2 Rutherford 2. Rutherford N. Bohr Rutherford 859 Kirchhoff Bunsen 86 Maxwell Maxwell 885 Balmer λ Balmer λ = 364.56 n 2 n 2 4 Lyman, Paschen 3 nm, n =3, 4, 5,
4 2 Rutherford 89 Rydberg λ = R ( n 2 ) n 2 n = n +,n +2, n = Lyman n =2 Balmer n =3 Paschen R Rydberg R = 9677.58 cm 896 Zeeman Zeeman Zeeman Lorentz Zeeman e /m =.6 C/kg 897 Thomson electron e /m =.7 C/kg 94Thomson 94 98 Rutherford 99 Geiger Marsden [] 9 Rutherford Rutherford [2] 93 Geiger Marsden Rutherford [3] Chadwick Rutherford Z Z 93 N. Bohr Planck 932 Chadwick [4]
2.2 Rutherford 5 2.2 Rutherford Coulomb 24 Po 2.2. Coulomb f(r) = a r 2, a =2Ze2 (2.) a =2Ze 2 m [ d 2 r dt 2 r ( ) ] dϕ 2 = f(r) dt r ( d dt mr 2 dϕ dt ) = (2.2) m Coulomb (2.2) L = mr 2 dϕ = (2.3) dt (2.2) r ϕ m d2 r dt 2 L2 mr 3 = f(r) = a r 2 (2.4) impact parameter r Coulomb f(r) v E = 2 mv2 (2.5) Coulomb E p = mv L = pb = mvb (2.6)
6 2 Rutherford b Coulomb 2. (2.5) (2.6) v b = L 2mE (2.7) b E L momentum p x b impact parameter z 2.: 24 Po E =5.4MeV mc 2 = 3727 MeV (2.8) v c =.54 v =.6 7 m/s (2.9) 2.2.2 (2.4) dr/dt t E ( ) dr 2 2 m + W (r) =E (2.) dt W (r) W (r) =V (r)+ L2 2mr 2, V(r) = a r 2 (2.)
2.2 Rutherford 7 V (r) Coulomb (2.) W (r) E > (dr/dt) 2 E W (r) (2.2) r r min r min 2.2 Coulomb r min r 2 a E r W (r min )=E (2.3) L2 2mE r2 2b r b 2 = (2.4) b b r min > r min = b = a 2E 24 Po b = 2Ze2 2E (2.5) ( a ) a 2 2E + + L2 2E 2mE = b + b 2 + b 2 (2.6) hc = Zα E =79 97.32 MeVfm = 2 fm (2.7) 37.36 5.4MeV α = e 2 /( hc) fine structure constant fm= 5 m 7fm 2.2 b =2fm 2.2.3 (2.4) r(t) r(ϕ) (2.23) r = u (2.8)
8 2 Rutherford 2 potential [ MeV ] 5 5 r min E 2 3 4 5 6 7 r [ fm ] 2.2: r r min Coulomb (2.4) dr dt = dr dϕ du = r2 dϕ dt dϕ L mr 2 = L m du dϕ (2.9) d 2 r dt 2 = d ( L ) du = L dt m dϕ m [ d dϕ ( )] du dϕ dϕ dt = L2 d 2 u m 2 dϕ 2 u2 (2.2) d 2 u dϕ 2 + u = am L 2 (2.2) u = am/l 2 (2.2) A ϕ u = A cos (ϕ ϕ ) am L 2 (2.22) ϕ + r ϕ = ϕ r r(ϕ) = L2 am ε cos (ϕ ϕ ) ε = cos ϕ > (2.23)
2.2 Rutherford 9 ϕ = ϕ 2.3 ϕ = ϕ b ϕ r min θ target 2.3: Rutherford θ ϕ = ϕ r min = L 2 am(ε ) (2.24) b b = lim ϕ r(ϕ) sin ϕ = L2 am ε sin ϕ = L 2 am tan ϕ (2.25) L 2 =2mEb 2 b (2.35) 2.2.4 (2.34) ϕ r dϕ dr = dϕ dt dt dr (2.26) r (2.3)
2 2 Rutherford (2.) dϕ dr = ± L mr 2 2 ( E W (r) m ) = ± L 2m r 2 E W (r) (2.27) ± dr/dt + ϕ = Coulomb ϕ =2ϕ 2ϕ r + r min + dr/dt 2.3 2ϕ = rmin L 2m dr r 2 E W (r) + r min L 2m dr r 2 E W (r) = 2L 2m r min dr r 2 E W (r) (2.28) b r = b u dr = b du (2.29) u2 E W (b/u) =E L2 u 2 2mb 2 au ( b = E u 2 a ) ( be u = E u 2 2b ) b u (2.3) 2ϕ = ( 2L ) du 2mE b b/r min (2b /b)u u 2 b/rmin du = 2 (2.3) (2b /b)u u 2 2ϕ =2 [ ] b/rmin tan u + b /b (2b /b)u u 2 = π 2 tan b b (2.32) r min (2.6) 2.3 ϕ =
