When does the structure function have parity symmetry?

I'm working on a magnetic system and in the calculations this function appears

$$\gamma(\vec{k})=\frac{1}{Z}\sum_{\vec{\mu}}e^{i \vec{\mu} . \vec{k}}$$

where $\vec{\mu}$ are the primitive vectors of a Bravais Lattice. In the book Spin waves: theory and applications by Anil Prabhakar and Daniel D. Stancil it is said that

For crystals with a center of symmetry $\gamma(\vec{k}) = \gamma(-\vec{k})$

What does this mean? When can I say that the lattice has this symmetry? By looking at the definition of $\gamma$ it seems to me that this symmetry only happens when for every $\vec{\mu}$ there is a $-\vec{\mu}$ in the base generating the lattice but I'm not sure.

By looking at the definition of $\gamma$ it seems to me that this symmetry only happens when for every $\vec{\mu}$ there is a $-\vec{\mu}$ in the base generating the lattice but I'm not sure.
It is exactly like that. If the crystal has a center of symmetry $O$, it means that for any vector $\vec{OP}$ connecting the center of symmetry to a lattice point, you will have a corresponding vector $-\vec{OP}$ connecting it to another lattice point.
For a perfect crystal, $\gamma(\vec k) = \gamma(-\vec k)$.
• Yes. For every perfect crystal, $\gamma(\vec k) = \gamma(- \vec k)$. I added this to the answer. Dec 27, 2017 at 15:16