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High-quality epitaxial LaCoO3 (LCO) thin films have been deposited on SrTiO3 (STO) substrates with pulsed laser deposition (PLD). We find that the LCO films undergo a typical ferromagnetic–paramagnetic (FM–PM) phase transition at ∼80 K. To understand the nature of magnetic phase transition, various methods, including the modified Arrott plot and critical isotherm analysis, were used to determine the critical exponents, which are β = 0.754(1) with a Curie temperature TC = 79.8(8) K and γ = 1.52(2) with TC = 79.9(2) K. The reliability of these critical exponents was confirmed using the Widom scaling relation and the scaling hypothesis. Further analysis revealed that the spin coupling within the LCO films exhibits two-dimensional (2D) long-range magnetic interaction and the magnetic exchange distance decays as J(r) ∼ r−(3.46). Our critical behavior analysis may shed new light on the further understanding of the origin of FM and the relatively fixed TC in LCO thin films.

We have studied the magnetic property as a function of temperature and magnetic field (μ0.H) and critical exponent on Ni–Mn-based Heusler alloys such as Ni46Mn41Sn13 (NMS), Ni46Mn41Sn10Si3 (NMSS), and Ni46Mn38Cu3Sn13 (NMCS). Magneto entropy change and the order of phase transition are estimated around the martensite and austenite phases of all these samples. Magnetic critical behavior of each sample is analyzed around second-order ferromagnetic (FM) austenite phase transition employing Arrott and Kouvel–Fisher plots. The reliability of estimated critical exponents is verified by Widom scaling relation and M3 vs H plot. Furthermore, one more local exponent n which depends on the magnetic state is also estimated through the isothermal magnetization of magnetic entropy change (∆SM) plots of all these samples. The obtained values for all these alloys are well close to the mean-field theoretical model and it is proposed that the existence of long-range magnetic interaction around austenite transition of these materials.

As a new antiperovskite nitride, ZnFe3N was synthesized and characterized by almost completely substituting iron atoms at corner positions of γ'-Fe4N. The magnetic interactions of the system with the space group Pm3[combining macron]m are fully investigated. The critical behavior was investigated based on the measured magnetic data around the ferromagnetic phase transition temperature. In this work, the values of critical exponents (β, γ and δ) were obtained systematically using the Kouvel-Fisher method in the critical region. The Widom scaling law (δ = 1 + γβ-1) and the scaling equation (m = f±(h)) were used to reveal the reliability of these values. The values of the critical exponents (β = 0.325, γ = 1.228, and δ = 4.778) are different from those predicted by the three-dimensional (3D) Heisenberg model and mean-field model, and are very close to those of the 3D-Ising model. Combined with ESR analysis, the spin clusters induced by changes in chemical bonds are considered to be the cause for the existence of an anisotropic short-range ordered state in this ferromagnetic system.

We report ferromagnetism in 2H-Co$_{0.22}$TaS$_2$ single crystals where Co atoms are intercalated in the van der Waals gap, and a systematic study of its magnetic critical behavior in the vicinity of $T_c \sim 28$ K. The obtained critical exponents $\beta$ = 0.43(2), $\gamma$ = 1.15(1), and $\delta = 3.54(1)$ fulfill the Widom scaling relation $\delta = 1+\gamma/\beta$ and follow the scaling equation. This indicates that the spin coupling in 2H-Co$_{0.22}$TaS$_2$ is of three-dimensional Hersenberg type coupled with long-range magnetic interaction, and that the exchange interaction decays with distance as $J(r)\approx r^{-4.69}$. 2H-Co$_{0.22}$TaS$_2$ exhibits a weak temperature-dependent metallic behavior in resistivity and negative values of thermopower with dominant electron-type carriers, in which obvious anomalies were observed below $T_c$ as well as the anomalous Hall effect (AHE). The linear scaling behavior between the modified anomalous Hall resistivity $\rho_{xy}/\mu_0H$ and longitudinal resistivity $\rho_{xx}^2M/\mu_0H$ implies that the origin of AHE in 2H-Co$_{0.22}$TaS$_2$ should be dominated by the extrinsic side-jump mechanism.

Structural, magnetic, critical behavior and magnetocaloric properties of La0.6Ca0.4−xSrxMnO3 (x = 0.0, 0.1 and 0.4) compounds have been investigated. Rietveld refinement of the X-ray diffraction patterns indicates that our samples are pure single phase adopting the rhombohedral structure (R-3c) for x = 0.0 and the orthorombic structure (Pbnm) for x = 0.1 and 0.4. Temperature dependence of magnetization curves exhibit a second order paramagnetic (PM)/ferromagnetic (FM) phase transition at Curie temperature Tc. The critical behavior has been determined through the isothermal magnetization measurements around the critical temperature Tc by means of various techniques such as modified Arrott plot (MAP), Kouvel-Fisher (KF) method and critical isotherm analysis (CIA). The results are fully satisfactory to the requirements of the Widom scaling relation and the universal scaling hypothesis confirming their accuracy. A phenomenological model is applied to describe the magnetocaloric effect (MCE) behavior of compounds under investigation. At $$ \mu_{0} H = 5T $$ , the obtained RCP values stand for about 98, 77 and 68% of that observed in pure Gd for x = 0.0, 0.1 and 0.4, respectively, making of these materials considered as promising candidates for magnetic refrigeration applications near room temperature. By analyzing the field dependence of the magnetic entropy change data as well as the relative cooling power, it was possible to evaluate the critical exponent values which were found not only agree with those deduced from (MAP), (KF) and (CIA) methods, but they also obey the scaling theory. Our findings confirm the good correlation between the critical behavior and the MCE properties in manganite systems.

The magnetic properties and critical behavior near the Curie temperature TC in the double perovskite RE2CoMnO6 (RE = La and Eu) were systematically investigated via the modified Arrott plots (MAP), the Kouvel-Fisher plot (K-F), and the critical isotherm analysis (CI). The reliable value of critical parameters (β, δ, and γ) obtained from K-F plot are to be 0.4704, 0.9978, and 3.12 with TC of 210.8 K for La2CoMnO6, and to be 0.6065, 0.9237, and 2.52 with TC of 130.3 K for Eu2CoMnO6, respectively. Moreover, the reliability of critical parameters of two samples was further verified by different methods including the Widom scaling relation and the magnetic equation. The obtained values of critical parameters are all close to the value of the mean-field theoretical model which manifested the presence of the long-range FM order for RE2CoMnO6 (RE = La and Eu) compounds.

