xIxΔ

Δ

I-spin180°パルス�

S
σ0 σ1σ2σ2σ1
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CS-term J-term refocusedx

IΔS-spin
180°パルス�

ΔxS
σ0 σ1σ2σ2σ1CS–Iy!180"Sx#J–Iy–Iy

CS
–Iy

J–Iy

–Iy
2IxSz

Ix

Ix
2IySz

2IxSz

–2IxSz–2IxSz–2IxSz–Iy–2IySz–2IySz
Ix

Ix

Ix

Ix

Ix

Ix
2IySz

Iy

Iy
–2IxSz

2IySz–2IySz–2IySz–2IySz
Ix

2IxSz

2IxSz
Iy

Ordered–Iy cos!" cos#J" cos!" cos#J"
2IxSz cos!" cos#J" cos!" sin#J"
Ix cos!" cos#J" sin!" cos#J"
2IySz cos!" cos#J" sin!" sin#J"–2IxSz cos!" sin#J" cos!" cos#J"–Iy cos!" sin#J" cos!" sin#J"–2IySz cos!" sin#J" sin!" cos#J"
Ix cos!" sin#J" sin!" sin#J"
Ix sin!" cos#J" cos!" cos#J"
2IySz sin!" cos#J" cos!" sin#J"
Iy sin!" cos#J" sin!" cos#J"–2IxSz sin!" cos#J" sin!" sin#J"–2IySz sin!" sin#J" cos!" cos#J"Ix sin!" sin#J" cos!" sin#J"
2IxSz sin!" sin#J" sin!" cos#J"
Iy sin!" sin#J" sin!" sin#J"

SortedIx cos!" cos#J" sin!" cos#J"
Ix cos!" sin#J" sin!" sin#J"
Ix sin!" cos#J" cos!" cos#J"
Ix sin!" sin#J" cos!" sin#J"–Iy cos!" cos#J" cos!" cos#J"–Iy cos!" sin#J" cos!" sin#J"
Iy sin!" cos#J" sin!" cos#J"
Iy sin!" sin#J" sin!" sin#J"
2IxSz cos!" cos#J" cos!" sin#J"–2IxSz cos!" sin#J" cos!" cos#J"–2IxSz sin!" cos#J" sin!" sin#J"
2IxSz sin!" sin#J" sin!" cos#J"
2IySz cos!" cos#J" sin!" sin#J"–2IySz cos!" sin#J" sin!" cos#J"
2IySz sin!" cos#J" cos!" sin#J"–2IySz sin!" sin#J" cos!" cos#J"

SummarizedIx 2 cos!" sin!" (cos2#J" + sin2#J") = Ix sin2!"–Iy (cos2!" – sin2!")(cos2#J" + sin2#J") = –Iy cos2!"
2IxSz (cos2!" – sin2!") (cos#J" sin#J" – cos#J" sin#J") = 0
2IySz 2 cos!" sin!" (cos#J" sin#J" – sin#J" cos#J") = 0

J-term refocusedCS-term not-refocusedCS-term not-refocused

x

Ix
Δ

I-spin/S-spin
180°パルス�

ΔxS
σ0 σ1σ2σ2σ1
CS–Iy!180"Ix + 180"Sx#J–Iy–IyCS

Iy

JIy

Iy
–2IxSz–Ix–Ix–2IySz2IxSz–2IxSz–2IxSz–2IxSz–Iy–2IySz–2IySz
Ix

Ix

Ix

Ix

Ix

Ix
2IySz

Iy

Iy
–2IxSz

2IySz

2IySz

2IySz

2IySz
–Ix

–2IxSz–2IxSz–IyOrdered
Iy cos!" cos#J" cos!" cos#J"–2IxSz cos!" cos#J" cos!" sin#J"–Ix cos!" cos#J" sin!" cos#J"–2IySz cos!" cos#J" sin!" sin#J"–2IxSz cos!" sin#J" cos!" cos#J"–Iy cos!" sin#J" cos!" sin#J"–2IySz cos!" sin#J" sin!" cos#J"
Ix cos!" sin#J" sin!" sin#J"
Ix sin!" cos#J" cos!" cos#J"
2IySz sin!" cos#J" cos!" sin#J"
Iy sin!" cos#J" sin!" cos#J"–2IxSz sin!" cos#J" sin!" sin#J"
2IySz sin!" sin#J" cos!" cos#J"–Ix sin!" sin#J" cos!" sin#J"–2IxSz sin!" sin#J" sin!" cos#J"–Iy sin!" sin#J" sin!" sin#J"

