2.2.2 Bethe-Bloch

1 Bethe-Bloch . Mott ( . ) *1 Bethe-Bloch . dE Z z 2 { ( 2me 2 v 2 Wmax ) }. = 2 Na re2 me c2 2 log 2 2. [MeV/cm] ( ). dx A I2. Z z 2 { ( 2me 2 c2 2 Wmax ) }. = 2 log 2 2. [MeV/cm]. A I2.. me . Z . A . [g/ cm3 ]. z . , , . Wmax . M . 2me c2 2 2. Wmax = ( ). 1+2 m M. e 1 + 2 2 + ( m e 2. M ). me M . Wmax 2me c2 2 2.. I .. {. I 12 + Z7 [eV] (Z < 13). = ( ). Z + [eV] (Z 13).. X0 , X1 , C0 , a, m .. 0 (log10 ( ) < X0 ). = + C0 + a(X1 X)m (X0 < log10 ( ) < X1 ) ( ).. + C0 (X1 < log10 ( ). *1 . ( ) 1 1 asuzuki/ 24 A5 ( ) http: . 1.. Bethe-Bloch . C . C ( ) . C =( 2 2 + 4 4 6 6 ) 10 6 I 2. ( ). + ( 2 2 4 4 + 6 6 ) 10 9 I 3.. 1 . Bethe-Bloch . dE Z z 2 { ( 2me 2 c2 2 Wmax ) C}. = 2 log 2 2. 2 [MeV/cm] ( ).)}

2 Dx A I2 Z. T . E 2 M 2 c4 1. 2 = =1 ( )2. E2 T +1. 2. Mc .. [MeV] . (i) Bethe-Bloch 2 . (ii) .. (iii) ( ) log .. (iv) .. Bethe-Bloch .. 2. dE . dx Bethe-Bloch . dE. = z 2 f ( ) ( ). dx . E 2 M 2 c4 1. 2 = =1 ( )2. E2 T +1. 2. Mc . = (T, M ).. dE. = z 2 f (T, M ) ( ). dx M1 , z1 M2 , z2 .. dE2 (M ) z 2 dE1 ( M1 ). (T ) = z22 f (T, M2 ) = z22 f . 1. T, M1 = 22 T ( ). dx M2 z1 dx M2.. 1 1 (M ). 1. (T, M2 ) = 1 ( ) 2 =1 ( ) 2 = T, M1. T M1 T M2. +1 +1. M2 c2 M2 M1 c2.. (ii) .. (Mass Stopping Power).. 1 dE Z. = z 2 f ( , I, , C) [MeV cm2 /g] ( ). dx A. Z Z I Bethe-Bloch . A. log C .. dE. dx Bethe-Bloch .. ( ) Xa11 Xann (X i Ai , Zi ) . Xi Bethe-Bloch . ( dE ) Zi ( Ci ). = z 2 i f , Ii , i , dx i Ai Zi 3.

3 N , Am = ai Ai . i=1.. ai Zi ( Ci ). n dE. = z2 f , Ii , i , dx i=1. Am Zi ( . n ai Ai i Zi ( Ci )). = z2 f , Ii , i , i=1. Am i Ai Zi ( n wi ( dE ) ). = . i=1. i dx i . 1 dE wi ( dE ). n = ( ). dx dx i i=1 i Xi . ai Ai wi = ( ). Am . Bethe-Bloch ( ) .. n Ze = ai Zi ( ). i=1.. n Ae = ai Ai ( ). i=1.. n ai Zi log Ii log Ie = ( ). i=1. Ze .. n ai Zi i e = ( ). i=1. Ze .. n Ce = ai Ci ( ). i=1.. Bethe-Bloch . ( ) Bete-Bloch . 100 [MeV] ) (4[GeV] ) . 26 .. Bethe-Bloch .. 4. < ( ) .. < Lindhard .. Bethe-Bloch .. zZa0 Ad c = ( ). 1670 .. ( . ) *2. (Range).. range number-distance curve .. 0 . ( .. ). 1 : = 2 : 1 ( ) .. 0 .. *2 ( ).. 5. T0 . T0 (. E0. dx dE ) 1. S(T0 ) = dE = dT ( ). mc2 dE 0 dx.

4 Bethe-Bloch . Bethe-Bloch Tmin . T0 ( dE ) 1. R(T0 ) = R0 (Tmin ) + dT ( ). Tmin dx R(T0 ) .. R T .. log R = D + b log T R Tb ( ). b = . dE . dx . T2 ( ) 1. dE2. R2 (T2 ) = (T2 ) dT2. 0 dx . z 2 T2 ( dE1 ( M1 )) 1. = 12 T2 dT2. z2 0 dx M2. M1. z 2 M2 M2 T2 ( dE1 ) 1 z 2 M2 ( M1 ). = 12 (T ) dT = 12 R1 T2 ( ). z2 M1 0 dx z2 M1 M2. i , Ai 2 .. R1 2 A1. = ( ). R2 1 A2. (Bragg-Kleeman ). Acomp Rcomp . Acomp Rcomp = ( ).. n ai Ai i=1. Ri Ri . 2cm .. 6.. = m c2 = 106[MeV] . ( 1 ). T = 1 m c2 = 272 ; 300[MeV]. 1 ( )2. [MeV cm2 /g] .. x dE. E = dx = 2 = [MeV]. 0 dx ( [g/ cm3 ] ). 300 [MeV]. [MeV] . [MeV] .. 600[MeV] 400[MeV] .. x 600 (. dE ) 1. x= dT. 400 dx N .. N (. ) 1 T. dE. x= (Ti ).

5 I=1. dx N.. 1 ( ( 1 dE ) 1 T ). N. x= (Ti ). i=1. dx N. *3 N = 10 Ti = 500, N = 1 .. 400 [MeV] . 400[MeV] . *3 NIST Stopping-Power and Range Tables for Electrons, Protons, and Helium Ions star/ . 7.