We use inelastic neutron scattering to study acoustic phonons and spin
excitations in single crystals of NaFeAs, a parent compound of iron pnictide
superconductors. NaFeAs exhibits a tetragonal-to-orthorhombic structural
transition at Ts≈58 K and a collinear antiferromagnetic (AF) order at
TN≈45 K. While longitudinal and out-of-plane transverse acoustic
phonons behave as expected, the in-plane transverse acoustic phonons reveal
considerable softening on cooling to Ts, and then harden on approaching
TN before saturating below TN. In addition, we find that spin-spin
correlation lengths of low-energy magnetic excitations within the FeAs layer
and along the c-axis increase dramatically below Ts, and show weak anomaly
across TN. These results suggest that the electronic nematic phase present
in the paramagnetic tetragonal phase is closely associated with dynamic
spin-lattice coupling, possibly arising from the one-phonon-two-magnon
mechanism