Biomedical Engineering Reference
In-Depth Information
same frequency and amplitude, but 90 phase difference (phase quadrature), to
obtain a left-circularly polarized field.
In the rotating reference frame (cf. Fig.
3.1
) the B
1
-field is given in r-direction as
B
1
¼
B
1
e
r
:
ð
3
:
25
Þ
The transverse amplitude B
1
is thus constant in terms of magnitude in the
rotating reference frame and is permanently (as long as the B
1
field is turned on)
tipping the magnetic moments away from the z-direction into the x-y-plane. This
process is not continuous (in terms of arbitrary continuous angles between the
z-axis and x-y-plane, as provided in classic mechanics) but discrete, as motivated
by quantum mechanics with two possible conditions for the proton: precession
about the z-axis
ð
B
1
¼
0
Þ
or precession about an axis situated in the x-y-plane,
referred to as 'spin-up' and 'spin-down', respectively. Due to quantum mechanics,
the rotation of the magnetic moments is often referred to as the flip-angle. (A flip-
angle of 90 flips the precession of the magnetic moment, initially along the z-axis,
into the plane transverse to the static field).
Based on (
3.9
), a relation between the gyromagnetic ratio c and the magnetic
field strength B
1
and the resulting spin angular precessional frequency x
1
of the
magnetic moments generated by the circular polarized field B
1
(assuming that no
additional field is apparent and B
1
is aligned along the rotating r-axis) around the
r-axis of the rotating reference frame can be established with
B
1
¼
x
1
c
e
r
:
ð
3
:
26
Þ
To determine the frequency x of the rf-field required to effectively rotate the
magnetic moment into the x-y-plane, the effective magnetic field is derived.
Generally, the time derivation of the magnetic moment vector l is represented
with respect to the rotating reference frame by
l
dl
dt
¼
d
G
l
dt
þ
X
l
ð
3
:
27
Þ
with the relative derivation d
G
l
=
dt
;
i.e. the time change of l with respect to the
rotating frame, the directional derivative X
l with the angular velocity vector X
describing the spatial rotation of the orthogonal base of the rotating frame.
Together with (
3.5
), the following is obtained
¼
!
d
G
l
dt
¼
l
þ
l
X
¼
cl
B
0
þ
l
X
cl
B
0
þ
1
c
X
cl
B
eff
with B
eff
:
¼
B
0
þ
1
c
X
:
ð
3
:
28
Þ
The 'magnetic'-term of the above vector product can be interpreted as the
''effective'' magnetic field B
eff
acting on the magnetic dipole moment, as observed
in the rotating reference frame.
