Pcristal = h.K
(where h is the reduced Planck's constant). Like regular momentum, crystal momentum frequently exhibits the property of being conserved, and is thus extraordinarily 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 V(x) such that
V(x+a) = V(x)
where a 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 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 a 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 T(a) commutes with the Hamiltonian, assuming a simple kinetic-plus-potential form.)
These conditions imply Bloch's theorem, which states in terms of equations that
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 ψ(x), finds its stationary state solutions in the form of a plane wave multiplied by a periodic function u(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 k, 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:
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 k, 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:
Pcristal = h.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, i.e., an electron can be described not only by the wave vector k, but also with any other wave vector k' such thatk* = k + K
where 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.
It is tempting to treat a phonon with wave vector as though it has a momentum, by analogy to photons and matter waves. This is not entirely correct, for is not actually a physical momentum; it is called the crystal momentum or pseudomomentum. This is because is only determined up to multiples of constant vectors, known as reciprocal lattice vectors. For example, in our one-dimensional model, the normal coordinates and are defined so that
It is usually convenient to consider phonon wave vectors which have the smallest magnitude in their "family". The set of all such wave vectors defines the first Brillouin zone. Additional Brillouin zones may be defined as copies of the first zone, shifted by some reciprocal lattice vector.
It is interesting that similar consideration is needed in analog-to-digital conversion where aliasing may occur under certain conditions.
It is interesting that similar consideration is needed in analog-to-digital conversion where aliasing may occur under certain conditions.
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