The magnetocaloric effect (MCE) and critical behavior in the REPtMg (RE = Tb, Dy and Ho) intermetallic compounds around Curie temperatures TC were investigated by magnetization measurements. All the REPtMg compounds exhibit pronounced MCE performance with the maximum magnetic entropy change ( − Δ S M max ) values up to 5.1, 7.2, 10.2 J/kg K under the magnetic field change ( Δ H ) of 0–5 T around TC of 58, 29, and 20 K for RE = Tb, Dy and Ho, respectively. The reasonable and accurate critical exponents for the present REPtMg series compounds for RE = Tb, Dy and Ho obtained by the Δ H dependence of Δ S M method are evaluated to be β = 0.4691, 0.4681, 0.4141, γ = 1.0412, 1.0482, 1.1067 and δ = 3.2196, 3.2393, 3.6728 around TC, which are verified by the Widom scaling law and scaling equations, respectively.

Nanosized La0.7Ca0.3Mn1-xNixO3 (x = 0, 0.02) manganites were synthesized via the auto-combustion method. The critical behavior of these samples was then carefully investigated by analyzing the results of static magnetic measurements near their critical temperatures. The samples exhibited second-order phase transition, and the effect of Ni doping was to lower the critical temperature. The analysis of the experimental data was carried out by means of various techniques, such as modified Arrott plots, the Kouvel-Fisher method, and critical isotherm analysis, which revealed that the values of the critical exponents β, γ, and δ are close to those theoretically predicted by mean-field theory. By using the Widom scaling relation and the universal scaling hypothesis, the dependability of the obtained values of the critical exponents was confirmed. Moreover, the temperature dependence of the spontaneous magnetization was estimated from the analysis of the variation of the magnetic entropy change with the magnetization within the framework of mean-field theory. Interestingly, the spontaneous magnetization determined from the entropy change and that obtained from classical extrapolation of the Arrott curves displayed high concordance. These results strongly suggested that the interactions among spins in the investigated compound are long-range.

In this work, the La0.8Ca0.2MnO3 nanoparticles of ∼30 nm were prepared by a sol-gel method, and their magnetic properties were investigated. The sample shows a paramagnetic-to-ferromagnetic phase transition at about 214 K. The Griffiths phase, which is identified by the downturn of the inverse susceptibility versus temperature plot from the Curie-Weiss law, is suppressed with the increase of the applied magnetic field strength such as 10 kOe. From the study of magnetic properties in terms of Arrot plots, the nature of magnetic transitions is found to be of second order. The critical behavior of La0.8Ca0.2MnO3 nanoparticles was studied around its Curie temperature. The estimated critical exponents were close to those expected for mean-field model. These critical exponents fulfill the Widom scaling relation, implying the reliability of our values.

In this work, we present the results of the magnetic, critical, and magnetocaloric properties of the rhombohedral-structured La0.55Bi0.05Sr0.4CoO3 cobaltite. Based on the modified Arrott plot, Kouvel–Fisher, and critical isotherm analyses, we obtained the values of critical exponents (β, γ, and δ) as well as Curie temperature (TC) for the investigated compound. These components were consistent with their corresponding values and they were validated by the Widom scaling law and scaling theory. The obtained critical exponents were close to the theoretical prediction of the mean-field model values, revealing the characteristic of long-range ferromagnetic interactions. The magnetic entropy, heat capacity, and local exponent n(T, μ0H) of the La0.55Bi0.05Sr0.4CoO3 compound collapsed to a single universal curve, confirming its universal behaviour. The estimated spontaneous magnetization value extracted through the analysis of the magnetic entropy change was consistent with that deduced through the classical extrapolation of the Arrott curves. Thus, the magnetic entropy change is a valid and useful approach to estimate the spontaneous magnetization of La0.55Bi0.05Sr0.4CoO3.

The critical behavior of La0.75Ca0.05Na0.20MnO3 was studied at around room temperature via magnetization measurements. From the relative slope, we deduced that the Tricritical Mean-Field model was the most suitable model. The estimated critical exponents were found to be β = 0.24 ± 0.004, γ = 0.98 ± 0.065 and δ = 5.08 at TC = 300 K. These critical exponents satisfied the Widom scaling relation δ = 1 + γ/β, implying the reliability of our values. Based on the critical exponents, the magnetization–field–temperature (M–μ0H–T) data around TC collapsed into two curves, obeying the single scaling equation with e = (T − TC)/TC being the reduced temperature. These results suggest short-range interaction in our sample.

We investigate the structural and critical properties of CrTe1−xSbx with 0.0 ≤ x ≤ 0.2. The XRD patterns revealed that Sb substitution resulted in a pure NiAs-hexagonal structure with P63/mmc (194) space group. Lattice refinement of the structure revealed little changes in the a-lattice parameter, along with a more pronounced reduction in the c-axis. The critical behavior in CrTe1−xSbx has been investigated using the magnetization isotherms near the ferromagnetic transition. The obtained critical exponents ( $$\beta , \,\gamma , \,{\text{and}}\, \delta$$ ) revealed that all samples (with 0.0 ≤ x ≤ 0.2) closely follow a mean field-like behavior with ferromagnetic Curie temperature near room temperature. The results from Widom scaling relation indicate self-consistency of the acquired values. Moreover, the magnetization isotherms near the Curie temperature follow a universal scaling behavior, giving further support for the obtained critical exponents.

The critical behavior of the manganese La0.75Ca0.25−xNaxMnO3 (x = 0.00; 0.05) was studied around the PM-FM phase transition. Various techniques are used to obtain the critical exponents of our samples; such as modified Arrott plot (MAP), Kouvel–Fisher (KF) method and the critical isotherm analysis (CI) around the Curie temperature (Tc). The experimental results indicated that our samples had a second-order magnetic phase transition. Based on the above methods, the critical exponents (β, γ and δ) were extracted in the low- and the high-magnetic fields from the magnetic isotherms data. The obtained critical exponents were close to those expected for the Tricritical Mean-Field model. These critical exponents obey to the Widom scaling relation δ = 1 + γ/β, which proves the reliability and the self-consistency of all critical exponents values. The magnetization–field–temperature (M–µ0H–T) falls into two curves, below and above TC. It follows the single scaling equation $$ M\left( {H,\varepsilon } \right) = \varepsilon^{\beta } f_{ \pm } \left( {\frac{H}{{\varepsilon^{\beta + \gamma } }}} \right) $$ with e = (T–TC)/TC is the reduced temperature. This confirms the reasonability of the critical exponents with the scaling hypothesis. Moreover, the analysis of the effective exponents βeff and γeff indicates that βeff (ɛ) for the two samples and γeff (ɛ) for x = 0.00 are self-consistency and in good agreements with the Tricritical mean-Field model, but γeff (ɛ) for x = 0.05 is heterogeneous with any class of predicted universality.

