Unravelling disorder in kagome Yb$_{0.5}$Co$_3$Ge$_3$

The presence of phonon instabilities that leads to the formation of charge modulated states in kagome metals has allowed their classification based on the atomic displacements. Here, we use diffuse and inelastic x-ray scattering, backed by Monte Carlo simulations to describe a type-I instability in the kagome metal Yb$_{0.5}$Co$_3$Ge$_3$. We find that the in-plane distortion of Co in the kagome plane drives a structural transition with the appearance of new Bragg peaks at odd $L$, which are surrounded by a hexagonal diffuse signal. The anisotropic diffuse scattering, characteristic of a highly frustrated triangular lattice was simulated following a combination of Ising hamiltonian and Lennard-Jones potential, and demonstrate that the structural phase transition in Yb$_{0.5}$Co$_3$Ge$_3$ is of an order-disorder transformation type. The inelastic spectra reveals no softening but an anomalous broadening of the $\Gamma-A$ low energy acoustic mode. Our results highlight the critical role of the geometric frustration in promoting ordering from disorder in kagome lattices and power of diffuse scattering to disentangle the internal atomic displacements and correlated disorder.