56 4 2 log N ( t ) 1 0 0 t 1/2 τ m 2τ m time t 4.1: λ decay rate λ = 1 τ m (4.8) A B b Γ = h τ m = hλ (4.9) A B + b (4.10) Q Q = M(B)+M(b) M(A) (4.11)

4 4.1 t N(t) t t +dt dn(t) N(t) dn(t) = λn(t)dt (4.1) dn(t) dt = λn(t) (4.2) t =0 N 0 = N(0) 4.1 N(t) =N 0 e λt (4.3) log N(t) = log N 0 λt (4.4) mean life half-life t N(t) τ m =1/λ 1/e τ m 1/2 t 1/2 T 1/2 : N(t + τ m ) = 1 N(t + t 1/2 ) = 1 (4.5) N(t) e N(t) 2 e e=2.718 281 828 459 045 235 (4.6) t 1/2 = τ m log e 2=0.693 τ m (4.7) 55

56 4 2 log N ( t ) 1 0 0 t 1/2 τ m 2τ m time t 4.1: λ decay rate λ = 1 τ m (4.8) A B b Γ = h τ m = hλ (4.9) A B + b (4.10) Q Q = M(B)+M(b) M(A) (4.11) Q Q = T B + T b (4.12) A T B + T b Q 4.2 exp (iht/ h) E R E = E R + iγ 2 (4.13)

4.1 57 count Γ Q T B + T b 4.2: exp (i Ēh ) [ ( t ER = exp i h + iγ )] = exp 2 h ( i E ) R h t exp ( Γ ) 2 h (4.14) ( exp i Ē ) 2 h t = exp ( Γ h ) t = exp ( λt) (4.15) (4.2) (4.9) Planck [1] h = 1.054 571 596 (82) 10 34 Js (4.16) = 6.582 118 89 (26) 10 22 MeV s τ m Γ 4.3 A =8 8 Be 8 Be 0 + 2 + 4 + J π =0 + Γ =8.8 ± 1.7 ev τ m =(7.5 ± 1.4) 10 17 s J π =2 + Γ =1.50 ± 0.02 MeV τ m =(4.39 ± 0.06) 10 22 s J π =4 + Γ 3.5 MeV τ m 1.9 10 22 s A =8 4.3 8 B β +

58 4 18 16 14 8 3Li β β 8 5B 12 10 α 4 + 8 6 4 2 0 [ MeV ] 4 4 2He + 2He α α 8 4Be 2 + 0 + 4.3: 8 Be disintegration proton decay

4.2 59 4.2 β β e ν e β + e + ν e β + EC: electron capture K β (A, Z) (A, Z +1)+e + ν e β + (A, Z) (A, Z 1) + e + + ν e (4.17) (A, Z)+e (A, Z 1) + ν e β β n p + e + ν e β + p n + e + + ν e (4.18) p + e n + ν e Q β Q β (A, Z) = M(A, Z) M(A, Z +1) = B(A, Z +1) B(A, Z)+(m n m H ) β + Q β +(A, Z) = M(A, Z) M(A, Z 1) 2m e = B(A, Z 1) B(A, Z) (m n m H ) 2m e (4.19) Q EC (A, Z) = M(A, Z) M(A, Z 1) = B(A, Z 1) B(A, Z) (m n m H ) β + 2m e (4.17) 2m e Q Q β + Q EC > 0 >Q β + 4.4 A =56 56 Fe 56 Mn β 56 Fe 56 Co β + /EC 56 Fe

60 4 5 4 3 2 1 Q β 2.579 h 56 25Mn β EC β 78.8 d 56 27Co Q β Q EC 0 [ MeV ] 56 26Fe 4.4: 56 Mn β 56 Co β + /EC Q Q β Q EC β + 2m e β β + 10 3 s Q 4.2 β +

4.3 61 4.3 separation energy S n S p mass excess S n (A, Z) = M(A 1,Z)+m n M(A, Z) = M(A 1,Z)+ M(1, 0) M(A, Z) = B(A, Z) B(A 1,Z) S p (A, Z) = M(A 1,Z 1) + m H M(A, Z) = M(A 1,Z 1) + M(1, 1) M(A, Z) = M(A, Z) B(A 1,Z 1) (4.20) S 2n (A, Z) = M(A 2,Z)+2m n M(A, Z) = M(A 2,Z)+2 M(1, 0) M(A, Z) = B(A, Z) B(A 2,Z) S 2p (A, Z) = M(A 2,Z 2) + 2m H M(A, Z) = M(A 2,Z 2) + 2 M(1, 1) M(A, Z) = B(A, Z) B(A 2,Z 2) (4.21) 0 S 2p S p drip line 0 proton drip line 0 neutron drip line Coulomb