In this study, we report a thorough examination of the critical behavior of the cubic Ni0.3−xCux Zn0.7Fe2O4 (x = 0.0, 0.1 and 0.2) spinel ferrite which are crystallized with different irregular shapes of particle size (spherical,polygonal and cubic). The magnetic transition is ferromagnetic to paramagnetic state for all Cu-doped compounds. Different theoretical analyses, as the modified Arrott plot, Kouvel–Fisher method, and critical isotherm, have been used to identify the values of the ferromagnetic transition temperature TC, and the critical exponents of β, γ and δ. All the obtained values of critical parameters were found to be close to the 3D-Heisenberg theoretical model, which implies short-range magnetic interactions in Cu-doped compounds. The reliability of critical parameters was further checked through different methods, such as the Widom scaling relationship and the magnetic equation.

In this research paper, our central focus is upon structural, optic, magnetic, magnetocaloric, and critical phenomena of La0.57Nd0.1Sr0.18Ag0.15MnO3 (LNSMAO) compound. This compound is crystallized in the rhombohedric system, with a $$R\mathop 3\limits^{ - } c$$ space group. The band gap energy of this powder was 2.5 eV. Therefore, the LNSAMO sample proved to be not only a photocatalysis material but also a photovoltaic application. The FTIR spectra confirmed the formation of the structure of orthorhombic perovskite. The second-order transition of the magnetic phase in the LNSMAO sample was also reported by the positive slopes of the Arrott plot. Beneath 5 T, the values of magnetic entropy change and relative cooling power, respectively, amounted to about 5.08 J kg−1 K−1 and 146.7 J/Kg. In perspective, this sample stands for an agreeable candidate for magnetic refrigeration. By analyzing the dependence on the field of the magnetic entropy change (ΔSM) data as well as the relative cooling power (RCP), it was possible to assess the values of the critical exponents. Moreover, the critical exponents of LNSMAO compound were estimated based on multiple techniques like Kouvel-Fisher (KF), modified Arrott plot (MAP) method, and critical isotherm analysis (CIA). The reliability of these critical behaviors was verified by the Widom scaling relation and the universal scaling hypothesis. Our results demonstrate an excellent correlation between magnetocaloric effect (MCE) properties in manganese systems and critical behavior.

In this manuscript, we have studied the critical fluctuations in the vicinity of phase transition temperature in the Y-doped La0.7Ca0.3MnO3 manganite compound. Details magnetization measurements were carried out to study the critical fluctuations involving modified Arrott plot, Kouvel-Fisher method, critical isotherm analysis and Widom scaling relation. The compound shows a second-order magnetic phase transition near the critical point. The reliability and self-consistency of the obtained critical exponents are further verified by the magnetic equation of state. The obtained value of critical exponents are indicative of mean field theory which suggest the presence of long range ordered ferromagnetic exchange interaction in the studied (La1-xYx)0.7Ca0.3MnO3 (x= 0.4) compound.

The critical behavior of ferromagnetic transition is investigated by bulk magnetization measurements for La0.85Co1.68As2 crystal with a ThCr2Si2-type structure, grown by using NaAs as a flux, in which the vacancies on the both La and Co sites exist. The critical exponents [ β = 0.388(3), γ = 1.44(2) and δ = 4.855(9)] with Curie temperature T c = 99.4 K are obtained, whose self-consistency and reliability are verified by the Widom scaling law and the scaling equation. The critical exponents of ferromagnetic transition occurring in La0.85Co1.68As2 belong to the three-dimensional (3D) Heisenberg model, and suggest that the magnetic interaction decays as J ( r ) ∼ r − 4 . 975 with a short range coupling. These results provide helpful information for understanding the mechanism of ferromagnetism emerging in La0.85Co1.68As2 compound.

This paper presents a detailed study on the critical properties of perovskite manganite La0.6-x Gdx Sr0.4 MnO3(x = 0, 0.05, 0.1) synthesized by Sol-gel method. The critical exponents near the PM-FM phase transition are determined through various methods such as modified Arrot plot, Kouvel-Fisher methods, and critical isothermal analysis based on the data of magnetic measurements recorded around the Curie temperature. These specimens are in agreement with their corresponding values proved by Widom scaling law, as well as the scaling theory. According to the values of critical exponents, the presence of the phase transition is found to be not relevant to the common transition classes but they can be attributed to some abnormal variations. It is suggested that the induced lattice disordering and magnetic disordering, as a result of Gadolinium incorporation, to be potential reasons for the presence of a large magnetocaloric effect, in addition to an anomalous ferromagnetic phase transition in the studied material.

We have performed a systematic study of the critical properties of Pr 0.8 K 0.2 MnO 3 manganite synthesized using two various methods in the vicinity of the ferromagnetic–paramagnetic phase transition. Our compound was successfully prepared by using the solid state reaction at high temperatures and Pechini sol-gel method. The X-ray diffraction pattern shows that all our samples adopt an orthorhombic structure with Pnma space group. Moreover , the critical exponents β, γ and δ are estimated through various techniques such as the modified Arrott plot, the Kouvel–Fisher method and the critical isotherm analysis founded on the data of the magnetic measurements on record near the Curie temperature. Compared to standard models, the estimated critical exponents are close to the theoretical values of 3D-Heisenberg model for the sample elaborated by solid state reaction and tricritical mean-field model for the sample elaborated by the sol-gel method. These results indicate the presence of a ferromagnetic short-range order in our samples. The calculated values using the Widom scaling equation are proximate to those obtained values from critical isotherm M (T C , µ 0 H). The accuracy of the critical exponents values was confirmed with the scaling hypothesis, the magnetization curves fall onto two sides independents below and above T C . Interestingly, the change of the universality class is due to the relevant the diminution of grain size. These results imply that the critical behavior of our samples depended sorely on the synthesis technique.

The critical behaviors of the perovskite La0.5Sr0.5MnO3 nanowires bundles around the ferromagnetic-paraferromagnetic (FM-PM) transition through modified Arrott plot and critical isothermal analysis is investigated. The obtained critical exponents are determined by modified Arrott Noakes equation ( H / M ) 1 / γ = ( T − T c ) / T c + ( M / M 1 ) 1 / β under a low external magnetic field of 0–0.8T, where β = 0.93 and γ = 2.25 were obtained at Tc ≈ 328 k. The availability of the parameters is proved by fulfilling the Widom scaling relation δ = 1 + γ / β which obeys the single scaling equation of M ( H , | e | ) = e β f ± ( H / | e | β + γ ) . The critical exponents determined in this system is unique and not following theories in any published literature. It may be attributed to the nano-size effect and surface magnetism of this system.

