Pulsed-field gradient nuclear magnetic resonance as a tool for ...

206
PRICE
ble if the source of the mismatch is due to eddy
currentgenerated gradients that are not parallel
to the applied gradients Ž 33. or nonconstant mismatch.
We also note that the MASSEY sequence
can be used to alleviate the phase-twist problem
Ž see Postprocessing . . The phase-twist problem is
considered in more detail in that section.
Eddy Currents and Perturbation of B 0
The rapid rise of the gradient pulses can generate
eddy currents in the surrounding conducting surfaces
around the gradient coils Že.g.,
probe housing,
cryostat, radiation shields, etc. . . The severity
of the eddy current problem is thus proportional
to dIdt and the strength of the gradient. Although
the generation of eddy currents is greatly
decreased through the use of shielded gradient
coils Ž see Gradient Coils . , they can still occur,
especially when using large, rapidly rising and
falling gradient pulses. The eddy currents, in turn,
generate additional magnetic fields and thus have
a close relationship to the problems discussed in
the previous section. It is the decay of the eddy
currents and their associated magnetic fields that
determine the minimum delay that must be left
between the end of the gradient pulse and the
start of spectral acquisition. Eddy currents can
have the following effects: Ž. a phase changes in
the observed spectrum and anomalous changes in
the attenuation, Ž b. gradient-induced broadening
of the observed spectrum, and Ž. c time-dependent
but spatially invariant B shift effects Ž
0
which appears
as ringing in the spectrum . .
We illustrate the effects of eddy currents using
the Stejskal and Tanner sequence as an example.
If the eddy current tail from the first gradient
pulse extends into the second -period, then the
total field gradient during the second evolution
period will not equal that in the first and the
situation is analogous to the case of mismatched
pulses see
Amplifier Noise, Earth Loops, and
Nonreproducible Ž Mismatched. Gradient Pulses .
Consequently, even if a spin has not moved in the
direction of the gradient during the sequence,
there will be a residual phase shift. As a result,
the point at which the maximum echo appears
will be shifted and its amplitude will be affected
Ž 34 . . Thus, assuming that signal acquisition is
begun as usual, at t 2 the eddy currents will
cause additional attenuation unrelated to diffusion,
and perhaps if the eddy currents have not
dissipated before acquisition begins, phase shifts
and spectral broadening.
To gain some insight into the effects on the
echo attenuation if the eddy currents generated
by the first gradient pulse have not dissipated
before the application of the pulse, and similarly,
if the disturbances generated by the second
gradient pulse have not dissipated prior to the
start of acquisition, assuming infinitely fast rise
and but exponential fall Žwith
exponential rate
constant k. of the gradient pulses Ž Table 2 . , we
can derive the echo attenuation equation for the
Stejskal and Tanner sequence using the same
method as before Žsee
Part 1, The Macroscopic
Approach . . An example program using the symbolic
algebra package Maple Ž 35. is given in the
Appendix Žn.b.,
the new definition of the function
F to allow for time-dependent gradients . . The
attenuation equation is given by
Ž 2 2 2 Ž . Ž .4.
Eexp g D 3 f t 10
Table 2 g( t) for the Stejskal and Tanner Sequence in Which Eddy Currents Generated
by the First Gradient Pulse Have Not Totally Decayed by the Time of Application of the
Pulse ( A Similar Situation is Depicted in Fig. 5, if te Is Shorter Than the Time Required
for the Eddy Current Effects to Dissipate) and Similarly the Eddy Currents from the
Second Gradient Pulse Extending into the Acquisition Period
Ž.
Subinterval of Pulse Sequence g t
0 t t 0
1
t t t g
1 1
t tt ge
1 1
t tt g
1 1
t t2 ge
1
k is the exponential rate constant.
kŽtt .
1
kŽtt .
1