Biomedical Engineering Reference
In-Depth Information
change in rest energy,
Q
=
M
1
+
M
2
-(
M
3
+
M
4
)
, is negative for the endothermic
reaction.
5)
The conservation of total energy requires that
(9.20)
E
1
=
E
3
+
E
4
-
Q
,
where
E
1
,
E
3
,and
E
4
are the kinetic energies of the moving particles. Conservation
of momentum gives
(9.21)
p
1
=
p
3
+
p
4
,
where
p
1
,
p
3
,and
p
4
are the magnitudes of the respective momenta. To calculate
the threshold energy
E
1
, we eliminate either
E
3
or
E
4
from these two equations
and solve for the other. This procedure will give the explicit condition that
E
1
must
fulfill.
We eliminate
E
4
. Using the relationship,
E
4
=
p
4
/2
M
4
, between energy and mo-
mentum, we write with the help of Eq. (9.21)
p
4
2
M
4
=
1
2
M
4
(
p
1
-
p
3
)
2
.
(9.22)
E
4
=
Substituting
p
1
=
(2
M
1
E
1
)
1/2
and
p
3
=
(2
M
3
E
3
)
1/2
gives
1
M
4
[
M
1
E
1
-2(
M
1
M
3
)
1/2
(
E
1
E
3
)
1/2
+
M
3
E
3
].
E
4
=
(9.23)
Using this expression in Eq. (9.20) and carrying out some algebraic manipulations,
one finds for
E
3
that
E
3
-
2(
M
1
M
3
E
1
)
1/2
M
3
+
M
4
E
3
-
(
M
4
-
M
1
)
E
1
+
M
4
Q
M
3
+
M
4
=
0.
(9.24)
This is a quadratic equation in
√
E
3
, having the form
E
3
-2
A
E
3
-
B
(9.25)
=
0,
where
A
and
B
are the coefficients that appear in Eq. (9.24). The two roots yield
E
3
=
B
+2
A
2
1
A
√
A
2
+
B
.
1
±
(9.26)
For
E
3
to be real,
A
2
+
B
0:
≥
(
M
3
+
M
4
)
2
+
(
M
4
-
M
1
)
E
1
+
M
4
Q
M
1
M
3
E
1
≥
0,
(9.27)
M
3
+
M
4
or
-
Q
1+
.
M
1
M
3
+
M
4
-
M
1
E
1
≥
(9.28)
energy
E
*
above the ground-state energy,
then one must replace
Q
by
Q
-
E
*
in this
discussion.
5 We assume that both particles are in their
ground states after the collision. If particle 3
or 4 is a nucleus in an excited state with
