## 2 Materials and Methods  
Ho\({}_{0.5}\)Ca\({}_{0.5}\)MnO\({}_{3}\) nanoparticles were obtained by the Pechini method [27]. In this sol-gel process, stoichiometric amounts of precursors salts (Ca(NO\({}_{3}\))\({}_{2}\)*4H\({}_{2}\)O (99.98%) Alfa Aesar, Ho(NO\({}_{3}\))\({}_{3}\)*5H\({}_{2}\)O (99.99%) Alfa Aesar, Mn(NO\({}_{3}\))\({}_{2}\)*4H\({}_{2}\)O (99.999%) Alfa Aesar) were dissolved in deionized water and mixed with ethylene glycol and citric acid, forming a stable solution. This solution was then heated on a thermal plate under constant stirring at 350 \({}^{\circ}\)C (625 K) to eliminate the excess of water and to accelerate the esterification reaction. After four hours, the solution underwent a drastic volume reduction, due to the formation of the gel. The gel was then calculated at 400 \({}^{\circ}\)C (673 K) for 4 h. The result was a dark amorphous material, with light and crumbly consistency. The product pieces were crushed in a marble mortar until a fine powder was obtained. Finally, the material was allowed to re-crystallize by thermal treatment at a maximum temperature of 700 \({}^{\circ}\)C (973 K) for 7 h. X-ray fluorescence analysis provided for the Ho:Ca ratio the value of 49:51, in very good agreement with the nominal one. A bulk polycrystalline sample, the synthesis and characterization of which can be found in [16; 28], was used as a reference for the magnetization measurements. Neutron powder diffraction patterns were collected for 10 K < T < 300 K at the Institut Laue Langevin (ILL) in Grenoble (France) on the D20 diffractometer, a fixed-wavelength 2-axis instrument that can provide different combinations of resolution and intensity of incident neutron flux, obtained by the use of different monochromators and their take-off angles. Knowing which parts of the diffraction pattern contain the information for which the experiment is carried out in the first place permits the most convenient choice. Since the main aim of our experiment was to detect low-intensity magnetic Bragg reflections, which occur at low scattering angles, and given the expected broad Bragg peaks of the nanoparticles, the instrument was used in the low-resolution, high-intensity mode. The wavelength was set at 2.41 A. Powdered samples were held in cylindrical vanadium containers with a nominal zero coherent scattering cross-section. Data were corrected for instrumental background and container scattering. The nuclear structure at different temperatures was refined according to the Rietveld method using FullProf software [29]; refinements were carried out using a file describing the instrumental resolution function, which is requested by FullProf to determine the average particle size. A discussion of the instrumental resolution function of neutron powder diffractometer can be found in [30], and specifically for D20 in [31]. An analysis of the size distribution of the manganite nanoparticle was also carried out by means of a transmission electron microscope (TEM) JEOL JEM 2010 operated at 200 kV and compared to the diffractometric result. DC and AC magnetization measurements were performed using a Quantum Design MPMS XL5 Superconducting Quantum Interference Device (SQUID) magnetometer, equipped with a superconducting magnet generating fields up to 50 kOe and calibrated using a Pd standard. The sensitivity for the magnetic moment is 10\({}^{-8}\) emu. The magnetometer is equipped with a system for setting to zero the applied field; the residual field during the zero field cooling is less than 10\({}^{-2}\) Oe. The dependence of DC magnetization on temperature was studied by zero field cooling (ZFC) and field cooling (FC) measurements in the interval 5 K < T < 300 K. For measuring the temperature dependence of the ZFC magnetization (M\({}_{\text{ZFC}}\)), the sample was cooled from 300 to 5 K in zero field, then the magnetic field was applied at 5 K and the magnetization was recorded on heating the sample. The FC magnetization (M\({}_{\text{FC}}\)) curves were recorded on cooling, keeping the sample in the same applied field.