![Quasimomentum](https://upload.wikimedia.org/wikipedia/commons/9/96/Quasimomentum.gif)
In
solid-state physics crystal momentum or quasimomentum is a
momentum-like
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
associated with
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no ...
s in a
crystal lattice. It is defined by the associated
wave vectors of this lattice, according to
:
(where
is the reduced
Planck's constant).
Frequently, 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
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
particles traveling through a fixed infinite periodic potential
such that
:
where
is an arbitrary
lattice vector
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an or ...
. Such a model is sensible because crystal
ions
An ion () is an atom or molecule with a net electrical charge.
The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conven ...
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
without changing any aspect of the problem, thereby defining a
discrete symmetry
In mathematics and geometry, a discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square' ...
. Technically, an infinite periodic potential implies that the lattice translation operator
commutes with the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
, assuming a simple kinetic-plus-potential form.
[
These conditions imply ]Bloch's theorem
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who d ...
, 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.[
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. For example, an electron can be described not only by the wave vector , but also with any other wave vector such that
:
where is an arbitrary ]reciprocal lattice
In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial fu ...
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
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
.
Physical significance
The phase modulation of the Bloch state
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who d ...
is the same as that of a free particle with momentum , i.e. 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 if we assign the particle an effective mass that's related to the curvature of the parabola.
Relation to velocity
Crystal momentum corresponds to the physically measurable concept of velocity according to[
:
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
In physics, a wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of diff ...
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
Electron scattering occurs when electrons are deviated from their original trajectory. This is due to the electrostatic forces within matter interaction or, if an external magnetic field is present, the electron may be deflected by the Lorentz f ...
'', are most commonly caused by crystallographic defect
A crystallographic defect is an interruption of the regular patterns of arrangement of atoms or molecules in crystalline solids. The positions and orientations of particles, which are repeating at fixed distances determined by the unit cell para ...
s, the crystal surface, and random thermal vibrations of the atoms in the crystal ( phonons).[
]
Response to electric and magnetic fields
Crystal momentum also plays a seminal role in the semiclassical model of electron dynamics, where it obeys the equations of motion (in cgs units):[
:
:
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.
]
Applications
Angle-resolved photo-emission spectroscopy (ARPES)
In angle-resolved photo-emission spectroscopy (ARPES), irradiating light on a crystal sample results in the ejection of an electron away from the crystal. Throughout the course of the interaction, one is allowed to conflate the two concepts of crystal and true momentum and thereby gain direct knowledge of a crystal's band structure. 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 is a single electron's mass. Because crystal symmetry in the direction normal to the crystal surface is lost at the crystal boundary, crystal momentum in this direction is not conserved. Consequently, the only directions in which useful ARPES data can be gleaned are directions parallel to the crystal surface.
References
{{DEFAULTSORT:Crystal Momentum
Electronic band structures
Moment (physics)
Momentum