durationt,
Iycosð (^) ItÞþIxsinð (^) ItÞ!
(^) IIzt
Iycos^2 ð (^) ItÞIxcosð (^) ItÞsinð (^) ItÞ
þIxsinð (^) ItÞcosð (^) ItÞþIysin^2 ð (^) ItÞ:
½ 2 : 270
Using the identity cos^2 þsin^2 ¼1, [2.270] can be written as
Iycosð (^) ItÞþIxsinð (^) ItÞ )
(^) IIzt
Iy: ½ 2 : 271
The overall effect of the spin echo segment, –t– 180x–t–, is seen to take
an initial stateIyand generate a final stateIy. Apart from a sign
change, no net evolution of the chemical shift occurs during the spin
echo sequence: evolution under the chemical shift Hamiltonian is
refocused. If a 180ypulse had been used for refocusing, then the sign
inversion would not have occurred.
The same result can be demonstrated more elegantly as follows.
The density operator at the end of the pulse sequence is given
by(t)¼U(0)U–1, with
U¼exp½i (^) ItIz exp½iIx exp½i (^) ItIz, ½ 2 : 272
in which each factor inUrepresents the propagator for one segment of
the spin echo sequence. Applying the identity of [2.121] yields
U¼exp½i (^) ItIz exp½iIx exp½i (^) ItIz
¼exp½i (^) ItIz exp½iIx exp½i (^) ItIz exp½iIx exp½iIx
¼exp½i (^) ItIz expi (^) IteiIxIzeiIx
exp½iIx
¼exp½i (^) ItIz exp½i (^) ItIz exp½iIx
¼exp½iIx: ½ 2 : 273
Therefore, thenetevolution during the spin echo sequence is given by
Iy!
Ix
Iy, ½ 2 : 274
in agreement with [2.271]. Considerable simplification of propagators for
pulse sequences containing 180 8 pulses is often possible by use of [2.121].
The same spin echo pulse sequence can be applied to a homonuclear
IStwo-spin system. The pulses are assumed to be nonselective and affect
both theIand theSspins equally. As for the isolated spin, the chemical
shift evolution of theIand theSspins is refocused over the spin echo
sequence and can be neglected. Therefore, evolution during the pulse
94 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY