Reciprocal Lattice Vector
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In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, the reciprocal lattice represents the Fourier transform of another
lattice (group) In geometry and group theory, a lattice in the real coordinate space \mathbb^n 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 poi ...
(usually a
Bravais lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial function in real space known as the ''direct lattice''. While the direct lattice exists in real space and is commonly understood to be a physical lattice (such as the lattice of a crystal), the reciprocal lattice exists in the space of
spatial frequencies In mathematics, physics, and engineering, spatial frequency is a characteristic of any structure that is periodic across position in space. The spatial frequency is a measure of how often sinusoidal components (as determined by the Fourier tra ...
known as reciprocal space or k space, where \mathbf refers to the
wavevector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
. In
quantum physics 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 ...
, reciprocal space is closely related to
momentum space In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension. Position space (also real space or coordinate space) is the set of all ''position vectors'' r in space, and h ...
according to the proportionality \mathbf = \hbar \mathbf, where \mathbf is the momentum vector and \hbar is the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
. The reciprocal lattice of a reciprocal lattice is equivalent to the original direct lattice, because the defining equations are symmetrical with respect to the vectors in real and reciprocal space. Mathematically, direct and reciprocal lattice vectors represent covariant and contravariant vectors, respectively. The reciprocal lattice is the set of all vectors \mathbf_m, that are wavevectors of plane waves in the Fourier series of a spatial function whose periodicity is the same as that of a direct lattice \mathbf_n. Each plane wave in this Fourier series has the same phase or phases that are differed by multiples of 2\pi at each direct lattice point (so essentially same phase at all the direct lattice points). The reciprocal lattice plays a fundamental role in most analytic studies of periodic structures, particularly in the theory of diffraction. In
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beh ...
,
helium Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas and the first in the noble gas group in the periodic table. ...
and
X-ray An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10  picometers to 10  nanometers, corresponding to frequencies in the range 30&nb ...
diffraction Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
, due to the Laue conditions, the momentum difference between incoming and diffracted X-rays of a crystal is a reciprocal lattice vector. The diffraction pattern of a crystal can be used to determine the reciprocal vectors of the lattice. Using this process, one can infer the atomic arrangement of a crystal. The
Brillouin zone In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice ...
is a Wigner-Seitz cell of the reciprocal lattice.


Wave-based description


Reciprocal space

Reciprocal space (also called -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 In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
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 Fourier transform. The domain of the spatial function itself is often referred to as real space. In physical applications, such as crystallography, both real and reciprocal space will often each be two or three dimensional. Whereas spatial dimensions of these two associated spaces will be the same, the spaces will differ in their units of length, so that when the real space has units of length L, its reciprocal space will have units of one divided by the length L so L−1 (the reciprocal of length). Reciprocal space comes into play regarding waves, both classical and quantum mechanical. Because a
sinusoidal plane wave In physics, a sinusoidal (or monochromatic) plane wave is a special case of plane wave: a field whose value varies as a sinusoidal function of time and of the distance from some fixed plane. For any position \vec x in space and any time t, the val ...
with unit amplitude can be written as an oscillatory term \cos, with initial
phase Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform * Phase space, a mathematic ...
\phi_0,
angular wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
k and
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
\omega, it can be regarded as a function of both k and x (and the time-varying part as a function of both \omega and t). This complementary role of k and x leads to their visualization within complementary spaces (the real space and the reciprocal space). The spatial periodicity of this wave is defined by its wavelength \lambda, where k \lambda = 2\pi; hence the corresponding wavenumber in reciprocal space will be k = 2\pi / \lambda. In three dimensions, the corresponding plane wave term becomes \cos, which simplifies to \cos at a fixed time t, where \mathbf is the position vector of a point in real space and now \mathbf=2\pi \mathbf / \lambda is the
wavevector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
in the three dimensional reciprocal space. (The magnitude of a wavevector is called wavenumber.) The constant \phi is the phase of the
wavefront In physics, the wavefront of a time-varying '' wave field'' is the set ( locus) of all points having the same '' phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal fr ...
(a plane of a constant phase) through the origin \mathbf=0 at time t, and \mathbf is a unit vector perpendicular to this wavefront. The wavefronts with phases \phi + (2\pi)n, where n represents any
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
, comprise a set of parallel planes, equally spaced by the wavelength \lambda.


