Reciprocal space

Reciprocal space (also called k-space) provides a way to visualize the results of the Fourier transform of a spatial function. It is similar in role to the frequency domain arising from the Fourier transform of a time dependent function; reciprocal space is a space over which the Fourier transform of a spatial function is represented at spatial frequencies or wavevectors of plane waves of the 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 the number of spatial dimensions of these two associated spaces will be the same, the spaces will differ in their quantity dimension, so that when the real space has the dimension length (L), its reciprocal space will of inverse length, 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 ${\displaystyle \cos(kx-\omega t+\varphi _{0})}$, with initial phase ${\displaystyle \varphi _{0}}$, angular wavenumber ${\displaystyle k}$ and angular frequency ${\displaystyle \omega }$, it can be regarded as a function of both ${\displaystyle k}$ and ${\displaystyle x}$ (and the time-varying part as a function of both ${\displaystyle \omega }$ and ${\displaystyle t}$). This complementary role of ${\displaystyle k}$ and ${\displaystyle x}$ 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 ${\displaystyle \lambda }$, where ${\displaystyle k\lambda =2\pi }$; hence the corresponding wavenumber in reciprocal space will be ${\displaystyle k=2\pi /\lambda }$.

In three dimensions, the corresponding plane wave term becomes ${\displaystyle \cos(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{0})}$, which simplifies to ${\displaystyle \cos(\mathbf {k} \cdot \mathbf {r} +\varphi )}$ at a fixed time ${\displaystyle t}$, where ${\displaystyle \mathbf {r} }$ is the position vector of a point in real space and now ${\displaystyle \mathbf {k} =2\pi \mathbf {e} /\lambda }$ is the wavevector in the three dimensional reciprocal space. (The magnitude of a wavevector is called wavenumber.) The constant ${\displaystyle \varphi }$ is the phase of the wavefront (a plane of a constant phase) through the origin ${\displaystyle \mathbf {r} =0}$ at time ${\displaystyle t}$, and ${\displaystyle \mathbf {e} }$ is a unit vector perpendicular to this wavefront. The wavefronts with phases ${\displaystyle \varphi +(2\pi )n}$, where ${\displaystyle n}$ represents any integer, comprise a set of parallel planes, equally spaced by the wavelength ${\displaystyle \lambda }$.

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. 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 ${\displaystyle \mathbf {k} }$ of plane waves in the Fourier series of any function ${\displaystyle f(\mathbf {r} )}$ 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 ${\displaystyle (2\pi )n}$ with an integer ${\displaystyle n}$) 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 ${\displaystyle \mathbf {R} =n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}}$, where the ${\displaystyle n_{i}}$ are integers defining the vertex and the ${\displaystyle \mathbf {a} _{i}}$ 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 ${\displaystyle \mathbf {R} =0}$ contains the direct lattice points at ${\displaystyle \mathbf {a} _{2}}$ and ${\displaystyle \mathbf {a} _{3}}$, and with its adjacent wavefront (whose phase differs by ${\displaystyle 2\pi }$ or ${\displaystyle -2\pi }$ from the former wavefront passing the origin) passing through ${\displaystyle \mathbf {a} _{1}}$. Its angular wavevector takes the form ${\displaystyle \mathbf {b} _{1}=2\pi \mathbf {e} _{1}/\lambda _{1}}$, where ${\displaystyle \mathbf {e} _{1}}$ is the unit vector perpendicular to these two adjacent wavefronts and the wavelength ${\displaystyle \lambda _{1}}$ must satisfy ${\displaystyle \lambda _{1}=\mathbf {a} _{1}\cdot \mathbf {e} _{1}}$, means that ${\displaystyle \lambda _{1}}$ is equal to the distance between the two wavefronts. Hence by construction ${\displaystyle \mathbf {a} _{1}\cdot \mathbf {b} _{1}=2\pi }$ and ${\displaystyle \mathbf {a} _{2}\cdot \mathbf {b} _{1}=\mathbf {a} _{3}\cdot \mathbf {b} _{1}=0}$.

