Classical Statistical Mechanics: Paramagnetism in the Canonical Ensemble

Paramagnetism 1. The Spin or intrinsic moments of the electrons. Paramagnetism occurs in substances where the individual atoms, ions or molecules possess a permanent magnetic dipole moment. The permanent magnetic moment is due to the contributions from: 1. The Spin or intrinsic moments of the electrons. 2. The Orbital motion of the electrons. 3. The Spin magnetic moment of the nucleus.

Some Paramagnetic Materials Metals. Atoms, & molecules with an odd number of electrons, such as free Na atoms, gaseous Nitric oxide (NO) etc. Atoms or ions with a partly filled inner shell: Transition elements, rare earth & actinide elements. Mn2+, Gd3+, U4+ etc. A few compounds with an even number of electrons including molecular oxygen.

Classical Theory of Paramagnetism Consider a material with N magnetic dipoles per unit volume, each with moment . In the presence of a magnetic field B, the potential energy of a magnetic dipole is:

This shows that dipoles tend to line up with B. B = 0, M = 0 B ≠ 0, M ≠ 0 This shows that dipoles tend to line up with B. The effect of temperature is to randomize the directions of the dipoles. The effect of these 2 competing processes is that some magnetization is produced.

The Canonical Ensemble Probability of Suppose that B is applied along the z-axis, so that  is the angle made by the dipole with the z-axis. The Canonical Ensemble Probability of finding the dipole along the  direction is proportional to the Boltzmann factor f(): The average value of z then has the form: The integration is carried out over the solid angle, whose element is d. The integration thus takes into account all possible orientations of the dipoles.

z =  cos & d = 2 sin d So: To do the integral, substitute which gives: Let cos = x, then sin d = - dx & limits -1 to +1 So:

The mean value of z is thus: “Langevin Function” L(a) So: z

Variation of L(a) with a. The mean value of z is thus: Langevin Function, L(a) z In most practical situations, a < < 1, so The magnetization is N = Number of dipoles per unit volume Variation of L(a) with a.

This is known as the “Curie Law” This is known as the “Curie Law”. The susceptibility is referred as the Langevin paramagnetic susceptibility. It can be written in a simplified form as: Curie constant 10