Crystal momentum
[3]: 139 Frequently[clarification needed], crystal momentum is conserved like mechanical momentum, making it useful to physicists and materials scientists as an analytical tool.
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 a model is sensible because crystal ions that form the lattice structure are typically on the order of tens of thousands of times more massive than electrons,[4] making it safe to replace them with a fixed potential structure, and 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.
Technically, an infinite periodic potential implies that the lattice translation operator
commutes with the Hamiltonian, assuming a simple kinetic-plus-potential form.
[3]: 134 These conditions imply Bloch's theorem, which states or 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.
[3]: 261–266 [5] 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 the Planck 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[6]), there are important theoretical differences.
[3]: 218 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.
if we assign the particle an effective mass that's related to the curvature of the parabola.
In a real crystal, an electron moves in this way—traveling in a certain direction at a certain speed—for only a short period of time, before colliding with an imperfection in the crystal that causes it to move in a different, random direction.
These collisions, called electron scattering, are most commonly caused by crystallographic defects, the crystal surface, and random thermal vibrations of the atoms in the crystal (phonons).
[3]: 216 Crystal momentum also plays a seminal role in the semiclassical model of electron dynamics, where it follows from the acceleration theorem[7][8] that it obeys the equations of motion (in cgs units):[3]: 218 Here perhaps the analogy between crystal momentum and true momentum is at its most powerful, for these are precisely the equations that a free space electron obeys in the absence of any crystal structure.
Crystal momentum also earns its chance to shine in these types of calculations, for, in order to calculate an electron's trajectory of motion using the above equations, one need only consider external fields, while attempting the calculation from a set of equations of motion based on true momentum would require taking into account individual Coulomb and Lorentz forces of every single lattice ion in addition to the external field.
That is to say, an electron's crystal momentum inside the crystal becomes its true momentum after it leaves, and the true momentum may be subsequently inferred from the equation by measuring the angle and kinetic energy at which the electron exits the crystal, where
