expression describing the pulses can be written asHt¼ Ixor Iy, for an
x-pulse ory-pulse, respectively, and is the flip angle of the pulse. Pulses
of arbitrary phase or that include the effects of resonance offset can
be obtained using composite rotations as in [2.120]. The transformations
for a pulse of phase xare given by
Ix!
I x
Ix, ½ 2 : 232 Iy!
I x
Iycos Izsin , ½ 2 : 233 Iz!
I x
Izcos Iysin , ½ 2 : 234 and, for a pulse of phase y,
Ix!I y
Ixcos Izsin , ½ 2 : 235 Iy!I y
Iy, ½ 2 : 236 Iz!I y
Izcos Ixsin : ½ 2 : 237 These transformations of the product operators are illustrated geomet-
rically in Fig. 2.6.
2.7.3.3 Practical Points The preceding rules enable description of a
wide variety of pulsed NMR experiments. Before examining some very
useful specific examples, some formalities and practical points will be
presented.
Cascades. During a period of free precession for the two-spin system
IandS, the evolution of the density operator is represented as
^1 )^4 :(^) IIztþ (^) SSztþ 2 JISIzSzt
½ 2 : 238
Because each term in this Hamiltonian commutes with the others, the
one evolution period can be divided into a series of rotations or a
cascade,
^1 )
(^) IIzt
^2 )
(^) SSzt
^3 )^4 :
2 JISIzSzt
½ 2 : 239
The order in which the rotations due to shift and coupling evolution are
applied is unimportant. Likewise, the effect of a nonselective pulse
applied to theIandSspins is written as
^1 )
ðIxþSxÞ
^3 : ½ 2 : 240
86 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY