3 3.1 3.1.1 kg m s J = kg m 2 s 2 MeV MeV [1] 1MeV=1 6 ev = 1.62 176 462 (63) 1 13 J (3.1) [1] 1MeV/c 2 =1.782 661 731 (7) 1 3 kg (3.2) c =1 MeV (atomic mass unit) 12 C u = 1 12 M(12 C) (3.3) 41
42 3 u = 931.494 13 (37) MeV/c 2 (3.4) [1] u amu m p m n [1] m H [2] m p = 938.272 (4) MeV/c 2 = 1.7 276 466 88 (13) u m n = 939.565 33 (4) MeV/c 2 = 1.8 664 915 78 (55) u m H = 938.782 98 (4) MeV/c 2 = 1.7 825 23 (55) u (3.5).1% m n m p = 1.293 331 8 (5) MeV/c 2 m n m H =.782 354 (2) MeV/c 2 (3.6) m e [1] m e =.51 998 92 (21) MeV/c 2 (3.7) 3.1.2 Z A Z M(A, Z) m n = M(1, ) m H = M(1, 1) M( 12 C) = M(12, 6) 1% binding energy Z N Zm H + Nm n B(A, Z) =Zm H + Nm n M(A, Z) (N = A Z ) (3.8) Z N B(A, Z)
3.1 43 Mass Excess A mass excess M(A, Z) =M(A, Z) Au (3.9) mass excess M(A, Z) = Zm H + Nm n B(A, Z) Au = B(A, Z)+Z (m H u)+n (m n u) (3.1) m H u =7.288 97 (5) MeV/c 2 m n u =8.71 32 (5) MeV/c 2 (3.11) mass excess m n m H > A =56 mass exces [2] 3.1 A =56 57 Z M(56,Z) B(56,Z) B/56 M(57,Z) B(57,Z) B/57 2 Ca -13.237 449.584 8.28-7.12 451.539 7.922 21 Sc -25.467 461.32 8.233-21.387 465.23 8.158 22 Ti -39.132 473.914 8.463-34.558 477.412 8.376 23 V -46.239 48.239 8.576-44.376 486.448 8.534 24 Cr -55.289 488.56 8.723-52.393 493.682 8.661 25 Mn -56.95 489.341 8.738-57.485 497.991 8.737 26 Fe -6.61 492.254 8.79-6.176 499.9 8.77 27 Co -56.35 486.95 8.695-59.34 498.282 8.742 28 Ni -53.9 483.988 8.643-56.75 494.234 8.671 29 Cu -38.61 467.97 8.355-47.35 484.682 8.53 3 Zn -25.728 454.251 8.112-32.686 469.281 8.233 31 Ga -4.741 432.482 7.723-15.91 451.713 7.925 MeV
44 3 3.2 Weizsäcker 3.2.1 [2,3] 1. B(A, Z) A 3.1 B(A, Z)/A 7.4 MeV 8.8 MeV 2. A 6 6 B(A, Z)/A 3.1 3. A Z B(A, Z) Z 3.2 4. 3. A A Z Z 3.2 5. Z = N 3.3 [ 4] 9. 8.5 B ( A,Z ) / A [ MeV ] 8. 7.5 7. 5 1 15 2 25 mass number A 3.1:
3.2 Weizsäcker 45 g.s. energy [ MeV ] g.s. energy [ MeV ] 4 35 3 25 2 15 1 5 35 3 25 2 15 1 5 A = 57 A = 56 21 22 23 24 25 26 27 28 29 3 atomic number 3.2: A =57 A =56 Z = N Stable Isotopes Z =2 Z =14 N =2 Z =2 N =2 Z =8 N =8 N =14 stable isotopes with Z = N = even 8 except Be 4 no stable isotopes for A = 5, 8 3 2 He : only stable for A > 1 above the Z = N line 3.3: Z 2
46 3 3.2.2 Weizsäcker mass formula [5] volume energy E 1 = b vol A (3.12) saturation property A A A C 2 = A(A 1)/2 A 2 surface energy σ R σ 4πR 2 E 2 =4πR 2 σ (3.13) 1/3 R = r A 1/3 (3.14) E 2 =4πr 2 σa 2/3 = b surf A 2/3 (3.15)
