Fourier component crystal potential - physical significance

[aaaa202]

I have done several exercises concering periodic potentials in crystal. Especially I did one, where I had to show that the Fourier component of the shortest reciprocal lattice vector (call this vector a) in the z-direction was zero. Now solving the problem was just about writing up the right equations (structure factor of basis etc.)
But it made me think? What is the physical interpretation of the crystal potentials Fourier component of some specific reciprocal lattice vector? And what is the physical interpretation of it vanishing?
I also had to show that the Fourier component of 2a was not zero, which was not hard. But why is it that this component does not vanish and the component of a does? I don't understand it physically..

 

[DrDu]

You should have a look at the specific structure you were considering. Try to sketch the arrangement of atoms in z-direction.

 

[aaaa202]

The specific structure is as indicated on the picture, and I am quite sure that the shortest vector parallel to the c-axis is the one connecting the two coloured atoms. But that still doesn't give me much intuition.

 

[DrDu]

That's a hexagonal close packed structure and I think the picture is quite obvious: It is a stacking of identical close packed sheets which are shifted to each other, i.e. ABABABAB... Now, if you take the Fourier transform with a vector parallel to the z-Axis, this involves averaging of the potential along the sheets and then multiplying each sheet with exp(ikx), with k=2π/a so schematically:
$\sum \bar{V_A}+\bar{V_B}\exp(i2(\pi/a)\cdot a/2)=\sum \bar{V_A}-\bar{V_A}=0$. In the second step I used that the average potential of the sheets is identical.