# Bandgap in a weak periodic potential

1. Nov 6, 2008

### thesage

Using the Schrodinger equation for an electron in a periodic potential where U(r +R) [R is the translation vector R=n1a1+n2a2+n3a3 and ni are intergers and ai are teh primitive lattice vectors, G is for reciprocal lattice G=n1b1+n2b2+n3b3 and ni are intergers and bi= (2PI*aj x ak)/(a1 . a2 x a3)]

a)Show that the periodic potential can be expanded as

U(r)=SUM (over G) exp(iG.r) . U(G)

show the potential is real and is reflection symmetric U(-r)=U(r)
show that implies U(-G)=U(complex conjiguate)(G)=U(G)
potential is chosen as U(G=0)=0

2. Relevant equations
I hope you know the SE for a periodic potential......

3. The attempt at a solution
can i jump straight to the fourier transform for the U(r+R)
f(x)=SUM(over m) exp(imx)f(m)
bcause after that it's just loosing the
exp (iR.G)=1

I don't get the complex conjugation of U(r) = U(r) unless the "i" in the exponential isn't changed then there wouldn't be a complex part......

U(-r) = U(r) cos it's jsut a translation in the real lattice which is stated in teh question.

It implies that U(-G)=U(complex conjiguate)(G)=U(G) cos the form of U(r) has U(G) in it