I am doing my undergrad in engineering at a small liberal arts school. Due to lack of research opportunities at my current school, I am looking at getting a research position at a state university nearby. My question is, how typical is it for professors to let students from other universities join their labs? I want to send an email to a professor that I am interested to do research with, but I am not sure how to approach him in my email.
I'm playing professor psychology here; may serve as another approach for you to think about.
If it's an experienced professor and the school is large, your cold calling e-mail may not stand out (unless you have some stellar skill sets or experiences.) If it's a relatively new professor, hiring someone outside his/her school may be a deterrence.
My suggestion is this: talk to your school to set up a cross-registration arrangement and take a course at the state university. Preferably, take a course that is taught by the professor with whom you would like to work.
This gives you a few edges: you'd get to see if commuting to two campuses work for you, you'd get to check out all the resources and feel the campus culture there, and you will get to know the professor better, and be able to make a personal contact with him/her. Get a good grade, and use the semester to build a relationship with the faculty there.
Once the professor got to know you and understand your aspiration and capability, it'd be a lot easier for the professor to engage and make a decision.
If the professor you propose to contact has relatively large labs with a number of researchers, your chances may be good. Less so if the students typically work as individuals.
In the former case a letter (maybe an email, but something more formal is better) to the professor is entirely appropriate. Detail your interest, your background and your skills.
However, before you send the letter get buy in from either your department head or other faculty on your idea. Then in your letter you can tell the prof to expect a supporting letter from Professor Xyzzy.
If the professor is interested, expect to be invited for a look around the lab, meeting other students and perhaps an interview. Be prepared for that. If your current faculty studied with this person, or otherwise knows him/her they can, perhaps prepare you for what to expect.
I doubt that, as an undergraduate, you would be asked to speak at a colloquium or other meeting, but it is (barely) possible.
Direct approach is good. Backup is also good.
Perhaps your contact might lead to a more formal arrangement that would benefit others. Your department head might be interested in that idea.
You should feel absolutely comfortable approaching professors you find interesting. Writing something as simple as:
I am looking for research opportunity and always was fascinated by [your field of work]. Can we discuss possibility of me working in your lab?
At the same time, talk to your academic advisor at your school. They will definitely have an idea how to connect you with the state school. On the same note, contact state school's department administration (academic advisor or something) in order to get details on paperwork and process.
Best case scenario is that you'll be able to get paid to work in the lab. Worst case scenario you'll do it for free.
Chances are that the professor will miss your email. That is why it's a two-step process: direct contact and via administration. Both are important