Reciprocal lattice

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
The computer-generated reciprocal lattice of a fictional monoclinic 3D crystal.
A two-dimensional crystal and its reciprocal lattice

In physics, the reciprocal lattice represents the Fourier transform of another lattice (usually a Bravais lattice). 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, 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 plays a very fundamental role in most analytic studies of periodic structures, particularly in the theory of diffraction. In neutron and X-ray diffraction, 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 is a Wigner-Seitz cell of the reciprocal lattice.

Wave-based description[edit]

Reciprocal space[edit]

Reciprocal space (also called k-space) provides a way to visualise 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. 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 these spatial dimensions will be the same, the spaces will differ in their units, so that when the real space has units of length L, its reciprocal space will have corresponding units of one divided by length 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 , with initial phase , angular wavenumber and angular frequency , it can be regarded as a function of both and (and the time-varying part as a function of both and ). This complementary role of and leads to their visualisation within complementary spaces. The spatial periodicity of this wave is defined by its wavelength , where ; hence the corresponding wavenumber in reciprocal space will be .

In three dimensions, the corresponding plane wave term becomes , which simplifies to at a fixed time , where is the position vector of a point in real space and now is the wavevector in the three dimensional reciprocal space. The constant is the phase of the wavefront (plane of constant phase) through the origin at time , and is a unit vector perpendicular to this wavefront. The wavefronts with phase comprise a set of parallel planes, equally spaced by the wavelength .

Reciprocal lattice[edit]

In general, a geometric lattice consists of an infinite, regular array of vertices in space, which can be modelled vectorially as a Bravais lattice. Some lattices may be skew, which means that their primary lines may not necessarily be at right angles. A reciprocal lattice is the periodic set of the wave vectors in reciprocal space that make up the Fourier series of any function 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 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 , where the are integers defining the vertex and the are linearly independent primitive vectors characteristic of the lattice. There is then a unique plane wave (up to a factor of negative one), whose wavefront through the origin contains the direct lattice points at and , and with its adjacent wavefront passing through . Its angular wavevector takes the form , where is the unit vector perpendicular to these two wavefronts and the wavelength must satisfy . Hence by construction and .

Cycling through the indices in turn, the same method yields three wavevectors with , where the Kronecker delta equals one when and is zero otherwise. The comprise a set of three primitive wavevectors for the reciprocal lattice, each of whose vertices takes the form , where the are integers. Simple algebra then shows that, for any plane wave with wavevector on the reciprocal lattice, the total phase shift between the origin and any point on the direct lattice is a (possibly zero) multiple of , so the phase will indeed be equal for every direct lattice vertex, in conformity with the reciprocal lattice definition above. (Although any wavevector 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 (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 pure mathematics, the dual space of linear forms and the dual lattice provide more abstract generalisations of reciprocal space and the reciprocal lattice.

Mathematical description[edit]

Demonstration of relation between real and reciprocal lattice. A real space 2D lattice (red dots) with primitive vectors and are shown by blue and green arrows respectively. Atop, plane waves of the form are plotted. From this we see that when is any integer combination of reciprocal lattice vector basis and (i.e. any reciprocal lattice vector), the resulting plane waves have the same periodicity of the lattice - that is that any translation from point (shown orange) to a point ( shown red), the value of the plane wave is the same. These plane waves can be added together and the above relation will still apply.

Assuming a two-dimensional Bravais lattice and labelling each lattice vector by the subscript

where .

Taking a function where is a vector from the origin to any position, if follows the periodicity of the lattice, e.g. the electronic density in an atomic crystal, it is useful to write as a multi-dimensional Fourier series

where now the subscript so this is a double sum.

As follows the periodicity of the lattice, translating by any lattice vector we get the same value, hence

Expressing the above instead in terms of their Fourier series we have

Because equality of two Fourier series implies equality of their coefficients, , which only holds when

where

This criterion restricts the values of to vectors that satisfy this relation. Mathematically, the reciprocal lattice is the set of all vectors that satisfy the above identity for all lattice point position vectors . As such, any function which exhibits the same periodicity of the lattice can be expressed as a Fourier series with angular frequencies taken from the reciprocal lattice.

This reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice, which reveals the Pontryagin duality of their respective vector spaces.

Two dimensions[edit]

For an infinite two-dimensional lattice, defined by its primitive vectors , its reciprocal lattice can be determined by generating its two reciprocal primitive vectors, through the following formulae,

Where,

Here represents a 90 degree rotation matrix, i.e. a quarter turn. The anti-clockwise rotation and the clockwise rotation can both be used to determine the reciprocal lattice: If is the anti-clockwise rotation and is the clockwise rotation, for all vectors . Thus, using the permutation

we obtain

Three dimensions[edit]

For an infinite three-dimensional lattice, defined by its primitive vectors , its reciprocal lattice can be determined by generating its three reciprocal primitive vectors, through the formulae

where the subscript in three dimensions, and for the scalar triple product :

Using column vector representation of (reciprocal) primitive vectors, the formulae above can be rewritten using matrix inversion:

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 comes naturally from the study of periodic structures. An equivalent definition, the "crystallographer's" definition, comes from defining the reciprocal lattice to be which changes the definitions of the reciprocal lattice vectors to be

and so on for the other vectors. The crystallographer's definition has the advantage that the definition of is just the reciprocal magnitude of in the direction of , dropping the factor of . 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.

Each point in the reciprocal lattice corresponds to a set of lattice planes in the real space lattice. The direction of the reciprocal lattice vector corresponds to the normal to the real space planes. The magnitude of the reciprocal lattice vector is given in reciprocal length and is equal to the reciprocal of the interplanar spacing of the real space planes.

dimensions[edit]

The formula for dimensions can be derived assuming an -dimensional real vector space with a basis and an inner product . The reciprocal lattice vectors are uniquely determined by the formula . Using the permutation

they can be determined with the following formula:

Here, is the volume form, is the inverse of the vector space isomorphism defined by and 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, and in two dimensions, , where is the rotation by 90 degrees (just like the volume form, the angle assigned to a rotation depends on the choice of orientation[1]).

Reciprocal lattices of various crystals[edit]

Reciprocal lattices for the cubic crystal system are as follows.

Simple cubic lattice[edit]

The simple cubic Bravais lattice, with cubic primitive cell of side , has for its reciprocal a simple cubic lattice with a cubic primitive cell of side ( 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[edit]

The reciprocal lattice to an FCC lattice is the body-centered cubic (BCC) lattice, with a cube side of .

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[edit]

The reciprocal lattice to a BCC lattice is the FCC lattice, with a cube side of .

It can be easily proven that only the Bravais lattices which have 90 degrees between (cubic, tetragonal, orthorhombic) have parallel to their real-space vectors.

Simple hexagonal lattice[edit]

The reciprocal to a simple hexagonal Bravais lattice with lattice constants c and a is another simple hexagonal lattice with lattice constants and 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. vectors a 1 = (a(3) 1/2 /2) i + (a/2) j ; a 2 = -(a(3) 1/2 /2) i + (a/2) j ve a 3 = a k

Arbitrary collection of atoms[edit]

Shadow of a 118-atom faceted carbon-pentacone's intensity reciprocal-lattice lighting up red in diffraction when intersecting the Ewald sphere.

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).[2] 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:

Here g = q/(2π) is the scattering vector q in crystallographer units, N is the number of atoms, fj[g] is the atomic scattering factor for atom j and scattering vector g, while rj 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 , where

when there are j=1,m atoms inside the unit cell whose fractional lattice indices are respectively {uj, vj, wj}. 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:

Here rjk 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) effects may be important to consider as well.

Generalization of a dual lattice[edit]

There are actually two versions in 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. 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 Q on V; if it is non-degenerate 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 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 locally compact topological group 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 Rn. 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 B to have columns as the linearly independent vectors that describe the lattice, then the matrix

has columns of vectors that describe the dual lattice.

See also[edit]

References[edit]

  1. ^ Audin, Michèle (2003). Geometry. Springer. p. 69.
  2. ^ B. E. Warren (1969/1990) X-ray diffraction (Addison-Wesley, Reading MA/Dover, Mineola NY).

External links[edit]