# What is the lattice sum and how can I calculate it for a general reciprocal lattice vector?

In lecture we were given the following definition:

Lattice sum $$\sum_{n=1}^N e^{-i \mathbf{Q}\mathbf{R}_n}$$

where $\mathbf Q=\mathbf k-\mathbf k'$ is the scattering vector.

First question: What exactly is the lattice sum? Is it something like the sum of all planes or the sum of all lattice points?

Using this expression I want to calculate the lattice sum for a general reciprocal lattice vector: $$\mathbf K_{hkl}=ha_x^*+ka_y^*+la_z^*=\frac{2\pi}{a}(h \mathbf i+k \mathbf i+j \mathbf k), \qquad h,k,l \in \Bbb Z$$

Can I just write: $$\sum_{n=1}^N e^{-i \mathbf K \mathbf R}=\sum_{n=1}^N e^{-i\frac{2\pi}{a}(h \mathbf i+k \mathbf i+j \mathbf k) \mathbf R_n}$$

or does this expression make no sense at all?

The summation given in the OP includes all points within a single unit cell; in this case there are $N$ such points. This looks like the expression for the structure factor, but the weighting for each point is unity, which implies that the crystal is made up of a single type of atom.