Environmental Engineering Reference
In-Depth Information
However, since this assumes a parabolic potential rather than Coulomb repulsive
potential, it will likely overestimate the fusion rate. The probability of D
-
D spacings
in the range 0
-
r
1
is the integral of
P
(
x
) over that range, which can be approximated as
1
=
2
exp
2
P
0
ð
r
1
s
1
0
r
1
Þ
ð
1
=pÞ
½ð
r
e
r
1
Þ=sÞ
ð
4
:
20
Þ
2
The result is
P
0
ð
¼ð
2
:
7
=
307
:
8
Þ
0
:
564 exp
½ð
648
2
:
7
Þ=
307
:
8
Þ
:
0
r
1
Þ
¼
8
:
77
10
3
0
:
0123
¼
6
:
08
10
5
0
:
564
:
10
12
s
1
,
(
v
/2
p
)
P
0
(0
The rate of fusion is then estimated as
R
¼
r
1
)
¼
6.43
corresponding to a lifetime
10
13
s
T
f
¼
1
:
55
:
ð
4
:
21
Þ
This is shorter than the observed lifetime, which is 1.5 ns.
The correct solution is dif
cult, but was described by Jackson [47]. While the full
solution is more advanced than we can present here, we suggest an approximation
that improves on what we have just done.
The improved approximation is to replace the harmonic oscillator potential with
the direct Coulomb potential, for the range between the classical turning point
R
r
1
. (In physical terms, this means that
we neglect the muon charge density in that region, removing screening of the
Coulomb potential.)
The corresponding Gamow transmission factor
¼
r
e
s ¼
340.2 f and the contact radius
R
¼
T
Gamow
¼
exp
2
c;
where
;
for
r
2
¼
340
:
2f
;
E
¼
4
:
233 keV
;
r
1
¼
2
:
7f
;
ð
4
:
22
Þ
1
=
2
1
=
2
c ¼ð
2
mE
Þ
=
h
½ðp=
2
Þ
r
2
2
ð
r
2
r
1
Þ
¼
6
:
78
10
6
. In this case, the integration of the oscillator probability
function is from 0 to
r
2
and
T
Gamow
¼
1.28
340.2 f and is about
P
0
¼
¼
0.023. The resulting rate of fusion
10
9
,so
(
v
/2
p
)
P
0
(0
R
¼
r
2
)
T
Gamow
¼
3.1
T
f
¼
0
:
32
10
9
s
:
ð
4
:
23
Þ
This is closer, but still below the observed value, which is 1.5 ns ([43], p. 26). We
attribute the difference to a reaction probability factor
T
0.21, which we can
interpret as the fraction of the collisions of the two deuterons at the contact point
that lead to fusion, an effect of the nuclear physics of the two particles. The estimate
here is of the same sense but numerically somewhat larger than the valueT¼
T
¼
¼
0.04 we
mentioned earlier.
The in
uence of the details of the nuclear reaction, suggested here, is supported by
the well-known large difference between this lifetime and the fusion lifetime of the
similar DT
10
13
s. This represents an increase in the reaction rate
R
by
a factor 21. (This is similar to the increase in the DT above the DD cross section in
m
ion,Tf¼
T
f
¼
7
