* 1 1 (i) (ii) Brückner-Hartree-Fock (iii) (HF, BCS, HFB) (iv) (TDHF,TDHFB) (RPA) (QRPA) (v) (vi) *
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1 * 1 1 (i) (ii) Brückner-Hartree-Fock (iii) (HF, BCS, HFB) (iv) (TDHF,TDHFB) (RPA) (QRPA) (v) (vi) *
2 1 ( ) ( ) MeV 1.2 ( ) ( ) s s? Λ s s
3 1.4 ( m ω) ψ(x) = f(x) exp( x 2 /2b 2 ) f(x) b 2 /mω (oscillator length) H(x/b) f(x) H(ξ) E = (n + 1/2) ω 1.5 Ψ(r) = ψ nx (x)ψ ny (y)ψ nz (z) (ω x = ω y = ω z ) 1.6 a (H = T +V ) n n T n = n V n n = (n x, n y, n z ) b n r 2 n c N = n x + n y + n z N max 2 A r 2 = A k=1 r2 k 1.7 e r2 /2b Schrödinger N = 2n + l Laguerre R nl (r) r l L l+1/2 n ((r/b) 2 ) s, p, s + d, p + f,
4 2 0 ˆV ext ( (response) ) (single-particle excitation) (collective excitation) ( ) (elementary excitation) (coherent) 0 n Ω n (random-phase approximation) (normal mode of excitation) n = Ω n 0. (1) H = h sp + V res = ɛ µ c µc µ µ V µνρσ c µc νc σ c ρ. (2) c µ µ ( ) h sp (Hartree-Fock ) (shell model) V res A µνρσ
5 A 0 = c 1 c 2 c A 0, (3) (1p1h) Ω ph = c pc h, (4) 2p2h Ω pp hh = c pc p c h c h, (5) np-nh n n 1p1h ( ph = Ω ph 0 ) 1p1h (ɛ = ɛ p ɛ h = const.) 1p1h (v 0 ) h sp ph = ɛ ph, (6) p h V res ph = v 0, (7) (3) 1p1h { ph } E = ɛ + (N ph 1)v 0 (8) ɛ v 0 N ph 1p1h (8) col ph col = ph C ph ph, (9) C ph 1/N ph N ph C ph v 0 ɛ(1p1h ) v 0 ( ) ( )
6 Ω col = ph C ph Ω ph (10) 1p1h (9) 1p1h N ph C ph 1p1h (Ω col )2 0 1/N ph 2.1 1p1h { ph } (2) (6) (7) 2 1 Q V res = κq Q, (11) (separable interaction) 1 1p1h ph Q 0 = q 0 p h Q ph = p h p h Q ph = 0 v 0 = κq0 2 col Q 0 2 = N ph q0, 2 (12) 1p1h ph Q 0 2 N ph 1 Q
7 1p1h Tamm-Dancoff 2.2 (12) Tamm-Dancoff 1p1h 0 (3) (ground-state correlation) Tamm-Dancoff (energy weighted sum rule) (self-consistent) (RPA) Tamm-Dancoff RPA 3 (8) v 0 ( ) N ph v 0 (11) κ (RPA) Ω col 0 0 Ω col 0 (condensation) (2) V res h sp V res h sp RPA
8 ( ) Hartree-Fock(HF) HF 2 + 1p1h ( ) 0 Q 20 = d 3 r ˆψ (r)r 2 Y 20 ˆψ(r) = µ r 2 Y 20 ν c µc ν (13) µν q = 0 Q 20 0 = 0 (14) ( 1(b)) q (order parameter) HF Hartree-Fock-Bogoliubov(HFB) HF+BCS ( ) Bohr Mottelson Pairing-plus-quadrupole interaction HF HF
9 (a) V(q) (b) V(q) q q 1 q ( ). (a). q = 0. (b). q 0. Brückner-Hartree-Fock Ψ V Φ G V Ψ = G Φ (15) G HF Ψ Φ ( ) { Φ } Ψ Brückner-Hartree-Fock
