Method for constructing a mineralogical composition from a measured sample of single components

Our aim is to infer the mineral composition from the chemical composition of a material. Several difficulties occur: Different mineral compositions can lead to the same chemical composition. Not all chemical compositions can be reached by compositions of certain minerals, while e.g. due to measurement errors impossible chemical compositions will typically be observed. In principle the dependency between mineral composition and chemical composition is linear. Inversion of the linear system however often leads to negative portions for some mineral components. Only the first problem is properly solved by state of the art linear end member calculation methods (see e.g. Tolosana et al. 2011).
Our algorithm computes the set of all mineral compositions leading to the chemical composition maximizing the likelihood of the observed concentrations. The measurement error for the chemical components are modelled as independent normal distributions with mean zero and standard deviation given by the measurement error reported by the lab. For any given mineral composition, the chemical composition can thus be computed by stoichiometric calculations and Maximum log-likelihood is than a weighted least squares problem. The parameter space is however constraint to the possible mineral compositions. The problem is transformed into a quadratic programming problem with linear equality and inequality constraints with a unique solution for the chemical composition. From this unique solution we can than compute all possible mineral compositions leading to the corresponding chemical composition by enumerating the corners of the simplex of equally fitting solutions.
As an example we will discuss a sample of waste material consisting of Rare Earth Elements (REE), since recycling of waste materials is getting more and more important. Rare Earth Elements (REE) are used in mostly all new technologies and until now, there is no environmentally friendly recycling-process for fluorescent phosphor. For the development of a suitable recycling method, it is important to know the composition of the materials. Due to company secrecy, in some cases only rough informations about the composition of the material can be found. In the case of fluorescent phosphor, which is collected during the recycling-process of energy saving bulbs, only the type of the dye but not the precise composition is known. Because of different restrictions in the analytical methods and the complex composition, only the elemental concentrations with relatively high measurement errors can be measured.