Reciprocal lattice

In general, a geometric lattice is an infinite, regular array of vertices (points) in space, which can be modelled vectorially as a
Bravais lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
. Some lattices may be skew, which means that their primary lines may not necessarily be at right angles. In reciprocal space, a reciprocal lattice is defined as the set of
wavevector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
s \mathbf of plane waves in the Fourier series of any function f(\mathbf) whose periodicity is compatible with that of an initial direct lattice in real space. Equivalently, a wavevector is a vertex of the reciprocal lattice if it corresponds to a plane wave in real space whose phase at any given time is the same (actually differs by (2\pi)n with an integer n) at every direct lattice vertex. One heuristic approach to constructing the reciprocal lattice in three dimensions is to write the position vector of a vertex of the direct lattice as \mathbf = n_\mathbf_ n_\mathbf_ n_\mathbf_, where the n_i are integers defining the vertex and the \mathbf_i are
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
primitive translation vectors (or shortly called primitive vectors) that are characteristic of the lattice. There is then a unique plane wave (up to a factor of negative one), whose wavefront through the origin \mathbf = 0 contains the direct lattice points at \mathbf_2 and \mathbf_3, and with its adjacent wavefront (whose phase differs by 2\pi or -2\pi from the former wavefront passing the origin) passing through \mathbf_1. Its angular wavevector takes the form \mathbf_1 = 2\pi \mathbf_1 / \lambda_, where \mathbf_1 is the unit vector perpendicular to these two adjacent wavefronts and the wavelength \lambda_1 must satisfy \lambda_1 = \mathbf_1 \cdot \mathbf_1, means that \lambda_1 is equal to the distance between the two wavefronts. Hence by construction \mathbf_1 \cdot \mathbf_1 = 2\pi and \mathbf_2 \cdot \mathbf_1 = \mathbf_3 \cdot \mathbf_1 = 0. Cycling through the indices in turn, the same method yields three wavevectors \mathbf_j with \mathbf_i \cdot \mathbf_j = 2\pi \, \delta_, where the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
\delta_ equals one when i=j and is zero otherwise. The \mathbf_j comprise a set of three primitive wavevectors or three primitive translation vectors for the reciprocal lattice, each of whose vertices takes the form \mathbf = m_\mathbf_ m_\mathbf_ m_\mathbf_, where the m_j are integers. The reciprocal lattice is also a
Bravais lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
as it is formed by integer combinations of the primitive vectors, that are \mathbf_1, \mathbf_2, and \mathbf_3 in this case. Simple algebra then shows that, for any plane wave with a wavevector \mathbf on the reciprocal lattice, the total phase shift \mathbf \cdot \mathbf between the origin and any point \mathbf on the direct lattice is a multiple of 2\pi (that can be possibly zero if the multiplier is zero), so the phase of the plane wave with \mathbf will essentially be equal for every direct lattice vertex, in conformity with the reciprocal lattice definition above. (Although any wavevector \mathbf on the reciprocal lattice does always take this form, this derivation is motivational, rather than rigorous, because it has omitted the proof that no other possibilities exist.) The
Brillouin zone In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice ...
is a primitive cell (more specifically a Wigner-Seitz cell) of the reciprocal lattice, which plays an important role in solid state physics due to
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 ...
. In pure mathematics, the dual space of linear forms and the
dual lattice In the theory of lattices, the dual lattice is a construction analogous to that of a dual vector space. In certain respects, the geometry of the dual lattice of a lattice L is the reciprocal of the geometry of L , a perspective which underlie ...
provide more abstract generalizations of reciprocal space and the reciprocal lattice.


