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Microscopic structures for fcc-based quaternary high-entropy alloys (HEA) inthermodynamically equilibrium state is examined based on first-principles (FP)calculation combined with our recently-developed theoretical approach. We findthat (i) hierarchical ordering tendency for whole quaternary, and ternary andbinary subsystems for the present five HEAs cannot be reasonably characterizedby conventional Goldschmidt atomic radius or by those obtained from unarysystem for FP calculation, but can be systematically characterized by atomicradius from specially fluctuated configuration. (ii) ordering tendency forwhole- and sub-systems can be simultaneously treated with individual definitionof atomic radius ratio, and that linear correlation between ordering tendencyand atomic radius ratio naturally decreases with decrease of number ofconstituents for whole (or sub-) system due to the counterbalance breaking oflike- and unlike-atom neighboring pair. We also find that by introducingappropriate normalization, ordering tendency for systems with different numberof constituents (here, quaternary and ternary) can also be simultaneouslytreated by the atomic radius ratio.

Universal Characterization of Hierarchical Ordering Tendency in High-Entropy Alloys from Configurational Geometry