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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 ...
, the reciprocal lattice represents the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
of another lattice (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 usually a periodic spatial function in real-space and is also known as the ''direct lattice''. While the direct lattice exists in real-space and is what one would commonly understand as a physical lattice (e.g., a lattice of a crystal), the reciprocal lattice exists in reciprocal space (also known as '' momentum space'' or less commonly as ''K-space'', due to the relationship between the Pontryagin duals momentum and position). 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 very 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 atomic nucleus, nuclei of atoms. Since protons and ...
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 In crystallography and solid state physics, the Laue equations relate incoming waves to outgoing waves in the process of elastic scattering, where the photon energy or light temporal frequency does not change by scattering, by a crystal lattice. T ...
, 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 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 A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
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 A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
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 with unit amplitude can be written as an oscillatory term \cos, with initial phase \phi_0, angular wavenumber 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 ti ...
\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 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 languag ...
, 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 wavevectors \mathbf of plane waves in the
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
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 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 will \mathbf 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 is a
primitive cell In geometry, biology, mineralogy and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessari ...
(more specifically a Wigner-Seitz cell) of the reciprocal lattice, which plays an important role in
solid state physics Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the ...
due to Bloch's theorem. In
pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
, the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
of
linear form In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the ...
s 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 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 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 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 \ ...
, 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 In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
: \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. 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: :\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. 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 ''� ...
; 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 to the real space planes. The magnitude of the reciprocal lattice vector \mathbf_m is given in
reciprocal length Reciprocal length or inverse length is a quantity or measurement used in several branches of science and mathematics. As the reciprocal of length, common units used for this measurement include the reciprocal metre or inverse metre (symbol: m− ...
and is equal to the reciprocal of the interplanar spacing of the real space planes.


n 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 vector space V with a basis (\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 In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
: \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 Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
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 In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties ...
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 In geometry, biology, mineralogy and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessari ...
of side a, has for its reciprocal a simple cubic lattice with a cubic primitive cell of side \textstyle \frac (or \textstyle \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 \textstyle \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 lattice, with a cube side of \textstyle \frac. It can be easily 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 o ...
\textstyle a and \textstyle c is another simple hexagonal lattice with lattice constants \textstyle \frac and \textstyle \frac rotated through 30° 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 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/(2π) 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 r ...
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 ''L'' in a real vector space ''V'', of finite dimension. The first, which generalises directly the reciprocal lattice construction, uses
Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph ...
. It may be stated simply in terms of Pontryagin duality. 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 In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
''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 definit ...
it allows an identification of the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
''V''* of ''V'' with ''V''. The relation of ''V''* to ''V'' is not intrinsic; it depends on a choice of
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, thou ...
(volume element) on ''V''. But given an identification of the two, which is in any case well-defined up to a scalar, 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 ''L'' in an
abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
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 Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuou ...
, 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'' (franchi ...
''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

*
Crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics ( condensed matter physics). The wor ...
* Dual basis * Ewald's sphere * Miller index * Powder diffraction * Kikuchi line * Brillouin zone * Zone axis


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 ...
-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