Chemistry Reference
In-Depth Information
D.
The Spin-Echo Calculation
The most significant difference between the spin-echo and simple bipolar pulse
sequences is the timing gap between the first and second rectangular gradients. We
follow the Stejskal and Tanner method [36] once again and employ
γ
2
D
F
(
t
)
θ
(
t
)
τ
)
F
(
τ
)
2
S
dS
dt
=−
−
2
θ
(
t
−
(156)
=
t
0
G
(
t
)
dt
and
θ
(
t
) is the unit step function. The 180
◦
where
F
(
t
)
refocusing
pulse is placed midway between the two gradient pulses, at
t
=
τ
. Each of the
gradient functions has duration
τ
and the second pulse begins at
t
=
. The solution
of Eq.(156) is
t
S
(0) exp
g
(
t
)
dt
S
(
t
)
=
−
(157)
0
where
τ
γ
2
D
τ
0
4
F
(
τ
)
τ
0
τ
)
F
2
(
τ
)
g
(
t
)
dt
=−
F
2
(
t
)
dt
−
F
(
t
)
dt
+
4(
τ
−
0
(158)
The fractional evaluation becomes more complicated, as we must integrate over
two arbitrary time intervals
t
1
and
t
2
, and therefore must define the two gradient
integrals,
F
(
t
1
) and
F
(
t
2
).
F
(
t
1
)
→
F
(
t
1
)
θ
(
t
1
)
−
2
θ
(
t
1
−
τ
)
F
(
τ
)
F
(
t
2
)
→
F
(
t
2
)
θ
(
t
2
)
−
2
θ
(
t
2
−
τ
)
F
(
τ
)
(159)
Recalling Eq.(146), we have
t
dt
2
t
1
0
kT
mβ
α
1
1
2
(
t
)
2
γ
2
=
t
2
)
2
−
α
F
(
t
1
)
F
(
t
2
)
dt
1
(160)
(
α
−
1)
(
t
1
−
0
Here the product of
F
(
t
1
) and
F
(
t
2
) can be presented as:
=
F
(
t
1
)
θ
(
t
1
)
τ
)
F
(
τ
)
F
(
t
2
)
θ
(
t
2
)
τ
)
F
(
τ
)
F
(
t
1
)
F
(
t
2
)
−
2
θ
(
t
1
−
−
2
θ
(
t
2
−
=
F
(
t
1
)
F
(
t
2
)
θ
(
t
1
)
θ
(
t
2
)
−
2
F
(
τ
)
F
(
t
2
)
θ
(
t
2
)
θ
(
t
1
−
τ
)
4
F
2
(
τ
)
θ
(
t
1
−
−
2
F
(
t
1
)
F
(
τ
)
θ
(
t
1
)
θ
(
t
2
−
τ
)
+
τ
)
θ
(
t
2
−
τ
)
(161)
The evaluation of integrals in Eq.(160) can then be performed in piecewise fashion
provided we can write
F
(
t
) explicitly in four intervals of integration, namely,
(0
,δ
)
,
(
δ,
)
,
(
,
+
δ
)
,
(
+
δ,
2
τ
)
