Darth Bandon said:
My idea is that particles are fired through matter, and the particles record the following:
The position of the atom
The element of the atom
Yes, these things are routinely done in physics labs.
Atomic positions (aka crystal structure) are determined by diffraction measurements (electron, neutron or x-ray diffraction) or by electron microscopy (look up TEM, STM). All of these methods involve shooting particles (electrons, neutrons or x-ray photons) at the sample of interest, and recording the effect of the sample on the emerging particles.
Elemental analysis is also possible with a standard XRD (x-ray diffraction) or TEM (tunneling/transmission electron microscopy) set-up. A simple diffractogram itself tells you what material you are analysing, but it's not great for telling you about components that make up less than 2% of the sample mass. Specifically for elemental analysis, these instruments are additionally outfitted with an EDS (for energy dispersive spectroscopy, sometime called EDAX - energy dispersive analysis of x-rays) or XPS (x-ray photoelectron spectroscopy) detector that tell you how much of different elements is present in the sample.
<Google the keywords above for more info>
The problem with applying these methods to the Earth is that...well it won't work. The Earth is not a crystalline solid (it's atoms do not have fixed, periodic positions), and it's too big - it'll absorb all the particles shot at it, and any sencdary emissions from its interior. Moreover, much of the Earth is liquid, semisolid or otherwise in continuous relative motion, and the atoms don't stay fixed in place. So, finding the atomic positions for the Earth is completely meaningless.
However, finding the elemental/chemical composition of the Earth is not meaningless, but only impossible. This of course, doesn't mean you can't concoct a scheme for a scifi story. You can either come up with some ultra-high energy photon or maybe some kind of neutrino-based analysis method that you don't go into the details of.
