Curie's Law - Derivation With Quantum Statistical Mechanics - Two-state (spin-1/2) Particles

Two-state (spin-1/2) Particles

To simplify the calculation, we are going to work with a 2-state particle: it may either align its magnetic moment with the magnetic field, or against it. So the only possible values of magnetic moment are then and . If so, then such a particle has only two possible energies

and

When one seeks the magnetization of a paramagnet, one is interested in the likelihood of a particle to align itself with the field. In other words, one seeks the expectation value of the magnetization :

leftlanglemurightrangle = mu Pleft(muright) + (-mu) Pleft(-muright) = {1 over Z} left( mu e^{ mu Bbeta} - mu e^{ - mu Bbeta} right) = {2mu over Z} sinh( mu Bbeta),

where the probability of a configuration is given by its Boltzmann factor, and the partition function provides the necessary normalization for probabilities (so that the sum of all of them is unity.) The partition function of one particle is:

Therefore, in this simple case we have:

This is magnetization of one particle, the total magnetization of the solid is given by

The formula above is known as the Langevin paramagnetic equation. Pierre Curie found an approximation to this law which applies to the relatively high temperatures and low magnetic fields used in his experiments. Let's see what happens to the magnetization as we specialize it to large and small . As temperature increases and magnetic field decreases, the argument of hyperbolic tangent decreases. Another way to say this is

this is sometimes called the Curie regime. We also know that if, then

so

with a Curie constant given by . Also, in the opposite regime of low temperatures or high fields, tends to a maximum value of, corresponding to all the particles being completely aligned with the field.

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