Cycling through the indices in turn, the same method yields three wavevectors ${\displaystyle \mathbf {b} _{j}}$ with ${\displaystyle \mathbf {a} _{i}\cdot \mathbf {b} _{j}=2\pi \,\delta _{ij}}$, where the Kronecker delta ${\displaystyle \delta _{ij}}$ equals one when ${\displaystyle i=j}$ and is zero otherwise. The ${\displaystyle \mathbf {b} _{j}}$ comprise a set of three primitive wavevectors or three primitive translation vectors for the reciprocal lattice, each of whose vertices takes the form ${\displaystyle \mathbf {G} =m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}+m_{3}\mathbf {b} _{3}}$, where the ${\displaystyle m_{j}}$ are integers. The reciprocal lattice is also a Bravais lattice as it is formed by integer combinations of the primitive vectors, that are ${\displaystyle \mathbf {b} _{1}}$, ${\displaystyle \mathbf {b} _{2}}$, and ${\displaystyle \mathbf {b} _{3}}$ in this case. Simple algebra then shows that, for any plane wave with a wavevector ${\displaystyle \mathbf {G} }$ on the reciprocal lattice, the total phase shift ${\displaystyle \mathbf {G} \cdot \mathbf {R} }$ between the origin and any point ${\displaystyle \mathbf {R} }$ on the direct lattice is a multiple of ${\displaystyle 2\pi }$ (that can be possibly zero if the multiplier is zero), so the phase of the plane wave with ${\displaystyle \mathbf {G} }$ will essentially be equal for every direct lattice vertex, in conformity with the reciprocal lattice definition above. (Although any wavevector ${\displaystyle \mathbf {G} }$ 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 generalizations of reciprocal space and the reciprocal lattice.

Two dimensions

For an infinite two-dimensional lattice, defined by its primitive vectors ${\displaystyle \left(\mathbf {a} _{1},\mathbf {a} _{2}\right)}$, its reciprocal lattice can be determined by generating its two reciprocal primitive vectors, through the following formulae,

${\displaystyle \mathbf {G} _{m}=m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}}$

where ${\displaystyle m_{i}}$ is an integer and

${\displaystyle {\begin{aligned}\mathbf {b} _{1}&=2\pi {\frac {-\mathbf {Q} \,\mathbf {a} _{2}}{-\mathbf {a} _{1}\cdot \mathbf {Q} \,\mathbf {a} _{2}}}=2\pi {\frac {\mathbf {Q} \,\mathbf {a} _{2}}{\mathbf {a} _{1}\cdot \mathbf {Q} \,\mathbf {a} _{2}}}\\[8pt]\mathbf {b} _{2}&=2\pi {\frac {\mathbf {Q} \,\mathbf {a} _{1}}{\mathbf {a} _{2}\cdot \mathbf {Q} \,\mathbf {a} _{1}}}\end{aligned}}}$

Here ${\displaystyle \mathbf {Q} }$ 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 ${\displaystyle \mathbf {Q} }$ is the anti-clockwise rotation and ${\displaystyle \mathbf {Q'} }$ is the clockwise rotation, ${\displaystyle \mathbf {Q} \,\mathbf {v} =-\mathbf {Q'} \,\mathbf {v} }$ for all vectors ${\displaystyle \mathbf {v} }$. Thus, using the permutation

${\displaystyle \sigma ={\begin{pmatrix}1&2\\2&1\end{pmatrix}}}$

we obtain

${\displaystyle \mathbf {b} _{n}=2\pi {\frac {\mathbf {Q} \,\mathbf {a} _{\sigma (n)}}{\mathbf {a} _{n}\cdot \mathbf {Q} \,\mathbf {a} _{\sigma (n)}}}=2\pi {\frac {\mathbf {Q} '\,\mathbf {a} _{\sigma (n)}}{\mathbf {a} _{n}\cdot \mathbf {Q} '\,\mathbf {a} _{\sigma (n)}}}.}$

Notably, in a 3D space this 2D reciprocal lattice is an infinitely extended set of Bragg rods—described by Sung et al.^[1]

