0.2). The rare stoichiometric endmember is known as the mineral troilite. Pyrrhotite is remarkable in that it has numerous polytypes, i.e. crystalline forms differing in symmetry (monoclinic or hexagonal) and composition (, , and others). These materials are always iron-deficient owing to the presence of lattice defects, namely iron vacancies. Despite those defects, the composition is usually expressed as a ratio of large numbers and the crystals symmetry is relatively high. This means the iron vacancies are not randomly scattered over the crystal, but form certain regular configurations. Those vacancies strongly affect the magnetic properties of pyrrhotite: the magnetism increases with the concentration of vacancies and is absent for the stoichiometric . Palladium hydrides Palladium hydride is a nonstoichiometric material of the approximate composition (0.02 < x < 0.58). This solid conducts hydrogen by virtue of the mobility of the hydrogen atoms within the solid. Tungsten oxides It is sometimes difficult to determine if a material is non-stoichiometric or if the formula is best represented by large numbers. The oxides of tungsten illustrate this situation. Starting from the idealized material tungsten trioxide, one can generate a series of related materials that are slightly deficient in oxygen. These oxygen-deficient species can be described as , but in fact they are stoichiometric species with large unit cells with the formulas , where n = 20, 24, 25, 40. Thus, the last species can be described with the stoichiometric formula , whereas the non-stoichiometric description implies a more random distribution of oxide vacancies. Other cases At high temperatures (1000 °C), titanium sulfides present a series of non-stoichiometric compounds. The coordination polymer Prussian blue, nominally and their analogs are well known to form in non-stoichiometric proportions. The non-stoichiometric phases exhibit useful properties vis-à-vis their ability to bind caesium and thallium ions. Applications Oxidation catalysis Many useful compounds are produced by the reactions of hydrocarbons with oxygen, a conversion that is catalyzed by metal oxides. The process operates via the transfer of "lattice" oxygen to the hydrocarbon substrate, a step that temporarily generates a vacancy (or defect). In a subsequent step, the missing oxygen is replenished by O2. Such catalysts rely on the ability of the metal oxide to form phases that are not stoichiometric. An analogous sequence of events describes other kinds of atom-transfer reactions including hydrogenation and hydrodesulfurization catalysed by solid catalysts. These considerations also highlight the fact that stoichiometry is determined by