- γ 1929 γ - SI γ 137 Cs 662 kev γ NaI active target NaI γ NaI 2 NaI γ NaI(Tl) γ 2 NaI γ γ γ

Transcription

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2 - γ 1929 γ - SI γ 137 Cs 662 kev γ NaI active target NaI γ NaI 2 NaI γ NaI(Tl) γ 2 NaI γ γ γ

3 1 3 2 γ γ γ γ Dirac M NaI(Tl) NaI(Tl) NIM/CAMAC NIM CAMAC γ

4 Cs Na Co ADC NaI(Tl) NaI γ A 100 B 102 C 105 D 107 2

5 γ γ NaI(Tl) 1. γ γ γ NaI(Tl) 4. γ NaI active target γ 3

6 2 γ γ 3 γ NaI 1 1 NaI active target 1 NaI 7 4

7 2 γ I 0 A n (x ) : N t N t = n Ad 5

8 2.2: 2.2 σ t σ t N t = σ t n Ad d σ t n Ad A = σ t nd (2.1) σ t N t A 1 (2.1) ( ) N y N y = I 0 (σ t nd) (2.2) σ t σ σ N y I 0 nd (2.3) I 0 nd L L = I 0 nd (2.3) ( ) 6

9 Ω Ω 2.3: Ω = A 2 l 2 dσ dω N y ( Ω) = dσ ΩL dω NaI(Tl) γ σ N y 2.4 x N y (2.2) dx di(x) di(x) = σni(x)dx I(x) = I 0 e σnx = I 0 e µx 7

10 µ µ = σn N y 2.4: N y = σn d 0 d dxi(x) = σni 0 e µx dx 0 = σn 1 e µd I 0 (2.4) µ = (1 e µd )I 0 (2.5) (2.5) σn = µ γ (2.4) (2.5) d dx (2.2) N y = I 0 (1 e µdx ) I 0 µdx = I 0 σndx Ω (2.4) σ dσ Ω dω N y ( Ω) = dσ e µd Ωn1 I 0 dω µ 8

11 2.2 γ γ K E e ( 2.5) E e = hν E b E b Z E b kev Z kev 2.5: Z 5 K L M K K 5/

12 γ hν hν = hν 1 + hν m e (1 cos θ) c 2 m e hν E e 2.6: hν E e = hν 1 + hν m e (1 cos θ) c 2 (2.6) θ = π hν E e (θ = π) = hν (2.7) hν m e c 2 (2.6) E e (θ = 0) = 0 (2.7) - [ ] 2 ] dσ dω = r2 e 1 [1 + cos 2 θ + α2 (1 cos θ) α(1 cos θ) 1 + α(1 cos θ) r e α = hν m e - c 2 σ c - [ 1 + α σ c = 2πre 2 α 2 ( 2(1 + α) 1 + 2α ) log(1 + 2α) log(1 + 2α) α ] α 2α (1 + 2α) 2 (2.8) Z 10

13 E e E e + E e + E e + = hν 2m e c 2 2m e c 2 = 1024 kev 2.7: 2.8: - Z

14 Z 2.8 vertex M Z σ σ M 2 Z 2 Z γ γ ( 2.9) ( 2.10) µ ρ µ ρ µ ( ) = µ ρ µ ( ) γ 300 kev 300 kev 7 MeV 7 MeV 2.10 γ γ 662 kev 0.1 cm 2 /g ρ lead 11.3 g/cm 3 d =5 cm γ e µd = e 0.1 ρlead d = % 12

15 2.9: (NaI) ( [1] ) ( )(cm 2 /g) γ (MeV) Photo τ/ρ Compton total σ/ρ Pair κ/ρ Total attenuation µ 0 /ρ Total absorption µ a /ρ (Compton scattering σ s /ρ) (Rayleigh σ r /ρ) γ Compton absorption σ a /ρ 13

16 2.10: (Pb) ( [1] )

17 3 1) 2) γ hν hν E e 3.1) 3.1: hν hν E e 3 hν c = hν c cos θ + γm ev e cos ϕ (3.1) 0 = hν c sin θ γm ev e sin ϕ (3.2) m e c 2 + hν = hν + γm e c 2 (3.3) 15

18 (3.1) (3.2) ( hν c hν ) 2 c cos θ = (γm e v e ) 2 cos 2 ϕ ) 2 = (γm e v e ) 2 sin 2 ϕ ( hν c sin θ ( ) 2 h [(ν ν cos θ) 2 + (ν sin θ) 2 ] = (γm e v e ) 2 c (3.3) [m e c 2 + h(ν ν )] 2 = 1 m 2 1 β ec 4 e 2 1 βe 2 m 2 = ec 4 [m e c 2 + h(ν ν )] 2 β 2 e = 1 = γ 2 m 2 ec 2 βe 2 = β2 e m 2 1 β ec 2 (3.4) e 2 m 2 ec 4 [m e c 2 + h(ν ν )] 2 β 2 e 1 β 2 e = 1 c 2 (2h(ν ν )m e c 2 + (h(ν ν )) 2 ) (3.5) (3.4) 2h(ν ν )m e c 2 + (h(ν ν )) 2 = (hν) 2 2hνhν cos θ + (hν ) 2 m e c 2 (E E ) = 2hνhν (1 cos θ) hν hν hν = hν m ec (1 cos θ) (3.6) E e E e = hν hν hν = hν hν m e (1 cos θ) + 1 c 2 16

19 662 kev 3.2 θ θ = 0 0 θ 200 kev 3.2: 662 kev θ θ = 0 0 θ 200 kev [2] [3] γ M 17

20 3.2.1 Dirac Dirac (p µ γ µ m)ψ = (p/ m)ψ = 0 (p/ = p µ γ µ ) (3.7) Dirac ψ ψ ψ γ 0 ψ Dirac u r (p), v r (p); (r = 1, 2) u r (p) = N v r (p) = N ( χ r σ p χ E+m r ) ( σ p E+m χ 3 r χ 3 r ) (3.8) (3.9) N N = E + m χ r σ Dirac ( ) ( ) 1 0 χ 1 = χ 2 = (3.10) 0 1 u r(p)u s (p) = 2Eδ rs = v r(p)v s (p) (3.11) u r(p)v s (p) = 0 = v ru s (p) (3.12) u r (p)ū r (p) = p/ + m (3.13) r=1,2 v r (p) v r (p) = p/ m (3.14) r=1,2 r, s A B γ TrAB = TrBA (3.15) {γ µ, γ ν } = 2g µν (3.16) 18

21 γ Tr[γ µ γ ν ] = 4g µν (3.17) Tr[γ ρ γ σ γ α γ β ] = 4(g ρσ g αβ + g ρβ g σα g ρα g σβ ) (3.18) Tr[γ µ 1 γ µ 2... γ µ 2n 1] = 0 }{{} (n = 1, 2,... ) (3.19) γ γ µ = γ 0 γ µ γ 0 (3.20) γ ( ) ( ) γ σ = γ = 0 1 σ 0 (3.21) 4 k µ = (ω; k) m 2 = E 2 p 2 m = 0 k µ k µ = ω 2 k 2 = E 2 p 2 = 0 (3.22) ϵ(k, λ) 4 k µ k µ ϵ µ (k, λ) = 0 (λ = 1, 2) (3.23) λ λ = 1, 2 (λ = 0) (λ = 3) ϵ µ (λ)ϵ µ (ρ) = g λρ (3.24) A µ A µ d 3 k [ = ϵ µ (k, λ)a(k, λ)e ikx + ϵ µ (k, λ)a (k, λ)e ikx] (3.25) (2π) 3 2ω λ ϵ µ (k, λ) a(k, λ), a (k, λ) [a(k, λ), a (k, λ )] = 2ω(2π) 3 δ λ,λ δ 3 (k k ) (3.26) 19

22 Dirac Dirac : d 3 p [ ψ = apr u (2π) 3 r (p)e ipx + b 2E prv r (p)e ipx] (3.27) r d ψ 3 p [ = a (2π) 3 2E pr ū r (p)e ipx + b pr v r (p)e ipx] (3.28) r a pr, a pr b pr, b pr 0 {a pr, a qs} = (2π) 3 2Eδ 3 (p q)δ rs (3.29) {b pr, b qs} = (2π) 3 2Eδ 3 (p q)δ rs (3.30) is F (x 1 x 2 ) 0 T (ψ(x 1 ) ψ(x 2 )) 0 (3.31) 0 ψ(x 1 ) ψ(x 2 ) 0 t 1 > t 2 = 0 ψ(x (3.32) 2 )ψ(x 1 ) 0 t 1 < t 2 T ( ) 20

