60 CHAPTER 6. DIAMAGNETISM
by we obtain, equating the magnetic force of Eq. (6.3) to mass times the change in
acceleration,or
The change in orbital angular velocity corresponds with a change in magnetic moment. If p
represents the orbital angular momentum of the electron before application of the magnetic
field, we may consider the equivalent magnetic shell and write
The change in the magnetic orbital moment due to the field is
This equation shows that there is a negative change of the magnetic moment that is
independent of the sign of and proportional to H.
If we consider a system consisting of N atoms, each containing i electrons with radii
we may write for the susceptibility
In the derivation of this equation, we have assumed that the orbital plane of the electrons
is perpendicular to the field direction. Instead of in Eq. (6.7), we should have used an
effective radius q of the orbit such that representing the average
of the square of the perpendicular distance of the electron from the field axis. The mean-
Usingsquare distance of the electrons from the nucleus is and since
for a spherical symmetrical charge distribution one has one finds that
instead of in Eq. (6.7), leads to
which is the classical Langevin formula for diamagnetism.
In the quantum-mechanical treatment, one has to consider that the electrons are
described by wave functions where at every point is the probability of finding the
electron. Alternatively, one may consider the electron as a charge cloud of intensity at
each point in space. It can be shown that the quantum-mechanical result is correctly given by
Eq. (6.8), provided one uses for the expectation value for the squared electron position
parameter
where the integration extends over the whole space.