Mathematical description

Assuming a three-dimensional
Bravais lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
and labelling each lattice vector (a vector indicating a lattice point) by the subscript n = (n_1, n_2, n_3) as 3-tuple of integers, :\mathbf_n = n_1 \mathbf_1 + n_2 \mathbf_2 + n_3 \mathbf_3 where n_1, n_2, n_3 \in \mathbb where \mathbb is the set of integers and \mathbf_i is a primitive translation vector or shortly primitive vector. Taking a function f(\mathbf) where \mathbf is a position vector from the origin \mathbf_n = 0 to any position, if f(\mathbf) follows the periodicity of this lattice, e.g. the function describing the electronic density in an atomic crystal, it is useful to write f(\mathbf) as a multi-dimensional Fourier series :\sum_m = f\left(\mathbf\right) where now the subscript m = (m_1, m_2, m_3), so this is a triple sum. As f(\mathbf) follows the periodicity of the lattice, translating \mathbf by any lattice vector \mathbf_n we get the same value, hence :f(\mathbf + \mathbf_n) = f(\mathbf). Expressing the above instead in terms of their Fourier series we have \sum_m = \sum_m = \sum_m . Because equality of two Fourier series implies equality of their coefficients, e^ = 1, which only holds when :\mathbf_m \cdot \mathbf_n = 2\pi N where N \in \mathbb. Mathematically, the reciprocal lattice is the set of all
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 ...
s \mathbf_m, that are wavevectors of plane waves in the Fourier series of a spatial function whose periodicity is the same as that of a direct lattice as the set of all direct lattice point position vectors \mathbf_n, and \mathbf_m satisfy this equality for all \mathbf_n. Each plane wave in the Fourier series has the same phase (actually can be differed by a multiple of 2\pi) at all the lattice point \mathbf_n. As shown in the section multi-dimensional Fourier series, \mathbf_m can be chosen in the form of \mathbf_m = m_1 \mathbf_1 + m_2 \mathbf_2 + m_3 \mathbf_3 where \mathbf_i \cdot \mathbf_j = 2\pi \, \delta_. With this form, the reciprocal lattice as the set of all wavevectors \mathbf_m for the Fourier series of a spatial function which periodicity follows \mathbf_n, is itself a Bravais lattice as it is formed by integer combinations of its own primitive translation vectors \left(\mathbf, \mathbf_2, \mathbf_3\right), and the reciprocal of the reciprocal lattice is the original lattice, which reveals the
Pontryagin duality In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numb ...
of their respective
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s. (There may be other form of \mathbf_m. Any valid form of \mathbf_m results in the same reciprocal lattice.)


Two dimensions

For an infinite two-dimensional lattice, defined by its primitive vectors \left(\mathbf_1, \mathbf_2\right), its reciprocal lattice can be determined by generating its two reciprocal primitive vectors, through the following formulae, :\mathbf_m = m_1 \mathbf_1 + m_2 \mathbf_2 where m_i is an integer and :\begin \mathbf_1 &= 2\pi \frac = 2\pi \frac \\ \mathbf_2 &= 2\pi \frac \end Here \mathbf represents a 90 degree
rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \en ...
, i.e. a ''q''uarter turn. The anti-clockwise rotation and the clockwise rotation can both be used to determine the reciprocal lattice: If \mathbf is the anti-clockwise rotation and \mathbf is the clockwise rotation, \mathbf\,\mathbf=-\mathbf\,\mathbf for all vectors \mathbf. Thus, using the permutation : \sigma = \begin 1 & 2 \\ 2 & 1 \end we obtain :\begin \mathbf_n = 2\pi \frac=2\pi \frac. \end Notably, in a 3D space this 2D reciprocal lattice is an infinitely extended set of Bragg rods—described by Sung et al.


