Structural and Magnetic Properties of Nanosized Half-Doped Rare-Earth Ho 0.5 Ca 0.5 MnO 3 Manganite

: We investigated the structural and magnetic properties of 20 nm-sized nanoparticles of the half-doped manganite Ho 0.5 Ca 0.5 MnO 3 prepared by sol-gel approach. Neutron powder diffraction patterns show Pbnm orthorhombic symmetry for 10 K < T < 290 K, with lattice parameters a, b, and c in the relationship c/ √ 2 < a < b, indicating a cooperative Jahn–Teller effect, i.e., orbital ordering OO, from below room temperature. In contrast with the bulk samples, in the interval 250 < T < 300 K, the ﬁngerprint of charge ordering (CO) does not manifest itself in the temperature dependence of lattice parameters. However, there are signs of CO in the temperature dependence of magnetization. Accordingly, below 100 K superlattice magnetic Bragg reﬂections arise, which are consistent with an antiferromagnetic phase strictly related to the bulk Mn ordering of a charge exchange-type (CE-type), but characterized by an increased fraction of ferromagnetic couplings between manganese species themselves. Our results show that in this narrow band half-doped manganite, size reduction only modiﬁes the balance between the Anderson superexchange and Zener double exchange interactions, without destabilizing an overall very robust antiferromagnetic state. FC magnetization (M FC ) curves were recorded on cooling, keeping the sample in the same applied ﬁeld. AC magnetization measurements were performed during warming, after zero ﬁeld cooling, using an AC ﬁeld with amplitude H AC = 3.5 Oe at 10, 100, and 1000 Hz frequencies, under zero DC bias ﬁeld. for used to the diameter distribution resulting in the average diameter of 21.2(3) nm with a standard deviation of 3 nm and a coefﬁcient of variation of

1:1 ordering of charge disproportionate Mn (3+δ)+ and Mn (4−δ)+ species, in which δ can be different from zero [7,8]. Such localization transition of the Mn 3+ e g electrons has been proposed to coincide with the establishment of a charge density wave (CDW), with amplitude determined by δ (with δ = 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 (acronymous 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 Ho0.5Ca0.5MnO3, orbital ordering is present already at room temperature in bulk samples, while charge ordering takes place at TCO = 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 TN 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 Pbnm 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 Tb0.5Ca0.5MnO3 [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]. A second argument is based on the known characteristic of antiferromagnetic nanoparticles of 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 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 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 Pbnm 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]. A second argument is based on the known characteristic of antiferromagnetic nanoparticles of having uncompensated surface spins, not sharing the core magnetic ordering, the importance of which is the greater the higher the surface to volume ratio [22][23][24][25]. Moreover, the increase of unit cell volume with a decreasing particle size was proposed to affect the Mn e g bandwidth, responsible for a changed balance between FM and AFM interactions [26] In this work, we studied how the structural and magnetic properties are modified by particle size reduction in the particular case of Ho 0.5 Ca 0.5 MnO 3 . Herewith, we shall consider the case of a very small particle size of about 20 nm.

