## 3 Results and Discussion  
Figure 2a shows a high-resolution transmission electron microscopy image of the nanoparticles. Despite accurate sonication, agglomeration of nanoparticles occurred, indicating high dipolar forces. The sample is constituted of ellipsoidal particles, mostly with a spherical shape. About 200 particles from several microscopy images were analyzed, and the average diameter was calculated measuring the minimal and maximal diameters for each one. A lognormal distribution was used to fit the diameter distribution resulting in the average diameter of 21.2(3) nm with a standard deviation of 3 nm and a coefficient of variation of 0.13. Figure 3 displays the neutron powder diffraction pattern collected at 300 K and refined by the Rietveld method, taking as the starting model the bulk _Pbnm_ crystal structure [16]. At the best fitting, the refined nanocrystalline lattice parameters were a = 5.291(6) A b = 5.435(6) A, c = 7.472(6) A, to be compared with the bulk values a = 5.3066(1) A, b = 5.4686(1) A, c = 7.4529(1) A. With respect to the bulk, a and b are slightly contracted while c is elongated. Lattice parameters are in the relation c/\(\surd{2}\cong\) a < b. This means that at room temperature the orthorhombic distortion is mainly due to steric effects, i.e., to the small size of Ho/Ca species at A site, while the additional deformation due to the cooperative Jahn-Teller effect, which leads to the bulk O\({}^{\prime}\) structure [32], is very weak. The diffractometric crystallite size was found to be 22 \(\pm\) 4 nm, which, if compared to the particle size determined by high-resolution transmission electron microscopy, indicates that most particles are monocrystalline. Figure 2: (a) High-resolution transmission electron microscopy image showing several agglomerates of nanoparticles. The white square indicated by the arrow highlights a single nanoparticle; (b) grain size distribution of the nanoparticles. The orthorhombic unit cell is shown in Figure 4 with selected bond lengths and angles. At room temperature, the c axis elongation in Ho\({}_{0.5}\)Ca\({}_{0.5}\)MnO\({}_{3}\) nanoparticles is accommodated not via octahedra deformation but through the increase of the Mn-O\({}_{\text{ap}}\)-Mn bond angle from the bulk value \(\theta=152.2\lx@math@degree\) to \(\theta=158.1\lx@math@degree\). This is interesting, since for a double-exchange system, the width W\({}_{\sigma}\) of the tight-binding band for itinerant Mn\({}^{3+}\) 3d e\({}_{\text{g}}\) electrons is strictly related to the bond bending angle (\(\pi-\theta\)) as W\({}_{\sigma}=\varepsilon_{\sigma}\lambda_{\sigma}{}^{2}\cos(\pi-\theta)\)\(<\)cos(\(\Theta_{\parallel}/2\))\(>\), in which \(\varepsilon_{\sigma}\) is the one-electron energy, \(\lambda_{\sigma}\) is the overlap integral, and \(\Theta_{\parallel}\) is the angle formed by the underlying localized Mn 3d t\({}_{\text{g}}\) (i.e., d(xy), d(xz), d(yz)) core spins [33]. With such an opened Mn-O\({}_{\text{ap}}\)-Mn angle, the e\({}_{\text{g}}\) electron, formally belonging to Mn\({}^{3+}\), might attain a certain degree of delocalization on Mn pairs along [001][34], leading to the formation of manganese dimers coupled ferromagnetically by double exchange Zener polarons [35] and opening a FM channel within the electron-localized AFM bulk-like matrix. Indeed, the value of 158\(\lx@math@degree\) is critical for the previous scenario to take place at the doping level x = 0.3 [36]. Other nanosize half-doped manganites, in which CO and antiferromagnetism are completely suppressed, have been reported to display even larger in-plane or out-of-plane bond angles [26]. The temperature variations of the refined orthorhombic cell parameters are shown in Figure 5a. All refinements were carried out in terms of a single _Pbnm_ phase, i.e., without attempting at using the lower monoclinic symmetry required in presence of CO to distinguish Mn\({}^{3+}\) and Mn\({}^{4+}\) sites [16], given the absence of superlattice reflections related to CO and the low data resolution.