This operator represents multiple-quantum coherence (containing more
than one transverse operator). The same result would be obtained if a
pulse is applied to theSspin alone,
2 IxSz!
2 Sx 2 I
xSy: ½^2 :^265 The two examples represented by [2.264] and [2.265] represent the
generation of multiple-quantum coherences in homonuclear and hetero-
nuclear spin systems, respectively.
2.7.7 EXAMPLES OFPRODUCTOPERATORCALCULATIONS
Some simple examples using product operators to follow evolution
during spin echo and polarization transfer pulse sequences will be
presented. Although these examples may appear trivial, each one plays
an important part as a component of more complicated pulse sequences.
These pulse sequence elements will be encountered in many of the
multidimensional NMR experiments discussed in Chapters 6 and 7.
2.7.7.1 The Spin Echo The spin echo pulse sequence must be
examined in three cases: (a) one spin, (b) two coupled spins of the same
nuclear type (homonuclear case), and (c) two coupled spins of different
nuclear types (heteronuclear case).
Starting from equilibrium magnetization proportional to Iz,an
initial 90xpulse yields
Iz!
2 Ix I
y: ½^2 :^266 The spin echo pulse sequence for an isolated spin is written as
t 180 xt: ½ 2 : 267 Evolution during the period of free precession,t, yields
Iy )(^) IIzt
Iycosð (^) ItÞþIxsinð (^) ItÞ: ½ 2 : 268
The 180xpulse converts this density operator to
Iycosð (^) ItÞþIxsinð (^) ItÞ!
Ix
Iycosð (^) ItÞþIxsinð (^) ItÞ: ½ 2 : 269
The 180x pulse inverts the Iy term but does not affect the Ix
term. The final part of the spin echo sequence is another delay of
2.7 PRODUCTOPERATORFORMALISM 93