10 Negele Skyrme-Hartree-Fock (Ring-Schuck ) ( ) ( Skyrme force Gogny force ) Hartree-Fock(-Bogoliubov) Strutinsky shell correction Nilsson Woods-Saxon HF ( Ring-Schuck ) Ẽ E sh E = Ẽ + E sh (16) Ẽ E sh (16) Ẽ E sh Ẽ E sh (shell structure) (shell energy) E sh
11 shell correction 0 0 h sp -Goldstone ( ) - 0 +, 2 +, 4 +, 180 (R) K = 0 ( Bohr-Mottelson ) 3.1 V (x, y, z) = Cr 2 + Dr 4 (C, D > 0) a m (r, θ, φ) V (r = r 0 ) (2 ) r = r 0 b x, y z V (x, y, z) x = y = 0 z = r 0 (x, y, z) = (0, 0, r 0 ) m (x, y, z r 0 ) 2 Schrödinger Morinaga Gugelot ( γ ) Coulomb
12 (16) Ẽ E sh :1 (superdeformed state) 90 3:1 hyperdeformed band ( ) ( 2000 ) Dy I = 68 J 85 2 MeV 1 4 (13) P = d 3 r ˆψ (r) ˆψ (r) (J π = 0 + p-p h-h ) ( ) BCS Ψ = Φ 0 P Φ 0 = 0 (17) ( 1 Ψ ) B = µ C µc µc µ
13 (gauge angle) ( ) ( ) Ψ = /g g BCS BCS ϕ = µ>0(u µ + e iϕ v µ c µc µ) 0 (18) (u 2 µ + v 2 µ = 1) φ(r) ( ) ( ) sin kf r r φ(r) = BCS ψ (r)ψ (0) BCS K 0 k F r ξ 0 π (K 0 (x) 2 Bessel ) ξ 0 = v F /π (coherence length) k F = 1.35 fm 1 = 1 MeV ξ 0 17 fm (mesoscopic system) 4.1 BCS (18) u µ, v µ ϕ BCS ϕ = exp(iϕ ˆN/2) BCS ϕ=0 (19) ˆN/2 4.2 Bogoliubov a µ = u µ c µ v µ c µ, a µ = u µ c µ + v µ c µ, (quasi-particle) {a µ, a ν } = 0, {a µ, a ν} = δ µν, BCS ϕ BCS ϕ Π k a k 0,
14 a µ BCS = 0 BCS (quasi-particle vacuum) HF BCS (u p = v h = 1, u h = v p = 0) BCS HF 1p1h Ω ph = c pc h 2 Ω µν = a µa ν H = k E ka k a k E k 2 (QRPA) QRPA ( -Goldstone ) 0 ( = 0) 1971 backbending Garrett backbending ω bb j ω bb 1.67 / j (20) ( ) ω de/di Inglis ( x ) H H ω rot J x (21)
15 H ω rot (21) H = h sp + V pair HFB 2 H routhian ω rot = 0 1 MeV 1 MeV (ω 1 = ω bb ) 2 j + (j 1) ω rot (j + (j 1)) 2 2 backbending ω bb ω bb (j 1/2) (20) ω rot ω bb ω bb gapless superconductivity (dynamical pairing correlation) BCS 1 (22)
16 2 routhian 164 Er ω 1 ω bb ( ) ( ) Ψ 1 (pairing vibration) 5
17 QRPA QRPA
18 6 1. P. Ring and P. Schuck, The Nuclear Many-Body Problem, Springer-Verlag (1980). 2. A. Bohr and B.R. Mottelson, Nuclear Structure Vol.1 & 2, World Scientific (1998). 3. J.-P. Blaizot and G. Ripka, Quantum Theory of Finite Systems, MIT Press (1986). 4. (2002). 5. (1997). 6. (1978) G. Do Dang, A. Klein, N.R. Walet, Phys. Rep. 335, 93 (2000).
多体問題
Many Body Problem 997 4, 00 4, 004 4............................................................................. 7...................................... 7.............................................
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