Journal of Statistical Physics

, Volume 167, Issue 3–4, pp 726–734 | Cite as

Frequency Estimate for Multicomponent Crystalline Compounds

  • M. WidomEmail author


Among crystal structures of N-component metal alloys, far fewer examples are known with \(N\ge 4\) 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. Specific estimates are developed through application of the empirical Miedema enthalpy model. Despite their rarity, we propose that the actual number of \(N=4\)- and 5-component alloys greatly exceeds the number that are currently known.


Multicomponent alloy Enthalpy Crystal structures Convex hull High entropy alloy 



I thank Alan Frieze and Wesley Pegden for a discussion on generalized triangle numbers. This paper is dedicated to the memory of Prof. Leo P. Kadanoff, my Ph.D. thesis advisor. Supported by the Department of Energy under Grant DE-SC0014506.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of PhysicsCarnegie Mellon UniversityPittsburghUSA

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