Frank's constant in the hexatic phase

Using videomicroscopy data of a two-dimensional colloidal system the bond-order correlation function ${G}_{6}$ is calculated and used to determine both the orientational correlation length ${\ensuremath{\xi}}_{6}$ in the liquid phase and the modulus of orientational stiffness, Frank's constant ${F}_{A}$, in the hexatic phase. The latter is an anisotropic fluid phase between the crystalline and the isotropic liquid phase. ${F}_{A}$ is found to be finite within the hexatic phase, takes the value $72∕\ensuremath{\pi}$ at the $\text{hexatic}\ensuremath{\leftrightarrow}\text{isotropic}$ liquid phase transition, and diverges at the $\text{hexatic}\ensuremath{\leftrightarrow}\text{crystal}$ transition as predicted by the Kosterlitz-Thouless-Halperin-Nelson-Young theory. This is a quantitative test of the mechanism of breaking the orientational symmetry by disclination unbinding.