I learned that at 0K atoms should stop moving. But do particles inside the nucleus still move ?
Where did you learn that?
Atoms do not move in a classical sense at 0 K: they do not change their place. They still have a non-zero expectation value for their kinetic energy. The same applies to all subatomic particles.
Two things. 1. Atoms still move due to zero point energy. 2. 0K is an idealization that can't quite be reached.
So there is no way to make unstable atoms stable by cooling down ?
Unstable nuclei? No, their stability does not depend on the temperature of the environment.
Unstable electronic configurations? If you have those, the system cannot be at 0 K.
I think the confusion lies in what 'temperature' is. Temperature is the average *kinetic* (motion) energy of a system. As the system cools, the total kinetic energy in the system drops to zero. But this is talking solely about objects as complete entities (ie: atoms as atoms, electrons as electrons when they're freely moving on their own, etc). So even if you could cool an atom to absolute zero - that would merely imply that the atom is completely stationary (in its local frame - I really don't want to get into relativity here :) ). The movement of its electrons, protons and neutrons do contribute to the kinetic energy of the atom as a whole.
However, even THAT is a problem. Because of quantum mechanics, you cannot know both the position and momentum of a particle with infinite accuracy. If your view of absolute zero were correct - you'd know both exactly (you'd have its position since it's not moving - and you'd know its moment perfectly - it's zero) - which would violate Heisenburg's Uncertainty. In fact, if you could stop a particle (reduce its momentum to zero), it's position would infinitely indeterminate - it would spread out until it's a little bit everywhere. Conversely, if you can hold it in one place, it would gain an effectively infinite momentum which would allow it to escape by tunnelling.
But neither of these things are caused by changes in kinetic energy so they can't be considered 'temperature' in any normal sense.
It is not.
Everything is stationary in "its local frame" by definition.
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