Magnetism: Part 7 - Isolated Magnetic Moments

diracdnb (55)in #science • 9 years ago (edited)

Previous posts: I, II, III, IV, V, VI

Welcome back, folks. In the last post, we met one of the three main types of magnetism, diamagnetism. We saw that this effect is largely characterised by a weak, negative susceptibility. The effect of an applied field to a diamagnetic material is to induce an internal magnetic moment that opposes the applied field; this gives rise to potential fun applications such as frog levitation! In this post we will consider isolated magnetic moments in slightly more detail by studying a single atom in a magnetic field. We will identify another type of magnetism from the Hamiltonain we will derive, and deduce the conditions necessary for this, so called, paramagnetic term to dominate, and bring about paramagnetic behaviour.

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Lets begin this discussion by thinking about an electron spin in a magnetic field which is parallel to the z-axis. This has an energy which is equal to

where we have g ~ 2, and m_s = +/- 1/2. We therefore have,

Now, in addition to spin angular momentum, electrons in an atom also posess orbital angular momentum. If the position of the i-th electron in the atom is r_i, and it has momentum p_i, then the total angular momentum is (hbar)L which can be written as,

where the sum is taken over all the electrons in an atom. Lets now consider an atom which has a Hamiltonian given by

Here, the sum is taken over the Z electrons in the atom, concerning the electronic kinetic energy and potential energy. We now make the assumption that the Hamiltonian has known eigenstates and known eigenvalues. Next, we add a magnetic field B given by

where A is the magnetic vector potential. Since have local gauge freedom, we choose a gauge such that

Then the kinetic energy must be altered according to the arguments made in post II/III. Since the charge on the electron is -e, the kinetic energy is given by the expression

which allows us to write the perturbed Hamiltonian as

It is usually the case that the dominant perturbation to the original Hamiltonian is given by the second term, but in certain cases it vanishes completely. This is the effect of the atom's own magnetic moment, and is known as the paramagnetic term.

So, under what conditions does the paramagnetic term vanish?

There exists a critical temperature, T_c; the Curie temperature, at which the magnetisation of a material which has a dominant paramagnetic Hamiltonian, vanishes. In a paramagnet below T_c, the effect of an external applied magnetic field is to align all spins within the material, generating a net magnetic moment. In this regime, we have symmetry breaking giving rise to long range magnetic order. Above T_c, thermal energy is sufficient to destroy magnetic order, with T_c characterising the temperature at which a transition of phase is brought about.

Thanks for reading. If anyone has a question about this post, or magnetism in general, then feel free to leave a comment, and I'll do my best to get back to you. In the next post, we're going to consider the case of paramagnetism from two approaches which have similar attributes in general, but are distinct.