Digital Signal Processing Reference
In-Depth Information
Fig. 2.1
Co-ordinate system
illustrating Bloch equation
dM
y
(
t
)
=−
γ
M
x
(
t
)
B
0
(2.3)
dt
dM
z
(
t
)
=
0
(2.4)
dt
M
Let the projection of the intial magnetic moment
(
0
)
with magnitude
M
0
kept
B
)
k
B
0
k
on the
XY
-plane is
at an angle
α
with the magnetic moment
(
t
)
=
B
z
(
t
=
the vector with magnitude
M
xy
(
0
)
=
M
0
sin
(α)
and it makes an angle
φ
(in the
anti-clock wise direction) with the
x
-axis.
Note that the intial values of the magnetic moment
M
(with initial magnitude
M
0
) projected on the three co-ordinates are mentioned as follows.
(
t
)
M
x
(
)
=
M
xy
(
)
(φ)
=
(α)
(φ)
0
0
cos
M
0
sin
cos
(2.5)
M
y
(
)
=
M
xy
(
)
(φ)
=
(α)
(φ)
0
0
sin
M
0
sin
sin
(2.6)
M
z
(
0
)
=
M
0
cos
(α)
(2.7)
=
√
−
To solve the Eq . (
2.1
), we assign
M
xy
(
t
)
=
M
x
+
M
y
j
, where
j
1. Rewriting
jointly the Eqs. (
2.2
) and (
2.3
), we get
dM
x
(
t
)
j
dM
y
(
t
)
+
=
γ
M
y
(
t
)
B
z
(
t
)
−
j
γ
M
x
(
t
)
B
z
(
t
)
dt
dt
dM
xy
(
t
)
⇒
=−
j
γ
B
z
(
t
)
M
xy
(
t
)
dt
Note that
B
z
(
t
)
is constant and is represented as
B
0
.
Ke
−
j
γ
B
0
t
⇒
M
xy
(
t
)
=
Applying the initial conditions (refer (
2.5
)-(
2.7
))
M
xy
(
0
)
=
M
0
sin
(α)
cos
(φ)
+
jM
0
sin
(α)
sin
(φ)
, we get
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