✢ ✓ ✎ ✟ ☛ ✌ ✔ ☎ ✝ ✔ ☎✆ ✄ ✁✂ � ✒ ✑ ✟ ✁ ✂ ☛ ✍ ✕ ✛✜ ✕ �✚ ✙ ✝ ☎ ✟ ✠ ✟ ✄ ✆ ✔ ✍ ☛ ☛ ✄ ✏ � � ✁✂ ✄ ☎✆ ✝ ✞✟ ✝ ✁✂ ✁ ✝ ☎ ✟ ☎✆ ☛ ✆ ✝ ✎ ✌✍ ✟ ✞✟ ☞ ✟ ✡☛ ☎ ✠ 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 110 Particle 100 drip lines 90 80 SLy4 ρ (volume) 70 60 Z or 50 20 SLy4 δρ (surface) 18 40 16 30 14 12 20 10 8 10 26 30 34 38 42 46 50 0 0 20 40 60 80 100 120 140 160 180 200 220 240 260 N 1.0 0.8 168 Sn with SLy4 ρ µ N → 0 v 2 ( eq ) µ N = − 0 . 608 MeV large density of 0.5 states around µ N � ∆ N � = 1 . 031 MeV 0.2 0.0 -60 -50 -40 -30 -20 -10 0 10 20 30 eq [MeV] ✑✖✕ ✑✘✗
✢ ✎ ✁✂ � ✒ ✑ ✟ ☛ ✄ ☛ ✏ ✆ ✞✟ ✌✍ ☎✆ ✟ ☛ ☞ ✡☛ ☎ ✠ ✟ ✝ ☎✆ � ☎ ✄ ✝ ✁ ✆ ✕ ✛✜ ✕ �✚ ✙ ✝ ✍ ☎ ✠ ✟ ✄ ✔ ✓ ✍ ☛ ✟ ✂ ✔ ✁ ✎ ✟ ☛ ✌ ✔ ☎ ✝ ✟ ✁✂ � ✝ ✞✟ ✝ ☎✆ ✄ ✁✂ Microscopic pairing Finite range (FR)and zero range (ZFR) � k |D ( k F , P, 0) | k ′ � = λv ( k ) h ( k F , P, 0) v ( k ′ ) → Density dependence: h ( k F , P, 0) Finite range Zero range approximation 7 C (k F ) = h (k F ,0,0) zr (k F ) C 6 n/2 => (5 terms) Fit in (k F ) n/2 => (2 terms) Fit in (k F ) 5 n => (3 terms) Fit in (ln k F ) n => (3 terms) Fit in (lnk F ) 4 C (k F ) 3 ∼ cste ∼ Surf. + Vol. 2 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 -1 ) -1 ) k F (fm k F (fm h (low density) � = DDDI ✑✖✕ ✑✘✗
✢ ✄ ☛ ✌ ✔ ☎ ✓ ✝ ☎✆ ✁✂ ✎ � ✒ ✑ ✟ ☛ ✄ ☛ ✏ ✟ ✁ ✞✟ ☎ ✕ ✛✜ ✕ �✚ ✙ ✝ ✍ ✠ ✔ ✟ ✄ ✆ ✔ ✍ ☛ ✟ ✂ ✆ ✎ ✌✍ ☎ ✄ ☎✆ ✝ ✞✟ ✝ ✁✂ ✟ ✁ ✝ � ✁✂ ☎✆ ✝ ✟ ✠ ☎ ✡☛ ☞ ☛ ✟ � Zero range effective interaction � γ � � � ρ ( r ) V pp eff ( r ) = t ′ 1 − η δ ( r ) 0 ρ 0 • η = 0 → “volume” pairing • η = 1 → “surface” pairing • η = 1 / 2 → “mixed” pairing • Divergence of E → cut-off E c DFT (V, S or M) pairing: mixed with E c = 60 MeV ( D obaczewski, F locard, T reiner, NPA ’84) ULB pairing: surface with E c = ± 5 MeV ✑✖✕ ✑✘✗
