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

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( ) [ ( )]
Table 4 g t for the Stejskal and Tanner Sequence for Sinusoidally Shaped Gradient Pulses see Fig. 10 B
Ž. 2
Subinterval of Pulse Sequence Sine-shaped g t Sine -shaped gŽ. t
0 t t1 0 0
Ž Ž . . Ž Ž . . 2
t1tt1 g sin Ntt1 gsin Ntt1
t1tt1 0 0
t tt g sinŽNtt Ž . . gsinŽNtt Ž . .
1 1 1 1
t1t2 0 0
2N Ž N is an integer. denotes the period of the gradient pulse. The corresponding echo attenuation equations are given by Eqs.
13 and 14 .
reference phase-angle evolution can then be subtracted
from all subsequent spectra obtained under
the same conditions to remove the effect of
B variation Ž 52 .
0
. Other variations based on deconvolution
using an experimental reference exist
Ž 53 . .
In work related to the helix picture of magnetization
subjected to a constant gradient Ž 29 . ,
Callaghan developed the MASSEY sequence
Ž 21. for minimizing phase instability in very-highgradient
NMR spectroscopy Ž Fig. 11 . . The method
also corrects for sample movement Žsee
Sample
Movement with Respect to the Gradient . . This
method incorporates a read gradient Ži.e.,
k space;
this usage of k is not to be confused with the
exponential rate constant used above . , G, into the
standard Stejskal and Tanner sequence; thus, in a
sense, it is also a pulse sequence solution and not
only a postprocessing solution. It is important to
realize that in this method the same gradient coil
is used for generating the gradient pulses and
also the read gradient. The addition of G allows
for the restoration of spatially dependent phase
shifts such as those caused by a mismatch in the
Ž .
q-space gradient pulses. To understand how this
method works, we need to consider the mathematics
behind the phase-twist problem. We start
from the average propagator representation of
the short gradient pulse approximation Žsee
Eq.
87 , Part 1 . , except we now include the effects of
a phase shift, , due to the effects of a gradient
mismatch see
Amplifier Noise, Earth Loops, and
Nonreproducible Ž Mismatched. Gradient Pulses ,
q, and of movement r of the entire Ž
o
i.e.,
rigid. sample Žsee
Sample Movement with Respect
to the Gradient. between the first and second
gradient pulses in the Stejskal and Tanner
sequence. Thus, we have
H
Ž . Ž . i2 qR
E q, P R, e dR 15
where PŽ R, . is the average propagator and R is
the dynamic displacement defined by r r Ž
1 0 the
starting and finishing positions of a spin with
respect to the first and second gradient pulses.
and
Ž . Ž .
2qR2 qq r Rr
0 o
qr . 16
o
Table 5 The ft ( ) Term in Eq. [ 12] for the various B0 gradient pulse shapes in the Stejskal and Tanner
Sequence ( see Fig. 10) given in Table 3
Ž.
Gradient Pulse Shape f t
Ramped rise and fall 3 2
30 6
2 k 2 2
Exponential rise and fall 2k Ž 1k . 4Ž ke .Ž 1k 2.
Ž 2 2k
2 k e . Ž 1k.
Exponential rise and fall with
overshoot and undershoot Ž 2 3 kŽ 2 2 2
8k 12k e 2 4 6 k
2 2 3 8k 12k 8k 12k .. g
kŽ 2 . Ž 2 . 2kŽ 2 2
4e k g k e 2 6 k
2 2 3. 2
4k 6k 2k 3k g
Ž . Ž 2 2
23k g k .
Sine rise and fall 2Ž . 2Ž . 2 3 3
4 2 8 3 64
Ž .
From Price and Kuchel 30 .
2