  In spite of their relatively high lattice thermal conductivity
$\kappa_{\ell}$, the XNiSn (X=Ti, Zr or Hf) half-Heusler compounds are good
thermoelectric materials. Previous studies have shown that $\kappa_{\ell}$ can
be reduced by sublattice-alloying on the X-site. To cast light on how the alloy
composition affects $\kappa_\ell$, we study this system using the phonon
Boltzmann-transport equation within the relaxation time approximation in
conjunction with density functional theory.The effect of alloying through
mass-disorder scattering is explored using the virtual crystal approximation to
screen the entire ternary Ti$_x$Zr$_{y}$Hf$_{1-x-y}$NiSn phase diagram. The
lowest lattice thermal conductivity is found for the Ti$_x$Hf$_{1-x}$NiSn
compositions; in particular, there is a shallow minimum centered at
Ti$_{0.5}$Hf$_{0.5}$NiSn with $\kappa_l$ taking values between 3.2 and 4.1 W/mK
when the Ti content varies between 20 and 80\%. Interestingly, the overall
behavior of mass-disorder scattering in this system can only be understood from
a combination of the nature of the phonon modes and the magnitude of the mass
variance. Mass-disorder scattering is not effective at scattering acoustic
phonons of low energy. By using a simple model of grain boundary scattering, we
find that nanostructuring these compounds can scatter such phonons effectively
and thus further reduce the lattice thermal conductivity; for instance,
Ti$_{0.5}$Hf$_{0.5}$NiSn with a grain size of $L= 100$ nm experiences a 42\%
reduction of $\kappa_{\ell}$ compared to that of the single crystal.
