## 3 Results and Discussion
It can be noted that, while c contracts non-linearly on cooling, the in-plane parameters a and b remain practically constant down to about 100 K, with only a modest initial reduction. This is different from what happens in bulk Ho\({}_{0.5}\)Ca\({}_{0.5}\)MnO\({}_{3}\), in which the arising of both OO and CO is evidenced by the typical, albeit much less pronounced, with respect to other CO half-doped systems, increase of both a and b on cooling in the range 300 K \(<\) T \(<\) 250 K, with a concomitant decrease of c [16]. Figure 3: Rietveld refinement plot for the neutron powder diffraction pattern collected at 300 K. The cyan solid line in the upper field represents the observed intensity data, while the calculated pattern is superposed and drawn as purple empty dots. The small black vertical bars indicate the positions of the allowed Bragg reflections for the orthorhombic phase. The difference between the observed and calculated patterns is plotted in the lower field as a green solid line. Rietveld discrepancy values are R-profile R\({}_{\text{p}}\) = 2.7 %, R-weighted profile R\({}_{\text{wp}}\) = 3.4%, goodness-of-fit GoF (i.e., \(\chi^{2}\)) = 1.3, R-Bragg R\({}_{\text{B}}\) = 3.4%. Therefore, in Ho\({}_{0.5}\)Ca\({}_{0.5}\)MnO\({}_{3}\) nanoparticles, there is no clear structural evidence for charge ordering in terms of unit cell metrics. Below room temperature, the relations among a, b, and c become compatible with orbital ordering (c/\(\sqrt{2}\) < a < b), with increasing cooperative Jahn-Teller distortion on cooling. The c axis contraction on cooling is accommodated by increasing the distortion of the MnO\({}_{6}\) units, while the Mn-O\({}_{\text{ap}}\)-Mn bond angle remains almost unaffected. The temperature evolution of such distortion can be evaluated in terms of the three (medium m, short s, long l) Mn-O bond distances. As shown in Figure 5b, the apical Mn-O\({}_{\text{ap}}\) distance remains intermediate between the two equatorial Mn-O\({}_{\text{eq}}\) ones in the analyzed temperature range, as expected for a Jahn-Teller distorted _Pbnm_ phase with OO. The Jahn-Teller effect in _Pbnm_ RE\({}_{0.5}\)Ca\({}_{0.5}\)MnO\({}_{3}\) phases is in fact dominated by the orthorhombic Q\({}_{2}\) active mode [37], which gives rise to alternating short and long s and l Mn-O distances in the (a,b) plane. The contribution of the typical tetragonal mode Q\({}_{3}\), quantified by the deviation of the m length from the average (l + s)/2, is shown in Figure 5c. It can be seen that m < (l + s)/2 in the whole temperature range, indicating a Q\({}_{3}\) contribution with apical compression of the MnO\({}_{6}\) units (Q\({}_{3}\) < 0) along the d\(x^{2}\)-type orbital direction. The Q\({}_{3}\)-type apical compression increases slightly on cooling. The major octahedra rearrangement takes place in the temperature range 100 K < T < 250 K. The previous results will be taken into account in the discussion of the magnetization measurements that will be presented in the following. The ZFC-FC magnetization curves M\({}_{\text{ZFC}}\)(T) and M\({}_{\text{FC}}\)(T) of the nanoparticles, measured in H = 25 Oe, are shown in Figure 6a. The analogous curves for a micrometric sample prepared according to [16], representative of the bulk behavior, are also reported for comparison (Figure 6b). Figure 4: Room temperature orthorhombic _Pbnm_ unit cell of Ho\({}_{0.5}\)Ca\({}_{0.5}\)MnO\({}_{3}\) nanoparticles. (**a**) Interconnection of MnO\({}_{3}\) octahedra through apical (O\({}_{\text{ap}}\)) and equatorial (O\({}_{\text{eq}}\)) oxygen atoms. Mn atoms are within the octahedra. The Ho/Ca sites are depicted as virtual balls composed of half Ho and half Ca of different blue shades. (**b**) Projection of the unit cell along the [010] orthorhombic direction. (**c**) Projection of the unit cell along the [001] orthorhombic direction.