  We measure the donor-bound electron longitudinal spin-relaxation time ($T_1$)
as a function of magnetic field ($B$) in three high-purity direct-bandgap
semiconductors: GaAs, InP, and CdTe, observing a maximum $T_1$ of
$1.4~\text{ms}$, $0.4~\text{ms}$ and $1.2~\text{ms}$, respectively. In GaAs and
InP at low magnetic field, up to $\sim2~\text{T}$, the spin-relaxation
mechanism is strongly density and temperature dependent and is attributed to
the random precession of the electron spin in hyperfine fields caused by the
lattice nuclear spins. In all three semiconductors at high magnetic field, we
observe a power-law dependence ${T_1 \propto B^{-\nu}}$ with ${3\lesssim \nu
\lesssim 4}$. Our theory predicts that the direct spin-phonon interaction is
important in all three materials in this regime in contrast to quantum dot
structures. In addition, the "admixture" mechanism caused by Dresselhaus
spin-orbit coupling combined with single-phonon processes has a comparable
contribution in GaAs. We find excellent agreement between high-field theory and
experiment for GaAs and CdTe with no free parameters, however a significant
discrepancy exists for InP.
