Finite-size effects on the evolution of magnetic correlations and magnetocaloric properties of Pr0.4Bi0.2Sr0.4MnO3

The effect of particle size reduction on the magnetic correlations of Pr0.4Bi0.2Sr0.4MnO3 nanoparticles prepared by top-down approach has been studied in detail. It was observed that as the milling time increases from 0 to 240 min, particle size decreases from 160 to 12 nm. Correspondingly it was observed that the ferromagnetic transition temperature (TC) drops (264 to 213 K) and saturation magnetization (MS) decreases (2.12–0.41 μB/f.u.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\upmu }_{\mathrm{B}}/\mathrm{f}.\mathrm{u}.$$\end{document}) while coercivity (HC) shows a monotonous increase (0.18–1.5 kOe) as the particle size decreases due to increase in milling. The magnetic entropy change (ΔS) also decreases (2.41–0.24 J/kg-K) as particle size decreases indicating a strong correlation between magnetism and particle size. The metamagnetic M–H response of the bulk sample, which signifies the magnetic phase coexistence, is suppressed, and the nature of magnetic interactions demonstrates a transition from long range to short range. The observed characteristics emphasizes that with particle size reduction there is an increase in the surface disorder which can be explained by considering the core–shell model for the nanoparticles.


Introduction
The understanding of magnetism at nanoscale is scientifically attractive not only because the quantum mechanical properties of the individual spins become significant, but also for the emergence of new phenomena from confinement and proximity, such as giant magnetoresistance (GMR), spintronics, and superparamagnetism [1][2][3][4]. The significant advances in the availability of strategies for the synthesis and characterization of materials have led to the massive interest in magnetic nanoparticles. Currently, magnetic nanostructures such as nanodisc, nanotubes, nanorods, nanowires, and nanocomposites are widely used as active components in magnetic refrigeration, magnetic diagnosis, drug delivery, catalysis, ferrofluids, sensors, energy storage, logic circuits, etc. [5][6][7][8]. Apart from the above, manipulating the size and shape of the nanostructures place them close to the biological entity such as gene, protein, and cell. This offers an upper hand for magnetic nanoparticles in biomedical applications such as magnetic hyperthermia, magnetic resonance imaging, tissue engineering, cell tracking, and bioseparation [9,10]. Additionally, the large surface of the magnetic nanostructures finds potential application in wastewater treatment as nanoabsorbent, energy harvesting, heat transfer, ferrofluids, photocatalysis, pigmentation, etc. [11][12][13][14][15][16].
The two key factors which dominate the magnetic properties of nanoparticles are the finite-size effects and the surface effects [5,7]. Finite-size effects result from the quantum confinement of electrons such as single-domain limit and superparamagnetic limit, while the surface effects are related to symmetry breaking of the crystal structure at the boundary. The large fraction of atoms residing at the surface of the nanoparticle control the surface interface effects [4,5,7].
In perovskite rare earth manganites, nanoparticles display exotic features such as superparamagnetism, spin glass behaviour, low-field saturation magnetization, low-field magnetoresistance, and large coercivity which are different from their bulk counterparts [17,18]. These striking characteristics make them suitable for application in magnetic hyperthermia, solid oxide fuel cells, magnetic memory devices, spintronic devices, and magnetic sensors [18]. The feasibility to tune the magnetic transition spread over a broad temperature range and the melting of the robust charge ordering (CO) state in the nanoparticles enable their potential for magnetic refrigeration application [18][19][20]. Recently tailoring the ferroelectricity in hexagonal manganite thin films has been realized to be suitable for photovoltaic application [21].
Among rare earth manganites, Pr 1-x Sr x MnO 3 has attracted extra attention owing to its resemblance to the prototype La 1-x Sr x MnO 3 (LSMO). The parent compound PrMnO 3 is an antiferromagnetic (AFM) insulator with T N ≈ 100 K. Systematic replacement of Pr 3+ by Sr 2+ (i.e. Pr 1-x Sr x MnO 3 ) changes the magnetic ground state and the system exhibits a second order paramagnetic (PM) to ferromagnetic (FM) transition for Sr 2+ content 0.20 ≤ x ≤ 0.45 [22][23][24]. In the present study, we focus our attention on Pr 0.6 Sr 0.4 MnO 3 (PSMO)-derived composition. PSMO is a room temperature metallic FM with T C = 310 K, T MI = 260 K and exhibits moderate magnetoresistance (MR) of 40% for H = 80 kOe [25,26]. It is reported to undergo a structural transition from orthorhombic structure (space group Pnma) to monoclinic (space group I2/a) structure at (T S ) = 88 K [27]. It shows both normal and inverse magnetocaloric effect (MCE) in the same sample which is interesting from a practical application point of view as magnetic cooling can be achieved from both adiabatic magnetization and demagnetization in different temperature ranges [26,28].
