Biomedical Engineering Reference
In-Depth Information
After the collision, the energy (
E
) carried by the positron-electron pair is trans-
ferred to the photons. The energy carried by positron-electron pair can be calcu-
lated using the relation
EMV
=
+
MV
(8.17)
1
light
2
light
where
V
light
is the light velocity. Net mass is calculated using relation,
2
10
−31
kg), and
v
is
the velocity of the pair's center of mass. If the pair's center of mass is fixed (
v
2
where
m
0
is the electron rest mass (9.109
×
mm
=
1/
−
vV
0
light
0),
the two photons fly apart in opposite directions (180° apart) with the same energy:
=
2
MV
=
MV
=
mV
=
511 keV
(8.18)
1
light
2
light
0
light
In the center-of-mass frame of the
positron-electron
pair, the total energy (
E
t
)
of the annihilation gamma rays is
EEE
=−
2
(8.19)
t
0
B
where
E
B
is the electron binding energy and
E
0
is the rest energy of the electron,
which is
2
EmV
=
(8.20)
0
0
light
However, when there is a net center of mass energy associated with the annihi-
lating pair, this total energy is not split equally to 511 keV between the two gamma
rays. One gamma ray is upshifted while the other is downshifted from the center
energy of
m
0
c
2
p
L
c
/2, where
p
L
is the lon-
gitudinal component of the electron-positron momentum along the direction of the
gamma ray emission. In the case of two-photon annihilation, measuring the devia-
tion angle q of the photons from 180
−
E
B
/2 by an amount equivalent to
Δ
E
=
E
of the annihilation
line (511 keV) makes it possible, at least in principle, to determine the momentum
of the electron-positron pair.
The lifetime of a positron is directly related to the electron density in the anni-
hilation site (i.e., the probability it will run into an electron); the higher the electron
density, the shorter the lifetime. The inverse of positron lifetime is referred as an-
nihilation rate,
°
or the Doppler shift
Δ
λ
D
is proportional to the effective electron density
n
e
sampled by the positron, namely,
λ
D
. Theoretically,
2
λπ
≈
rV
n
D
e
light
e
where
r
e
is the electron radius. The process of gamma emission is also characterized
with the annihilation rate
λ
D
is dependent on the type of isotope used, and
each positron emitter has a different
E
max
for positrons and hence a different range,
this imposes an isotope-dependent limitation on spatial resolution in PET imaging.
λ
D
. Since
