Fig. 1: Summary of the CEF states and multipoles under non-crystallographic point group symmetries for given half-integer values of J.
From: Unveiling multipole physics and frustration of icosahedral magnetic quasicrystals

a J = 7/2 (Yb3+) (b) J = 9/2 (Nd3+) and (c) J = 15/2 (Dy3+). Under the icosahedral symmetry Ih, the CEF states are split into several multiplets. Particularly, note that J = 7/2 which is the case for Yb3+ has a Kramers doublet and it hosts pure magnetic octupoles without magnetic dipoles or quadrupoles, as shown in the red box. Whereas, for C5v where the mirror reflection of Ih is absent, all the multiplets are split into Kramers doublets. In this case, the z-component of the pseudospin, Σz is dipole for any Kramers doublets. In contrast, the x, y-components, Σx,y, could be not only dipoles but also octupoles and dotriacontapoles. Depending on the energy scale of the breaking of the mirror symmetry, the order of energy could change. d Dipoles (black-copper), octupoles (red-white) and dotriacontapoles (red-yellow). See the main text for more details.