In view of the contrasting properties of PSMO and BSMO, in the present work, the effect of particle size reduction on the magnetic properties of 20% Bi 3+ -doped Pr 0.6 Sr 0.4 MnO 3 (PSMO) is investigated in detail. 20% Bi 3+ when substituted in the case of FM La 0.7 Sr 0.3 MnO 3 , retains dominant FM state both in the bulk form and in the nanoparticles [43]. However, in the case of FM LaCaMnO 3 [37,44] and PrSrMnO 3 [39], AFM and FM phase separation has been noticed. Previous reports on Bi-substituted PSMO [38,39,45] reveal that for partial Bi 3+ substitution T C reduces and MR improves. For a higher concentration of Bi 3+ , multiple magnetic interactions are observed [39]. In our previous studies on magnetic phase coexistence, the Bi 3+ -substituted LSMO nanoparticles reveal an overall drop in net magnetization with size reduction while the metamagnetic M-H behaviour of the system remains significantly unaltered [46][47][48]. Contrary to that, we find in the present study the suppression in the metamagnetic M-H response with the decrease in particle size. Therefore, a detailed study of the present system was warranted, and we here present our analysis of X-ray diffraction studies and magnetization data discuss the results to reveal the role of particle size reduction on the structure, magnetization, magnetic correlations, and magnetocaloric effect (MCE) of Pr 0.4 Bi 0.2 Sr 0.4 MnO 3 nanoparticles. These oxides were mixed thoroughly using agate mortar pestle with the help of isopropyl alcohol, to get a homogeneous mixture. The obtained mixture was calcined at 800 °C, 900 °C, and 1000 °C, respectively, for 24 h each with intermediate grinding. The powders were then pressed into rectangular pellets and were sintered at 1100 °C for 24 h.

Experimental details
In the second step, the pellets were then crushed into a fine powder and were subjected to high-energy planetary ball milling to prepare nanoparticles. The powders were taken in tungsten carbide jars of 50 ml volume along with 10 mm tungsten carbide balls such that the ball mass-tosample mass ratio was maintained as 20:1. The ball milling was performed at the main disc speed of 300 rpm. The milling time was varied from 0 to 240 min, and a small quantity of sample was taken out a certain interval of time up to 240 min.
The obtained nanoparticles are characterized for their crystal structure and phase purity using room temperature X-ray diffraction (XRD) measurements taken on a M/s. Bruker D2 Phaser powder X-ray diffractometer with Cu-Kα radiation in the 2θ range of 20 °-80 ° with a step size of 0.02°. Magnetic measurements were taken on a M/s quantum design, superconducting quantum interference device (SQUID)-based vibrating sample magnetometer (VSM) in magnetic fields up to ± 70 kOe and in the temperature range 2-380 K. Figure 1a shows the room temperature XRD patterns of bulk and ball-milled PB20SMO samples. The XRD peaks display a systematic broadening with increase in milling time (t m ), suggesting reduction in particle size (D) and concomitant inducement of microstrain (ε) in the lattice. The full width at half maximum (FWHM) of the selected reflections, i.e. (1 2 1), (2 0 2), and (0 4 2), plotted as a function of t m (Fig. 1b) shows a sudden rise in the value during the beginning of ball milling compared to higher t m . This indicates a substantial drop in D occurs during the initial phase of ball milling compared to higher t m .

Room temperature X-ray diffraction
To derive the structural parameters, the obtained XRD patterns have been subjected to Rietveld refinement method using the FullProf program [49]. Figure 1c shows the XRD profiles for selected PB20SMO-t (t = 0, 20, 120, and 240 min) along with the pattern calculated through Rietveld refinement method. The XRD patterns have been indexed considering the orthorhombic structure in Pnma space group. As all the Bragg peaks could be indexed using this structure, contamination in the sample during ball milling can be ruled out. This indicates that all the samples studied here are of single phase. The unit cell parameters and reliability values of Rietveld refinement are summarized in Table 1.