3.2 Weizsäcker 47 R 3 R 2 Coulomb Coulomb energy e Coulomb R Ze Coulomb E 3 = Z(Z 1) 2 ρ dr ρ dr r r ρ 4πR 3 3 = e (3.16) Z(Z 1)/2 Z r r R E 3 = Z(Z 1)e2 2 6 5 1 R = 3 e 2 Z(Z 1) 5 r A 1/3 b Coul Z 2 A 1/3 (3.17) Z(Z 1) Z 2 Coulomb symmetry energy Coulomb Coulomb ( ) N Z 2 ( ) A 2Z 2 (A 2Z) 2 E 4 = c A = c A = b sym N + Z A 2A (3.18) N/Z A Z = N N Z pairing energy Z N 2 H 6 Li 1 B 14 N δ(a) Z =, N= E 5 = (A) = Z + N = δ(a) Z =, N= (3.19)
48 3 4 3 proton pairing energy [ MeV ] 2 1 4 3 neutron 2 1 5 1 15 2 25 mass number A 3.4: 12/ A δ(a) = 12 A MeV (3.2) 3.4 [2] Weizsäcker B(A, Z) = b vol A b surf A 2/3 b Coul Z 2 A 1/3 b sym (A 2Z) 2 A Coulomb (A) (3.21) Green[6] MeV b vol =15.56, b surf =17.23, b Coul =.697, b sym =23.29 (3.22)
3.3 49 3.3 3.3.1 Weizsäcker Z Coulomb A Z B(A, Z) Z = (3.23) A= Z = A 2+ b CoulA 2/3 b sym (3.24) b Coul /b sym Z = N Coulomb Z Z = N A Z 3.5 β β-stability line 1 proton number Z 8 6 4 2 Stable Isotopes 2 4 6 8 1 12 14 16 neutron number N 3.5: (3.24)
5 3 3.6 [2] A 9 14 21 3.7 Coulomb 8MeV 3.3.2 [4] 287 27 A =1 A = 29 A =5 A =8 232 Th 234,235,238 U [4] 1) 235 U 238 U 2) 4 K 3) 3 H 7 Be 1 Be 14 C 22 Na 32 P 35 S 36 Cl 3.2 4 K 1.28 1 9 y β, EC 148 S m 7 1 15 y α 87 Rb 4.8 1 1 y β 152 Gd 1.1 1 14 y α 113 Cd 9 1 15 y β 176 Lu 3.6 1 1 y β 115 In 4.4 1 14 y β 174 Hf 2. 1 15 y α 123 Te 1.3 1 13 y EC 187 Re 5 1 1 y β 138 La 1.3 1 11 y β, EC 186 Os 2 1 15 y α 144 Nd 2.4 1 15 y α 19 Pt 6. 1 11 y α 147 S m 1.6 1 11 y α
3.3 51 9. 8.5 B ( A,Z ) / A [ MeV ] 8. 7.5 7. 5 1 15 2 25 mass number A 3.6: B ( A,Z ) / A [ MeV ] 16 volume energy 14 12 1 8 6 4 surface energy Coulomb energy symmetry energy 2 5 1 15 2 25 mass number A 3.7: Weizsäcker
52 3 1.5 1 8 3.3.3 shell effect Weizsäcker 3.6 A = 9, 14, 21 E(A, Z) =B(A, Z) B(A, Z) (3.25) 3.8 E E Z, N = 28, 5, 82 E N = 126
3.3 53 2 15 5 82 E [ MeV ] 1 5 28-5 5 1 proton number Z 2 82 E [ MeV ] 15 1 5 28 5 126-5 5 1 15 neutron number N 3.8: Weizsäcker
54 3 3.4 3 1. D.E. Groom et al., European Physical Journal C15 (2) 1, available on the Particle Data Group WWW page (URL http://pdg.lbl.gov ) 2. The 1995 Update to the Atomic Mass Evaluation, available on http://www.nndc.bnl.gov/nndcscr/masses/ 3. Table of Isotopes, Eighth Edition, R.B. Firestone, Ed. V.S. Shirley, (John Wiley and Sons, Inc., New York, 1996) 4. 21 5. C.F. von Weizsäcker, Z. Phys. 96 (1935) 431 H.A. Bethe, Rev. Mod. Phys. 8 (1936) 82 6. A.E.S. Green and D.F. Edwards, Phys. Rev. 91 (1953) 46; A.E.S. Green, Phys. Rev. 95 (1954) 15