2 0.1 Introduction NMR 70% 1/2

Y. Kondo 2010 1 22

2 0.1 Introduction NMR 70% 1/2

3 0.1 Introduction......................... 2 1 7 1.1.................... 7 1.2............................ 11 1.3................... 12 1.4.......................... 12 2 15 2.1........................ 15 2.2............................ 15 2.3.................... 17 2.4 Si Ge.................... 18 3 21 3.1 SI........................... 21 3.2...................... 22 3.3...................... 23 4 27 4.1......................... 27 4.2............ 28

4 4.3..................... 29 4.4............ 30 4.5.............................. 31 5 33 5.1.................... 33 5.2 I.......................... 35 5.3......................... 37 5.4 II.......................... 39 6 45 6.1.................. 45 6.2..................... 49 6.3............... 52 7 55 7.1......................... 55 7.2......................... 57 8 71 8.1............................ 71 8.2.......................... 72 8.3.......................... 77 9 NMR 83 9.1.............................. 83 9.2........................... 84 9.3........................... 85 9.4........................ 87

5 9.5......................... 89 9.6 NMR.................... 90 10 NMR 95 10.1.................... 95 10.2...................... 96 10.3.................. 97 10.4................ 98 10.5.................... 98 10.6...................... 99 10.7........................ 99 101

7 1 1.1 16 *1 electrum electrica 17 18 *1 Wikipedia

8 1 1.1 1.2 *2 *3 18 *2 *3, 20

1.1 9 18 1791 1.3 1899 *4 *4

10 1 19 *5 1.4 19 1864 1888, 2(H++e-) H2 *5

1.2 11 1.5 1.2 2 1.6 2 1.6

12 1 1 e = 1.602 10 19 C (1.1) 1.3 10 18 Ωm 10 6 Ωm 10 3 Ωm 1.4 *6 1.7 1.7 *6

1.4 13 1.7

15 2 2.1 n n = 1, 2, 3,... 2.1 2.2 2.1 2? 2.2 2

16 2 E = 0 n = 3 n = 2 n = 1 2.1 ŒÇ µ ½Œ Žq QŒÂ ŠŒÝ ì p µ Ä éœ Žq QŒÂ 2.2 2 2 2

2.3 17 E = 0 ½ Ì ŠŒÝ ì p µ Ä éœ Žq 2.3 2.3

18 2 2.4 Si Ge Si Ge 4 Si Ge 3 2 2.4 ³ E Ž R džq Si â Ge Ì ÅŠOŠk Ì džq ^ «¼ ± Ì 2.4

2.4 Si Ge 19 Si Ge P As A 2.5 5 ß è È džq džq Ì s «Ž Œ^ ¼ ± Ì Œ^ ¼ ± Ì 2.5 n p B Al Ga 3

21 3 3.1 SI SI EB SI kg s m A I [I] = A [ ]=

22 3 3.2 3.2.1 I Q Q = It t [t] = s C C = [Q] = [It] = s A (3.1) 3.2.2 F = EQ [ F ] = [ E][Q] [ F ] = m kg s 2 E [ E] = m kg s 3 A 1 (3.2) 3.2.3 ϕ ϕ = E

3.3 23 = ( x, y, z ) [ ] = m 1 [ϕ] = m 2 kg s 3 A 1 (3.3) 3.2.4 F = 1 Q 1 Q 2 r 4πϵ 0 r 2 r (3.4) ϵ 0 [ F = kg m s 2 [ ] Q1 Q 2 r r 2 = A2 s 2 r m 2 [ϵ 0 ] = m 3 kg 1 s 4 A 2 (3.5) ϵ 0 ϵ 0 = 1 4πc 2 107 c = 2.99792458 10 8 m/s 3.3

24 3 3.3.1 H d r = i I i H [ H] = m 1 A (3.6) 3.3.2 EB H Q m F F = HQ m Q m [Q m ] = m 2 kg s 2 A 1 (3.7) [Q m ] = Wb 3.3.3 F = 1 Q m1 Q m2 r 4πµ 0 r 2 r (3.8)

