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Re: Induced dipole moments



Here's my New Year's gift to Ludwik:

The linear polarizability of a nonpolar molecule is understandable in
terms of a simple model which every student who has taken the
introductory physics course can readily recognize understand. I shall
take as my physical model a system of electrons, imagined to be
smoothly distributed over space and/or time on a scale of 10^-11 cm*,
and positively charged nuclei which can be thought of as point
charges on the same scale. In static equilibrium and in the absence
of an electric field these nuclear point charges sit in electrostatic
potential minima. By hypothesis the medium is unpolarized, so these
electrostatic potential minima are isotropic, regardless of the
complexity of structure on an atomic scale.

Here's where the frog jumps into the pond: the isotropy of the
electrostatic potential well means that it can be modeled as a well
at the center of some spherically symmetric charge distribution**.
Thus our general problem of a nucleus in a nonpolar system with all
its physical complexity on the atomic scale is seen to be equivalent
to the problem of a nucleus near the center of a spherically
symmetric charge distribution on the scale of several tens of nuclear
diameters.

The electrostatic field E(r) at a distance r from the center of a
spherically symmetric charge distribution (and the potential
derivable from it) is a function of the total charge q(r) contained
within the concentric sphere of radius r. Also, by hypothesis, the
electronic charge density rho(r) = rho can be taken as constant in
the vicinity of the equilibrium position of the nucleus. Hence

4 pi r^3 rho rho
E(r) = ------------------- = -------- r
3 * 4 pi eps0 r^2 3 eps0

The electric field is directly proportional to the distance from the
center of the potential well***. Hence the restoring force on a
nucleus displaced from its equilibrium position is harmonic.

Q.E.D.

Now comes the "aha" part. When I took introductory physics at
Sacramento Junior College I was assigned a multi-part problem
relating to the motion of a particle moving under gravity in a
frictionless, straight tunnel through an earth of hypothetically
uniform density. Even if this technically inaccessible tunnel did not
go through the center of this improbable planet, the motion was seen
to be simple harmonic with exactly the same period in all cases. The
problem of the polarizability of a nonpolar medium has been shown to
be formally identical, an it should be conceptually identical, to the
old chestnut from Sears and Zemansky****.

Leigh

* 10^-11 cm = 100 fm = 1/1000 Å

** I take this point to be self-evident, but perhaps someone else
will wish to construct a proof.

*** I have used SI in deference to modern custom, however misguided
that may be.

**** This result must have been known to Newton as well.