filename=decay-text141118.tex made by R.Okamoto, Emeritus Prof., Kyushu Inst.Tech. * 1, 320 265 radioactive ray ( parent nucleus) ( daughter nucleus) disintegration, decay 2 1. 2. 4 ( 4 He) 3. 4. X 5., 1
6. [1] (cal) (J) (kwh) ( 1 1 100 1 ( 3 t N(t) dt dn ( ) dn N dt N 0 = λ dt (3.1) N(t) = N 0 e λt (3.2) λ (decay constant), λ [λ] = 1/s 1947 2
N 0 N(t) = N 0 (1 e λt ) (3.3) (3.2) N 0 t N(t) (3.2) t e λt t t + dt p(t)dt t e λt t t + dt λdt p(t)dt = e λt λdt t 0 p(t)dt = 0 e λt λdt = 1 (3.4) 1 4 t dt dn dn/n 0 mean life τ τ 0 t ( 1) dn dt 1 dt = λ t e λt dt. (4.5) N 0 0 f(x), g(x) f (x)g(x)dx = f(x)g(x) f(x)g (x)dx 0 t e λt dt = t d [ ] [ ] 1 t 1 0 dt λ e λt dt = λ e λt 1 0 0 λ e λt dt = 1 [ ] e λt λ = 1 (4.6) 2 0 λ 2 τ τ = 1 λ (4.7) τ λ (half life)(t, T 1/2 ) τ N(t + T ) = 1 2 N(t) T = ln 2 λ = 0.693 = 0.693 τ. (4.8) λ 3
5 Ci( ) Bq( ) (1978 5 ) Ci 1. Bq( ) (Bq) Bq Bq 1/s. (5.9) 1kBq = 10 3 Bq = 1000, 1MBq(1 ) = 10 6 Bq =, 1GBq(1 ) = 10 9 Bq = 10. 2. Ci( ) Ci g Ra-226 3.7 10 10 Ci 3.7 10 10 Bq Ci Ci 3.7 10 10 Bq, (5.10) 1mCi 10 3 Ci, 1µCi 10 6 Ci, 1nCi 10 9 Ci, 1pCi 10 12 Ci. (5.11) 6 6.1 (t) (t) λn(t) (6.12) [2][3][4] (t) [(t)] = 1/s M T m N a (6.12). (t) = ln 2 T m M N a. (6.13) [2] 4
1. t N(t) N 0 e λt dn/dt = λn(t) 2., B, C, B N B (t) e λ Bt B B (t) λ B N B (t) 3. (6.12) 4. t = 0 1. R : dn(t) dt = R λn(t) (6.14) N 0 0 λn 0 N(t) = N 0 e λt + R λ (1 e λt ), (6.15) (t) λn(t) = 0 e λt + R(1 e λt ) (6.16) 2. R(t) dn(t) dt = R(t) λn(t), (6.17) N(t) = N 0 e λt + (t) λn(t) = 0 e λt + λ R(t) t 0 t R(t ) e λ(t t ) dt, (6.18) 0 R(t ) e λ(t t ) dt. (6.19) 5
6.2 a b c N a, N b, N c, λ a, λ b a N 0 dn a dt dn b dt dn c dt = λ a N a, = λ b N b + λ a N a, = λ b N b (6.20) N a (t) = N 0 e λat, (6.21) ) ( λa N b (t) = λ b λ a ( N c (t) = N 0 N 0 (e λ at e λ bt ), (6.22) 1 λ b λ b λ a e λat + λ ) a e λ bt λ b λ a (6.23), a (t) = λ a N a (t), b (t) = λ b N b (t), c (t) = λ c N c (t) 6.3 a b λ a, λ b λ eff λ eff = λ a + λ b, τ eff = dn = λ a Ndt λ b Ndt λ eff Ndt (6.24) 1 τ eff = 1 τ a + 1 τ b, 1 T eff = 1 T a + 1 T b, (6.25) τ a τ b τ a + τ b, T eff = T at b T a + T b (6.26) τ a, τ b, τ eff (T a, T b, T eff ) a,b (t) λ eff N(t) = λ eff N 0 e λ efft 6
7 specific radioactivity S m S m (7.27) S Bq/g Bq/Kg SI N M N N = N /M, = λn (7.27) S = λn M (7.28) T 1/2 (y, year) (s, second) S = 1.32 10 16 4.17 1023 Bq/g, S = Bq/g (7.29) (T 1/2 /y)(m/g) (T 1/2 /s)(m/g) 8 α 2 2 (a) Y ZX N 4 Z 2Y N 2 + α( 4 2He 2 ) + Q. (8.30) Q Q = (M X M Y M α )c 2, (8.31) M E α Y V y, V α Q = 1 2 M yv 2 y + 1 2 M αv 2 α (8.32) = 1 2 M αvα 2 (1 + M yvy 2 ). (8.33) M α Vα 2 7