3.3 25 µ 0 [ F = kg m s 2 [ ] Qm1 Q m2 r r 2 = m 2 kg 2 s 4 A 2 r [µ 0 ] = m kg s 2 A 2 (3.9) µ 0 µ 0 = 4π 10 7 c 2 ϵ 0 µ 0 = 1 3.1 D = ϵ 0 E B = µ0 H 3.2 Q = CV V = L d I C, L dt C, L

27 4 4.1

28 4 4.2 t x(t) δt x(t + δt) x(t) t f (x(t), u(t)) x(t + δt) x(t) = f (x(t), u(t)) δt + o(δt 2 ) (4.1) u(t) o(δt 2 ) δt 2 δt 0 d x(t) = f (x(t), u(t)) (4.2) dt u(t) y(t) x(t), u(t) t y(t) = g (x(t), u(t)) (4.3) y(t) = S (u(t)) (4.4)

4.3 29 S 4.1 *1 u(t) S y(t) 4.1 S t = τ L τ (u(t)) = { u(t), t τ 0, t > τ (4.5) S u(t) s.t. L τ (u(t)) = 0 L τ S (u(t)) = 0 (4.6) s.t. such that L τ (u(t)) = 0 u(t) L τ S (u(t)) = 0 4.3 S(α 1 u 1 + α 2 u 2 ) = α 1 S(u 1 ) + α 2 S(u 2 ) (4.7) *1 TV TV

30 4 t s.t. y(t) = S(u(t)) τ y(t + τ) = S(u(t + τ)) (4.8) 4.4 V V I A R V = RI (4.9) V/A Ω I = u(t) V = y(t) S(u(t)) = R u(t) I V y(t) u(t) e(t) x(t) v(t) e(t) v(t) = R ( C d dt v(t)) (4.10)

4.5 31 RC = τ d u(t) x(t) x(t) = dt τ (4.11) y(t) = u(t) x(t) (4.12) u(t) = e(t) t = 0 0 e 0 t 0 x(t) = (x(0) e 0 )e t/τ + C (4.13) *2 C *3 t x(t) x(0) *4 4.5 *2 z(t) = x(t) e 0 d z(t) z(t) = z(t) = z(0)e dt τ t/τ z(0) = x(0) e 0 *3 t = 0 x(t) C = e 0 *4 u(t) = e(t) x(t) = x(0)e t/τ + 1 τ t 0 e t t τ u(t )dt (4.14)

32 4 4.2

33 5 5.1

34 5 5.1.1 V V I A R V = RI (5.1) V/A Ω 5.1.2 R L m S m 2 T K R(T ) Ω R(T ) = ρ(t ) L S (5.2) ρ(t ) Ω m ρ(t ) ρ(t ) = ρ(t 0 ){1 + α(t T 0 )} (5.3) α T 0 *1 *2 *1 *2

5.2 I 35 http://www.phys.kindai.ac.jp/users/kondo 5.2 I 5.2.1 5.2.2 5.1

36 5 5.2 5.2.3 v i j i v i j i

5.3 37 v i j i = 0 i 5.3 5.3.1 R 1 R 2 V 1 V 2 I V V = V 1 + V 2 (5.4) V i = R i I V R 1 R 2 V V 1 2 5.3 5.3.2 R 1 R 2 I 1 I 2

38 5 V I I = I 1 + I 2 (5.5) I i = V/R i V I I 1 2 R 1 R 2 5.4 5.3.3 R 1, R 2,... R = R 1 + R 2 +... 1 R = 1 R 1 + 1 R 2 +... 5.3.4

5.4 II 39 5.5(a) 2 5.5(b) 4 5.5 2 4 5.4 II 5.4.1

40 5 5.4.2 - v i j i R i 2 *3 *4 - R *5 v R 0 v i = R + R 0 = 5.6 *3 port *4 *5

5.4 II 41 R 0 5.7 v R 0 v R 0 = + v v ï R R 5.7 R 0 R 0 R 0 R

42 5 j R 0 v 0 = RR 0 R + R 0 j 5.8 R 0 v j 0 j 0 = R 0 R + R 0 1/R 0 v j j = v/r j 0 = 1/R + 1/R 0 R + R 0 R 0 R 0 v j R R 0 R R 0 = 5.8 5.1 5.9 1. 2. 5.2 5.10 R 5-1. R 5

