Critical Isotherm Analysis Research Articles

Critical behavior in itinerant ferromagnet [formula omitted]

SrRuO3 presents a rare example of ferromagnetism among the 4d based oxides. While the nature of magnetic state in SrRuO3 is mostly believed to be of itinerant type, recent studies suggest a coexistence of both itinerant and localized model of magnetism in this material. Here, we have investigated the evolution of magnetic state in doped SrRu1-xTixO3 through studying the critical behavior using standard techniques such as, modified Arrott plot, Kouvel-Fisher plot and critical isotherm analysis across the magnetic transition temperature Tc. The substitution of nonmagnetic Ti4+ (3d0) for Ru4+ (4d4) would simply dilute the magnetic system apart from modifying the electron correlation effect and the density of states at Fermi level. Surprisingly, Tc does not change with x. Moreover, our analysis show the exponent β related to spontaneous magnetization increases while the exponents γ and δ related to initial inverse susceptibility and critical magnetization, respectively decrease with Ti substitution. The estimated exponents do not match with any established theoretical models for universality classes, however, the exponent obey the Widom relation and the scaling behavior. Interestingly, this particular evolution of exponents in present series has similarity with that in isoelectronic doped Sr1-xCaxRuO3. We believe that site dilution by Ti leads to formation magnetic clusters which causes this specific changes in critical exponents.

Adiabatic temperature change, magnetic entropy change and critical behavior near the ferromagnetic–paramagnetic phase transition in La0.7(Ca,Sr)0.3MnO3 perovskite

ABSTRACTA systematic study of the magnetocaloric effect in La0.7Ca0.15Sr0.15MnO3 perovskite has been performed by both direct and indirect measurement techniques. In particular, the adiabatic temperature change has been measured by a purpose-built probe suitable for laboratory-scale samples, while magnetic measurements were carried out by standard magnetometric techniques. The observation of the effective behavior of the studied systems in adiabatic conditions and the significant value of the isothermal entropy variation indicate that La0.7Ca0.15Sr0.15MnO3 could be considered as a potential candidate for application in magnetic refrigeration around room temperature. Moreover, the critical exponents (, and ) determined using the Kouvel–Fisher method, the modified Arrott plot and the critical isotherm analysis are in good agreement with the theoretical prediction of the 3D-Ising model, indicating that the magnetic interactions are of short-range nature.

Critical behavior and magnetocaloric effect in monoclinic Cr5Te8

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 Tc= 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.

Magnetocaloric effect and critical behaviors of R2NiMnO6 (R=Eu and Dy) double perovskite oxides

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 (|ΔSMmax|) 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 |ΔSM| 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,ε)=εβf±(H/εβ+γ) where ε = (T – TC)/TC, f+ (T > TC) and f- (T < TC) are the regular analytic functions. Therefore, the deduced critical exponents are reasonably accurate. Moreover, temperature variation in effective critical exponents resemble with those for disordered ferromagnet, which matches with the Griffiths-like phase behavior observed from inverse magnetic susceptibility just above TC.

Critical behavior of magnetization in URhAl: Quasi-two-dimensional Ising system with long-range interactions

The critical behavior of dc magnetization in the uranium ferromagnet URhAl with the hexagonal ZrNiAl-type crystal structure has been studied around the ferromagnetic transition temperature T_C. The critical exponent beta for the temperature dependence of the spontaneous magnetization below T_C, gamma for the magnetic susceptibility, and delta for the magnetic isotherm at T_C have been obtained with a modified Arrott plot, a Kouvel-Fisher plot, the critical isotherm analysis and the scaling analysis. We have determined the critical exponents as beta = 0.287 +- 0.005, gamma = 1.47 +- 0.02, and delta = 6.08 +- 0.04 by the scaling analysis and the critical isotherm analysis. These critical exponents satisfy the Widom scaling law delta=1+gamma/beta. URhAl has strong uniaxial magnetic anisotropy, similar to its isostructural UCoAl that has been regarded as a three-dimensional (3D) Ising system in previous studies. However, the universality class of the critical phenomenon in URhAl does not belong to the 3D Ising model (beta = 0.325, gamma = 1.241, and delta = 4.82) with short-range exchange interactions between magnetic moments. The determined exponents can be explained with the results of the renormalization group approach for a two-dimensional (2D) Ising system coupled with long-range interactions decaying as J(r)~r^-(d+sigma) with sigma = 1.44. We suggest that the strong hybridization between the uranium 5f and rhodium 4d electrons in the U-Rh_I layer in the hexagonal crystal structure is a source of the low dimensional magnetic property. The present result is contrary to current understandings of the physical properties in a series of isostructural UTX uranium ferromagnets (T: transition metals, X: p-block elements) based on the 3D Ising model.