The critical behavior of Heusler alloy $\mathrm{Co}{}_{2}\mathrm{TiSn}$ is investigated by bulk magnetization study around the paramagnetic to ferromagnetic transition. The precise value of Curie temperature ($T{}_{c}=358\phantom{\rule{4pt}{0ex}}\mathrm{K}$) as well as the critical exponents ($\ensuremath{\beta}=0.527\ifmmode\pm\else\textpm\fi{}0.003, \ensuremath{\gamma}=1.229\ifmmode\pm\else\textpm\fi{}0.002$, and $\ensuremath{\delta}=3.33\ifmmode\pm\else\textpm\fi{}0.002$) were determined by means of different analytical methods such as modified Arrott plot analysis, the Kouvel-Fisher method, and critical isotherm analysis. With these critical exponents the isotherm $M(H)$ curves below and above $T{}_{c}$ collapse into two universal branches, fulfilling the single scaling equation $m={f}_{\ifmmode\pm\else\textpm\fi{}}h$, where $m$ and $h$ are normalized magnetization and field, respectively. The reliability of the critical exponents were confirmed by Widom scaling hypothesis $\ensuremath{\delta}=\ensuremath{\gamma}{\ensuremath{\beta}}^{\ensuremath{-}1}$. Apart from the slight increase in $\ensuremath{\beta}$ and $\ensuremath{\gamma}$, the deduced critical exponents were consistent with the theoretical prediction of the mean-field model, indicating the long range magnetic interaction in $\mathrm{Co}{}_{2}\mathrm{TiSn}$. Additionally, it is obtained that spin interaction decays as $J(r)\ensuremath{\sim}{r}^{\ensuremath{-}4.7}$. We suggest that the competition between localized majority spins and itinerant minority spins magnetic interaction could be responsible for critical behavior in $\mathrm{Co}{}_{2}\mathrm{TiSn}$.

Nanosized La0.7Sr0.3Mn0.9Cu0.1O3 manganite was synthesized via the sol–gel procedures. The structural, morphological, magnetic and magnetocaloric properties as well as the critical phenomena were investigated. The sample was described by X-ray diffraction (XRD) technique at room temperature which confirms the single phase with a rhombohedral structure belonging to the R$$\overline{3}$$c space group. The magnetic measurements indicated the presence of a second-order ferromagnetic–paramagnetic phase transition at 320 K. According to hysteresis cycles, at 10 K a typical soft FM behavior with a low coercive field was observed. The critical exponents values (β, γ, and δ) as well as the Curie temperature TC were determined using different approaches such as the extrapolation of Arrott plots, Kouvel–Fisher (KF) method and critical isotherm analysis. The critical parameters values (β = 0.489 ± 0.003 with TC = 323.41 K ± 0.23, γ = 1.014 ± 0.020 with TC = 324.32 K ± 0.48 and δ = 3.074) are located near those theoretically predicted by mean-field theory, indicating the long-range ferromagnetic coupling in this system. The reliability of these critical parameters was confirmed by both the universal scaling hypothesis and the Widom scaling relation. Additionally, to estimate the spontaneous magnetization MS (T), we used a process which depends on the analysis of the magnetic entropy change (− ΔSM). A satisfactory agreement was found between the values of the spontaneous magnetization determined from the entropy change ((− ΔSM) vs. M2) and those obtained from the extrapolation of the Arrott curves (μ0H/M vs. M2).

The critical behavior of La0.6Sr0.4Mn0.9Al0.1O3 compound was investigated based on the data of static magnetic measurements in the vicinity of its critical temperature TC. Through various techniques such as modified Arrott plot, Kouvel-Fisher method and critical isotherm analysis, the estimated critical exponents were found to be β = 0.45, γ = 1.1, δ = 3.44 at TC = 321 K which is close to Mean-Field model values. The reliability of the critical exponents’ values was confirmed by the Widom scaling relation and the universal scaling hypothesis. In order to estimate the spontaneous magnetization MS (T) at a given temperature, we used a process based on the analysis, in the mean-field theory, of the magnetic entropy change (−ΔSM) versus the magnetization data. An excellent agreement was found between the spontaneous magnetization determined from the entropy change ((−ΔSM) vs. M2) and the classical extrapolation from the Arrott curves (µ0H/M vs. M2). Moreover, the local exponent n was studied in terms of the field dependence of the magnetic entropy change. It was noted that n evolves with field in the entire studied temperature range of our sample, indicating the validity of the mean-field theory for our compound.

The critical behavior and magnetocaloric effects of the monoclinic Cr5Te8 single crystal have been studied by using magnetization measurements. Experimental results show that the paramagnetic (PM) to ferromagnetic (FM) phase transition is a second-order phase transition at T c = 222 K. The detailed critical behavior analysis indicates that the values of critical exponents β, γ, and δ determined by the modified Arrott plots, the Kouvel-Fisher method as well as the critical isotherm analysis agree well with each other and fulfill the Widom scaling law. The critical exponents are close to those expected for the three-dimensional Ising model and the ferromagnetic interaction is at the boundary between long-range type and short-range one. In addition, the magnetocaloric property of monoclinic Cr5Te8 is consistent with the analysis of critical behavior.

We have investigated the magnetocaloric effects and critical behaviors of Eu2NiMnO6 and Dy2NiMnO6 double perovskite oxides. For a magnetic field varying from 0 to 7 T, the maximum values of the magnetic entropy change ( | Δ S M max | ) reach 4.0 J/kg K and 5.2 J/kg K, and the relative refrigeration capacity (RCP) values are 241.5 J/kg and 385.1 J/kg for Eu2NiMnO6 and Dy2NiMnO6, respectively. The relatively high values of | Δ S M | and RCP suggest that these double perovskite oxides can be used as candidates of magnetic refrigerants working in a wide temperature range. Various techniques such as modified Arrott plot, Kouvel-Fisher method and critical isotherm analysis are used to estimate the critical exponents (β, γ and δ) as well as TC for Eu2NiMnO6. These critical exponents not only obey the Widom scaling relation δ = 1 + γ / β , but also fulfill the scaling equation M ( H , e ) = e β f ± ( H / e β + γ ) where e = (T – TC)/TC, f+ (T > TC) and f- (T

Microscopic magnetic interactions and magnetism can be probed by studying the critical paramagnetic–ferromagnetic phase transition of magnetic materials. In this work, the critical behavior of the lightly Ga-doped manganites Nd0.55Sr0.45Mn0.98Ga0.02O3 (NSMGO) has been investigated. Scaling analysis reveals the critical exponents β = 0.308, γ = 1.197, and δ = 4.916, where their self-consistency and reliability are verified by the Widom scaling law and the scaling equation. These exponents indicate that the magnetic phase transition in NSMGO belongs to the anisotropic 3D-Ising universality class. Nevertheless, the asymptotic critical exponents βeff and γeff are found to approach to the values of 3D-Heisenberg model, implying the existence of isotropic short-range magnetic interactions. We suggest that, although the Nd-based manganites possess a strong magnetocrystalline anisotropy, this effect can be partly averaged out in view of the present polycrystalline NSMGO. Therefore, in the asymptotic critical region, the phase transition reveals the crossover from the scaling of 3D-Ising toward 3D-Heisenberg universality class.