2.2 Rutherford 2 ϕ = π ϕ =2ϕ θ θ = π 2ϕ (2.33) θ = 2 tan a 2Eb (2.34) θ E b 2.4 (2.34) 2.4: Rutherford b = a 2E tan (θ/2) (2.35) E θ b r min θ r min (θ) = a ( ) + (2.36) 2E sin (θ/2) 2.2.5 m
22 2 Rutherford z z N b b +db dn =2πbdbN (2.37) db b 2.5: b b +db (2.34) θ θ +dθ θ θ +dθ dn =2πb db dθn (2.38) dθ N [ ] dn [ ] R θ θ +dθ 2π sin θr R dθ R 2 θ θ +dθ z dω = 2π sin θr R dθ R 2 =2π sin θ dθ (2.39) θ π 4π
2.2 Rutherford 23 R dθ θ z 2.6: θ θ +dθ (2.38) dω dn = b sin θ N dn dσ dσ = dn N = b sin θ db dωn (2.4) dθ db dω (2.4) dθ dn N σ σ cross section dσ dω = b db (2.42) sin θ dθ differential cross section Rutherford (2.42) Coulomb db/dθ (2.35) db/dθ Coulomb ( ) dσ a 2 dω = 4E sin 4 (θ/2) (2.43) Rutherford Z =79 E =5.4MeV Rutherford 2.7 barn/sr barn = 28 m 2 sr Rutherford θ = Coulomb /r
24 2 Rutherford 5 4 dσ/dω [barn/sr] 3 2 3 6 9 2 5 8 scattering angle θ [degree] 2.7: Rutherford Coulomb 2.2.6 Rutherford Rutherford Coulomb 99 Geiger Marsden Rutherford 24 Po 9 Rutherford Ze Z 79 (2.36) θ = 8
Z =79 E =5.4MeV r min (θ = π) = a 2E 2.2 Rutherford 25 = 42 fm (2.44) 42fm 5 fm E r min E θ E Rutherford Coulomb Coulomb L = 2mE b E =5.4MeV m = 3727 MeV/c 2 L [MeV s]=6.7 22 [MeV s/fm ] b fm (2.45) θ =9 b b = b =2fm L =.4 2 MeV s (2.46) L Planck h L 2 h
26 2 Rutherford 2.3 Rutherford 2.3. Schrödinger V (r) m Schrödinger ( h2 2m + V (r) ) ψ(r) =Eψ(r) (2.47) ψ(r) V (r) lim rv(r) = (2.48) r Coulomb Schrödinger E k k 2 = 2mE 2mV (r) h 2 U(r) = h 2 (2.49) Schrödinger ( + k 2) ψ(r) =U(r) ψ(r) (2.5) z 2.8 ψ(r) (2π) 3/2 [ exp(ikz)+ f(θ) ] exp(ikr) r ( r ) (2.5) f(θ) scattering amplitude θ (2π) 3 exp[ i(k k ) r ]dr = δ(k k ) (2.52) 2π 2π 2π (2π) 3 dx dy dz exp(ikz) 2 = (2.53) 2π
2.3 Rutherford 27 outgoing spherical wave r θ z incoming plane wave 2.3.2 Green 2.8: Schrödinger (2.5) Helmholtz ( + k 2) χ(r) =U(r) ψ(r) (2.54) ( + k 2) G (r) =δ(r) (2.55) G (r) (2.54) Green G (r) (2.54) χ(r) = dr G (r r ) U(r )ψ(r ) (2.56) ( + k 2) χ(r) = dr δ(r r ) U(r )ψ(r )=U(r)ψ(r) (2.57) χ(r) Green G (r) G (r) δ(r) Fourier G (r) = dk G (k ) exp (ik r) δ(r) = (2.58) (2π) 3 dk exp (ik r)
28 2 Rutherford (2.54) ( k 2 k 2 ) G (k )= (2π) 3 (2.59) G (r) Fourier G (r) G (r) = (2π) 3 dk exp (ik r) k 2 k 2 (2.6) k 2.9 r z G (r) = (2π) 3 k 2 dk π sin θ dθ exp (ik r cos θ ) k 2 k 2 2π dϕ (2.6) ϕ 2π θ G (r) = k sin k r 2π 2 r k 2 k 2 dk (2.62) z k r θ k x ϕ y 2.9: k k sin k r exp (±ik r) G (r) = 4π 2 r I ± = k sin k r k 2 k 2 dk = 6π 2 ir 2 [ I + I ] (2.63) ( k + k + ) k exp (± ik r)dk k (2.64) k = k k = k k k 2.
