Talk:Unit cell

Extra math expressions

A given real-space cubic lattice will have dimensions:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(a,b,c\right)}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{q}_{hkl} = 2\pi \left\langle \frac{h}{a} , \frac{k}{b} , \frac{l}{c} \right\rangle }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=b=c} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{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{alignat} }

TBD: Reciprocal vector components

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{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 \\ a b \sin{\gamma} \end{bmatrix} \\ \end{alignat} }