Learning matplotlib

I've been doing some calculations today on the structure of some heavy transactinide nuclei.  This has involved running a code (ev8, from W. Ryssens et al, Comput. Phys. Commun. 187 175 (2015)) and manipulating the output data, which I would like to plot in the kind of sector plot that nuclear physicists use when plotting the potential energy surfaces of nuclei using Hill–Wheeler coordinates.

Here's an example of such a plot, which I've taken from the arxiv paper 1809.04406 (also by W. Ryssens et al., though the al is slightly different to the first paper above):

I have all the necessary data and have mapped it to the right sort of coordinates, but I am struggling to get matplotlib to encase it in the right circular sector.  So far, I've got it looking like this:

Any suggestions on how to mask this into a sector, would be most welcome.  I'd quite like to add a labelled axis along the gamma=60º line, too, if you have any ideas.  And add in those grey axis lines.  I can handle changing all the colours and font sizes that I need to do.

I should probably be posting this to stackexchange, right?

edit: This blog post might prove helpful, so I link to it for my future reference: http://blog.rtwilson.com/producing-polar-contour-plots-with-matplotlib/