5.4 II 43 R 0 R 3 E 1 R 1 E 3 E 2 5.9 2. R 5 3. R 5 R 1 R 3 E R 5 R 4 R 2 5.10

45 6 6.1 V (t) = V 0 cos ωt ω f = ω/2π R L C

46 6 6.1.1 *1 I(t) V (t) = V 0 cos ωt = RI(t) I(t) = V 0 cos ωt R 6.1.2 V (t) L di dt = V 0 cos ωt L di dt = 0 I(t) = V 0 cos ωtdt = V 0 cos(ωt π/2) L ωl *2 6.1.3 V (t) Q/C = 0 I(t) = dq dt = C dv dt = C dv 0 cos ωt dt = V 0 ωc cos(ωt + π/2) *1 *2 t t = t π/(2ω) V 0 cos ωt

6.1 47 *3 I II III R L C - q q 6.1 6.1.4 R V (t) = V 0 cos ωt V 2 (t)/r < V 2 (t) R > = 1 T = V 0 2 R = 1 2 T 0 1 T V 2 0 R V 2 0 R cos2 ωtdt T 0 1 + cos 2ωt dt 2 T V e = V 0 2 Ve 2 R I e = I 0 2 *3 t t = t + π/(2ω) V 0 cos ωt

48 6 π/2 π/2 ϕ cos ϕ I e V e cos ϕ 6.1.5 C Q 0 S I(t) I(t) = dq(t) dt Q(t) S Q L - Q C R 6.2 0 = L di(t) dt = L d2 Q(t) dt 2 + RI(t) + Q(t) C + R dq(t) dt + Q(t) C

6.2 49 R = 0 Q(t) d 2 Q(t) dt 2 = 1 LC Q(t) Q(t) = Q 0 cos(ω 0 t + δ) 1 ω 0 = LC Q(t) = Ae αt cos(ω t + δ) α = R 2L 1 ω = LC R2 4L 2 6.2 6.1 ϕ(t) = ϕ 0 cos(ωt + α) (6.1)

50 6 { I(t) = I0 cos(ωt + β) (6.2a) Q(t) = Q 0 cos(ωt + γ) (6.2b) ϕ 0 = ϕ 0 e iα Ĩ(t) = I 0 e (ωt+β) iωt = Ĩ0e (6.3c) ϕ(t) = ϕ 0 e i(ωt+α) = ϕ 0 e iωt Ĩ 0 = I 0 e iβ Q(t) = Q 0 e (ωt+γ) = Q 0 e iωt Q 0 = Q 0 e iγ (6.3a) (6.3b) (6.3d) (6.3e) (6.3f) L dĩ(t) dt Q + RĨ(t) + C = ϕ(t) {L di(t) dt + RI(t) + Q C } + i{ldi (t) dt + RI (t) + Q C } = ϕ(t) + iϕ (t) L, R, C { } 6.2.1 L di(t) dt + RI(t) + Q = ϕ(t) (6.4) C

6.2 51 6.2 d Q(t) dt = iω Q 0 e iωt = Ĩ0e iωt iω Q = Ĩ dĩ(t) = iωĩ0e iωt dt iωlĩ + RĨ + Ĩ iωc = ϕ e iωt Ĩ = ϕ Z Z = R + i ( ωl 1 ωc ) (6.5) (6.6) R ω ωl 1 ωc = 0 Z 6.6 ω 0 ω 0 = 1 LC (6.7) Z

52 6 6.3 6.1 j(t) v(t) j(t) v(t) v(t) = Z(ω)j(t) 6.1 Z(ω) = 1 iωc Z(ω) = iωl *4 6.1 5 *5-5 6.1 6.3 1. 2. *4 Z(ω) = R *5

