# Building up a body centered cubic lattice

  200 lightgrey Body_centered_cubic_lattice.xyz  A body centered cubic lattice
• Consider:
1. a cubic simulation box whose sides are of length ${\displaystyle \left.L\right.}$
2. a number of lattice positions, ${\displaystyle \left.M\right.}$ given by ${\displaystyle \left.M=2m^{3}\right.}$, with ${\displaystyle m}$ being a positive integer
• The ${\displaystyle \left.M\right.}$ positions are those given by:
${\displaystyle \left\{{\begin{array}{l}x_{a}=i_{a}\times (\delta l)\\y_{a}=j_{a}\times (\delta l)\\z_{a}=k_{a}\times (\delta l)\end{array}}\right\}}$

where the indices of a given valid site ${\displaystyle (i_{a},j_{a},k_{a})}$ must fulfill:

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

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

## Atomic position(s) on a cubic cell

• Number of atoms per cell: 2
• Coordinates:

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

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

Cell dimensions:

• ${\displaystyle a=b=c=l}$
• ${\displaystyle \alpha =\beta =\gamma =90^{0}}$