Can I move the atom nucleus only? I was wondering if it is possible to move the atom nucleus and leave behind the electrons? I can imagine that the electrons will follow the nucleus. But what if the speed of the nucleus is almost the same as the speed of the electrons or faster. where will the electrons go?
If it is not possible, do we have a theory I can read to explain what could happen?

(Edit: as of the comments, "core" actually refers to "nucleus" -changed)
 A: To access the nucleus separately from the atom, you need to affect it by a "high-energy probe", one that has a good enough spatial resolution. For example, you may hit the nucleus with another nucleus or X-rays etc. When you do so, it's like hitting it with a small bullet, and if you hit the nucleus with a small bullet, the electrons will continue in their motion almost undisturbed. However, because the nucleus will be kicked away rather quickly, the electrons find out they are no longer parts of a bound state, the atom. So the atom will be ionized: the electrons will be "liberated". It's almost an inevitable consequence of the high-energy manipulation because the energies needed to manipulate with nuclei are multiples of 1 MeV or so, about 1 million times greater than atomic bound state energies that are comparable to 1 eV.
A: Not only is it possible to remove the nucleus from an atom, but the RHIC does it every day!
The RHIC collides heavy nuclei like gold to measure the properties of nuclear matter at high densities. Gold atoms have their electrons stripped off in the Tandem van de Graaff accelerator. The atoms are subjected to such strong electric fields that the positive nuclei and negative electrons are pulled apart.
Response to comment:
See http://isnap.nd.edu/research/facility/accelerator/ for a few more details on how the atoms can have their electrons stripped off (this is a different accelerator from the one at the RHIC). You start with singly ionised atoms. These are easily made e.g. by shining ultraviolet light on the atoms. The singly ionised atoms are accelerated to a high speed than crashed into a very thin carbon sheet. The heavy nuclei plough straight through while the electrons are scattered, and the nuclei are then accelerated away with a second electric field.
A: There is a nucleus and there is a core--- they mean two different things. You can knock a nucleus out of an atom by smashing it with a heavy particle, or with a fast neutron, or if it undergoes fission, or in many other ways. This is adressed in the other answers.
But the core of an atom is the nucleus plus all the electrons in inner shells. If you were to remove the core of the atom, you would leave behind only the valence electrons, It might seem that it is possible to knock the core out of an atom, because the inner electrons are more tightly bound than the outer electrons by a factor of order unity.
But there is no real mechanism for doing so suddenly without disturbing the atom. The reason is that a fast particle can knock out the nucleus, but it is very unlikely to knock out the remaining core electrons so that they move along with the nucleus--- the phase space is very small.
This has a practical effect--- it means that the energy required to knock a nucleus out of a heavy-atom crystal by collision is much larger than the energy required to completely ionize away the outermost electrons of the atom in vacuum. This means that K-shell vacancies in a heavy metal can only knock out H-isotopes from their lattice positions, where there is no core, not heavy atoms, because to knock out the heavy atom nucleus requires making two K-shell (innermost S-shell) vacancies, and this is at least twice the energy of the K-shell excitation. So although a vacant K-shell has enough energy to knock out the core of the heavy metal atom, the coherent process which does this is unavailable since it requires a conspiracy which simultaneously knocks out the nucleus and the two K-shell electrons at the same time into nearly the same k, and this is a small corner of the outgoing phase space.
This is not really said in the literature, but it is certainly understood in atomic physics. The knock-out interactions are taken particle by particle, and do not involve the atomic cores.
In your question you use "core" to mean "nucleus", so the other answers are more appropriate, but I think it is interesting to adress the question as stated.