6.3 53 E 1, E 3 E i (t) = E i,0 e iωit L C R E 1 E 2 E 3 6.3 6.2 6.4 R 5 - E = E 0 e iωt 1. R 5 2. R 5 3. R 5 L 1 C 3 E R 5 R 4 C 2 6.4

55 7 6.2 7.1 7.1.1 u(t) = e iωt *1 *1 6.2

56 7 y(t) = G(iω)e iωt G(iω) ω y(t) = S(e iωt ) y(t + τ) = S(u(t + τ)) = S(u(t)u(τ)) = u(t)s(u(τ)) *2 τ = 0 y(t) = S(u(0))e iωt S(u(0)) t G(iω) ω u(t) y(t) G(iω) y(t) = G(iω)u(t) ( ) k d a k y(t) = ( l d b l u(t) (7.1) dt dt) k l u(t) = e iω, y(t) = G(iω)e iωt G(iω) k a k (iω) k = l b l (iω) l (7.2) *2 S(ab) = as(b) a = u(t), b = u(τ)

7.2 57 G(iω) k G(iω) = b k (iω) k l a l (iω) l (7.3) G(iω) 7.1.2 u(t) u(t) = 1 2π u ω e iωt dω y(t) = 1 G(iω)u ω e iωt dω 2π y(t) y ω G(iω)u ω 7.2 7.2.1 f(t) L(f(t)) = 0 f(t)e st dt (7.4)

58 7 f(t) = L(α 1 f 1 (t) + α 2 f 2 (t)) L(f(αt)) = G(s) 0 (α 1 f 1 (t) + α 2 f 2 (t))e st dt = α 1 L(f 1 ) + α 2 L(f 2 ) (7.5) L(f(αt)) = f(αt)e st dt 0 αt t = f(t )e (s/α)t dt /α 0 = 1 α G ( s α ) (7.6) f(t) = L( d dt f(t)) 0 ( d dt f(t))e st dt = [ e st f(t) ] ( d 0 dt e st )f(t)dt = f(0) + s L(f(t)) 0 0 e st f(t)dt L( d f(t)) = f(0) + sl(f(t)) (7.7) dt ( t ) L f(t )dt = 0 0 0 ( t ) f(t )dt e st dt

7.2 59 = [ 1s t ] e st f(t )dt 0 0 ( d t dt = 1 s 0 0 0 e st f(t)dt f(t )dt ) 1 s e st dt ( t ) L f(t )dt = 1 L(f(t)) (7.8) 0 s 7.1 7.1 (s > 0) δ(t) 1 1 1/s t 1/s 2 e t 1/(s 1) cos t s/(s 2 + 1) sin t 1/(s 2 + 1) 7.2.2 δ x 0 δ(x) = 0 δ(x)dx = ϵ ϵ δ(x)dx = 1 f(x)δ(x)dx = f(0)

60 7 δ(x) = δ( x) δ(x 2 a 2 ) = 1 2a (δ(x a) + δ(x + a)) δ(ax) = 1 a δ(x) a 0 δ(x) = d dxθ(x) Θ(x) f(x)δ(x a)dx = f(a) *3 δ(x) = lim n φ n(x) n 2 φ n (x) = π e nx *4 n 1 ; t > 0 1 u(t) = 2 ; t = 0 0 ; t < 0 1 δ(t) = lim (u(t) u(t w)) w 0 w δ 1 (7.9) L (δ(t)) δ(t)e st dt 0 ϵ ( 0 ϵ L (δ(t)) = δ(t)e st dt = e s 0 = 1 *3 δ *4 φ(0) = n/π n x 0 1

7.2 61 φ n (t) φ n (t) 0 ϵ ϵ > 0 n 2 π e nt e st dt = 1 ( ) s 2nϵ /4n Erfc 2 es2 2 n ϵ Erfc n 2.0 1.5 1.0 0.5 10 5 5 10 7.1 7.1 L(δ(t)) = 1 lim (u(t) u(t w)) w ( ) 1 L (u(t) u(t w)) = 1 ( 1 w w s 1 ) s e wt = 1 ( 1 e wt ) ws = 1 ws +... 2! w 0 1 w 0 L(δ(t)) = 1