23 (3.11) (3.12) (3.13) (3.14) (3.32) is F (x 1 x 2 ) d 3 p = 0 T ( (a (2π) 3 pr u r (p)e ipx 1 + b 2E prv r (p)e ipx 1 ) r d 3 q (a (2π) 3 2E qsū(q)e iqx 2 + b qs v r (q)e iqx 2 ) 0 s = d 3 p d 3 q 0 (2π) 3 2E (2π) 3 2E (a pra qsu r (p)ū s (q)e iqx 2 ipx 1 + b prb qs v r (p) v s (q)e ipx 1 iqx 2 ) 0 θ(t 1 t 2 ) r,s d 3 p d 3 q 0 (2π) 3 2E (2π) 3 2E (a qsa pr u r (p)ū s (q)e iqx 2 ipx 1 + b qs b prv r (p) v s (q)e ipx 1 iqx 2 ) 0 θ(t 2 t 1 ) r,s = d 3 p d 3 q 0 (2π) 3 2E (2π) 3 2E (a pra qsu r (p)ū s (q)e iqx 2 ipx 1 ) 0 θ(t 1 t 2 ) r,s d 3 p d 3 q 0 (2π) 3 2E (2π) 3 2E (b qsb prv r (p) v s (q)e ipx 1 iqx 2 ) 0 θ(t 2 t 1 ) ( a 0 = 0, b 0 = 0) r,s = (2π) 3 2Eδ 3 d 3 p d 3 q (p q)δ rs 0 (2π) 3 2E (2π) 3 2E u r(p)ū s (q)e iqx 2 ipx 1 ) 0 θ(t 1 t 2 ) r,s d 3 p d 3 q 0 (2π) 3 2E (2π) 3 2E (a qsa pr u r (p)ū s (q)e iqx 2 ipx 1 ) 0 θ(t 1 t 2 ) r,s (2π) 3 2Eδ 3 d 3 p (p q)δ rs 0 (2π) 3 2E + r,s 0 d 3 p (2π) 3 2E r,s d 3 q (2π) 3 2E (v r(p) v s (q)e ipx 1 iqx 2 ) 0 θ(t 2 t 1 ) d 3 q (2π) 3 2E (b prb qs v r (p) v s (q)e ipx 1 iqx 2 ) 0 θ(t 2 t 1 ) ( (3.29), (3.30)) d 3 p = (2π) 3 2E ( u r (p)ū r (p)e ipx 2 ipx 1 θ(t 1 t 2 ) v r (p) v r (p)e ipx 1 ipx 2 θ(t 2 t 1 )) r r d 3 p = (2π) 3 2E ((p/ + m)eipx 2 ipx 1 θ(t 1 t 2 ) (p/ m)e ipx 1 ipx 2 θ(t 2 t 1 )) = i [ ] d 4 pe ip(x 1 x 2 ) p/ + m (2π) 4 p 2 m 2 + iϵ Dirac B 21

24 3.3: p µ M ( dσ = M fi 2 (2π) 4 δ 4 p 1 + p 2 ) p f 2E 1 2E 2 v 12 f f ( d 3 p f (2π) 3 2E f ) (3.33) p 1 p 2 p f E 1 E 2 E f v m 1 = m 3 = 0 E 1 = p 1, E 3 = p 3 (3.34) p 2 = 0 E 2 = m p 2 2 = m 2 (3.35) 2E 1 E 2 v 12 ( p1 2E 1 2E 2 v 12 = 2E 1 E 2 p ) 2 E 1 E ( 2 p1 = 4E 1 m 2 p ) 2 E 1 E 2 p 1 = 4E 1 m 2 ( p E 2 = 0) 1 = 4m 2 p 1. ( p E = γmv γm ( (3.35)) = v, γ ) 22

25 ( 1 (2π) 4 δ 4 p 1 + p 2 ) ( ) d 3 p f p f 2E 1 2E 2 v 12 (2π) 3 2E f f f ( 1 = 4m 2 p 1 (2π)4 δ 4 p 1 + p 2 ) ( ) d 3 p f p f (2π) 3 2E f f f ( ) ( ) 1 d = 4m 2 p 1 (2π)4 δ (E 1 + E 2 E 3 E 4 ) δ 3 3 p 3 d 3 p 4 (p 1 + p 2 p 3 p 4 ) (2π) 3 2E 3 (2π) 3 2E 4 ( ) ( ) 1 d = 4m 2 p 1 (2π)4 δ (E 1 + m 2 E 3 E 4 ) δ 3 3 p 3 d 3 p 4 (p 1 p 3 p 4 ) (2π) 3 2E 3 (2π) 3 2E 4 (3.36) d 3 p 4 ( ) ( ) 1 d 3 p 3 1 4m 2 p 1 (2π)4 δ (E 1 + m 2 E 3 E 4 ) (2π) 3 2E 3 (2π) 3 2E 4 (3.37) p 4 = p 1 p 3 E 4 E 4 = M 2 + p 2 4 = M 2 + (p 1 p 3 ) 2. (3.37) d 3 p 3 = p 3 2 d p 3 dω d p 3 p 3 δ(f(x)) = 1 f (x 0 ) δ(x x 0). (3.38) E 1 + m 2 E 3 E 4 E 1 + m 2 E 3 E 4 = E 1 + m 2 E 3 M 2 + (p 1 p 3 ) 2 = E 1 + m 2 p 3 M 2 + p p 1 p 3 cos θ + p 3 2 (E 1 + m 2 E 3 E 4 ) p 3 = 1 2 p 1 cos θ + 2 p 3 2E 4 = E 4 + E 3 p 1 cos θ E 4 = E 2 + E 1 p 1 cos θ E 4 = m 2 + p 1 (1 cos θ) E 4. (3.39) p 1 p 3 = p 4 p 2 (p 1 p 3 ) 2 = 2p 1 p 3 = 2(E 1 E 3 p 1 p 3 cos θ) = 2 p 1 p 3 (1 cos θ) (p 4 p 2 ) 2 = p p 2 2 2p 4 p 2 = 2m 2 2 2E 4 m 2 23

26 (3.39) p 1 (1 cos θ) = m2 2 + E 4 m 2. (3.40) p 3 (E 1 + m 2 E 3 E 4 ) p 3 = m 2 p 3 m 2 + E 4 E 4 p 3 = m 2 p 1 m 2 + m 2 E 4 p 3 = m 2 p 1 E 4 p 3. (3.41) (3.37) (3.38) (3.41) ( ) ( ) 1 p3 2 d p 3 dω 1 dσ = 4m 2 p 1 (2π)4 δ (E 1 + m 2 E 3 E 4 ) (2π) 3 2E 3 (2π) 3 2E 4 ( ) ( ) 1 1 p3 2 dω 1 = 4m 2 p 1 (2π)4 (E 1 +m 2 E 3 E 4 ) (2π) 3 2E 3 (2π) 3 2E 4 p 3 ( ) ( ) 1 E 4 p 3 p3 2 dω 1 = 16(2π) 2 m 2 p 1 m 2 p 1 E 3 E 4 ( ) 2 1 p3 = dω (3.42) 64π 2 m 2 2 p 1 γ + e γ + e (3.43) 3.4 vertex vertex H I (x)h I (y) = j µ (x)a µ (x)j ν (y)a ν (y) (3.44) it fi = S fi δ fi = ( ie)2 d 4 xd 4 y f T (j µ (x)a µ (x)j ν (y)a ν (y)) i. (3.45) 2! 24

27 3.4: y off-shell x y - on-shell off-shell x 25

28 x, y 2! it fi = ( i) 2 e 2 d 4 xd 4 y f T (j µ (x)a µ (x)j ν (y)a ν (y)) i (3.46) = ( i) 2 e 2 d 4 xd 4 yt [ k A µ (x) 0 p j µ (x)j ν p 0 A ν (y) k ] (3.47) = ( i) 2 e 2 d 4 xd 4 yϵ µ (k ) p T (j µ (x)j ν (y)) p ϵ ν (k)e ik x iky. (3.48) 0 A ν (y) k = 0 = 0 = 0 d 3 k 1 (2π) 3 2ω d 3 k 1 (2π) 3 2ω d 3 k 1 (2π) 3 2ω = ϵ ν (k, λ)e iky [ ϵ ν (k 1, λ 1 )a(k 1, λ 1 )e ik1y + ϵ ν (k 1, λ 1 )a (k 1, λ 1 )e ] ik1y k, λ 1 [ ϵ ν (k 1, λ 1 )a(k 1, λ 1 )e ] ik1y a (k, λ) 0 ( 0 a = 0) λ 1 [ ϵ ν (k 1, λ 1 )e ] ik1y (2π) 3 2ωδ 3 (k 1 k)δ λ1 λ 0 λ 1 j µ = ϕ(x)γ µ ϕ(x) p T (j µ (x)j ν (y) p = p T ( ψ(x)γ µ ψ(x) ψ(y)γ ν ψ(y)) p (3.49) Dirac (3.29) (3.30) p ψ(x)γ µ ψ(x) ψ(y)γ ν ψ(y) p = p d 3 q 1 (a (2π) 3 q 2E 1 r1ūr 1 (q 1)e iq 1 x + b q 1 r 1 v r1 (q 1)e iq 1 x )γ µ r 1 d 3 q 2 (a (2π) 3 q 2E 2 r 2 u r2 (q 2)e iq 2 x + b q 2 r v r 2 2 (q 2)e iq 2 x ) r 2 d 3 q 1 (a (2π) 3 q 2E 1 s 1 ū s1 (q 1 )e iq1y + b q1 r 1 v s1 (q 1 )e iq1y )γ ν s 1 d 3 q 2 (a (2π) 3 q2 s 2E 2 u s2 (q 2 )e iq2y + b q 2 r 2 v s2 (q 2 )e iq2y ) p. (3.50) s 2 16 p a aa a p, p baa b p 0, 0 26

