Kramers Trimer Clusters and Time-Reversal Symmetry

Abstract

In this chapter is shown that structural distortions of trihomonuclear Kramers clusters occur as a consequence of the time-reversal symmetry. These distortions are caused by the fact that the time-reversal operator $\mathbf {T}$ is not a separate element of the magnetic point group $G^{\prime }$, but forms a combined $\mathbf {T}g$ element with other g elements of the original point group G. The order of the element g cannot be an odd number, otherwise it would lead to $\mathbf {T}\in G^{\prime }$, which contradicts the definition of the group $G^{\prime }$. Forbidden rotations by $2\pi /3$ and $2\pi /5$ in the group $G^{\prime }$ lead to the fact that metal ions of homonuclear magnetic clusters in a magnetic ordered state cannot occupy the vertices of an equilateral triangle or those of a regular pentagon. The distortion of the regular triangle or pentagon occurs only if the time-reversal operator $\mathbf {T}$ is taken into account. Hence these structural distortions are due to the time-reversal symmetry. There are presented and discussed experimental data concerning the anomalous behavior of chromium(II), iron(III), copper(II), vanadium(II) and cobalt(II) trihomonuclear clusters determined by their four-color symmetry.