62 7 ϵ L( d δ(t)) = δ( ϵ) + sl(δ(t)) dt δ( ϵ) = 0 L(δ(t)) = 1 L( d dt δ(t)) = s L( dn δ(t)) = sn dtn 7.1 7.1 7.2.3 E t = 0 L d dt i + Ri = E 7.1 τ t = τt L τ d dt i + Ri = E L τ i(0) + L τ sl(i) + RL(i) = E s

7.2 63 t = 0 i(0) = 0 L(i) L(i) = E s(ls/τ + R) = E ( ) 1 R s 1 s + τ(r/l) τ = L/R L(i) = E ( 1 R s 1 ) s + 1 7.1 i(t ) = E R (1 e t ) t = τt t i(t) = E R (1 e t/τ ) 7.2.4 f(t), g(t) (f g) (f g)(t) = t L (f(t)) L (g(t)) L (f(t)) L (g(t)) = f(x)e sx dx g(y)e sy dy 0 0 : x + y t ( ) = f(t y)e st g(y)dt dy 0 y 0 f(τ)g(t τ)dτ (7.10)

64 7 7.2 ( t ) = f(t y)e st g(y)dy dt = 0 0 0 e st ( t = L ((f g)(t)) 0 ) f(t y)g(y)dy dt t t y y 7.2 7.2.5 t t = τ u(t)δ(t τ) y δ (t τ) t 0 u(τ)y δ (t τ)dτ (7.11) τ = 0 t y δ (t τ) t τ τ u(t) y(t) y δ (t) y δ u

7.2 65 L (y δ (t)) L (u(t)) = L ((u y δ )(t)) L (y δ (t)) 7.2.6 ( ) k d a k y(t) = ( l d b l u(t) (7.12) dt dt) k u(t) = δ(t) y(t) dn dt n y(x)] t=0 = 0 y δ (t) a k s k L (y δ (t)) = k k l b k s k (7.13) y δ (t) L (y δ (t)) = G(s) = l b ls l k a ks k (7.14) 7.2.7 L di(t) dt + q(t) C = E(t)

66 7 τ L di(t ) τ dt + q(t ) C = E(t ) L τ sl(i) + 1 L(q) = L(E) + Li(0) C i(t ) = 1 τ dq/dt L(q) = τ(l(i) + q(0))/s E = E 0 δ(t) L(E) = E 0 L(i) = E 0 L/τ τ 2 = LC 7.1 t s s 2 + τ 2 LC L(i) = E 0 s L/τ s 2 + 1 i(t ) = E 0 cos t L/τ i(t) = E 0 L/τ cos t/τ = E 0 ω 0 L cos ω 0t ω 2 0 = 1/LC 7.2.8 t f(t) L(f(t)) f(t)

7.2 67 f(t) = 1 2πi γ+i γ i F (s)e st ds t > 0 0 t < 0 (7.15) L 1 L 1 (F (s)) = f(t) F (s) R(s) > γ 7.3 C C s 0 F (s 0 ) = 1 F (s) ds (7.16) 2πi C s s 0 s F (s) 0 (7.17) iω C s' s 0 0 γ σ 7.3

68 7 0 F (s 0 ) = 1 γ i F (s) ds 2πi s s 0 = 1 2πi γ+i γ+i γ i F (s) s 0 s ds t > 0 t [ e t(s0 s) 1 dt = e t(s 0 n) s s 0 0 1/(s 0 s) F (s) = 1 2πi = 1 2πi = 1 2πi γ+i ( F (s) γ i ( γ+i 0 γ i ( γ+i 0 γ i 0 ] 0 = 1 s 0 s ) e t(s0 s) dt ds ) F (s)e t(s0 s) ds dt ) F (s)e ts ds e ts0 dt (7.18) F (s)e t(s 0 s) 7.18 7.2.9 F (s) = 1 f(t) s s 1/s 0 t > 0 7.4 C 1 *5 s = 0 *5 e st t

