Precipitate diameter from fraction and spacing ($d=L(6f/\pi {)}^{1/3}$)

For one spherical particle per representative volume $L^3$, a volume fraction $f$ implies $d=L(6f/\pi {)}^{1/3}$. A mean nearest-neighbor distance in a random population does not determine this number density exactly, and an atomic thermodynamic phase fraction requires a molar-volume conversion before it is used as $f$.