Stable icosahedral quaicrystals (
i-QC) have been identified in a large number alloys. Stability of the
i-QCs has been understood in terms of Hume-Rothery rules
i.e., atomic size factor and valency electron concentration (
e/a) . The stable
i-QCs are electron compounds which have strict
e/a, and hence most stable
i-QCs were discovered along the criterion of
e/a. According to the structures of icosahedral clusters derived from the corresponding approximants, the stable
i-QCs are classified into three classes ; Al-Mn-Si, Zn-Mg-Al and Cd-Yb. All the
i-QCs commonly have their corresponding
e/a, but the dependence of stability on the atomic size factor is different among the three classes. Qualitatively, the most relevant criterion for the formation of the stable
i-QC is that the alloys should have a definite
e/a. On top of this, a higher stability of the
i-QC is expected when the atomic size factor is favored.
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