Matrices of simple face and cubic centered cubic lattice

[Petar Mali]

S=<br /> \begin{bmatrix} a &amp; 0 &amp; 0 \\ 0 &amp; a &amp; 0 \\ 0 &amp; 0 &amp; a \end{bmatrix}<br />

for simple cubic


I=<br /> \begin{bmatrix} -\frac{a}{2} &amp; \frac{a}{2} &amp; \frac{a}{2} \\ \frac{a}{2} &amp; -\frac{a}{2} &amp; \frac{a}{2} \\ \frac{a}{2} &amp; \frac{a}{2} &amp; -\frac{a}{2} \end{bmatrix}<br />

for volume centered cubic

F=<br /> \begin{bmatrix} 0 &amp; \frac{a}{2} &amp; \frac{a}{2} \\ \frac{a}{2} &amp; 0 &amp; \frac{a}{2} \\ \frac{a}{2} &amp; \frac{a}{2} &amp; 0 \end{bmatrix}<br />

for face centered cubic

I don't see any logic for this matrices? How can I get this? I can axcept that simple is P because minimum distance between neighbors is a. But But what with the other two matrices?

 

[Mapes]

How are these matrices defined?

 

[kanato]

Those matrices have the lattice vectors as rows (or columns, since they are symmetric). If you have a cube with side length a and one corner at the origin, then integer combinations of those vectors give you the lattice points. I'm not sure what your question is. Look at FCC. You have one atom at each corner, so points like (a,0,0) and (0,a,0), etc. Then at the center of each face, so points like (a/2,a/2,0) and (a/2,0,a/2). Those points come from adding those vectors together.