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 ${\displaystyle \mathbf {k} }$ of this lattice, according to $${\displaystyle \mathbf {p} _{\text{crystal}}\equiv \hbar \mathbf {k} }$$ (where ${\displaystyle \hbar }$ 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.

A common method of modeling crystal structure and behavior is to view electrons as quantum mechanical particles traveling through a fixed infinite periodic potential ${\displaystyle V(x)}$ such that $${\displaystyle V(\mathbf {x} +\mathbf {a} )=V(\mathbf {x} ),}$$ where ${\displaystyle \mathbf {a} }$ 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 ${\displaystyle \mathbf {a} }$ without changing any aspect of the problem, thereby defining a discrete symmetry. Technically, an infinite periodic potential implies that the lattice translation operator ${\displaystyle T(a)}$ commutes with the Hamiltonian, assuming a simple kinetic-plus-potential form.

These conditions imply Bloch's theorem, which states $${\displaystyle \psi _{n}(\mathbf {x} )=e^{i\mathbf {k} \cdot \mathbf {x} }u_{n\mathbf {k} }(\mathbf {x} ),\qquad u_{n\mathbf {k} }(\mathbf {x} +\mathbf {a} )=u_{n\mathbf {k} }(\mathbf {x} ),}$$ or that an electron in a lattice, which can be modeled as a single particle wave function ${\displaystyle \psi (\mathbf {x} )}$, finds its stationary state solutions in the form of a plane wave multiplied by a periodic function ${\displaystyle u(\mathbf {x} )}$. 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 ${\displaystyle \mathbf {k} }$, 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: $${\displaystyle \mathbf {p} _{\text{crystal}}=\hbar \mathbf {k} .}$$

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 ${\displaystyle \mathbf {k} }$, but also with any other wave vector ${\displaystyle \mathbf {k} '}$such that $${\displaystyle \mathbf {k'} =\mathbf {k} +\mathbf {K} ,}$$ where ${\displaystyle \mathbf {K} }$ 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.

The phase modulation of the Bloch state ${\displaystyle \psi _{n}({\mathbf {x} })=e^{i\mathbf {k} \cdot \mathbf {x} }u_{n\mathbf {k} }(\mathbf {x} )}$ is the same as that of a free particle with momentum ${\displaystyle \hbar k}$, i.e. ${\displaystyle k}$ gives the state's periodicity, which is not the same as that of the lattice. This modulation contributes to the kinetic energy of the particle (whereas the modulation is entirely responsible for the kinetic energy of a free particle).

In regions where the band is approximately parabolic the crystal momentum is equal to the momentum of a free particle with momentum ${\displaystyle \hbar k}$ if we assign the particle an effective mass that's related to the curvature of the parabola.

Crystal momentum corresponds to the physically measurable concept of velocity according to $${\displaystyle \mathbf {v} _{n}(\mathbf {k} )={\frac {1}{\hbar }}\nabla _{\mathbf {k} }E_{n}(\mathbf {k} ).}$$ This is the same formula as the group velocity of a wave. More specifically, due to the Heisenberg uncertainty principle, an electron in a crystal cannot have both an exactly defined k and an exact position in the crystal. It can, however, form a wave packet centered on momentum k (with slight uncertainty), and centered on a certain position (with slight uncertainty). The center position of this wave packet changes as the wave propagates, moving through the crystal at the velocity v given by the formula above. 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).