## 3 Results and Discussion
The mean values of the intermediate, short, and long Mn-O bond distances (respectively \(m\) Mn-O\({}_{\text{ap}}\), \(s\) Mn-O\({}_{\text{eq}}\), \(l\) Mn-O\({}_{\text{eq}}\)) and of the Mn-O\({}_{\text{ap}}\)-Mn and Mn-O\({}_{\text{eq}}\)-Mn bond angles are: Mn-O\({}_{\text{eq}}\)-Mn = 151.4\({}^{\circ}\), Mn-O\({}_{\text{ap}}\)-Mn = 158. 1\({}^{\circ}\), \(m\) Mn-O\({}_{\text{ap}}\) = 1.91 Å, \(s\) Mn-O\({}_{\text{eq}}\) = 1.87 Å, \(l\) Mn-O\({}_{\text{eq}}\) = 2.05 Å. In the bulk, CO manifests itself as a bump in M(T) dependence--both FC and ZFC--with T\({}_{\rm CO}\approx 250\) K, which is absent in the nanoparticles. On cooling, both the micro and nanometric samples enter into a canted AFM phase. The bulk Neel point is easily identified at about 110 K, where the ZFC Curie-type magnetization has a clear inflection and the ZFC-FC irreversibility begins. In the absence of such ZFC anomaly, the Neel point of the  
Figure 5: (**a**) Temperature dependence of the orthorhombic _Pbmm_ lattice parameters. (**b**) Temperature dependence of the Mn-O distances and of the mean (s + l)/2 of the two equatorial short (s Mn-O\({}_{\rm eq}\)) and long (l Mn-O\({}_{\rm eq}\)) ones. (**c**) Temperature dependence of the Q\({}_{3}\)-type apical octahedra compression (see text for details). Figure 6: (**a**) Zero-field-cooled (ZFC, blue empty dots), field-cooled (FC, red full dots) magnetization curves of the nanometric sample recorded at H = 25 Oe. Green arrows and ellipses are guides for the eye for the identification of the transitions. Inset displays the inverse ZFC branch. The straight red line is a guide for the eye to evidence the deviation from the linear behavior on cooling. (**b**) Corresponding ZFC-FC magnetization curves for the micrometric sample. nanoparticles is tentatively set at about 105 K, corresponding to the onset of the ZFC-FC irreversibility. The lower transition at about 40 K, rather well marked in the ZFC branch of the bulk, is much less visible in the nanoparticles, but it appears in the plot of inverse ZFC magnetization as a function of temperature (Inset of Figure 6a) as a deviation from linear behavior. Overall, apart from the absence of a clear signature related to CO, the sequence of transitions at the nanometric length scale seems to be bulk-like. To go deeper into the analysis, we considered the temperature dependence of the FC magnetization of the nanoparticles at 25 Oe, after subtraction of the strong Ho\({}^{3+}\) contribution (with its molar susceptibility \(\chi\) defined as M/H and modeled as X(T) = \(\frac{\text{N}_{A}}{2}\frac{(10.6)^{2}}{3\text{k}_{\text{B}}\text{T}}\)\(\mu_{B}^{2}\) (N\({}_{A}\) = 6.023 \(\times\) 10\({}^{23}\) mol\({}^{-1}\) Avogadro number, k\({}_{\text{B}}\)= 1.38 \(\times\) 10\({}^{-16}\) eV/K Boltzmann constant, 10.6 theoretical effective magnetic moment of free Ho\({}^{3+}\) expressed in Bohr magnetons \(\mu_{\text{B}}\) = 9.274 10\({}^{-21}\) emu). In this way, we intended to put under evidence the behavior of Mn ions (too heavily masked in nanoparticles) that are the only ones responsible for the CO phenomenon and the magnetic transitions. The details of the subtraction are shown in Figure 7a, in which we display the experimental FC M(T) curve at 25 Oe, the aforementioned contribution of paramagnetic Ho\({}^{3+}\) and the Curie-type M(T) magnetization for the sum of paramagnetic Mn and Ho species, with effective magnetic moment calculated from the free ions values as \(\left\{1/2\left[\mu_{\text{eff}}^{2}\Big{(}\text{Mn}^{4+}\Big{)}+\mu_{\text{ eff}}^{2}\Big{(}\text{Mn}^{3+}\Big{)}+\mu_{\text{eff}}^{2}\Big{(}\text{Ho}^{3+} \Big{)}\right]\right\}^{1/2}\), in the absence of any magnetic transition.