29 p a q 1 r 1 a q 2 r 2 a q 1 s 1 a q2 s 2 p = 0 a p ua q 1 r 1 a q 2 r 2 a q 1 s 1 a q2 s 2 a pt 0 = (2π) 3 2Eδ 3 (p q 1)δ ur1 0 a q 2 r 2 a q 1 s 1 a q2 s 2 a pt 0 0 a q 1 r 1 a p ua q 2 r 2 a q 1 s 1 a q2 s 2 a pt 0 = (2π) 3 2Eδ 3 (p q 1)δ ur1 0 a q 2 r 2 a q 1 s 1 a q2 s 2 a pt 0 = ((2π) 3 2E) 2 δ 3 (p q 1)δ ur1 δ 3 (q 2 p)δ s2 t 0 a q 2 r 2 a q 1 s 1 0 = ((2π) 3 2E) 3 δ 3 (p q 1)δ ur1 δ 3 (q 2 p)δ s2 tδ 3 (q 2 q 1 )δ r2 s 1 (3.50) d 3 q 1 ū (2π) 3 u (p )e ip x γ µ u s1 (q 1 )e iq1xū s1 (q 1 )e iq1y γ ν u t (p)e ipy 2E s 1 = ū u (p )e ip x γ µ d 3 q 1 (2π) 3 2E (q / 1 + m)e iq1( x+y) γ ν u t (p)e ipy (3.51) p b q 1 r 1 a q 2 r 2 a q 1 s 1 b q 2 s 2 p = 0 a p ub q 1 r 1 a q 2 r 2 a q 1 s 1 b q 2 s 2 a pt 0 = 0 b q 1 r 1 b q 2 s 2 a p ua q 2 r 2 a q 1 s 1 a pt 0 = (2π) 3 2Eδ 3 (q 1 q 2 )δ r1 s 2 0 a p ua q 2 r 2 a q 1 s 1 a pt 0 = ((2π) 3 2E) 2 δ 3 (q 1 q 2 )δ r1 s 2 δ 3 (q 2 q 1 )δ r2 s 1 0 a p ua pt 0 (2π) 3 2Eδ 3 (q 1 q 2 )δ r1 s 2 0 a p ua q 1 s 1 a q 2 r 2 a pt 0 = ((2π) 3 2E) 3 δ 3 (q 1 q 2 )δ r1 s 2 δ 3 (q 2 q 1 )δ r2 s 1 δ 3 (p p)δ ut 0 0 ((2π) 3 2E) 3 δ 3 (q 1 q 2 )δ r1 s 2 δ 3 (p q 1 )δ us1 δ 3 (q 2 p)δ r2 t 0 0 = const. ((2π) 3 2E) 3 δ 3 (q 1 q 2 )δ r1 s 2 δ 3 (p q 1 )δ us1 δ 3 (q 2 p)δ r2 t 0 0 ((2π) 3 2E) 3 δ 3 (q 1 q 2 )δ r1 s 2 δ 3 (p q 1 )δ us1 δ 3 (q 2 p)δ r2 t ū u (p )e ip y γ ν (q 2 ) d 3 q 2 (2π) 3 2E ( q / 2 + m)e iq2(x y) γ µ u t (p)e ipx (3.52) (3.52) T x y µ ν ū u (p )e ip x γ µ d 3 q 2 (q 2 ) (2π) 3 2E ( q / 2 + m)e iq2(y x) γ ν u t (p)e ipy (3.53) 27

30 (3.51) (3.53) ū u (p )e ip x γ µ d 3 q 1 (2π) 3 2E (q / 1 + m)e iq1( x+y) γ ν u t (p)e ipy θ(t x t y ) ū u (p )e ip x γ µ d 3 q 2 (q 2 ) (2π) 3 2E ( q / 2 + m)e iq2(y x) γ ν u t (p)e ipy θ(t y t x ) = ū u (p )e ip x γ µ d 3 q 1 (2π) 3 2E {(q / 1 + m)e iq1( x+y) ( q/ 1 + m)e iq1( y+x) }γ ν u t (p)e ipy id = ū u (p )e ip x γ µ 4 q 1 (2π) { q/ 1 + m 4 q1 2 m 2 + iϵ }γν u t (p)e ipy. id it fi = ( i) 2 e 2 d 4 xd 4 yϵ µ (k )ū u (p )e ip x γ µ 4 q 1 (2π) { q/ 1 + m 4 q1 2 m 2 + iϵ }γν u t (p)e ipy ϵ ν (k)e ik x iky. = (2π) 4 δ 4 (k + p k p )( i) 2 e 2 ϵ µ (k )ū u (p )γ µ i(p/ + k/ + m) { (p + k) 2 m 2 + iϵ }γν u t (p)ϵ ν (k). M 1 = e 2 ϵ µ (k )ū u (p )γ µ p/ + k/ + m { (p + k) 2 m 2 + iϵ }γν u t (p)ϵ ν (k). (3.54) M 2 = e 2 ϵ ν (k)ū u (p )γ ν p/ k / + m { (p k ) 2 m 2 + iϵ }γµ u t (p)ϵ µ (k ) M M M [ M = M 1 + M 2 = e 2 ū s (p (p/ + k/ + m) ) ϵ/ (p + k) 2 m ϵ/ + ϵ/ (p/ ] k/ + m) 2 (p k ) 2 m 2 ϵ/ u r (p) (3.55) ϵ/ = ϵ µ γ µ, k/ = k µ γ µ p = (m; 0) ϵ = (0; ϵ), ϵ = (0; ϵ ) 28

31 ϵ p = (0; ϵ) (m; 0) = 0 (3.56) ϵ p = (0; ϵ ) (m; 0) = 0 (3.57) ϵ k = (0; ϵ) (ω; k) = ϵ k = 0 (3.58) ϵ k = (0; ϵ ) (ω ; k ) = ϵ k = 0 (3.59) ϵ (p + k ) = ϵ (p + k) = ϵ p + ϵ k = 0 ϵ = ϵ k (3.60) ϵ (p k) = ϵ (p k ) = ϵ p ϵ k = 0 (3.61) ϵ ϵ = ϵ ϵ = 1 ( (λ = 1, 2) (3.24) ). (3.62) p + k = p + k ϵ/p/ = ϵ µ γ µ p ν γ ν = ϵ µ p ν (2g µν γ ν γ µ ) = ϵ p p/ϵ/ = p/ϵ/ (3.63) ϵ / p/ = ϵ µγ µ p ν γ ν = ϵ µp ν (2g µν γ ν γ µ ) = ϵ p p/ϵ / = p/ϵ / (3.64) ϵ/k/ = ϵ µ γ µ k ν γ ν = ϵ µ k ν (2g µν γ ν γ µ ) = ϵ k k/ϵ/ = k/ϵ/ (3.65) ϵ / k / = ϵ µγ µ k νγ ν = ϵ µk ν(2g µν γ ν γ µ ) = ϵ k k / ϵ / = k / ϵ / (3.66) p k = p k (3.22) (3.63) (3.64) (p + k) 2 = (p + k ) 2 m 2 + 2p k + k 2 = m 2 + 2p k + k 2 p k = p k ( (3.22)) (3.67) (p + k ) 2 = (p k) 2 m 2 2p k + k 2 = m 2 2p k + k 2 p k = p k ( (3.22)) (3.68) (p + k) 2 m 2 = p 2 + 2p k + k 2 m 2 = 2p k (3.69) (p k ) 2 m 2 = p 2 2p k + k 2 m 2 = 2p k (3.70) (p/ + m)ϵ/u(p) = ( ϵ/p/ + mϵ/)u(p) = ϵ/(p/ m)u(p) = 0 (3.71) (p/ + m)ϵ / u(p) = ( ϵ/ p/ + mϵ / )u(p) = ϵ / (p/ m)u(p) = 0 (3.72) 29

