22 2 24
W 1983 W ± Z 0 3 10 cm 10 cm 50 MeV TAC - ADC 65000 18 ADC [ (µs)] = 0.0207[] 0.0151 (2.08 ± 0.36) 10 6 s 3 χ 2 2 1 20 µ + µ 8 = (1.20 ± 0.1) 10 5 (GeV) 2 G µ ( hc) 3 1
1 7 1.1............................. 7 1.2 β..................................... 10 2 12 2.1............................... 12 2.2.................................. 13 2.2.1..... 14 2.2.2........................... 15 3 17 3.1................................ 17 3.2...................................... 18 4 21 4.1........................ 21 4.1.1.................... 21 4.1.2............................. 21 4.2 NIM................................ 21 4.2.1 Discriminator............................. 21 4.2.2 Coincidence.............................. 21 4.2.3 Delay................................. 21 4.2.4 TPHC................................. 22 4.2.5 ADC.................................. 23 4.2.6 MCA.................................. 23 5 24 5.1 TAC ADC.......................... 24 5.1.1.................................. 24 5.1.2....................... 24 5.1.3................................ 25 5.1.4.................................. 26 5.2........................ 27 2
5.2.1.................................. 27 5.2.2....................... 29 5.2.3................................ 30 5.2.4.................................. 32 5.2.5.............................. 36 5.3 µ + µ............................. 45 6 48 3
1.1 β.......................... 8 1.2 β................................ 9 1.3 β............................ 9 1.4 β.......................... 10 1.5 β +.......................... 11 2.1 [19]............................ 12 2.2 µ µ...................... 14 2.3 µ........................... 16 3.1 [18]....................... 18 3.2................... 19 3.3 ()...... 19 3.4........... 20 4.1 TAC............................. 22 5.1................................. 24 5.2 (ADC)................................ 25 5.3........................ 26 5.4....... 27 5.5 TAC............... 28 5.6........................ 28 5.7................................. 29 5.8........................ 30 5.9 ()............ 31 5.10 2.5µs10.45µs)...................... 32 5.11 ()..................... 34 5.12 ()............................ 35 5.13 p 0................................ 36 5.14 p 1................................ 37 5.15 p 2................................ 38 5.16 p 0................................ 39 4
5.17 p 1................................ 40 5.18 p 2................................ 40 5.19 A χ 2 p 1.................... 42 5.20 µ + µ........................... 45 5.21 µ + µ ()...................... 46 5
