Environmental Engineering Reference
In-Depth Information
m
e
c
2
and
P
γ
P
e
·
P
e
=
·
P
γ
=
0
, i.e.
P
γ
+
P
e
+
P
γ
2
P
γ
·
+
2
P
γ
·
P
e
−
(
P
γ
+
P
e
)
m
e
c
2
.
=
Now is a good time to expand out the four-vectors:
2
E
m
e
−
2
E
E(
1
m
e
c
2
cos
θ)/c
2
m
e
c
2
,
+
2
Em
e
−
−
=
which can be rearranged to give
1
E
=
1
E
+
1
−
cos
θ
m
e
c
2
and this is Eq. (7.44). Clearly the four-vector formalism allows us to see our way to
the answer in a much more elegant manner. Note also that the elegance would be lost
if we had rushed into component form at too early a stage. As in the manipulations
of ordinary vectors in three dimensions it is usually wise to try and stay in vector
notation for as long as possible.
π
0
Example 12.3.2
Consider the particle physics process γ
p in which a
photon (γ ) collides with a proton (p) to produce a neutral pion (π
0
) and a proton.
If the initial proton is at rest what is the minimum energy (called the 'threshold
energy') that the photon must have in order for the process to occur? [The proton
has mass 938 MeV/c
2
and the pion has mass 135 MeV/c
2
.]
+
p
→
+
Solution 12.3.2
As usual it is a good idea to start with a sketch illustrating the
process, like that shown in Figure 12.2. Since the sum of the rest masses in the final
state is greater than that in the initial state it is clear that the photon must deliver
0
p
g
Laboratory
frame
p
p
Before
After
0
p
Zero momentum
frame
g
p
p
The reaction
γ
+
p
→
π
0
Figure 12.2
+
p
viewed in two different inertial frames.
