Negative thermal expansion and valence fluctuation in quasi-one-dimensional platinum compounds

Negative thermal expansion observed along the chain direction in [Pt(${\mathrm{NH}}_3$ ${)}_4$][Pt$X_2$(${\mathrm{NH}}_3$ ${)}_4$]) ${}_4{}^{}{}_{}{}^{}$] (${\mathrm{HSO}}_4$ ${)}_4$, where X=Cl or Br, is explained from the viewpoint of the valence fluctuation of Pt atoms. The present crystals are considered essentially ionic system with the mean valencies of ${\mathrm{Pt}}^{3+\mathrm{\delta }}$ and ${\mathrm{Pt}}^{3\mathrm{-}\mathrm{\delta }}$, and the valence fluctuation is treated in two different approaches: slow and rapid fluctuation. In both cases, minimization of the (free) energy provides the contraction of ${\mathrm{Pt}}^{3\mathrm{-}\mathrm{\delta }}$-X distance and expansion of ${\mathrm{Pt}}^{3+\mathrm{\delta }}$-X distance with increasing temperature leading to the overall negative thermal expansion, in agreement with experiment.