Chapter 6: Product operators - The James Keeler Group

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