✢ ☛ ✓ ✝ ☎✆ ✄ ✁✂ � ✒ ✑ ✟ ✄ ✔ ☛ ✏ ✆ ✞✟ ✎ ✌✍ ✟ ☛ ☞ ✡☛ ☎ ✌ ✟ ✟ ✕ ✛✜ ✕ �✚ ✙ ✝ ✍ ☎ ✠ ✄ ☛ ✆ ✔ ✍ ☛ ✟ ✂ ✔ ✁ ✎ ✟ ☎ ✠ ✝ � ☎✆ � ☎ ✝ ✁ ✟ ✁✂ ✝ ✞✟ ✝ ☎✆ ✄ ✁✂ Regularization Cf. A. Bulgac : nucl-th/0109083, nucl-th/0302007 1 • V pp ∝ δ ( r ) = ⇒ E p = + ∞ ⇐ = ˜ ρ ( r 1 , r 2 ) ∝ | r 1 − r 2 | r 1 → r 2 • Infinite matter ρ ( r 1 , r 2 ) ˜ − → + ∞ r 1 → r 2 m ∆ e ikF | r 1 − r 2 | = ρ reg ( r 1 , r 2 ) ˜ + 4 π � 2 | r 1 − r 2 | < + ∞ + ∞ • Nuclei ∆( r ) = t ′ ρ reg ( r ) ≡ t ′ 0 ˜ 0 , eff [ ρ ] ˜ ρ ( r ) = ⇒ more complex density dependence Regularized DFT pairing : “RDFT” (V, S or M) ✑✖✕ ✑✘✗
✢ ✒ ✔ ☎ ✓ ✝ ☎✆ ✄ ✁✂ � ✑ ☛ ✟ ☛ ✄ ☛ ✏ ✆ ✞✟ ✎ ✌✍ ✌ ✟ ☛ ✠ ✕ ✛✜ ✕ �✚ ✙ ✝ ✍ ☎ ✟ ✎ ✄ ✆ ✔ ✍ ☛ ✟ ✂ ✔ ✁ ✟ ☞ ✡☛ ✁ ✄ ☎✆ ✝ ✞✟ ✝ ✁✂ ✟ ✝ � ☎ � ☎✆ ✝ ✟ ✠ ☎ ✁✂ Link to an effective pairing interaction 1 S 0 • V ≡ V sep → ... Cf. Thomas(D. & L.)’s presentations ... ∆ m � ∆ i = − � i ¯ ı |T (0) | m ¯ m � 2(1 − ρ m ) ρ m 2 E m m ∆ m � = − � i ¯ ı |D (0) | m ¯ m � 2 ρ m 2 E m m cut-off → 2 v 2 V eff ( ρ q ) m “ZFR” pairing = lim α → 0 FR(range α ) K + E p < + ∞ , but K and E p diverge ✑✖✕ ✑✘✗
✢ ☎ ✟ ☛ ✄ ☛ ✏ ✆ ✞✟ ✎ ✌✍ ✟ ☛ ☞ ✡☛ ✠ ✒ ✟ ✝ ☎✆ � ☎ ✝ ✁ ✟ ✁✂ ✝ ✞✟ ✝ ☎✆ ✄ ✑ � � ✍ ✕ ✛✜ ✕ �✚ ✙ ✝ ✍ ☎ ✠ ✟ ✄ ✆ ✔ ☛ ✁✂ ✟ ✂ ✔ ✁ ✎ ✟ ☛ ✌ ✔ ☎ ✓ ✝ ☎✆ ✄ ✁✂ Convergence 2000 40 E = -1017.862 MeV 20 1500 RDFT 0 1000 -20 500 RDFT E p = -11.798 MeV SLy5 -40 0 40 2000 DFT E p = -12.845 MeV 120 Sn 20 1500 E tot (E max ) - E tot (130 MeV) [keV] 0 E p (E max ) - E p (130 MeV) [keV] 1000 -20 500 E = -1018.356 MeV DFT -40 0 2000 40 E = -1019.470 MeV ZFR 20 1500 0 1000 ZFR -20 500 E p = -21.992 MeV -40 0 40 2000 E = -1018.968 MeV E p = -13.912 MeV 20 1500 0 1000 FR -20 FR 500 -40 0 10 30 50 70 90 110 130 10 30 50 70 90 110 130 E max [MeV] E max [MeV] ✑✖✕ ✑✘✗