7.2 69 iω C 1 C 2 0 γ σ 7.4 f(t) = 1 γ+ 1 2πi γ s est ds = 1 1 2πi C 1 s est ds = e 0 = 1 t < 0 C 2 *6 f(t) = 0 F (s) = 1/s F (s) = 1 s 2 f(t) + 1 s 1/(s 2 + 1) 0 F (s) s = ±i 7.4 γ > 1 t > 0 C 1 s = ±i *6 e st t

70 7 f(t) = 1 γ+ 1 2πi γ s 2 + 1 est ds = 1 γ+ ( 1 1 2πi γ 2i s i 1 ) e st ds s + i = 1 ( 1 γ+ e st 2i 2πi γ s i ds 1 γ+ e st ) 2πi γ s + i ds = 1 ( e it e it) 2i = sin t t < 0 C 2 f(t) = 0

71 8 8.1 8.1.1 2

72 8 8.1.2 3 2 3 8.2 FET z É z É ƒoƒšƒbƒh ƒq [ƒ^ @ @ @ @ A É ƒq [ƒ^ @ @ @ @ A É (a) Q ÉŠÇ @ @ @ @ @ @ @ @(b) R ÉŠÇ 8.1 2 3

8.2 73 8.2.1 PN p n PN PN n p 0 p n n p n p *1 n p p pn n p *2 *1 *2

74 8 - - - - + + + + ó R w Œ^ ¼ ± Ì @ @ @ @ @ ŽŒ^ ¼ ± Ì ƒtƒfƒ ƒ~ ˆÊ Ö Ñ (a) dˆ³ ª Á Ä È Žž @ @ - - - - + + + + ó R w - - - - + + + + ó R w ƒtƒfƒ ƒ~ ˆÊ ƒtƒfƒ ƒ~ ˆÊ Ö Ñ (b) ûœüƒoƒcƒaƒx @ @ @ @(c) t ûœüƒoƒcƒaƒx Ö Ñ 8.2 8.2.2 NPN n p PN NP -

8.2 75 ŽŒ^ ¼ ± Ì @ @ @ @ @ @ ŽŒ^ ¼ ± Ì E ƒgƒ~ƒbƒ^ Œ^ ¼ ± Ì B ƒx [ƒx C ƒrƒœƒnƒ^ ó R w 8.3 PN - PN p *3 - *3

76 8 *4 PNP 8.2.3 Field Effect Transistor; FET (Junction-type FET; JFET) (metal-oxide-semiconductor FET; MOSFET) MOSFET MOSFET p n S( ) D( ) G p SB SB S G D 8.4 FET *4 100 100 1

8.3 77 n n 8.3 IC 8.3.1 2 ( +, - ) 1 2 = V + V V- V + - + V o 8.5

78 8 V O = A(V + V ) A (operational amplifier; ) A ( ) 2 8.3.2 R 2 R 2 V I R 1 - V o R 1 - V o + V I + 8.6 + (0 V) V I R 1 - - R 2 A V O V + V = 0 *5 V = 0 *5 V O /A = V + V A V + V = 0 A V + V 0

8.3 79 - R 1 R 2 V O = R 2 R 1 V I R 2 - + R 1 I I = V I /R 1 - I = (V O V I )/R 2 V O = R 1 + R 2 V I R 1 8.3.3 V i V I V i + - A H V o + ÁŽZ ð s È 8.7 H 0 < H < 1

80 8 V O = AV i V i = V I HV O V i G A 1 G = V O A = V I 1 + AH 1 H 1/H A 8.3.4 R 2-1 - 1-1 + + + 8.8

8.3 81 H - 1-1 + + ƒ [ƒpƒx EƒtƒBƒ ƒ^ [ @ @ƒnƒcƒpƒx EƒtƒBƒ ƒ^ [ 8.9

83 9 NMR NMR NMR 9.1 H 0 B 0 M 0 H 0 *1 z H 0 = (0, 0, H 0 ) x - y - B H B *1

84 9 NMR z M 0 y x H 0 9.1 H 0 M 0 9.2 B 0 M d M dt = γ M B 0, (9.1) γ γ > 0 M B 0 M 9.1 z *2 ω 0 = γb 0 *3 42.59 MHz/T 10.71 MHz/T *4 z M *2 *3 γb 0 ω 0 > 0 *4 1 T 42.59 MHz 10.71 MHz