32 M [ ] ϵ/ M = e 2 ū r (p k/ϵ/ ) 2p k + ϵ/k/ ϵ / u 2k s (p) (3.73) p (3.42) 2 M 2 = e4 [ ] { [ ] ϵ/ ū r (p k/ϵ/ ) 2 2p k + ϵ/k/ ϵ / ϵ/ u 2k s (p) ū r (p k/ϵ/ ) p 2p k + ϵ/k/ ϵ / u 2k s (p)} (3.74) p r,s {ū r (p )γ µ γ ν γ ρ u s (p)} = u s (p) γ ρ γ ν γ µ ū r (p ) = u s (p) γ 0 γ 0 γ ρ γ 0 γ 0 γ ν γ 0 γ 0 γ µ γ 0 u r (p ) ( (γ 0 ) 2 = 1 ) = ū s (p)(γ 0 γ ρ γ 0 )(γ 0 γ ν γ 0 )(γ 0 γ µ γ 0 )u r (p ) = ū s (p)γ ρ γ ν γ µ u r (p ) ( (3.20)) (3.74) M 2 = e4 [ ϵ/ ū r (p k/ϵ/ ) 2 2p k + ϵ/k/ ϵ / 2k p r,s [ { = e4 ϵ/ 2 Tr (p/ k/ϵ/ + m) 2p k + ϵ/k/ ϵ / 2k p ] [ ϵ/k/ϵ/ u s (p)ū s (p) } (p/ + m) k Tr[(p / + m)ϵ / k/ϵ/(p/ + m)ϵ/k/ϵ / ] = Tr[p / ϵ / k/ϵ/p/ϵ/k/ϵ / ] + m 2 Tr[ϵ / k/ϵ/ϵ/k/ϵ / ] ] u r (p ) 2p k + ϵ / k/ ϵ/ 2k p { ϵ/k/ϵ/ 2p k + ϵ / k/ }] ϵ/ 2k p ( (3.13 )) ( m γ (3.19) 0) (3.75) = Tr[p / ϵ / ϵ/k/p/k/ϵ/ϵ / ] + m 2 Tr[ϵ / ϵ/k/k/ϵ/ϵ / ] ( (3.65)) (3.76) m 2 k/k/ 0 A, B Tr[Ak/k/B] = Tr[Ak µ γ µ k ν γ ν B] = Tr[Ak µ k ν (2g µν γ ν γ µ )B] ( (3.15)) = 2k µ k µ Tr[AB] Tr[Ak/k/B] = Tr[Ak/k/B] ( (3.22) k µ k µ = 0) 30

33 (3.76) Tr[p / ϵ / ϵ/k/p/k/ϵ/ϵ / ] = 2k ptr[p / ϵ / ϵ/k/ϵ/ϵ / ] + Tr[p / ϵ / ϵ/p/k/k/ϵ/ϵ / ] = 2k ptr[p / ϵ / ϵ/k/ϵ/ϵ / ] ( (3.77)) = 2k ptr[p / ϵ / ϵ/( ϵ/k/)ϵ / ] ( (3.65)) Tr[Ak/k/B] = 0 (3.77) k/p/ = 2k p p/k/ (3.78) = 2k ptr[p / ϵ / k/ϵ / ] ( (3.24) ϵ/ϵ/ = 2ϵ ϵ ϵ/ϵ/ = 2 ϵ/ϵ/ ϵ/ϵ/ = 1) = 2(k p)p ρϵ σk α ϵ β4(g ρσ g αβ + g ρβ g σα g ρα g σβ ) ( (3.18)) = 2(k p)4{(p ϵ )(k ϵ ) + (p ϵ )(ϵ k) (p k)(ϵ ϵ )} = 8(k p){2(p ϵ )(k ϵ ) + (p k)} ( (3.62)) = 8(k p){2(k ϵ ) 2 + (p k )} ( (3.60)). Tr[(p / + m)ϵ / k/ϵ/(p/ + m)ϵ/k/ϵ / ] = 8(k p){2(k ϵ ) 2 + (p k)}. (3.79) (3.75) k 2 k k Tr[ϵ/k / ϵ / (p/ + m)ϵ / k / ϵ/(p / + m)] = Tr[ϵ/k / ϵ / p/ϵ / k / ϵ/p / ] + m 2 Tr[ϵ/k / ϵ / ϵ / k / ϵ/] 1 = Tr[ϵ/k / ϵ / p/ϵ / k / ϵ/p / ] m 2 Tr[ϵ/k / k / ϵ/] = Tr[ϵ/k / ϵ / p/ϵ / k / ϵ/p / ] = 2(ϵ p )Tr[ϵ/k / ϵ / k / ϵ/p / ] Tr[ϵ/k / p/ϵ / ϵ / k / ϵ/p / ] 2(ϵ p )Tr[ϵ/k / ϵ / k / ϵ/p / ] = 2(ϵ p )Tr[ϵ/( ϵ / k / )k/ ϵ/p / ] = 0 31

34 2 Tr[ϵ/k / p/ϵ / ϵ / k / ϵ/p / ] = Tr[ϵ/k / p/k / ϵ/p / ] = 2(k p)tr[ϵ/k / ϵ/p / ] + Tr[ϵ/p/k / k / ϵ/p / ] = 2(k p)tr[ϵ/k / ϵ/p / ] = 2(k p)4{(ϵ k )(ϵ p ) + (ϵ p )(ϵ k ) (ϵ ϵ)(k p )} = 2(k p)4{2(ϵ p )(ϵ k ) + (k p )} = 2(k p)4{ 2(ϵ k ) 2 + (k p )} = 8(k p){ 2(ϵ k ) 2 + (k p)} ( (3.67)) Tr[ϵ/k / ϵ / (p/ + m)ϵ / k / ϵ/(p / + m)] = 8(k p){ 2(ϵ k ) 2 + (k p)} (3.80) k k Tr [(p/ + m) {ϵ/ k/ϵ/} (p/ + m) {ϵ/ k/ ϵ/}] = Tr [p/ ϵ/ k/ϵ/p/ϵ / k/ ϵ/] + m 2 Tr [ϵ/ k/ϵ/ϵ / k/ ϵ/] ( γ 0) = Tr [p/ ϵ/ ϵ/k/p/k/ ϵ / ϵ/] + m 2 Tr [ϵ/ ϵ/k/k/ ϵ / ϵ/] ( (3.65) (3.66)) = (Tr [p/ϵ/ ϵ/k/p/k/ ϵ / ϵ/] + m 2 Tr [ϵ/ ϵ/k/k/ ϵ / ϵ/]) + Tr [k/ϵ/ ϵ/k/p/k/ ϵ / ϵ/] Tr [k/ ϵ/ ϵ/k/p/k/ ϵ / ϵ/] ( p = p + k k ) (3.81) γ (3.81) Tr [p/ϵ/ ϵ/k/p/k/ ϵ / ϵ/] + m 2 Tr [ϵ/ ϵ/k/k/ ϵ / ϵ/] p/p/ = 1 2 {p/, p/} = 1 2 p µp ν 2g µν = p p = m 2 (3.82) = 2(k p)tr [p/ϵ/ ϵ/k/ ϵ / ϵ/] Tr [p/ϵ/ ϵ/p/k/k/ ϵ / ϵ/] + m 2 Tr [ϵ/ ϵ/k/k/ ϵ / ϵ/] ( (3.78)) = 2(k p)tr [p/ϵ/ ϵ/k/ ϵ / ϵ/] Tr [ϵ/ p/p/ϵ/k/k/ ϵ / ϵ/] + m 2 Tr [ϵ/ ϵ/k/k/ ϵ / ϵ/] ( (3.63) (3.64)) = 2(k p)tr [p/ϵ/ ϵ/k/ ϵ / ϵ/] m 2 Tr [ϵ/ ϵ/k/k/ ϵ / ϵ/] + m 2 Tr [ϵ/ ϵ/k/k/ ϵ / ϵ/] ( (3.82)) = 2(k p)tr [p/ϵ/ ϵ/k/ ϵ / ϵ/] = 2(k p)2(ϵ ϵ)tr [p/k/ ϵ / ϵ/] + 2(k p)tr [p/ϵ/ϵ/ k/ ϵ / ϵ/] (3.83) 32

