Classical angular momentum and magnetic dipole moment

1.

Reminder: classical angular momentum and magnetic dipole moment

2.

Classical angular momentum
I
F
I
F
r
m
right hand grip rule
I r mV
D r F
dI
– angular momentum
– torque
torque changes the anglular momentum
dI
D
dt

3.

Magnetism
geographic north pole
magnetic south pole
magnet
north
pole
B
compass
N
Earth
E
W
S
magnet
south
pole
geographic south pole
magnetic north pole

4.

Classical magnetic dipole moment
E B B cos B cos
B
B
E
0
0o
E B
90 o
E 0
180o
E B
B
2
0
oscillations of magnetic moment in magnetic field
B
dI
D B
dt
z
dI
I
x
dI
y
3
4

5.

Classical magnetic dipole with angular momentum
, I
z
N
S
B
I
– gyromagnetic ratio
dI
d
,I
y
x
precession with B
Joseph Larmor
1857-1942
Cambridge University
– Larmor frequency

6.

Magnetic field of the pulse
2 B cos ref t
1
z z
B1
0
B1 B1
0
z
B cos ref t
z 1
B1 B1 sin ref t
0
B1 z
B1 cos ref t
B1 sin ref t
0
y
z
B1
B1
x
z
B1
y

7.

Pulse
laboratory frame
z
B1 cos ref t
B1 sin ref t
0
M
z
ref
u
y
z
B1
y
x
x
rotating frame
z
B1
0
0
z
B1
u
M
z
u

8.

longitudinal and transverse relaxation
laboratory frame
z
z
y
x
z
y
x
z
y
y
x
x
rotating frame
z
z
u
z
u
z
u
u

9.

Free Induction Decay
z
z
0
0 ref
y
x
u
laboratory frame
rotating frame
M x M e
M y M e
t
t
T2
T2
sin 0t
cos 0t
M u M e
M M e
t
t
T2
sin 0 ref t
T2
cos 0 ref t

10.

Free Induction Decay
voltage in coil
voltage in coil
x
u
0
0 ref
t
t
phases depend on B1
voltage in coil
y
voltage in coil
0
0 ref
t
laboratory frame
t
rotating frame

11.

Spectral lines
M
T2
2 2
0
2
2
T2
1
T2
L 0
0
0
2
1
L 0 d 2 2 M 0
Lorentz lineshape
D 0 Dispersion lineshape