(product-operator) I I cos ω ( t sin ω ( t x x ) + Iy )

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2 (product-operator) I I cos( t) + I sin( t) x x y

3 z 2π (rad) y 1 y t x = 2πν x t (rad)

4 sin t Iy# cos t t Ix# Ix# (t ) z Ix# Iy# Ix# (t ) z Ix cos (t ) + Iy sin (t )

5 -x -y t y I-y# I-y# (t ) z (t ) z x I-y# Ix# I-y cos (t ) + Ix sin (t )

6 J- I I cos( πjt) + 2I S sin( πjt) x x y z

7 ± J 2 (Hz) = ± πj ( rad s ) z Ix Sβ Ix Sα z Ix Sβ# y y πjt# x J Hz = 2πJ rad/s x πjt# Ix Sα# IxSβ# IxSβ cos (πjt) IySβ sin (πjt)# IxSα# IxSα cos (πjt) + IySα sin (πjt)#

8 J - + Ix Sα# Ix Sβ# Ix (Sα + Sβ) Ix Sα cos (πjt) + Iy Sα sin (πjt)# Ix Sβ cos (πjt) Iy Sβ sin (πjt)# Ix (Sα + Sβ) cos (πjt) + Iy (Sα Sβ) sin (πjt)# Sα + Sβ = 1 Sα - Sβ = 2 Sz Ix# Ix cos (πjt) + 2 Iy Sz sin (πjt)# Ix Sα S Sβ# Sα# = 1 = 2 Sz S z Ix Sβ

9 -2IxSz -Ix -Iy Iy# -2IySz 2IySz πjt πjt 2IxSz# 2IxSz# 2IxSz# (πjt) 2IzSz (πjt) 2IzSz Ix 2IxSz# Iy# 2IxSz cos (πjt) + Iy sin (πjt)

10 = z + [ I cos( t + φ) + I sin( t φ) ] = I z rf 1 x y

11 z 0 : Larmor z γb 0 = 0 γb = # y γ(b 0 -B) = 0 -# y y x γb = # γβ1 t = 1 t : x γβ1 = 1# 1 :

12 0 -# : # γ(b 0 -B) = 0 -# z θ 2 2 ( ) : cosθ = sinθ = y 2 2 ( ) ( ) γβ1 = 1# x 1 :

13 z ( ) γ(b 0 -B) = 0 -# θ γβ1 = 1# y x

14 z ( ) γ(b 0 -B) = 0 -# x θ γβ1 = 1# y

15 x z y θ θ γ(b 0 -B) = 0 -# γβ1 = 1# ( ) cos θ + = ( ) ( ) ( ) sin θ + = ( ) sin θ + = ( ) ( ) ( ) cos θ + = ( )

α α 800 MHz NMR Co (176 ppm) Cα (56 ppm) 800 * 0.

16 α α 800 MHz NMR Co (176 ppm) Cα (56 ppm) 800 * * (176-56) = Hz = Hz = 35.9 µs 2

17 z ppm = Hz 2 3 y = Hz x 180

18 I 0 exp( i I t ) 2 exp( i S t ) 1 =I 0 cos I t 2 { ( ) + isin( I t )}{ 2 cos( S t ) 1 + isin( S t )} 1 =I 0 cos I t 2 { ( ) + isin( I t )} 2 cos( S t ) 1 +ii 0 cos I t HSQC --- { ( ) + isin( I t )} 2 sin( S t ) 1

19 ( πj ) δ = δ π/4 ( + πj ) δ = δ + π/4 I S J δ 1 4J δ y π/4 π/4 π/4 π/4 y I x I Sα# Sβ# Sα# Sβ# x S Sα# Sβ# y S -2IxSz sin (πj2δ)

20 x I J 4J y 6 S x 7 6 z I y I z S y S x I x S 2IzSz 7 z I Iα# y I y S 5 x I Iβ# -2IzSy x S

21 8 z I Iα# S t 1 2 x I Iβ# y I y S 9 z I Iα# x S x I Iβ# y I y S I S 7 8 t1 9 x S

22 α β β α S t 1 10 z I Iα# y S y I x I Iβ# x S sin I S t 1 10 cos cos sin

23 10 2IzSy cos ( S t 1 ) 11 2IzSz cos ( S t 1 ) -2IzSx sin ( S t 1 ) -2IzSx sin ( S t 1 ) z S sin cos S t 1 cos sin S t 1 y S cos ( S t 1 ) sin ( S t 1 ) cos ( S t 1 ) sin ( S t 1 ) y S x S x S I S t

24 12-2IySz cos ( S t 1 ) +2IySx sin ( S t 1 ) MQ S 90y z S y I Sα cos ( S t 1 ) x I x S Sβ cos ( S t 1 ) y S I x/y 12 S t 1

25 reverse INEPT I S J δ J δ π/4 π/4 π/4 π/4 y I x I Sα cos ( S t 1 ) Sα cos ( S t 1 ) Sβ cos ( S t 1 ) Sβ cos ( S t 1 ) Ix cos ( S t 1 )