35 (3.83) 1 4(k p)(ϵ ϵ)tr [p/k/ ϵ / ϵ/] = 4(k p)(ϵ ϵ)4{(p k )(ϵ ϵ) + (p ϵ)(k ϵ ) (k ϵ)(p ϵ )} 2 = 16(k p)(p k )(ϵ ϵ) 2 ( (3.56) (3.57)) 2(k p)tr [p/ϵ/ϵ / k/ ϵ / ϵ/] = 2(k p)tr [ϵ/p/ϵ / k/ ϵ / ϵ/] (3.83) = 2(k p)tr [ϵ/ϵ/p/ϵ/ k/ ϵ / ] ( (3.15)) = 2(k p)tr [p/ϵ / k/ ϵ / ] ( ϵ/)ϵ/ = 1) = 2(k p)4((p ϵ )(k ϵ ) + (p ϵ )(ϵ k ) (p k )(ϵ ϵ )) = 8(k p)(k p) ( (3.56) (3.57)) Tr [p/ϵ/ ϵ/k/p/k/ ϵ / ϵ/] + m 2 Tr [ϵ/ ϵ/k/k/ ϵ / ϵ/] = 16(k p)(p k )(ϵ ϵ) 2 8(k p)(k p) (3.81) Tr [k/ϵ/ ϵ/k/p/k/ ϵ / ϵ/] = Tr [k/ϵ/ k/ϵ/p/k/ ϵ / ϵ/] ( (3.65)) = 8(k p)(p k ){2(ϵ ϵ) 2 1} (3.84) = 2(k ϵ )Tr [k/ϵ/p/k/ ϵ / ϵ/] + Tr [ϵ/ k/k/ϵ/p/k/ ϵ / ϵ/] = 2(k ϵ )Tr [k/ϵ/p/k/ ϵ / ϵ/] ( (3.15) = 2(k ϵ )Tr [ϵ/k/ϵ/p/k/ ϵ / ] ( (3.65)) = 2(k ϵ )Tr [k/ϵ/ϵ/p/k/ ϵ / ] ( ϵ/)ϵ/ = 1) = 2(k ϵ )Tr [k/p/k/ ϵ / ] = 2(k ϵ )4((k p)(k ϵ ) + (k ϵ )(p k ) (k k )(p ϵ )) = 8(k ϵ ) 2 (p k ) ( (3.57) (3.59)) 33

36 (3.81) Tr [k/ / ϵ ϵ/k/p/k / ϵ / ϵ/] = Tr [k/ ϵ / ϵ/k/p/ϵ / k / ϵ/] ( (3.66)) = 2(k ϵ)tr [k/ / ϵ ϵ/k/p/ϵ / ] Tr [k/ ϵ / ϵ/k/p/ϵ / ϵ/k / ] = 2(k ϵ)tr [k/ / ϵ ϵ/k/p/ϵ / ] ( Tr [k/ ϵ / ϵ/k/p/ϵ / ϵ/k / ] = Tr [k/ k / ϵ / ϵ/k/p/ϵ / ϵ/] = 0) = 2(k ϵ)tr [ϵ/ k / ϵ / ϵ/k/p/] = 4(k ϵ)(k ϵ )Tr [ϵ/ ϵ/k/p/] 2(k ϵ)tr [k/ ϵ / ϵ / ϵ/k/p/] = 2(k ϵ)tr [k/ ϵ / ϵ / ϵ/k/p/] ( (3.59)) = 2(k ϵ)tr [k/ ϵ/k/p/] = 2(k ϵ)4((k p)(ϵ k) + (k ϵ)(k p) (k k)(ϵ p)) = 8(k ϵ) 2 (k p) (3.81) Tr [(p/ + m) {ϵ/ k/ϵ/} (p/ + m) {ϵ/ k/ ϵ/}] = 8(k p)(p k )(2(ϵ ϵ) 2 1) 8(k ϵ ) 2 (p k ) + 8(k ϵ) 2 (k p) M M 2 = e 4 8(p k) 8(k p){2(k 2 ϵ ) 2 + (p e 4 k)} + 8(p k ) 2 8(k p){ 2(ϵ k ) 2 + (k p)} e 4 + 4(p k)(p k ) (8(k p)(p k )(2(ϵ ϵ) 2 1) 8(k ϵ ) 2 (p k ) + 8(k ϵ) 2 (k p)) = e 4 [ 2(k ϵ ) 2 + (p k) p k = e 4 [ p k p k + k p k p + 4(ϵ ϵ) (ϵ k ) 2 + (k p) k p ] (3.85) + 2{2(ϵ ϵ) 2 1) (k ϵ ) 2 p k + (k ϵ) 2 p k } p k = (m; 0) (ω; k) = mω, (3.86) p k = (m; 0) (ω ; k ) = mω (3.87) ] [ mω M 2 = = e 4 mω + mω ] mω + 4(ϵ ϵ) 2 2 [ ω = e 4 ω + ω ] ω + 4(ϵ ϵ)

37 3.5: k, k ϵ λ ϵ 1 ϵ 2 k θ ϵ 1 ϵ 2 k ϵ 2 ϵ 2 ϵ 1 θ ϵ 1 (3.42) ( ) dσ 1 k dω = e 4 2 [ ω LAB 64π 2 m 2 e k ω + ω ] ω + 4(ϵ ϵ) 2 2 (3.88) ( ) e 4 ω 2 [ ω = 64π 2 m 2 e ω ω + ω ] ω + 4(ϵ ϵ) 2 2. (3.89) dσ dω = 1 1 LAB 2 64π 2 m 2 λ,λ e = 1 64π 2 m 2 e ( ) ω e 4 2 [ ω ω ω + ω ] ω + 4(ϵ λ ϵ λ )2 2 ( ) ω e 4 2 [ω 2 ω ω + ω ω + (ϵ λ ϵ λ )2 2 λ,λ ] (3.90). (3.91) 3.5 ϵ 1 ϵ 2 k θ ϵ 1 ϵ 2 k ϵ 2 ϵ 2 ϵ 1 θ ϵ 1 ϵ 2 ϵ 2 = 1 ϵ 1 ϵ 1 = ϵ 1 ϵ 1 cos θ = cos θ dσ dω = LAB 1 32π 2 m 2 e ( ) ω e 4 2 [ ω ω ω + ω ] ω + (1 + cos2 θ) 2. (3.92) (3.92) SI 35

38 2 1 m 2 e 1 ħ2 c 2 [MeV 2 cm 2 ] = ħ2 m 2 ec 4 [MeV 2 ] m 2 ec 2 (3.93) e 4 e 4 [C 4 ] ϵ 2 0[C 4 /(MeV 2 cm 2 )]ħ 2 c 2 [MeV 2 cm 2 ] (3.94) m 2 e SI (3.92) e4 32π 2 m 2 e 1 1 e 4 1 ( ) ( ) ħ 2 e 4 32π 2 m 2 e 32π 2 m 2 ec 2 ϵ 2 0ħ 2 c 2 = e 4 32π 2 ϵ 2 0m 2 ec 4 r e = e 2 4πϵ 0 m ec 2 e 4 32π 2 ϵ 2 0m 2 ec = r2 e 4 2 ω ω hν (3.6) ω ω = hν hν 1 = hν m ec (1 cos θ) = α(1 cos θ) + 1 α = (3.92) ω ω + ω ω + (1 + cos2 θ) 2 = (3.92) SI [ dσ dω = r2 e LAB 2 hν m e c 2 (3.95) α(1 cos θ) + (1 + α(1 cos θ)) + (1 + cos2 θ) 2 = (1 + cos 2 θ) α(1 cos θ) + α(1 cos θ) + α2 (1 cos θ) α(1 cos θ) = 1 + cos 2 θ + α2 (1 cos 2 θ) 1 + α(1 cos θ) α(1 cos θ) ] 2 [ ] 1 + cos 2 θ + α2 (1 cos 2 θ). (3.96) 1 + α(1 cos θ) 36

39 662 kev 3.6 (θ < 90 ) (θ < 90 ) 8 θ θ = 0 θ = 90 θ θ = : 662 kev (θ < 90 ) (θ < 90 ) θ θ 37

40 4 4.1 NaI(Tl) γ γ : NaI(Tl) Z γ NaI(Tl) NaI(Tl) NaI Tl

41 4.2: : (PMT) PMT 39

42 NaI(Tl) NaI(Tl) 88/DM NaI 5.08 cm(2 inch) 2.54 cm(1 inch) /DM NaI(Tl) 4.5 NaI 4.6 NaI NaI PMT 4.4: 88/DM NaI(Tl) NaI PMT( ) NaI PMT 2 inch NaI 40

43 図 4.5: 88/DM 型の NaI(Tℓ) 検出器の正面図 NaI 結晶と検出器のケースの大きさが見 えるようにその他の部分を省いている NaI 結晶は直径 2 inch 高さ 2 inch の円柱形で ある 図 4.6: 使用した NaI 検出器の画像 NaI 結晶と PMT が一つのアルミケースに入ってい る 左に接続されている黒いソケットは PMT に適切な電圧をかけるための Divider 回路 である 銀色のアルミケースの内部の左側に PMT がある 右側には NaI 結晶がある 41

44 NaI の密度 ρnai は 3.67 g/cm3 であり モル質量 ANaI は 150 g/mol である そのため 単位体積あたりの原子数 n 個/cm3 は n= ρnai NA = (個/cm ) ANaI 150 (4.1) である ナトリウム Na の電子数は 11 ヨウ素 I の電子数は 53 であるから 単位体積あ たりの電子数は 3 n ( ) (個/cm ) である 4.2 NIM/CAMAC モジュール 本実験の同時計測回路およびその読み取りで使用した NIM と CAMAC のモジュール について説明する 図 はモジュールを実際に接続している様子である 図 4.7: 使用した NIM モジュール HVPS は別の場所にある Gate and Delay Generator は 2 台用いている このクレートの下に CAMAC のクレート (図 4.8) がある 42