9.3 85 M z z x,y ω M ω 0 ω (0, 0, B 0 ω/γ) 9.3 B 0 = (0, 0, B 0 ) B 1 = B 1 (cos (ω rf t ϕ), sin (ω rf t ϕ), 0) (9.2) B 0 B 1 9.1 ω rf B 1 = ( B 1 cos ϕ, B 1 sin ϕ, B 0 ω rf /γ) M (a) z z (b) z M H M 0 y y y x x x 9.2 (a) M ω 0 = γb 0 (b) M

86 9 NMR γ B 2 1 + (B 0 ω rf /γ) 2 z φ x y 9.3 B 1 = ( B 1 cos ϕ, B 1 sin ϕ, B 0 ω rf /γ) ω rf (= ω 0 ) ω rf M (cos ϕ, sin ϕ, 0) ω 1 = γb 1 t p (0, 0, M) M β = ω 1 t p β = π/2 π/2-90 - (0, 0, M) x-y (M sin ϕ, M cos ϕ, 0) 90 ϕ 90 0 90 π/2 90 π 90 3π/2 90 x 90 y 90 x 90 y β = π π- (, 180 - ) M M 2B 1 (cos (ω rf t ϕ), 0, 0) 2B 1 (cos (ω rf t ϕ), 0, 0)

9.4 87 (a) z (b) z y y x x 9.4. (a) 90 x- y (b) 180 x- = B 1 (cos (ω rf t ϕ), sin (ω rf t ϕ), 0) B 1 (cos (ω rf t ϕ), sin (ω rf t ϕ), 0), ω rf ω rf 2ω rf NMR t p 1/(γB 1 ) 1/(γB 0 ) 1/ω rf 9.4 M 0 = (0, 0, M 0 ) dm = γm dt B 0 Γ( M M 0 ), (9.3) 1/T 2 0 0 Γ = 0 1/T 2 0 0 0 1/T 1

88 9 NMR z z (a) (b) y y x x (c) z y (d) z y x x 9.5 T 2 (a) t = 0 M x (b d) M xy (d) T 1 T 2 2 Γ( M M 0 ) T 2 T 1 NMR t = 0 M(0) = M 0 (cos χ, sin χ, 0) M(t) T 2 T 1 t T 2 M(t) = 0 xy M 0 T 1 T 2 T 2 t = 0 x M 9.5 xy M xy

9.5 89 z z (a) (b) y y x x (c) z y (d) z y x x 9.6 (a) M i 90 x- y-axis (b) r τ M i xy (c) 180 y M i y (d) τ M i y 9.5 NMR ω0 i M i *5 B 1 B0 i ω rf /γ 90 x- M i y τ M i xy 180 y y M i τ *5

90 9 NMR M i y 180 - M i 9.6 NMR NMR 9.7 Pulse Generator test tube Oscillator Power Amp. z' ADC ADC cos ω reft sin ω LPF LPF ref t Pre.Amp. x' s A s B Mixer s A xs B Directional Coupler 9.7 NMR Oscillator Pulse Generator test tube LPF ADC - Directional Coupler mixer

9.6 NMR 91 xy M = (M x, M y, 0) = M(cos χ, sin χ, 0) M(t) = M(cos χ, sin χ, 0) exp( t/t 2 ), T 2 T 1 T 2 T 1 NMR M (t) = M(cos(ω 0 t χ), sin(ω 0 t χ), 0) exp( t/t 2 ). ω 0 x M cos(ω 0 t χ) exp( t/t 2 ) * 6 Free Induction Decay (= FID) FID cos ω ref t M cos(ω 0 t χ) exp( t/t 2 ) cos ω ref t = 1 2 M (cos( ω t χ) + cos(( ω + 2ω ref)t χ)) exp( t/t 2 ), ω ref > 0 ω = ω 0 ω ref *6 x dm x dt = Mω 0 sin(ω 0 t χ) exp( t/t 2 ), ω 0 1/T 2 exp( t/t 2 ) ω 0 x M cos(ω 0 t χ) exp( t/t 2 )

