Chemistry Reference
In-Depth Information
that is,
2
δ
23
120
Dγ
2
G
2
δ
3
F
2
dt
=−
(72)
0
G.
Summary of the Preceding Classical Results
In Sections II.C and D we have seen that the Bloch equations as originally formu-
lated [10, 26] ignore the Brownian motion of the liquid nuclei and that numerous
attempts to incorporate this phenomenon in them have been made [11, 37]. One of
the best known being that of Carr and Purcell [29] leading
inter alia
to Eqs. (60)
and (64). Their treatment (effectively Einstein's theory of the Brownian motion
[14, 15] adapted to the phase variable
above) is based on the notion that a
nucleus executes a discrete-time random walk. The walk is due to the cumulative
effect of very large numbers of impacts of the surrounding particles on a nucleus,
so that the fluctuating displacement
r
(
t
) and thus the phase is a sum of random
variables, each having arbitrary distributions. In other words, the random walker
executes a
discrete
jump of
finite
jump-length variance in a
fixed
time interval (i.e.,
the elementary steps of the walk are taken at
uniform
intervals in time to one of
the
nearest
-neighbor sites) so that the only random variable is the direction of the
walker. The direction of the jump-length vector [15] has finite variance and the
waiting time between jumps has a finite mean. The problem is always to find the
probability that the walker will be in state
n
at some time
t
given that it was in a state
m
at some earlier time, giving rise in general to a difference equation [13, 17, 18].
However, by the central limit theorem [15] (since one is dealing with a sum of
centered random variables each having arbitrary distributions) the dephasing ef-
fect due to the Brownian motion may be calculated explicitly in the continuum
limit of extremely small mean-square displacements in infinitesimally short times.
In such a process, any jump-length distribution of
finite
jump-length variance and
any jump-time distribution with
finite
average jump-time will (in the continuum
limit of a very large number of small steps of short duration) as a consequence of
the central limit theorem, give rise to the same dephasing effect.
We have also seen that the above analysis was later much simplified by Torrey
[28]. He avoided the problem, encountered by Carr and Purcell [29], of explic-
itly passing to the continuum limit of the discrete time random walk by simply
adding (adapting a method originally described by Einstein, see [15], Chapter 1)
a magnetization diffusion term to the transverse magnetization in the Bloch equa-
tions, resulting in a partial differential equation. Boundary conditions may also be
incorporated in this equation allowing one to study phase diffusion in a confined
space [30]. The Bloch-Torrey equation may be easily solved [28, 38] for nuclei
diffusing freely in an infinite reservoir (Section II.F). Thus he obtained the fol-
lowing expression for the dephasing following the application of a step gradient