45 図 4.8: 使用した CAMAC モジュールおよび Delay このクレートの上に NIM のクレー トがある ADC は 2 つ接続されているが使用するのは上の 1 つだけである NIM モジュール NIM とは Nuclear Instrument Modules の略で原子核 高エネルギー物理のためのモ ジュールの規格である High Voltage Power Supply 光電子増倍管に電圧をかけるために高電圧電源 (HVPS) を用いた 本実験では 林栄 精器株式会社の RPH-030 型 -3.2 kv 4ch 高圧電源を用いた Gate and Delay Generator CAMAC の ADC に Gate 信号を送るのに用いる NIM 信号の幅を広げたり 信号のタ イミングを遅らせることができる 本実験で使用したのは テクノランドコーポレーショ ン社の N-TM 307 型 2ch Gate and delay である Discriminator 光電子増倍管から出力されたアナログ信号を NIM のデジタル信号に変換するために用 いる Discriminator はアナログ信号が設定したスレッショルドを超えたときにデジタル 信号を出力する装置である 本実験ではテクノランドコーポレーション社の N-TM 405 型 8 ch Discriminator を用いた 43

46 Coincidence Coincidence VETO VETO N-TM 103 3ch 4-Fold 1-VETO Coincidence Delay Delay Delay N Hoshin 16ch 300 ns fixed delay TDC 300 ns N-TS ns delay delay ADC Fan in / Fan out Fan In / Fan Out Phillips Scientific NIM Model 740 quad linear Fan-In/Out CAMAC CAMAC Computer Automated Measurement And Control NIM Scaler Scaler C-SE 113 ADC CAMAC ADC(Analog to Digital Converter) 4096 ch Gate C-QV

47 TDC TDC(Time to Digital Converter) Start Stop C-TS 105 CC-USB CAMAC (Crate controller CC) Wiener CC-USB CC-USB USB 4.3 γ γ α β γ α 4 2α 241 Am A ZX A 2 Z 2 Y +4 2 α β A ZX A Z+1 Y + e + ν e A ZX A Z 1 Y + e + + ν e (β ) (β + ) β 3 β

48 4.9: β n p d W u W e ν e 4.10: β + p n u W + d W + e + ν e β γ X X + γ γ β γ γ t N(t) N(t) dn(t) dt = λn(t) (4.2) λ τ τ = 1 λ (4.2) N(t) = N 0 exp( λt) 46

49 N 0 t = 0 t 1/2 t 1/2 = log 2 λ dn dt dn ( dt = dn ) dt e λt = t=0 = τ log 2 t=0 = λn 0 ( dn dt ) t=0 e t log 2 t 1/ Cs 137 Cs β 137 Ba 662 kev γ 137 Cs Cs t 1/ Cs 4.11: 137 Cs [4] µci(±3.7%) (Bq) exp ( ) log 2 = (Bq) Cs 3 4 mm 1 mm 47

50 図 4.12: 137 Cs 線源の画像 線源は 23 mm 10 mm の長方形のケースに入っている 放 射性物質は 1 mm 程度の幅をもつ Na Na は崩壊して 22 Ne になり 1275 kev の γ 線を放出する 22 Na 原子の崩壊図を図 4.13 に示す 22 Na 線源によって観測される γ 線は γ 崩壊による 1275 kev の γ 線の他に 511 kev の γ 線もある 22 Na 原子核は β + 崩壊して陽電子を放出するが この陽電子が物質 中の電子と対消滅して 511 kev の γ 線を 2 つ放出するためである e + e+ 2γ このとき 陽電子は物質中でエネルギーを失ってから対消滅するため γ 線のエネルギー は電子と陽電子の静止質量エネルギーを 2 つにわけた 511 kev となるのである 22 Na の 半減期 t1/2 は 年である 本実験で使用した 22 Na 線源の 2014 年 2 月 12 日時点で Bq であった そのため 2016 年 1 月 20 日時点では ( ) (Bq) exp log 2 = (Bq) である 放射能は JRIA(日本アイソトープ協会) 校正値であり 誤差は ±2 5% であるが 今回は ±5 % と見積もった 48

51 4.13: 22 Na [4] Co 60 Co 60 Ni 1333 kev 1173 kev γ 60 Co Co t 1/ : 60 Co [4] 60 Co Bq ( (Bq) exp ) log 2 = (Bq) JRIA( ) ±2 5% ±5 % 49

52 Discriminator1 2 Coincidence Gate and Delay Generator 5.1: NIM CAMAC ADC TDC Scaler CC-USB 5.2 Discriminator 1 2 Coincidence Gate and Delay Generator Discriminator mv Discriminator mv γ Discriminator width 40 nsec Coincidence 3 nsec 50

53 5.2: CAMAC TDC Scaler CC-USB CAMAC ADC 5.1 Discriminator 1 2 Coincidence Gate and Delay Generator CAMAC Scaler ADC TDC ADC Gate Coincidence Gate and Delay Generator 5.3 Gate NaI 1 NaI 2 Gate and Delay Generator Coincidence Veto Gate Gate and Delay Generator Gate and Delay Generator Veto 200 µsec CAMAC 51

54 5.3: ADC Gate PMT ADC Gate NaI 1 NaI 2 Gate 5.2 ADC CAMAC ADC 5.1 Coincidence Discriminator 1 Discriminator sec NaI NaI Root 5.3 ADC Channel 1000 ch 40 K γ D ADC Channel 0 ch pedestal Pedestal ADC Gate 52

55 5.4: NaI Cs 662 kev 5.5: NaI 1 22 Na 511 kev 5.6: NaI 1 22 Na 1275 kev 53

56 5.7: NaI 1 60 Co 1173 kev 5.8: NaI 1 60 Co 1333 kev Energy (kev) = p 0 + p 1 (ADC Channel) Root p 0 p 1 σ p0 σ p1 E 2 E = σ 2 p 0 + (ADC Channel) 2 σ 2 p 1 cov(p 0, p 1 ) NaI Energy (kev) = ( ± 6.469) + (1.419 ± ) (ADC-1 Channel) (5.1) NaI Energy (kev) = (12.58 ± 1.963) + (1.448 ± ) (ADC-2 Channel) (5.2) reduced χ 2 NaI 1 12 NaI

57 5.9: NaI : NaI NaI(Tl) 5.11 NaI 2 γ NaI γ I s NaI γ 55

58 5.11: NaI 2 N det Ω ϵ abs N det = ϵ abs I s Ω ϵ int ϵ abs = Ω 4π ϵ int ϵ ϵ int Root ) A (x µ)2 exp ( + p 2πσ 2σ p 1 x x µ, σ, A A A 4 ch 1 bin 1/4 22 Na 511 kev 2 γ 1/ : NaI Cs 56

59 5.13: NaI 2 22 Na 5.14: NaI 2 22 Na 5.15: NaI 2 60 Co 5.16: NaI 2 60 Co ϵ Ω γ E γ ϵ Ω = p 0 (E γ ) p 1 (p 0, p 1 ) Root NaI 2 ϵ Ω 5.17 ϵ Ω = E γ 57

60 5.1: γ (MeV) ϵ Ω p 0 p 1 σ p0 = σ p1 = cov(p 0, p1) = σ Is σ Ndet 2 ϵ Ω = (ϵ Ω) 2 ( σ 2 Ndet N 2 det + σ2 I s I 2 s ) (5.3) σ p0 σ p1 ϵ Ω ϵ Ω ( ) 2 (ϵ Ω) 2 ϵ Ω = σp 2 p ( (ϵ Ω) p 1 ) 2 ( (ϵ Ω) σp p 1 ) ( (ϵ Ω) p 0 ) cov(p 0, p 1 ) = σ 2 p 0 E 2p 1 γ + σ 2 p 1 (p 0 E p 1 γ log E γ ) 2 + 2cov(p 0, p 1 )p 0 E 2p 1 γ log E γ (5.4) 58

61 5.17: NaI 2 ϵ Ω Ω NaI 35 cm ϵ (5.3) σ Is σ Ndet ϵ Ω 2 ϵ Ω (5.4) γ ϵ Ω 59

62 6 6.1 NaI 1 active target γ NaI 1 NaI 2 NaI 1 NaI NaI Cs γ 1 cm NaI 1 NaI 1 θ NaI NaI 6.1 Coincidence NaI 2 NaI Cs ( ) 300 sec 6.5 ADC-1 channel 6.6 ADC-1 channel 6.6 γ 465 ch γ 662 kev 662 kev γ γ

