42 3 u = (37) MeV/c 2 (3.4) [1] u amu m p m n [1] m H [2] m p = (4) MeV/c 2 = (13) u m n = (4) MeV/c 2 =

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1 kg m s J = kg m 2 s 2 MeV MeV [1] 1MeV=1 6 ev = (63) 1 13 J (3.1) [1] 1MeV/c 2 = (7) 1 3 kg (3.2) c =1 MeV (atomic mass unit) 12 C u = 1 12 M(12 C) (3.3) 41

2 42 3 u = (37) MeV/c 2 (3.4) [1] u amu m p m n [1] m H [2] m p = (4) MeV/c 2 = (13) u m n = (4) MeV/c 2 = (55) u m H = (4) MeV/c 2 = (55) u (3.5).1% m n m p = (5) MeV/c 2 m n m H = (2) MeV/c 2 (3.6) m e [1] m e = (21) MeV/c 2 (3.7) 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 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 = (5) MeV/c 2 m n u = (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 Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga MeV

4 Weizsäcker [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 A Z B(A, Z) Z A A Z Z Z = N 3.3 [ 4] B ( A,Z ) / A [ MeV ] mass number A 3.1:

5 3.2 Weizsäcker 45 g.s. energy [ MeV ] g.s. energy [ MeV ] A = 57 A = 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, He : only stable for A > 1 above the Z = N line 3.3: Z 2

6 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)

7 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)e 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)

8 proton pairing energy [ MeV ] neutron 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)

9 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 Stable Isotopes neutron number N 3.5: (3.24)

10 [2] A Coulomb 8MeV [4] 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 K y β, EC 148 S m y α 87 Rb y β 152 Gd y α 113 Cd y β 176 Lu y β 115 In y β 174 Hf y α 123 Te y EC 187 Re y β 138 La y β, EC 186 Os y α 144 Nd y α 19 Pt y α 147 S m y α

11 B ( A,Z ) / A [ MeV ] mass number A 3.6: B ( A,Z ) / A [ MeV ] 16 volume energy surface energy Coulomb energy symmetry energy mass number A 3.7: Weizsäcker

12 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

13 E [ MeV ] proton number Z 2 82 E [ MeV ] neutron number N 3.8: Weizsäcker

14 D.E. Groom et al., European Physical Journal C15 (2) 1, available on the Particle Data Group WWW page (URL ) 2. The 1995 Update to the Atomic Mass Evaluation, available on 3. Table of Isotopes, Eighth Edition, R.B. Firestone, Ed. V.S. Shirley, (John Wiley and Sons, Inc., New York, 1996) C.F. von Weizsäcker, Z. Phys. 96 (1935) 431 H.A. Bethe, Rev. Mod. Phys. 8 (1936) A.E.S. Green and D.F. Edwards, Phys. Rev. 91 (1953) 46; A.E.S. Green, Phys. Rev. 95 (1954) 15

1 1 H Li Be Na M g B A l C S i N P O S F He N Cl A e K Ca S c T i V C Mn Fe Co Ni Cu Zn Ga Ge As Se B K Rb S Y Z Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb T e

1 1 H Li Be Na M g B A l C S i N P O S F He N Cl A e K Ca S c T i V C Mn Fe Co Ni Cu Zn Ga Ge As Se B K Rb S Y Z Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb T e No. 1 1 1 H Li Be Na M g B A l C S i N P O S F He N Cl A e K Ca S c T i V C Mn Fe Co Ni Cu Zn Ga Ge As Se B K Rb S Y Z Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb T e I X e Cs Ba F Ra Hf Ta W Re Os I Rf Db Sg Bh

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