Digital Signal Processing Reference
In-Depth Information
2.3.1 Disturbance to obtain Non-Zero
α
Va l u e
The external field (apart from the strong constant magnetic moment
B
0
) is applied for
the short duration (
) in such away that the resultant magneticmoment
E
τ
(
t
)
is rotating
M
exactly with the larmor frequency of the magnetic moment
(
t
)
to be disturbed. It
M
is noted that the macro magnetic moment
obtained using the hydrogen atoms
that are aligned in the
z
-direction due to the availibility of strong magnetic moment
B
0
in the
z
-direction. The interaction between the magnetic moment
(
t
)
M
(
t
)
aligned
E
in the
z
-direction with magnitude
M
0
and the rotating magnetic moment
on
the transverse plane is described by the bloch equations as described below. The
strength of the magnetic moment
(
t
)
E
(
t
)
is strong compared with the natural magnetic
M
moment
available in the human body that are aligned initially in the
z
-direction.
Rewriting the bloch equation using
M
(
t
)
and
E
(
t
)
(
t
)
, we get the following.
→
dJ
(
t
)
M
E
=
(
t
)
×
(
t
)
(2.11)
dt
⎡
⎤
i j k
00
M
z
(
M
E
⎣
⎦
⇒
⇒
(
t
)
×
(
t
)
=
t
)
E
x
(
t
)
E
y
(
t
)
0
dM
x
(
t
)
=
γ
E
y
(
t
)
M
0
=
γ
E
0
cos
(
−
γ
B
0
t
+
θ)
M
0
(2.12)
dt
dM
y
(
t
)
=
γ
E
x
(
)
M
0
=
γ
(
−
γ
+
θ)
t
E
0
sin
B
0
t
M
0
(2.13)
dt
dM
z
(
t
)
=
0
(2.14)
dt
Solving the Eqs. (
2.12
)-(
2.14
) as described in the Sect.
2.1
, we still get the resultant
magnetic moment
M
lies only in the
z
-direction. It is noted from the equations
that the transverse magnetic moment is zero due to the initial conditions
M
x
(
(
t
)
0
)
=
,
M
y
(
)
=
0
0.
But in practice, due to the external field, there is the disturbance in the resultant
magnetic moment and there exist very low magnitude
M
x
and
M
y
component that
rotates in the larmor frequency due to the existance of strong field
B
0
as described in
the Sect.
2.1
. Now consider the interaction between the magnetic fields
E
0
(
t
)
(which
components) and
B
has
E
x
(
t
)
and
E
y
(
t
)
(
t
)
(which has
B
z
(
t
)
=
B
0
component) on the
M
magnetic moment
(
t
)
which have all the three components.
→
dM
(
)
t
M
E
B
=
γ
(
t
)
×
(
(
t
)
+
(
t
))
(2.15)
dt
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