2.3 Rutherford 29 I + C + I C I ± =2πi exp (ikr) (2.65) (2.63) Green G (r) = exp (ikr) (2.66) 4πr C + k k k k C 2.: k I + I z Schrödinger (2.5) ψ(r) = exp (ik r r ) exp (ikz) (2π) 3/2 4π r r U(r ) ψ(r )dr (2.67) Schrödinger ψ(r) Schrödinger f(θ) ψ(r) (2.5) (2.67) r r V r r n r r r r r = r 2 + r 2 2(n r ) r r (n r ) (2.68) r r r r r [ +(n r ) ] r (2.69)
3 2 Rutherford (2.67) [ ψ(r) (2π) 3/2 exp (ikz) { exp (ikr) + (2π)3/2 r 4π exp ( ik r ) U(r ) ψ(r )dr }] (2.7) k k = k n r (2.5) f(θ) f(θ) = (2π)3/2 4π exp ( ik r ) U(r ) ψ(r )dr (2.7) 2.3.3 N φ = exp (ikz) (2.72) (2π) 3/2 z N = j z = [ φ h φ 2m i z h φ ] i z φ (2.73) (2.72) j z = (2π) 3 hk m = v =( hk)/m χ = v (2π) 3 (2.74) exp (ikr) (2π) 3/2 f(θ) (2.75) r r j r = [ χ h χ 2m i r h χ ] i r χ (2.76) r 2 j r = v (2π) 3 f(θ) 2 r 2 (2.77)
2.3 Rutherford 3 ds dn dn = j r ds = v (2π) 3 f(θ) 2 dω (2.78) dω ds dσ = dn N = f(θ) 2 dω (2.79) dσ dω = f(θ) 2 (2.8) 2.3.4 Born f(θ) = (2π)3/2 4π exp ( ik r ) U(r ) ψ(r )dr (2.8) ψ(r ) ψ(r ) ψ(r ) φ(r )= exp (ikz) (2.82) (2π) 3/2 Born f(θ) = exp ( ik r ) U(r ) exp (ikz )dr (2.83) 4π k r n θ q k z V ( r ) 2.:
32 2 Rutherford θ 2. k k hq = hk hk (2.84) k = k hq = h q =2 hk sin θ 2 (2.85) q f(θ) = exp( iq r ) U(r )dr (2.86) 4π q z r f(θ) = r sin qr U(r )dr = j (qr ) U(r ) r 2 dr (2.87) q j (qr ) Bessel V (r ) f(θ) = 2m h 2 = j (qr ) V (r ) r 2 dr (2.88) [ dσ 2m ] 2 dω = h 2 j (qr ) V (r ) r 2 dr (2.89) 2.3.5 Coulomb V (r) = a r aρ(r ) r r dr (2.9) ρ ρ(r)dr =4π ρ(r) r 2 dr = (2.9) (2.9) Coulomb /r r
2.3 Rutherford 33 (2.9) Born (2.87) f(θ) = 4π 2ma h 2 [ I + I 2 ] (2.92) exp (iq r ) ρ(r I = r dr ) exp (iq r ) I 2 = r r dr dr (2.93) g(r) = exp (iq r ) r r dr (2.94) I = g() I 2 = ρ(r ) g(r )dr (2.95) (2.94) g(r) exp (iq r) g(r) g(r) = 4π exp (iq r) (2.96) g(r) =4π exp (iq r) q 2 (2.97) I = 4π q 2 I 2 = (4π)2 q 2 j (qr) ρ(r) r 2 dr (2.98) I 2 f(θ) = 2ma +F (q) h 2 q 2 (2.99) F (q) Fourier F (q) =4π j (qr) ρ(r) r 2 dr (2.) q 2 =4k 2 sin 2 θ 2 = 4m2 v 2 f(θ) = h 2 sin 2 θ 2 a 4E [ +F (q)] sin 2 (θ/2) ( ) dσ a 2 dω = [ +F (q)] 2 4E sin 4 (θ/2) (2.) (2.2) (2.3)
