Hehe, well, thanks for the correction. Yes on all of these things, I'm sure.
I should have said "the exact solution for the most basic justifiable approximation". Honestly, it's been 9 years since I last did quantum seriously, so that was the extent that I remembered.
Deriving Bragg's law in front of my quals committee and refusing to use "mirror planes" despite it being a Materials Science Ph.D. was just viscerally satisfying. It may not have been absolutely correct math, but at least it was way more justifiable than the typical accepted "solution" in my field =)
No, this is only under the 1st born approximation, in the far-field and without the 'dish' shape intrinsic to propagation. A lot of times this model doesn't work. Also now you are in the momentum space which doesn't map immediately to the x,y,z coordinates you might have started with. Also the evanescent fields that the Ewald sphere didn't include...