Environmental Engineering Reference
In-Depth Information
Figure 2.3 (a) Sketch of incoming proton upon
Coulomb barrier, arrow terminates at classical
turning point. The tunneling probability is T to
reach inner radius R
o
, and the further probability
is TTto achieve the deuteron-bound state (b).
Note that (b) is described by
wavefunctions (2.18) and (1.19a), matched
together at R
o,
The matching in amplitude and
slope, in fact, determines the bound state
energy, shown as 2.22MeV. (From [25],
Figure 6.4).
A simpli
ed view of what may happen is suggested by the known lifetime for decay
of the free neutron, 880 s, into a proton, electron, and neutrino. If one assumes that
the decay of the proton (into neutron, positive electron, and neutrino) has a similar
time as the known decay of a neutron into a proton, electron, and neutrino, then the
chance of this occurring before the excited pp state decays can be estimated as
T¼D
t
/
t
is the lifetime of the excited pp or
2
He state.
How can we estimate the lifetime
880, where
D
t
of the unstable excited state?
One estimate might be the oscillation period of the proton crossing the
2
He
protodeuteron, 4
1.51 f/(0.5
10
6
m/s)
¼
1.21
10
20
s. In this case, we get
T¼
1.38
10
23
.
The second estimate might be from the uncertainty principle. There are two forms
of the uncertainty principle, both originate in the wave aspect of particle behavior. The
more familiar form is
D
x
h
/2, where
p
and
x
refer to the momentum and
position of the particle. The less familiar form is
D
p
D
E
h
/2, where
t
and
E
are time
and energy, respectively. We can apply the second form to estimate the lifetime as
D
D
t
D
t ¼
h
/(2
D
E
), wherewe can take
D
E¼
1.293 keV
þ
2.22MeV (see Figure 2.3). In this
t ¼
2.95
10
22
s, and probability
T¼
2.95
10
22
s/880 s
¼
3.35
10
25
. These estimates of
T
are quite close to our earlier estimateT¼
T¼
9.6
10
25
, and
the consensus from the literature as summarized by Atzeni.
So from this point of view, our simple analysis is reasonably accurate! The
implication is that our simpli
ed method might work quite well in cases where
case, we
nd
D