Critical behaviors of Fe1.5−xCoxRh0.5Mo3N ( x = 0.5 , 1.0 , 1.2 ) are investigated, which are well known as a new family hosting skyrmion phase. The critical exponents β, γ, and δ, which correspond to the magnetization, are obtained by the modified Arrott plot and critical isothermal analysis. The reliability of these critical exponents are testified by the Widom scaling law and scaling equation. The critical exponents of Fe1.5−xCoxRh0.5Mo3N ( x = 0.5 , 1.0 , 1.2 ) are close to the universality class of 3D-Heisenberg model, which suggests that the magnetic coupling in this system is of an isotropic short-range type.

In this work, we report the structural and magnetic properties of La0.75Ca0.25MnO3 nanoparticles synthesized by the sol-gel route. Rietvield refinement of X-ray powder diffraction confirms that our sample is single phase and crystallizes in orthorhombic system with Pnma space group. The facts that effective magnetic moment is large and the inverse susceptibility deviates from the Curie Weiss lawn indicate the presence of Griffiths-like cluster phase. The critical exponents have been estimated using different techniques such as modified Arrott plot, Kouvel-Fisher plot and critical isotherm technique. The critical exponents values of La0.75Ca0.25MnO3 are very close to those found out by the mean-field model, and this can be explained by the existence of a long-range interactions between spins in this system. These results were in good agreement with those obtained using the critical exponents of magnetic entropy change. The self-consistency and reliability of the critical exponent was verified by the Widom scaling law and the universal scaling hypothesis. Using the Harris criterion, we deduced that the disorder is relevant in our case. The maximum magnetic entropy change (ΔSM) calculated from the M-H measurements is 3.47 J/kg K under an external field change of 5 T. The ΔSM-T curves collapsed onto a single master curve regardless of the composition and the applied field, confirming the magnetic ordering is of second order nature. The obtained result was compared to ones calculated based on the Arrott plot and a good concordance is observed. Moreover, the spontaneous magnetization obtained from the entropy change is in excellent agreement with that deduced by classically extrapolation the Arrott curves. This result confirms the validity of the estimation of the spontaneous magnetization using the magnetic entropy change.

In this paper, we investigate the critical behavior and the magnetocaloric effect of the polycrystalline MnCo1-xTixGe samples with x = 0.02, 0.04, 0.06 and 0.08. For such materials, we show that the maximum entropy change SM depends on applied magnetic field H2/3. In order to present a detailed study on the critical behaviors, various techniques such as modified Arrott plot, Kouvel-Fisher method, and critical isotherm analysis are used to determine the critical exponent β, γ and δ. The exponents calculated from the critical magnetization isotherm are found to obey the Widom scaling relation remarkably well. Moreover, we also report the accuracy of the three methods using the so-called relative slopes (RS) approach. These critical exponents are found similarly to the theoretical values of the mean-field model, consistent with the existence of long-range ferromagnetic interactions. Suitable magnetocaloric effect and good tunable TC responses are good candidates for magnetic refrigeration materials room temperature.

In this study, the critical behaviors around the paramagnetic or the antiferromagnetic to ferromagnetic martensite transition in normal Ni49Mn39Sb12 alloy by bulk magnetization were investigated. The critical exponents (β, γ, and δ) have been obtained by different methods such as modified Arrott plot, Kouvel-Fisher method, and critical isotherm analysis. By these critical exponents, the data of magnetization, field and temperature around T C M collapse into two curves, obeying the single scaling equation M ( H , e ) = ( e ) β f ± [ H / e ( γ + β ) ] . These critical exponents are also confirmed by the Widom scaling law δ = 1 + γ / β . These exponents are close to those theoretically predicted by mean-field model, indicating that the magnetic interactions are long-range-like ferromagnetism around TC. In addition, the exchange distance has been obtained as J ( r ) = r − 4.52 , which also supports the long-range field-mean-like behaviors.

In the present work, we report a detailed study of the structural, magnetic, magnetocaloric properties as well as critical behavior in La 0.65 Dy 0.05 Sr 0.3 MnO 3 ceramic synthetized by the conventional solid-state reaction. Rietveld refinement of the X-ray diffraction patterns indicates that our sample is pure single phase adopting the rhombohedral structure ( R 3 ¯ c space group). Temperature dependence of magnetization exhibits a paramagnetic (PM) to ferromagnetic (FM) transition at Curie temperature T C = 265 K. Close to this transition, the magnetic entropy change reaches a maximum of about 2.19 J/kgK and the relative cooling power (RCP) value is of 241 J/kg, when a magnetic field of 5 T is applied. Additionally, Banerjee's criteria, Landau theory and the universal behavior of phase transitions were also studied to assess magnetic phase ordering in this material, such investigations indicate that the phase transition from the PM to FM states is of second order. Through various techniques such as modified Arrott plot (MAP), Kouvel–Fisher (KF) method and critical isotherm (CI) analysis, the critical parameters (T C , β, γ, and δ) have been determined. The estimated critical exponents (β = 0.515, γ = 1.082 and δ = 2.932 at T C = 264.85 K) are found to be close to the mean-field model. This proves the existence of long range ferromagnetic interactions in this system. The reliability of the critical exponent values was confirmed by the Widom scaling relation and the universal scaling hypothesis.

