nope. The size in meters of a particle has nothing to do with how many electrons/protons get ionized, that only has to do with the charge of the particle that flies through matter, to first order. And yes, it is definitely not every atom that gets ionised, the probabilities are really small.
But charge is much easier to measure than displacement. That is exactly why we don't use the size. It is much much MUCH more likely that an atom is ionised some way than that a particle actually hits (and then displaces significantly) a proton, neutron in an atom.
I did find an error though, those 10000s of ionisations (which depend heavily on material that the particle flies though) should be per mm, not cm. But that is a factor 10, not millions
(ps: don't mix SI and imperial units)
You didn't say it was an ionized cloud you said it was a solid.. 1 cm cu. which 1 cm ^3 is 1000mm^3
Thats 3 decimal places there.....Your math is really bad....Almost as bad as my writing!
That being said...
I understand that electrons have huge orbital paths around the nuclei containing protons... and atoms do not go into each others space....
But they are to small to see, and I have worked with microscopes for silicon wafer handling inspection magnifications of 100,000x ...
and have seen 100,000 nand gates in one layer of a chip in a square mm on a single layer....I'm certainly not looking through an electron microscope, just a plain old leitz glass industrial clean room microscopt.
So you claim of a few thousand particles is what makes up cm^3 or even a cubic mm^3 ....
IS SPECIOUS ....
Where do you teach? Obviously not college...