  Interactions between nematic fluctuations, magnetic order and
superconductivity are central to the physics of iron-based superconductors.
Here we report on in-plane transverse acoustic phonons in hole-doped
Sr$_{1-x}$Na$_x$Fe$_2$As$_2$ measured via inelastic X-ray scattering, and
extract both the nematic susceptibility and the nematic correlation length. By
a self-contained method of analysis, for the underdoped ($x=0.36$) sample,
which harbors a magnetically-ordered tetragonal phase, we find it hosts a short
nematic correlation length $\xi$ ~ 10 $\AA$ and a large nematic susceptibility
$\chi_{\rm nem}$. The optimal-doped ($x=0.55$) sample exhibits weaker phonon
softening effects, indicative of both reduced $\xi$ and $\chi_{\rm nem}$. Our
results suggest short-range nematic fluctuations may favor superconductivity,
placing emphasis on the nematic correlation length for understanding the
iron-based superconductors.

Superconductivity;Strongly Correlated Electrons