Environmental Engineering Reference
In-Depth Information
This value is larger than observed. The predicted ef
ciency from an energy release
point of view is (3.5MeV/115 keV)
10
4
. These numbers are
approximately applicable to the deuteron experiment [45], but they predict a higher
yield than was observed. From these results, con
rmed by Naranjo
et al.
[45], there
seems not much hope for making a practical fusion reactor from an energetic beam
hitting a solid or liquid target, because of the rapid energy loss, as was earlier
predicted by Ruggiero [46].
10
6
4.7
¼
1.43
4.1.1
Electric Field Ionization of Deuterium (Hydrogen)
Ionizing an atom, as in the compact apparatus of Figure 4.1, is done with an electric
field, and the question is how large a
field is required? The same question arises in a
Tokamak device where it is necessary to ionize the deuterium gas to form a plasma.
So the rate of
eld ionization, which is governed by an electron tunneling process,
is important.
The electronic properties of deuterium are the same as hydrogen H, which differs
only by the additional (uncharged) neutron mass in the nucleus. Thus, the simplest
estimate of the required electric
field,
E
to ionize deuteriummight be the
field of the
proton at the Bohr radius in
H
, which is
E
k
C
e
/
a
o
2
514.5 V/nm.
This is an overestimate: a much smaller electric
field,
E
¼
¼
25 V/nm, quickly
10
15
s) removes the electron by a process of tunneling.
In an electric
field,
E
, the potential energy of the electron has a term
U
(
x
)
(
¼
eEx
,so
that at a spacing
x
¼
E
o
/
eE
, the electron will have the same energy as in its ground
state, and can tunnel out. Here
E
o
is the electronic binding energy, 13.6 eV for
deuterium, as well as for hydrogen. How quickly will this happen?
The electron can tunnel through the barrier that extends from
x
a
o
to
x
. In detail
this is a dif
cult problem to solve, but a simpli
ed treatment is possible. The earliest
estimate of the lifetime of H against
field ionization was given by Oppenheimer in
1928. Oppenheimers notable estimate [49] is that for H in a
field of 1000 V/m the
lifetime
t
is (10
10
)
10
s.
We can use a simpli
¼
ed, one-dimensional, model to estimate the
E
-
field ionization
rate of H. We will
field ionization lifetime of hydrogen
and deuterium is on the order of 10
15
s, while, at
E
find that for
E
¼
25 V/nm, the
¼
2.5 V/nm,
t
is extremely long,
10
33
s or 1.9
10
26
y.
estimated as 6
The ground state with
E
¼
0 is spherically symmetric, depending only on
r
.Withan
electric
field,
E
x
¼
E
, approximate the situation as depending only on
x
: electron
potential energy
k
C
e
2
U
ð
x
Þ¼
=
x
eEx
;
ð
4
:
4
Þ
10
9
Nm
2
/C
2
, with
E
in V/m. Assume that the ground-state energy is
unchanged by the electric
field, and remains
where
k
C
¼
9
E
o
¼
13.6 eV.
The electron barrier potential energy is
V
B
ð
x
Þ¼
U
ð
x
Þ
E
o
:
ð
4
:
4a
Þ