63 6.1: 137 Cs γ NaI 1 γ NaI 1 θ NaI 2 l 1 NaI 137 Cs 6.2: 5 cm γ γ NaI 61

64 6.3: ( ) γ 1 cm 6.4: NaI NaI NaI 1 D 2 NaI 1 NaI NaI D 3 62

65 6.5: 137 Cs NaI ADC-Channel 6.6: 137 Cs NaI 1 63

66 465 ch ± (kev) 662 kev ch γ γ kev ( ) 330 ch ± (kev) ch γ kev 184 kev 120 ch ± (kev) ch γ γ ch 1 bin counts counts/sec 64

67 6.7: 137 Cs NaI ADC-Channel D NaI γ γ NaI γ D D D = D l 1 l (cm) = 1 (cm) = (cm) 6.9 NaI 1 θ ( 6.9 ) γ γ 65

68 6.8: D = 1 cm γ D 6.9: NaI θ γ θ d θ γ γ 66

69 NaI d L(θ) Ω Ω = D (l 1 θ) l 2 1 = D θ l 1 γ γ I s I s Ω 4π = I s D θ 4πl 1 (2.4) L(θ) = n 1 e µd Ω I s µ 4π d θ d = 2 R 2 l1 2 sin 2 θ L(θ) γ θ L n Is 4π = θmax θ min L(θ) n Is 4π = θmax θ min 1 e µd µ D θ l 1 I s n µ 2.9 NaI 662 kev γ µ = 0.075/3.67 = cm 1 L n Is 4π L n Is 4π = (6.1) N y σ N y = Lσ = L n I s 4π I s nσ 4π 67

70 137 Cs I s = Bq (6.1) γ σn σn = = 99, 83 (counts/sec) ( ) L Is 4π n Is 4π (counts/sec) π (cm 1 ) NaI n (4.1) σ σ = (cm 1 ) (cm 3 ) = (cm 2 ) = 5.79 (barn) σn ρ = (cm 1 ) 3.67 = (cm 2 /g) 2.9 γ 662 kev τ ρ = (cm2 /g) σ ρ = (cm2 /g) 2 γ 2.9 γ 68

71 NaI 2 Coincidence NaI 1 active target NaI 1 2 θ γ θ ADC 6.10: θ = 30 ADC ADC-1 Channel ADC-2 Channel NaI 1 NaI 2 2 θ = NaI 1 NaI kev NaI 1 γ NaI 2 ( 6.16) γ 662 kev 69

72 6.11: θ = 45 ADC 6.12: θ = 60 ADC 70

73 6.13: θ = 75 ADC 6.14: θ = 90 ADC 71

74 6.15: θ = 45 ADC γ NaI 1 NaI 2 NaI 1 NaI 2 γ NaI 1 NaI : 6.15 NaI 1 γ NaI 2 NaI 1 NaI 1 NaI kev 72

75 6.17: 6.15 NaI 1 NaI 2 2 γ NaI NaI 1 NaI kev ( ) θ γ NaI 1 NaI NaI 1 NaI 2 γ NaI 1 NaI 2 2 ( 6.17) γ NaI 1 NaI θ ADC-1 Channel+ADC-2 Channel NaI 662 kev kev ADC-1 Channel - ADC-2 Channel

76 6.18: θ = 45 ADC-1+ADC-2 Channel 6.19: θ = 45 ADC-1+ADC-2 Channel ADC-1 ADC-2 Channel ADC-Channel 6.1 ADC-1 Channel ADC-2 Channel 74

77 6.20: ADC-1+ADC-2 Channel ADC 1 bin ADC-1+ADC-2 Channel x θ = : θ ADC Channel θ ( ) ADC-1 + ADC-2 channel ADC-1 ADC-2 channel

78 6.21: ADC-1 ADC-2 Channel ADC ADC-1+ADC-2 Channel ADC-1 ADC-2 Channel 1 bin ADC-1 ADC-2 Channel x θ = 45 76

79 6.22: NaI γ E mean E mean = i E in i i bin E i n i bin θ ADC- 1 ADC-2 Channel 6.22 Emean σ ni p 0 + p 1 Channel i E mean = i E in i = 2 E mean = σ 2 p 0 + = σ 2 p 0 + i n i i n i i (p 0 + p 1 Channel i ) i n i i = p 0 + p Channel i n i 1 i n i ( i Channel ) 2 i n i σp ni i ( i Channel i n i i n i p 1 i Channel i n i i n i n i ) 2 ( σp p 2 i Channel2 i n i 1 ( i n ( i) 2 2 ( n i ) 2 i Channel ) i n i ) 2 ( i n i) 3 77

80 6.22 NaI NaI 2 γ NaI 1 NaI 1 NaI 2 NaI 2 NaI kev TDC NaI 1 NaI 2 Coincidence Discriminator width ±37 nsec TDC nsec ±37 nsec Coincidence( ) 200 µsec TDC TDC TDC θ 30 TDC-0 Channel TDC-1 Channel 90 TDC-0 Channel TDC-1 Channel NaI 1 NaI 2 NaI 1 NaI 2 78

81 θ θ Coincidence NaI 1 NaI 2 θ = 30 θ = NaI 1 NaI θ = 30 θ = 90 θ = 30 Discriminator θ = 30 NaI 2 NaI 1 79

82 6.23: θ = 30 tdc-1 tdc-2 channel 6.24: θ = 45 tdc-1 tdc-2 channel 6.25: θ = 60 tdc-1 tdc-2 channel 6.26: θ = 75 tdc-1 tdc-2 channel 6.27: θ = 90 tdc-1 tdc-2 channel 80

83 6.28: θ = 30 tdc-1 tdc-2 channel 6.29: θ = 45 tdc-1 tdc-2 channel 6.30: θ = 60 tdc-1 tdc-2 channel 6.31: θ = 75 tdc-1 tdc-2 channel 6.32: θ = 90 tdc-1 tdc-2 channel 81

84 6.33: θ = 30 tdc-1 tdc-2 channel 6.34: θ = 45 tdc-1 tdc-2 channel 6.35: θ = 60 tdc-1 tdc-2 channel 6.36: θ = 75 tdc-1 tdc-2 channel 6.37: θ = 90 tdc-1 tdc-2 channel 82

85 6.38: θ = 90 Coincidence NaI 1 NaI ns 1 10 mv mv 6.39: θ = 30 Coincidence NaI 1 NaI ns 1 10 mv mv 83

86 6.3.3 NaI NaI 2 N det N det = L dσ dω Ω ϵ (6.2) Ω NaI 1 NaI 2 ϵ NaI 2 γ NaI (6.2) f(θ) N det = L dσ dω Ω ϵ f(θ) (6.3) 6.40 x γ x γ y γ x γ γ x γ γ y l l = y ( D 2 D 2 + D 3 ) sin θ x γ I scat y I scat y D x γ γ I scat D D 2 D 2 e µl = I scat D D 2 D 2 exp ( µ y ( D D )) D 3 sin θ = I scat sin θ (1 e µd sin θ )e µ( D µd 2 D 2+D 3 ) 84

87 6.40: x γ θ NaI γ y l γ NaI 6.41: x γ θ 6.40 γ f(θ) 85

88 f(θ) f(θ) = sin θ µd (1 e µd sin θ )e µ( D 2 D 2+D 3 ) µ γ E γ θ µ γ 1 MeV µ = µ photo + µ comp µ comp (2.8) µ comp = Zσ c N A A ρ N A A NaI Z NaI (Z = Z Na + Z I ) ρ NaI τ [5] [ ( 3 mc τ = 2 φ 0α 4 Z 5 2 hν [ ( 4 γ(γ 2) + 3 γ + 1 )] ( mc 2 1 hν ) 4 (γ 2 1) γ γ 2 1 log ( γ + ))] γ 2 1 γ γ 2 1 ν1 exp( 4ξcot 1 ξ) 2π ν 1 exp( 2πξ) hν γ E γ mc 2 Z Z I ϕ 0 = 8π 3 ( e2 mc 2 )2 α = hν + mc2 γ = mc 2 hν 1 = (Z 0.03) 2 mc 2 α2 2 ν1 ξ = ν ν 1 µ photo µ photo = τ N A A ρ 86

89 γ NaI 1 NaI ADC Channel θ 6.2 (6.3) : θ ADC Channel θ ( ) n (counts) : γ θ (cm 2 /str) 87

90 6.4: θ θ ( ) θ min ( ) θ max ( ) I s σ Is ϵ Ω σ ϵ Ω n σ n σ dσ/dω σ 2 dσ/dω = ( ) { 2 (σis dσ dω I s ) 2 + } ( ) 2 ( σϵ Ω σn ) 2 + ϵ Ω n (6.4) I s σ Is % 5.3 (5.4) n n (%) : θ ϵ Ω θ ( ) ϵ Ω (%) n (%) θ θ 6.43 θ max θ min θ γ θ 6.4 NaI 2 2 (6.4)

