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

Question

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?

Answer 1

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.

Lattice sum, as given here, means the sum over all lattice points in the unit cell. You can also sum over the entire crystal for calculations of the total intensity of the diffracted beam.