5.1 ROOT............................ 33 5.2...................................... 33 5.3.................................. 38 5.4 ROOT............................ 39 5.5.................................. 41 5.6 p 1................................ 42 5.7 ROOT............................ 44 5.8 µ + µ..................... 45 6
1 10 cm TAC(-) 3 65000 18 PC 3 2 1 1.1 β β β W. A. Pauli 1931 1935 E. Fermi 4 4-7
1.1: β β 1.25 V-A β µ µ 3 1948 G. Puppi V-A 1935 π β β W 1983 W Z 0 8
1.2: β 1.3: β 9
1.2 β β β Z N 1 β A Z (A, Z) (A, Z + 1) + e + ν e (1.1) n p + e + ν e (1.2) 1.4: β β + (A, Z) (A, Z 1) + e + + ν e (1.3) p n + e + + ν e (1.4) 10
1.5: β + 11
2 2.1 2.1: [19] 1911 Hess 10 20 ev 12
+ + + (2.1) + (2.2) K + + + (2.3) K + (2.4) 1 cm 2 1 144 cm 2 144 2 2.2 m = 105.658369 0.000009 MeV [7] 1/2 ±1 µ + µ µ e + e + ν µ (2.5) µ + e + + ν e + µ 13
2.2: µ µ 2.2.1 m µ c 2 = m e c 2 + m ν c 2 + m ν c 2 + E e + E ν + E ν (2.6) 0 = P µ = P e + P ν + P ν (2.7) (2.7) 3 4 E e E ν E ν P e P ν P ν 12 E 2 = p 2 +m 2 0 E e 055 MeV E e = (m µ m e ) 2 (m ν m ν ) 2 2m µ c 2 (2.8) 14
2.2.2 t Ndecay(t) N0 Γ V-A [1] Ndecay(t) = N0(1 e t/ ) (2.9) Γ µ = h τ = G 2 192π 3 ( hc) 6(m µc 2 ) 5 (1 + ϵ) (2.10) G ( hc) 3 = 1.16637 10 5 [(GeV ) 2 ] τ m µ () ϵ m µ [7] Γ µ = 2, 996 10 10 [s 1 ] m = 105.658369 0.000009 MeV (2.11) τ = (2.19703 ± 0.00004) 10 6 s (2.12) a) 100 b) 15
K + p + n (2.13) C τ µ = 2.020 ± 0.020 µs ( 8.05 )H τ µ = 2.194903 ± 0.000066 µs ( 0.097 ) [11] Γ tot = Γ tot = h τ µ h τ µ = 3.26 10 10 s 1 for C (2.14) = 2.999 10 10 s 1 for H (2.15) Γ free = h τ µ,free = 2.996 10 10 Γ cap = Γ cap = h τ µ h τ µ Γ tot = Γ cap + Γ free (2.16) = 2.52 10 11 s 1 for C (2.17) = 3 10 13 s 1 for H (2.18) ( Γ cap, Γ free ) Γ tot Γ tot = (7.7, 91.9 ) for C (2.19) = (0.001, 99.8 ) for H (2.20) 2.3: µ 16
3 3.1 Bethe-Bloch x [g/cm 2 ] [16] de dx = m e ze V n I z 2 e 4 n ln[ m ev 2 ] (MeV/cm) (3.1) 4πϵ 02 V 2 m e I 17
3.1: [18] 3.2 E 0 [MeV] R(E 0 ) [16] R(E 0 ) = E0 E 0 0 ( de dx ) 1 de (cm) (3.2) 18
R (cm) 90 80 70 60 50 40 30 20 10 0 µ 0 50 100 150 200 250 粒子の入射時の運動エネルギー (MeV) 3.2: 2.267 [g/cm 3 ] (CH) n 3.3: () 10 cm 0 50 MeV 19
30 3.4: 20
4 4.1 4.1.1 18 8 cm 2 6 mm 2 16 8 cm 2 10 cm 1 4.1.2 H7195 RD0734 4.2 NIM 4.2.1 Discriminator Le Croy 4608C 4.2.2 Coincidence 2 N017 4.2.3 Delay N-TM205 21
4.2.4 TPHC Time to Pulse Height Converter(-) TAC(Time to Amplitude Converter) TAC ORTEC 467 4.1: TAC 22
4.2.5 ADC Analog to Digital Converter(-) 510 4.2.6 MCA Multi Channel Analyzer ADC 500 23
5 5.1 TAC ADC 5.1.1 1.1 10 4 Hz 16 ns 1.2 V Discriminator TAC delay 1246 µs TAC TAC ADC MCA 100 5.1.2 5.1: 24
HV 2000 V TAC 10 µs 5.1.3 5.2: (ADC) 25
5.1.4 7 time (μs) 6 5 4 3 2 1 0 0 50 100 150 200 250 300 350 5.3: channel ADC y(µs) = 0.0207x 0.0151 (5.1) 26
5.2 5.2.1 1 18 8 cm 2 6 mm 2 16 8 cm 2 10 cm 3 18 8 cm 2 6 mm 5.4: 3 2 2 1 3 1 2 3 2 Discriminator Coincidence 3 VETO 3 TAC 2 TAC ADC MCA 65000 18 27
5.5: TAC 5.6: 28
