Digital Signal Processing Reference
In-Depth Information
e
−
j
γ
B
0
t
⇒
M
xy
(
t
)
=
(
M
0
sin
(α)
cos
(φ)
+
jM
0
sin
(α)
sin
(φ))
e
j
φ
e
−
j
γ
B
0
t
⇒
M
xy
(
)
=
(α)
t
M
0
sin
⇒
M
x
(
t
)
=
M
0
sin
(α)
cos
(φ
−
γ
B
0
t
)
=
M
0
sin
(α)
cos
(
−
γ
B
0
t
+
φ)
(2.8)
M
y
(
t
)
=
M
0
sin
(α)
sin
(φ
−
γ
B
0
t
)
=
M
0
sin
(α)
sin
(
−
γ
B
0
t
+
φ)
(2.9)
M
z
(
t
)
=
M
0
cos
(α)
(2.10)
2.2 Comment on the Equations 2.8-2.10
M
When the weak initial magnetic moment
(
0
)
with magnitude
M
0
is kept at an
B
B
0
k
, due to
bloch equation, magnetic moment in the
z
-direction remains unchanged. But the
magnetic moment in the
x
-direction and the
y
-direction oscillates with the angular
frequency of
)
k
angle
α
with the strong constant magnetic moment
(
t
)
=
B
z
(
t
=
B
0
radians or
γ
B
0
. Thus at
any particular time instant, the magnitude of the resultant magnetic moment on the
X-Y
plane is constant and is equal to
M
0
sin
γ
Hz with maximum amplitude
M
0
sin
(α)
2
π
. Also note that at any particular time
instant
t
, the resultant magnetic moment on the
XY
-plane is making an angle with
magnitude (
(α)
−
γ
+
φ
) with the
x
-axis measured in the anti-clock wise direction.
As time goes, the magnitude of the angle is increasing. This implies the resultant
vector on the
XY
-plane rotates in the anti-clock wise direction (when viewed in the
z
direction) with the frequency
B
0
t
γ
B
0
. This frequecy is called larmor frequeny in radians
γ
B
0
2
and it is computed as
in Hz. For the constant strong magnetic moment
B
0
,the
larmor frequency purey depends on the gyromagnetic ratio of the magnetic moment
B
π
. Note that the magnitude of the resultant magnetic moment in the
XY
-plane
(transverse plane) is directly proportional to the angle
(
t
)
. Note:Clock wise direction
is identified with respect to the view point in the direction of
α
−
zaxis(referFig.
2.1
).
2.3 The Larmor Frequency and the Tip Angle
α
In general, resultant magnetic moment (without externel strong magnetic moment)
obtained in the macroscopic level in the human body is zero. When the human body
is kept under the constant strong magnetic moment of
B
0
in the
z
-direction. The
resultant magnetic moment in the macroscopic level is aligned to the direction of
the external strong magnetic moment (i.e)
z
-direction. When it is disturbed to bring
the resultant magnetic moment to make an angle
(measured anti-clock-wise direc-
tion) with the
z
-axis, there exists the resultant anti-clock-wise rotating magnetic
moment in the transverse plane (due to bloch equations), that rotates with the fre-
quency 42.58Mhz, when
B
0
is 1Tesla.
α
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