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    Describe the complications you would run into if you would try to use perturbation theory to study a system with a nucleus of general shape without having made a multipole expansion first.

    I think to use perturbation theory, we need to know the Hamiltonian responsible for the perturbation. For general nuclei, this is difficult to obtain exactly, and that is why we need the multipole expansion. Each multipole term gives us a bit more info about the nucleus and gives us a breakdown of e.g. total charge, its shape (deformation or not) et cetera, as seen before. We can then treat each term of that series seperately, simplifying the problem. I would also tend to think certain terms have symmetries which can simplify computations, while it may be that the total shape no longer has these symmetries, but this is just a vague idea. Otherwise, the Hamiltonian (if it can even be known) will be really complicated. We can also choose when to stop the expansion, i.e. choose how well we want our approximation to be, how many multipole terms we want to include.

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