Lecture 4:
Motion of spins in magnetic field
Lecture aims to explain:
1. Motion of isolated spins – classical treatment
2. Rotating coordinate system
Motion of isolated spins: classical
treatment
The equation of motion
The equation of motion is found by
equating the torque with the rate of
change of angular momentum:
Using
H
μ = γJ
θ
we will obtain:
dJ
= μ × H0
dt
dμ
= μ × (γH0 )
dt
μ
dμ
= μ × Ω0
dt
This equation describes precession of the
magnetic moment around the magnetic field:
where
Ω 0 = γH 0
Note that this is the same frequency which was needed for the
magnetic resonance absorption
Rotating coordinate system
Rotating coordinate system
z
Rotation will be described by:
k
y
i
x
j
di
= Ω×i
dt
It’s possible to show that for a
vector function of time F(t):
dF δF
=
+Ω×F
dt
δt
Here δF/δt is the time rate of change of F with respect
to coordinate system i, j, k
Equation of the magnetic moment motion in a
rotating frame
δμ
= μ × (γH + Ω)
δt
The actual magnetic field can
be replaced with an effective
magnetic field:
For a static field H0 if
we take
We will obtain:
This equation works for
rotation with any Ω.
H eff = H + Ω / γ
Ω = −γH 0k
δμ
=0
δt
z
k
y
i
x
This means in this rotating frame the magnetic
moment effectively does not “feel” any magnetic field
j
Effect of the alternating magnetic field in the rotating
frame
H x = H x 0 cos ωt
Basic approach: break the
oscillating field in two rotating
components, one rotating
clockwise, and the other
anticlockwise:
H R = H1 ( i cos ωt + jsinωt)
H L = H1 ( i cos ωt - jsinωt)
The counter-rotating component HL will be
neglected in the following
By converting to a
frame rotating with
HR we obtain:
Or in terms of the
effective magnetic field:
δμ
= μ × [ k (ω + γH 0 ) + iγH 1 ]
δt
H eff = k ( H 0 − ω / γ ) + iH 1
Rotation of the magnetic moment around the
oscillating field
H eff = k ( H 0 − ω / γ ) + iH 1
If resonant condition is fulfilled
H0 − ω / γ = 0
magnetic moment will be rotating
around the oscillating field
The angle of rotation is
then defined by:
θ = γH 1t
Example 4.1
Find a duration of a π/2- and π-pulses for hydrogen in resonant
conditions if the oscillating magnetic field magnitude is 1 mT. The
gyromagnetic ratio for 1H is 26.75×107 rad s-1 T-1.
Example 4.2
Find the number of revolutions which the spin of a hydrogen
nucleus makes in the laboratory frame around the external field
while a π/2- rotation around H1 is being carried out (H0=1T)
d
μ
Motion of isolated spins:
= μ × Ω0
dt
SUMMARY
where
Ω 0 = γH 0
Rotating coordinate system: very useful for description of the effect of
the oscillating field. On resonance the oscillating field is replaced with a
static field (rotating with the spin in the laboratory frame)
A combination of an alternating field H x
and a static field H0 along z-axis in the
frame rotating with angular
velocity ω is given by
eff
H
= H x 0 cos ωt
= k ( H 0 − ω / γ ) + iH 1
If resonant condition is fulfilled
H0 − ω / γ = 0
magnetic moment will be rotating around the oscillating field
(in the rotating frame).
The angle of rotation is defined by:
1
θ = γH t