Editing Reciprocal lattice (section)

==Mathematical description==

[[File:Lattice waves.png|thumb|Illustration of a real-space lattice, with plane waves whose wave-vectors are from the reciprocal lattice are overlaid. A real space 2D lattice (red dots) with primitive vectors <math>a_1</math> and <math>a_2</math> are shown by blue and green arrows respectively. Plane waves of the form <math>e^{iG\cdot r}</math> are plotted. From this we see that when <math>G</math> is any integer combination of reciprocal lattice vector basis <math>b_1</math> and <math>b_2</math> (i.e. any reciprocal lattice vector), the resulting plane waves have the same periodicity of the lattice – that is that any translation from point <math>r</math> (shown orange) to a point <math>R+r</math> (<math>R</math> 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 three-dimensional [[Bravais lattice]] and labelling each lattice vector (a vector indicating a lattice point) by the subscript <math>n = (n_1, n_2, n_3)</math> as [[Tuple|3-tuple]] of integers,

:<math>\mathbf{R}_n = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3</math> where <math>n_1, n_2, n_3 \in \mathbb{Z}</math>

where <math>\mathbb{Z}</math> is the set of integers and <math>\mathbf{a}_i</math> is a primitive translation vector or shortly primitive vector. Taking a function <math>f(\mathbf{r})</math> where <math>\mathbf{r}</math> is a position vector from the origin <math>\mathbf{R}_n = 0</math> to any position, if <math>f(\mathbf{r})</math> follows the periodicity of this lattice, e.g. the function describing the electronic density in an atomic crystal, it is useful to write <math>f(\mathbf{r})</math> as a [[Fourier series#Multidimensional|multi-dimensional Fourier series]]{{Broken anchor|date=2024-12-29|bot=User:Cewbot/log/20201008/configuration|target_link=Fourier series#Multidimensional|reason= The anchor (Multidimensional) [[Special:Diff/1052516823|has been deleted]].}}

:<math>\sum_m f_m e^{i \mathbf{G}_m \cdot \mathbf{r}} = f\left(\mathbf{r}\right) </math>

where now the subscript <math>m = (m_1, m_2, m_3)</math>, so this is a triple sum.

As <math>f(\mathbf{r})</math> follows the periodicity of the lattice, translating <math>\mathbf{r}</math> by any lattice vector <math>\mathbf{R}_n</math> we get the same value, hence

:<math>f(\mathbf{r} + \mathbf{R}_n) = f(\mathbf{r}).</math>

Expressing the above instead in terms of their Fourier series we have
<math display="block">\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}}.
</math>

Because equality of two Fourier series implies equality of their coefficients, <math> e^{i \mathbf{G}_m \cdot \mathbf{R}_n} = 1</math>, which only holds when

:<math>\mathbf{G}_m \cdot \mathbf{R}_n = 2\pi N</math> where <math>N \in \mathbb{Z}.</math>

Mathematically, the reciprocal lattice is the set of all [[vector (geometric)|vector]]s <math>\mathbf{G}_m</math>, that are [[Wave vector|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 <math>\mathbf{R}_n</math>, and <math>\mathbf{G}_m</math> satisfy this equality for all <math>\mathbf{R}_n</math>. Each plane wave in the Fourier series has the same phase (actually can be differed by a multiple of <math>2\pi</math>) at all the lattice point <math>\mathbf{R}_n</math>.

As shown in the section [[Fourier series#Multidimensional|multi-dimensional Fourier series]]{{Broken anchor|date=2024-12-29|bot=User:Cewbot/log/20201008/configuration|target_link=Fourier series#Multidimensional|reason= The anchor (Multidimensional) [[Special:Diff/1052516823|has been deleted]].}}, <math>\mathbf{G}_m</math> can be chosen in the form of <math>\mathbf{G}_m = m_1 \mathbf{b}_1 + m_2 \mathbf{b}_2 + m_3 \mathbf{b}_3</math> where {{nowrap begin}}<math>\mathbf{a}_i \cdot \mathbf{b}_j = 2\pi \, \delta_{ij}</math>{{nowrap end}}. With this form, the reciprocal lattice as the set of all wavevectors <math>\mathbf{G}_m</math> for the Fourier series of a spatial function which periodicity follows <math>\mathbf{R}_n</math>, is itself a Bravais lattice as it is formed by integer combinations of its own primitive translation vectors <math>\left(\mathbf{b_{1}}, \mathbf{b}_2, \mathbf{b}_3\right)</math>, and the reciprocal of the reciprocal lattice is the original lattice, which reveals the [[Pontryagin duality]] of their respective [[vector space]]s. (There may be other form of <math>\mathbf{G}_m</math>. Any valid form of <math>\mathbf{G}_m</math> results in the same reciprocal lattice.)

