## 3 Results and Discussion
The development of AFM correlations and long-range ordering in the Mn sublattice produce an increasing difference between the experimental and calculated total curve on cooling below 140 K. However, even under the assumption that Mn ions do not contribute anymore to M(T) below their AFM transition, one can see that the Ho\({}^{3+}\) contribution is overestimated on further cooling. This may indicate that the Ho\({}^{3+}\) 4f electronic levels, split in the low symmetry crystal field, are affected by the exchange field of the ordered Mn sublattice, with a complex variation of the apparent Curie constant on cooling. Coupling among 4f and 3d electrons has been already discussed for other CMR RE\({}_{1-x}\)Ca\({}_{x}\)MnO\({}_{3}\) systems, in particular for Pr\({}_{1-x}\)Ca\({}_{x}\)MnO\({}_{3}\) at various doping levels [36; 38]. Aware of the low-temperature issues, we display in Figure 7b the holimium subtraction limited to the temperature range of interest for the manganese transitions. It appears that the ordering of Mn moments starts around 110 K, with a steep rise of M(T), with a small discontinuity at about 80 K and a subsequent increase up to a maximum, which, however, could depend on the mentioned onset of Ho-Mn interaction. The features of the paramagnetic region (T > 110 K) emerge instead in the inverse magnetization as a function of temperature shown in Inset of Figure 7b. A single Curie-Weiss law cannot fit the entire temperature region above T\({}_{\text{N}}\) (indicatively above 120 K), clearly, a net upward of 1/M(T) occurs distinguishing between a high temperature (HT) and a low temperature (LT) region above and below this upward trend, respectively. Linear fits in the HT and LT regions show strong antiferromagnetic correlations (i.e., high negative Weiss temperature) developing on cooling below about 220 K. The obtained ratio between the Curie constants C\({}_{\text{LT}}\)/C\({}_{\text{HT}}\) of 2.23 largely exceeds the value of 1.62 expected for the formation, below T\({}_{\text{CO}}\), of Zener polarons, according to the \"bond centered\" description of CO given by [PERSON] et al. [38], alternative to the site-centered model of [PERSON] [10]. Indeed, the two very different equatorial short and long Mn-O bond distances obtained from the structural analysis are a straightforward indicator by their own of two crystallographically well-differentiated valence states of manganese atoms, resulting in a OO/CO pattern matching rather the bulk-like CO solution [7]. Figure 8 displays the hysteresis cycles measured at 5 K for the nanometric and micrometric samples. The open M(H) loops confirm a switchable magnetization of a canted phase or the presence of a small ferromagnetic component related to surface effects, not unexpected in such small particles. The coercive field H\({}_{\text{c}}\) is 60 Oe for bulk and 110 Oe for the nano-sample, larger at the nanoscale possibly due to surface anisotropy effect [39]. In the light of the previous results, we now discuss the neutron diffraction experiments as it concerns the detection of magnetic reflections on cooling. Figure 9 shows the low  
Figure 8: (**a**) Low magnetic field region of the hysteresis cycle of the nanometric sample measured at 5 K. (**b**) Low magnetic field region of the hysteresis cycle of the micrometric sample measured at 5 K. Insets show the complete hysteresis loops for the two samples. Figure 7: (**a**) Temperature dependence of the field-cooled magnetization recorded at H = 25 of Ho\({}_{0.5}\)Ca\({}_{0.5}\)MnO\({}_{3}\) nanoparticles (red points), calculated paramagnetic contribution of Ho\({}^{3+}\) (blue continuous line) and total paramagnetic contribution of Ho plus Mn species (green continuous line), both treated as free ions (see text). (**b**) Temperature dependence of the field-cooled magnetization and of its inverse (shown in Inset) after subtraction of the strong Ho\({}^{3+}\) paramagnetic contribution.