Time-Reversal Symmetry pp 143-154 | Cite as

# 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.