Three dimensions

For an infinite three-dimensional lattice ${\displaystyle \mathbf {R} _{n}=n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}}$, defined by its primitive vectors ${\displaystyle \left(\mathbf {a_{1}} ,\mathbf {a} _{2},\mathbf {a} _{3}\right)}$ and the subscript of integers ${\displaystyle n=\left(n_{1},n_{2},n_{3}\right)}$, its reciprocal lattice ${\displaystyle \mathbf {G} _{m}=m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}+m_{3}\mathbf {b} _{3}}$ with the integer subscript ${\displaystyle m=(m_{1},m_{2},m_{3})}$ can be determined by generating its three reciprocal primitive vectors ${\displaystyle \left(\mathbf {b_{1}} ,\mathbf {b} _{2},\mathbf {b} _{3}\right)}$

$${\displaystyle {\begin{aligned}\mathbf {b} _{1}&={\frac {2\pi }{V}}\ \mathbf {a} _{2}\times \mathbf {a} _{3}\\[8pt]\mathbf {b} _{2}&={\frac {2\pi }{V}}\ \mathbf {a} _{3}\times \mathbf {a} _{1}\\[8pt]\mathbf {b} _{3}&={\frac {2\pi }{V}}\ \mathbf {a} _{1}\times \mathbf {a} _{2}\end{aligned}}}$$

where

$${\displaystyle V=\mathbf {a} _{1}\cdot \left(\mathbf {a} _{2}\times \mathbf {a} _{3}\right)=\mathbf {a} _{2}\cdot \left(\mathbf {a} _{3}\times \mathbf {a} _{1}\right)=\mathbf {a} _{3}\cdot \left(\mathbf {a} _{1}\times \mathbf {a} _{2}\right)}$$

is the scalar triple product. The choice of these ${\displaystyle \left(\mathbf {b_{1}} ,\mathbf {b} _{2},\mathbf {b} _{3}\right)}$ is to satisfy ${\displaystyle \mathbf {a} _{i}\cdot \mathbf {b} _{j}=2\pi \,\delta _{ij}}$ 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 ${\displaystyle e^{i\mathbf {G} _{m}\cdot \mathbf {R} _{n}}=1}$ mathematically derived above. Using column vector representation of (reciprocal) primitive vectors, the formulae above can be rewritten using matrix inversion:

${\displaystyle \left[\mathbf {b} _{1}\mathbf {b} _{2}\mathbf {b} _{3}\right]^{\mathsf {T}}=2\pi \left[\mathbf {a} _{1}\mathbf {a} _{2}\mathbf {a} _{3}\right]^{-1}.}$

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 ${\displaystyle 2\pi }$ comes naturally from the study of periodic structures. An essentially equivalent definition, the "crystallographer's" definition, comes from defining the reciprocal lattice ${\displaystyle \mathbf {K} _{m}=\mathbf {G} _{m}/2\pi }$. which changes the reciprocal primitive vectors to be

${\displaystyle \mathbf {b} _{1}={\frac {\mathbf {a} _{2}\times \mathbf {a} _{3}}{\mathbf {a} _{1}\cdot \left(\mathbf {a} _{2}\times \mathbf {a} _{3}\right)}}}$

and so on for the other primitive vectors. The crystallographer's definition has the advantage that the definition of ${\displaystyle \mathbf {b} _{1}}$ is just the reciprocal magnitude of ${\displaystyle \mathbf {a} _{1}}$ in the direction of ${\displaystyle \mathbf {a} _{2}\times \mathbf {a} _{3}}$, dropping the factor of ${\displaystyle 2\pi }$. 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.

${\displaystyle m=(m_{1},m_{2},m_{3})}$ is conventionally written as ${\displaystyle (h,k,\ell )}$ or ${\displaystyle (hk\ell )}$, called Miller indices; ${\displaystyle m_{1}}$ is replaced with ${\displaystyle h}$, ${\displaystyle m_{2}}$ replaced with ${\displaystyle k}$, and ${\displaystyle m_{3}}$ replaced with ${\displaystyle \ell }$. Each lattice point ${\displaystyle (hk\ell )}$ in the reciprocal lattice corresponds to a set of lattice planes ${\displaystyle (hk\ell )}$ 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 ${\displaystyle \mathbf {K} _{m}}$ is given in reciprocal length and is equal to the reciprocal of the interplanar spacing of the real space planes.