✢ ✁✂ ✌ ✔ ☎ ✓ ✝ ☎✆ ✄ � ✟ ✒ ✑ ✟ ☛ ✄ ☛ ✏ ✆ ☛ ✎ ✎ ✠ ✕ ✛✜ ✕ �✚ ✙ ✝ ✍ ☎ ✟ ✁ ✄ ✆ ✔ ✍ ☛ ✟ ✂ ✔ ✞✟ ✌✍ ✟ ✝ ✄ ☎✆ ✝ ✞✟ ✝ ✁✂ ✟ ✁ ☎ � � ☎✆ ✝ ✟ ✠ ☎ ✡☛ ☞ ☛ ✁✂ Summary and recipes • FR → E max ∼ 30 MeV Forces with regularization: • ULB → E max ∼ E F + 5 MeV • DFTx → E max ∼ E c + 30 MeV ∼ 90 MeV (= E c if direct integration) • RDFTx → E max ∼ 60 MeV (staggering) • ZFR → E max ∼ 90 MeV � � � ∆ N � κ − � ∆ N � (5) � Strengths adjusted to minimize � for 120 Sn, 198 Pb and 212 Pb. ✑✖✕ ✑✘✗
✢ ✏ ☎✆ ✄ ✁✂ � ✒ ✑ ✟ ☛ ✄ ☛ ✆ ✓ ✞✟ ✎ ✌✍ ✟ ☛ ☞ ✡☛ ☎ ✠ ✟ ✝ ☎ � ✄ ✕ ✛✜ ✕ �✚ ✙ ✝ ✍ ☎ ✠ ✟ ✆ ✔ ✔ ✍ ☛ ✟ ✂ ✔ ✁ ✎ ✟ ☛ ✌ ✝ ☎✆ ☎ � ✝ ✁ ✟ ✁✂ ✝ ✞✟ ✝ ☎✆ ✄ ✁✂ Computation time 80 FR 70 ZFR ULB 60 DFT 50 RDFT Time [s] 40 30 20 10 300 0 10 30 50 70 90 110 130 E max [fm] 250 FR ZFR ULB 200 DFT Time [s] RDFT 150 100 50 0 20 30 40 R box [fm] ✑✖✕ ✑✘✗
✢ ✂ � ✁✂ ✄ ☎✆ ✝ ✓ ☎ ✔ ✌ ☛ ✟ ✎ ✁ ✔ ✟ ✟ ✍ ✕ ✛✜ ✕ �✚ ✙ ✝ ☎ ☛ ✠ ✟ ✄ ✆ ✔ ✍ ✒ ✑ ☛ ✝ � ✁✂ ✄ ☎✆ ✝ ✞✟ ✝ ✁✂ ✟ ✁ ✝ ☎ � ☎✆ ✟ ✌✍ ✄ ☛ ✏ ✆ ✞✟ ✠ ✎ ✟ ☛ ☞ ✡☛ ☎ 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... zr (k F ) C n/2 => (2 terms) Fit in (k F ) n => (3 terms) Fit in (lnk F ) ZFR very strong at low density 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 -1 ) k F (fm γ ρ ( r ) use of the same density dependence V pp eff ≡ t ′ 1 − η δ ( r ) 0 ρ 0 for each regularization scheme: ULB = cut off (narrow window) DFT = cut off (wide window) “R” = Bulgac & Yu. “2 v 2 ” = same as ZFR γ < 1 ⇒ pairing enhancement at low density ✑✖✕ ✑✘✗
✢ ✄ ✝ ☎✆ ✄ ✁✂ � ✒ ✑ ✟ ☛ ☛ ☎ ✏ ✆ ✞✟ ✎ ✌✍ ✟ ☛ ☞ ✡☛ ☎ ✓ ✔ ✟ ✄ ✕ ✛✜ ✕ �✚ ✙ ✝ ✍ ☎ ✠ ✟ ✆ ✌ ✔ ✍ ☛ ✟ ✂ ✔ ✁ ✎ ✟ ☛ ✠ ✝ ☎✆ � � ☎ ✝ ✁ ✟ ✁✂ ✝ ✞✟ ✝ ☎✆ ✄ ✁✂ Density dependences – η = 1 / 2 1200 800 f[ (r)] [MeV] 600 1000 400 f( ) [MeV] 800 In a nucleus 200 0 600 0 5 10 r [fm] 400 = 1 f ZFR = 1/2 200 = 1/4 = 1/6 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 [fm -3 ] 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) ✑✖✕ ✑✘✗
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