91 6.43: θ θ max θ min θ γ 6.44: I s σ Is ϵ Ω σ ϵ Ω n σ n 89

92 6.4 θ θ 6.45: γ 2 γ NaI 2 NaI : γ γ γ 6.45 γ 2 γ NaI 2 NaI 2 γ NaI γ NaI 2 γ γ NaI 1 NaI 2 ( ) γ NaI 1 NaI 2 NaI 1 3 NaI 2 2 γ 6.45 NaI NaI 90

93 6.5: θ θ ( ) θ ROOT TRandom3 class [6] γ γ NaI 1 NaI NaI 1 NaI 2 γ γ 6.53 γ 6.54 γ l 1 γ NaI 1 l 3 γ NaI 1 l 4 NaI 1 γ g(θ) g(θ) = 1 e µ 662 l 3 e µ hν l4 {ϵ Ω(hν )} l 1 2 µ kev γ NaI µ hν γ NaI ϵ Ω(hν ) γ γ g(θ) g(θ) θ 91

94 6.47: θ = : θ = : θ = : θ = : θ = 90 92

95 6.52: NaI 1 γ γ NaI γ θ 1 θ θ 6.53: γ 93

96 6.54: γ 6.55: θ = 30 g(θ) ( ) θ x g(θ) y 2 ( ) NaI 1 θ g(θ) e µ hν l 4 µ hν l

97 6.56: θ = 45 g(θ) ( ) θ x g(θ) y 2 ( ) 6.57: θ = 60 g(θ) ( ) θ x g(θ) y 2 ( ) 6.58: θ = 75 g(θ) ( ) θ x g(θ) y 2 ( ) 95

98 6.59: θ = 90 g(θ) ( ) θ x g(θ) y 2 ( ) 6.6: g(θ) θ θ ( ) NaI 2 γ θ ±

99 7-2 M M SI γ 137 Cs 662 kev γ 2 NaI(Tl) NaI 1 active target NaI γ NaI 2 2 NaI γ γ γ 1 97

100 CAMAC CC-USB NaI 98

101 [1] Robley Dunglison Evans, The atomic nucleus, McGraw-Hill, New York, United States, [2] Michael Edward Peskin, An introduction to quantum field theory, Perseus, Reading, Mass., United States, [3],,, [4] Richard B. Firestone et al., Table of Isotopes, Wiley, New York, 8th edition, [5] Kai Siegbahn, Alpha-, beta- and gamma-ray spectroscopy, North-Holland, Amsterdam, Netherlands, [6] ROOT homepage: TRandom3 class reference, classtrandom3.html, 2016/2/13. [7] B. Povh, K. Rith, C. Scholtz, F. Zesche,, Particles and nuclei : an introduction to the physical concepts,,, [8] Glenn F. Knoll,,,, 4, [9] William. R. Leo, Techniques for Nuclear and Particle Physics Experiments, Springer- Verlag Berlin Heiderberg GmbH, second revised edition, [10] I,

102 A i E mean = E in i i n i E mean = i n ie i ( i n i) 2 (A.1) n i A.1: E i E i n i Channel E i N = i n i (A.2) 100

103 X µ σ 2 ( ) e ixt = ) e ixt f(x)dx = exp (iµt σ2 2 t2 f(x) f(x) = 1 ( ) (X µ) 2 exp. 2πσ 2σ 2 (A.3) E mean ( e ie mean t i = exp i E ) in i N t ( = exp i E ) in i N t i = ( ( )) Ei t exp in i N i ( n i ) (A.4) n i n i n i = n i σ 2 = n i ( e in i t = exp in i t n ) it 2 (A.4) (A.5) t E it N ( i ( ( )) Ei t exp in i = N i exp ( ) Ei t in i N ( ( i = exp i n ie i N (A.5) n i 2 ) t t2 2 ( ) ) 2 Ei t N ( )) i E2 i n i N 2 (A.3) E mean i µ(e mean ) = n ie i σ 2 i (E N mean ) = n ie i N 101

104 B i d 4 pe ip(x 1 x 2 ) p/ + m (2π) 4 p 2 m 2 + iϵ d 3 p { = (p/ + m)e ip(x 1 x 2 ) θ(t (2π) 4 1 t 2 ) (p/ m)e ip(x 2 x 1 ) θ(t 2 t 1 ) } (B.1) 2E p/+m p 0 p 2 m 2 +iϵ (p 0 m 2 + p 2 + iϵ)(p 0 + m 2 + p 2 iϵ) = (p 0 ) 2 ( m iϵ) 2 = (p 0 ) 2 (m 2 + p 2 ) + 2iϵ m 2 + p 2 + O(ϵ 2 ) 2ϵ m 2 + p 2 ϵ (p 0 m 2 + p 2 + iϵ)(p 0 + m 2 + p 2 iϵ) p 2 m 2 + iϵ m 2 + p 2 + iϵ m 2 + p 2 + iϵ E E = m 2 + p 2 E + iϵ E + iϵ 2 B.1 2 E = m 2 + p 2 i dp 0 d 3 p/ + m p (2π) 4 C 1 (p 0 m 2 + p 2 + iϵ)(p 0 + m 2 + p 2 iϵ) e ip 0(t 1 t 2 ) e ip (x 1 x 2 ) = i (2π) 2πi d 3 p p/ + m p 0 =E e ie(t 1 t 2 ) e ip (x 1 x 2 ) 4 2E = 1 d 3 p(p/ + m)e ip(x 1 x 2 ) (2π) 3 2E 102

105 B.1: p 0 C 1 C 2 C R1 i dp 0 d 3 p/ + m p (2π) 4 C R1 (p 0 ) 2 p 2 m 2 + iϵ e ip 0(t 1 t 2 ) e ip (x 1 x 2 ) 0 ire iθ dθ R (t 1 t 2 ) θ= π R 2 e ireiθ 0 = dθe R sin θ(t 1 t 2 ) π = dθe R sin θ(t 1 t 2 ) 1 R θ= π 0 (R ) θ=0 π 0 e r sin θ dθ < π r R(t 1 t 2 ) > 0 t 1 t 2 > 0 C R1 0 = C 1 C R1 = i d 4 pe ip(x 1 x 2 ) p/ + m (2π) 4 p 2 m 2 + iϵ θ(t 1 t 2 ) = C 1 d 3 p (2π) 4 2E (p/ + m)e ip(x 1 x 2 ) θ(t 1 t 2 ) (B.2) 103

106 p 0 = E + iϵ i dp 0 d 3 p/ + m p (2π) 4 C 2 (p 0 m 2 + p 2 + iϵ)(p 0 + m 2 + p 2 iϵ) e ip 0(t 1 t 2 ) e ip (x 1 x 2 ) = i (2π) 2πi d 3 p (p/ + m) p 0 = E e ie(t 1 t 2 ) e ip (x 1 x 2 ) 4 2E = i (2π) 2πi d 3 ( p) ( Eγ 0 i pi γ i + m) e ie(t 1 t 2 ) e ip (x 1 x 2 ) (B.3) 4 2E 1 = d 3 p( p/ + m)e ip(x 2 x 1 ) (B.4) (2π) 3 2E (B.3) p p C R2 i dp 0 d 3 p/ + m p (2π) 4 C R 2 (p 0 ) 2 (m 2 + p 2 ) + iϵ e ip 0(t 1 t 2 ) e ip (x 1 x 2 ) π ire iθ dθ R (t 1 t 2 ) θ=0 R 2 e ireiθ π = dθe R sin θ(t 1 t 2 ) 1 R θ=0 0 (R ) R(t 1 t 2 ) < 0 t 1 t 2 < 0 C R2 R 0 = C 2 C R2 C 2 (B.5) (B.4) (B.5) i (2π) 4 d 4 pe ip(x 1 x 2 ) p/ + m p 2 m 2 + iϵ θ( t 1 + t 2 ) = d 3 p (2π) 4 2E (p/ m)e ip(x 2 x 1 ) θ(t 2 t 1 ) (B.6) (B.2) (B.6) (B.1) 104

107 C [3] [2] C.1 vertex C.2 M C.3 2 C.3 M 1 M 2 ( ) M 1 = M 2 = d 4 xd 4 yϵ ν(k, λ )e ik xū(p )e ip x ieγ ν i(p/ + k/ + m) (p + k) 2 m 2 + iϵ ieγµ u(p)e ipy ϵ µ (k, λ)e iky d 4 xd 4 yϵ ν (k, λ)e ikx ū(p )e ip x ieγ ν i(p/ k / + m) (p k ) 2 m 2 + iϵ ieγµ u(p)e ipy ϵ µ(k, λ )e ik y d 4 xd 4 y C.1: vertex vertex 105

108 C.2: vertex p u(p) v(p) x vertex 4 C.3: 106

109 D [4] D.1-D.3 D.4 40 K 40 K γ 1461 kev D.1: 137 Cs D.2: 60 Co 107

110 D.3: 22 Na D.4: 40 K 108