## 3 Results and Discussion
The solid lines in Inset represent the linear fits in the high-temperature (HT) region (red) and in the low-temperature (LT) region (blue). angle range of the diffraction pattern of the nanoparticles taken at different temperatures. Magnetic superlattice reflections arise below 100 K, indicating the transition towards the antiferromagnetic state. A first intense magnetic peak, at 20 = 25.9\({}^{\circ}\), rises between 100 K and 80 K, while a second less intense one, at 20 = 18.6\({}^{\circ}\), becomes visible at 60 K. Some considerations can be made with reference to the bulk CE-type AFM spin arrangement and its modified version called pseudo-CE-type. Both structures are formed by zigzag ferromagnetic chains in the (001) _Pbmm_ plane, with in-plane antiferromagnetic inter-chain coupling, as discussed in Introduction (cfr Figure 1e). However, the stacking of the planar pattern along the [001] direction is antiferromagnetic in the CE-type structure, but ferromagnetic in the pseudo-CE-type one [40; 41; 42]. In both orderings, the result is a four-fold magnetic supercell 2a \(\times\) 2b \(\times\) c when referring to the _Pbmm_ unit cell, with decoupled magnetic reflections for the Mn\({}^{3+}\) and Mn\({}^{4+}\) sublattices. Reflections coming from Mn\({}^{3+}\) are indexed as (h/2 k l) with odd integer h, while those related to Mn\({}^{4+}\) as (h/2 k/2 l) with odd integer h, k. Index 1 is an odd integer for CE-type and even integer for pseudo-CE-type. The two magnetic reflections shown in Figure 9 are indexed in the nuclear cell as CE-type (\(\frac{1}{2}\)\(\frac{1}{2}\) 1) (the one at higher 20 angle) and pseudo-CE-type (\(\frac{1}{2}\)\(\frac{1}{2}\) 0) (the one at lower 20 angle). The low-intensity bump that appears between them--the only one that refers to Mn\({}^{3+}\) in this scheme--is indexed as a CE-type (\(\frac{1}{2}\) 0 1). Figure 9 also shows the extent of long-range correlated AFM regions obtained by the Scherrer formula. Magnetic reflections with the l index of mixed parity have already been reported for charge ordered half-doped manganitesand explained, in most cases, in terms of strain driven phase-separation into two slightly different lattices (i.e., more or less compressed along c), associated to distinct OO/CO patterns, as in the case of Pr\({}_{0.5}\)Ca\({}_{0.5}\)Mn\({}_{0.97}\)Ga\({}_{0.03}\)O\({}_{3}\)[43] and Pr\({}_{0.5}\)Ca\({}_{0.5}\)Mn\({}_{1-x}\)Ti\({}_{x}\)O\({}_{3}\)[44]. This seems appealing for nanoparticles, since it immediately refers to a core-shell model, with the inner and outer parts of the particles undergoing distinct transitions. However, the bulk-like transition sequence suggests instead a single-phase scenario. In this context, we can mention the single-phase solution discussed for Nd\({}_{0.5}\)Ca\({}_{0.5}\)MnO\({}_{3.02}\) in presence of mixed l indexes [45], in which the in-plane components m\({}_{x}\) and m\({}_{y}\) of the magnetic moment in the (001) planes remain arranged according to a CE-type structure, while the component m\({}_{x}\) in the [001] direction change to a pseudo-CE-type AFM order. Within the interval of confidence of the measurements, we show in Figure 10 the evolution of the Mn moment direction of Ho\({}_{0.5}\)Ca\({}_{0.5}\)MnO\({}_{3}\) nanoparticles, when diffraction data are treated as a single nuclear and magnetic phase. One may see that a gradual reorientation of the moment direction occurs between 80 and 20 K, with an increase of the out-of-plane m canting for both Mn\({}^{3+}\) and Mn\({}^{4+}\) crystallographic species with respect to the CE-type bulk structure. Two final questions arise. The first one is whether the canted phase features glassy properties below 40 K, as proposed in [16] and discussed in detail for Sm\({}_{0.5}\)Ca\({}_{0.5}\)MnO\({}_{3}\) in [25]. We guess it does not, at least in the bulk. Indeed, Figure 11 shows the real part of the complex susceptibility of the micrometric Ho\({}_{0.5}\)Ca\({}_{0.5}\)MnO\({}_{3}\) sample, in which no frequency dependence can be detected in the whole temperature range.