Editing Reciprocal lattice (section)

== Reciprocal lattices of various crystals ==

Reciprocal lattices for the [[Cubic (crystal system)|cubic crystal system]] are as follows.

===Simple cubic lattice===

The simple cubic [[Bravais lattice]], with cubic [[primitive cell]] of side <math>a</math>, has for its reciprocal a simple cubic lattice with a cubic primitive cell of side <math display="inline">\frac{2\pi}{a}</math> (or <math display="inline"> \frac{1}{a}</math> 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 <math display="inline"> \frac{4\pi}{a}</math>.

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 [[Cubic crystal system|BCC]] lattice is the [[Cubic crystal system|FCC]] lattice, with a cube side of <math display="inline"> 4\pi/a</math>.

It can be proven that only the Bravais lattices which have 90 degrees between <math>\left(\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\right)</math> (cubic, tetragonal, orthorhombic) have primitive translation vectors for the reciprocal lattice, <math>\left(\mathbf{b}_1, \mathbf{b}_2, \mathbf{b}_3\right)</math>, parallel to their real-space vectors.

===Simple hexagonal lattice===

The reciprocal to a simple hexagonal Bravais lattice with [[lattice constants]] <math display="inline"> a</math> and <math display="inline"> c</math> is another simple hexagonal lattice with lattice constants <math display="inline"> 2\pi/c</math> and <math display="inline"> 4\pi/(a\sqrt3)</math> 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
<math display="block">
\begin{align}
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{align} </math><ref>{{Cite book|last=Kittel|first=Charles | title=Introduction to Solid State Physics | publisher=John Wiley & Sons, Inc | year=2005 | isbn=0-471-41526-X | edition=8th | pages=44}}</ref>