Building up a face centered cubic lattice

• Consider:

1. a Cubic Simulation box of length $\left.L\right.$
2. a number of lattice positions, $\left.M\right.$ given by:

$\left.M=4m^{3}\right.$

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 are integer number that must fulfill:

• $0\leq i_{a}<m$
• $0\leq j_{a}<m$
• $0\leq k_{a}<m$,
• and the sum: 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. i_a + j_a + k_a \right. } must be, for instance, an even number.

with 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. \delta l = L/(2m) \right. }