34 2 Rutherford E = 2 mv2 F (q) = ( ) dσ a 2 dω = 4E sin 4 (θ/2) (2.4) Rutherford F (q) = rv (r) (r ) F (q) = Born Born F (q) = F(q) = Born Rutherford V (r) =a/r Schrödinger
2.4 35 2.4 2.4. 4 Coulomb Coulomb Ze 2 ρ(r ) V (r) = r r dr (2.5) ρ(r) ρ(r)dr =4π ρ(r) r 2 dr = (2.6) Ze θ Dirac αz =(Ze 2 )/( hc) dσ dω = dσ dω F (q) 2 (2.7) Mott Mott [5] Ze ( ) dσ Ze 2 2 cos 2 (θ/2) dω = Mott 2E sin 4 (2.8) (θ/2) F (q) ρ(r) Fourier F (q) =4π ρ(r) j (qr) r 2 dr (2.9)
36 2 Rutherford j (qr) Bessel q q q = 2E sin (θ/2) (2.) hc F(q) = F (q) = F (q) ρ(r) Fourier ρ(r) = 2π 2 (2.9) F (q) j (qr) q 2 dq (2.) ρ(r) qr Bessel j (qr) = sin qr qr 6 (qr)2 + 2 (qr)4 (2.2) (2.9) F (q) = 4π ρ(r) r 2 dr q2 6 4π ρ(r) r 4 dr + q4 2 4π ρ(r) r 6 dr = 6 q2 r 2 + 2 q4 r 4 (2.3) r 2 =4π r 2 ρ(r) r 2 dr (2.4) r 2 E reduced de Broglie λ 2π = hc 2 MeV fm E E (2.5) fm E 2 MeV m e =.5 MeV/c 2 p = c E 2 (m e c 2 ) 2 E c (2.6) E = 2 MeV θ =6 q =fm
2.4 37 E = 2 MeV b =fm L = 2m e Eb=4.8 23 MeV s=.7 h (2.7) Coulomb Rutherford (2.7) Mott (2.8) Rutherford cos 2 (θ/2) /2 Dirac σ p/ p 2.4.2 (2.) q F (q) r q 2.2 F (q) 2 Fermi q F (q) 2 Mott q q ρ(r) Fourier (2.9) F (q)
38 2 Rutherford ρ / ρ.2..8.6.4.2. 2 4 6 8 radius r [fm] - -2 F ( q ) 2-3 -4-5 -6..2.4.6.8..2.4 momentum transfer q [fm - ] 2.2: Fermi ρ ρ(r) = ( ) (2.8) r R + exp a Fermi ρ a Fermi 2.3 r = R ρ(r) =ρ /2 a.6 fm r = R.5 fm
2.4 39 9 % % t t = a log e 9=2.2a R..9 t t t = 2.2 a ρ / ρ.5.. 2 3 4 5 6 7 8 9 radius r [fm] 2.3: Fermi R =6fm 5 fm 4fm Fermi R a Fermi ρ.7 fm 3 (2.9) 6fm 3 ρ m mρ =2.8 4 g cm 3 (2.2)
4 2 Rutherford 2.5 2. H. Geiger and E. Marsden, Proc. Roy. Soc. (London), A82 (99) 495 2. E. Rutherford, Phil. Mag. 2 (9) 669 3. H. Geiger and E. Marsden, Phil. Mag. 25 (93) 64 4. J. Chadwick, Nature, 29 (932) 32, Proc. Roy. Sco. (London), A36 (932) 692 5. N.F. Mott, Proc. Roy. Soc. (London) A35 (932) 429, A24 (929) 426 6. 977