Prediction of hexagonal lattice parameters of various apatites by artificial neural networks

In this study, the hexagonal lattice parameters of apatite compounds, M₁₀(TO₄)₆X₂, where M is Na⁺, Ca²⁺, Ba²⁺, Cd²⁺, Pb²⁺, Sr²⁺, Mn²⁺, Zn²⁺, Eu²⁺, Nd³⁺, La³⁺ or Y³⁺, T is As⁺⁵, Cr⁺⁵, P⁵⁺, V⁵⁺ or Si⁺⁴, and X is F^-, Cl^-, OH^- or Br^-, were predicted from their ionic radii by artificial neural networks. A multilayer perceptron network was used for training and the best results were obtained with a Bayesian regularization method. Four neurons were used in the hidden layer, utilizing a tangent sigmoid activation function, while one neuron was used in the output layer with a pure linear function. The results of the training showed that the correlation coefficients for the hexagonal lattice parameters were 0.991 for the training data set, which is very close to unity, demonstrating that the learning process was successful. In addition, the average errors of the predicted lattice parameters were less than 1% for the data set prepared with single ions at the M, T and X sites, as well as for apatites with coupled substitutions involving up to three different ions at each site. Simple mathematical formulae were derived for the prediction of lattice parameters using average ionic radii as independent variables.