text: There is an inverse relationship between delta yield strength and grain size to some power, x.
  where k is the strengthening coefficient and both k and x are material specific. Assuming a narrow monodisperse grain size distribution in a polycrystalline material, the smaller the grain size, the smaller the repulsion stress felt by a grain boundary dislocation and the higher the applied stress needed to propagate dislocations through the material.
  The relation between yield stress and grain size is described mathematically by the Hall–Petch equation:
  where σ is the yield stress, σ is a materials constant for the starting stress for dislocation movement (or the resistance of the lattice to dislocation motion), k is the strengthening coefficient (a constant specific to each material), and d is the average grain diameter.  It is important to note that the H-P relationship is an empirical fit to experimental data, and that the notion that a pileup length of half the grain diameter causes a critical stress for transmission to or generation in an adjacent grain has not been verified by actual observation in the microstructure.
  Theoretically, a material could be made infinitely strong if the grains are made infinitely small. This is impossible though, because the lower limit of grain size is a single unit cell of the material. Even then, if the grains of a material are the size of a single unit cell, then the material is in fact amorphous, not crystalline, since there is no long range order, and dislocations can not be defined in an amorphous material. It has been observed experimentally that the microstructure with the highest yield strength is a grain size of about , because grains smaller than this undergo another yielding mechanism, grain boundary sliding. Producing engineering materials with this ideal grain size is difficult because only thin films can be reliably produced with grains of this size. In materials having a bi-disperse grain size distribution, for example those exhibiting abnormal grain growth, hardening mechanisms do not strictly follow the Hall–Petch relationship and divergent behavior is observed.