## 1 Introduction
Different OO patterns at different doping levels are associated with typical AFM structures, the most common of which, displayed in Figure 1b-d, are known as G-type (no OO at all), A-type (JT compressed octahedra), and C-type (JT elongated octahedra) [8]. CO superimposed to OO characterizes instead the half-doped (x = 0.5) or nearly half-doped compositions that contain nearly equal amounts of Jahn-Teller active Mn\({}^{3+}\) and not-active Mn\({}^{4+}\) (t\({}_{\text{2g}}\)\({}^{3}\)e\({}_{\text{6}}^{0}\)) ions. In its most common description, CO is a real space 1:1 ordering of charge disproportionate Mn\({}^{(3+\delta)+}\) and Mn\({}^{(4-\delta)+}\) species, in which \(\delta\) can be different from zero [7; 8]. Such localization transition of the Mn\({}^{3+}\) e\({}_{\text{g}}\) electrons has been proposed to coincide with the establishment of a charge density wave (CDW), with amplitude determined by \(\delta\) (with \(\delta\) = 0 corresponding to the maximum amplitude) and allowed to be commensurate or incommensurate with the crystal lattice [9; 10; 11]. The CO transition is also a structural phase transformation with martensitic character: it is in fact a first-order orthorhombic to monoclinic ferroelastic transition at which CO twin domains nucleate and grow within the charge disordered (CD) matrix [12]. Very interestingly, especially for the analogy with a sliding CDW, the insulating AFM state with CO can be converted to electrically conductive FM by various means, which include the application of a magnetic field [13], electric field [14], or pressure [15]. The melting of the electronically localized CO state under the effect of an external magnetic field is indeed the phenomenon of colossal magnetoresistance itself. The coexistence of OO and CO in half-doped manganites produces a complex AFM phase called the CE-type (acronyms of charge exchange-type). In such a structure (Figure 1e), Jahn-Teller distorted and undistorted octahedra alternate, leading to a checkerboard ordering of Mn\({}^{3+}\) and Mn\({}^{4+}\) in the (x,y) Cartesian planes, with planar zig-zag chains of ferromagnetically interacting Mn\({}^{3+}\)-O-Mn\({}^{4+}\) ions, with interchain AFM coupling (Figure 1e). The CE-type AFM structure is obtained by the stacking along the \(z\)-axis of such planes, with AFM interplanar coupling. In the case of Ho\({}_{0.5}\)Ca\({}_{0.5}\)MnO\({}_{3}\), orbital ordering is present already at room temperature in bulk samples, while charge ordering takes place at T\({}_{\rm CO}\) = 270 K, even if in most samples there is no special feature in the temperature dependence of magnetization M(T) [16; 17]. Antiferromagnetic correlations between Mn species develop at about 140 K, followed by a transition to a canted AFM phase with T\({}_{\rm N}\) of about 105 K. The AFM supercell, refined from neutron diffraction data, is compatible with a CE-type structure, with the aforementioned checkerboard pattern of two not equivalent Mn ions in the (a,b) plane of the paramagnetic \(Pbmm\) phase. A further magnetic transition is found at 40 K, interpreted as the passage to a glassy state [16] for the marked thermomagnetic irreversibility between the zero field cooled (ZFC) and field cooled (FC) magnetization M(T), similar to what was proposed for Tb\({}_{0.5}\)Ca\({}_{0.5}\)MnO\({}_{3}\)[18]. In the bulk, a fraction estimated of about 8% of the entire sample remains untransformed charge disordered (CD) and it is responsible for a ferromagnetic neutron Bragg scattering superimposed to the AFM one [16]. Later on, it has been shown that the destabilization of the CO CE-type AFM state of half-doped manganites, in favor of Zener ferromagnetism and conductive state, can also be achieved in polycrystalline samples through particle size reduction [19; 20]. Different explanations have been reported for this size effect, among which the fundamental fact that, in nanoparticles, it may become difficult for the CD matrix to accommodate the long-range strain field associated with the transformed CO regions [21].