* 1 1 (i) (ii) Brückner-Hartree-Fock (iii) (HF, BCS, HFB) (iv) (TDHF,TDHFB) (RPA) (QRPA) (v) (vi) *1 2004 1
1 ( ) ( ) 1.1 140 MeV 1.2 ( ) ( ) 1.3 2.6 10 8 s 7.6 10 17 s? Λ 2.5 10 10 s 6 10 24 s
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 2 1.8 Schrödinger N = 2n + l Laguerre R nl (r) r l L l+1/2 n ((r/b) 2 ) s, p, s + d, p + f,
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 + 1 4 µ V µνρσ c µc νc σ c ρ. (2) c µ µ ( ) h sp (Hartree-Fock ) (shell model) V res A µνρσ
A 0 = c 1 c 2 c A 0, (3) 0 1-1 (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 ( ) ( )
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
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
( ) Hartree-Fock(HF) HF 2 + 1p1h 2 + 2 + ( ) 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
(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
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
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
(16) Ẽ E sh 60 70 2:1 (superdeformed state) 90 3:1 hyperdeformed band ( ) ( 2000 ) 3.2 152 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
(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,
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)
H ω rot (21) H = h sp + V pair HFB 2 H routhian ω rot = 0 1 MeV 1 MeV 2 2 2 (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)
2 routhian 164 Er ω 1 ω bb ( ) ( ) Ψ 1 (pairing vibration) 5
QRPA QRPA
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). 7. 2000 http://www.nt.phys.kyushu-u.ac.jp/shimizu/download/natsuk.ps.gz 1 4 5 8. G. Do Dang, A. Klein, N.R. Walet, Phys. Rep. 335, 93 (2000).