Environmental Engineering Reference
In-Depth Information
on the outside, while an outgoing wave will be large on the inside, but the penetration
probability will be the same. So the same formalism applies to our incoming wave
case as to the outgoing wave case represented by the alpha particle decay mentioned
above.
With
V
B
(
r
)
¼ k
c
e
2
/
r
and
m
r
¼m
p
/2, for two protons, with incoming energy
E¼
k
c
e
2
/
r
2
the resulting formula is
2
ð
r
2
1
1
=
2
d
r
h
1
1
=
2
1
=
2
c ¼
h
ð
2
m
r
EÞ
1
=
r
1
ðr
2
=
r
1
Þ
ð
2
m
r
EÞ
½pr
2
=
2
2
ðr
1
r
2
Þ
;
for
r
1
<
r
2
:
ð
2
:
24
Þ
(This formula is reached by substituting
r ¼ r
2
sin
2
u
in
Ð
r
2
r
2
1
=
2
ðr
2
=
r
1
Þ
d
r
to get
(
r
2
(
p
/2
sin
1
(
r
1
/
r
2
)
1/2
)
(
r
1
(
r
2
r
1
))
1/2
). For
r
1
<
r
2
, using the small-angle formula
sin
x x
, one gets (2.24) [22]).
In this expression for
c
h
1
(2
m
r
E
)
1/2
(
p r
2
/2
2(
r
1
r
2
)
1/2
), for two protons
r
2
¼ k
c
e
2
/
E
, and also that the
p r
2
/2 term in the square bracket dominates. Thus,
c
is nearly
proportional to
E
1/2
, and the literature has adopted the notation 2
c¼
(
E
G
/
E
)
1/2
with
E
G
called the Gamow energy, so the fusion rate will be expressed as exp(
(
E
G
/
E
)
1/2
).
One can easily see the factors that appear in
E
G
, by inspecting the formula
c
h
1
(2
m
r
E
)
1/2
(
p r
2
/2
2(
r
1
r
2
)
1/2
), noting that in general
r
2
¼ k
c
Z
1
Z
2
e
2
/
E
and neglecting
the term in
r
1
.
Let us apply this formula to the two protons approaching and assume
E¼
1.293 keV, so
r
2
¼
1113 f. (It has been carefully explained that the correct energy is
E¼ kT
, in this case 1.293 keV, which corresponds to the peak in the velocity
distribution in the center of mass frame of the two nucleons (Section 3.5 inRef. [25])).
The reduced mass is
m
p
/2,
r
1
¼
2.4 f. Then,
c ¼
h
1
1
=
2
1
=
2
ð
2
m
r
EÞ
½pr
2
=
2
2
ðr
1
r
2
Þ
1
=
2
10
15
6
10
34
67
10
27
6
10
19
¼½
2
p=
6
:
½
1
:
1293
1
:
1
=
2
½
1113
p=
2
2
ð
1113
2
:
4
Þ
¼
9
:
21
:
ð
2
:
25
Þ
So T
¼
exp(
18.42)
¼
1.0
10
8
, which we may refer to as the tunneling
probability or the
Gamow probability
, not to be confused with the temperature, for
the p
-
p reaction, evaluated at
E¼ k
B
T¼
1.293 KeV.
The rate of geometric collisions per proton can be estimated from the mean free
path
represents how far it will go before making a
collision, of the type we have described, with another proton. The basic formula for
the mean free path is
L
. If we imagine one proton,
L
=ðnpr
2
Þ;
L ¼
1
=ðnsÞ¼
1
ð
2
:
26
Þ
where
n¼N
p
is the number per unit volume of scattering centers (protons) and
s
is
an area if interceptedwill lead to a geometric collision.We need a further probabilityT
T
that a geometric collision actually leads to a fusion reaction (this is sometimes called
the astrophysical
S
factor). This
T
is expected to be small based on the complicated
nature of the p
-
p reaction, which needs to generate a positron and neutrino. Per
