Mean-Field Critical Behavior in the Sr$_{1-x}$(Ca$_{0.5}$Ba$_{0.5}$)$_{x}$RuO$_{3}$ (0 $\le $ x $\le$ 0.35)

Orthorhombic SrRuO$_{3}$ is a metallic ferromagnet with T$_{c} \quad \approx$ 160 K where mean-field (MF) critical behavior has been observed. [1] Recently, we have shown that cubic BaRuO$_{3}$ belongs to the 3D Heisenberg universality class with short-range exchange interactions. [2] The partial chemical substitution by either smaller Ca$^{2+}$ or larger Ba$^{2+}$ in Sr$_{1-x}$A$_{x}$RuO$_{3}$ which changes the bond angle $<$Ru-O-Ru$>$ and introduces an A-cation size variance $\sigma ^{2}$ $\equiv \quad <$r$_{A}^{2}> \quad - \quad <$r$_{A}>^{2 }$, causes a clear deviation from the MF behavior. In order to distinguish effects of $<$Ru-O-Ru$>$ versus $\sigma ^{2}$, we have synthesized Sr$_{1-x}$(Ca$_{0.5}$Ba$_{0.5})_{x}$RuO$_{3 }$(0 $\le $ x $\le $ 0.35) under 1000\r{ }C and 10 GPa in a Walker-type multianvil; these samples have the same average $<$Ru-O-Ru$>$ as that of SrRuO$_{3}$, but a different $\sigma ^{2}$. All samples exhibit perfect MF critical behaviors, which indicates that the peculiar bond angle $<$Ru-O-Ru$> \quad \approx $ 163\r{ } plays an essential role in determining the MF critical behavior of SrRuO$_{3}$.\\[0pt] [1] D. Kim, \textit{et al}., Phys. Rev. B \textbf{67}, 100406 (2003).\\[0pt] [2] J.-S. Zhou, \textit{et al}., Phys. Rev. Lett. \textbf{101}, 077206 (2008).