92 9 NMR ( ω + 2ω ref ) 1 2 M cos( ω t χ) exp( t/t 2). 2 ω ref FID sin ω ref t 1 2 M sin( ω t χ) exp( t/t 2), ω ref ω 0 100 MHz, ω 10 khz 1/T 2 1 Hz s(t) = M (cos( ωt χ) + i sin( ωt χ)) exp( t/t 2 ) = M exp( iχ) exp(i ωt) exp( t/t 2 ) t < 0 s(t) = 0 s(t) S(ω) = = M exp( iχ) s(t) exp( iω t)dt 0 1/T 2 i(ω ω) = M exp( iχ) (1/T 2 ) 2 + (ω ω) 2. exp(i ωt) exp( t/t 2 ) exp( iω t)dt χ = 0, S(ω) ω R(S(ω)) = M/T 2 (1/T 2 ) 2 + (ω ω) 2. ω = ω MT 2 R(S(ω)) > MT 2 /2 FWHH 1/πT 2 T 2

9.6 NMR 93 Absorptive (Real) Spectrum M T 2 M T 2 2 ω FWHH = 1 π T 2 ω Dispersive (Imaginary) Spectrum ω ω 9.8 ω = ω 0 ω ref M T 2 (FWHH) M S(ω) M(ω ω) I(S(ω)) = (1/T 2 ) 2 + (ω ω) 2. χ 0

95 10 NMR NMR NMR 10.1 II *1 47 µt ω H = 2π 2 10 3 rad s 1 *2 *1 γ C = 2π 10.71 10 6 s 1 T *2 2 khz

96 10 NMR h = 1.055 10 34 J s k B = 1.38 10 23 J K 1 µ 0 = 4π 10 7 N A 2 γ H = 2π 42.58 10 6 s 1 T N A = 6.02 10 23 mol 1 ρ Cu = 1.7 10 8 Ωm 10.1 SI 10.2 sd sl sd = 25.0 10 3 m sl = 45.0 10 3 m sv sv = πsd 2 sl/4 = 2.209 10 5 m 3 1 18 10 3 kg 1 m 3 1000 kg 1 2 1.11 10 5 mol m 3 sa sa = sv 1.11 10 5 mol m 3 = 2.452 mol

10.3 97 T = 300 K B 0 = 30 mt *3 1 µ H µ H = ( hγ H) 2 4k B B 0 T = 1.44 10 33 A m 2 M H M H = sa µ H N A = 2.13 10 9 A m 2 Φ H Φ H = µ 0 M H sv (πsd2 /4) = µ 0 M H /sl = 5.95 10 14 Wb 1 V H V H = dφ H dt = ω H Φ H = 7.48 10 10 V 10.3 ϕ = 0.5 10 3 m N L = 10 N t = (sl/ϕ)n L L π(sd/2) 2 L = A n µ 0 sl N 2 t = 7.64 10 3 H A n A n = 0.688 R πsd R = ρ Cu π(ϕ/2) 2 N t = 6.12 Ω *3 1 A 30 mt

98 10 NMR Q Q = ω HL R = 15.7 V H N t Q = 1.1 10 5 V 100 µv ω H C C = 1 ω 2 H L = 8.29 10 7 F 10.4 (M H /sv) 2 2µ 0 sv = 8 10 8 J 1 (V H N t Q) 2 R 2π ω H = 9.1 10 15 J 10.5 1 mv 10 µv 100 2 khz 1000

10.6 99 1000 10.6 H H = ni n n = N L /ϕ B 0 = µ 0 N L ϕ I 30 mt I = 1.2 A 10.7 2 khz FID NMR

101 [1] I,II. [2] http://homepage2.nifty.com/eman/electromag/contents.html [3] EH EB EB EH B EH EB http://www.fdenshi.com/000tokiwajpn/32denjk/010elc.html monograph http://ir.lib.hiroshimau.ac.jp/metadb/up/81936204/refunit43w.pdf [4] [5] G. Arfken, Mathematical Method for Physicist.