For better estimation of D and ε using XRD, the instrumental resolution file (IRF) was obtained by measuring a NIST standard sample (corundum), before starting the Rietveld refinement of PB20SMO. For subsequent PB20SMO samples, only the quantities influenced by the sample properties were refined. The TCH pseudo-Voigt peak shape function [50] was considered for both standard and our samples. On supplying the IRF file during Rietveld refinement, Fullprof generates a volume average of particle size (D) and microstrain (ε) through the integral breadth method in a microstructure file (*.mic file) [50]. Figure 2 shows the variation in D and ε as a function of t m . A sudden drop in the D has been noticed as the t m changes from 0 to 60 min while the ε exhibits monotonous increment up to 150 min of ball milling and then tends to saturate. The substantial drop in D could be attributed to the enhanced rate of dislocation densities (δ) and defects induced due to the constant collision between the sample mass and balls with the walls of the jar [51,52]. From D, an estimation of δ can be obtained from the relation = 1 D 2 [53]. Inset in Fig. 2 shows the variation in with t m . As t m increases from 0 to 240 min, shows substantial increase from 10 13 to 10 15 lines/m 2 . Also, since the dimension of the feed size before ball milling is larger compared to after each milling  (Table 1). Thus, based on the estimated D from XRD, four samples obtained after t = 0, 20, 120, and 240 min of ball milling having particle size D = 160, 36, 17, and 12 nm, respectively, were chosen to study, in detail, the influence of particle size reduction on the magnetic properties.   well-defined PM-to-FM transition (T C ) which drops from 264 K for D = 160 nm to 210 K for D = 12 nm (inset (ii) in Fig. 3a). Compared to PSMO which has T C = 308 K [28,54], 20% Bi 3+ substitution reduces the T C to 264 K. Also, the low-temperature structural transition which was seen as a drop in ZFC and FC curve for PSMO [26,28] is suppressed with Bi 3+ substitution. However, the bulk PB20SMO-160 (D = 160 nm) sample shows a hump in FC at T ˂ 200 K which could be attributed to the disordered antiferromagnetic (AFM) interactions in the system. Similar low-temperature hump in FC curve has been noticed in case of La 0.7-x Bi x Sr 0.3 MnO 3 (x = 0.30 and 0.35) [46,48,55]. The decrease in T C with Bi 3+ substitution can be explained considering the increase in unit cell volume. As seen from the XRD analysis, the unit cell volume increases from 229.214 to 231.011 Å 3 as x changes from 0 to 0.20 [54]. This elongates the Mn-O-Mn chains, thus reducing the hopping of e g electrons. The double exchange coupling weakens, thereby resulting in the decrease of T C . Also, the highly covalent 6s lone pair character of Bi 3+ is expected to create localization of electrons around the Bi 3+ -rich region which leads to AFM coupling, thereby decreasing the FM character of PSMO. Further, the sharp magnetic transition recorded for the bulk sample broadens with the decrease in D which can be seen as the increase in the FWHM of dM/dT curve (inset (ii) in Fig. 3a). This may be due to the weak magnetic interaction among the spins on the surface of the nanoparticles. Also, a large bifurcation between ZFC and FC curves noticed in all samples specifies the presence of inhomogeneity in the system. The strength of the inhomogeneity can be quantified as M FC -M ZFC (inset in Fig. 3b) which is higher for PB20SMO-36 and PB20SMO-17 samples.