Sorted–Ix cos!" cos#J" sin!" cos#J"
Ix cos!" sin#J" sin!" sin#J"
Ix sin!" cos#J" cos!" cos#J"–Ix sin!" sin#J" cos!" sin#J"
Iy cos!" cos#J" cos!" cos#J"–Iy cos!" sin#J" cos!" sin#J"
Iy sin!" cos#J" sin!" cos#J"–Iy sin!" sin#J" sin!" sin#J"–2IxSz cos!" cos#J" cos!" sin#J"–2IxSz cos!" sin#J" cos!" cos#J"–2IxSz sin!" cos#J" sin!" sin#J"–2IxSz sin!" sin#J" sin!" cos#J"–2IySz cos!" cos#J" sin!" sin#J"–2IySz cos!" sin#J" sin!" cos#J"
2IySz sin!" cos#J" cos!" sin#J"
2IySz sin!" sin#J" cos!" cos#J"

Summarized–Ix cos!" sin!" (cos2#J" – sin2#J" – cos2#J" + sin2#J") = 0
Iy (cos2!" + sin2!")(cos2#J" – sin2#J") = Iy cos2#J"–2IxSz (cos2!" + sin2!") 2 sin#J" cos#J" = –2IxSz sin2#J"–2IySz 2 cos!" sin!" (cos#J" sin#J" – cos#J" sin#J") = 0

CS-term refocused
J-term not-refocused
J-term not-refocused

x

Ix
Δyx

x

Δxx
Δxx

FID

Δ = 1/(4JIS)x
decouplingt1

S
σ0 σ1σ2 σ3σ4 σ5σ6σ2σ1
CS
–Iy

HSQC

Δ!180"Ix + 180"Sx#J–Iy–IyCSIyJIy

Iy–2IxSz–Ix–Ix–2IySz2IxSz–2IxSz–2IxSz–2IxSz–Iy–2IySz–2IySz
Ix

Ix

Ix

Ix

Ix

Ix
2IySz

Iy

Iy
–2IxSz

2IySz

2IySz

2IySz

2IySz
–Ix

–2IxSz–2IxSz–Iy

OrderedIy cos!" cos#J" cos!" cos#J"–2IxSz cos!" cos#J" cos!" sin#J"–Ix cos!" cos#J" sin!" cos#J"–2IySz cos!" cos#J" sin!" sin#J"–2IxSz cos!" sin#J" cos!" cos#J"–Iy cos!" sin#J" cos!" sin#J"–2IySz cos!" sin#J" sin!" cos#J"
Ix cos!" sin#J" sin!" sin#J"
Ix sin!" cos#J" cos!" cos#J"
2IySz sin!" cos#J" cos!" sin#J"
Iy sin!" cos#J" sin!" cos#J"–2IxSz sin!" cos#J" sin!" sin#J"
2IySz sin!" sin#J" cos!" cos#J"–Ix sin!" sin#J" cos!" sin#J"–2IxSz sin!" sin#J" sin!" cos#J"–Iy sin!" sin#J" sin!" sin#J"

Sorted–Ix cos!" cos#J" sin!" cos#J"
Ix cos!" sin#J" sin!" sin#J"
Ix sin!" cos#J" cos!" cos#J"–Ix sin!" sin#J" cos!" sin#J"
Iy cos!" cos#J" cos!" cos#J"–Iy cos!" sin#J" cos!" sin#J"
Iy sin!" cos#J" sin!" cos#J"–Iy sin!" sin#J" sin!" sin#J"–2IxSz cos!" cos#J" cos!" sin#J"–2IxSz cos!" sin#J" cos!" cos#J"–2IxSz sin!" cos#J" sin!" sin#J"–2IxSz sin!" sin#J" sin!" cos#J"–2IySz cos!" cos#J" sin!" sin#J"–2IySz cos!" sin#J" sin!" cos#J"
2IySz sin!" cos#J" cos!" sin#J"
2IySz sin!" sin#J" cos!" cos#J"

Summarized–Ix cos!" sin!" (cos2#J" – sin2#J" – cos2#J" + sin2#J") = 0
Iy (cos2!" + sin2!")(cos2#J" – sin2#J") = Iy cos2#J" = 0–2IxSz (cos2!" + sin2!") 2 sin#J" cos#J" = –2IxSz sin2#J" = –2IxSz–2IySz 2 cos!" sin!" (cos#J" sin#J" – cos#J" sin#J") = 0(" = 1/4J)(" = 1/4J)

xIx
Δyxx

ΔΔxxxx

FID

Δ = 1/(4JIS)x
decouplingt1S
σ0 σ1σ2 σ3σ4 σ5σ6σ4σ3CS–2IzSyHSQCΔ!180"Ix#J–2IzSy–2IzSy

