Environmental Engineering Reference
In-Depth Information
Thus, the peak height of the barrier is
1
=
2
V
BMAX
¼
E
o
2
e
ð
k
C
eE
Þ
:
ð
4
:
9a
Þ
Next, we evaluate
f
escape
for
E
¼
25 V/nm, using
=
1
2
f
escape
¼
f
approach
exp
½
2
ð
2
mV
BMAX
=
2
Þ
D
x
=
h
;
ð
4
:
6
Þ
taking the average barrier as
V
BMAX
/2. To evaluate the square bracket term one
nds,
at 25 V/nm,
h
i
1
=
2
2
ð
mV
BMAX
=
2
Þ
D
x
=
h
¼
2
:
35
:
ð
4
:
10
Þ
The escape frequency is then
f
escape
¼
10
15
exp(
2.35) s
1
10
14
s
1
, and the ionization time is
6.5
¼
6.2
10
15
s
t ¼
:
;
for
E
¼
=
Þ:
ð
:
Þ
1
61
25 V
nm
4
11
If we repeat the analysis of
f
escape
from the working formula for
E
¼
2.5 V/nm, we
find that the square bracket is
118.8. The lifetime is about
10
33
s
t ¼
6
ð
for
E
¼
2
:
5V
=
nm
Þ
ð
4
:
12
Þ
10
26
y. The simple tunneling model predicts a sharp cutoff in the
ionization rate for electric
fields falling from 25 V/nm to 2.5 V/nm, as observed.
Tunneling rates are sensitively dependent on the parameters related to the barrier.
This is observed both experimentally and in model calculations like the one we
have given.
or about 1.9
4.2
Deuterium Fusion Demonstration Based on Muonic Hydrogen
This section is based on Equations 3.36
-
3.42 in Chapter 3. The hydrogen molecule
ion has the same de
ning equations as themolecular ion based on two deuterons and
one muon, which we now describe.
The muon has charge
s, the
properties of the corresponding muonic hydrogen and deuterium atoms are well
known. The reduced mass
m
r
e
and mass 207
m
e
. Although its lifetime is only 2.2
m
M
) enters the Bohr radius in the denom-
inator, and therefore multiplies the Bohr energy. Focusing on the D
¼
mM
/(
m
þ
m
formed by a
the reduced mass is
m
r
¼
þ
muon and a deuteron D,
207
2
1836/[207
2(1836)]
¼
196
m
e
. The equivalent Bohr radius is 270 f and the binding energy is
196
atom is smaller by a factor 196 than hydrogen and
has a binding energy 196 times larger. If a beam of muons passes through a gas of
hydrogen atoms or hydrogen molecules, the electrons will be ejected with 2.66 keV
energy and the muonic atoms will form. In dense hydrogen exposed in this way, it is
observed that muonic DD
13.6 eV
¼
2.666 keV. The D
m
m
þ
molecule ions form. For example,
mþ
DeeD
!
DD
mþ
2e
:
