Chemistry Reference
In-Depth Information
Now, using the property of the gamma function, that
(
z
+
1)
=
z
(
z
), Eq.(149)
may be simplified to
kT
mβ
α
2(
α
+
1)
2
(
t
)
γ
2
G
2
3)
t
α
+
2
=
(150)
(
α
+
exp
3)
t
α
+
2
e
−
2
kT
mβ
α
(
α
+
1)
e
i
γ
2
G
2
=
=
−
(151)
2
(
α
+
For the
bipolar
gradient-echo experiment (consisting of two step functions, with
opposite polarity), the normalized signal decay becomes
ln
S
(
t
)
S
0
2
γ
2
G
2
3)
t
α
+
2
kT
mβ
α
(
α
+
1)
=
exp
−
(152)
(
α
+
which can be rewritten as:
ln
S
(
t
)
S
0
2
γ
2
G
2
D
t
α
+
2
=
−
exp
(153)
(2
+
α
)
(1
+
α
)
(See Fig. 16.) In the normal diffusion limit,
α
→
1, Eq.(153) reduces to Eq.(65),
namely,
ln
S
(
t
)
S
0
2
3
γ
2
G
2
t
3
=−
(154)
For short times (
tβ
1), we have
t
4
1
.
γ
2
G
2
kT
4
m
8
β
α
t
2
−
α
2
(
t
)
≈
−
α
)
+···
(155)
(6
−
α
)
(5
−
The leading term of the expansion (155) coincides with the purely kinematic result,
Eq.(131). Again
is a linear transformation of a Gaussian random variable so
that the characteristic function, Eq.(133), yields the dephasing.
The fractional Brownian motion we have just discussed assumes that the driving
force
λ
(
t
) is Gaussian so that the characteristic function Eq.(133) still applies as
in the normal Brownian motion. Hence, the decay of the phase remains (albeit
stretched) exponential. Thus the phase as calculated from the fractional Brownian
motion unlike fractal time relaxation (more precisely the diffusion limit of the
continuous time random walk, where unlike the discrete random walk considered
by Einstein [15] no mean waiting time between jumps of the walker exists) does not
exhibit the characteristic long time tail often associated with anomalous diffusion
processes. This banding signifies that the diffusion process depends strongly on
