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

228
PRICE
We set k and consider the integral
a
J Ž x.
1
Ž . ikx
F e dx 61 H x
ikx Ž . Ž .
which, by recalling that e cos kx i sin kx ,
becomes
J Ž x.
1
FŽ . cosŽ kx. dx
H x
J Ž x.
1
i sinŽ kx. dx 62 H x
We note that the first integrand is even and the
second integrand is odd and therefore goes to
zero, and so Eq. 62 becomes
J Ž x.
1
FŽ . 2 cosŽ kx. dx 63 H x
0
Next, we note the standard integral Že.g.,
6.693.2
in Ref. 104.
J Ž x.
cos xdx
0
1
cos arcsin
ž /
cos
2
2 2
H x
ž ' /
Re 0 64
in the present case 1, 1, and k.
Thus, Eq. 59 becomes
ž /
ei 2
SŽ . cos arcsinž gr gr /
gr 65
and equals 0 otherwise.
As an aside, we consider Eq. 58 when t 2.
² 2 We can calculate : a using the methods de-
scribed previously Ži.e.,
see Part 1, The GPD
Approximation, especially Eq. 59 ,
and set the
limits of integration in accordance with the spin-
.
echo sequence with a constant gradient to be
and noting
4
² 2: 2 2 3
a g D 66 3
ž /
2J ŽgŽ 2t. r.
1
lim 1 67 gŽ 2t. r
t2
and so Eq. 58 becomes
4
2 2 3
g D
0
Ž .
3
S 2 exp
2
2J ŽgŽ 2t. r.
1
lim
t2ž
gŽ 2t. r /
ž /
2
2 2 3 exp g D 68 3
as expected for the Hahn spin-echo sequence
excluding the effects of spinspin relaxation.
Case 2: Gradient Directed along the Cylinder.
Nowadays, we are more likely to be using a superconducting
magnet with the gradient direction
along the z axis, i.e., g g z.
In this case, we
define
1 hŽ z . 0 l
0 z 0 l
z 0 l
69 Ž .
i.e., 1l is the normalization factor and, performing
the same procedure as in the previous
subsection, we have
H
SŽ 2. SŽ 2. gŽ . cos d
g0 1 1 1
H 0 0 0
hŽ z . cosŽgŽ 2 t. z . dz 70
We first consider the integral over z in Eq. 70 ,
0
H hŽ z . cosŽgŽ 2 t. z .
0 0 dz0
1 l
cosŽgŽ 2 t. z . dz
H 0 0
l 0
sinŽgŽ 2 t. l.
71 gŽ 2t. l