thumb|There are an infinite number of sinusoidal oscillations that perfectly fit a set of discrete oscillators, making it impossible to define a k-vector unequivocally. This is a relation of inter-oscillator distances to the spatial [[Nyquist frequency of waves in the lattice. See also for more on the equivalence of k-vectors.]]
In solid-state physics, crystal momentum or quasimomentum is a momentum-like vector associated with electrons in a crystal lattice. It is defined by the associated wave vectors <math>\mathbf{k}</math> of this lattice, according to
<math display="block">\mathbf{p}_{\text{crystal \equiv \hbar \mathbf{k}</math>
(where <math>\hbar</math> is the reduced Planck constant).
In systems with discrete translation symmetry, crystal momentum is conserved like mechanical momentum, making it useful to physicists and materials scientists as an analytical tool.
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 <math>V(x)</math> such that
<math display="block">V(\mathbf{x} + \mathbf{a}) = V(\mathbf{x}),</math>
where <math>\mathbf{a}</math> is an arbitrary lattice vector. 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,
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 <math> \mathbf{a}</math> without changing any aspect of the problem, thereby defining a discrete symmetry. Technically, an infinite periodic potential implies that the lattice translation operator <math>T(a)</math> commutes with the Hamiltonian, assuming a simple kinetic-plus-potential form.
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 <math>\mathbf{k}</math>, 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:
<math display="block">\mathbf{p}_{\text{crystal = \hbar \mathbf{k}.</math>
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. For example, an electron can be described not only by the wave vector <math>\mathbf{k}</math>, but also with any other wave vector <math>\mathbf{k}'</math>such that
<math display="block">\mathbf{k'} = \mathbf{k} + \mathbf{K},</math>
where <math>\mathbf{K}</math> is an arbitrary reciprocal lattice vector. that it obeys the equations of motion (in cgs units):