Chemistry Reference
In-Depth Information
relation and their lowest moments are all equal with the exception of the second
moment of the velocity [2, 15].
C.
Extension to Fractional Diffusion for the Gradient-Echo Case
Proceeding for the case of a step field gradient and noting Eq.(130), we have the
mean-square phase,
2
γ
2
G
2
t
0
t
1
t
1
t
2
X
(
t
1
)
X
(
t
2
)
dt
1
dt
2
2
(
t
)
=
(146)
0
t
t
1
t
1
t
2
E
2
−
α
−
t
2
)
2
−
α
dt
1
dt
2
2
γ
2
G
2
kT
m
=
β
α
(
t
1
−
0
0
In the noninertial limit, (i.e., at long times,
tβ
1), again noting the asymptotic
behavior of the Mittag-Leffler function
E
2
−
α
−
β
α
t
2
−
α
∼
1
β
α
t
2
−
α
(
α
−
1)
(147)
we have
t
t
1
2
γ
2
G
2
kT
m
1
2
(
t
)
=
t
1
t
2
1)
dt
1
dt
2
t
2
)
2
−
α
(
α
β
α
(
t
1
−
−
0
0
t
t
1
kT
mβ
α
1
t
1
t
2
(
t
1
−
2
γ
2
G
2
=
t
2
)
2
−
α
dt
1
dt
2
(148)
(
α
−
1)
0
0
The value of the first integral is
t
1
t
1
α
(
α
t
2
t
2
)
2
−
α
dt
2
=
(
t
1
−
−
1)
0
While,
t
t
1
α
(
α
t
α
+
2
(
α
1
1)
t
1
dt
1
=
−
α
(
α
−
1)
+
2)
0
By substituting this result back into Eq.(148), we obtain
t
α
+
2
(
α
kT
mβ
α
1
1
2
(
t
)
2
γ
2
G
2
=
(149)
(
α
−
1)
α
(
α
−
1)
+
2)
