Environmental Engineering Reference
In-Depth Information
state and the
final state. The large increase in cross section, approximately the inverse
of the Gamow factor
T
Gamow
¼
0.067 found for 57.5 keV in connection with Equa-
tion 4.2, namely, 14.9, is expected for resonant tunneling,
when the incoming particle
energy matches that of the resonant state
.
In the physics analogy, one has two barriers separated by a potential well. The
interesting effect comes when the well has a bound state at the energy of the input
tunneling particle. In this case, the transmission probability (across the whole
system) becomes 1, which corresponds to setting the Gamow factor to 1 as we
mentioned. There is thus a strong peak in the transmission probability
T
(
E
) at the
resonant energy
E
¼
E
0
. In more detail, near the energy of the resonant state
E
0
2
2
2
T
ð
E
Þ¼C
=½C
þð
E
E
0
Þ
:
ð
4
:
24
Þ
) symbols shown in Figure 4.6 as calculated by Li
et al.
[50] are in good
agreement with the measurements and establish the resonance state interpretation
of the large enhancement of the DTcross section over theDD cross section peaking at
114 keV. In Equation 4.24, the energy width is of the form
The (
þ
'¼
h
/
t
where
t
is the
lifetime of the resonant state. The lifetime is the inverse of the rate of tunneling out of
the state,
v¼v
attack
T
Gamow
, so for
2
E
0
)
2
we get
C
(
E
2
E
2
T
ð
E
Þð
h
v
attack
T
Gamow
Þ
=
:
ð
4
:
24a
Þ
This is not a familiar formula, but does indicate that the transmission factor doubly
enters the rate as (
T
Gamow
)
2
away from the resonance, when the barrier width is
effectively doubled.
4.2.1
Catalysis of DD Fusion by Mu Mesons
Our discussion leaves no doubt that molecular ions of DD
will be strongly
bound until destroyed by fusion or by the decay of the mu meson, after 2200 ns.
Fusion occurs in these ions on timescales about 1.5 ns for DD and 0.7 ps, for DT. In
both cases the muon is released, following a fusion event, and, after some delay,
forms a new DD
m
and DT
m
ion. The role of the muon is as a catalyst.
The detailed cyclic process ([43], p. 26) for DT is shown in Figure 4.7. A dense gas
mixture of D and T is bombarded with muons, momentarily forming D
m
or DT
m
m
and T
m
atoms.
These atoms formmolecules andmolecular ions, as shown in Figure 4.7. The DT
m
molecule formed in a time about 1 ns, and promptly undergoes fusion in 0.7 ps. After
fusion, themuon is released, and again available to catalyze further fusions. The cycle
described takes place in about 5 ns, so in the muon lifetime about 440 fusion events
could occur. However, as suggested in Figure 4.7, a small fraction, about 0.006, of the
fusionmuons are not released, but remain attached to the helium fusion product. On
this basis, the number of fusions per muon is estimated as120. Experiments have
shown up to 200 fusion events per muon.
