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The Relative Angle Distribution Function in the Langevin Theory of Dilute Dipoles
1
The Relative Angle Distribution Function in the Langevin Theory of
Dilute Dipoles
Robert D. Nielsen
ExxonMobil Research and Engineering Co., Clinton Township, 1545 Route 22 East,
Annandale, NJ 08801
robert.nielsen@
2
Abstract
The Langevin theory of the polarization of a dilute collection of dipoles by an external
field is often included in introductory solid state physics and physical chemistry
curricula. The average polarization is calculated assuming the dipoles are in thermal
equilibrium with a heat bath. The heart of the polarization calculation is a derivation of
the average dipole-field projection, whose dependence on the external field is given by
the Langevin function. The Langevin problem is revisited, here, and the average
projection of any given dipole onto any other dipole from the collection is derived in
terms of the Langevin function. A simple expression is obtained for the underlying
dipole-dipole angular distribution function.
I. Introduction
A single magnetic dipole μ in an external magnetic field H has a potential energy:
HV μ= ? ? ( )H cosμ θ= ? ? .1 While formulating a theory of magnetism, Langevin
considered a collection of dipoles in an external magnetic field. 2 The concentration of
the dipoles was assumed to be sufficiently diluted that dipole-dipole interactions could be
neglected, leaving only the sum over the individual dipole-field potential energies for the
total energy. Langevin developed the equilibrium average value of the dipole projection
on to the external field, ( )cos θ , by assuming that the dipoles were in contact with a
heat bath. The distribution function, which allows the equilibrium averages to be
calculated, is the Boltzmann distribution:
( )
cosV kT F
L
e e
Z Z F
θ? ?
=
3
where HF kTμ= ?
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