The present paper investigates the influence of Ca- doping on the structural, magnetic, magnetocaloric properties and the critical behavior of La0.6Bi0.1Sr0.3−xCaxMn0.9Cu0.1O3 specimens (with x = 0.1 and 0.15). Powder X-ray diffraction revealed the coexistence of rhombohedral (R-3c) and tetragonal (I4/mcm) phases. The temperature dependence of magnetization suggests that samples underwent a ferromagnetic-paramagnetic transition in the vicinity of the Curie temperature (TC). According to the Banerjee criterion, the magnetic phase transition is of a second-order type. The maximum of the magnetic entropy change ( - Δ S M max ) was found to be 3.66 J Kg−1 K−1 and 3.74 J Kg−1 K−1 for × = 0.1 and × = 0.15 respectively, under a field change of 5 T. The modified Arrott plot, the Kouvel-Fisher procedure and the critical isotherm were used to determine the critical parameters values of β, γ and δ. All the obtained values of the critical exponent turned out to be close to the value of the theoretical 3D-Ising model. The reliability of the critical exponent was further checked through different methods, such as the Widom scaling relationship and the magnetic equation.

This paper presents a detailed study on the structural, magnetic, and critical behavior and magnetocaloric effect of La0.7Sr0.3Mn0.94Cu0.06O3 prepared by solid-state reaction method. X-ray diffraction patterns indicate that Cu-doped manganite crystallizes to a single phase with a rhombohedral structure. Critical components β, γ, and δ were determined by using modified Arrott plot, Kouvel–Fisher method, and critical isotherm analyses. These components are consistent with their corresponding values and were validated by Widom scaling law and scaling theory. The deduced critical exponents (β = 0.456, γ = 0.9912, and δ = 3.206) are also close to their corresponding theoretical values predicted by mean-field model, indicating that long-range interaction dominates the critical behavior of the LSMCO system. Spontaneous magnetization was determined by two methods. One method is based on magnetic entropy change under different magnetic fields, and the other method is based on classical extrapolation from the Arrott plot. The two methods are in excellent agreement, confirming the validity of the deduction of spontaneous magnetization using the magnetic entropy change.

The critical behavior associated with the magnetic phase transition has been investigated by magnetization isotherms in La 0.6 Ca 0.4− x Sr x MnO 3 nano-particle compounds for x = 0 (S0C1), 0.05 (SC. 4-4) and 0.4 (S1C0). The critical exponents are estimated by various techniques, such as the Modified Arrott plot, Kouvel–Fisher plot and critical isotherm techniques. Thus, the average values of the critical exponent and the critical temperature obtained by the different methods are ( β m o y = 0.47 ; γ m o y = 1.02 ; δ m o y = 3.00 ; T C = 230.6 K ) for S0C1, ( β m o y = 0.47 ; γ m o y = 1.00 ; δ m o y = 3.21 ; T C = 361.9 K ) for S1C0, and ( β m o y = 0.26 ; γ m o y = 1.02 ; δ m o y = 4.92 ; T C = 286.4 K ) for SC. 4-4. These values are in good agreement with those given by the theoretical models: Mean-Field model ( β = 0.5, γ = 1 and δ = 3) for S0C1 and S1C0 compounds; and Tricritical mean-Field model ( β = 0.25, γ = 1 and δ = 5) for SC. 4-4 one. The reliability of the critical exponent values was confirmed by the Widom scaling relation and the universal scaling hypothesis.

We have studied the influence of Ce, Pr and Nd elements substitution on the magnetic and magnetocaloric properties of La 0.6 A 0.2 Sr 0.2 MnO 3 (A = Pr, Nd, Ce) polycrystalline manganites. X-ray diffraction patterns show that the samples crystallize in the rhombohedral structure with R-3C space group. The Pr and Nd doped samples are single phase, while Ce doped sample has a CeO 2 secondary phase. The temperature dependent magnetization exhibits a sharp paramagnetic (PM)–ferromagnetic (FM) transition at 333, 252 and 243 K for Ce, Pr and Nd doped samples, respectively. The results show that the saturation magnetization of the pure and Nd doped samples are higher than Pr doped sample, while the Ce doped sample exhibits lowest magnetization. The magnetic entropy change (ΔS M ) was calculated by applying the Maxwell equation and the estimated values in H = 25 kOe were found to be 3.20, 2.63, 3.34 and 3.50 J/kgK for pure, Pr, Nd and Ce doped samples, respectively. Also, improved relative cooling power was observed for the Pr and Nd doped samples. By using three different methods including analyzing magnetization data using Banerjee's criterion, normalizing entropy change versus reduced temperature and using ΔS M (T) data which is related to the local exponent (n L ), it is found that all samples display a second-order magnetic phase transition. The critical exponents of the second-order magnetic phase transition for the samples has been calculated, using scaling law and Widom scaling relation and their reliability has also been confirmed with the scaling hypothesis.

The scaling critical behaviors of Gd12Co7 compound around TC were investigated based on the M-H curves in a magnetic field change of 0–2 T. The critical exponents β and γ determined by modified Arrott plot (MAP) and Kouvel–Fisher (KF) methods are [β=0.479(5) and γ=1.004(2)] and [β=0.473(2) and γ=0.983(3)], respectively. The exponents δ derived from Widom scaling relation ( M T = T C = 163 K = D H 1 δ ) and universal relation of the relative cooling power ( R C P ∝ H 1 + 1 δ ) are δ=3.032(8) and δ=2.903(1). The average values of critical exponent (β=0.476(3), γ=0.993(7), and δ=2.967(9)) are very close to mean-field model (β=0.5, γ=1, and δ=3), which indicates that the magnetic interactions in Gd12Co7 compound are long–range interactions. The average value of critical exponent n for MAP (0.649(1)), KF (0.638(3)), and | Δ S M | ∝ H n (0.714(8)) at T C is 0.667(4) and well in agreement with mean field long–range interaction model ( n = 2 / 3 ). The plot M 1 β vs. ( H / M ) 1 γ constructed by above critical exponents fall into two distinct branches above and below T C and completely complies with the scaling hypothesis. At the same time, the normalized curve of magnetic entropy change shows that renormalized magnetic entropy change Δ S ′ of Gd12Co7 is mainly determined by a=1.548(1) and b=1.549(3) in Lorentz function.

The quantum phase transition and thermodynamics of a periodic Anderson-like polymer chain in a magnetic field are investigated by Green’s function theory. The T - h phase diagram is explored, wherein a crossover temperature T ∗ denoting the gapless phase crossover into quantum critical regimes, smoothly connects near the critical fields to the universal linear line T ∗ ∼ ( h − h c,s ), and ends at h c,s , providing a new route to capture quantum critical point (QCP). The quantum critical scaling around QCPs is demonstrated by analyzing magnetization, specific heat and Gruneisen parameter Γ h , which provide direct access to distill the power-law critical exponents ( β , δ and α ) obeying the critical scaling relation α + β (1 + δ ) = 2, analogous to the quantum spin system. Furthermore, scaling hypothesis equations are proposed to check the scaling analysis, for which all the data collapse onto a single curve or two independent branches for the plot against an appropriate scaling variable, indicating the self-consistency and reliability of the obtained critical exponents.

