Building up a diamond lattice

[EN CONSTRUCCION]

• 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 $\left.M=8m^{3}\right.$,

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 numbers that must fulfill the following criteria

• $0\leq i_{a}<4m$
• $0\leq j_{a}<4m$
• $0\leq k_{a}<4m$,
• the sum of 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. } can have only the values: 0, 3, 4, 7, 8, 10, ...

i.e, 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 = 4 n \right. } ; OR; 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 + z_a = 4 n + 3 \right. } , 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 n } is any integer number

• the indices 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 either all even or all odd.

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/(4m) \right. }