  We study the problem of heat transport in one-dimensional (1D) spin-chain
systems weakly coupled to three-dimensional phonons and impurities. We consider
the limit of fast spin excitations and slow phonons, applicable to a number of
compounds of the cuprates family, such as Sr2CuO3, where the superexchange J is
much larger than the Debye energy. In this case the Umklapp scattering among
the spin excitations is strongly suppressed for all relevant temperatures. We
argue that the leading scattering mechanism for the spin excitations at not too
low temperatures is the "normal" (as opposed to the Umklapp) spin-phonon
scattering in which the non-equilibrium momentum is transferred from the spin
subsystem to phonons where it quickly relaxes through the "phonon bath".
Because of the lower dimensionality of the spin excitations it is only the
momentum along the chains which is conserved in such a scattering. We find that
this effect leads to a particular momentum- and temperature-dependence of the
spin-phonon relaxation rate valid for the broad class of low-dimensional spin
systems. Subsequently we demonstrate that the spin-phonon relaxation mechanism
is insufficient for the low-energy, long-wavelength 1D spin-chain excitations,
which make the thermal conductivity diverge. We complete our consideration by
taking into account the impurity scattering, which in 1D cuts off the
quasi-ballistic spin excitations and renders the thermal conductivity finite.
Our results compare very well with the existing experimental data for Sr2CuO3.
Using our microscopic insight into the problem we propose further experiments
and predict an unusual impurity concentration dependence for a number of
quantities.
