Crystal Momentum - Lattice Symmetry Origins

Lattice Symmetry Origins

A common method of modeling crystal structure and behavior is to view electrons as quantum mechanical particles traveling through a fixed infinite periodic potential such that

where is an arbitrary lattice vector. Such a model is sensible because (a) crystal ions that actually form the lattice structure are typically on the order of tens of thousands of times more massive than electrons, making it safe to replace them with a fixed potential structure, and (b) the macroscopic dimensions of a crystal are typically far greater than a single lattice spacing, making edge effects negligible. A consequence of this potential energy function is that it is possible to shift the initial position of an electron by any lattice vector without changing any aspect of the problem, thereby defining a discrete symmetry. (Speaking more technically, an infinite periodic potential implies that the lattice translation operator commutes with the Hamiltonian, assuming a simple kinetic-plus-potential form.)

These conditions imply Bloch's theorem, which states in terms of equations that

,

or in terms of words that an electron in a lattice, which can be modeled as a single particle wave function, finds its stationary state solutions in the form of a plane wave multiplied by a periodic function . The theorem arises as a direct consequence of the aforementioned fact that the lattice symmetry translation operator commutes with the system's Hamiltonian.

One of the notable aspects of Bloch's theorem is that it shows directly that steady state solutions may be identified with a wave vector, meaning that this quantum number remains a constant of motion. Crystal momentum is then conventionally defined by multiplying this wave vector by Planck's constant:

While this is in fact identical to the definition one might give for regular momentum (for example, by treating the effects of the translation operator by the effects of a particle in free space ), there are important theoretical differences. For example, while regular momentum is completely conserved, crystal momentum is only conserved to within a lattice vector, i.e., an electron can be described not only by the wave vector, but also with any other wave vector k' such that

where is an arbitrary reciprocal lattice vector. This is a consequence of the fact that the lattice symmetry is discrete as opposed to continuous, and thus its associated conservation law cannot be derived using Noether's theorem.