Phase Cycling and Gradient Pulses - The James Keeler Group

of the gradient pulse the vectors representing transverse magnetization in all
these discs are aligned, but after some time each vector has precessed through a
different angle because of the variation in Larmor frequency. After sufficient
time the vectors are disposed in such a way that the net magnetization of the
sample (obtained by adding together all the vectors) is zero. The gradient pulse
is said to have dephased the magnetization.
It is most convenient to view this dephasing process as being due to the
generation by the gradient pulse of a spatially dependent phase. Suppose that
the magnetic field produced by the gradient pulse, B g
, varies linearly along the
z-axis according to
B Gz
g =
where G is the gradient strength expressed in, for example, T m –1 or G cm –1 ; the
origin of the z-axis is taken to be in the centre of the sample. At any particular
position in the sample the Larmor frequency, ω L
(z), depends on the applied
magnetic field, B 0
, and B g
ωL = γ( B0 + Bg)= γ( B0
+ Gz ) ,
where γ is the gyromagnetic ratio. After the gradient has been applied for time
t, the phase at any position in the sample, Φ(z), is given by Φ()= z γ ( B0 + Gz)
t .
The first part of this phase is just that due to the usual Larmor precession in the
absence of a field gradient. Since this is constant across the sample it will be
ignored from now on (which is formally the same result as viewing the
magnetization in a frame of reference rotating at γB 0
). The remaining term γGzt
is the spatially dependent phase induced by the gradient pulse.
If a gradient pulse is applied to pure x-magnetization, the following
evolution takes place at a particular position in the sample
γGztIz
I ⎯ →cos γGzt I sin γGzt I .
x
⎯⎯ ( ) + ( )
The total x-magnetization in the sample, M x
, is found by adding up the
magnetization from each of the thin discs, which is equivalent to the integral
M x ()= t
1
r max
x
1
2 r max
∫
– 1 2 r max
cos( γGzt)dz
where it has been assumed that the sample extends over a region ± 1 2
r max
.
Evaluating the integral gives an expression for the decay of x-magnetization
during a gradient pulse
( )
sin Gr t
Mx( 1 2
γ
max
t)=
1
2
γGrmaxt
The plot below shows M x (t) as a function of time
y
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