Using the BWA (Bertaut-Warren-Averbach) Method to Optimize the Technology of Crystalline Powders such as LiFePO4

Abstract

The average sizes \ {\bar{L}}_{i\ } and their dispersion W_i along the i-th axis of crystallites in powders are used to determine the X-ray diffraction sizes, D_{i\ XRD}, averaged over crystallite columns within the BWA method. Numerical calculations have been carried out for an orthorhombic lattice of crystallites, such as LiFePO4, NMC, having Lamé's g-type superellipsoid shape. For Lognormal distributions, the analytical expression for the normalized coefficient Kn is found and includes two constants. The first K_{g,0} is a constant at W→0, the second K_g is a constant depending on the g -type shape. The dependences of D_{i\ XRD} are also calculated for Normal distribution. A fairly simple equation can be obtained as a result of analytical transformations in the framework of experimentally validated approximations. However, a simpler way is to carry out numerical computer calculations with subsequent approximation of the calculated curves. Using the obtained analytical expressions to control technologies from nuclear fuel to cathode materials will improve the efficiency of flexible energy network especially storage in autonomous and standby power plants.