Editing Reciprocal lattice (section)

===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>