The epitaxial Bain paths (EBP's) of tetragonal Fe are found by minimizing t
he total energy with respect to c at each a using first-principles total-en
ergy calculations with the local-spin-density approximation plus a generali
zed gradient approximation and relativistic corrections. The energy minima
along the EBP's give the equilibrium states of the ferromagnetic, nonmagnet
ic and two antiferromagnetic phases-the type-I phase, called here AF1, and
a phase in which pairs of ferromagnetically coupled (001) layers alternate
in moment, called here AF2. The A.F2 phase is found to be lower in energy t
han AF1 when each is in equilibrium and also when the (001) plane lattice c
onstant is strained to the Cu(001) lattice constant. Calculations of the si
x elastic constants of AF2 at equilibrium inserted in the four tetragonal s
tability conditions show that it is unstable for [100] and [010] shears in
the (001) plane, but is stabilized by epitaxy on Cu(001). The AF2 phase is
indicated to be the phase of the bulk of epitaxial Fe films on Cu(001). The
bulk structure of Fe films on Cu(001) found by quantitative low-energy ele
ctron diffraction analysis is shown to be close to the EBP's of the AF1 pha
se, but farther from the AF2 phase. On substrates with slightly larger latt
ice constants than Cu(001), the ferromagnetic phase is found to be more sta
ble than AF2.