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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 ...
That is, (hkℓ) simply indicates a normal to the planes in the basis of the primitive reciprocal lattice vectors. Because the coordinates are integers, this normal is itself always a reciprocal lattice vector. The requirement of lowest terms means that it is the shortest reciprocal lattice vector in the given direction.
The translational invariance of a crystal lattice is described by a set of unit cell, direct lattice basis vectors (contravariant [1] or polar) called a, b, and c, or equivalently by the lattice parameters, i.e. the magnitudes of the vectors, called a, b and c, and the angles between them, called α (between b and c), β (between c and a), and γ (between a and b).
While the Bragg formulation assumes a unique choice of direct lattice planes and specular reflection of the incident X-rays, the Von Laue formula only assumes monochromatic light and that each scattering center acts as a source of secondary wavelets as described by the Huygens principle. Each scattered wave contributes to a new plane wave given by:
The boundaries of this cell are given by planes related to points on the reciprocal lattice. The importance of the Brillouin zone stems from the description of waves in a periodic medium given by Bloch's theorem, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone.
If it does span , then is called the dual basis or reciprocal basis for the basis . Denoting the indexed vector sets as B = { v i } i ∈ I {\displaystyle B=\{v_{i}\}_{i\in I}} and B ∗ = { v i } i ∈ I {\displaystyle B^{*}=\{v^{i}\}_{i\in I}} , being biorthogonal means that the elements pair to have an inner product equal to 1 if the indexes ...
Fig. 1: A hexagonal sampling lattice and its basis vectors v 1 and v 2 Fig. 2: The reciprocal lattice corresponding to the lattice of Fig. 1 and its basis vectors u 1 and u 2 (figure not to scale). The concept of a bandlimited function in one dimension can be generalized to the notion of a wavenumber-limited function in higher dimensions.
For each Bravais lattice vector we define a translation operator ^ which, when operating on any function () shifts the argument by : ^ = (+) Since all translations form an Abelian group, the result of applying two successive translations does not depend on the order in which they are applied, i.e. ^ ^ = ^ ^ = ^ + In addition, as the Hamiltonian ...