Crystal momentum ( $ℏ k$ )

Crystal momentum is the translation quantum number

ℏ k

of a Bloch state. Since

k

and

k + G

give the same translation character whenever

G

belongs to the reciprocal lattice, crystal momentum is defined modulo reciprocal-lattice vectors.

Table of contents
- Switch-on drift in a circularly driven parabolic band Crystal momentum
- Crystal momentum conservation Crystal momentum

Switch-on drift in a circularly driven parabolic band ( $k = (e E_{0} /ℏ ω) (cos ω t - 1, - sin ω t)$ )

A charge

- e

starting at a band minimum when the field

E_{0} (sin ω t, cos ω t)

is applied acquires the displayed crystal momentum. Its orbit is a shifted circle. For masses

m_{x}, m_{y}

, the real-space coordinates are proportional to

(sin ω t - ω t) / m_{x}

and

(cos ω t - 1) / m_{y}

: an elliptic oscillation plus a uniform drift. Omitting the integration constant changes the physical initial condition and reduces the maximum light-axis kinetic energy by a factor of four.

Crystal momentum conservation ( $k_{1} + k_{2} = k_{3} + k_{4} (mod G)$ )

A lattice-translation-invariant interaction conserves total crystal momentum modulo a reciprocal lattice vector

G

. On a periodic chain of

M

sites with unit spacing, the discrete Fourier transform supplies

\sum_{j = 1}^{M} e^{i (k_{1} + k_{2} - k_{3} - k_{4}) j} = M δ_{k_{1} + k_{2}, k_{3} + k_{4} (mod 2 π)}

. This Fourier orthogonality proves the selection rule directly; it permits transfer by a reciprocal lattice vector rather than requiring equality of unrestricted momenta.

 Ancestors (6)

 Incoming links (55)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (1)

Crystal momentum by

Wikipedia Bot  1

Crystal momentum is a concept used in solid-state physics that refers to the effective momentum of particles (such as electrons) in a crystalline solid. It arises from the periodic potential of the crystal lattice in which the particles reside. In quantum mechanics, particles exhibit wave-like properties, leading to the concept of wave vectors.

 Read the full article

  See all articles in the same topic Create my own version