Three dimensions

For an infinite three-dimensional lattice \mathbf_n = n_1 \mathbf_1 + n_2 \mathbf_2 + n_3 \mathbf_3, defined by its primitive vectors \left(\mathbf, \mathbf_2, \mathbf_3\right) and the subscript of integers n = \left( n_1, n_2, n_3 \right), its reciprocal lattice \mathbf_m = m_1 \mathbf_1 + m_2 \mathbf_2 + m_3 \mathbf_3 with the integer subscript m = (m_1, m_2, m_3) can be determined by generating its three reciprocal primitive vectors \left(\mathbf, \mathbf_2, \mathbf_3\right) \begin \mathbf_1 &= \frac \ \mathbf_2 \times \mathbf_3 \\ \mathbf_2 &= \frac \ \mathbf_3 \times \mathbf_1 \\ \mathbf_3 &= \frac \ \mathbf_1 \times \mathbf_2 \end where V = \mathbf_1 \cdot \left(\mathbf_2 \times \mathbf_3\right) = \mathbf_2 \cdot \left(\mathbf_3 \times \mathbf_1\right) = \mathbf_3 \cdot \left(\mathbf_1 \times \mathbf_2\right) is the
scalar triple product In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector- ...
. The choice of these \left(\mathbf, \mathbf_2, \mathbf_3\right) is to satisfy \mathbf_i \cdot \mathbf_j = 2\pi \, \delta_ as the known condition (There may be other condition.) of primitive translation vectors for the reciprocal lattice derived in the heuristic approach above and the section multi-dimensional Fourier series. This choice also satisfies the requirement of the reciprocal lattice e^ = 1 mathematically derived above. Using column vector representation of (reciprocal) primitive vectors, the formulae above can be rewritten using
matrix inversion In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplicati ...
: :\left mathbf_1\mathbf_2\mathbf_3\right\mathsf = 2\pi\left mathbf_1\mathbf_2\mathbf_3\right. This method appeals to the definition, and allows generalization to arbitrary dimensions. The cross product formula dominates introductory materials on crystallography. The above definition is called the "physics" definition, as the factor of 2 \pi comes naturally from the study of periodic structures. An essentially equivalent definition, the "crystallographer's" definition, comes from defining the reciprocal lattice \mathbf_m = \mathbf_m / 2\pi. which changes the reciprocal primitive vectors to be : \mathbf_1 = \frac and so on for the other primitive vectors. The crystallographer's definition has the advantage that the definition of \mathbf_1 is just the reciprocal magnitude of \mathbf_1 in the direction of \mathbf_2 \times \mathbf_3, dropping the factor of 2 \pi. This can simplify certain mathematical manipulations, and expresses reciprocal lattice dimensions in units of
spatial frequency In mathematics, physics, and engineering, spatial frequency is a characteristic of any structure that is periodic across position in space. The spatial frequency is a measure of how often sinusoidal components (as determined by the Fourier tra ...
. It is a matter of taste which definition of the lattice is used, as long as the two are not mixed. m = (m_1, m_2, m_3) is conventionally written as (h, k, l) or (hkl), called
Miller indices Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices. In particular, a family of lattice planes of a given (direct) Bravais lattice is determined by three integers ''h'', ''k'', and ''â ...
; m_1 is replaced with h, m_2 replaced with k, and m_3 replaced with l. Each lattice point (hkl) in the reciprocal lattice corresponds to a set of lattice planes (hkl) in the real space lattice. (A lattice plane is a plane crossing lattice points.) The direction of the reciprocal lattice vector corresponds to the
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
to the real space planes. The magnitude of the reciprocal lattice vector \mathbf_m is given in reciprocal length and is equal to the reciprocal of the interplanar spacing of the real space planes.


dimensions

The formula for n dimensions can be derived assuming an n-
dimensional In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
vector space V with a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
(\mathbf_1,\ldots,\mathbf_n) and an inner product g\colon V\times V\to\mathbf. The reciprocal lattice vectors are uniquely determined by the formula g(\mathbf_i,\mathbf_j)=2\pi\delta_. Using the permutation : \sigma = \begin 1 & 2 & \cdots &n\\ 2 & 3 & \cdots &1 \end, they can be determined with the following formula: : \mathbf_i = 2\pi\fracg^(\mathbf_\,\lrcorner\ldots\mathbf_\,\lrcorner\,\omega)\in V Here, \omega\colon V^n \to \mathbf is the
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of th ...
, g^ is the inverse of the vector space isomorphism \hat\colon V \to V^* defined by \hat(v)(w) = g(v,w) and \lrcorner denotes the inner multiplication. One can verify that this formula is equivalent to the known formulas for the two- and three-dimensional case by using the following facts: In three dimensions, \omega(u,v,w) = g(u \times v, w) and in two dimensions, \omega(v,w) = g(Rv,w), where R \in \text(2) \subset L(V,V) is the rotation by 90 degrees (just like the volume form, the angle assigned to a rotation depends on the choice of orientation).


Reciprocal lattices of various crystals

Reciprocal lattices for the cubic crystal system are as follows.


Simple cubic lattice

The simple cubic
Bravais lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
, with cubic primitive cell of side a, has for its reciprocal a simple cubic lattice with a cubic primitive cell of side \frac (or \frac in the crystallographer's definition). The cubic lattice is therefore said to be self-dual, having the same symmetry in reciprocal space as in real space.