M y V y = M α V α. (8.34) E α 1 2 M αv 2 α Q = E α (1 + M α M y ) M y E α = Q( ). (8.35) M α + M y (b) Y Y ZX N 4 Z 2YN 2 + α( 4 2He 2 ) + Q. (8.36) Q ( E y) Q = Q E y ( 1928 Gamov, Condon, Gurney r 2(Z 2)/(4πε 0 r) r R 2(Z 2)/(4πε 0 R) 8.6MeV 4.2MeV 9 β Z N M(, Z) M (, Z), Z B e M(, Z) = M (, Z) + Z m e + B e /c 2 (9.37) 8
B e M(, Z) = M (, Z) + Z m e (9.38) 1. β n p + e + ν (T = 1000s) (9.39) β n p + e + ν (9.40) 3 1H 3 2He + e + ν (T = 12y ), (9.41) 32 15P 32 16S + e + ν (T = 14d ) (9.42) ν anti-neutrino β ZX N Z+1 Y N 1 + e + ν + Q(β ). (9.43) β Q(β ) ZX N Z+1 Y N 1 + e + ν + Q (β ) (9.44) Q (β ) Q(β ) E ex (Q (β ) = Q(β ) E ex ) Q (β ) < Q(β ) Q(β ) [M (, Z) M (, Z + 1) m e ] c 2 > 0. (9.45) Q(β ) [M(, Z) M(, Z + 1)] c 2 > 0 (9.46) Q(β ) [m n m p ] c 2 = 0.5 MeV > 0 (9.47) 9
2. β + β + p n + e + + ν (9.48) 10 6 C 10 5 B + e + + ν (T = 19.4s ), (9.49) 11 6 C 11 5 B + e + + ν (T = 20.3min ) (9.50) e + e 0.5 MeV γ 2 ( ) β +, ZX N Z 1 Y N+1 + e + + ν + Q(β + ). (9.51) β + ZX N Z 1 Y N+1 + e + + ν + Q (β + ) (9.52) Q (β + ) Q Q (β + ) < Q(β + ) β Q(β + ) Q(β + ) [M (, Z) M (, Z 1) m e ] c 2 > 0 (9.53) Q(β + ) [M(, Z) M(, Z + 1) 2m e ] c 2 > 0 (9.54) 3. electron capture, EC : p + e n + ν (9.55) 10
7 4Be + e 7 3Li + ν (T = 53.6d) (9.56) ZX N e Z 1 Y N+1 + ν + Q. (9.57) Q(EC) [M (, Z) + m e M (, Z 1)] c 2 I > 0 (9.58) I (ionization energy) Q(EC) [M(, Z) M(, Z 1)] c 2 I > 0 (9.59) I ev Q(EC) 10eV K K (K ) K K X uger process uger electron 4. β 5. β β BE(, Z) = c v c s 2/3 c a (N Z) 2 Z 2 c c + δ(, Z), (9.60) 1/3 c v = 15.826MeV, c s = 17.907MeV, c a = 23.517MeV, c c = 0.7183MeV, (9.61) 11
11.2 MeV (Z, ) 1/2 δ(z, ) = 0 ( ) 11.2 MeV (Z, ) 1/2 (9.62), Z M(, Z)c 2 = [M H Z + ( Z)m n ] c 2 BE(, Z) = [M H Z + ( Z)m n ] c 2 c v + c s 2/3 [(/2) Z] 2 + c a Z 2 +c c δ(, Z). (9.63) 1/3 M(, Z) M H m n [?]) Z 2 Z β 0 = M(, Z) Z = (M H m n )c 2 4c a + 8c a Z + 2c c 1/3 Z Z β = 2c a + (m n M H )c 2 (9.64) 4c a + c c 2/3 1 2 + ( c (9.65) c 2c a ) 2/3 Z β = 1 1.98350 + 0.015272/3 (9.66) 1 2 + 0.0152/3 (9.67) N(= Z) Z β Heisenberg, β (Heisenberg ) = 63 Z β 28.15, = 135 Z β 56.37 10 γ 12
gamma decay X X X + γ (10.68) E i, E f, λ, ν E i E f hν(= h hc ). (10.69) ω P ( recoil) E i = E f + P 2 P hν c 2M + hν, (10.70) = 0, (10.71) E i E f = ( hν c )2 + hν (10.72) 2M MeV 940MeV R.Meyer 15 10 10 s metastable state isomeric transition 103 Rh 103 Rh m 57 [1] 1983 pp.6 9 [2] 1998 p.60 p.62 (3-101) t T t ( 13
[3] J. R. ( ) 1995 p 13, pp.22-22. [4] J. R. ( ) 2003 pp.22-26 14