Biomedical Engineering Reference
In-Depth Information
When the incident particle is an electron or positron, the special circumstance
arises in which its mass is the same as that of the struck particle:
M
=
m
.Equa-
tion (5.4) then implies that
Q
max
=
E
, and so its entire energy can be transferred in
a single, billiard-ball-type collision. As already mentioned, electrons and positrons
can thus experience relatively large energy losses and deflections, which contribute
to their having tortuous paths in matter. The next particle more massive than the
electron is the muon, having a mass
M
=
207
m
.
1)
The maximum fraction of energy
that a muon can transfer in a single collision is, from Eq. (5.4),
Q
max
E
=
4
m
(207
m
)
(208
m
)
2
4
208
=
=
0.0192.
(5.5)
Thus, the muon (and all heavy charged particles) travel essentially straight paths in
matter, except for occasional large-angle deflections by atomic nuclei.
The exact relativistic expression for the maximum energy transfer, with
m
and
M
denoting the rest masses of the electron and the heavy particle, is
2
mV
2
2
γ
Q
max
=
m
/
M
+
m
2
/
M
2
,
(5.6)
1+2
γ
where
γ = 1/
1-
β
2
,
β
=
V
/
c
,and
c
is the speed of light (Appendix C). Except at
extreme relativistic energies,
γ
m
/
M
1
, in which case (5.6) reduces to
2
mV
2
2
mc
2
2
,
Q
max
= 2γ
= 2γ
β
(5.7)
which is the usual relativistic result.
Example
Calculate the maximum energy that a 10-MeV proton can lose in a single electronic
collision.
Solution
For a proton of this energy the nonrelativistic formula (5.4) is accurate. Neglecting
m
compared with
M
,wehave
Q
max
=
10
-2
MeV
4
mE
/
M
=
4
×
1
×
10/1836
=
2.18
×
=
21.8 keV, which is only 0.22% of the proton's energy.
Example
Use the relativistic formula (5.7) to calculate the maximum possible energy loss in a
single collision of the 10-MeV proton in the last example.
Solution
We first find
. Since the proton rest energy is
Mc
2
938 MeV (Appendix A), we can
use the formula in Appendix C for the relativistic kinetic energy,
T
=
γ
=
10 MeV, to write
1 The muon (
M
207
m
), pion (270
m
), and
kaon (967
m
) are unstable particles with
masses intermediate between those of the
electron (
m
) and proton (1836
m
). They occur
with cosmic radiation and can also be
generated in particle accelerators.
=