5.2.2 測定器の配置 各種設定 各種設定 装置 光電子増倍管の HV Discriminator スレッショルド TAC フルスケール 図 5.7: 検出器の配置 29 値 2000 V 15 mv 10 µs
5.2.3 497 10 bin 1 200 ns t = 84 ns 10.45 ns 51 counts per channel 2000 1800 1600 1400 1200 1000 800 600 400 200 0 0 2 4 6 8 10 time (sec) -6 10 5.8: 30
counts per channel 200 180 160 140 120 100 80 60 40 20 0 0 2 4 6 8 10 time (sec) -6 10 5.9: () 31
5.2.4 dn decay (t) = p 0 e t/p 1 + p 2 (5.2) dt 2.5 µs 10.45 µs 39 5.1 counts per channel 200 180 160 140 120 100 80 60 40 20 0 0 2 4 6 8 10 time (sec) -6 10 5.10: 2.5µs10.45µs) 32
5.1: ROOT 39 χ 2 38.05 p 0 281.3 ± 70.1 p 1 (2.079 ± 0.357) 10 6 p 2 37.41 ± 3.092 5.2: 0 µs 10.45 µs 2.5 µs 10.45 µs 7591 2384 0.117 /s 0.037 /s 65000 s dn decay(t) 37.41 dt 37.41 33
counts per channel 3 10 10 2 10 2 4 6 8 10 time (sec) -6 10 5.11: () 34
counts per channel 3 10 10 2 10 2 4 6 8 10 time (sec) -6 10 5.12: () 35
5.2.5 ROOT ROOT p 0 p 1 p 2 3 3 2 1 χ 2 1 χ 2 = 39.05 p 0 p 1 p 2 ROOT χ 2 : χ 2 = (y i f(x i )) 2 i ( = (y i f(x i )) 2 (5.3) y i ) 2 i y i dn decay(t) = p dt 0 e (t A)/p 1 + p 2 3 2 A=0 χ 2 = 1 p 0 p 1 p 2 A = 2.5 10 6 s χ 2 = 1 p 0 p 1 p 2 A = 3.545 10 6 s χ 2 = 1 p 1 A=0 a) p 1 = 2.079 10 6 p 2 = 37.41 p 0 260270281.3290300 χ 2 41 40.5 χ^2 40 39.5 39 38.5 38 37.5 255 260 265 270 275 280 285 290 295 300 305 p0 5.13: p 0 χ 2 = 1 ±13 36
b) p 0 = 281.3p 2 = 37.41 p 1 2 10 6 2.05 10 6 2.079 10 6 2.1 10 6 2.15 10 6 χ 2 5.14: p 1 χ 2 = 1 ±0.51 10 7 37
c) p 0 = 281.3p 1 = 2.079 10 6 p 2 3636.537.4138.539 χ 2 40 χ^2 39.5 39 38.5 38 37.5 35.5 36 36.5 37 37.5 38 38.5 39 39.5 p2 5.15: p 2 χ 2 = 1 ±1.39 ROOT 5.3: parameter ROOT 2 p 0 ±70.1 ± 13 p 1 ±3.57 10 7 ± 0.51 10 7 p 2 ±3.092 ± 1.39 p 1 ROOT 2 7 p 0 p 1 dn decay (t) dt = p 0 e (t 2.5 10 6 )/p 1 + p 2 (5.4) 38
t = 2.5 10.45 µs 2.5 µs 5.4: ROOT 39 χ 2 38.05 p 0 84.53 ± 6.126 p 1 (2.079 ± 0.357) 10 6 p 2 37.41 ± 3.092 ROOT 5.1 a) p 1 = 2.079 10 6 p 2 = 37.41 p 0 78808384.53868890 χ 2 41 40.5 χ^2 40 39.5 39 38.5 38 37.5 76 78 80 82 84 86 88 90 92 p0 5.16: p 0 χ 2 = 1 ±4.47 39
b) p 0 = 84.53p 2 = 37.41 p 1 1.9 10 6 1.95 10 6 2 10 6 2.079 10 6 2.1 10 6 2.15 10 6 2.2 10 6 2.25 10 6 χ 2 40.5 40 χ^2 39.5 39 38.5 38 37.5 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2 2.25 2.3 p1 (μs) 5.17: p 1 χ 2 = 1 ±1.351 10 6 c) p 0 = 84.53p 1 = 2.079 10 6 p 2 35.6363737.41383939.2 χ 2 χ^2 41 40.5 40 39.5 39 38.5 38 37.5 34 36 38 40 p2 5.18: p 2 40
χ 2 = 1 ±1.39 5.5: parameter ROOT 2 p 0 ±6.126 ± 4.47 p 1 ±3.57 10 7 ± 1.31 10 7 p 2 ±3.092 ± 1.39 p 1 ROOT 2 2.7 dn decay(t) = p dt 0 e t/p 1 + p 2 2.5 3 ROOT 41
dn decay(t) = p dt 0 e (t A)/p 1 +p 2 A = 2.53.545 10 6 2 p 0 p 2 p 1 χ 2 A = 02.53.545 10 6 5.19: A χ 2 p 1 A = 2.5 4 10 6 A = 4 10 6 A = 3.5 4 10 6 5.6: p 1 ROOT A = 0 A = 2.5 10 6 A = 3.5 10 6 p 1 ±3.57 10 7 ±0.51 10 7 ±1.31 10 7 ±1.86 10 7 42
