I did my bachelor at university A in which Prof. X supervised my graduation project. I asked him for a recommendation when I was applying to university B for masters and he wrote me a great one. That was a year and a half ago, and now I am applying for PhD at university C. I need his recommendation again but I feel bad for not emailing him for a long time. I don't know how to write my email. I don't want to jump directly for the recommendation. Could you share your suggestions on what to write ??
After a decade of teaching, I get these quite often. While people use a variety of styles, the one that I am most responsive to is:
Once you get the ok, make it as easy as possible for me to write the letter. If there are things you want me to emphasize, be blunt about saying it ("Please don't mention the folly of FTL dog-walking" or "Please emphasize my familiarity with temporal dilation and astrophysics").
If someone has already written you a recommendation letter in the past and you want them to write another one, the key question is: has the old letter become obsolete, or can it (in essential content, anyway) still be used?
If the answer to the above question is yes, I think you're golden: just thank the professor for writing the recommendation before and ask them if they are willing to write an updated copy. They will have the same question as above, so be clear to them that you are not looking for them to say anything they haven't already said.
If the answer is no -- significant updating is required -- then it becomes a "bigger ask". If the additional information is something that you feel like you can faithfully supply to them in the form of written documents -- e.g. updated CV, new papers -- then you should enclose them along with your email request. If you really need a qualitatively different letter, then an in-person visit is probably in order, and if that seems impractical and/or doesn't go well, then perhaps you should really be looking for someone more au courant to be writing your letter.
In your case, I suspect that the affirmative is more likely to apply: at least in my field and location (mathematics in the United States) applying for a master's program is essentially identical to applying for a PhD program. Even assuming you did a master's degree, getting a letter from someone who only interacted with you as an undergrad and who only addresses your undergraduate career would be totally appropriate. (You should however try to get at least one letter from your most recent program or job, if at all possible.) In this case, the letter is likely to begin by mentioning that the professor's interactions with you were limited to the undergraduate program at his university and mention that he can't speak to your master's program but that someone else will. After that he can really cut and paste the former letter, if he likes.
If I have learned one thing in my time on this site, it is that the world is a big place and there are relatively few universal tenets of academic culture. But in the part of academia I am familiar with (again, mathematics in the United States), your request for an additional letter is absolutely standard. I would not view a year and a half as a "long time" -- as I mentioned above, the determining factor is not the number of years that have passed but to what extent the letter needs / is desired to be updated: that's what takes up the professor's time and effort -- nor do I see lack of contact in the interim as anything to be embarrassed about in the slightest. By not jumping directly to the recommendation request you risk taking up a busy person's valuable time by giving them more text to read. I would begin with a short paragraph of the "Hello there? Remember me? I hope you are well" variety -- two or three sentences would be sufficient. And then yes -- jump to the recommendation request. Writing these letters -- and rewriting them in the intervening years -- is part of our job, after all.
How about (drum roll) the truth?
Paraphrasing you: "Thanks for the recommendation letter you wrote me when I was applying to university B for masters. [Tell him how it went and what you did.] Now I am applying for PhD at university C, and I want to ask your recommendation again, but I feel bad for not emailing you for a long time."
Continue from then on. He may be good people and help you out. What's the big deal? Piece of advice: stop thinking of (lowercase) professors as the aristocrats of knowledge.