Biomedical Engineering Reference
In-Depth Information
On the other hand, it can be shown that due to the structure of (
3.5
), the relative
derivation d
G
l
=
dt must vanish (this can be demonstrated by scalar multiplication
of (
3.5
) with l and presenting the vectors in any arbitrary basis) such that (
3.6
)
degenerates to
l
¼
x
l
l
x
ð
3
:
7
Þ
(note in (
3.7
) the anti-commutativity of the cross product). By substitution of (
3.7
)
in (
3.5
) it follows after rearrangement
l
x
þ
cl
B
0
l
ð
x
þ
cB
0
Þ¼
0
:
ð
3
:
8
Þ
Since, in general, l is not aligned parallel to B
0
and x
;
the bracket term in (
3.8
)
must vanish for l
6¼
0
:
This, however, immediately leads to the (coordinate-
invariant) L
AMOR
-equation (Joseph L
AMOR
, Irish physicist: *1857-1942)
x
¼
cB
0
and
x
L
¼
cB
0
ð
3
:
9
Þ
with x
L
termed L
ARMOR
-'Frequency'. It represents the response of a single proton,
i.e. single magnetic moment, in an external field without considering any inter-
action with the environment, i.e. the magnetic fields produced by the spin of
neighbouring protons. The L
ARMOR
-Frequency depends on the specific elementary
particle species, and it varies with the strength of the external magnetic field.
The torque T induced by the external magnetic field B
0
(see (
3.3
)) acting
perpendicular to the spin axis on the proton's magnetic dipole moment l thus leads
to a precession with angular frequency x
L
about the external magnetic field's
longitudinal axis on basis of the law of conservation of angular momentum. In case
of a static field, the rotation is a constant precession.
Proceeding from microscopic magnetization resulting from one single proton to
a finite number N of protons contained in a certain volume element (voxel, abbrev.
for volumetric pixel) with volume V, the vector sum of the magnetic dipole
moments leads to a macroscopic net magnetization M in the direction of the
external field, i.e. the magnetic dipole moment per unit volume V (this is done by
introducing l
i
;
into (
3.6
) and summing both sides over i and dividing by V)
X
N
1
V
M
¼
:
l
i
ð
3
:
10
Þ
i
¼
1
where the order of the series expansion N is the number of protons contained in
V. Due to incoherent precession, the vector sum of magnetization in the x-y-
direction vanishes.
Using (
3.5
) and the summed magnetic dipole moments (
3.10
), results in a
differential equation in terms magnetization
M
¼
cM
B
0
:
ð
3
:
11
Þ
