Talk:Unit cell

Extra math expressions

A given real-space cubic lattice will have dimensions:

$\left(a,b,c\right)$

Such that the position of any particular cell within the infinite lattice is:

$\mathbf {r} _{hkl}=\left\langle ah,bk,cl\right\rangle$

Where h, k, and l are indices. The corresponding inverse-space lattice would be:

$\mathbf {q} _{hkl}=2\pi \left\langle {\frac {h}{a}},{\frac {k}{b}},{\frac {l}{c}}\right\rangle$

$q_{hkl}=2\pi {\sqrt {\left({\frac {h}{a}}\right)^{2}+\left({\frac {k}{b}}\right)^{2}+\left({\frac {l}{c}}\right)^{2}}}$

In the case where $a=b=c$:

${\begin{alignedat}{2}q_{hkl}&=2\pi {\sqrt {\left({\frac {h}{a}}\right)^{2}+\left({\frac {k}{a}}\right)^{2}+\left({\frac {l}{a}}\right)^{2}}}\\&={\frac {2\pi }{a}}{\sqrt {h^{2}+k^{2}+l^{2}}}\end{alignedat}}$

TBD: Reciprocal vector components

${\begin{alignedat}{2}\mathbf {u} &=...\\\mathbf {v} &=...\\\mathbf {w} &={\frac {\mathbf {a} \times \mathbf {b} }{\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )}}\\&={\frac {1}{V}}\mathbf {a} \times \mathbf {b} \\&={\frac {1}{V}}{\begin{bmatrix}0\\0\\ab\sin {\gamma }\end{bmatrix}}\\\end{alignedat}}$