Materials and Methods
Ho 0.5 Ca 0.5 MnO 3 nanoparticles were obtained by the Pechini method [27]. In this sol-gel process, stoichiometric amounts of precursors salts (Ca(NO 3 ) 2 *4H 2 O (99.98%) Alfa Aesar, Ho(NO 3 ) 3 *5H 2 O (99.99%) Alfa Aesar, Mn(NO 3 ) 2 *4H 2 O (99.999%) Alfa Aesar) were dissolved in deionized water and mixed with ethylene glycol and citric acid, forming a stable solution. This solution was then heated on a thermal plate under constant stirring at 350 • C (625 K) to eliminate the excess of water and to accelerate the esterification reaction. After four hours, the solution underwent a drastic volume reduction, due to the formation of the gel. The gel was then calcinated at 400 • C (673 K) for 4 h. The result was a dark amorphous material, with light and crumbly consistency. The product pieces were crushed in a marble mortar until a fine powder was obtained. Finally, the material was allowed to re-crystallize by thermal treatment at a maximum temperature of 700 • C (973 K) for 7 h. X-ray fluorescence analysis provided for the Ho:Ca ratio the value of 49:51, in very good agreement with the nominal one. A bulk polycrystalline sample, the synthesis and characterization of which can be found in [16,28], was used as a reference for the magnetization measurements.
Neutron powder diffraction patterns were collected for 10 K < T < 300 K at the Institut Laue Langevin (ILL) in Grenoble (France) on the D20 diffractometer, a fixed-wavelength 2-axis instrument that can provide different combinations of resolution and intensity of incident neutron flux, obtained by the use of different monochromators and their take-off angles. Knowing which parts of the diffraction pattern contain the information for which the experiment is carried out in the first place permits the most convenient choice. Since the main aim of our experiment was to detect low-intensity magnetic Bragg reflections, which occur at low scattering angles, and given the expected broad Bragg peaks of the nanoparticles, the instrument was used in the low-resolution, high-intensity mode. The wavelength was set at 2.41 Å. Powdered samples were held in cylindrical vanadium containers with a nominal zero coherent scattering cross-section. Data were corrected for instrumental background and container scattering. The nuclear structure at different temperatures was refined according to the Rietveld method using FullProf software [29]; refinements were carried out using a file describing the instrumental resolution function, which is requested by FullProf to determine the average particle size. A discussion of the instrumental resolution function of neutron powder diffractometer can be found in [30], and specifically for D20 in [31].
An analysis of the size distribution of the manganite nanoparticle was also carried out by means of a transmission electron microscope (TEM) JEOL JEM 2010 operated at 200 kV and compared to the diffractometric result.
DC and AC magnetization measurements were performed using a Quantum Design MPMS XL5 Superconducting Quantum Interference Device (SQUID) magnetometer, equipped with a superconducting magnet generating fields up to 50 kOe and calibrated using a Pd standard. The sensitivity for the magnetic moment is 10 −8 emu. The magnetometer is equipped with a system for setting to zero the applied field; the residual field during the zero field cooling is less than 10 −2 Oe. The dependence of DC magnetization on temperature was studied by zero field cooling (ZFC) and field cooling (FC) measurements in the interval 5 K < T < 300 K. For measuring the temperature dependence of the ZFC magnetization (M ZFC ), the sample was cooled from 300 to 5 K in zero field, then the magnetic field was applied at 5 K and the magnetization was recorded on heating the sample. The FC magnetization (M FC ) curves were recorded on cooling, keeping the sample in the same applied field. AC magnetization measurements were performed during warming, after zero field cooling, using an AC field with amplitude H AC = 3.5 Oe at 10, 100, and 1000 Hz frequencies, under zero DC bias field. Figure 2a shows a high-resolution transmission electron microscopy image of the nanoparticles. Despite accurate sonication, agglomeration of nanoparticles occurred, indicating high dipolar forces. The sample is constituted of ellipsoidal particles, mostly with a spherical shape. About 200 particles from several microscopy images were analyzed, and the average diameter was calculated measuring the minimal and maximal diameters for each one. A lognormal distribution was used to fit the diameter distribution resulting in the average diameter of 21.2(3) nm with a standard deviation of 3 nm and a coefficient of variation of 0.13. zero field, then the magnetic field was applied at 5 K and the magnetization was recorded on heating the sample. The FC magnetization (MFC) curves were recorded on cooling, keeping the sample in the same applied field. AC magnetization measurements were performed during warming, after zero field cooling, using an AC field with amplitude HAC = 3.5 Oe at 10, 100, and 1000 Hz frequencies, under zero DC bias field. Figure 2a shows a high-resolution transmission electron microscopy image of the nanoparticles. Despite accurate sonication, agglomeration of nanoparticles occurred, indicating high dipolar forces. The sample is constituted of ellipsoidal particles, mostly with a spherical shape. About 200 particles from several microscopy images were analyzed, and the average diameter was calculated measuring the minimal and maximal diameters for each one. A lognormal distribution was used to fit the diameter distribution resulting in the average diameter of 21.2(3) nm with a standard deviation of 3 nm and a coefficient of variation of 0.13.  Figure 3 displays the neutron powder diffraction pattern collected at 300 K and refined by the Rietveld method, taking as the starting model the bulk Pbnm crystal structure [16]. At the best fitting, the refined nanocrystalline lattice parameters were a = 5.291(6) Å b = 5.435(6) Å, c = 7.472(6) Å, to be compared with the bulk values a = 5.3066(1) Å, b = 5.4686(1) Å, c = 7.4529(1) Å. With respect to the bulk, a and b are slightly contracted while c is elongated. Lattice parameters are in the relation c/√2 ≅ a < b. This means that at room temperature the orthorhombic distortion is mainly due to steric effects, i.e., to the small size of Ho/Ca species at A site, while the additional deformation due to the cooperative Jahn-Teller effect, which leads to the bulk O' structure [32], is very weak. The diffractometric crystallite size was found to be 22 ± 4 nm, which, if compared to the particle size determined by high-resolution transmission electron microscopy, indicates that most particles are monocrystalline.

