Environmental Engineering Reference
In-Depth Information
T¼
1 as in the case of DT (deuterium-tritium) fusion of interest for terrestrial fusion
machines. The p
-
p reaction is really the hardest one to understand.
To return to and to emphasize our main interest in this analysis,
in classical physics
the Gamow tunneling factor
T
would be zero, there would be no fusion!
So by explaining
how the sun generates energy we have shown the necessity for Schrodingers wave
treatment of matter particles, which we will extend to atoms and solids.
You can see that we came out quite well in this simpli
ed calculation. To compare
with a more standard approach, amenable to a wide variety of fusion reactions, we
mention that the major de
ciency in our analysis has been to overlook the
distribu-
tions
of speeds and tunneling probabilities (cross sections) by replacing themby their
most probable values. The rate of fusion is proportional to
v s
, and in a more
accurate analysis the calculated property is
hvsi
, which has units m
3
/s. We have seen
that there is a distribution of speeds
v
, and the cross section will vary as the speed
varies. So integrations over variables are needed. A standard framework for carrying
this out gives specifically [23] for the p
-
p reaction
56
10
43
T
2
=
3
exp
ð
14
T
1
=
3
03
10
4
T
2
m
3
hvsi¼
1
:
:
94
=
Þ½
1
þ
0
:
044
T þ
2
:
=
s
;
ð
2
:
29
Þ
where temperature
T
is expressed in keV. Evaluating this for
T¼
1.293 keV we
nd,
for the p
-
p reaction at 1.5
10
7
K,
56
10
43
hvsi¼
1
:
0
:
843
exp
½
14
:
94
=
1
:
089
456
10
49
54
10
49
m
3
¼
1
:
½
1
þ:
057
þ:
0003
¼
1
:
=
s
:
(Note that the exponential factor here is exp[
14.94/1.089]
¼
exp[
13.71]
¼
1.107
10
6
as compared to tunneling probability
T¼
10
8
in the simpli
ed
analysis. Compared to the simpli
ed analysis, the tunneling probability is 111 times
larger, which indicates that the optimal fusion events involve higher energy particles,
which, however, are fewer in number, suggested in Equations 1.8, 1.9).
We can compare this with our simpli
ed result, by setting
hvsi¼vs:
ð
2
:
30
Þ
r
2
¼
9
Using our
earlier numbers, we
nd,
taking
s ¼TT
p
,
:
6
10
25
0
10
8
p r
2
¼
3
75
10
56
m
2
1
:
:
where
r
2
¼
1113 f and
v ¼
0.498
10
6
m/s, so
498
10
6
m
75
10
57
m
2
86
10
50
m
3
vs ¼
0
:
=
s3
:
¼
1
:
=
s
:
ð
2
:
31
Þ
So our simplified result is only 0.121
¼
1/8.3 of the Angulo formula, setting
hvsi¼vs
. If we take the view that our initial approximate result is low by a factor of
8.3, it implies that the reaction constant
T
Tshould be increased by the same factor, 8.3,
to get
T¼
8.0
10
24
. This value is understood as the factor by which the crucial p
-
p
reaction is slowed down by the necessity of turning a proton into a neutron,
rst
suggested by Bethe and Critch
eld [24].
We can pause for a moment to summarize what we have learned, by looking at
Figure 2.4.
