In this chapter we examine the magnetic behavior in different types of solids. Magnetic order in a solid, induced either by an external field or by inherent properties of the structure, may be destroyed by thermal effects, that is, the tendency to increase the entropy by randomizing the direction of microscopic magnetic moments. Thus, magnetic phenomena are typically observed at low temperatures where the effect of entropy is not strong enough to destroy magnetic ordering.

Up to this point we have treated electrons in solids as essentially independent particles and solved the appropriate single-particle Schrödinger equations, exploiting only the symmetries imposed by the presence of the crystal lattice of ions. Given the strong and long-range interaction between electrons, that is, the Coulomb repulsion, and the fact that they are indistinguishable particles, we would expect a more complicated behavior, including some degree of correlation in the motion of these interacting particles. Thus, a better justification for the single-particle picture is required. We provide a comprehensive justification in this chapter.

In thewe discussed in detail the effects of lattice periodicity on the single-particle wavefunctions and the energy eigenvalues. We also touched on the notion that a crystal can have symmetries beyond the translational periodicity, such as rotations around axes, reflections on planes, and combinations of these operations with translations by vectors that are not lattice vectors, called “non-primitive” translations. All these symmetry operations are useful in calculating and analyzing the physical properties of a crystal. There are two basic advantages to using the symmetry operations of a crystal in describing its properties. First, the volume in reciprocal space for which solutions need to be calculated is further reduced, usually to a small fraction of the first Brillouin zone, called the “irreducible” part; for example, in the FCC crystals with one atom per unit cell, the irreducible part is $1/48$ of the full BZ.

Up to this point we have been dealing with the ground-state properties of electrons in solids. Even in the case of doped semiconductors, the presence of extra electrons or of holes relative to the undoped ideal crystal was due to the introduction of additional electrons (or removal of some electrons) as a result of the presence of impurities, with the additional charges still corresponding to the ground state of the solid. Some of the most important applications of materials result from exciting electrons out of the ground state. These include the optical properties of solids and the dielectric behavior (shielding of external electric fields). These phenomena are also some of the more interesting physical processes that can take place in solids when they interact with external electromagnetic fields. We turn our attention to these issues next.