Chapter 6: Product operators - The James Keeler Group

a rotation about x. Overall, the effect of the 180° pulse is thenπIx − cosΩτ I + sin Ωτ I ⎯⎯ →cosΩτ I + sin ΩτI [6.3]yxAs was shown using the vector model, the y-component just changes sign. Thenext stage is the evolution of the offset for time τ. Again, each term on the rightof Eqn. [6.3] is considered separatelyΩτIzcosΩτ I ⎯⎯→cosΩτcosΩτ I −sin ΩτcosΩτIyΩτIzsin Ωτ I ⎯⎯ →cosΩτsin Ωτ I + sin ΩτsinΩτIxCollecting together the terms in I xand I ythe final result iscosΩτcosΩτ + sin Ωτsin Ωτ I cosΩτsin Ωτ sin ΩτcosΩτI( ) + ( −)yThe bracket multiplying I xis zero and the bracket multiplying I yis =1 because ofthe identity cos2 θ + sin2 θ = 1. Thus the overall result of the spin echo sequencecan be summarisedIz90° ( x)– − 180° x − −⎯⎯⎯⎯⎯ τ ( ) τ⎯→IIn words, the outcome is independent of the offset, Ω, and the delay τ, eventhough there is evolution during the delays. The offset is said to be refocused bythe spin echo. This is exactly the result we found in section 3.8.In general the sequence– τ – 180°(x) – τ – [6.4]refocuses any evolution due to offsets; this is a very useful feature which ismuch used in multiple-pulse NMR experiments.One further point is that as far as the offset is concerned the spin echosequence of Eqn. [6.4] is just equivalent to 180°(x).xyyyxyxxJ 12J 12Ω 1Ω 2The spectrum from two coupledspins, with offsets Ω 1 and Ω 2(rad s –1 ) and mutual couplingJ 12 (Hz).6.3 Operators for two spinsThe product operator approach comes into its own when coupled spin systemsare considered; such systems cannot be treated by the vector model. However,product operators provide a clean and simple description of the importantphenomena of coherence transfer and multiple quantum coherence.6.1.1 Product operators for two spinsFor a single spin the three operators needed for a complete description are I x, I yand I z. For two spins, three such operators are needed for each spin; anadditional subscript, 1 or 2, indicates which spin they refer to.spin 1: I I I spin 2 : I I I1x 1y 1z 2x 2y 2zI 1zrepresents z-magnetization of spin 1, and I 2zlikewise for spin 2. I 1xrepresents x-magnetization on spin 1. As spin 1 and 2 are coupled, thespectrum consists of two doublets and the operator I 1xcan be further identifiedwith the two lines of the spin-1 doublet. In the language of product operators I 1xis said to represent in-phase magnetization of spin 1; the description in-phase6–6