26 --- pulsed field gradient --- ( ) = γb 0 t + γ G( τ) φ t t 0 z( τ)dτ

27 Z - gradient - +3 ΔG +2 ΔG +1 ΔG 0-1 ΔG -2 ΔG -3 ΔG 0 3 ΔG -2 ΔG

28 gradient 30 G/cm for 1 ms ( ) 30 G = T T * (100 MHz / T) = MHz ( 1 H) MHz * 1 ms = 128 (/cm) 1 1cm 128 Gr 30 G/cm 1 cm 1 ms

29 Gradientecho (1) I Gr δ δ 2 3

30 Gradientecho (2) I Gr δ 6 δ 4 5

31 Gradientecho (3) I Gr δ δ 6 7

32 Gradientecho (4) I Gr δ δ 8 y δ Gr x +y

33 ( ) = S t S ( )exp( it)dt

34 FID NMR

35 7 7 x t# Sin 1 t# FID y 1 Cos 8 7 t# 6 5

38 ( ) = S t S ( )exp( it)dt = S(t) exp(it) dt

39 ( ) = S t S ( )exp( it)dt = S(t) exp(it) dt

40 ( ) = S t S ( )exp( it)dt = S(t) exp(it) dt

41 ( ) = S t S ( )exp( it)dt = S(t)exp(i( )t)dt

42 ( ) = S t S ( )exp( it)dt = S(t)exp(i( )t)dt

43 d = γ Iγ S! 2 r 3 A = I Z S Z 1 3cos 2 θ ( A + B + C + D + E + F) ( ) B = 1 ( 4 I + S + I S + )( 1 3cos 2 θ) C = 3 ( 2 I + S Z + I Z S + )sinθ cosθ exp( iϕ) D = 3 ( 2 I S Z + I Z S )sinθ cosθ exp ( +iϕ) E = 3 4 I + S + sin 2 θ exp( 2iϕ) F = 3 4 I S sin 2 θ exp ( +2iϕ)

44 % d = µ I r 3 ( µ r ( I ' )r * ' 3 r & 5 * µ S ) µ I = γ I!I = γ I! I X, I Y, I Z,. - ( ) = γ I! I + + I r = ( r sinθ cosϕ, r sinθ sinϕ, r cosθ) 2, I + I 2i /, I Z 1 0 B 0 I + = I X + ii Y S d = γ Iγ S! 2 r 3 I = I X ii Y ( A + B + C + D + E + F) I φ# θ# r r sinθ r cosθ

45 z = B0 µ S µ % d = µ I r + 3 ( µ r ( I ' )r * ' 3 r & 5 * µ S ) B 0 I r S $ D I = µ I r + 3 ( µ r ' I & )r ) & 3 r % 5 ) ( µ

46 $ D I = µ I r + 3 ( µ r ' I & )r ) & 3 r % 5 ) ( µ I ( ) r re = µ I r + 3 µ I r cosα 3 r 5 = µ I + 3µ I cosα re r 3 µ I I α µ I cosα re µ I µ I + 3µ I cosα re r S r 3 µ I + 3µ I cosα re

47 S µ I r r µ I + 3µ I cosα re = 2µ I µ I µ I I α = 0 µ I r µ I µ I I r S α = 90! µ I + 3µ I cosα re = µ I

48 µ I r magic- r S µ I µ I 54.7 I

49 B 0 dipole-dipole interaction Sx Iα Iβ Iα Ωs B 0 Iβ Iα Iα Sx Ωs

50 B 0 magic angle Sx Iβ Iα Iα Ωs B 0 B o Iβ Iα Iα Sx Ωs

51 B 0 Iz Sz Iz 2IzSz J(0) S Sz I S I S T 2 I S

52 Ι Σ 0 I x S y x S I y x y x 0 y y y x x y 2I + S + J (2 0 ) x x y

53 % d = µ I r 3 ( µ I r ( ' )r * ' 3 r & 5 * µ S ) µ I = γ I!I = γ I!( I X, I Y, I Z ) r = ( rcosθ X, rcosθ Y, rcosθ Z ) d = γ Iγ S! 2 ( ) r I X S X + I Y S Y + I Z S Z 3I X S X cosθ X cosθ X 3I X S Y cosθ X cosθ Y 3I X S Z cosθ X cosθ Z θ x # B 0 θ z # r θ y # I S 3I Y S X cosθ Y cosθ X 3I Y S Y cosθ Y cosθ Y 3I Y S Z cosθ Y cosθ Z 3I Z S X cosθ Z cosθ X 3I Z S Y cosθ Z cosθ Y 3I Z S Z cosθ Z cosθ Z θ θ θ

54 d = γ Iγ S! 2 ( ) r I X S X + I Y S Y + I Z S Z... 3I Z S Y cosθ Z cosθ Y µ I z I Z 2. cos(θz) 3. µ S y S Y 4. y cos(θy) 3 { µ } I cosθ Z re Z S 3 { µ I } cosθ Z cosθ Y Z { µ } I Z θz I θy θz { µ } I Z I Z S Y ( 3cosθ Z cosθ Y ) { µ I } + 3 µ I Z { } cosθ Z cosθ Z Z I Z S Z ( +1 3cosθ Z cosθ Z )

55 ?? 94 35

56 R 360 R 360 φ R 360 sin φ ( ) φ φ

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