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I too calculated the nearest neighbour for the atomic radius. I think that's probably a more important measurement than the unit cell volume/theoretical density. As for why it prefers a BCC phase a room temp, I have no idea. As you say, a calculation of the free energy is the real answer, but even then you have to know what to account for! I'll try and follow up with those metallurgists to see if they have an indication of what the real driving force is (bond energy/length? configurational entropy? magnetisation?).
hamrack-
So far i like most the 'empty d orbitals' argument from one of the links in your first post. I'm less keen on the 'magnets' explanation because the Curie temperature doesn't line up with the transition to FCC, so what's holding it BCC between the Curie point and the phase transition?
Maybe it's something like this: at low temperature iron's d orbitals favor the BCC structure, but its metalness objects to the poor packing efficiency and the atoms get squashed in closer to each other than they'd really like to be. As it gets hotter the atoms are jiggling so much that the d orbitals aren't lining up enough of the time to claim the energy benefit from it, so close packing wins. Hotter still and the d-orbitals conspire with entropy (which favors things being spread out) to switch back to BCC before it melts.
moth
I can't quite tell if you're taking account of FCC unit cells having more atoms per cell than BCC. (8 8ths + 6 halfs = 4 for FCC vs. 8 8ths + 1 = 2 for BCC.) So FCC is denser despite the larger lattice parameter. I did the maths on the nearest neighbor distances from your link above, and got 0.258 nm for FCC and 0.248 nm for BCC. (The BCC number is room temperature, but the 'Steels' book doesn't say what temp the FCC number relates to.)
Still thinking what to make of this all. I fear we'll need to dig deep into how to calculate the phases' free energy to get much further, and that might need both more quantum mechanics and computing time than i have access to.