  Evidence for broken time reversal symmetry (TRS) has been found in the
superconducting states of the $R_5$Rh$_6$Sn$_{18}$ (R = Sc, Y, Lu) compounds
with a centrosymmetric caged crystal structure, but the origin of this
phenomenon is unresolved. Here we report neutron diffraction measurements of
single crystals with $R$=Lu, as well as measurements of the temperature
dependence of the magnetic penetration depth using a self-induced tunnel
diode-oscillator (TDO) based technique, together with band structure
calculations using density functional theory. Neutron diffraction measurements
reveal that the system crystallizes in a tetragonal caged structure, and that
one of nominal Lu sites in the Lu$_5$Rh$_6$Sn$_{18}$ structure is occupied by
Sn, yielding a composition Lu$_{5-x}$Rh$_6$Sn$_{18+x}$ ($x=1$). The low
temperature penetration depth shift $\Delta\lambda(T)$ exhibits an exponential
temperature dependence below around $0.3T_c$, giving clear evidence for fully
gapped superconductivity. The derived superfluid density is reasonably well
accounted for by a single gap $s$-wave model, whereas agreement cannot be found
for models of TRS breaking states with two-component order parameters.
Moreover, band structure calculations reveal multiple bands crossing the Fermi
level, and indicate that the aforementioned TRS breaking states would be
expected to have nodes on the Fermi surface, in constrast to the observations.