Critical behavior of two-dimensional intrinsically ferromagnetic semiconductor CrI3

CrI3, which belongs to a rare category of two-dimensional (2D) ferromagnetic semiconductors, is of great interest for spintronic device applications. Unlike CrCl3 whose magnetism presents a 2D-Heisenberg behavior, CrI3 exhibits a larger van der Waals gap, smaller cleavage energy, and stronger magnetic anisotropy which could lead to a 3D magnetic characteristic. Hence, we investigate the critical behavior of CrI3 in the vicinity of magnetic transition. We use the modified Arrott plot and Kouvel-Fisher method and conduct critical isotherm analysis to estimate the critical exponents near the ferromagnetic phase transition. This shows that the magnetism of CrI3 follows the crossover behavior of a 3D-Ising behavior with mean field type interactions where the critical exponents β, γ, and δ are 0.323 ± 0.006, 0.835 ± 0.005, and 3.585 ± 0.006, respectively, at the Curie temperature of 64 K. We propose that the crossover behavior can be attributed to the strong uniaxial anisotropy and inevitable interlayer coupling. Our experiment demonstrates the applicability of crossover behavior to a 2D ferromagnetic semiconductor.

Three-dimensional magnetic critical behavior in CrI3

CrI$_3$ is a promising candidate for the van der Waals bonded ferromagnetic devices since its ferromagnetism can be maintained upon exfoliating of bulk crystals down to single layer. In this work we studied critical properties of bulk CrI$_3$ single crystals around the paramagnetic to ferromagnetic phase transition. Critical exponents $\beta$ = 0.260(4) with a critical temperature $T_c$ = 60.05(13) K and $\gamma$ = 1.136(6) with $T_c$ = 60.43(4) K are obtained by the Kouvel-Fisher method, whereas $\delta$ = 5.32(2) is obtained by a critical isotherm analysis at $T_c$ = 60 K. The critical exponents determined in bulk CrI$_3$ single crystals suggest a three-dimensional long-range magnetic coupling with the exchange distance decaying as $J(r)\approx r^{-4.69}$.

Critical phenomena and estimation of the spontaneous magnetization from a mean field analysis of the magnetic entropy change in La0.7Ca0.1Pb0.2Mn0.95Al0.025Sn0.025O3.

In the present work, we have studied the universal critical behavior in the perovskite-manganite compound La0.7Ca0.1Pb0.2Mn0.95Al0.025Sn0.025O3. Experimental results revealed that all samples exhibit a second-order magnetic phase transition. To study the critical behavior of the paramagnetic–ferromagnetic transition, various techniques such as modified Arrott plots, the 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 critical exponent values for our sample are consistent with the prediction of the mean field model (β = 0.46, γ = 1.0, and δ = 2.94 at an average TC = 302.12 K), indicating that the magnetic interactions are long-range. The reliability of the critical exponent values was confirmed by the Widom scaling relation (WSR) and the universal scaling hypothesis. Moreover, to estimate the spontaneous magnetization of La0.7Ca0.1Pb0.2Mn0.95Al0.025Sn0.025O3 perovskite, we used the magnetic entropy change (ΔSM), obtained from isothermal magnetization. The results obtained through this approach are compared to those obtained from classical analysis using Arrott curves. An excellent agreement is found between this approach and the one obtained from the extrapolation of the Arrott curves.

