Building up a body centered cubic lattice

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A body centered cubic lattice

• Consider:

1. a cubic simulation box whose sides are of length $\left.L\right.$
2. a number of lattice positions, $\left.M\right.$ given by $\left.M=2m^{3}\right.$, with $m$ being a positive integer

• The $\left.M\right.$ positions are those given by:

$\left\{{\begin{array}{l}x_{a}=i_{a}\times (\delta l)\\y_{a}=j_{a}\times (\delta l)\\z_{a}=k_{a}\times (\delta l)\end{array}}\right\}$

where the indices of a given valid site Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (i_{a},j_{a},k_{a})} must fulfill:

• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i_{a},j_{a},k_{a}} must be either all odd or all even.
• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 0\leq i_{a}\leq 2m}
• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 0\leq j_{a}\leq 2m}
• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 0\leq k_{a}\leq 2m}

and $\left.\delta l=L/(2m)\right.$

Atomic position(s) on a cubic cell

• Number of atoms per cell: 2
• Coordinates:

Atom 1: $\left(x_{1},y_{1},z_{1}\right)=\left(0,0,0\right)$

Atom 2: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( x_2, y_2, z_2 \right) = \left( l/2, l/2, l/2 \right) }

Cell dimensions:

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=b=c = l }

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = \beta = \gamma = 90^0 }