Direct Relationship between Magnetism and ${\mathrm{MnO}}_6$ Distortions in ${\mathrm{La}}_{1-\mathit{x}}{\mathrm{Ca}}_{\mathit{x}}{\mathrm{MnO}}_3$

Measured distortions of the $\mathrm{Mn}-\mathrm{O}$ bond length distribution from x-ray-absorption fine-structure measurements are found to relate linearly to the doped hole concentration $x$ at room temperature in ${\mathrm{La}}_{1-x}{\mathrm{Ca}}_x{\mathrm{MnO}}_3$. Comparison of the distortions above and below $T_c$ for colossal magnetoresistor (CMR) samples gives an estimate of the number of delocalized holes $n_{\mathrm{dh}}$, and we find that $\ln \left(n_{\mathrm{dh}}\right)\propto M$ (magnetization). These results are complementary to resistance measurements that show that $\ln \left(\rho \right)\propto -M$. We have thus established the functional relationship between the electronic, spin, and lattice degrees of freedom in the CMR perovskites.