Frequency estimate for multicomponent crystalline compounds

Among crystal structures of N-component metal alloys, far fewer examples are known with N>=4 than with N=2 or 3, in apparent contradiction to the exponentially growing number of possible combinations of elements. Two effects contribute to this shortfall. Since the N-component composition space resides within a d-dimensional simplex with d=N-1, the vanishing volume in high dimensions reduces the distinct N-component compositions. Additionally, the increasing surface area makes it more probable that stable structures reside on the surface of the simplex (containing fewer than N components) as opposed to its interior. Despite their rarity, we propose that the actual number of N=4- and 5-component alloys greatly exceeds the number that are currently known. Specific estimates are developed through application of the empirical Miedema enthalpy model.