Plane waves and sites distance in a lattice

[AdeBlackRune]

Hi, i would ask you an opinion about a (maybe stupid) doubt.
Let us think of a 1D lattice whose sites distant from each other "a"; a plane wave
in the lattice is given by $e^{ikja}$ where k is the momentum and j an
integer label for each site. Now, we modify the lattice in this way: between sites j
and j+1 we put a distance that is a+c (instead of a). Then, what is the effect of this
alteration on the plane wave? It is right to say that the only effect is a shift in the
phase of the wave before and after the sites j,j+1?

 

[AdeBlackRune]

no one has a little suggestion ç_ç?

 

[Bill_K]

You've introduced a lattice defect. As you say, there will be a phase shift added to waves on the right.

Now consider a wave incident from the left. When it gets to the defect, it will have a problem. The only way to satisfy the matching conditions (both ψ and ψ' continuous) is to add another wave traveling to the left. Thus the defect will cause incident waves to be partially transmitted and partially reflected.

 

[AdeBlackRune]

Thanks! So such a defect introduce a non zero reflection amplitude and unlukily it is not the answer i hoped to hear :(
The real problem I'm trying to resolve is to physically justify the introduction of a global phase factor in the S-matrix. I hoped that such a defect could help me but, as you say, I'm wrong. Do you know any way to introduce a global phase in a scattering matrix?