===Two dimensions===
For an infinite two-dimensional lattice, defined by its [[primitive cell|primitive vector]]s <math>\left(\mathbf{a}_1, \mathbf{a}_2\right)</math>, its reciprocal lattice can be determined by generating its two reciprocal primitive vectors, through the following formulae,

:<math>\mathbf{G}_m = m_1 \mathbf{b}_1 + m_2 \mathbf{b}_2</math>

where <math>m_i</math> is an integer and

:<math>\begin{align}
  \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{align}</math>

Here <math>\mathbf{Q}</math> represents a 90 degree [[rotation matrix]], i.e. a ''q''uarter turn. The anti-clockwise rotation and the clockwise rotation can both be used to determine the reciprocal lattice: If <math>\mathbf{Q}</math> is the anti-clockwise rotation and <math>\mathbf{Q'}</math> is the clockwise rotation, <math>\mathbf{Q}\,\mathbf{v}=-\mathbf{Q'}\,\mathbf{v}</math> for all vectors <math>\mathbf{v}</math>. Thus, using the [[Permutation#Two-line notation|permutation]]
: <math>\sigma = \begin{pmatrix}
  1 & 2 \\
  2 & 1
\end{pmatrix}</math>
we obtain
:<math>
\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)}}.
</math>

Notably, in a 3D space this 2D reciprocal lattice is an infinitely extended set of Bragg rods&mdash;described by Sung et al.<ref name="hovden2019">{{Cite journal| last1=Sung|first1=S.H.| last2=Schnitzer|first2=N.| last3=Brown|first3=L.| last4=Park|first4=J.| last5=Hovden|first5=R.| date=2019-06-25|title=Stacking, strain, and twist in 2D materials quantified by 3D electron diffraction| journal=Physical Review Materials|volume=3|issue=6| pages=064003| doi=10.1103/PhysRevMaterials.3.064003| bibcode=2019PhRvM...3f4003S| arxiv=1905.11354| s2cid=166228311}}</ref>

===Three dimensions===
For an infinite three-dimensional lattice <math>\mathbf{R}_n = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3</math>, defined by its [[primitive cell|primitive vector]]s <math>\left(\mathbf{a_{1}}, \mathbf{a}_2, \mathbf{a}_3\right)</math> and the subscript of integers <math>n = \left( n_1, n_2, n_3 \right)</math>, its reciprocal lattice <math>\mathbf{G}_m = m_1 \mathbf{b}_1 + m_2 \mathbf{b}_2 + m_3 \mathbf{b}_3</math> with the integer subscript <math>m = (m_1, m_2, m_3)</math> can be determined by generating its three reciprocal primitive vectors <math>\left(\mathbf{b_{1}}, \mathbf{b}_2, \mathbf{b}_3\right)</math>
<math display="block">\begin{align}
  \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{align}</math>
where <math display="block">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)</math> is the [[scalar triple product]]. The choice of these <math>\left(\mathbf{b_{1}}, \mathbf{b}_2, \mathbf{b}_3\right)</math> is to satisfy {{nowrap begin}}<math>\mathbf{a}_i \cdot \mathbf{b}_j = 2\pi \, \delta_{ij}</math>{{nowrap end}} as the known condition (There may be other condition.) of [[Bravais lattice|primitive translation vectors]] for the reciprocal lattice derived in the [[Reciprocal lattice#Reciprocal lattice|heuristic approach above]] and the section [[Fourier series#Multidimensional|multi-dimensional Fourier series]]{{Broken anchor|date=2024-12-29|bot=User:Cewbot/log/20201008/configuration|target_link=Fourier series#Multidimensional|reason= The anchor (Multidimensional) [[Special:Diff/1052516823|has been deleted]].}}. This choice also satisfies the requirement of the reciprocal lattice <math> e^{i \mathbf{G}_m \cdot \mathbf{R}_n} = 1</math> mathematically derived [[Reciprocal lattice#Mathematical description|above]]. Using column vector representation of (reciprocal) primitive vectors, the formulae above can be rewritten using [[matrix inversion]]:

