Pulsed-field gradient nuclear magnetic resonance as a tool for ...

318
PRICE
probability of a spin starting from r 0 and moving
to r in time is given by Žrecall Eq. 24. 1
Ž . Ž .
r P r ,r , . 73
0 0 1
The NMR signal is proportional to the vector
sum of the transverse components of the magnetization,
and so the signal from one spin is given
by
Ž . Ž . igŽr 1r 0.
r P r ,r , e . 74
0 0 1
But in NMR, the signal results from the ensemble
of spins, and thus we must integrate over all
possible starting and finishing positions, and finally
we arrive at our result Ž 55, 56.
EŽ g,. HH Ž r . PŽ r ,r ,.
0 0 1
igŽr e 1r0. dr dr . 75 0 1
Thus, the total signal is a superposition of signals
Ž transverse magnetizations . , in which each phase
term is weighted by the probability for a spin to
begin at r 0 and move to r1 during .
As an example, let us rederive Eq. 51 using
the SGP approximation Ži.e., Eq. 75 . . From Eq.
75 and 27 ,
we have
1 2
Žr 1r 0.
4 D
HH 0 32
EŽ g,. Ž r .
e
Ž 4D.
igŽr 1r 0.
e dr dr . 76
1 0
In the present case, we set g g Ž z we will drop
the subscript z, however . , and R r1r 0.Us-
ing spherical polar coordinates Ži.e.,
R is the
radius and and are the polar and azimuthal
angles, respectively . , we note that dR
R2 sin dR d d; also, since is the angle between
R and g we have
1 2 2 R 4 D 2
32
H H
EŽ g,.
Ž 4D.
0
d
0
e R
igR cos H e sin d dR
0
77 As there is no dependence, it can be integrated
out
2 2 R 4 D 2
e R
32
H
Ž 4D.
0
igR cos H e sin d dR
0
78 then, by noting that d cos sin d, we get
2 2
1
R 4 D 2 igR
32H
H
Ž . 0 1
4D
e R e d dR.
79
The integral over is then performed and evalu-
ated using Eq. 11 , resulting in
4
2 R 4 D Ž .
32
H
g Ž 4D.
0
Re sin gR dR
80
We note from a table of standard integrals e.g.,
Ž 53. Eq. 3.952 1. that
2 2 a 2 2
p x Ž . a 4p
xe sin ax dx e . 81 4p
'
H 3
0
Ž . 12
In our case, x R, p 4D , and a g,
and thus we obtain the final result
Ž . Ž 2 2 2 .
E g, exp g D . 82
This is the same as Eq. 51 ,
but in the limit of
0; thus the 3 term which accounts for the
finite width of the gradient pulse is absent. When
we consider restricted diffusion, the evaluation of
Eq. 75 proceeds in exactly the same manner,
except that we must substitute the relevant Ž r . 0
and PŽ r , r , t . 0 1 . However, as the confining geometry
becomes more complicated, so does the
mathematical complexity.
The Analogy between PFG Measurements and Scattering.
Returning back to the derivation of the
SGP approximation, and in particular, Eq. 72 ,
because we are using a constant Žcommonly
termed ‘‘linear’’ . gradient, what is important is
not the actual starting and finishing positions of
the spin but the net displacement between the
two points in the direction of the gradient. As
before, we can write R r1r0 and, analogously
to Eq. 73 ,
the probability that a particle
that starts at r 0 displacing a distance R during
is given by Žrecall Eq. 24. Ž . Ž .
r P r ,r R, . 83
0 0 0