CS
2IzSy

J
2IzSy

2IzSy
–Sx

–2IzSx–2IzSx
–SySx

Sx

Sx

Sx
2IzSy

Sy

Sy
–2IzSx

2IzSx

2IzSx

–2IzSx–2IzSx–2IzSx–Sy–2IzSy–2IzSy
Sx

SySy

Sy

Sy
–2IzSx–Sx–Sx–2IzSyOrdered
2IzSy cos!" cos#J" cos!" cos#J"–Sx cos!" cos#J" cos!" sin#J"–2IzSx cos!" cos#J" sin!" cos#J"–Sy cos!" cos#J" sin!" sin#J"
Sx cos!" sin#J" cos!" cos#J"
2IzSy cos!" sin#J" cos!" sin#J"
Sy cos!" sin#J" sin!" cos#J"–2IzSx cos!" sin#J" sin!" sin#J"–2IzSx sin!" cos#J" cos!" cos#J"–Sy sin!" cos#J" cos!" sin#J"–2IzSy sin!" cos#J" sin!" cos#J"
Sx sin!" cos#J" sin!" sin#J"
Sy sin!" sin#J" cos!" cos#J"–2IzSx sin!" sin#J" cos!" sin#J"–Sx sin!" sin#J" sin!" cos#J"–2IzSy sin!" sin#J" sin!" sin#J"

Sorted–Sx cos!" cos#J" cos!" sin#J"
Sx cos!" sin#J" cos!" cos#J"
Sx sin!" cos#J" sin!" sin#J"–Sx sin!" sin#J" sin!" cos#J"–Sy cos!" cos#J" sin!" sin#J"
Sy cos!" sin#J" sin!" cos#J"–Sy sin!" cos#J" cos!" sin#J"
Sy sin!" sin#J" cos!" cos#J"–2IzSx cos!" cos#J" sin!" cos#J"–2IzSx cos!" sin#J" sin!" sin#J"–2IzSx sin!" cos#J" cos!" cos#J"–2IzSx sin!" sin#J" cos!" sin#J"
2IzSy cos!" cos#J" cos!" cos#J"
2IzSy cos!" sin#J" cos!" sin#J"–2IzSy sin!" cos#J" sin!" cos#J"–2IzSy sin!" sin#J" sin!" sin#J"

Summarized–Sx (cos2!" – sin2!") (cos#J" sin#J" – sin#J" cos#J") = 0–Sy 2 cos!" sin!" (cos#J" sin#J" – sin#J" cos#J") = 0–2IzSx 2 cos!" sin!" (cos2#J" + sin2#J") = –2IzSx sin2!" = –2IzSx sin! t12IzSy (cos2!" – sin2!")(cos2#J" + sin2#J") = 2IzSy cos2!" = 2IzSy cos! t1(" = t1/2)(" = t1/2)

x

Ix
Δyxx

ΔΔxxxx

FID

Δ = 1/(4JIS)x
decouplingt1

S
σ0 σ1

σ2 σ3

σ4 σ5σ6σ6σ5CS–2IySz

HSQC

Δ!180"Ix + 180"Sx#J–2IySz–2IySz

CS–2IySz

J–2IySz–2IySz
Ix

2IxSz

2IxSz
Iy

Ix

Ix

Ix

Ix
2IySz

Iy

Iy
–2IxSz2IxSz

2IxSz–2IxSz–2IxSz–2IxSz–Iy–2IySz–2IySzIx

Iy–Iy–Iy–Iy
2IxSz

Ix

Ix
2IySz

Ordered–2IySz cos!" cos#J" cos!" cos#J"
Ix cos!" cos#J" cos!" sin#J"
2IxSz cos!" cos#J" sin!" cos#J"
Iy cos!" cos#J" sin!" sin#J"
Ix cos!" sin#J" cos!" cos#J"
2IySz cos!" sin#J" cos!" sin#J"
Iy cos!" sin#J" sin!" cos#J"–2IxSz cos!" sin#J" sin!" sin#J"–2IxSz sin!" cos#J" cos!" cos#J"–Iy sin!" cos#J" cos!" sin#J"–2IySz sin!" cos#J" sin!" cos#J"
Ix sin!" cos#J" sin!" sin#J"–Iy sin!" sin#J" cos!" cos#J"
2IxSz sin!" sin#J" cos!" sin#J"
Ix sin!" sin#J" sin!" cos#J"
2IySz sin!" sin#J" sin!" sin#J"

