The Doctor of Science (ScD) degree is 'the highest degree awarded by the University for distinguished research in science', according to the very helpful web pages of the Degree Committee of the Faculty of Physics and Chemistry. It is only open to candidates who already hold a Cambridge degree, and of order of a dozen or so are awarded each year as far as I can tell.
When I was planning to apply for the degree it was quite difficult to assess what the standard was. Applicants who currently work in a Cambridge institution can probably find someone to advise them, but if you're now outside the system, it's more difficult. So I'm making this brief description of the process as I experienced it publicly available in the hope that it may be of some use to others.
I applied in 2017, 20 years after I completed my PhD and 21 years after my first refereed publication. Typical successful applicants, judging from web searches, seem to be in their 40s and up.
I submitted around 110 publications, out of a little over 200 that I had at that time. I only submitted, according to the application form, "refereed publications where I was the lead author or, for a student project, was the originator and supervisor of the project or made a substantial contribution to the paper through data analysis, software development or leading an aspect of the work reported." — but I submitted all such publications without any other selection. I was the first author (lead author, by astronomy convention) on a little over half the papers I submitted, and stated contributions to the papers ranging from 20% to 100% (for single-author papers). Papers based on my PhD work were included.
The process took almost exactly a calendar year from submission to approval for the degree. The acknowledgement that you have a prima facie case comes quite quickly; the delay is in waiting for the external referees. Having applied in September 2017, I graduated in January 2019.
I was told at graduation, though obviously I haven't seen referee reports and can't tell how polite my presenter was being, that the application had been a strong one, which I take to mean that the standard is such that I didn't just scrape through and could have applied sooner or with fewer publications. If you rely on this and are disappointed, don't blame me; it's just an impression.
The picture to the right was taken by my wife at graduation. No comment necessary, except that, of course, I am wearing in the picture the undress gown and the hood of the highest Cambridge degree I held as I walked into the Senate House, i.e. the PhD. The ScD festal gown is far more impressive but you don't wear it at graduation.
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