Magnetization studies
The high-temperature inverse magnetic susceptibility ( −1 ) has been analysed using the Curie-Weiss (CW) law [56], = C T− p , C is Curie's constant, and θ p is Curie-Weiss temperature. Figure 4 shows the −1 vs T plots for PB20SMO-D. From the slope of the linear fit in the temperature range 310 ˂ T ˂ 380 K, the values of θ p have been determined. Inset in Fig. 4 shows the θ p vs D −1 plot. The positive values of θ p demonstrate the dominant FM interactions in the system. With a decrease in D, the θ p value decreases suggest a weakening of FM interaction with size reduction. Additionally, it is interesting to note that all samples show a deviation from the CW behaviour exhibiting a downturn, below a certain temperature (T G ) in each sample's case. This may be accounted as the presence of short-range FM correlations above T C . Most reports assign this anomaly to the Griffith's like singularity in the system [57][58][59][60] while some literature suggests the presence of FM polarons above T C could result in −1 (T) downturn [61][62][63]. In general, the existence of GP like behaviour is characterized by a downturn in −1 (T), which is sharp for lower applied fields and subdue with an increase in the magnitude of applied magnetic field. Also, the low field −1 shows a temperature dependence given by the power law [57], where T R C is the critical temperature of a random ferromagnet and the exponent determines the deviation from CW law. The temperature at which −1 (T) deviates from linearity is known as Griffith's temperature, T G . The T G in the present case decreases with a decrease in particle size. Figure 5 shows the −1 vs T plots in different applied magnetic fields for PB20SMO-D. A systematic softening of the downturn has been noticed with an increase in the magnitude of applied magnetic field. This may be understood as in higher applied fields the magnetic contribution from PM matrix enhances and dominates the embedded FM clusters, hence driving towards linear variation in −1 vs T. Using the procedure described in Ref [57], the value of has been estimated from the plot of −1 vs [(T/ T R C )-1] in double logarithmic scale (inset in each panel of Fig. 5). The slope of the linear fit gives the value of tabulated in Table 2. The estimated value of is in good agreement with the reported literature [57][58][59][60], thereby suggesting the applicability of GP model to explain the deviation in −1 (T). Also, the magnitude of is found to increase with particle size reduction implying the enhancement in the strength of GP like singularity. On the contrary, Souza et al. [61] suggested that GP like treatment for accounting the −1 (T) downturn is inappropriate and proposed the presence of FM polarons, intrinsic to manganites due to structural and chemical disorder to be responsible to −1 (T) deviation from CW behaviour. Later, Rozenberg et. al. [64,65] and Aga Shahee et al. [66] supported the above hypothesis. Since conductivity in manganites above T MI is accounted by the hopping of small polarons, it appears as if both the proposed models hold. However, the magnitude of resistivity and polaron activation energy E a of a ferromagnetic manganite is found to increase with a decrease in D [67][68][69], thus suspecting the applicability of the FM polarons hypothesis.
To gain deeper insights into the magnetic interactions, field-dependent isothermal magnetization loops were recorded for PB20SMO-D at different temperatures ( Fig. 6a-d). For PB20SMO-160, the M-H behaviour at 300 K is nonlinear and unsaturated suggesting the presence of weak magnetic interactions in the system. On lowering the temperature to T ˂ T C , i.e. 200 K, the M-H response shows a significant spontaneous moment (M 0 ) with unsaturated magnetization suggesting the coexistence of FM and AFM interactions in the system. At T = 100 K and 50 K, the M-H   The magnetization in the virgin loop remains unsaturated even in the field as high at 70 kOe with notable spontaneous moment signifying the alignment of residual FM domains parallel to the applied magnetic field. In the reverse field sweep (demagnetization curve), the sample remains in a high state down to 10 kOe and then rapidly drops to zero at H = 0 kOe. In the negative field sweep, a response similar to the positive field sweep has been noticed; however, the area under the loop along with the forward and reverse field sweep is less compared to the positive field sweep. The observed M-H response implies the presence of metamagnetic magnetization with the ramping field. This suggests field-induced AFM-to-FM transition in the system. At T = 3 K, a clear step like feature has been observed in the virgin curve at critical field H CR = 50 kOe. Along the demagnetization path, the system remains in a high magnetic state down to 9 kOe field and then drops to zero at zero field. When the field is increased in the negative direction, the magnetization curve is similar to the demagnetization curve of the positive field and follows a similar pathway with further field sweep. This indicates that the field-induced transition from the AFM to FM state is irreversible and the induced FM phase is stable at this temperature. The net magnetization at 3 K is less compared to that at 50 K and 100 K and indicates an increase in the volume fraction of AFM interactions at 3 K. For PB20SMO-36, the recorded magnetization with the ramping field is similar to that of PB20SMO-160 but a small drop in the open-loop area has been noticed. In the case of PB20SMO-17 and PB20SMO-12, the M-H is highly unsaturated signifying the increased surface disorder that leads to the reduced magnetic interaction. Also, it is interesting to note that the difference between the virgin curve and loop which was clearly identified in the bulk sample tend to merge in ball-milled sample implying the increase in the strength of AFM interaction or surge in the surface disorder due to ball milling that brings down the strength of FM interaction in the nanoparticles. Further, the net magnetization has been found to systematically increase with a decrease in temperature contrary to PB20SMO-160 and PB20SMO-36. Figure 7 shows the temperature evolution of (a) saturation magnetization (M S ) and (b) coercivity (H C ). On lowering the temperature from 300 K, an overall increase in the value of M S and H C is noticed. The elevation in M S with a decrease in T corroborates the PM-to-FM transition in all the samples while a small drop in M S at a lower temperature for D = 160 and 36 nm ratifies the hump recorded in the FC curve below T = 200 K for PB20SMO-160 (inset (ii) in Fig. 3a). The H C values plotted as a function D (inset in Fig. 7b) display a rise with a drop in D due to the increased surface anisotropy because of the lattice strain and broken bonds as the D comes down [70]. In general, with a decrease in D, H C increases and attains a maximum value for a critical particle size D C where nanoparticles show a transition from multidomain to single-domain state. For further decrease in D, H C drops. Inset in Fig. 7b shows the variation in H C vs D −1 fitted to H C = m + n/D, where m and n are constants. Linear variation in H C vs D −1 (inset in Fig. 7b) suggests the multidomain nature of the nanoparticles [43,71].