Analysis based on scaling relations of critical behaviour at PM-FM phase transition and universal curve of magnetocaloric effect in selected Ag-doped manganites.

The critical behaviour of Pr0.5Sr0.5−xAgxMnO3 (0 ≤ x ≤ 0.2) samples around the paramagnetic–ferromagnetic phase transition is studied based on isothermal magnetization measurements. The assessments based on Banerjee's criteria reveal the samples undergoing a second-order magnetic phase transition. Various techniques such as modified Arrott plot, Kouvel–Fisher method, and critical isotherm analysis were used to determine the values of the ferromagnetic transition temperature TC, as well as the critical exponents of β, γ and δ. The values of critical exponents, derived from the magnetization data using the Kouvel–Fisher method, are found to be (β = 0.43 ± 0.002, 0.363 ± 0.068 and 0.328 ± 0.012), (γ = 1.296 ± 0.007, 1.33 ± 0.0054 and 1.236 ± 0.012) for x = 0.0, 0.1 and 0.2, respectively. This implies that the Pr0.5Sr0.5−xAgxMnO3 with 0 ≤ x ≤ 0.2 systems does not belong to a single universality class and indicates that the presence of magnetic disorder in the system must be taken into account to fully describe the microscopic interaction of these manganites. With these values, magnetic-field dependences of magnetization at temperatures around TC can be well described following a single equation of state for our samples. From magnetic entropy change (ΔSM), it was possible to evaluate the critical exponents of the magnetic phase transitions. Their values are in good agreement with those obtained from the critical exponents using a modified Arrott plot (MAP). We used the scaling hypotheses to scale the magnetic entropy change and heat capacity changes to a universal curve respectively for Pr0.5Sr0.5−xAgxMnO3 samples.

3D-Heisenberg magnetic coupling in the skyrmion system Fe1.5−xCoxRh0.5Mo3N ([formula omitted

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.

Critical behavior, magnetic and magnetocaloric properties of melt-spun Ni50Mn35Sn15 ribbons

Microstructure, structural and magnetic phase transitions of the melt spun Ni50Mn35Sn15 ribbons have been examined by means of scanning electron microscopy, X-ray diffraction, differential scanning calorimetry and magnetic measurements. The melt spun ribbons exhibit a single phase cubic L21 austenite structure at room temperature with a space group Fm−3m, and lattice parameter a = 5.956 Å. The DSC results reveal the first order reverse and forward martensitic transition (Ms = 147.4 K, Mf = 133.7 K, As = 155 K and Af = 171 K) with a thermal hysteresis of about 21.3 K around the martensitic transition between heating and cooling. The thermomagnetic measurements show that the melt spun ribbons undergo a second order magnetic transition at a Curie temperature TC = 310 K and a first order martensitic transition at TM = 160 K. The critical behavior associated with the magnetic phase transition has been investigated through the isothermal magnetization measurements around TC. The critical exponents have been estimated by several methods such as the modified Arrott plots, Kouvel-Fisher method and critical isothermal analysis. The critical exponents values β=0.456, γ=0.88 and δ=2.929 are close to those predicted from the mean field model revealing a dominated long-range order of magnetic interactions. For an applied magnetic field of 5 T, the maximum magnetic entropy change (ΔSMmax) and the relative cooling power (RCP) values around TC are of about 2.105 J/kg.K and 132.5 J/kg, respectively. The melt-spun Ni50Mn35Sn15 alloy is a good candidate for magnetic refrigeration near room temperature.

Critical Phenomena and Estimation of Spontaneous Magnetization by Magnetic Entropy Analysis of La0.7Sr0.3Mn0.94Cu0.06O3

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.

Critical behavior and magnetocaloric effect near room temperature in MnCo1-xTixGe alloys

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.

