How Can We Visualize the Primitive Vectors of a Basic Cubic Lattice?

[Petar Mali]

Do yoh have some nice picture to show why the primitive vectors of basic cubic lattice are

$$\vec{a}_1=\frac{a}{2}(-\vec{e}_x+\vec{e}_y+\vec{e}_z)$$

$$\vec{a}_2=\frac{a}{2}(\vec{e}_x-\vec{e}_y+\vec{e}_z)$$

$$\vec{a}_3=\frac{a}{2}(\vec{e}_x+\vec{e}_y-\vec{e}_z)$$

Thanks!

 

[kanato]

This is the body centered cubic system. It's the middle picture if you scroll down a bit here:
http://en.wikipedia.org/wiki/Cubic_crystal_system

 

[aanidaani]

I'm afraid I don't have a diagram to show you, but it's pretty easy to visualize the primitive vectors by just thinking about it. A primitive vector simply connects two identical lattice points.

So, for instance, the a₃ vector translates one-half a lattice parameter in the +x direction, one-half a lattice parameter in the +y direction, and one-half a lattice parameter in the -z direction. If you start at a corner atom in the BCC structure, this will take you to the body center.

I don't know if this helps at all--and it doesn't really answer your question--but it was useful to me when I first learned crystal structures.