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Advice for requesting letters of recommendation

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Do you want a recommendation letter from me or someone else for a PhD program? Follow the excellent advice of a Brown CS professor (transposing CS to physics, of course, but not much else):
http://cs.brown.edu/~sk/Memos/Reco-From-Me/
Note especially his description of a "brag sheet" and its purpose. Consider supplementing this by saying to a particular writer "I hope you will be able to speak to my experience at mentoring other undergrad researchers" or "I hope you will be able to remark on my ability to design and build an experimental setup" (just examples.)

The same principles hold for letters for other purposes (internships, coterm programs, postdocs, faculty positions, industry jobs) though obviously what's expected at different stages and for different roles is different.

See also that same Prof's excellent advice to letter-writers for PhD program applications. Read it, follow its guidance on selecting a complementary collection of letter-writers rather than ones who all speak to just one aspect of your capabilities, and consider sharing it with your writers (in a non-pushy way), especially if they may not be experienced at writing for this purpose or this level of institution. 

http://cs.brown.edu/~sk/Memos/Grad-School-Recos/

A personal story: When I was applying for Ph.D. programs and fellowships, I had a research advisor with whom I had done substantial work. I knew my relationship with him was not perfect, but I asked him for a letter -- I couldn't imagine not asking him. He replied that he expected others could write me more effective letters. Decades later I remain deeply grateful to him: I got into an excellent Ph.D. program and even was awarded fellowships. Probably none of these would have happened had he written one of my three letters luke-warmly (or worse.) I was fortunate to have others who could step in to write for me. I take from this several lessons I'd like to share.

1. When someone asks me for a letter and I don't feel I can write a letter that will help them toward their goal, if possible I suggest that they consider having someone else write for them. This scenario doesn't happen often, but my practice gives people a chance to be judged by their best sides, which is important especially early in studies and career. Of course, if one of my primary Ph.D. thesis advisees or postdoc advisees is applying for a faculty position, it may be practically necessary for them to get a letter for me. This is only one reason to maintain good communications between advisor and advisee.

2. If you want a letter from someone, you can't rely on this degree of candor from your proposed letter-writer. Instead of asking "Can you write me a letter?", consider instead "I'm considering applying to XXX. Do you think you could write me a letter that would help me be selected / be seriously considered?" This gives a reasonably gracious way for them to suggest that you ask someone else. They'll probably even appreciate that -- few of us like the feeling of writing a less-than-positive letter. In my experience, if the rest of one's portfolio is strong one can be successful in obtaining Ph.D. admission, postdoc positions, and even interviews for faculty positions based on one or two stellar letters, combined with other letters that are positive but more generic.

3. Months or even years before you'll want a letter, think about who might be possible letter-writers. Develop and maintain the relevant relationships -- not primarily for the sake of the letters!