A = 4 10 6 ±1.91 10 7 A = 5 10 6 ±1.66 10 7 dn decay(t) = p dt 0 e (t A)/p 1 + p 2 p 0 e (t A)/p 1 + p 2 t = A p 0 + p 2 p 1 p 0 e (t A)/p 1 + p 2 p 0 e A/p 1 e t/p 1 + p 2 p 0 e t/p 1 + p 2 p 0 p 0 e A/p 1 p 1 p 1 p 2 p 2 Be t/τ t = 0 Be t/τ dt = [B( τ)e t/τ ] 0 = Bτ (5.5) 0 Bτ = B 0 B = B 0 τ Be t/τ = B 0 τ e t/τ B 0 B 0 τ t = t 1 t 2 t2 t 1 Be t/τ dt = [B( τ)e t/τ ] t 2 t 1 = Bτ(e t 1/τ e t 2/τ ) (5.6) Bτ(e t 1/τ e t 2/τ ) = B 0 B = B 0 τ(e t 1 /τ e t 2 /τ ) Be t/τ = B 0 τ(e t 1/τ e t 2/τ ) e t/τ (5.7) B 0 t = t 1 t 2 τ Be t/τ + C t = t 1 t 2 t2 t 1 (Be t/τ + C)dt = [B( τ)e t/τ + Ct] t 2 t 1 = Bτ(e t 1/τ e t 2/τ ) + C(t 2 t 1 ) (5.8) Bτ(e t 1/τ e t 2/τ ) + C(t 2 t 1 ) = D B = D C(t 2 t 1 ) τ(e t 1 /τ e t 2 /τ ) Be t/τ + C = D C(t 2 t 1 ) τ(e t 1/τ e t 2/τ ) e t/τ + C (5.9) D t = t 1 t 2 τ C 43
dn decay(t) dt = p 0 p 2 (t 2 t 1 ) p 1 (e t 1 /p 1 e t 2 /p 1 ) e t/p 1 +p 2 5.7: ROOT 39 χ 2 38.05 p 0 (4.317 ± 0.09337) 10 4 p 1 (2.079 ± 0.356) 10 6 p 2 37.41 ± 3.082 p 0 p 0 = 0.25 70 281 p 0 = 0.09 4.3 = 0.02 5.1 p 0 p 0 = D p 0 4 10 4 B C counts per channel counts per 200 n second time bin τ 44
5.3 µ + µ µ + µ dn decay(t) dt 159.35 5.8: µ + µ µ + dn decay (t) = 121.94e t/2.197 10 6 + 37.41 dt µ dn decay (t) = 121.94e t/2.020 10 6 + 37.41 dt µ + + µ dn decay (t) = 121.94e t/2.197 10 6 + 121.94e t/2.020 10 6 + 74.82 dt 350 counts per channel 300 250 200 150 100 μ+ μ μ++μ 50 0 0 0.000002 0.000004 0.000006 0.000008 0.00001 0.000012 time (s) 5.20: µ + µ 45
1000 counts per channel 100 10 μ+ μ μ++μ 1 0 0.000002 0.000004 0.000006 0.000008 0.00001 0.000012 time (s) 5.21: µ + µ () 46
[1] Γ µ = h G 2 = τ µ 192π 3 ( hc) 6(m µc 2 ) 5 (1 + ϵ) (5.10) τ µ = (2.079 ± 0.357) 10 6 s G ( hc) 3 = (1.20 ± 0.1) 10 5 (GeV) 2 (5.11) 47
6 10 cm 50 MeV TAC(-) ADC [ (µs)] = 0.0207[] 0.0151 τ (2.08 ± 0.36) 10 6 s 17 20 G = (1.20 ± 0.1) 10 5 (GeV) 2 ( hc) 3 µ + µ µ + µ µ + µ 8 17 () µ 880 ns( 60 ) µ + µ 48
ROOT 49
[1] B. /K. /C. /F. (, 2001) [2] (, 1998) [3] (, 1992) [4] (, 1994) [5] (, 1997) [6] --(, 1998) [7] Caso et al., Particle Data Group The European Physical Journal C(Springer, 1998), http://pdg.lbl.gov/ [8] R. Brun et al.,root Users Guide 4.04(CERN LIBRARY, 2005) [9] [10] [11] T. Suzuki and D.F.Measday Total Nuclear Capture Rates For Negative Muons, Phys. Rev. C Vol. 35, No. 6, pp. 2212-2224 (1987) [12] D.F.Measday The Nuclear Physics of Muon Capture, Phys. Rep. 354, pp. 243-409 (2001) [13] Donald E.Groom Muon Stopping Power And Range Tables 10MeV-100TeV, Atomic Data and Nuclear Data Tables, 78, pp. 183-356 (2001) [14] Shuhei Tsuji et al., Measurements of Muons at Sea Level, J. Phys. G: Nucl. Part. Phys. 24, pp. 1805-1822 (1998) [15] (2009) [16] (2008) 50
[17] (2009) [18] (2008) [19] http://www.rist.or.jp/atomica/data/fig pict.php?pictno=09-02-07-09-05 51