We review the different approaches to establish the boundary between the liquid and the quasi-gas fluid above the critical point. Some experimental data by inelastic X-ray scattering demonstrates different dynamical regimes: gas-like and liquid-like reminiscent of the subcritical domains. Theoretical models based on the Lennard-Jones and square well interparticle potentials together with phenomenological Van der Waals model confirm similar picture for the Widom line in supercritical domain. To minimize uncertainty experimental data we calculated extrema of response functions via fundamental equations of state. The shift of critical points under influences of impurities and nanoparticle adding is analyzed.

Based on the modified Arrott plots and the critical isotherm analysis, the critical exponents of both La0.5Ca0.5MnO3 and La0.5Ca0.4Ag0.1MnO3 systems have been comprehensively studied and investigated near the paramagnetic-ferromagnetic (PM-FM) phase transition. Critical exponents, β, γ, and δ, which are corresponding to the thermal variation of the spontaneous magnetization, initial susceptibility, and the isothermal magnetization, respectively, obeyed the Widom scaling law δ = 1 + γ/β. It indicates thus the validity the obtained results. We performed then another analysis of the critical behavior via the magnetocaloric effect presenting an accord with the analysis of magnetization measurements. Relative cooling power (RCP) follows the evolution of the magnetic field as RCP ∼ H 1+1/δ . The acquired critical exponents were in conformity with those gotten from the modified Arrott plots and Widom scaling law.

Magnetic phase transitions and the magnetocaloric effect of the polycrystalline Mn1-xSnxCoGe samples with x = 0.02 , 0.04, and 0.06 have been investigated. The transformation temperature, room temperature magnetocaloric effect and magnetic orders of these alloys could be effectively tuned by Sn substitution. A detailed study on the critical behavior was performed by various techniques, such as modified Arrott plot, Kouvel–Fisher method, and critical isotherm analysis, to determine the critical exponents (β, γ, and δ). Those exponents calculated from the critical magnetization isotherm were found to obey the Widom scaling relation remarkably well and the validity of the obtained critical exponents was also verified by the prediction of the scaling theory in the critical region. All the results are similar to the theoretical values of the mean-field model, indicating the presence of long-range ferromagnetic coupling in this system. Moreover, we made an attempt to contrast the universal curve for magnetocaloric effect in the present system, finding that rescaled entropy data for all alloys collapse into a universal curve.

In this paper, we studied in detail the critical behavior of La0.7Ca0.2Ba0.1MnO3 manganite sample synthesized by the solid state reaction. The critical properties near the ferromagnetic–paramagnetic phase transition have been analyzed from the magnetic data, using various techniques such as the modified Arrott plots, the Kouvel fisher method, critical isotherm analysis and the field dependence of magnetocaloric results. Moreover, the validity of the critical exponents using the various methods has been confirmed by the scaling equation of state and all data fall on two distinct branches, one for T TC, indicating that the critical exponents obtained in this work are accurate. The exponents determined in this study have revealed that the β value is in between the theoretically expected values for 3D-Ising and tricritical mean field model. While, the γ value is close to the mean field theory. The third exponent δ was determined independly from the critical isotherms satisfying the Widom scaling relation. Our systematic result reflects the coexistence of a long range and short range ferromagnetic order in our sample. The temperature variation in the effective exponents βeff and γeff resemble with those for disordered ferromagnet around Curie temperature.

The critical properties of La0.7Ca0.2Sr0.1MnO3 manganite elaborated using two different methods have been investigated around the ferromagnetic-paramagnetic phase transition. Our sample was successfully synthesized by solid-state reaction (S1) and sol-gel route (S2). The X-ray diffraction characterizations show that (S1) crystallized in orthorhombic structure with Pnma space group without any detectable impurity. Though, (S2) crystallized in rhombohedra structure with R-3C space group. Various techniques such as modified Arrott plots, Kouvel-Fisher method and critical isotherm analysis were used to determine the values of the Curie temperature TC, as well as the critical exponents β (corresponding to the spontaneous magnetization), γ (corresponding to the initial susceptibility) and δ (corresponding to the critical magnetization isotherm). The estimated results are close to those expected by the mean-field model for sample S2; while for sample S1, the exponents values are close to those expected using the tricritical model. The obtained values from critical isotherm M (TC, μ0H) are close to those determined using the Widom scaling relation, and also there are found to follow scaling equation with the magnetization data scaled into two different curves below and above TC, which implies the reliability and accuracy of the exponents. The change of the universality class can be explained by the reduction of the grain size. These results show that the critical behavior of our sample depend havely on the synthesis technique.

We have considered a generalized version of a model consisting of self-propelled particles, where the speed is treated as an independent stochastic variable described by a Gamma-distribution, and the motion direction is adjusted in according to two behavioral rules: repulsion and orientation. We used the finite-size scaling theory to estimate the static critical exponents β , γ and ν as a function of the density ρ . We have obtained different values of the critical exponents for different values of the density ρ .

We report the magnetic, magnetocaloric properties and critical behavior near transition temperature in La 0.78 Pb 0.22 MnO 3 perovskite manganite, prepared by solid state reaction method. Temperature dependence of magnetization exhibits a ferromagnetic to paramagnetic transition (T C ) at 292 K . Close to this transition, a maximum magnetic entropy change ( − Δ S M max ) of 4.31 J/kg K and a relative cooling power (RCP) of 238 J/K are determined at a field of 5 T. Experimental results revealed that this sample underwent a second-order phase transition. Using the modified Arrott plot, Kouvel-Fisher method and critical isotherm analysis, the critical parameters (T C , β, γ, and δ) have been determined. The estimated critical exponents (β=0.246, γ=0.929 and δ=4.49 at T C =290.01 K) are found to be close to the tricritical mean-field model. These critical exponents fulfill the Widom scaling relation δ = 1 + γ / β , implying the reliability of our values. On the basis of the obtained critical exponents, the magnetization-field -temperature (M-μ 0 H-T) data around T C are found to collapse into two curves by obeying the single scaling equation M ( μ 0 H , e ) = ( e ) β f ± ( μ 0 H / e β + γ ) where e = T − T C / T C , f + above and f − below T C , and confirm that the obtained values of β, γ and δ are reasonably accurate and unambiguous.