The law of approach to saturation (LAS) fit [72], given to the demagnetization curves in the first quadrant provide more insights into the magnetic behaviour. The term M S in the above relation refers to the saturation magnetization, a/H corresponds to the structural defects while b/H 2 is related to the uniaxial magneto-crystalline anisotropy given as [73] K = √ 15 4 bM 2 s (erg/cm) and p H is the high field susceptibility. Figure 8 shows the LAS fit given to the M-H curves at 3 K and the fitting parameters are summarized in Table 2. A rise in the a/H values specifies an increase in the structural defects due to ball milling. The estimated K values are of the order 10 5 erg/cm 3 and decrease with particle size reduction due to disorder in the core of the nanoparticles. Further a substantial drop in M S from 2.1 μ B /f.u. to 0.2 μ B / f.u. has been noticed with a drop in D (Fig. 9). M S plotted as a function of D −1 (inset in Fig. 9) shows a linear variation Arrott's plots) [57,66]. If the linear interpolation of the high-field-magnetization region of M 2 vs H/M yields a positive slope, it represents the presence of spontaneous moment with long-range FM order while the negative slope represents short-range FM correlations in the system. As shown in Fig. 10, the intercept value for PB20SMO-160 and PB20SMO-36 is negative at T = 300 K ˃ T C and positive for T ≤ 200 K ˂ T C , demonstrating that the magnetic interactions above T C are short-ranged while those below T C the system exhibit long-range FM order. Similarly, for PB20SMO-17, the intercept has negative value at T ˃ 200 K and transforms to be positive T ˂ 100 K signifying a crossover in the range of FM interactions across magnetic ordering. However, for PB20SMO-12, the intercept is negative at all temperatures demonstrate the FM interactions to be short-ranged due to increased surface disorder because to ball milling.
The magnetic entropy change ΔS has been determined from iso-field magnetization using Maxwell's thermodynamic relation [74], where M is the magnetization, H is the applied magnetic field, and T is the temperature. According to Maxwell relation, the change in magnetic entropy upon application of magnetic field is related to magnetization with respect to the temperature through Most frequently magnetization isotherms recorded across the T C /T N and specific heat measurements are used to estimate the ΔS values [74]. However, some recent reports suggest the estimation of ΔS from iso-field magnetization [75,76]. A close overlap in the estimation of ΔS from isothermal and iso-field magnetization validates the application of the latter approach to estimate the MCE. Figure 11 a-d shows the variation of ΔS as a function of temperature in different applied fields (i.e. H = 10 kOe and 50 kOe). As expected, a large change in ΔS has been observed in the vicinity of T C . The bulk PB20SMO shows a maximum entropy change ΔS M of 2.43 J/kg-K for H = 50 kOe which drops to 0.24 J/ kg-K as particle size reduces (table. 3) along with broadening of ΔS curve. The relative cooling power (RCP) defined as RCP = ΔS M × T FWHM , where T FWHM is the full width half maximum of ΔS curve, also drops with a decrease in the particle size (table 3).
