I think the idea is that when an ion acquires a more positive oxidation state (it loses electron density), it becomes smaller. So when atomic volume contraction is observed, it may be that the atoms in the Zintl ions increased in oxidation state. That's really the only reason why I would think that could happen.
You can actually get the physical volume, when you assume an atom is perfectly spherical and you acquire its radius.
Sometimes people perform X-ray experiments, such as X-ray diffraction (XRD) or X-ray Crystallography (XRC). Those often allow you to acquire atomic radii.
- For XRD, you'd receive data these days in the form of
#h#, #k#, and #l#lattice coordinates, as well as a Bragg diffraction angle, and other info. At my university, I literally just tightly packed a powder into an insert, and placed it into the X-ray diffraction instrument. I haven't seen an actual chemical structure from these particular experiments.
- I don't know how XRC works because I've never done it before, but it would give you a crystal stucture in the end, and it's one of the best ways to acquire the structure of a compound so that you can analyze its atomic radii, bond lengths, etc.
From that info, there's a way to solve for atomic radii. I don't remember exactly how to do it, but XRD and other X-ray techniques generally allow you to do so, and it's specifically great for atoms.