Write a letter to a professor , who you found to be working in your research area, seeking him for a 2 month student internship during your summer vacation.
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Answer:
The first paragraph. Introduce yourself saying your name, where are you studying, which year of which program and your major/minor program, etc. End the paragraph by stating the purpose of your email.
(3) The second (and may be the third too) paragraph(s): This is crucial. You should write a short “story” to convince why you are interested in the sort of research work that the professor is doing. How did you get to know about their work, any previous reference like I attended your talk at so and so conference, I spoke to you briefly on that corridor, or how you found the website of that profesor, read one of his/her paper (that is one of the most convincing ways to demonstrate your interest) and found the work interesting. Do you have past experience in research? Or do you have your own idea? Do mention and explain it briefly (or may be even attach a short write up).
Of course, the level and quality of the story expected of you will vary depending on whether you are a first year UG student and a final year student, your background (an ecology student writing to ecologist can be expected to write differently compared to a mathematics student writing to get exposure to ecology). But the main point is, is your story compelling? You should present a convincing story of how given your background and interest, the lab you are applying is an appropriate one.
(5) Logistics: How long do you want the project to be? Tentative dates (note that lot of faculty travel in summer)? Do you have your own funding (Are you any of the KVPY, INSPIRE, IAS fellows)? If not, are you expecting the professor to fund your stay or are you self-funded?
(5) CV: Attach your updated CV (in pdf, not MS word). And mention that CV has been attached!
End your email thanking the professor for their time, and that you look forward to their reply.