Chapter 6: Product operators - The James Keeler Group

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where Ω 1and Ω 2are the offsets of spin 1 and spin 2 respectively. Evolutionunder this Hamiltonian can be considered by applying the two terms sequentially(the order is immaterial)HfreetI1x⎯ ⎯→Ω1tI1z+Ω2tI2zI1x⎯⎯⎯ ⎯→Ω1tI1zΩ2tI2zI1x⎯⎯⎯→⎯⎯⎯→The first "arrow" is a rotation about zΩ1tI1zΩ2tI2zI ⎯⎯⎯ →cosΩtI + sin Ω tI ⎯⎯⎯→1x1 1x1 1yThe second arrow leaves the intermediate state unaltered as spin-2 operatorshave not effect on spin-1 operators. Overall, thereforeΩ1tI1z+Ω2tI2zI ⎯⎯⎯ ⎯ →cosΩtI + sin Ω tI1x1 1x1 1yA second example is the term 2I 1xI 2zevolving under a 90° pulse about the y-axis applied to both spins. The relevant Hamiltonian isH = ω I + ω I1 1y1 2yThe evolution can be separated into two successive rotationsω1tI1yω1tI2y2I 1 xI2 z⎯ →⎯⎯ ⎯⎯⎯→The first arrow affects only the spin-1 operators; a 90° rotation of I 1xabout ygives – I 1 z(remembering that ω 1t = π/2 for a 90° pulse)ω tI1x2z1 1x 2z 1 1z 2z1 1y1 2y2I I ⎯⎯⎯→cosωt2I I −sinωt2I I ⎯⎯⎯→1y2y2I I ⎯⎯⎯→−2I I ⎯⎯⎯→1x2zπ 2 I1z2zThe second arrow only affects the spin 2 operators; a 90° rotation of z about ytakes it to xπ 2 I1y2y2I1xI2z⎯⎯⎯→−2I1zI2z⎯⎯⎯→−2I1zI2xThe overall result is that anti-phase magnetization of spin 1 has been transferredinto anti-phase magnetization of spin 2. Such a process is called coherencetransfer and is exceptionally important in multiple-pulse NMR.6.1.3 Evolution under couplingThe new feature which arises when considering two spins is the effect ofcoupling between them. The Hamiltonian representing this coupling is itself aproduct of two operators:H = 2 JπJ12 I1zI2zwhere J 12is the coupling in Hz.Evolution under coupling causes the interconversion of in-phase and antiphasemagnetization according to the following diagramsπ 2 Iπ 2Iω tI6–8
IVxVy–yzzzyzxzzz–xz–xangle = πJtFor example, in-phase magnetization along x becomes anti-phase along yaccording to the diagram d2πJ tI zIzI ⎯⎯⎯⎯ 12 1 2 →cosπJ tI + sinπJ t 2 I I1x–y12 1x 12 1y 2znote that the angle is πJ 12t i.e. half the angle for the other rotations, I–III.Anti-phase magnetization along x becomes in-phase magnetization along y;using diagram V:2πJ12tI1 zI2z2I I ⎯⎯⎯⎯ →cosπJ t2I I + sinπJ tI1x2z12 1x 2z 12 1yThe diagrams apply equally well to spin-2; for example2πJ12tI1 zI2z−2I I ⎯⎯⎯⎯ →– cosπJ t2I I + sinπJ tI1z2y12 1z 2 y 12 2 xComplete interconversion of in-phase and anti-phase magnetization requires adelay such that πJ12t= π 2 i.e. a delay of 1/(2J 12). A delay of 1/J 12causes inphasemagnetization to change its sign:2πJ12tI1 zI2 z t= 1 2J12 2πJ12tI1 zI2 z t=1 J12I ⎯⎯⎯⎯⎯⎯→2I I I ⎯⎯⎯⎯⎯⎯→−I1x1y 2z 2y2 y6.4 Spin echoesIt was shown in section 6.2.6 that the offset is refocused in a spin echo. In thissection it will be shown that the evolution of the scalar coupling is notnecessarily refocused.6.4.1 Spin echoes in homonuclear spin systemIn this kind of spin echo the 180° pulse affects both spins i.e. it is a non-selectivepulse:– τ – 180°(x, to spin 1 and spin 2) – τ –At the start of the sequence it will be assumed that only in-phase x-magnetizationon spin 1 is present: I 1x. In fact the starting state is not important to the overalleffect of the spin echo, so this choice is arbitrary.It was shown in section 6.2.6 that the spin echo applied to one spin refocusesthe offset; this conclusion is not altered by the presence of a coupling so theoffset will be ignored in the present calculation. This greatly simplifies things.For the first delay τ only the effect of evolution under coupling need beconsidered therefore:2πJ12τI1 zI2 zI ⎯⎯⎯⎯ →cosπJ τ I + sinπJ τ 2I I1x12 1x 12 1y 2zThe 180° pulse affects both spins, and this can be calculated by applying the6–9
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: means that the two lines of the spi
Page 11 and 12: 6.1.3 Spin echoes in heteronuclear
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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