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

218
PRICE
Figure 15 The ratio of the signal obtained from the
stimulated echo Ž STE. sequence to that obtained from
the Stejskal and Tanner spin-echo Ž SE. sequence versus
T1T2 in the absence of gradients. The simulations
were calculated using Eqs. 32 and 33 .
In performing
these simulations, we have assumed that 2 T1 in
the case of the Stejskal and Tanner sequence and that
2 12T1 in the case of the STE sequence. The
simulations were performed for the cases of 122 Ž . and that 4 Ž ---- .
1 2 . The solid horizontal line
indicates the boundary above which the STE sequence
gives better signal-to-noise than the Stejskal and Tanner
sequence. As expected, when T1T21, the stimulated
echo sequence gives only half the intensity of the
spin-echo sequence.
nal magnetic gradients and are not an inherent
property of the species being measured Ži.e.,
T2 and not T . 2 . Thus, in addition to simply using the
STE sequence, benefit may be had by using specialized
sequences to counteract the effects of the
internal gradients, and these form the subject of
the following subsection.
Internal Gradients. Ideally, the only magnetic gradients
present during the performance of a PFG
sequence would be the purposely applied constant
gradients. In practice, the B field is never
0
perfect Že.g.,
imperfect shimming and the proximity
of the thermocouple to the sample. and nonhomogeneous
internal gradients are common
within many samples Že.g.,
red blood cells, metal
hydrides, colloids, and porous media. owing to
differences in magnetic susceptibility. For exam-
ple, it is estimated that red blood cells have
2 1 Ž .
gradients up to 2 10 T m 67, 68 . Even
minute air bubbles in an apple can lead to large
background gradients Ž 69 . . In fact, in hydride
samples, such internal gradients can be of the
1 order of 0.5 T m Ž 70 . . In this section, we
consider the effects of constant and nonconstant
Ž i.e., nonuniform. background gradients Ži.e.,
nonuniform both in direction and magnitude
throughout the sample. on diffusion measurements.
These background gradients result in a
decrease in the observed T2 through the effects
of translational diffusion of nuclear spins Ž 7179 . .
This can be easily understood by considering the
effects of a gradient on the Hahn spin-echo sequence
Ž see the discussion below . , although in
fact there is no exact theory for treating the
effects of diffusion in a general nonuniform gradient
Ž 76 . .
Let us begin our investigation of the effect of
background gradients by considering the Stejskal
and Tanner sequence, but in the presence of a
uniform constant background gradient of strength
g Ž Fig. 16 .
0 . Using the theory developed in the
first article Žsee
Part 1, The Macroscopic Approach,
and the Maple program in the Appendix.
to calculate the diffusion related part of the attenuation,
we can derive the echo signal amplitude
including the effects of the background
Figure 16 The Stejskal and Tanner Pulse sequence in
the presence of a background gradient g 0.
Simplistically,
it is assumed that the background gradient is
uniform in magnitude and direction throughout the
entire sample during the sequence.