  The structure of the quasi two dimensional heavy fermion antiferromagnet
CeRhIn$_{5}$ has been investigated as a function of pressure up to 13 GPa using
a diamond anvil cell under both hydrostatic and quasihydrostatic conditions at
room (T=295 K) and low (T=10 K) temperatures. Complementary resonant ultrasound
measurements were performed to obtain the complete elastic moduli. The bulk
modulus ($B\approx 78$ GPa) and uniaxial compressibilities ($\kappa_{a}$
$=3.96\times $ $10^{-3}$ GPa$^{-1}$ and $% \kappa_{c}$ $=4.22\times $ $10^{-3}$
GPa$^{-1}$) found from pressure-dependent x-ray diffraction are in good
agreement with the ultrasound measurements. Unlike doping on the Rh site where
$T_{c}$ increases linearly with the ratio of the tetragonal lattice parameters
$c/a,$ no such correlation is observed under pressure; instead, a double peaked
structure with a local minimum around 4-5 GPa is observed at both room and low
temperatures.
