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On the basis of the point charge model, we formulate the crystallineelectronic field (CEF) Hamiltonian $H_{\rm CEF}$ in the rare-earth basedquasicrystal (QC) and approximant crystal (AC) with ligand ions located atpseudo 5-fold configurations by using the operator equivalent method. Bysetting the total angular momentum $J=7/2$, the CEF in the quantum critical QCAu$_{51}$Al$_{34}$Yb$_{15}$ and the 1/1 AC Au$_{51}$Al$_{35}$Yb$_{14}$ isanalyzed with consideration for the effect of Al/Au mixed sites. We find thatthe ratio of the valences of ligand ions $x=Z_{\rm Al}/Z_{\rm Au}$ plays animportant role in characterizing the CEF ground state. As $x$ decreases from$x=3$, the 4f wave function of the CEF ground state with the flat shape lyingin the mirror plane is deformed around $x\approx 0.8$ to the flat shapeperpendicular to the pseudo 5-fold axis at $x=0$. The formulated $H_{\rm CEF}$by $J$ is generally applicable to rare-earth-based QCs and ACs, which is usefulto analyze the CEF.

Crystalline Electronic Field in Rare-Earth Based Quasicrystal and Approximant: Analysis of Quantum Critical Au-Al-Yb Quasicrystal and Approximant