62 4 4.5 Z =9 15 MeV 19 F N =10 Z =9 15 separation energies Sx [ MeV ] 10 5 0 S 2p S p S n F isotopes S 2n 4 6 8 10 12 14 16 18 20 neutron number N 4.5: Z =9 A =5 A =8 8 Be A =8 A =5 A =5 5 He 5 Li 4 He 4.6 4.7 A =5 5 He Q =0.89 MeV 5 Li Q =1.97 MeV 5 He 4 He + p Γ 0.60 MeV (4.22) 5 Li 4 He + n Γ 1.5 MeV 10 22 s

4.3 63 B ( A,Z ) / A [ MeV ] 8 7 6 5 4 3 2 4 He He Li Be B C O N 1 0 H 2 4 6 8 10 12 14 16 18 20 22 mass number A 4.6: Z 8 10 8 6 1/2 1/2 4 2 0 [ MeV ] 4 2He + n 5 2He 3/2 5 3Li 3/2 4 2He + p 4.7: A =5 pairing energy J π =0 + P n (A, Z) = 1 4 ( 1)A Z+1 [ S n (A +1,Z)+S n (A 1,Z) 2S n (A, Z)] P p (A, Z) = 1 4 ( 1)Z+1 [ S p (A +1,Z +1)+S p (A 1,Z 1) 2S p (A, Z)] (4.23)

64 4 4.4 A 200 α S α (A, Z) = M(A 4,Z 2) + M(4, 2) M(A, Z) = B(A, Z) B(A 4,Z 2) B(4, 2) (4.24) M(4, 2) = M( 4 He) Q Q α (A, Z) = S α (A, Z) (4.25) B(4, 2) = B( 4 He) = 28.296 MeV (4.26) 4.6 S α (A, Z) < 0 4.8 Q α (A, Z) = S α (A, Z) > 0 A 90 214 Po 210 Pb 214 Po 210 Pb + α (4.27) 214 Po S = 210 Pb α 214 Po 2 (4.28) spectroscopic factor Coulomb 4.9

4.4 65 120 100 proton number Z 80 60 40 20 0 0 20 40 60 80 100 120 140 160 180 neutron number N 4.8: S α (A, Z) < 0 Coulomb potential 0 S α α r potential quantum mechanical penetration of the α wave function through the Coulomb barrier nuclear potential 4.9: Coulomb

66 4 Coulomb R V (r) Coulomb V (r) > 0 Coulomb 4.9 S α (> 0) V (r) ( S α ) > 0 L =0 s Schrödinger Coulomb [ 8m exp h V (r) E dr ] (4.29) V (r) 4.9 E = S α V (r) E 0 Coulomb 210 Po 0.298 µs 10 15 y 232 Th 1.41 10 10 y 25

4.5 67 4.5 (A, Z) (A 1,Z 1 )+(A 2,Z 2 ) { A = A1 + A 2 Z = Z 1 + Z 2 (4.30) Q Q = B(A 1,Z 1 )+B(A 2,Z 2 ) B(A, Z) (4.31) A 1 /A = Z 1 /Z A 2 /A = Z 2 /Z A 1 = A 2 Q Q >0 Z 2 /A Z2 /A > 0.35 (4.32) 2b surf /b Coul 50 Z 2 /A > 18 4.10 half life [ y ] 10 20 232 Th 230 spontaneous fission 234 10 15 238 U 236 232 242 10 10 238 Pu 244 240 236 Cm Z = 90 Thorium 246 242 10 5 244 240 Z = 92 Uranium 250 248 Z = 94 Plutonium 246 Z = 96 Curium 252 Cf 10 0 Z = 98 Californium 254 254 Z = 100 Fermium Fm 10-5 256 34 35 36 37 38 39 40 Z 2 / A 4.10: 90 Z 100

68 4 4.6 10 MeV M(Eλ) = ρ(r) r λ Y λµ ( r)dr 1 (4.33) M(Mλ) = j(r) (r ) r λ Y λµ ( r)dr c(λ +1) M(Eλ) ρ(r) M(Mλ) c 1/2 J 1 J 2 hcq T (E(M)λ) = B(E(M)λ) = 8π(λ +1) q 2λ+1 λ[(2λ + 1)!!] 2 h B(E(M)) (4.34) 1 2J 1 +1 J 2 M(E(M)λ) 2 B(E(M)λ) λ J 1 J 2 λ J 1 + J 2 (4.35) Eλ ( 1) λ Mλ ( 1) λ+1 λ 1 + 2 E1 M2 E3 E1 1 + 2 + M1 E2 M3 M3

4.7 69 4.7 10 22 s 6000 4.11 [4] 140 120 100 proton number Z 80 60 40 20 0 0 20 40 60 80 100 120 140 160 180 200 neutron number N 4.11:

70 4 [ 5-20 ] [4]

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