Chemistry Reference
In-Depth Information
we have the mean-square value of the phase in terms of integrals involving the
velocity correlation function and field gradient only
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
)
=
(131)
0
t
t
1
2
γ
2
G
2
kT
m
t
1
t
2
e
−
β
(
t
1
−
t
2
)
dt
1
dt
2
=
0
0
γ
2
G
2
kT
3
β
4
m
6
tβ
)
t
2
β
2
(2
tβ
6
e
−
βt
(1
=
+
−
3)
−
+
which reduces to the Carr-Purcell-Torrey result, Eq. (60), for long times specified
by
tβ
1. For short times, such that
tβ
1, we have the purely kinematic result
γ
2
G
2
kT
4
m
2
t
4
(
t
)
=
(132)
is a linear transformation of a Gaussian random variable so that by
the properties of characteristic functions
e
i
=
Again
e
−
2
/
2
(133)
Hence, Eq. (131) yields the inertia corrected dephasing [1] for a step gradient.
In general, an infinity of fast relaxation modes will be generated due to the dou-
ble transcendental nature of this function and one dominant much slower mode
that is associated with the slow diffusive motion (c.f. Eq. (100)). An obvious
generalization of the right-hand side of Eq.(100) for arbitrary gradient shapes
defined by
t
G
(
t
)
dt
F
(
t
)
=
(134)
0
is
2
γ
2
2
(
t
)
X
(
t
1
)
X
(
t
2
)
F
(
t
1
)
F
(
t
2
)
dt
1
dt
2
=
(135)
t
t
1
Hence, in order to calculate the dephasing for a Gaussian process all that is
required is a knowledge of the velocity correlation function and the precise form
of the field gradients.
In this chapter, we have shown how the dephasing magnetization in MRI, aris-
ing from the Brownian motion of the nuclei, may be determined by simply writ-
ing the Langevin equation for the phase random variable and then calculating its
characteristic function. The method yields in transparent fashion, from the prop-
erties of the characteristic function of Gaussian random variables, the classical
