  An atomistic method of calculating the spin-lattice relaxation times ($T_1$)
is presented for donors in silicon nanostructures comprising of millions of
atoms. The method takes into account the full band structure of silicon
including the spin-orbit interaction. The electron-phonon Hamiltonian, and
hence the deformation potential, is directly evaluated from the
strain-dependent tight-binding Hamiltonian. The technique is applied to single
donors and donor clusters in silicon, and explains the variation of $T_1$ with
the number of donors and electrons, as well as donor locations. Without any
adjustable parameters, the relaxation rates in a magnetic field for both
systems are found to vary as $B^5$ in excellent quantitative agreement with
experimental measurements. The results also show that by engineering electronic
wavefunctions in nanostructures, $T_1$ times can be varied by orders of
magnitude.