Results and Discussion
The orthorhombic unit cell is shown in Figure 4 with selected bond lengths and angles. At room temperature, the c axis elongation in Ho0.5Ca0.5MnO3 nanoparticles is accommodated not via octahedra deformation but through the increase of the Mn-Oap-Mn bond angle from the bulk value θ = 152.2° to θ = 158.1°. This is interesting, since for a double-exchange system, the width Wσ of the tight-binding band for itinerant This means that at room temperature the orthorhombic distortion is mainly due to steric effects, i.e., to the small size of Ho/Ca species at A site, while the additional deformation due to the cooperative Jahn-Teller effect, which leads to the bulk O' structure [32], is very weak. The diffractometric crystallite size was found to be 22 ± 4 nm, which, if compared to the particle size determined by high-resolution transmission electron microscopy, indicates that most particles are monocrystalline.
Zener polarons [35] and opening a FM channel within the electron-localiz bulk-like matrix. Indeed, the value of 158° is critical for the previous scenari place at the doping level x = 0.3 [36]. Other nanosize half-doped manganites, in w and antiferromagnetism are completely suppressed, have been reported to disp larger in-plane or out-of-plane bond angles [26]. The orthorhombic unit cell is shown in Figure 4 with selected bond lengths and angles. At room temperature, the c axis elongation in Ho 0.5 Ca 0.5 MnO 3 nanoparticles is accommodated not via octahedra deformation but through the increase of the Mn-O ap -Mn bond angle from the bulk value θ = 152.2 • to θ = 158.1 • . This is interesting, since for a double-exchange system, the width W σ of the tight-binding band for itinerant Mn 3+ 3d e g electrons is strictly related to the bond bending angle (π − θ) as W σ = ε σ λ σ 2 cos(π − θ)<cos(Θ ij / 2 )>, in which ε σ is the one-electron energy, λ σ is the overlap integral, and Θ ij is the angle formed by the underlying localized Mn 3d t 2g (i.e., d(xy), d(xz), d(yz)) core spins [33]. With such an opened Mn-O ap -Mn angle, the e g electron, formally belonging to Mn 3+ , might attain a certain degree of delocalization on Mn pairs along [001] [34], leading to the formation of manganese dimers coupled ferromagnetically by double exchange Zener polarons [35] and opening a FM channel within the electron-localized AFM bulklike matrix. Indeed, the value of 158 • is critical for the previous scenario to take place at the doping level x = 0.3 [36]. Other nanosize half-doped manganites, in which CO and antiferromagnetism are completely suppressed, have been reported to display even larger in-plane or out-of-plane bond angles [26].
The temperature variations of the refined orthorhombic cell parameters are shown in Figure 5a. All refinements were carried out in terms of a single Pbnm phase, i.e., without attempting at using the lower monoclinic symmetry required in presence of CO to distinguish Mn 3+ and Mn 4+ sites [16], given the absence of superlattice reflections related to CO and the low data resolution. 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]. 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/ √ 2 < a < b), with increasing cooperative Jahn-Teller distortion on cooling.   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 dz 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 ZFC (T) and M 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). happens in bulk Ho0.5Ca0.5MnO3, 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]. Therefore, in Ho0.5Ca0.5MnO3 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/√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 MnO6 units, while the Mn-Oap-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-Oap distance remains intermediate between the two equatorial Mn-Oeq ones in the analyzed temperature range, as expected for a Jahn-Teller distorted Pbnm phase with OO. The Jahn-Teller effect in Pbnm RE0.5Ca0.5MnO3 phases is in fact dominated by the orthorhombic Q2 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 Q3, 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 Q3 contribution with apical compression of the MnO6 units (Q3 < 0) along the dz 2 -type orbital direction. The Q3-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 MZFC(T) and MFC(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). In the bulk, CO manifests itself as a bump in M(T) dependence-both FC and ZFC-with TCO ≈ 250 K, which is absent in the nanoparticles. On cooling, both the micro and nanometric samples enter into a canted AFM phase. The bulk Néel 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 Néel point of the 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  In the bulk, CO manifests itself as a bump in M(T) dependence-both FC and ZFCwith T CO ≈ 250 K, which is absent in the nanoparticles. On cooling, both the micro and nanometric samples enter into a canted AFM phase. The bulk Néel 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 Néel point of the 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 sus- , in the absence of any magnetic transition. 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 holmium 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 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 LT /C HT of 2.23 largely exceeds the value of 1.62 expected for the formation, below T CO , of Zener polarons, according to the "bond centered" description of CO given by Daoud Aladine et al. [38], alternative to the site-centered model of Goodenough [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 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 angle range of the diffraction pattern of the nanoparticles taken at 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 angle range of the diffraction pattern of the nanoparticles taken at 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 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 2θ = 25.9 • , rises between 100 K and 80 K, while a second less intense one, at 2θ = 18.6 • , 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) Pbnm 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 × 2b × c when referring to the Pbnm 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 l 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 ( 1 2 1 2 1) (the one at higher 2θ angle) and pseudo-CE-type ( 1 2 1 2 0) (the one at lower 2θ 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 ( 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 z 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. Attempts at analogous measurements for the nanopowders failed, owing to the high noise affecting the sample response. The second one is whether the presence of small amounts of secondary phases, which do not appear in diffraction analyses, might be responsible for the anomaly around 40 K in the low-field temperature-dependent magnetization (cfr inset of Figure 6a). Actually, 40 K is a temperature scale common to many magnetically ordered crystalline systems with Mn-O-Mn superexchange interactions, like hausmannite Mn 3 O 4 (ferrimagnetic with T N = 43 K) [46], perovskite-type Mn 2 O 3 (T N = 100 K and 49 K) [47], hexagonal, and orthorhombic HoMnO 3 themselves (spin reorientation at Tsr = 42 K and T N = 40 K, respectively) [3][4][5]). The same temperature scale may well characterize doped manganite perovskites too. However, the question of magnetic secondary phases, raised for instance for La 0.7 Ce 0.3 MnO 3 and La 0.8 Hf 0.2 MnO 3 [48,49] and discussed in that case by reference to undetected MnO 2 , should not be underestimated.

