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 = 2 m^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 \le i_a \le 2 m
  •  0 \le j_a \le 2 m
  •  0 \le k_a \le 2 m

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

Atomic position(s) on a cubic cell[edit]

  • 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