Environmental Engineering Reference
In-Depth Information
This is a treatment applicable to the trapped alpha particle inside the nucleus, the
left portion of Figure 1.6, an exact solution of Schrodingers equation at the level used
originally by Gamow to explain alpha particle decay!
2.3
Protons and Neutrons and Their Binding
What is a deuteron? From a simple point of view a deuteron is a proton con
ned in a
spherical
finite potential well of radius (Eq. 2.1)
a¼
1.2 f 2
1/3
¼
1.51 f. The potential
well
V
o
is generated by the two-body attraction, in a useful simpli
cation. So, we can
describe the deuteron as an example of Equations 2.16
-
2.19.We can estimate the
minimum
barrier height as
V
o
¼
h
2
/8
ma
2
taking radius
a ¼
1.51 f. This gives
2
6
10
34
51
10
15
67
10
27
6
10
19
V
o
¼ð
6
:
=
2
p
1
:
Þ
=½
8
1
:
1
:
¼
2
:
26 MeV
:
(In fact, the
binding energy
of the deuteron is known to be 2.2245MeV, while the
simple estimate we just performed would correspond to a bound state at near-zero
binding energy
E
o
0. We could go through Equations 2.16
-
2.19 again to
nd
V
o
(a larger number would be needed) such that the bound state energy
E
o
¼
2.2245
MeV. The binding energy is the result of the strong or nuclear force, whose claim
on existence is indeed provided by the known nuclei, the deuteron being the
smallest nucleus.
The binding energy of a nucleon when surrounded by other nucleons, say six in a
cubic local environment, will be amultiple of the energy here estimated for a nucleon
in contact with one other nucleon. If there are six nearest neighbors, then we would
estimate 2.26MeV
6
¼
13.6MeV per nucleon. As we will see, this is a reasonable
value for the binding energy per nucleon in nuclear matter.
The proton
proton reaction that we have discussed can be put into this framework.
Two protons approaching de
nitely experience the Coulomb barrier. The nuclear
reaction is known to produce D
þ
e
þ
þ n
e
suggesting that an initial doubly charged
nucleus
2
He is formed, which in some cases decays to a deuteron, a positron, and a
neutrino. This complicated decay process (inwhich a proton somehowmutates into a
neutron, positron, and neutrino) makes the whole reaction less likely. In most cases
the
2
He reverts to two separate protons, but the fact that this reaction occurs indicates
that the Gamow tunneling process operates.
-
2.4
Gamow
s Tunneling Model Applied to Fusion in the Sun
s Core
To consider the fusion interaction of two protons, the incoming distant proton can be
treated as a spherical wave exp(
ikr
)/
r
, which is valid beyond the classical turning
point
r
2
. In the region between the outer turning point
r
2
¼ r
tp
and the point of contact
r
1
¼ r
n
(see Figure 1.6) is the
forbidden barrier region
where the solution to the
Schrodinger equation, for kinetic energy less than potential energy, is a real decaying