Conclusions
To summarize, we studied nanoparticles of the half-doped CMR m Ho0.5Ca0.5MnO3, of about 20 nm in size, to investigate the role of reduced dim the stability of the charge and orbital ordered CE-type antiferromagnetic characteristic of the bulk. We conclude that, contrary to what happens in most h rare earth CMR manganites at this length scale, charge ordering and antiferrom in Ho0.5Ca0.5MnO3 are not suppressed. The proposed magnetic structure at the n based on a supercell metrically coincident with the bulk one, however, hostin fraction of Zener double-exchange ferromagnetic interactions among Mn sp result counts Ho0.5Ca0.5MnO3 among those few half-or nearly half-do manganites, such as Sm1−x Ca x MnO3 (x ~ 0.5) [50], in which the antiferromagne state is so robust that the Zener double exchange cannot emerge as a l interaction even by reducing the grain size to the nanoscale. An open issu concerning the nature of the transition seen around 40 K, which Ho0.5 nanoparticles have in common with their bulk counterpart and with sev half-doped CMR phases. The question of whether such a transition is intrinsic be triggered by the rare-earth-manganese interaction, or if it is due to impurities-deserves further investigations.  Figure 11. Real part of the AC magnetization as a function of temperature at three selected frequencies for the micrometric sample. Data were collected using a null DC magnetic field and an AC magnetic field H AC = 3.5 Oe.

Conclusions
To summarize, we studied nanoparticles of the half-doped CMR manganites Ho 0.5 Ca 0.5 MnO 3 , of about 20 nm in size, to investigate the role of reduced dimensions on the stability of the charge and orbital ordered CE-type antiferromagnetic structure characteristic of the bulk. We conclude that, contrary to what happens in most half-doped rare earth CMR manganites at this length scale, charge ordering and antiferromagnetism in Ho 0.5 Ca 0.5 MnO 3 are not suppressed. The proposed magnetic structure at the nanosize is based on a supercell metrically coincident with the bulk one, however, hosting a larger fraction of Zener double-exchange ferromagnetic interactions among Mn species. This result counts Ho 0.5 Ca 0.5 MnO 3 among those few half-or nearly half-doped CMR manganites, such as Sm 1−x Ca x MnO 3 (x~0.5) [50], in which the antiferromagnetic ground state is so robust that the Zener double exchange cannot emerge as a long-range interaction even by reducing the grain size to the nanoscale. An open issue remains concerning the nature of the transition seen around 40 K, which Ho 0.5 Ca 0.5 MnO 3 nanoparticles have in common with their bulk counterpart and with several other half-doped CMR phases. The question of whether such a transition is intrinsic-if it may be triggered by the rare-earth-manganese interaction, or if it is due to magnetic impurities-deserves further investigations.

Data Availability Statement:
The data presented in this study are available upon request from the corresponding authors.