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In crystallography, a lattice plane of a given Bravais lattice is any plane containing at least three noncollinear Bravais lattice points. Equivalently, a lattice plane is a plane whose intersections with the lattice (or any crystalline structure of that lattice) are periodic (i.e. are described by 2d Bravais lattices). [1]
This is based on the fact that a reciprocal lattice vector (the vector indicating a reciprocal lattice point from the reciprocal lattice origin) is the wavevector of a plane wave in the Fourier series of a spatial function (e.g., electronic density function) which periodicity follows the original Bravais lattice, so wavefronts of the plane wave ...
The structure of a crystal is often represented by the pole figure of its crystallographic planes. A plane is chosen as the equator, usually the (001) or (011) plane; its pole is the center of the figure. Then, the poles of the other planes are placed on the figure, with the Miller indices for each pole. The poles that belong to a zone are ...
Vectors and planes in a crystal lattice are described by the three-value Miller index notation. This syntax uses the indices h, k, and ℓ as directional parameters. [4] By definition, the syntax (hkℓ) denotes a plane that intercepts the three points a 1 /h, a 2 /k, and a 3 /ℓ, or some multiple thereof. That is, the Miller indices are ...
The attitude of a lattice plane is the orientation of the line normal to the plane, [12] and is described by the plane's Miller indices. In three-space a family of planes (a series of parallel planes) can be denoted by its Miller indices (hkl), [13] [14] so the family of planes has an attitude common to all its constituent planes.
Reciprocal space (also called k-space) provides a way to visualize the results of the Fourier transform of a spatial function. It is similar in role to the frequency domain arising from the Fourier transform of a time dependent function; reciprocal space is a space over which the Fourier transform of a spatial function is represented at spatial frequencies or wavevectors of plane waves of the ...
Tetragonal crystal lattices result from stretching a cubic lattice along one of its lattice vectors, so that the cube becomes a rectangular prism with a square base (a by a) and height (c, which is different from a).
In geometry and group theory, a lattice in the real coordinate space is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point.