Biomedical Engineering Reference
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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
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.
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