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.
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