Reciprocal lattice is a concept associated with solids with translational symmetry which plays a major role in many areas such as X-ray and electron diffraction as well as the energies of electrons in a solid. It emerges from the Fourier transform of the lattice associated with the arrangement of the atoms. The direct lattice or real lattice is a periodic function in physical space, such as a crystal system (usually a Bravais lattice). The reciprocal lattice exists in the mathematical space of spatial frequencies or wavenumbers k, known as reciprocal space or k space; it is the dual of physical space considered as a vector space. In other words, the reciprocal lattice is the sublattice which is dual to the direct lattice. The reciprocal lattice is the set of all vectors G m {\displaystyle \mathbf {G} _{m}} , that are wavevectors k of plane waves in the Fourier series of a spatial function whose periodicity is the same as that of a direct lattice R n {\displaystyle \mathbf {R} _{n}} . Each plane wave in this Fourier series has the same phase or phases that are differed by multiples of 2 π {\displaystyle 2\pi } , at each direct lattice point (so essentially same phase at all the direct lattice points). 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 Brillouin zone is a Wigner–Seitz cell of the reciprocal lattice.
Wave-based description
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 spatial domain or 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 have 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 cos ( k x − ω t + φ 0 ) {\displaystyle \cos(kx-\omega t+\varphi _{0})} , with initial phase φ 0 {\displaystyle \varphi _{0}} , angular wavenumber k {\displaystyle k} and angular frequency ω {\displaystyle \omega } , it can be regarded as a function of both k {\displaystyle k} and x {\displaystyle x} (and the time-varying part as a function of both ω {\displaystyle \omega } and t {\displaystyle t} ). This complementary role of k {\displaystyle k} and x {\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 λ {\displaystyle \lambda } , where k λ = 2 π {\displaystyle k\lambda =2\pi } ; hence the corresponding wavenumber in reciprocal space will be k = 2 π / λ {\displaystyle k=2\pi /\lambda } . In three dimensions, the corresponding plane wave term becomes cos ( k ⋅ r − ω t + φ 0 ) {\displaystyle \cos(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{0})} , which simplifies to cos ( k ⋅ r + φ ) {\displaystyle \cos(\mathbf {k} \cdot \mathbf {r} +\varphi )} at a fixed time t {\displaystyle t} , where r {\displaystyle \mathbf {r} } is the position vector of a point in real space and now k = 2 π e / λ {\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 φ {\displaystyle \varphi } is the phase of the wavefront (a plane of a constant phase) through the origin r = 0 {\displaystyle \mathbf {r} =0} at time t {\displaystyle t} , and e {\displaystyle \mathbf {e} } is a unit normal vector to this wavefront. The wavefronts with phases φ + ( 2 π ) n {\displaystyle \varphi +(2\pi )n} , where n {\displaystyle n} represents any integer, comprise a set of parallel planes, equally spaced by the wavelength λ {\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 k {\displaystyle \mathbf {k} } of plane waves in the Fourier series of any function f ( r ) {\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 ( 2 π ) n {\displaystyle (2\pi )n} with an integer n {\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 R = n 1 a 1 + n 2 a 2 + n 3 a 3 {\displaystyle \mathbf {R} =n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}} , where the n i {\displaystyle n_{i}} are integers defining the vertex and the a i {\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 R = 0 {\displaystyle \mathbf {R} =0} contains the direct lattice points at a 2 {\displaystyle \mathbf {a} _{2}} and a 3 {\displaystyle \mathbf {a} _{3}} , and with its adjacent wavefront (whose phase differs by 2 π {\displaystyle 2\pi } or − 2 π {\displaystyle -2\pi } from the former wavefront passing the origin) passing through a 1 {\displaystyle \mathbf {a} _{1}} . Its angular wavevector takes the form b 1 = 2 π e 1 / λ 1 {\displaystyle \mathbf {b} _{1}=2\pi \mathbf {e} _{1}/\lambda _{1}} , where e 1 {\displaystyle \mathbf {e} _{1}} is the unit vector perpendicular to these two adjacent wavefronts and the wavelength λ 1 {\displaystyle \lambda _{1}} must satisfy λ 1 = a 1 ⋅ e 1 {\displaystyle \lambda _{1}=\mathbf {a} _{1}\cdot \mathbf {e} _{1}} , means that λ 1 {\displaystyle \lambda _{1}} is equal to the distance between the two wavefronts. Hence by construction a 1 ⋅ b 1 = 2 π {\displaystyle \mathbf {a} _{1}\cdot \mathbf {b} _{1}=2\pi } and a 2 ⋅ b 1 = a 3 ⋅ b 1 = 0 {\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 b j {\displaystyle \mathbf {b} _{j}} with a i ⋅ b j = 2 π δ i j {\displaystyle \mathbf {a} _{i}\cdot \mathbf {b} _{j}=2\pi \,\delta _{ij}} , where the Kronecker delta δ i j {\displaystyle \delta _{ij}} equals one when i = j {\displaystyle i=j} and is zero otherwise. The b j {\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 G = m 1 b 1 + m 2 b 2 + m 3 b 3 {\displaystyle \mathbf {G} =m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}+m_{3}\mathbf {b} _{3}} , where the m j {\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 b 1 {\displaystyle \mathbf {b} _{1}} , b 2 {\displaystyle \mathbf {b} _{2}} , and b 3 {\displaystyle \mathbf {b} _{3}} in this case. Simple algebra then shows that, for any plane wave with a wavevector G {\displaystyle \mathbf {G} } on the reciprocal lattice, the total phase shift G ⋅ R {\displaystyle \mathbf {G} \cdot \mathbf {R} } between the origin and any point R {\displaystyle \mathbf {R} } on the direct lattice is a multiple of 2 π {\displaystyle 2\pi } (that can be possibly zero if the multiplier is zero), so the phase of the plane wave with G {\displaystyle \mathbf {G} } will essentially be equal for every direct lattice vertex, in conformity with the reciprocal lattice definition above. (Although any wavevector G {\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.
Mathematical description
Assuming a three-dimensional Bravais lattice and labelling each lattice vector (a vector indicating a lattice point) by the subscript n = ( n 1 , n 2 , n 3 ) {\displaystyle n=(n_{1},n_{2},n_{3})} as 3-tuple of integers,
R n = n 1 a 1 + n 2 a 2 + n 3 a 3 {\displaystyle \mathbf {R} _{n}=n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}} where n 1 , n 2 , n 3 ∈ Z {\displaystyle n_{1},n_{2},n_{3}\in \mathbb {Z} }
where Z {\displaystyle \mathbb {Z} } is the set of integers and a i {\displaystyle \mathbf {a} _{i}} is a primitive translation vector or shortly primitive vector. Taking a function f ( r ) {\displaystyle f(\mathbf {r} )} where r {\displaystyle \mathbf {r} } is a position vector from the origin R n = 0 {\displaystyle \mathbf {R} _{n}=0} to any position, if f ( r ) {\displaystyle f(\mathbf {r} )} follows the periodicity of this lattice, e.g. the function describing the electronic density in an atomic crystal, it is useful to write f ( r ) {\displaystyle f(\mathbf {r} )} as a multi-dimensional Fourier series
∑ m f m e i G m ⋅ r = f ( r ) {\displaystyle \sum _{m}f_{m}e^{i\mathbf {G} _{m}\cdot \mathbf {r} }=f\left(\mathbf {r} \right)}
where now the subscript m = ( m 1 , m 2 , m 3 ) {\displaystyle m=(m_{1},m_{2},m_{3})} , so this is a triple sum. As f ( r ) {\displaystyle f(\mathbf {r} )} follows the periodicity of the lattice, translating r {\displaystyle \mathbf {r} } by any lattice vector R n {\displaystyle \mathbf {R} _{n}} we get the same value, hence
f ( r + R n ) = f ( r ) . {\displaystyle f(\mathbf {r} +\mathbf {R} _{n})=f(\mathbf {r} ).}
Expressing the above instead in terms of their Fourier series we have
∑ m f m e i G m ⋅ r = ∑ m f m e i G m ⋅ ( r + R n ) = ∑ m f m e i G m ⋅ R n e i G m ⋅ r . {\displaystyle \sum _{m}f_{m}e^{i\mathbf {G} _{m}\cdot \mathbf {r} }=\sum _{m}f_{m}e^{i\mathbf {G} _{m}\cdot (\mathbf {r} +\mathbf {R} _{n})}=\sum _{m}f_{m}e^{i\mathbf {G} _{m}\cdot \mathbf {R} _{n}}\,e^{i\mathbf {G} _{m}\cdot \mathbf {r} }.}
Because equality of two Fourier series implies equality of their coefficients, e i G m ⋅ R n = 1 {\displaystyle e^{i\mathbf {G} _{m}\cdot \mathbf {R} _{n}}=1} , which only holds when
G m ⋅ R n = 2 π N {\displaystyle \mathbf {G} _{m}\cdot \mathbf {R} _{n}=2\pi N} where N ∈ Z . {\displaystyle N\in \mathbb {Z} .}
Mathematically, the reciprocal lattice is the set of all vectors G m {\displaystyle \mathbf {G} _{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 R n {\displaystyle \mathbf {R} _{n}} , and G m {\displaystyle \mathbf {G} _{m}} satisfy this equality for all R n {\displaystyle \mathbf {R} _{n}} . Each plane wave in the Fourier series has the same phase (actually can be differed by a multiple of 2 π {\displaystyle 2\pi } ) at all the lattice point R n {\displaystyle \mathbf {R} _{n}} .
