Biomedical Engineering Reference
In-Depth Information
where
J
is the angular momentum and g is known as the
geomagnetic ratio. For nuclei with an odd number of
protons or an odd number of neutrons, there is never
a net angular momentum equal to zero because of op-
posite spin states (Pauli exclusion principle that the
angular momentum of each proton in a nucleus must
assume opposite spin states). Many biological nuclei
usually have odd numbers of protons and spins; exam-
ple of such are
1
H,
13
C,
19
F,
23
N, and
31
P. U s u a l l y
1
H
is the most significant nucleus in most MRI imaging
becauseitispartofthewatermoleculeandithas
the highest NMR sensitivity with a geomagnetic ratio
g/2p
¼
42.58. Let's consider a magnetic flux
B
0
in the
z
direction.
The proton can assume two positions: with its
z
component of the magnetic moment aligned with the
external field or with its
z
component of the magnetic
moment opposed to the external field. Both states are
stable, but the energy associated with the latter is higher.
This can be observed in
Figure 6.1-8
and the angle q
0
can
be determined by the value of the magnetic moment
M
and its
z
component
M
z
, which are given by quantum
mechanics as
M ¼
g
h
3
Therefore the frequency y is given by
y
¼
2
M
z
h
B
0
(6.1.13)
It can be observed that the frequency is directly pro-
portional to the magnetic field.
The angular momentum of a nucleus is linearly related
to its magnetic moment
d
!
dt
¼
g
!
!
(6.1.14)
0
Representing this equation in terms of scalar compo-
nents and combining such equations we obtain
MðtÞ¼a
x
ðM
x
cos
ð
g
B
0
Þt þM
y
sin
ð
g
B
0
ÞÞ
þ a
y
ðM
y
cos
ð
g
B
0
Þt M
x
sin
ð
g
B
0
ÞÞ þ a
z
M
z
This last expression of the magnetic moment has
a frequency component given by
f ¼
vB
0
2p
(6.1.15)
p
M
z
¼
g
h
4p
(6.1.10)
which is known as the Larmor resonant frequency. It can be
easily shown that the frequency of the radiation is the same
as the precessional frequency of the magnetic moment.
When a magnetic field is applied to a bulk mass, each
individual magnetic moment must align itself either with
or against the external field. Alignment with the magnetic
field involves a lower energy and is called the parallel
state (a). Alignment against the magnetic field involves
a higher energy and it is called the antiparallel state (b).
Boltzmann's law is given by
4p
where
h
is the Planck constant (
h ¼
6.629
10
34
Js).
Therefore
q
0
¼
cos
1
M
z
¼
cos
1
1
p
z54
:
7
(6.1.11)
M
The difference in energy between the two states
D
E ¼
2
M
z
B
0
¼ h
y
(6.1.12)
¼
exp
D
E
KT
P
a
P
b
(6.1.16)
B
0
where
P
a
denotes the probability that a nucleus is found in
the parallel state (a),
P
b
denotes the probability that
a nucleus is found in the antiparallel state (b),
K
is Boltz-
mann's constant (
K ¼
1.3800
10
23
J/K),
T
is the ab-
solute temperature of the sample. For protons at 20
C,
D
E
is on the order of 10
26
Jand
KT
is on the order of 10
21
J.
Therefore the previous equation can be expressed as
Z
M
θ
D
E
2
KT
Y
P
a
P
b
z
(6.1.17)
If we estimate,
P
a
y
P
b
y
2
:
The net magnetic
moment per unit volume, known as magnetization, is
given by the expression
X
D
E
2
KT
nM
z
a
z
M ¼ðP
a
P
b
ÞnM
z
a
z
z
(6.1.18)
Figure 6.1-8 The magnetization vector.
