Do atomic nuclei have fixed orientation and position with respect to the electron cloud? Does the nucleus of an atom of any element (besides protium) maintain a particular orientation in relation to its electron cloud? Does it bounce around within its cloud? Can it be ejected from its cloud outright by the impact of a high-energy particle or sudden entry into a strong negative electric field?
 A: The position and orientation of a nucleus behave rather differently.


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*In general, the position of the nucleus is at the centre of the electron cloud, essentially by definition. To get the dynamical equations that govern the atom, it is necessary to factor away the position of the centre of mass, so that the internal degrees of freedom get separated from the overall motion in space. This is done by taking as a degree of freedom the coordinates of all the particles with respect to the centre of mass. However, because nuclei are much heavier than electrons, this means that the centre of mass is essentially frozen to the nucleus, and that therefore the electron cloud will follow it around pretty faithfully. That is, the nucleus is always in the centre of the cloud unless it accelerates very rapidly.

*The orientation of the nucleus, on the other hand, is very hard to even define. Because the nuclear dynamics are invariant under rotations, typical nuclei tend to have definite angular momentum but not definite orientation. Those two variables are exactly like position and momentum in quantum mechanics, and they are governed by a corresponding Uncertainty Principle. This means that unless you make specific state preparations, you can't even speak of the nuclear orientation: it is pointing in all directions at once, (i.e. it is in a (specific) superposition state of all  the possible orientations).
On the other hand, nuclei tend to spin. This can come from the individual spin of the protons and neutrons, or from their orbital motion inside the nucleus, or from a combination of both. Either way, we can't really tell what the state inside looks like, but we do know that the ground states of different nuclei have different angular momenta.
Thus, for example, in hydrogen 1 (protium), the nucleus has spin 1/2. In contrast, the ground state for deuterium has zero angular momentum because the two spins cancel out. Certain large nuclei can have angular momentum quantum numbers of 5 or more. For zero angular momentum, the nucleus is completely isotropic, and it looks exactly the same if you rotate it. 
The higher angular momenta do have some sense of "direction", i.e. the axis the nucleus spins around. It is possible to control this direction, using strong magnetic or electric fields, very low temperatures, and sophisticated sequences of laser pulses, but in general it will be pointing in an arbitrary direction with respect to the outside world and the electron cloud around it. (As you get more comfortable with quantum physics, hyperfine structure is the term to look up.)
It is in general very hard to 'eject' a nucleus from its cloud of electrons using a collision, because if the collision is energetic enough to do this, it is also likely to initiate a nuclear reaction which will essentially break the nucleus in pieces. Once that happens, though, the pieces are very likely to fly away at such high speeds that only a few electrons, at most, can cling to them.
Sudden entry into an electric field is not really possible, because electric fields are constrained in how fast they can grow in space. The equivalent is to try and yank out the nucleus using a very strong laser field, but then you're simply going to strip away most of the electrons and be left with a nucleus and a small electron cloud.
