We don't “cut an atom” and get its parts, as an atom is by definition non-cuttable (“a-tomos” in Greek). What we do is, we raise the energy of the atom at a level to which it's not stable anymore, and then we see what elementary particules we get from the destruction of the atom.

Which is to say that the elementary particules before we break the atom, were not existing on the first place and in the physical world. They were more like a “potential” that may or may not be realised into those particules. And what realized this potential into concrete reality was the energy we gave it.

We call those potentials “fields” or “quantum fields", and they don't look like anything we know in the physical world in everyday's experience. We use lots of Mathematics to deal with them, and those Maths are really and annoyingly abstract, so that it's very very hard to talk anymore correctly about them.