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 $(i_{a},j_{a},k_{a})$ must fulfill:

• $i_{a},j_{a},k_{a}$ must be either all odd or all even.
• $0\leq i_{a}\leq 2m$
• $0\leq j_{a}\leq 2m$
• $0\leq k_{a}\leq 2m$

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: $\left(x_{2},y_{2},z_{2}\right)=\left(l/2,l/2,l/2\right)$

Cell dimensions:

• $a=b=c=l$

• $\alpha =\beta =\gamma =90^{0}$