Chemistry Reference
In-Depth Information
The remaining two terms are evaluated from
τ
to 2
τ
only. The second term can
be evaluated as:
f
=
F
(
τ
)
=
Gδ
4
f
2
τ
τ
4
Gδ
Gδ
τ
G
+
δ
2
Gδ
2
τ
dt
−
Fdt
=−
dt
+
(
t
−
+
δ
)
dt
+
+
δ
4
f
2
τ
τ
4
Gδ
6
τ
)
1
2
Gδ
(
δ
−
Fdt
=−
−
+
2
−
4
f
2
τ
τ
2
G
2
δ
2
(
δ
−
Fdt
=
+
2
−
6
τ
)
(68)
The final term is simply,
4
f
2
τ
4
G
2
δ
2
τ
=
These terms combine to produce the Stejskal-Tanner expression [36] for the
spin-echo experiment,
Dγ
2
G
2
δ
2
3
ln
S
(2
τ
)
δ
S
(0)
=−
−
(69)
For gradient-echo experiments with gradient shapes other than a simple rectan-
gular gradient shape, the same technique must be applied where the phase state at
the end of one integral must be taken into account in the calculation of the follow-
ing integral. In the gradient-echo case, however, the contribution of the last two
terms to the Stejskal-Tanner expression are zero. The experimental application of
two other gradient configurations will be described in later sections, and so their
corresponding expressions for the reduction in magnetization amplitude will also
be outlined here.
The first was a ramped gradient configuration, as illustrated in Fig. 8. The
gradient strength was increased from zero to the maximum at the beginning of the
bipolar gradient, and decreased continuously, passing through the origin, to the
same maximum value with the opposite polarity (i.e., the gradient strength was 0
at
t
δ
).
As this is essentially one continuous gradient, the expression could be calculated
using the single integral,
=
δ
). The slope of this gradient was therefore
G
(
t
)
=
(
G/δ
)(
t
−
2
δ
2
δ
t
0
−
δ
)
dt
2
ln
S
(2
δ
)
G
δ
(
t
F
2
dt
S
(0)
=
=
−
dt
0
0
ln
S
(2
δ
)
4
15
Dγ
2
G
2
δ
3
S
(0)
=−
(70)