Higher dimensions

The formula for ${\displaystyle n}$ dimensions can be derived assuming an ${\displaystyle n}$-dimensional real vector space ${\displaystyle V}$ with a basis ${\displaystyle (\mathbf {a} _{1},\ldots ,\mathbf {a} _{n})}$ and an inner product ${\displaystyle g\colon V\times V\to \mathbf {R} }$. The reciprocal lattice vectors are uniquely determined by the formula ${\displaystyle g(\mathbf {a} _{i},\mathbf {b} _{j})=2\pi \delta _{ij}}$. Using the permutation

${\displaystyle \sigma ={\begin{pmatrix}1&2&\cdots &n\\2&3&\cdots &1\end{pmatrix}},}$

they can be determined with the following formula:

${\displaystyle \mathbf {b} _{i}=2\pi {\frac {\varepsilon _{\sigma ^{1}i\ldots \sigma ^{n}i}}{\omega (\mathbf {a} _{1},\ldots ,\mathbf {a} _{n})}}g^{-1}(\mathbf {a} _{\sigma ^{n-1}i}\,\lrcorner \ldots \mathbf {a} _{\sigma ^{1}i}\,\lrcorner \,\omega )\in V}$

Here, ${\displaystyle \omega \colon V^{n}\to \mathbf {R} }$ is the volume form, ${\displaystyle g^{-1}}$ is the inverse of the vector space isomorphism ${\displaystyle {\hat {g}}\colon V\to V^{*}}$ defined by ${\displaystyle {\hat {g}}(v)(w)=g(v,w)}$ and ${\displaystyle \lrcorner }$ 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, ${\displaystyle \omega (u,v,w)=g(u\times v,w)}$ and in two dimensions, ${\displaystyle \omega (v,w)=g(Rv,w)}$, where ${\displaystyle R\in {\text{SO}}(2)\subset L(V,V)}$ is the rotation by 90 degrees (just like the volume form, the angle assigned to a rotation depends on the choice of orientation^[2]).

Simple cubic lattice

The simple cubic Bravais lattice, with cubic primitive cell of side ${\displaystyle a}$, has for its reciprocal a simple cubic lattice with a cubic primitive cell of side ${\textstyle {\frac {2\pi }{a}}}$ (or ${\textstyle {\frac {1}{a}}}$ 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 ${\textstyle {\frac {4\pi }{a}}}$.

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. 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 ${\textstyle 4\pi /a}$.

It can be proven that only the Bravais lattices which have 90 degrees between ${\displaystyle \left(\mathbf {a} _{1},\mathbf {a} _{2},\mathbf {a} _{3}\right)}$ (cubic, tetragonal, orthorhombic) have primitive translation vectors for the reciprocal lattice, ${\displaystyle \left(\mathbf {b} _{1},\mathbf {b} _{2},\mathbf {b} _{3}\right)}$, parallel to their real-space vectors.

Simple hexagonal lattice

The reciprocal to a simple hexagonal Bravais lattice with lattice constants ${\textstyle a}$ and ${\textstyle c}$ is another simple hexagonal lattice with lattice constants ${\textstyle 2\pi /c}$ and ${\textstyle 4\pi /(a{\sqrt {3}})}$ 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

$${\displaystyle {\begin{aligned}a_{1}&={\frac {\sqrt {3}}{2}}a{\hat {x}}+{\frac {1}{2}}a{\hat {y}},\\[8pt]a_{2}&=-{\frac {\sqrt {3}}{2}}a{\hat {x}}+{\frac {1}{2}}a{\hat {y}},\\[8pt]a_{3}&=c{\hat {z}}.\end{aligned}}}$$

^[3]