My father died of cancer earlier this year. I am trying to finish a math PhD thesis.
I informed my advisor almost immediately (by email) and told that I would not be available for some time. This was not questioned. I should have told of the situation earlier, but the end can come quickly. I did not have teaching duties at the moment. My relationship with my advisor is good. If you have a troubled relationship with your advisor, then things may be more complicated.
I was away from the university for maybe two weeks - I don't recall exactly. After that performance has been spotty. The sorrow returns every now and then, even if it usually stays out of consciousness.
- If you have a precise set of courses you should take at precise times, then you should try to negotiate them with whoever is appropriate. Ask for help freely and explain your situation honestly. This may involve crying.
- Also try negotiating any teaching duties you may have. Around here the common way is to swap some teaching with someone else; you might want to arrange this before you are in hurry. Select someone who can replace you and whom you trust. Tell them what is going on and ask if they would teach in your place for some days on short notice, if necessary.
- If possible, try to stop your research at a suitable point. You can write an introduction to an article without your full mental capacity. You can also submit a paper to arXiv and to journals and consider changes suggested by referees. Finding more references is also not terribly taxing. Try to have such easier, but still relevant, tasks to return to. Also, inform any co-authors about having personal problems and not doing anything with the paper for at least a month or two.
You may want to find out how a formal break from PhD studies works at your university.
At least in Finland there is a fair amount of paperwork to do (hopefully you are not doing them alone) after a death. If you need to sell a house or apartment, there will be even more. Do not underestimate the emotional side of dealing with your mother's possessions and home.