The Relative Angle Distribution Function in the Langevin Theory of Dilute Dipoles.pdf

The Relative Angle Distribution Function in the Langevin Theory of Dilute Dipoles.pdf

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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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