100 R 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 1.4 Thomas-Fermi 1.2 Quantal 1 ∆ (µ) MeV 0.8 0.6 0.4 0.2 0 -40 -30 -20 -10 0 10 20 30 40 50 µ (MeV) X.Vinyas, P.S., PRL 107

Thomas-Fermi approach to pairing. TF-BCS Gap equation ∆ n ′ � � n | v | n ′ ¯ n ′ � ∆ n = (1) ( e n ′ − µ ) 2 + ∆ 2 � 2 n ′ n ′ Time reversal invariance: n | rr ′ � ≡ � r | n �� n | r ′ � ≡ ρ n ( r , r ′ ) � n ¯ TF: ρ n ( r , r ′ ) → δ ( E n − p 2 2 m − U ( R )) TF-gap equation � ∆( E ′ ) E ′ g ( E ′ ) V ( E , E ′ ) ∆( E ) = ( E ′ − µ ) 2 + ∆ 2 ( E ′ ) � 2 dE κ ( E ) δ ( E − p 2 � + O ( � 2 ) κ n = u n v n → κ ( r , p ) = 2 m − U ( R ))

HFB calculation in double HO-potential r=0.1 r=0.2 r=0.4 r=1 0.8 E qp 0.4 10 15 20 25 30 µ 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 ∆ LCS (a) ∆ LCS 1.5 1.5 < ∆ uv> < ∆ uv> ∆ exp exp n [MeV] n [MeV] Mo SLy4 1 Ca SLy4 1 (e) ∆ ∆ 0.5 0.5 20 40 60 80 100 40 60 80 100 120 N N Vertical broken line corresponds to drip-line. SLy4 force is used. Pastore, Margueron, P.S., Vinyas, PRC 88.

LDA can become very bad ... 2 1,8 1,6 LDA 1,4 A = 500 Z = 50 1,2 ∆ (R) (MeV) 1 0,8 0,6 0,4 TF-BCS 0,2 0 0 5 10 15 20 25 30 35 40 45 R (fm) Wigner-Seitz cell; SLy4 + pairing

HFB BCS + TF-BCS 3 3 1800 Sn 1800 Sn LOC (R) [MeV] LOC (R) [MeV] 2 2 1100 Sn 1100 Sn 1 1 500 Zr 500 Zr n n ∆ ∆ 250 Zr 250 Zr 0 10 20 30 40 50 0 10 20 30 40 50 R [fm] R [fm] 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 ... 20 18 HFB BCS 16 14 E qp [MeV] 12 10 8 6 4 2 0 80 120 160 200 µ F [MeV] Deep Woods-Saxon potential from steep violet to soft (HO-like) red

What is reason for failure? U eff . = U ( R ) − µ + ∆ 2 ( R ) E qp 0 0 µ =120 MeV µ =120 MeV µ =140 MeV µ =140 MeV µ =160 MeV µ =160 MeV -50 U eff (R) [MeV] U eff (R) [MeV] µ =180 MeV µ =180 MeV -50 µ =200 Mev µ =200 Mev -100 -100 -150 0 4 8 12 16 20 0 4 8 12 16 20 R [fm] R [fm] Pockets of different Woods-Saxon potentials with varying width parameters a1 = 1, 11

µ=24 0.4 µ=26 0.2 v nl (R) 0 -0.2 r=0.1 -0.4 0 5 10 15 20 R

Anomalous density. Wigner-Kirkwood � -expansion: κ = κ 0 ≡ κ LDA + κ 2 (2) � h 4 E 3 − 3∆ 2 4 E 5 + 5 h 2 ∆ 2 � 2 p 2 � � 2 ∇ 2 ∆ κ 2 = − 12 E 7 m 4 m 4 E 5 − 5 h ∆ 3 � 2 p 2 � � 2 ( ∇ ∆) 2 � 3 h ∆ ∆ + 4 E 7 + 12 E 5 m 4 m � ∆ � � 2 4 E 3 − 3∆ 3 4 m 2 ∇ 2 U + 8 E 5 � h ∆ � � 2 2 E 5 − 5 h ∆ 3 � 2 p 2 ( ∇ U ) 2 + +1 m ∇ 2 U � � − 4 E 7 4 m 3 � 1 � � 2 2 E 3 − 5 h 2 ∆ 2 � � + ∇ U · ∇ ∆ (3) , 2 E 7 4 m

Strinati � e k − µ ( r ) � 2 ∆( r ) − 1 2 E ( r , 0 , k ) + 1 ∆( r ) � � dk � dk 2 E 3 ( r , 0 , k ) ∇ 2 g ∆( r ) = (4) . r (2 π ) 3 2 (2 π ) 3 4 m 1.2 (k F a F ) -1 = -1 (k F a F ) -1 = 0 (k F a F ) =-1 (k F a F ) =0 (k F a F ) =1 39 (k F a F ) -1 = 1 1 T=0.9T c T=0.9T c T=0.9T c 43 1 19 0.8 j( ρ )/j max 0.0002 Θ / Θ cl 0.6 0.00015 0.5 Θ / Θ cl 3 0.0001 0.4 5e-05 0 0.2 0 0.01 0.02 0.03 0.04 0 (k F a F ) -1 =-1 (k F a F ) -1 =0 (k F a F ) -1 =1 Ω 1 T=0.5T c T=0.5T c T=0.5T c 0 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Ω j( ρ )/j max 0.9 0.8 0.5 0.7 0.6 0 (k F a F ) -1 =-1 (k F a F ) -1 =0 (k F a F ) -1 =1 0.5 T=0 T=0 T=0 y 1 0.4 0.3 j( ρ )/j max 0.2 0.5 0.1 0 x 0 0 5 10 15 20 0 2 4 6 8 0 2 4 6 k F ρ

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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