I'm currently working on a taught MA in history. I've identified a potential PhD supervisor at a university, but I'm not sure how to go about contacting him. I only have a general idea of what I want to study for a PhD and I know I need to narrow it down in order to make a specific research proposal.

My MA supervisor said I should send an email to the potential PhD supervisor and that he might be able to help me focus my research ideas. But I'm not sure how to phrase the email. So far I have:

"I am interested in applying for a PhD in history at the University of ____. My research interest is ________. I am currently researching for my MA dissertation on ___________. I would be interested in expanding that type of research to other figures of the period. Would you be able to supervise a dissertation on this topic?"

Is it okay to ask a potential supervisor if they can help you focus your ideas? Also the university I'm interested in doesn't require that you identify a supervisor in the application. The application deadline to apply for the program and funding is May 27th so I have just over 2 months.