Atoms don't vibrate, at least not in any way relevant to an atomic clock. And lattice vibrations in solids are thermal and don't have a well defined frequency. If you read the wiki link V50 provided, it would be clear that atomic clocks don't work the way you appear to think, and your question as written is meaningless.
Electrons in atomic orbitals can absorb energy in various ways and get kicked into higher energy states. They can then decay back into lower energy states, emitting the energy as a photon. Because the energy, ##E##, of a photon is related to its frequency, ##f##, by ##E=hf## where ##h## is Planck's constant, the frequency of the emitted photon is defined by the energy difference between the states. You can then use the photons' oscillating electromagnetic fields as "pendulums" in a very precise clock. (Glossing over a lot of technical details here.) So it's the electromagnetic field oscillating, not the atom.
Caesium has one transition in particular that was convenient to work with and convenient for defining a second that was consistent with the historical definition of the second, but is otherwise fairly unremarkable. All atoms have electrons; all can be kicked into higher energy states and decay. I don't know if there's a list of them all anywhere. The Lyman, Ballmer and Paschen series of hydrogen transitions are quite famous. The sodium D line (wavelength 589nm) is also quite famous.