Influence of insolubility of silver on the Hirshfeld surface analyses and magnetic behavior of La0.5Ca0.1Ag0.4MnO3 compound

In the present study, we report a detail investigation on the effect of insolubility of Ag 40% doping on the structural and magnetic properties of LCMO-Ag compound prepared by solid-state reaction. The analysis of X-ray diffraction for our samples using Rietveld refinement showed that they crystallize in the orthorhombic structure with Pnma space group. We have shown that this compound has a secondary phase (Ag metallic phase). The detailed analyses of Hirshfeld surface and fingerprint plots provide insight into the nature of intermolecular interactions in the title compound. Magnetization measurements versus temperature at different magnetic fields have shown the presence of two magnetic transitions. This result was confirmed previously in the crystallographic study. This phase exhibits magnetic properties, which has a Curie temperature (Tc) which is lower than the Curie temperature of the FM-PM transition. The critical properties of perovskite manganite LCMO-Ag around of transition with Tc = 230 K is investigated through various techniques such as modified Arrott plot, Kouvel-Fisher method, and critical isotherm analysis. Moreover, the critical exponents also obeyed the single scaling equation of M(H,ɛ)|ɛ|−β=f±(H|ɛ|-(β+γ)). We find that the critical exponent for LCMO-Ag is close to that theoretically predicted for mean field tricritical model.

A Coexistence of Short- and Long-Range Ferromagnetic Interactions in La1–xKxMnO3 Compounds

In this paper, we present a detailed study on the critical behavior around the ferromagnetic (FM)–paramagnetic phase transition in La1− x K x MnO3 (LKMO) compounds with $x = 0.05 - 0.20$ . Our results pointed out the easiness in controlling the value of $T_{C}$ from 212 to 306 K by increasing K-doping concentration (x) from 0.05 to 0.20. Analyzing temperature and magnetic field dependences of magnetization M(T, H) at temperatures around $T_{C}$ reveals all the samples undergoing the second-order phase transition. Using the modified Arrott plots, the Kouvel–Fisher, and the critical isotherm analysis methods, we have found that the values of exponent $\beta $ in the range of 0.435–0.493, which deviate from the mean-field theory (MFT) for the long-range FM interactions, indicating a coexistence of short- and long-range FM orders. We also pointed out that the critical exponents ( $\beta $ and $\delta)$ depend strongly on K-concentration, shifting from values approaching those of the MFT ( $\beta = 0.5$ and $\delta = 3$ ) to values approaching those of the 3-D-Heisenberg model ( $\beta = 0.365$ and $\delta = 4.80$ ). In contrast, the value of the exponent $\gamma (=1.222 - 1.249)$ is almost independent of K-concentration. It suggests that K-doping favors establishing a short-range FM order in LKMO compounds.

Critical behavior of the van der Waals bonded ferromagnet Fe3−xGeTe2

The critical properties of the single-crystalline van der Waals bonded ferromagnet Fe$_{3-x}$GeTe$_2$ were investigated by bulk dc magnetization around the paramagnetic (PM) to ferromagnetic (FM) phase transition. The Fe$_{3-x}$GeTe$_2$ single crystals grown by self-flux method with Fe deficiency $x \approx 0.36$ exhibit bulk FM ordering below $T_c = 152$ K. The M\"{o}ssbauer spectroscopy was used to provide information on defects and local atomic environment in such crystals. Critical exponents $\beta = 0.372(4)$ with a critical temperature $T_c = 151.25(5)$ K and $\gamma = 1.265(15)$ with $T_c = 151.17(12)$ K are obtained by the Kouvel-Fisher method whereas $\delta = 4.50(1)$ is obtained by a critical isotherm analysis at $T_c = 151$ K. These critical exponents obey the Widom scaling relation $\delta = 1+\gamma/\beta$, indicating self-consistency of the obtained values. With these critical exponents the isotherm $M(H)$ curves below and above the critical temperatures collapse into two independent universal branches, obeying the single scaling equation $m = f_\pm(h)$, where $m$ and $h$ are renormalized magnetization and field, respectively. The exponents determined in this study are close to those calculated from the results of the renormalization group approach for a heuristic model of three-dimensional Heisenberg ($d = 3, n = 3$) spins coupled with the attractive long-range interactions between spins that decay as $J(r)\approx r^{-(3+\sigma)}$ with $\sigma=1.89$.

