1
PHYS 6720  Lecture X
1
Lecture X
MRI
1.
Magnetism of nuclear spin
2.
A single spin in a DC field
3.
A spin system in a DC field
4.
Detection with an RF field
5.
Magnetic Resonance Imaging
PHYS 6720  Lecture X
2
• The source of magnetic field: electric current or EM fields.
• The forms of magnetic source: magnetic dipole
m
, quadruple Q, …
• The origin of
m
in an atom:
(i) spin:
nuclear spin
and electronic spin;
 its existence can only be explained by QM.
(ii) orbital motion of electrons.
 can be modeled by either classical EM or QM.
X1
Magnetism of nuclear spin
Magnetism and nuclear spin
PHYS 6720  Lecture X
3
• Magnetic dipole moment
m
(could also be written as
μ
) and angular
momentum
j
are related as
m
=
γ
j
γ
= gyromagnetic ratio
where
j
can be either the classical angular momentum for electornic
orbital motion (=
r
x
p
) or quantum angular momentum for nuclear
spin
• The term MRI evolves from Nuclear Magnetic Resonance (NMR)
which detects magnetic dipole moment
m
in atomic nuclei of tissues
X1
Magnetism of nuclear spin
Magnetism and nuclear spin
PHYS 6720  Lecture X
4
The rules for determining nuclear spin quantum number:
(i) If the number of neutrons
and
the number of protons are both
even, then the nucleus has
NO
spin.
(ii) If the number of neutrons
plus
the number of protons is odd,
then the nucleus has a halfinteger spin (1/2, 3/2, 5/2, …)
(iii) If the number of neutrons
and
the number of protons are both
odd, then the nucleus has an integer spin (1, 2, 3, …)
Æ
Hydrogen nucleus has only one proton so the spin quantum
number is 1/2
X1
Magnetism of nuclear spin
Magnetism and nuclear spin
PHYS 6720  Lecture X
5
Nuclear spins measured by NMR
X1
Magnetism of nuclear spin
Magnetism and nuclear spin
nucleus
1
H
13
C
17
O
19
F
23
Na
31
P
Spin #
1/2
1/2
5/2
1/2
3/2
1/2
Isotopic
abundance
100%
1.11%
0.04%
100%
100%
100%
Relative
physiologic
concentration
100
50
4x10
6
8x10
2
7.5x10
2
PHYS 6720  Lecture X
6
• MRI image signals are acquired from tissues that are magnetized
• Understanding MRI requires one to consider (i) how does a single
nuclear spin
m
interact with an “magnetization” field B
0
and (ii) how
does its environment affects this interaction
• For a single spin in an external field
B
0
, the equation of motion for
the angular momentum of the spin is given by
or
X2
A single spin in a DC field
Motion of a spin in a constant field
B
0
0
d
dt
=
×
j
mB
0
d
dt
γ
=×
m
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PHYS 6720  Lecture X
7
• Spin precession and Lamor frequency f
0
Let use a cartesian coordinate system with zaxis being parallel to
the magnetic field direction:
m
=(m
x
, m
y
, m
z
) and
B
0
=(0, 0, B
0
).
Then the equation of motion becomes
X2
A single spin in a DC field
Motion of a spin in a constant field
B
0
0
ˆˆ
ˆ
ˆ
()
00
xyz
x
y
z
x
yz
d
mx my mz
m
m
m
dt
B
γ
++
=
PHYS 6720  Lecture X
8
• Spin precession and Lamor frequency f
0
Separate the equation into different components, take time
derivative on both sides and substitute the 1
st
order differential
equations on x and y components into each other, assuming
B
0
is a
static field, we find
X2
A single spin in a DC field
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 Spring '10
 HU
 Current, Magnetism, Magnetic resonance imaging, Nuclear magnetic resonance, RF field

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