  We unravel theoretically a key intrinsic relaxation mechanism among the
low-lying singlet and triplet donor-pair states in silicon, an important
element in the fast-developing field of spintronics and quantum computation.
Despite the perceived weak spin-orbit coupling (SOC) in Si, we find that our
discovered relaxation mechanism, combined with the electron-phonon and
inter-donor interactions, dominantly drives the transitions in the two-electron
states over a large range of donor coupling regime. The scaling of the
relaxation rate with inter-donor exchange interaction $J$ goes from $J^5$ to
$J^4$ at the low to high temperature limits. Our analytical study draws on the
symmetry analysis over combined band, donor envelope and valley configurations.
It uncovers naturally the dependence on the donor-alignment direction and
triplet spin orientation, and especially on the dominant SOC source from donor
impurities. While a magnetic field is not necessary for this relaxation, unlike
in the single-donor spin relaxation, we discuss the crossover behavior with
increasing Zeeman energy in order to facilitate comparison with experiments.