G m {\displaystyle \mathbf {G} _{m}} can be chosen in the form of G m = m 1 b 1 + m 2 b 2 + m 3 b 3 {\displaystyle \mathbf {G} _{m}=m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}+m_{3}\mathbf {b} _{3}} where a i ⋅ b j = 2 π δ i j {\displaystyle \mathbf {a} _{i}\cdot \mathbf {b} _{j}=2\pi \,\delta _{ij}} . With this form, the reciprocal lattice as the set of all wavevectors G m {\displaystyle \mathbf {G} _{m}} for the Fourier series of a spatial function which periodicity follows R n {\displaystyle \mathbf {R} _{n}} , is itself a Bravais lattice as it is formed by integer combinations of its own primitive translation vectors ( b 1 , b 2 , b 3 ) {\displaystyle \left(\mathbf {b_{1}} ,\mathbf {b} _{2},\mathbf {b} _{3}\right)} , and the reciprocal of the reciprocal lattice is the original lattice, which reveals the Pontryagin duality of their respective vector spaces. (There may be other form of G m {\displaystyle \mathbf {G} _{m}} . Any valid form of G m {\displaystyle \mathbf {G} _{m}} results in the same reciprocal lattice.)
Two dimensions For an infinite two-dimensional lattice, defined by its primitive vectors ( a 1 , a 2 ) {\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,
G m = m 1 b 1 + m 2 b 2 {\displaystyle \mathbf {G} _{m}=m_{1}\mathbf {b} _{1}+m_{2}\mathbf {b} _{2}}
where m i {\displaystyle m_{i}} is an integer and
b 1 = 2 π − Q a 2 − a 1 ⋅ Q a 2 = 2 π Q a 2 a 1 ⋅ Q a 2 b 2 = 2 π Q a 1 a 2 ⋅ Q a 1 {\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 Q {\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 Q {\displaystyle \mathbf {Q} } is the anti-clockwise rotation and Q ′ {\displaystyle \mathbf {Q'} } is the clockwise rotation, Q v = − Q ′ v {\displaystyle \mathbf {Q} \,\mathbf {v} =-\mathbf {Q'} \,\mathbf {v} } for all vectors v {\displaystyle \mathbf {v} } . Thus, using the permutation
σ = ( 1 2 2 1 ) {\displaystyle \sigma ={\begin{pmatrix}1&2\\2&1\end{pmatrix}}}
we obtain
b n = 2 π Q a σ ( n ) a n ⋅ Q a σ ( n ) = 2 π Q ′ a σ ( n ) a n ⋅ Q ′ a σ ( n ) . {\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.
Three dimensions For an infinite three-dimensional lattice R n = n 1 a 1 + n 2 a 2 + n 3 a 3 {\displaystyle \mathbf {R} _{n}=n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}} , defined by its primitive vectors ( a 1 , a 2 , a 3 ) {\displaystyle \left(\mathbf {a_{1}} ,\mathbf {a} _{2},\mathbf {a} _{3}\right)} and the subscript of integers n = ( n 1 , n 2 , n 3 ) {\displaystyle n=\left(n_{1},n_{2},n_{3}\right)} , its reciprocal lattice G