Biomedical Engineering Reference
In-Depth Information
The smallest possible value of
E
1
, satisfying the equality, is the threshold energy
for the incident particle:
E
th
= -
Q
1+
.
M
1
M
3
+
M
4
-
M
1
(9.29)
The smaller the incident-particle mass
M
1
is, compared with the total mass
M
3
+
M
4
, the more nearly the threshold energy is equal to
-
Q
. It differs from
-
Q
by the
energy needed to keep the center of mass in motion (Fig. 9.9).
Example
Calculate the threshold energy for the reaction
32
S(n, p)
32
P.
Solution
For a neutron (mass
M
1
) incident on a sulfur atom at rest in the laboratory, we write
1
0
n+
3
16
S →
1
1
H+
3
15
P.
(9.30)
Since the number of electrons is the same on both sides of the arrow, we can use the
atomic mass differences
given in Appendix D to find
Q
. Taking the values in the
order in which they occur in (9.30), we obtain
Q
=
8.0714 - 26.013 - (7.2890 - 24.303)
=
-0.9276.
(9.31)
The threshold energy is, from Eq. (9.29),
0.9276
1+
1
1+32-1
E
th
=
=
0.957 MeV.
(9.32)
Nuclear cross sections for neutron reactions with a threshold usually increase
steadily from zero at
E
th
to a maximum and then decline at higher energies. The
activity induced in a foil is indicative of the relative number of neutrons over a
range of energies above the threshold. The energy at which the cross section has
approximately its average value is called the
effective threshold energy
, which is larger
than
E
th
.For
32
S, the effective threshold energy for neutron activation is about
3.2 MeV.
For a nuclear reaction to be induced by a positively charged projectile, the re-
pulsive Coulomb barrier must be overcome. For example, the reaction
14
N(
,p)
17
O
α
has the value
Q
-1.19 MeV. Equation (9.29) implies that the threshold energy for
the reaction is
E
th
=
=
1.53 MeV. However, the Coulomb barrier for the alpha particle
and the nitrogen nucleus is 3.9 MeV. The effective threshold energy for the reaction
to occur at an appreciable rate is about 4.6 MeV.
9.9
Neutron Activation
The time dependence of the activity induced by neutron capture can be described
quantitatively. If a sample containing
N
T
target atoms with cross section
σ
is ex-
