  Time-of-flight inelastic neutron scattering measurements have been carried
out on polycrystalline samples of the 4$d^3$ and 5$d^3$ double pervoskite
antiferromagnets La$_2$LiRuO$_6$ and La$_2$LiOsO$_6$. These reveal the
development of an inelastic spin gap in La$_2$LiRuO$_6$ and La$_2$LiOsO$_6$ of
$\sim$ 1.8(8) meV and 6(1) meV, below their respective ordering temperatures,
$T_N$, $\sim$ 23.8 K and 30 K. The bandwidth of the spin excitations is shown
to be $\sim$ 5.7(9) to 12(1) meV, respectively, at low temperatures. Spin gaps
are surprising in such magnets as the t$_{2g}$ levels of Ru$^{5+}$ or Os$^{5+}$
are expected to be half-filled, resulting in an anticipated orbital singlet for
both materials. We compare these results in monoclinic double perovskites
La$_2$LiRuO$_6$ and La$_2$LiOsO$_6$ with those in cubic Ba$_2$YRuO$_6$ and
Ba$_2$YOsO$_6$, as well as with those in the more strongly monoclinic
La$_2$NaRuO$_6$, La$_2$NaOsO$_6$, and Sr$_2$ScOsO$_6$, and model the inelastic
magnetic scattering with linear spin wave theory using minimal anisotropic
exchange interactions. We discuss the possible role of the distortion of the
face-centered cubic, double perovskite structure on the spin gap formation and
geometric frustration in these materials, and show that $T_N$ scales with the
top of the spin wave band in all members of these families that display long
range order.
