4 - Exchange Interactions and Magnetism in Solids

Summary

The first interaction between magnetic moments, which is expected to play a role in magnetism, is of course the interaction between two magnetic dipoles μ₁ and μ₂ separated by a distance r. The energy of this system can be expressed as:

The order of magnitude of the effect of dipolar interaction for two moments each of μ ≈ 1μB separated by a distance of r ≈ 1 ^˚A can be estimated to be approximate μ² /4πr³ ∽10⁻²³ J, which is equivalent to about 1 K in temperature. This dipolar interaction is too weak to explain the magnetic ordering observed in many materials at much higher temperatures, even around 1000 K.

Coupling between Spins

Before we look for suitable interactions between two magnetic moments to explain the magnetic ordering observed in various materials, we shall first discuss the coupling of two spins. We now consider two interacting spin- 1/2 particles represented by a Hamiltonian:

Here S and S represent the spin operators of the two particles. Combining the two particles as a single entity, the total spin operator can be expressed as:

This leads to:

A combination of two spin-1/2 particles gives rise to a single entity with quantum number s = 0 or 1. This leads to the eigenvalue of (S^Total)2 as s(s + 1), which is 0 for s = 0 and 2 for s = 1. Now the eigenvalues of both (S)² and (S)² are 3/4 [4]. Hence, from Eqn. 4.4 we can write:

The system has two energy levels for s = 1 and 0 with energies as follows:

Each state will have a degeneracy given by (2s + 1). The s = 0 state is a singlet and the z-component of the spin ms of this state takes the value 0. On the other hand, s = 1 state is a triplet and ms takes one of the three values -1, 0, and 1.

The eigenstates of this two interacting spin-1/2 particles can be represented as linear combinations of the following basis states: |↑↑〉, |↑↓〉, |↓↑〉and |↓↓〉, where the first (second) arrow corresponds to the z-component of the spin labelled by a(b). The possible eigenstates are presented in Table 4.1.