Critical behaviors of a second order ferromagnetic transition in martensitic state of nominal Ni49Mn39Sb12 alloy

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 TCM collapse into two curves, obeying the single scaling equation M(H,ε)=(ε)βf±[H/ε(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.

Critical behaviour and filed dependence of magnetic entropy change in K-doped manganites Pr0.8Na0.2−xKxMnO3 (x = 0.10 and 0.15)

The orthorhombic Pr0.8Na0.2−xKxMnO3 (x = 0.10 and 0.15) manganites are prepared by using the solid state reaction at high temperatures. The critical exponents (β, γ, δ) are investigated through various techniques such as modified Arrott plot, Kouvel-Fisher method and critical isotherm analysis based on the data of the magnetic measurements recorded around the Curie temperature. The critical exponents are derived from the magnetization data using the Kouvel-Fisher method, are found to be β = 0.32(4) and γ = 1.29(2) at TC ~ 123K for x = 0.10 and β = 0.31(1) and γ = 1.25(2) at TC ~ 133K for x = 0.15. The critical exponent values obtained for both samples are comparable to the values predicted by the 3D-Ising model, and have also been verified by the scaling equation of state. Such results demonstrate the existence of ferromagnetic short-range order in our materials. The magnetic entropy changes of polycrystalline samples with a second-order phase transition are investigated. A large magnetic entropy change deduced from isothermal magnetization curves, is observed in our samples with a peak centered on their respective Curie temperatures (TC). The field dependence of the magnetic entropy changes are analyzed, which show power law dependence ΔSmax≈a(µ0H)n at transition temperature. The values of n obey to the Curie Weiss law above the transition temperature. It is shown that for the investigated materials, the magnetic entropy change follow a master curve behaviour. The rescaled magnetic entropy change curves for different applied fields collapse onto a single curve for both samples.

Structural characterization, magnetic, magnetocaloric properties and critical behavior in lacunar La0.5Eu0.2Ba0.2□0.1MnO3 nanoparticles

The lacunar La0.5Eu0.2Ba0.2□0.1MnO3 nanoparticles where □ is a barium deficiency has been prepared using the sol-gel method. The M-T data suggests that the sample undergoes a paramagnetic (PM) to ferromagnetic (FM) transition upon cooling. Deviation of the temperature dependence of inverse susceptibility curves from the Curie Weiss law and less than unity susceptibility exponents indicating the existence of a Griffiths singularity. The magnetocaloric effect and the critical behavior of the sample are systematically studied using bulk magnetization M (H, T) in the vicinity of the Curie temperature. The critical exponents have been obtained using different methods such as modified Arrott plot, critical isotherm analysis and field dependence of magnetocaloric effect. Though the self-consistency and reliability of obtained critical exponents of β = 0.581 ± 0.022, γ = 0.165 ± 0.008 and δ = 3.12 ± 0.03 are verified by the Widom scaling law and scaling equations, the critical parameters do not obey magnetocaloric effect scaling laws, indicating that the nanometer size effect play a significant role on the magnetic phase transition for La0.5Eu0.2Ba0.2□0.1MnO3 nanocrystalline.

Structural, magnetic, magnetocaloric and critical behavior investigations of La0.65Dy0.05Sr0.3MnO3 manganite

In the present work, we report a detailed study of the structural, magnetic, magnetocaloric properties as well as critical behavior in La0.65Dy0.05Sr0.3MnO3 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 (R3¯c space group). Temperature dependence of magnetization exhibits a paramagnetic (PM) to ferromagnetic (FM) transition at Curie temperature TC = 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 (TC, β, γ, and δ) have been determined. The estimated critical exponents (β = 0.515, γ = 1.082 and δ = 2.932 at TC = 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.