SortedIx cos!" cos#J" cos!" sin#J"
Ix cos!" sin#J" cos!" cos#J"
Ix sin!" cos#J" sin!" sin#J"
Ix sin!" sin#J" sin!" cos#J"
Iy cos!" cos#J" sin!" sin#J"
Iy cos!" sin#J" sin!" cos#J"–Iy sin!" cos#J" cos!" sin#J"–Iy sin!" sin#J" cos!" cos#J"
2IxSz cos!" cos#J" sin!" cos#J"–2IxSz cos!" sin#J" sin!" sin#J"–2IxSz sin!" cos#J" cos!" cos#J"
2IxSz sin!" sin#J" cos!" sin#J"–2IySz cos!" cos#J" cos!" cos#J"
2IySz cos!" sin#J" cos!" sin#J"–2IySz sin!" cos#J" sin!" cos#J"
2IySz sin!" sin#J" sin!" sin#J"

SummarizedIx (cos2!" + sin2!") 2 sin#J" cos#J" = Ix sin2#J" = Ix(" = 1/4J)
Iy 2 cos!" sin!" (cos#J" sin#J" – cos#J" sin#J") = 0
2IxSz cos!" sin!" (cos2#J" – sin2#J" – cos2#J" + sin2#J") = 0–2IySz (cos2!" + sin2!")(cos2#J" – sin2#J") = –2IySz cos2#J" = 0(" = 1/4J)

x

Ix
Δy
Δx

Δ

y…

xx
Δxx
Δy

FID…

TROSYΔx

Δ = 1/(4JIS)x…

t1

Sσ0 σ1σ2 σ3 σ4σ5σ4σ52IzSx

2IxSx

CS(t1)J(t1) (90!Iy)

Syy…

σ6σ7σ6J(2!)

Sy

σ8σ7(90!Ix + 90!Sy)
2IxSx–Sy

2IzSx

σ8J(2!)–2IxSz–Sy

2IySz(90!Sx)
2IxSz–2IxSyIy

Iy

SySz–2IxSz

2IxSy

2IySz

–2IySy–Ix–Ix

2IzSy

2IxSy

–2IxSy–2IxSy

2IxSy

2IxSz

–Sx–Sx–SxSz–Sz

Sy

2IzSy

–2IySy–2IySy–2IySzObserved
Iy cos"t1 cos#Jt1 sin#J2! = Iy cos"t1 cos#Jt1–Ix cos"t1 sin#Jt1 sin#J2! sin#J2! = –Ix cos"t1 sin#Jt1
2IxSz sin"t1 cos#Jt1–2IySz sin"t1 sin#Jt1 sin#J2! = –2IySz sin"t1 sin#Jt1

Phase cycling is necessary for TROSY!

I/S

2-spin system

Product Operator Calculation Guide

Pulses! "#$%&'( !)*+,-.#/01)*+,-2(34.5678(!9Ix)
Iz(!9Iy)
Iz cos!

Iz

Iz cos!

—Iy sin!

Ix sin!

(!9Ix)
Ix(!9Iy)Ix

Ix

Ix cos!
—Iz sin!

(!9Ix)
Iy(!9Iy)
Iy cos!

Iy

Iy

Iz sin!

Chemical Shift! "#:;<=>'( !)*+,-.#/01)*+,-2(34.5678("It Iz)
Iz("It Iz)
Iz

Ix("It Iz)
Ix cos"ItIy

Iy cos"ItIy sin"It—Ix sin"It

J-coupling? '( !)*+,-01)*+,-@A.5678(#JISt IzSz)
Iz

Iz

2IxSx

2IxSx

2IxSy

2IxSy

2IySx

2IySx

2IySy

2IySy

2IxSx

2IxSx

2IzSz

2IzSz

(#JISt IzSz)
Ix

Ix cos#JISt(#JISt IzSz)
Iy

2IySz sin#JISt
2IxSz

2IxSz cos#JISt—2IxSz sin#JISt
2IySz

Iy sin#JIStSxSx cos#JISt

2IzSx cos#JISt
Sy sin#JISt

2IySz cos#JISt—Ix sin#JISt

Sy

2IzSy sin#JISt
2IzSx

Iy cos#JISt

Sy cos#JISt—2IzSx sin#JISt

2IzSy

2IzSy cos#JISt—Sx sin#JISt