which is correct? Half hour or Half an hour

In my own scholarly writing, I use both (but usually involving distance rather than time).  If I want to represent an approximate distance, I will use "The grain warehouse was half a mile south of downtown." If I want to indicate more precision, I will use "The junction is one-half mile past the bridge."

Jonathon,  you  misplaced your hyphen in your example: One half-hour after the suspect ...

Two rules of hyphenization come into play. The first -- compounds formed with fractions -- says that when the compound (half hour) functions as an adjective before a noun, it is hyphenated. When serves as a noun, it is left open.  a half hour versus a half-hour session. In Jonathon's example, half hour should be left open because it does not modify a following noun..

The second rule involves simple fractions with numbers. Noun, adjective, and adverb forms are hyphenated except when the second element is already part of a hyphenated compound. So one-half hour, but a one half-hour session.  

Both are correct. It depends on the context. Usually in formal writing or speech, "half-hour" is used:

One half-hour after the suspect was seen fleeing the area, she was apprehended by Constable Grayson.

... whereas "half an hour" would be used in a more casual setting:

Where were you? I've been waiting for half an hour!