Face-centered cubic (FCC) lattice

The reciprocal lattice to an FCC lattice is the body-centered cubic (BCC) lattice, with a cube side of \frac. Consider an FCC compound unit cell. Locate a primitive unit cell of the FCC; i.e., a unit cell with one lattice point. Now take one of the vertices of the primitive unit cell as the origin. Give the basis vectors of the real lattice. Then from the known formulae, you can calculate the basis vectors of the reciprocal lattice. These reciprocal lattice vectors of the FCC represent the basis vectors of a BCC real lattice. Note that the basis vectors of a real BCC lattice and the reciprocal lattice of an FCC resemble each other in direction but not in magnitude.


Body-centered cubic (BCC) lattice

The reciprocal lattice to a BCC lattice is the
FCC The Federal Communications Commission (FCC) is an independent agency of the United States federal government that regulates communications by radio, television, wire, satellite, and cable across the United States. The FCC maintains jurisdictio ...
lattice, with a cube side of \frac. It can be proven that only the Bravais lattices which have 90 degrees between \left(\mathbf_1, \mathbf_2, \mathbf_3\right) (cubic, tetragonal, orthorhombic) have primitive translation vectors for the reciprocal lattice, \left(\mathbf_1, \mathbf_2, \mathbf_3\right), parallel to their real-space vectors.


Simple hexagonal lattice

The reciprocal to a simple hexagonal Bravais lattice with
lattice constants A lattice constant or lattice parameter is one of the physical dimensions and angles that determine the geometry of the unit cells in a crystal lattice, and is proportional to the distance between atoms in the crystal. A simple cubic crystal has ...
a and c is another simple hexagonal lattice with lattice constants \frac and \frac rotated through 90° about the c axis with respect to the direct lattice. The simple hexagonal lattice is therefore said to be self-dual, having the same symmetry in reciprocal space as in real space. Primitive translation vectors for this simple hexagonal Bravais lattice vectors are a_1 = (3^a/2)\hat + (a/2)\hat, a_2 = - (3^a/2)\hat + (a/2)\hat, and a_3 = c \hat.


Arbitrary collection of atoms

One path to the reciprocal lattice of an arbitrary collection of atoms comes from the idea of scattered waves in the Fraunhofer (long-distance or lens back-focal-plane) limit as a Huygens-style sum of amplitudes from all points of scattering (in this case from each individual atom).B. E. Warren (1969/1990) ''X-ray diffraction'' (Addison-Wesley, Reading MA/Dover, Mineola NY). This sum is denoted by the
complex amplitude Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
F in the equation below, because it is also the Fourier transform (as a function of spatial frequency or reciprocal distance) of an effective scattering potential in direct space: :F
vec Vec may mean: Mathematics: * vec(''A''), the vectorization of a matrix ''A''. * Vec denotes the category of vector spaces over the reals. Other: * Venetian language (Vèneto), language code. * Vecuronium, a muscle relaxant. * vec, a sentient mora ...
= \sum_^N f_j\!\left vec\righte^. Here g = q/(2Ï€) is the scattering vector q in crystallographer units, ''N'' is the number of atoms, ''f''''j'' ''gis the
atomic scattering factor In physics, the atomic form factor, or atomic scattering factor, is a measure of the scattering amplitude of a wave by an isolated atom. The atomic form factor depends on the type of scattering, which in turn depends on the nature of the incident ...
for atom ''j'' and scattering vector g, while r''j'' is the vector position of atom ''j''. Note that the Fourier phase depends on one's choice of coordinate origin. For the special case of an infinite periodic crystal, the scattered amplitude F = M Fhkl from M unit cells (as in the cases above) turns out to be non-zero only for integer values of (hkl), where :F_ = \sum_^m f_j\left _\righte^ when there are j=1,m atoms inside the unit cell whose fractional lattice indices are respectively . To consider effects due to finite crystal size, of course, a shape convolution for each point or the equation above for a finite lattice must be used instead. Whether the array of atoms is finite or infinite, one can also imagine an "intensity reciprocal lattice" I ''g which relates to the amplitude lattice F via the usual relation I = F*F where F* is the complex conjugate of F. Since Fourier transformation is reversible, of course, this act of conversion to intensity tosses out "all except 2nd moment" (i.e. the phase) information. For the case of an arbitrary collection of atoms, the intensity reciprocal lattice is therefore: :I
vec Vec may mean: Mathematics: * vec(''A''), the vectorization of a matrix ''A''. * Vec denotes the category of vector spaces over the reals. Other: * Venetian language (Vèneto), language code. * Vecuronium, a muscle relaxant. * vec, a sentient mora ...
= \sum_^N \sum_^N f_j \left vec\rightf_k \left vec\righte^. Here r''jk'' is the vector separation between atom ''j'' and atom ''k''. One can also use this to predict the effect of nano-crystallite shape, and subtle changes in beam orientation, on detected diffraction peaks even if in some directions the cluster is only one atom thick. On the down side, scattering calculations using the reciprocal lattice basically consider an incident plane wave. Thus after a first look at reciprocal lattice (kinematic scattering) effects, beam broadening and multiple scattering (i.e.
dynamical In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...
) effects may be important to consider as well.


