WOLFRAM|DEMONSTRATIONS PROJECT

Magnetic Resonance and Bloch Equations

This Demonstration visualizes the dynamics in the process of magnetic resonance in which the macroscopic magnetization
M
of an ensemble of paramagnetic particles is exposed to the common action of a static magnetic field
B
0
and a weak magnetic field
B
1
(

B
1
=
Ω
R
/γ
) that rotates with a frequency
ω
around
B
0
. The motion of
M
is governed by the Bloch equations. The effect of
B
1
on
M
is maximized when the rotational frequency
ω
is equal to the Larmor free precession frequency
ω
0
=γ
B
0

(i.e. detuning
ω-
ω
0
≈0
).
Many interesting cases are registered as bookmarks, which you can activate by clicking the small cross at the upper-right corner. For example:
• Free Larmor precession, which occurs for
B
1
=0
, so that magnetization precesses around
B
0
at the Larmor frequency.
• A
π/2
-pulse, for which
B
1
rotates at the Larmor frequency (detuning = 0) and for which
B
1
is switched on for a time duration
τ
π/2
, such that
Ω
R
τ
π/2
=π/2
. As a result, the magnetization is flipped by
π/2
into the
x
-
y
plane.
• A
π
-pulse, for which
B
1
rotates at the Larmor frequency (detuning = 0) and for which
B
1
is switched on for a time duration
τ
π
, such that
Ω
R
τ
π
=π
. As a result, the magnetization is flipped by
π
to the
-

z
direction.
• Magnetic resonance with relaxation, for which the magnetization reaches a steady state. You can play with the value of the frequency detuning
ω-
ω
0
. If
ω≪
ω
0
or
ω≫
ω
0
, the effect of
B
1
is small and the steady state is close to the equilibrium magnetization along
z
. However, when
ω≈
ω
0
, the effect of
B
1
is important and the steady state is reached far from the
z
axis.
• Adiabatic following occurs for
ω≪
ω
0
, in which case the magnetization precesses rapidly around the magnetic field and therefore follows the direction of
B
adiabatically.