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Let's say I sum an exponential function $e^{\imath \left(k-k^{\prime}\right) x_{i}}$ over a chain system where every member of the chain is of the same type, e.g.,

A-A-A-...-A-A (total of N sites)

I would obtain (typical textbook example)

$$\sum_{i=1}^{N} e^{\imath \left(k-k^{\prime}\right) x_{i}} = \frac{1}{N} \delta _{k;k^{\prime}}$$

When we consider alternating site types and sum only over sites of type A, i.e.,


what would the sum look like?

$$\sum_{i\in A} e^{\imath \left(k-k^{\prime}\right) x_{i}} = ?$$

Note: The sum is now over $N_{A}=\frac{N}{2}$ sites. Also, the fact that it's a chain is just an example, I could also consider 2D or 3D grids like in crystals. The only difference would be that $k$, $k^{\prime}$ and $x_{i}$ are vectors then.

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Hint: how do you sum geometrical series? – Piotr Migdal Jul 11 '12 at 15:04
I realized after trying to figure it out that I asked the wrong question... – Robert Hoeppner Jul 12 '12 at 18:01

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