:<math>\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}.</math>

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

:<math>
  \mathbf{b}_1 = \frac{\mathbf{a}_2 \times \mathbf{a}_3}{\mathbf{a}_1 \cdot \left(\mathbf{a}_2 \times \mathbf{a}_3\right)} 
</math>
and so on for the other primitive vectors. The crystallographer's definition has the advantage that the definition of <math>\mathbf{b}_1</math> is just the reciprocal magnitude of <math>\mathbf{a}_1</math> in the direction of <math>\mathbf{a}_2 \times \mathbf{a}_3</math>, dropping the factor of <math>2 \pi</math>. 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.

<math>m = (m_1, m_2, m_3)</math> is conventionally written as <math>(h, k, \ell)</math> or <math>(hk\ell)</math>, called [[Miller index|Miller indices]]; <math>m_1</math> is replaced with <math>h</math>, <math>m_2</math> replaced with <math>k</math>, and <math>m_3</math> replaced with <math>\ell</math>. Each lattice point <math>(hk\ell)</math> in the reciprocal lattice corresponds to a set of lattice planes <math>(hk\ell)</math> in the [[space|real space]] lattice. (A lattice plane is a plane crossing lattice points.) The direction of the reciprocal lattice vector corresponds to the [[Surface normal|normal]] to the real space planes. The magnitude of the reciprocal lattice vector <math>\mathbf{K}_m</math> 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 <math>n</math> dimensions can be derived assuming an <math>n</math>-[[dimension (vector space)|dimensional]] [[real number|real]] vector space <math>V</math> with a [[basis (linear algebra)|basis]] <math>(\mathbf{a}_1,\ldots,\mathbf{a}_n)</math> and an inner product <math>g\colon V\times V\to\mathbf{R}</math>. The reciprocal lattice vectors are uniquely determined by the formula <math>g(\mathbf{a}_i,\mathbf{b}_j)=2\pi\delta_{ij}</math>. Using the [[Permutation#Two-line notation|permutation]]
: <math>\sigma = \begin{pmatrix}
  1 & 2 & \cdots &n\\
  2 & 3 & \cdots &1
\end{pmatrix},</math>
they can be determined with the following formula:
:<math>
     \mathbf{b}_i = 2\pi\frac{\varepsilon_{\sigma^1i\ldots\sigma^ni}}{\omega(\mathbf{a}_1,\ldots,\mathbf{a}_n)}g^{-1}(\mathbf{a}_{\sigma^{n-1}i}\,\lrcorner\ldots\mathbf{a}_{\sigma^1i}\,\lrcorner\,\omega)\in V
</math>
Here, <math>\omega\colon V^n \to \mathbf{R}</math> is the [[volume form]], <math>g^{-1}</math> is the inverse of the vector space isomorphism <math>\hat{g}\colon V \to V^*</math> defined by <math>\hat{g}(v)(w) = g(v,w)</math> and <math>\lrcorner</math> denotes the [[Interior product|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, <math>\omega(u,v,w) = g(u \times v, w)</math> and in two dimensions, <math>\omega(v,w) = g(Rv,w)</math>, where <math>R \in \text{SO}(2) \subset L(V,V)</math> is the [[rotation]] by 90 degrees (just like the volume form, the angle assigned to a rotation depends on the choice of orientation<ref>{{cite book | last=Audin|first=Michèle | title=Geometry | publisher=Springer | year=2003|page=69}}</ref>).