Chapter 6: Product operators - The James Keeler Group

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180° rotation to each in successionπI1xπI2xcosπJ τ I + sinπJ τ 2I I ⎯⎯→⎯⎯→12 1x 12 1y 2zwhere it has already been written in that ω 1t p= π, for a 180° pulse. The 180°rotation about x for spin 1 has no effect on the operator I 1xand I 2z, and it simplyreverses the sign of the operator I 1yπI1 xπI2xcosπJ τI + sinπJ τ2I I ⎯⎯→cosπJ τI −sinπJ τ2I I ⎯⎯→12 1x 12 1y 2z12 1x 12 1y 2zThe 180° rotation about x for spin 2 has no effect on the operators I 1xand I 1 y, butsimply reverses the sign of the operator I 2 z. The final result is thusπI1xcosπJ τI + sinπJ τ2I I ⎯⎯→cosπJ τI −sinπJ τ2I I12 1x 12 1y 2z12 1x 12 1y 2zπI2x⎯⎯ →cosπJ τI + sinπJ τ2I I12 1x 12 1y 2zNothing has happened; the 180° pulse has left the operators unaffected! So, forthe purposes of the calculation it is permissible to ignore the 180° pulse andsimply allow the coupling to evolve for 2τ. The final result can therefore just bewritten down:τ− 180° ( x)−τI ⎯⎯⎯⎯ →cos2πJ τ I + sin 2πJ τ 2I I1x12 1x 12 1y 2zFrom this it is easy to see that complete conversion to anti-phase magnetizationrequires 2πJ 12τ = π/2 i.e. τ = 1/(4 J 12).The calculation is not quite as simple if the initial state is chosen as I 1y, but thefinal result is just the same – the coupling evolves for 2τ:τ− 180° ( x)−τI ⎯⎯⎯⎯→− cos 2πJ τ I + sin 2πJ τ 2I I1yIn fact, the general result is that the sequence12 1y 12 1x 2z– τ – 180°(x, to spin 1 and spin 2) – τ –is equivalent to the sequence– 2τ – 180°(x, to spin 1 and spin 2)in which the offset is ignored and coupling is allowed to act for time 2τ.6.1.2 Interconverting in-phase and anti-phase statesSo far, spin echoes have been demonstrated as being useful for generating antiphaseterms, independent of offsets. For example, the sequence90°(x) – 1/(4J 12) – 180°(x) – 1/(4J 12) –generates pure anti-phase magnetization.Equally useful is the sequence– 1/(4J 12) – 180°(x) – 1/(4J 12) –which will convert pure anti-phase magnetization, such as 2I 1xI 2zinto in-phasemagnetization, I 1y.6–10
6.1.3 Spin echoes in heteronuclear spin systemsIf spin 1 and spin 2 are different nuclear species, such as 13 C and 1 H, it ispossible to choose to apply the 180° pulse to either or both spins; the outcome ofthe sequence depends on the pattern of 180° pulses.Sequence a has already been analysed: the result is that the offset is refocusedbut that the coupling evolves for time 2τ. Sequence b still refocuses the offset ofspin 1, but it turns out that the coupling is also refocused. Sequence c refocusesthe coupling but leaves the evolution of the offset unaffected.aspin 1spin 2bspin 1spin 2ττττSequence bIt will be assumed that the offset is refocused, and attention will therefore berestricted to the effect of the coupling2πJ12τI1 zI2 zI ⎯⎯⎯⎯ →cosπJ τ I + sinπJ τ 2I I1xThe 180°(x) pulse is only applied to spin 112 1x 12 1y 2zπI1xcosπJ τ I + sinπJ τ 2I I ⎯⎯→cosπJ τ I − sinπJ τ 2I I [6.5]12 1x 12 1y 2z12 1x 12 1y 2zThe two terms on the right each evolve under the coupling during the seconddelay:2πJ12τI1 zI2zcosπJ12τI1x⎯⎯⎯⎯→cosπJ τcosπJ τ I + sinπJ τcosπJ τ 2I I12 12 1x 12 12 1y 2z2πJ12τI1 zI2z−sinπJ12τ2I1yI2z⎯⎯⎯⎯→− cosπJ12τsinπJ12τ 2I1yI2 z+ sinπJ12τ sinπJ12τI1xCollecting the terms together and noting that cos2 θ + sin2 θ = 1 the final result isjust I 1 x. In words, the effect of the coupling has been refocused.cspin 1spin 2τThree different spin echosequences that can be appliedto heteronuclear spin systems.The open rectangles represent180° pulses.τSequence cAs there is no 180° pulse applied to spin 1, the offset of spin 1 is not refocused,but continues to evolve for time 2τ. The evolution of the coupling is easy tocalculate:2πJ12τI1 zI2 zI ⎯⎯⎯⎯ →cosπJ τ I + sinπJ τ 2I I1xThis time the 180°(x) pulse is applied to spin 212 1x 12 1y 2zπIxcosπJ τ I + sinπJ τ 2I I ⎯⎯2 →cosπJ τ I −sinπJ τ 2I I12 1x 12 1y 2z12 1x 12 1y 2zThe results is exactly as for sequence b (Eqn. [6.5]), so the final result is thesame i.e. the coupling is refocused.SummaryIn heteronuclear systems it is possible to choose whether or not to allow theoffset and the coupling to evolve; this gives great freedom in generating andmanipulating anti-phase states which play a key role in multiple pulse NMRexperiments.6–11
Page 1 and 2: 6 Product Operators †The vector m
Page 3 and 4: and z-components of the magnetizati
Page 5 and 6: ( ) −{ } ( )≡ { − }+ { }exp
Page 7 and 8: means that the two lines of the spi
Page 9: IVxVy-yzzzyzxzzz-xz-xangle = πJtFo
Page 13 and 14: ( I I + I I )= ( I + iI )( I + iI )
Page 15 and 16: 6.9 Multiple -quantum coherence6.9.

Chapter 6: Product operators - The James Keeler Group

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