Redfield non-secular master equation governing relaxation of a spin in weak
interaction with a thermal bath is studied. Using the fact that the relaxation
follows the exponential law, we prove that in most cases the semi-secular
approximation is sufficient to find the system relaxation rate. Based on this,
a "secular" form of the non-secular master equation is for the first time
developed which correctly set up one of most fundamental equations in
relaxation investigation. This key secular form allows us to derive a general
formula of the phonon-induced quantum tunneling rate which is valid for the
entire range of temperature regardless of the basis. In incoherent tunneling
regime and localized basis, this formula reduces to the ubiquitous incoherent
tunneling rate. Meanwhile, in eigenstates basis, this tunneling rate is
demonstrated to be equal to zero. From this secular form, we end the
controversy surrounding the selection of basis for the secular approximation by
figuring out the conditions for using this approximation in localized and
eigenstates basis. Particularly, secular approximation in localized basis is
justified in the regime of high temperature and small tunnel splittings. In
contrast, a large ground doublet's tunnel splitting is required for the secular
approximation in eigenstates basis. With these findings, this research lays a
sound foundation for any treatments of the spin-phonon relaxation under any
conditions provided that the non-secular master equation is relevant.Comment: 9 pages, 0 figure