# Talk:Lattices

## To Do

B.M. Mladek, B.M.; Mladek, B.M.; Fornleitner, J.; Martinez-Veracoechea, F.C.; Dawid, A.; Frenkel, D. Procedure to construct a multi-scale coarse-grained model of DNA-coated colloids from experimental Soft Matter 2013, 9, 7342-7355 doi: 10.1039/C3SM50701G

## Extra math

A given real-space lattice will have dimensions:

${\displaystyle \left(a,b,c\right)}$

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

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

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

${\displaystyle \mathbf {q} _{hkl}=2\pi \left\langle {\frac {h}{a}},{\frac {k}{b}},{\frac {l}{c}}\right\rangle }$
${\displaystyle q_{hkl}=2\pi {\sqrt {\left({\frac {h}{a}}\right)^{2}+\left({\frac {k}{b}}\right)^{2}+\left({\frac {l}{c}}\right)^{2}}}}$

In the case where ${\displaystyle a=b=c}$:

{\displaystyle {\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}}}