Page 98 - Electrical Properties of Materials

80 Bonds

How will these energies vary with d, the distance between the protons?
What is A anyway? A has come into our equations as a coupling term. The
A
Energy E E o larger A, the larger the coupling, and the larger the split in energy. Hence A
must be related to the tunnelling probability that the electron may get through

exponentially with distance—we have talked about this before when solving
d the potential barrier between the protons. Since tunnelling probabilities vary
Schrödinger’s equation for a tunnelling problem—A must vary roughly in the
way shown in Fig. 5.6.
Fig. 5.6 Now what is E 0 ? It is the energy of the states shown in Fig. 5.5. It consists of
The variation of E 0 and A with the the potential and kinetic energies of the electron and of the potential energies
interproton separation, d. E 0 is the of the protons (assumed immobile again). When the two protons are far away,
energy when the states shown in their potential energies are practically zero, and the electron’s energy, since
Fig. 5.5 are uncoupled. A is the it is bound to a proton, is a negative quantity. Thus, E 0 is negative for large
coupling term.
interproton distances but rises rapidly when the separation of the two protons
is less than the average distance of the fluctuating electron from the protons. A
plot of E 0 against d is also shown in Fig. 5.6.
We may now obtain the energy of our states by forming the combinations
E 0 ± A. Plotting these in Fig. 5.7, we see that E 0 – A has a minimum, that is,
at that particular value of d a stable configuration exists. We may also argue in
Energy terms of forces. Decreasing energy means an attractive force. Thus, when the
E + A
0 protons are far away, and we consider the state with the energy E 0 – A, there is
an attractive force between the protons. This will be eventually balanced by the
d Coulomb repulsion between the protons, and an equilibrium will be reached.
Thus, in order to explain semi-quantitatively the hydrogen molecular ion,
E – A
0
we have had to introduce a number of new or fairly new quantum-mechanical
Fig. 5.7 ideas.
Summing the quantities in Fig. 5.6
to get E 0 + A and E 0 – A. The latter 5.5 Nuclear forces
function displays all the
characteristics of a bonding curve. Feynman in his Lectures on Physics goes on from here and discusses a large
number of phenomena in terms of coupled modes. Most of the phenomena
are beyond what an engineering undergraduate needs to know; so with regret
we omit them. (If you are interested you can always read Feynman’s book.) But
I cannot resist the temptation to follow Feynman in saying a few words about
nuclear forces. With the treatment of the hydrogen molecular ion behind us,
we can really acquire some understanding of how forces between protons and
neutrons arise.
∗ If you permit us a digression in a di- It is essentially the same idea that we encountered before. A hydrogen atom
gression, I should like to point out that and a proton are held together owing to the good services of an electron. The
Yukawa, a Japanese, was the first non-
European ever to make a significant con- electron jumps from the hydrogen atom to the proton converting the latter into
tribution to theoretical physics. Many a hydrogen atom. Thus, when a reaction
civilizations have struck independently
upon the same ideas, as for example the H, p → p, H (5.39)
virgin births of gods or the command-
ments of social conduct, invented in- takes place, a bond is formed.
dependently useful instruments like the Yukawa proposed in the middle of the 1930s that the forces between nuc-
∗
arrow or the wheel, and developed in-
dependently similar judicial procedures leons may have the same origin. Let us take the combination of a proton and a
and constitutions, but, interestingly, no neutron. We may say again that a reaction
civilization other than the European one
bothered about theoretical physics. p, n → n, p (5.40)