Environmental Engineering Reference
In-Depth Information
The authors [45] measured 130 neutrons/s at peak, when the accelerating voltage
was actually 115 kV, rather than 80 kV. To adjust the tunneling probability
T
Gamow
value to 115 keV, note (text near Equation 2.24) that the energy
E
dependence of
c
is
approximately
E
1/2
. This gives tunneling probability
T
Gamow
¼
(
E
G
/
E
)
1/2
].
For the D
-
D reaction in the compact chamber, this would imply 3.239
exp[
(
E
G
/
E
)
1/2
¼
at
E
¼
40 keV, so that
E
G
¼
419.6 keV. At (115/2) keV, then,
T
Gamow
¼
0.067. (We
neglect the small change in
r
2
.)
From our simpli
ed approach in Chapter 2, the cross section for fusion is, for
r
2
¼
36 f, and
T
Gamow
¼
0.067,
s ¼ pr
2
T
Gamow
¼
10
28
m
2
2
:
728
¼
2
:
728 barn
:
ð
4
:
2
Þ
This cross section value, for an energy near 100 keV, exceeds the maximum value
quoted in Table 4.1, which is 0.11
10
28
m
2
at an energy of 1750 keV. We can
attribute this discrepancy to a reaction probability
T
0.04, having to do with the
details of the nuclear reaction, recalling that an analogous factor
T
¼
10
24
was
¼
8.0
found for the p
-
p reaction on the sun.
From a more general point of view, it may be useful to look at the expected energy
dependence of the cross section shown in Figure 4.3, in the right-most curve.
Figure 4.4 shows the dominant energy-dependent functions for nuclear reactions
between charged particles in a thermalized plasma, as compared to a
xed energy
beam as in the compact device of Figure 4.1. While both the energy distribution
Figure 4.3 Right curve represents tunneling
probability T
Gamow
(Equation 2.21) as a function
of center of mass energy. The compact fusion
device provides D at a fixed energy 40 keV,
rather than a thermal Maxwell
-
Boltzmann
distribution of energies as occurs on the sun or
in a Tokamak reactor. The Gamow peak
represents the optimal overlap of the energy
distribution and the tunneling probability curves
([48], Figure 4.6, p. 159.