Generalization of a dual lattice

There are actually two versions in
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
of the abstract dual lattice concept, for a given
lattice 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 orna ...
''L'' in a real vector space ''V'', of finite dimension. The first, which generalises directly the reciprocal lattice construction, uses Fourier analysis. It may be stated simply in terms of
Pontryagin duality In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numb ...
. The dual group ''V''^ to ''V'' is again a real vector space, and its closed subgroup ''L''^ dual to ''L'' turns out to be a lattice in ''V''^. Therefore, ''L''^ is the natural candidate for ''dual lattice'', in a different vector space (of the same dimension). The other aspect is seen in the presence of a quadratic form ''Q'' on ''V''; if it is
non-degenerate In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'' ) given by is not an isomorphism. An equivalent defin ...
it allows an identification of the dual space ''V''* of ''V'' with ''V''. The relation of ''V''* to ''V'' is not intrinsic; it depends on a choice of Haar measure (volume element) on ''V''. But given an identification of the two, which is in any case
well-defined In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A func ...
up to a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
, the presence of ''Q'' allows one to speak to the dual lattice to ''L'' while staying within ''V''. In mathematics, the dual lattice of a given
lattice 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 orna ...
''L'' in an abelian locally compact
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
''G'' is the subgroup ''L''∗ of the dual group of ''G'' consisting of all continuous characters that are equal to one at each point of ''L''. In discrete mathematics, a lattice is a locally discrete set of points described by all integral linear combinations of dim = ''n'' linearly independent vectors in R''n''. The dual lattice is then defined by all points in the linear span of the original lattice (typically all of R''n'') with the property that an integer results from the inner product with all elements of the original lattice. It follows that the dual of the dual lattice is the original lattice. Furthermore, if we allow the
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
''B'' to have columns as the linearly independent vectors that describe the lattice, then the matrix A = B\left(B^\mathsf B\right)^ has columns of vectors that describe the dual lattice.


See also

*
Brillouin zone In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice ...
* Crystallography *
Dual basis In linear algebra, given a vector space ''V'' with a basis ''B'' of vectors indexed by an index set ''I'' (the cardinality of ''I'' is the dimension of ''V''), the dual set of ''B'' is a set ''B''∗ of vectors in the dual space ''V''∗ with the ...
*
Ewald's sphere The Ewald sphere is a geometric construction used in electron, neutron, and X-ray crystallography which demonstrates the relationship between: :* the wavevector of the incident and diffracted x-ray beams, :* the diffraction angle for a given ref ...
* Kikuchi line *
Miller index Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices. In particular, a family of lattice planes of a given (direct) Bravais lattice is determined by three integers ''h'', ''k'', and ''â ...
*
Powder diffraction Powder diffraction is a scientific technique using X-ray, neutron, or electron diffraction on powder or microcrystalline samples for structural characterization of materials. An instrument dedicated to performing such powder measurements is call ...
*
Zone axis Zone axis, a term sometimes used to refer to "high-symmetry" orientations in a crystal, most generally refers to ''any'' direction referenced to the direct lattice (as distinct from the reciprocal lattice) of a crystal in three dimensions. It is ...


References


External links

* http://newton.umsl.edu/run//nano/known.html -
Jmol Jmol is computer software for molecular modelling chemical structures in 3-dimensions. Jmol returns a 3D representation of a molecule that may be used as a teaching tool, or for research e.g., in chemistry and biochemistry. It is written in the ...
-based electron diffraction simulator lets you explore the intersection between reciprocal lattice and Ewald sphere during tilt.
DoITPoMS Teaching and Learning Package on Reciprocal Space and the Reciprocal Lattice


{{DEFAULTSORT:Reciprocal Lattice Crystallography Fourier analysis Lattice points Neutron-related techniques Synchrotron-related techniques Diffraction Condensed matter physics