A sudden variation in ΔS across T C could be accounted for by considering the change in magnetization and the  [77][78][79][80][81][82][83][84]. The ΔS M value in the present study is less compared to PSMO which indicates that Bi 3+ addition suppresses the net magnetization of the system. From the above results, it can be noticed that with an increase in t m, the dislocation density and microstrain increase while the degree of crystallinity drops. Correspondingly a decrease in T C , M S , K, and ΔS has been observed while H C values display a monotonous increase. Also, a transition in the range of magnetic interactions from long range to short range has been noticed. A similar variation in the magnetic properties with a decrease in particle size has been previously reported in the case of manganites [28,71,73,[85][86][87][88] and other oxides [89][90][91][92]. The observed characteristic could be accounted for by considering the core-shell structure of the nanoparticles [93], according to which each nanoparticle has a FM/AFM grain surrounded by a nonmagnetic shell comprised of broken bonds and randomly oriented magnetic spins.
Assuming the net magnetization of the shell to be zero, an estimate of shell thickness (t) can be attained from the relation [94], where D is the particle size, M S (Bulk) and M S (Nano) corresponds to the saturation magnetization of bulk and nanoparticles, respectively. The M S (Bulk) value is obtained from the intercept of M S vs D −1 plot (inset in Fig. 10). The estimated obtained values of t are tabulated in Table 2.   shows the variation of t/D (i.e. shell thickness/particle size) as a function of D. The t/D ratio increases monotonously with a drop in D. A linear dependence of t/D with D −1 suggests a direct impact of size reduction on the thickness of the non-magnetic shell that strongly controls the magnetic properties of the nanoparticles. Since it has been well understood that with a decrease in D, the surface effects dominate compared to the core. The growth in dislocation density, vacancies, lattice defects, and low crystallinity adds to the surface defects which increases the separation between the FM core. This, in turn, reduces the magnetic interaction among them resulting in the drop of net magnetization, T C and ΔS M with broad magnetic transitions. However, the metamagnetic magnetization loops noticed for bulk and ball-milled PB20SMO demonstrate the observed behaviour to be a characteristic of magnetic interaction among the cores. Since it has been well established that PSMO is a metallic FM [12] and BSMO is a CO-AFM [16], PB20SMO represents a solid solution in both parent phases, thus showing the magnetic state of both phases. Similar coexistence of FM and AFM phases has been reported in case of Bi-substituted LaCaMnO 3 [22,34], LaSrMnO 3 [20,41], NdSr/CaMnO 3 [25,26] for a particular concentration of Bi 3+ . Even though Bi 3+ is diamagnetic and the Mn 3+ / Mn 4+ concentration is unaltered with Bi 3+ substitution, the presence of 6s lone pair electrons of Bi 3+ leads to the localization of charges across the Bi-O bonds. The antiparallel spin orientation between localized electrons and Mn 3+ /Mn 4+ results in AFM superexchange coupling between Mn-O-Mn chains around Bi 3+ -rich regions, thereby driving the system towards AFM.
Further, the scientific conflict in assigning the sharp downturn observed in χ −1 (T) to either GP like singularity or the presence of FM polarons demands the presence of quenched disorder in the system [43][44][45][46][47][48][49][50][51][52][53]. Generally, in most of the manganites, A-site substitution that leads to the structural distortion due to the tilting of MnO 6 octahedra is expected to be the cause of quenched disorder while some reports suggest the competitive coexistence of the FM and AFM phases [44] and the magnetic interactions among the different magnetic ions due to B-site substitution [43] also results in quenched disorder. In the present case, the tilting of MnO 6 octahedra along with FM and AFM phase coexistence due to Bi 3+ ions could be considered as the source of quenched disorder in bulk PB20SMO. Additionally, the broken Mn-O-Mn network due to ball milling also adds to the quenched disorder in PB20SMO nanoparticles. The local FM fluctuations resulted from the random spatial variation  in the magnetic exchange interactions due to particle size reduction result in the formation of short-range FM domains embedded in the PM matrix.

Conclusions
Systematic analysis of the magnetization data has been carried out to understand the role of particle size reduction on the magnetic correlations and magnetocaloric properties of Pr 0.4 Bi 0.2 Sr 0.4 MnO 3 . As the particle size decreases from 160 to 12 nm, T C drops from 264 to 213 K along with a drastic reduction in the net magnetization from 2.12 to 0.41 μ B /f.u. and ΔS M from 2.43 to 0.24 J/kg-K. Correspondingly a suppression in the metamagnetic M-H loops has been observed which suggests a reduced magnetic interaction among the nanoparticles. The observed properties can be explained due to the formation of a non-magnetic shell around the FM/ AFM core. Additionally, a sharp downturn noticed in the inverse susceptibility suggests the presence of short-range FM correlations above T C .
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