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Detailed experimental methods for Part 1 (length and volume measurements from SPIM) and Part 2 (cross-section diameter and cristae density from TEM), and calculation of mid-piece volume assuming the volume of a truncated cone.

Lattice sites and collective ordering of oxygen atoms in HgBa$_2$CaCu$_2$O$_{6+\delta}$ were studied using the perturbed angular correlation (PAC) technique at ISOLDE/CERN. The electric field gradients (EFG) at $^{199m}$Hg nuclei have been measured as functions of oxygen doping on the Hg planes, above and below T$_c$. In comparison with the results obtained for oxygen and fluorine doping in Hg-1201, the analysis shows a different oxygen ordering exhibited by Hg-1212. Moreover, for all studied cases, the experimental results show that at a local scale there is non-uniform oxygen distribution. A series of ab initio EFG calculations allowed to infer that at low concentrations, regions without oxygen coexist with regions where O$_{2\delta}$ dumbbell molecules are located at the center of the Hg mesh. On the other side, at high concentrations, O$_{2\delta}$ dumbbell molecules coexist with single O$_\delta$ atoms occupying the center of the Hg mesh. The present results suggest that oxygen sits on the Hg planes in the form of a molecule and not as a single atom.

The cubic spinel CdCr$_2$S$_4$ gained recently a vivid interest, given the relevance of relaxor-like dielectric behavior in its paramagnetic phase. By a singular combination of local probe techniques namely Pair Distribution Function and Perturbed Angular Correlation we firmly establish that the Cr ion plays the central key role on this exotic phenomenon, namely through a dynamic off-centering displacement of its coordination sphere. We further show that this off centering of the magnetic Cr-ion gives rise to a peculiar entanglement between the polar and magnetic degrees of freedom, stabilizing, in the paramagnetic phase, short range magnetic clusters, clearly seen in ultra-low field susceptibility measurements. Moreover, the Landau theory is here used to demonstrate that a linear coupling between the magnetic and polar order parameters is sufficient to justify the appearance of magnetic cluster in paramagnetic phase of this compound. These results open insights on the hotly debated magnetic and polar interaction, setting a step forward in the reinterpretation of the coupling of different physical degrees of freedom.