Critical behavior and magnetic properties of La0.67Sr0.33MnO3 (symbolized as LS33MO) have been studied by means of Monte Carlo simulation (MCS). By using the heat-bath algorithm and the Ising model up to third nearest neighbor exchange interactions, we have computed the magnetic and magneto-caloric properties. The effects of the temperature (T) and the external magnetic field (h) on the magnetic behavior of LS33MO have been discussed. The results demonstrate that the Curie temperature (TC) is close to the experimental value. The magnetic entropy shows a maximum value around TC which increases linearly with the rise of the external field h. For h = 5T, the maximum entropy ($$ \Delta {S}_m^{\mathrm{max}} $$) is about 3.67 J K−1 kg−1 which is in good agreement with the experimental results. The relative cooling power (RCP) has been estimated and it has been found that it increases with respect to the external field. The predicted critical behavior has been studied by isothermal magnetization of LS33MO nanoparticle compound. The critical exponents (Crt.Exps) are determined by various approaches, such as the modified Arrott plot (AP) and critical isotherm ways. Therefore, the Crt.Exp values obtained by different techniques are β = 0.306, γ = 1.178, and δ = 4.025 respectively. The trustworthiness of the values of the Crt.Exp was approved by the Widom scaling equation and the universal scaling hypothesis.

We have investigated the critical phenomenon associated with the magnetic phase transition in the half-metallic full-Heusler ${\mathrm{Co}}_{2}\mathrm{TiGe}$. The compound undergoes a continuous paramagnetic-to-ferromagnetic phase transition below the Curie temperature ${T}_{C}=371.5$ K. The analysis of magnetization isotherms in the vicinity of ${T}_{C}$, following a modified Arrott plot method, the Kouvel-Fisher technique, and a critical isotherm plot, yields the asymptotic critical exponents $\ensuremath{\beta}=0.495, \ensuremath{\gamma}=1.324$, and $\ensuremath{\delta}=3.67$. The self-consistency and reliability of the obtained exponents are further verified by the Widom scaling relation and scaling equations of state. The mean-field-like value of the critical exponent $\ensuremath{\beta}$ suggests a long-range nature of the exchange interactions, whereas the values of the critical exponents $\ensuremath{\gamma}$ and $\ensuremath{\delta}$ imply sizable critical spin fluctuations. The half-metallic itinerant character of ${\mathrm{Co}}_{2}\mathrm{TiGe}$ in the presence of magnetic inhomogeneity may result in such a strong deviation from the three-dimensional Heisenberg values ($\ensuremath{\beta}=0.369, \ensuremath{\gamma}=1.38$, and $\ensuremath{\delta}=4.8$) of the critical exponents toward the mean-field values ($\ensuremath{\beta}=0.5, \ensuremath{\gamma}=1$, and $\ensuremath{\delta}=3)$. The results suggest the complex nature of exchange couplings that stabilize the long-range ferromagnetic ordering in the system and are consistent with the earlier theoretical studies on the exchange mechanism in ${\mathrm{Co}}_{2}\mathrm{TiGe}$.

The critical behavior of manganite perovskite type La0.8-x·xNa0.2-x·xMnO3 (with x = 0 and 0.1) was studied based on the experimental magnetic measurements in the paramagnetic ferromagnetic transition region using different techniques such as the modified Arrott plot, the Kouvel-Fisher method and critical isothermal analysis. Experimental results revealed that the two samples show a second-order magnetic phase transition. The deduced critical exponents are close to a 3D Ising model for La0.8Na0.2MnO3 and a 3D tricritical mean field model for La0.7·0.1Na0.1·0.1MnO3. The reliability of these critical exponents was confirmed by the Widom scaling relation and the universal scaling hypothesis. Moreover, an excellent agreement was found between the spontaneous magnetization determined from the entropy change and from the extrapolation of the modified Arrott plots, confirming the universal behavior of the magnetocaloric effect in these systems.

We study the magnetocaloric effect and critical behavior of Co$_2$Cr$_{1-x}$Mn$_x$Al ($x=$ 0.25, 0.5, 0.75) Heusler alloys across the ferromagnetic (FM) transition (T$_{\rm C}$). The Rietveld refinement of x-ray diffraction patterns exhibit single phase cubic structure for all the samples. The temperature dependent magnetic susceptibility $\chi$(T) data show a systematic enhancement in the Curie temperature and effective magnetic moment with Mn concentration, which is consistent with the Slater-Pauling behavior. The M(H) isotherms also exhibit the FM ordering and the analysis of $\chi$(T) data indicates the nature of the phase transition to be a second order, which is further supported by scaling of the entropy curves and Arrott plot. Interestingly, the Mn substitution causes an increase in the magnetic entropy change and hence large relative cooling power for multi-stage magnetic refrigerator applications. In order to understand the nature of the magnetic phase transition we examine the critical exponents $\beta$, $\gamma$, $\delta$ for the $x=$ 0.75 sample by the modified Arrott plot and the critical isotherm analysis, which is further confirmed by Kouvel-Fisher method and Widom scaling relation, respectively. The estimated values of $\beta=$ 0.507, $\gamma=$ 1.056, $\delta=$ 3.084 are found to be close to the mean field theoretical values. The renormalized isotherms (m vs h) corresponding to these exponent values collapse into two branches, above and below T$_{\rm C}$ that validates our analysis. Our results suggest for the existence of long-range FM interactions, which decays slower than power law as $J(r)\sim r^{-4.5}$ for a 3 dimensional mean field theory., Comment: originally submitted on 26/3/2019

We study the critical exponents in amorphous FexNi80-xB12Si8 alloys with (x = 2.4, 8 and 16) around ferromagnetic (FM) to paramagnetic (PM) phase transition temperature. The melt spinning method has been used to synthesize the amorphous alloy. The magnetic transition nature undergoes a second-order magnetic phase transition from ferromagnetic to paramagnetic states. The critical exponents were analyzed utilizing the modified Arrott plot technique (MAP), the Widom scaling relation (WSR), and the critical isotherm analysis (CIA). Besides, a separate analysis of the critical performance is presented in terms of the magnetocaloric effect (MCE). It was found to be in line with the conclusion drawn from magnetization data analysis. The determined critical exponent values were found to be consistent and comparable with those predicted by mean field approach. This result emphasizes that the ferromagnetic exchange interaction is of the long range type.

A model for the description of thermodynamic properties of hard/soft magnetic bilayer is proposed. We perform a thorough research of critical phenomena of this model by the Monte-Carlo method. Critical exponents of the heat capacity α , magnetization β , susceptibility γ , and correlation radius ν are calculated using the finite-size scaling theory. Obtained values of critical exponents agree with the theoretical predictions for the XY-model.