Building up a face centered cubic lattice

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A face centered cubic lattice

• Consider:

1. a cubic simulation box whose sides are of length $L$
2. a number of lattice positions, $M$ given by $M=4m^3$,

with $m$ being a positive integer

• The $M$ positions are those given by:

$\{\begin{array}{c}{x}_a=i_a\times (\delta l)\\ y_a=j_a\times (\delta l)\\ z_a=k_a\times (\delta l)\end{array}\}$

where the indices of a given valid site are integer numbers that must fulfill the following criteria

• $0\le i_a<2m$
• $0\le j_a<2m$
• $0\le k_a<2m$,
• the sum of $i_a+j_a+k_a$ must be, for instance, an even number.

with $\delta l=L/(2m)$

Atomic position(s) on a cubic cell

• Number of atoms per cell: 4
• Coordinates:

Atom 1: $(x_1,y_1,z_1)=(0,0,0)$

Atom 2: $(x_2,y_2,z_2)=(0,\frac{l}{2},\frac{l}{2})$

Atom 3: $(x_3,y_3,z_2)=(\frac{l}{2},0,\frac{l}{2})$

Atom 4: $(x_4,y_4,z_2)=(\frac{l}{2},\frac{l}{2},0)$

Cell dimensions:

• $a=b=c=l$

• $\alpha =\beta =\gamma =90^0$