Expanding the periodic potentials

[hokhani]

Could one always write the periodic potentials in the form:
v(r)=Ʃf(r-G)
where the sum is over G (reciprocal lattice vectors)?

 

[nasu]

r-G does not make sense. They are vectors in different spaces. They have different units.

 

[hokhani]

Excuse me. I wrote wrong.
Can we write v(r)=Ʃf(r-R) where the sum is over lattice vectors R?

 

[nasu]

This looks like superposition. Adding the potentials of each atom, in the coordinate system with origin in the specific atom. Periodicity does not seem to be necessary for this.
f(r-R) will be the potential "produced" by the atom at location R.

 

[hokhani]

Ok, thank you. But I like to know that could we earn this formula merely by having periodicity condition (without regarding atoms)?

 

[cgk]

yes, if the potential v is periodic, then it can be written in this form. This is the most general form of a function v(r) which fulfills v(r+R) = v(r) for all lattice vectors R (i.e., which is periodic).

 

[hokhani]

yes, if the potential v is periodic, then it can be written in this form. This is the most general form of a function v(r) which fulfills v(r+R) = v(r) for all lattice vectors R (i.e., which is periodic).

Could you please prove it or give me a reference which has proved it?

 

[cgk]

Consider a large, finite lattice with periodic boundary conditions. Note that then v(r)=Ʃf(r-R) hold if you put in f(r) = v(r)/N_R itself, where N_R is the total number of lattice points. This also still holds if you set f(r) = v(r) if r is in one single unit cell, and 0 if not.